"n...
I. u I

r‘

an"

rm".
. .

..X .
[lirae A

mm... 3a..
s.) 5.. . Iﬂh. . ‘
:3 an. Via
a... .9 “#3.
19%?»

. v! .
v). .I an...
3:. 1..

. .2

Jam. 2 y
E I ‘4. ~
..v i

:7... t 2, aﬁangﬂvrw .7
r. 5.4 an, :5 a...“ ..

r
.x,
x.‘ 1‘

,.

. (ll)
33!: 1....

i J.\ 7.3
150‘ a:

 

l.

THESIS
2.

.I'

I f}. _’( '

p)
1

\al

llllllllllllilllllIlllllllllllllilllllllll

3 1293 01420 3321

This is to certify that the

dissertation entitled
TO HOME AND BACK:

THE INFLUENCE OF STUDENTS' CONVERSATIONS
ON THEIR COMPLETION OF SCHOOL MATHEMATICS TASKS
presented by

James Weldon Reineke

has been accepted towards fulﬁllment
of the requirements for

 

 

 

Ph.D. degree in Educational Psychology
/ WW
U I / Major professor

/ /
Date /’Od. /‘),1/?q)

 

MSU is an Afﬁrmative Action/Equal Opportunity Institution 042771

 

 

LIBRARY l
Michigan Statel
University i

 

 

 

PLACE IN RETURN BOXto rumour-thin chockomfrom your record.
TO AVOID FINES Mum on or baton m duo.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Afﬂnndivo Action/Equal Opportunity Institution
mm:

 

 

. ..__. ._—_——.

 

TO HOME AND BACK:
THE INFLUENCE OF STUDENTS’ CONVERSATIONS
on THEIR COMPLETION OF SCHOOL MATHEMATICS TASKS
by

James Weldon Reineke

ADISSERTATION
Submitted to Michigan State University
in partial fulfillment ot the requirements
for the degree of

DOCTOROFPi-llLOSOPl-IY

Department of Counseling, Educational Psychology,
and Special Education

1995

ABSTRACT
TOHOMEAND BACK:
THE INFLUENCE OF STUDENTS’ CONVERSATIONS
ON THEIR COMPLETION OF SCHOOL MATHEMATICS TASKS
By
James Weldon Reineke

This study explored the influence of various practices on students' mathematics
learning. Sociocultural theories of learning hold that people develop skills and
knowledge as they carry out routine everyday activities or participate in socially-
defined practices. As people move among practices it is possible that skills and
knowledge helpful in one practice may conﬂict with the requirements of other practices.
Mathematics instruction, students’ homework conversations, and other out-of-school
practices such as grocery shopping represent potentially conflicting practices.
Embedding various practices in students' classwork, juxtaposing traditional
elementary-school mathematics assignments and those consistent with the reforms in
mathematics education, and watching students participate in conversations at home and
in school provided an opportunity to explore the influence of various practices on
students’ completion of their mathematics schoolwork.

To explore the inﬂuence of these practices, ongoing whole-class instruction was
video taped, small group interactions were video taped or audio taped, and students and
their parents recorded their conversations at home. Students' interactions in each of
these settings were transcribed and analyzed to trace the development of ideas,
strategies, and answers as the students moved among the conversations.

The students' and their parents' quickly completed tasks that were consistent with
their elementary mathematics experience. When the tasks were inconsistent wim their
experience, no one practice determined what or how students learned. Instead, students,
their parents, and classmates brought together different conceptions of mathematics and

other experiences to form an amalgamated practice. Students conversations at home

often had a greater influence than those in school. In school, students sought to prove
that their answers were correct and others were not and their arguments often rested on
warrants that were not mathematical. In light of these findings, mathematics educators
can no longer assume that school provides students' sole exposure to mathematical ideas
or that students naturally make connections among mathematical ideas. Implementing

the reforms in mathematics education may need to include helping parents understand

reform mathematics.

Copyright by
JAMES WELDON' REINEKE
1 9 9 5

To
June, Caitlin, and Rebekka

ACKNOWLEDGMENTS

One week from today and seven years from the start of my doctoral studies, I will
ﬁle this dissertation. Over those seven years I have been influenced by many things. My
professors at Michigan State, teachers in local elementary schools, fellow graduate
students, and the writings of past masters, among other things have been tremendously
influential. Nothing, however, has influenced me as much as completing this
dissertation. From the original idea, through two small pilot studies and a series of
refinements, a data collection period spent clenching my teeth waiting for students to
return tapes of conversations recorded at home, and the ever daunting task of making
sense of it all, this study has provided endless intellectual and logistic challenges that I
could not have met alone. In this space I want to thank some of the people who
contributed to this study. ‘

First, I want to thank Ms. Smith, the teacher in whose class this study was
conducted, and her students. Ms. Smith’s dedication to this study and our three-year
collaboration proved invaluable in devising tasks, communicating with parents, and
collecting data. Her students' willingness to discuss mathematical ideas while being
taped was essential to the success of this study.

There is one group of students and parents that I particularly need to thank. These
six families recorded their conversations at home. Asking them to record their
conversations was an imposition; agreeing to record their conversations was a sacrifice.
Taking on another responsibility—one that does not contribute to an otherwise smooth-
running system-~is risky business. Things may be inadvertently recorded or

conversation may be awkward with recording equipment staring you in the face. These

vi

families accepted the risks and responsibilities of recording their conversations and
they, alone, are responsible for documenting the conversations at home.

I also want to thank my program and dissertation committee: Ralph Putnam,
Deborah Ball, King Beach, Jim Gavelek, and Penelope Peterson. Each of these people
contributed greatly to my education and to the completion of this study. By providing
research opportunities and guidance, thoughtfully prepared courses, thoughtful
critiques of early conceptions of this study, and constructive comments on early drafts of
my dissertation and other writing projects, they contributed to a exceptional educational
experience at Michigan State. Although each of these members have been influential, I
want to thank two members of my committee in particular.

First, I want to thank Ralph Putnam, my program and dissertation chair. Ralph
served as a mentor in all aspects of academia. He apprenticed me in conducting
educational research, helped me develop the ability to turn criticism into constructive
change, and to have confidence in my ideas. His thoughtful and timely comments, both
substantive and technical, have helped me write more clearly and precisely. Without his
friendship, assistance, prodding, and support I would not have finished this dissertation.

I also want to thank Deborah Ball for introducing me to the wonders of mathematics
and of exploring children's thinking about mathematics. Watching her teach and talking
with her about her teaching, more than anything else, has influenced my understanding
of good pedagogy and thoughtful interaction with children. Throughout my time at
Michigan State it was Deborah who asked the tough questions and then worked patiently
with me to address those questions. Her contributions are evident throughout this
dissertation and without her friendship, support, and assistance I would not have
finished.

Finally, and most importantly, I want to thank the members of my family. My wife
June, in particular, for loving me when I may not have been deserving, supporting me in

stressful times, and not letting me quit the many times I threatened to. She supported

vii

me in too many ways to include here, but perhaps most importantly, I want to thank her
for being the greatest mother possible to our children. During the time I spent writing
this dissertation, she has accepted the lion's share of our parenting responsibilities and
maintained her own busy schedule. Though at times I am sure she wished I was doing
something else, she constantly supported and encouraged me. Without her love, support,
and acceptance this study would not have been possible.

I also want to thank our two daughters, Caitlin and Rebekka, for their unwitting
support. I want to thank Caitlin, who has seen me through this project from the
beginning, for her unparalleled ability to make people smile in the toughest of times.
Her energy, unconditional love, and youthful insight into life constantly reassured me of
the importance of my work. I also want to thank Rebekka, who was born during the last
stages of this study, for providing the ﬁnal burst of energy I needed to finish. To them, I
promise more bedtime stories and Saturday mornings at the park and I dedicate my

future work to improving their educational experiences.

viii

TABLEOFOONTBWS

UST OF TABLES ............................................................................................................... .x
UST OF FIGURES ............................................................................................................. x i
CHAPTER 1
INTRODUCTION ................................................................................................................. 1
Challenge 1: Finding a Niche ................................................................................ 4
Challenge 2: Data Collection Methods ................................................................... 1 0
Goals and Purposes of This Study ......................................................................... 1 1
Enculturation into Practices .................................................................... 1 1
The Influence of Socially-Defined Practices ........................................... 1 2

Embedded Tasks: Bringing the Out-of-School World into

School ........................................................................................... 1 3
Evolving Practices: Reforms in Mathematics Education .............. 1 3
Moving among Practices ........................................................................... 1 5
Summary .................................................................................................. 1 5
Overview of the Dissertation ............................................................................... 1 5
Chapter 2 .................................................................................................. 1 5
Chapter 3 .................................................................................................. 1 7
Chapter 4 .................................................................................................. 1 8
Chapter 5 .................................................................................................. 1 8
Chapter 6 .................................................................................................. 1 9
Chapter 7 ................................................................................................. .2 0
CHAPTER2 '
A REVIEW OF THE LITERATURE ....................................................................................... .2 1
Negotiation within the ZPD ................................................................................. .2 3
Sociocultural Influence on the Direction of Development ................................... 2 8

ix

The Home-School Relationship ............................................................................ 4 O

The Influence of the Home on the Development of Literacy Skills ........... 4 1

The family as Educator ................................................................. 4 4

The Resilient Family .................................................................... 4 5

Parent-School Partnership ......................................................... 4 6

Summary ...................................................................................... 4 7

The Discontinuity between Home and School ............................................ 4 8

Homework ............................................................................................................. 5 3

Families as Educators ........................................................................................... 5 8

Summary .............................................................................................................. 6 2
CHAPTER 3

METHODOLOGY AND METHODS .......................................................................................... 6 4

Methodology .......................................................................................................... 6 4

Methods ................................................................................................................. 7 0

The Classroom and the Teacher ................................................................ 7 O

The Students and their Families ............................................................... 7 3

Kathy ............................................................................................ 7 4

Tony .............................................................................................. 7 6

Karen ............................................................................................ 7 7

Pete ............................................................................................... 7 9

Shaundra ....................................................................................... 8 0

Ronnie ........................................................................................... 8 1

The Tasks .................................................................................................. 8 3

Task 1 ........................................................................................... 8 4

Instructional Considerations ............................................ 8 4

Mathematical Considerations ........................................... 8 5

Research Considerations .................................................. 8 6

Task 2 ........................................................................................... 8 7

Instructional Considerations ............................................ 8 7

Mathematical Considerations ........................................... 8 7
Research Considerations .................................................. 8 8
Task 3 ........................................................................................... 8 9
Instructional Considerations ............................................ 8 9
Mathematical Considerations ........................................... 8 9
Research Considerations .................................................. 9 1
Task 4 ........................................................................................... 9 1
Instructional Considerations ............................................ 91
Mathematical Considerations ........................................... 9 2
Research Considerations .................................................. 9 3
The Data Collection Path ........................................................................... 9 3
Methods of Analysis .................................................................................. 9 5
CHAPTER 4
BRINGING THE OUT -OF-SCI-IOOL WORLD INTO SCHOOL ................................................... 9 8
Task 1 ................................................................................................................... 9 9
Question 1: The Rules of Practice ............................................................ 9 9
Question 2: Situation Definitions ............................................................. 1 O 8
Question 3: Differing Assumptions .......................................................... 1 16
Question 4: Accepted Answers .................................................................. 1 21
Summary .................................................................................................. 125
The Practice of Math Homework in Kathy's Household ............................ 125
Summary and Conclusions .................................................................................... 131
CHAPTER 5
THE EFFECT OF EVOLVING PRACTICES ON STUDENTS’ COMPLETION OF THE
TASKS ............................................................................................................................... 1 34
Task 2 ................................................................................................................... 1 37
The Conversations around Task 2 ............................................................. 143
Task 3 A .................................................................................................... 1 5 2

xi

Question 1 ..................................................................................... 1 5 2

Question 2 ..................................................................................... 1 53
Question 3 ................................................................................... _..1 53
Task 3b ..................................................................................................... 1 54
Question 1 ..................................................................................... 1 54
Question 2 ..................................................................................... 154
Question 3 ..................................................................................... 155
Question 4 ..................................................................................... 1 56
The conversations about Task 3 ............................................................... 1 5 7
Summary and Conclusions .................................................................................... 1 66
CHAPTER 6
TOI-IOMEANDBACIQLOOKINGACROSSOONVERSATIONS ................................................. 169
It’s Two Something ............................................................................................... 1 73
At Home ..................................................................................................... 1 73
In School ................................................................................................... 1 78
Summary .................................................................................................. 1 84
Reconstructing a Tool ........................................................................................... 1 86
At Home ..................................................................................................... 1 86
In School ................................................................................................... 1 88
Back at Home ............................................................................................ 1 89
Back in School ........ 193
Summary .................................................................................................. 2 01
Task 4
Making Connections Across Tasks ............................................................ 2,0 3
Summary ............................................................................................................. .2 O 8
CHAPTER 7
SUMMARY AND IMPLICATIONS ........................................................................................ .2 1 2
Students' Conversations ....................................................................................... 213

xii

Conversations at Home ............................................................................ .2 1 3

Consistent Tasks ........................................................................... 2 1 4

Inconsistent Tasks ........................................................................ 2 1 5

Contributions to Theory ............................................................... 2 1 6

Conversations in School .......................................................................... .2 2 1
Contributions to Theory ............................................................... 2 23

Looking across Conversations .................................................................. 2 2 5
Contributions to Theory ............................................................... 2 2 7

Summary ................................................................................................. .2 3 0
Methodology and Methods ...................................................................................... 2 3 2
Implications for Practice ..................................................................................... 2 37
Summary and Conclusions .................................................................................... 2 3 9
UST OF REFERENCES ....................................................................................................... 2 4 2

xiii

UST OF TABLES

Table 3.1

Characteristics of Student Participants and their parents ............................................. 75
Table 3.2

Returned Tapes ................................................................................................................. 9 5
Table 4.1

Solution Table for Task 1, Question 1 .............................................................................. 1 02
Table 4.2

Solution Table for Task 1, Question 2 .............................................................................. 1 10
Table 4.3

Solution Table for Task 1, Question 3 .............................................................................. 1 17
Table 4.4

Solution table for Task 1, Question 4 .............................................................................. 1 21
Table 4.5

Solutions Accepted by Ms. Smith's Class ......................................................................... 1 24
Table 5.1

Solution Table for Task 2 ................................................................................................. 139
Table 5.1 (cont.)

Solution Table for Task 2. ................................................................................................ 140
Table 5.2

Solution Table for Task 3a ............................................................................................... 1 50
Table 5.2 (cont.)

Solution Table for Task 3a ............................................................................................... 151
Table 5.3

Solution Table for Task 3b ............................................................................................... 1 55
Table 5.3 (cont.)

Solution Table for Task 3b ............................................................................................... 1 56
Table 6.1

Solution Table for Task 4 ................................................................................................. 2 05

xiv

UST OF FIGURES

Figure 3.1
Task 1 Assignment Sheet .................................................................................................. 8 5

Figure 3.2
Task 2 Assignment Sheet .................................................................................................. 8 8

Figure 3.3
Task 3 Assignment Sheet .................................................................................................. 9 0

Figure 3.4
Task 4 Assignment Sheet .................................................................................................. 9 2

Figure 3.5
The Data Collection Path ................................................................................................... 9 4

Figure 4.1
Task 1 Assignment Sheet .................................................................................................. 1 0 0

Figure 4.2
Kathy's Diagram ............................................................................................................... 1 26

Figure 5.1
Task 2 Assignment Sheet ................................................................................................. 1 3 8

Figure 5.2
Task 3 Assignment Sheets ................................................................................................ 1 4 8

Figure 6.1
Task 4 Assignment Sheet .................................................................................................. 2 0 4

Figure 6.2
Tony's Matrix ................................................................................................................... 2 O7

XV

CHAPTER 1
INTRODUCTION

In this study I investigated the influence of students' participation in conversations
at home and in school on their completion of a series of elementary school mathematics‘
tasks. Investigating this phenomenon offeredan opportunity to address issues raised
both by theorists traditionally interested in human Ieaming (e.g., educational
psychologists, anthropologists, and sociologists) and by mathematics educators who have
traditionally focused on providing the best possible classroom instruction in
mathematics.

Recent research on learning (Beach, 1993; Beach, 1990; Lave, 1988; Lave &
Wenger, 1991; Scribner, 1984) has suggested that what and how people learn is
heavily inﬂuenced by the chores, vocations, and avocations that mark their daily lives.
Each of these activities require that participants know certain things that allow them to
be effective participants in the activities. Many of the activities people engage in
involve mathematics. The participants in these activities construct mathematical
systems that allow them more efficiently to fulﬁll the requirements of the tasks they are
performing. Dairy workers (Scribner, 1984), candy sellers (Carraher, Carraher, &
Schliemann, 1985), and grocery shoppers (Lave, Murtaugh, & de la Rocha, 1984), for
instance, have all been shown to construct mathematical systems that allow them to
assess situations they encounter and carry out the necessary mathematical operations.
But, at the same time they are able to compute complex problems in the ﬁeld, their
knowledge of school mathematics is limited. In several studies, the participants could
not compute school-like mathematics problems that included the same mathematical

ideas with which they were fluent in the field. The researchers concluded from these

 

1I use the term “elementary school mathematics" to distinguish between what
mathematicians do in their professions and what students in elementary mathematics
classes do. Whereas mathematicians construct new mathematical knowledge,
mathematics students are learning what past mathematicians have done.

1

2
findings that the mathematics people learn is inextricably tied to the activities they

perform routinely and that these activities should become the focus of psychological
investigation (Scribner, 1984).

This research has not been lost on mathematics educators (Cobb, Wood, & Yackel,
1993; Schoenfeld, 1992). Accepting the idea that the mathematics people learn is
shaped by the things they do and where they do them, mathematics educators began
looking at classrooms as having a similar influence on the mathematics students learn
(Bishop, 1991; Cobb et al., 1993; Lampert, 1990). Students in classrooms in which
computation was emphasized learned to compute; students in classrooms where
mathematical thinking was emphasized Ieamed to reason through Situations using
mathematics as a system of inquiry. The latter of these characterizations is the focus of
recent reforms in mathematics education (National Council of Teachers of Mathematics,
1989; National Research Council, 1989; National Research Council, 1990). As a result
of participating in a community of practice in which students are required to assess
situations mathematically, flexibly apply mathematical tools, and justify their
responses, it is believed that students will develop the ability to “think mathematically”
and solve problems (National Council of Teachers of Mathematics, 1989; National
Research Council, 1990).

Both the psychological research and the work in mathematics education focused on
how the immediate surroundings in which the people were participating influenced what
and how they Ieamed. Recent theorizing, however, has suggested that learning is not
restricted to the immediate situation. Learning, it is posited, includes extending what is
Ieamed beyond the immediate situation to other situations where what was Ieamed can
be beneficial (Lave, 1993). As a result, to understand what a person has Ieamed we
need to watch them as they participate in various practices and attempt to document the

ﬂow of ideas as they move among the practices.

3
In investigations of learning in mathematics classrooms, looking beyond the

immediate situation might include looking at how students construct and use

mathematical knowledge in out-of-school settings. In particular, looking at how

students are inﬂuenced by both their classroom instruction and by working on their
school work with their parents at home would provide a glimpse at the mutual inﬂuence
of these two practices. At the same time that studying this phenomenon can inform
mathematics education, it also provides an opportunity to explore the larger theoretical
issues posited in recent psychological theory. A study of this type has not been attempted
in the past. As a result, new methodology and methods needed to be devised. These three
things are represented in the questions I posed at the beginning of this study:

0 How do students' interactions in school and at home inﬂuence their performance on a
series of school math tasks and what are the implications of this inﬂuence for our
understanding of children’s mathematics Ieaming?

0 What are the implications of simultaneous participation in the home and school for
our understanding of Ieaming and development and educational practices?

0 Will a methodology tracing the inﬂuences of interactions in school and at home
contribute to our understanding of how children learn?

In addressing these questions I accepted two challenges. First, addressing two
audiences-educational psychologists and mathematics educators--presents a
substantial challenge of providing adequate explanations to each group. Scholars
representing these two ﬁelds of inquiry bring different assumptions to the study of
Ieaming in mathematics classrooms that makes an attempt to address both audiences
particularly challenging. Historically, the relationship between mathematics education
and educational psychology has been characterized by tensions that have made it difﬁcult
to bring the two ﬁelds together. Although successful research agendas have bridged the
two fields (Fuson, 1988; Leinhardt & Greeno, 1986; Leinhardt 8. Smith, 1985; Nesher,

1989; Romberg & Carpenter, 1986) tensions have remained. Educational

. 4
psychologists’ recent interest in the sociocultural influences on human learning and

mathematics education, however, provides a reason to attempt another union between the
two fields of study. Second, to investigate the influence of participating In conversations
at home and in school I relied on the subjects to record the home conversations. The
reliance on subjects to record their conversations created a potentially incomplete
record of the conversations students had with their parents and, perhaps, created
inauthentic conversations. In this chapter I look at these two challenges and how I
attempted to address them in this study.
Challenge 1: Finding a Niche

Research on teaching and learning in mathematics classrooms has its roots in the
disciplines of mathematics and psychology. At the turn of the century, as they began to
contemplate what ought to be included in mathematics instruction, scholars from ihese
two disciplines drew on the accumulated work of their disciplines. Translating ﬁndings
or beliefs about subject matters into prescriptions for practice is a difficult task. As
they transcribe their beliefs, scholars cautiously commit the “naturalistic fallacy” of
moving from “is” statements (statements that represent the scholars’ current
explanation for the phenomena they study) to “ought” statements (statements that
represent prescriptions for practice) (Kohlberg, 1971; Bronfenbrenner, Kessel,
Kessen, & White, 1986). The assumption held by scholars is that their beliefs--their
is statements--are true and should be reﬂected in practice. Even within disciplines,
however, scholars disagree about what is true and what practice should include. The
scholars that began researching teaching and Ieaming in mathematics classrooms came
from disparate backgrounds, all of which had their own set of is statements they wanted
the newly constructed field to reﬂect.

Psychologists brought is statements that reflected beliefs about human learning or
behavior that transcended subject matters and ought statements that were instructional

interventions to help people develop appropriate behaviors. Having modeled

5
psychological inquiry after the natural sciences where the goal is to discover basic,

universal laws, the psychologists believed that people learn everything in the same way
and the study of Ieaming can be “content independent;" that is, you could replace
mathematics with any other subject and people would learn it the same way. As Kohlberg
(1971) described this belief,

The critical category of the Stimulus-response approach was “Ieaming” not

“knowing,” where the concept of “learning” did not imply “knowing.”

Accordingly. S-R theory assumes that the process of learning truths is the same as

the processes of learning lies or illusions. It explains the Ieaming of logical

operations or “truths” in terms of the same processes as those involved in Ieaming

a social dance step (which is cognitively neutral), or those involved in “Ieaming” a

psychosis or a pattern of maze errors (which are cognitively erroneous). (pp.

1 51 -1 52)

The psychologists also believed that only observable behavior was worthy of
psychological study. As a result, they needed to deﬁne mathematics, or at least what they
would consider evidence of having learned mathematics, behaviorally. Students’
performance on written computation exercises that could be reduced to a series of
discrete behaviors taught independent of each other became the definition of
mathematics.

Mathematicians started with a different set of beliefs. The mathematicians’ is
statements focused on mathematical processes--both mental and mechanicaluthat
mathematicians used while doing mathematics. To develop his is statements they looked
to practicing mathematicians and how they constructed new mathematical knowledge.
Mathematicians’ ought statements focused on the content and activities that should be
emphasized in mathematics instruction to help students develop the processes associated
with doing mathematics. Mathematicians argued that instruction needed to include ways
of thinking mathematically, rather than focusing exclusively on computation (Hadamard,
1945/1954; Kilpatrick, 1992).

Bringing the psychologists’ behavioral conceptions of Ieaming and mathematicians’

conceptions of mathematics together into one coherent agenda of research on teaching and

Ieaming in mathematics classrooms proved difﬁcult, if not impossible. The researchers

6
could not agree on what should be investigated or how it should be investigated. As a

result, two parallel fields of study emerged: educational psychology and mathematics
education. Educational psychologists were devoted to the application of psychological
principles to classroom instruction (e.g., Thorndike and Judd). Although mathematics
was used as a site for much of their research (Kilpatrick, 1992), these psychologists
believed their conclusions and subsequent prescriptions were generic and contributed to
their universal understanding of human Ieaming.

In contrast, mathematics educators continued to emphasize mathematical content.
Although they disagreed on what aspects of mathematics ought to be represented in
classroom instruction, they agreed that the study of teaching and learning mathematics
needed to be speciﬁc to the discipline. They rejected their psychological
contemporaries’ belief that all things were Ieamed in the same way and raised the
question: “Where’s the math?” in response to research in educational psychology.

Over time, however, psychological thought, and consequently educational
psychology, changed. Following the cognitive revolution of the 1950s, there were
reasons to believe that educational psychology and mathematics education had something
to offer each other. Psychologists had opened behaviorism’s “black box” and had begun
investigating how people process information as they encounter new situations.
Psychologists came to believe that, although the basic processes of encoding, storing, and
retrieving information were generic (Shuell, 1986), the organization of knowledge was
specific to the discipline being studied (Resnick, 1987). In response to these beliefs,
many researchers began successful agendas looking at the cognitive requirements of
teaching and learning mathematics. Researchers compared and contrasted the knowledge
of expert and novice mathematics teachers (Leinhardt & Greeno, 1986; Leinhardt &
Smith, 1985). They studied expert and novice problem-solvers (Schoenfeld, 1983),
searched for the best instructional representations for mathematical ideas (Nesher,

1989) and sought to define how a person’s knowledge of speciﬁc domains ought to be

7
organized (Carpenter, Peterson, Chiang, & Loef, 1988; Romberg, 1982). Each of these

research agendas contributed to a growing understanding of teaching and Ieaming in
mathematics classrooms.

But even the seminal research that grew out of the cognitive revolution did not
resolve all of the differences between the psychological and mathematical viewpoints.
Indeed some of the same concerns-~the differing perspectives on what mathematics is,
how it should be studied, and what instruction should ‘include--remained. Long,
Meltzer, and Hilton (1970) summarized the remaining tension this way:

There seems to be a consensus that cognitive psychology at the present ﬁme is not

oriented toward problems or approaches which are likely to be useful to

mathematics education. . . . A substantial problem seems to be to find psychologists

(mathematicians) who understand mathematics (psychology) in any but a shallow

manner. (p, 451)

Recently, however, many educational psychologists, drawing largely from research
in developmental psychology (Laboratory of Comparative Human Cognition, 1983;
Rogoff, 1990; Wertsch, 1991), anthropology (Lave, 1988; Lave & Wenger, 1991),
and sociology (Eckert, 1989), have broadened their view of Ieaming to account for the
social structures within which people work, play, and study. Recent theorizing in these
disciplines has suggested that what and how people learn is greatly inﬂuenced by their
participation in particular forms of socially-defined practices. Socially-defined
practices are the routine activities in which people participate every day or nearly
every day. They have their own histories and can be discussed-pr studied-
independent of the people who participate in them. They encompass bodies of knowledge,
selected technologies, and sets of niles that guide or shape me participants’ actions and
interactions. Practices can be as large as the practice of law or as narrow as the
practice of writing a memo and one may be nested within another.

Researchers interested in the inﬂuence of socially defined practices on cognition

argue that, as participants in the practices become more experienced, they gradually

adopt accepted ways of thinking and behaving in the practice. Part of this adoption

8
includes developing ways of communicating that assume a shared understanding of the

problems and solutions in the practice. As a result, practitioners are enculturated into
ways of assessing situations for their worth as problems, and determining intelligent
ways of solving the identified problems.

The inﬂuence of practice on a person’s thoughts and actions can be seen in many
areas. Scientists working within a given paradigm look upon their subject matter in
specific ways. The unique way of looking shapes the questions the scientists ask, how
they go about investigating phenomena, and how they interpret their results. Trades-
and craftspeople also learn to work within specific boundaries that define participation
in their practices. In the construction trades, which require a working knowledge of
contemporary technology and local building codes, or recreational activities such as
technical rock climbing where a sophisticated knowledge of the rock on which they are
climbing is imperative to the climbers’ safety, the knowledge necessary to be successful
is embedded in the practice or activity in which the person is participating.

Mathematics educators have taken notice of this research in situated cognition and
have begun thinking of classrooms as communities of practice (Bishop, 1991; Cobb,
1986; Cobb et al., 1993; Schoenfeld, 1992); that is as groups of people simultaneously
engaged in a practice. Mathematics classrooms, and the way mathematics is done in the
classrooms, represent socially-deﬁned practices that function the same way as other
communities of practice. Participation in mathematics classrooms not only results in
Ieaming mathematics, but in Ieaming about mathematics. Students in mathematics
classrooms learn conceptions, origins, and uses of mathematics as well as mathematical
skills and abilities. These things include whether mathematics is thought to be a
collection of memorized facts and procedures or the “science of patterns” emphasized in
recent reform documents that emphasize seeking solutions, exploring patterns, and

formulating conjectures (National Research Council, 1990). Students in classrooms in

9
which one of these conceptions of mathematics is emphasized will emerge with a set of

beliefs consistent with that conception (Schoenfeld, 1992; Skemp, 1976).

Both the research on situated cognition and on classroom communities of practice
have assumed that practices are discrete entities. The practice in which people are
participating, it is assumed, shapes--in a rather behavioral use of the term--what and
how the participants Ieam. Although most of the situated cognition research has
compared the mathematics used in practice to the mathematics usually taught in western
schools, the emphasis is on the differences between the two mathematical systems not on
the relationship between the two practices. In mathematics education, it has long been
believed that what students are exposed to in their classrooms is their primary, if not
sole, exposure to mathematics (Kilpatrick, 1987; Lampert, 1990). The emphasis on
how practices shape participants’ thoughts and actions has led situated Ieaming
researchers to argue that “The practices themselves need to become objects of cognitive
analysis.” (Scribner, 1984, p. 14). Mathematics educators have also focused on
classroom practices. Cobb asked the mathematics education question this way: “With
regard to school mathematics, the question that arises is how do social Interactions in the
classroom inﬂuence the overall goals that students establish and thus their beliefs?"
(Cobb, 1986, p. 6).

If knowledge was determined solely by the practices in which we participate,
however, people would be involved in a never ending string of “code switches” (Edwards
& Mercer, 1987) among a series of reproductionist communities of practice. As we
entered each different practice we would leave behind all our experiences and become an
unquestioning member of the community--all carpenters would think and do the same
things as would all psychologists and all teachers. But all members of a community are
not the same. People bring ways of knowing and speciﬁc bodies of knowledge with them
as they begin to participate in practices. It would seem that their participation would

depend as much on what they bring as on what is demanded by the practice. Indeed, their

1 0
participation shapes the practice at the same time the practice shapes their thoughts and

actions. This idea has not escaped mathematics educators. In 1993, Cobb and his
colleagues wrote:

We ﬁnd ourselves in agreement with a basic tenet of Leont’ev’s activity theory as

interpreted by Minick (1989): that an individual’s psychological development is

profoundly inﬂuenced by his or her participation in particular forms of social
practice. We do, however, reject any analysis that takes a particular social

practice such as schooling or the mathematics education of young children as

pregiven . . . [and] we question . . . [the] metaphor of either students or teachers
being embedded or included in a social practice. Such metaphors, tend to reify social
practices, whereas we believe that they do not exist apart from and are

interactively constituted by the actions of actively interpreting individuals. (Cobb

et al., 1993, p. 96)

The agreement between mathematics educators and situated learning theorists that
an investigation of Ieaming must strive to understand what individual participants bring
to the practice and the social structure in which they participate offers another
opportunity for addressing the fields of psychology and mathematics education at the
same time. Watching a person participate-in separate, but related, socially-defined
practices can help us understand what a person brings to a practice. In mathematics
education, watching students work on school math tasks both in scth and at home (i.e.,
in different practices) may shed light on the how students come to complete and
understand their school math work and, as a result, contribute to classroom practice. At
the same time, and in agreement with Bob Davis (1967) and others (Moll 8. Whitrnore,
1993), classroom practice often surpasses theory and an understanding of classroom
practice may contribute to the growing theory of situated Ieaming.

Challenge 2: Data Collection Methods

Investigating the inﬂuence of participating in conversations in school and at home
also presents a methodological challenge. To document completely students’ participation
it might be necessary to follow them as they move among practices. It was not possible,
however, to follow the students from their classroom to their home and back to school in

person. As a result, students were asked to record their conversations at home

themselves. The resulting conversations may not be authentic or complete. The students

1 1
and their parents may have talked about the tasks without turning on the tape recorder

or they may have had conversations that would not otherwise have occurred. In either
case, the conversations might not represent authentic participation in the practice of
doing homework in their homes.

Beyond the concern over authenticity,lhowever, it is still possible to see ideas as
they emerge in one conversation and reappear in subsequent conversations. As ideas
emerge in the conversations, it is possible to infer experiences that may have given rise
to the idea or shaped it in some way. Tracing ideas from conversation to conversation
also provides a glimpse of how students’ ideas are formed by their participation in
subsequent conversations-whether or not these conversations represent authentic
participation in socially-defined practices.

Goals and Purposes of This Study

The purpose of this study was to investigate the influence of various practices on
students’ completion of school math tasks. Investigating the inﬂuence would elaborate
contextual theories of learning and help explain the relationship between the individual
and social structure in elementary school mamematics classrooms and in socially-
defined practices in general. This goal required that I ﬁrst look at what students take
away from the practices in which they participate; that is, how are they enculturated
into a given practice.

Enculturation into Practices

A hallmark of Vygotsky’s sociohistorical theory (Vygotsky, 1978) is that more
knowledgeable others-teachers, parents, or peers-gradually transfer the authority
for completing the task to the learners. As students gradually develop their ability to
complete the tasks independently, teachers or parents contribute less. The transfer of
authority can occur within a conversation or across conversations in which participants

work on similar tasks.

1 2
At the end of this process, students have appropriated ways of completing the task

that are consistent with the practice in which they are participating and, in effect, are
enculturated into the practice. At the same time, the rules used to complete similar
tasks might change to reflect the experiences of the participants.

With respect to elementary mathematics, I believed that the conversations students
and their parents have at home would change over time to reflect the transfer of
authority from the adult to the child. In tasks that contain similar questions, this
transfer might occur in the conversation, or the transfer of authority might also be seen
in later tasks that include similar questions.

In light of the recent calls for reform in mathematics education, the parents may
also change their participation in the practice of homework. Their knowledge of
mathematics may not include an understanding of the tasks with which they are asked to
help their children--at least not the first time they are exposed to them. Including
similar questions in later tasks would, I hoped, provide a glimpse at parents’ changing
participation in their children’s’ math homework.

The Influence of Socially-Deﬁned Practices

To investigate the inﬂuence of participating in various practices on students'
completion of their school math tasks, different practices were brought together with
the goal of illuminating the inﬂuence each had on students’ completion of the tasks. The
practices were brought together in three ways, each of which is described in more detail
below. First, potentially conflicting practices were embedded in the tasks presented to
students. Second, conﬂict among practices can also result from the evolution of a
practice. Students in this study were presented with tasks that represented both
traditional mathematics instruction and tasks that represented teaching in the spirit of
the recent calls for reform in mathematics education. Third, the tasks moved back and

forth between practices--from the classroom to students’ homes back to the classroom.

1 3
Embedded Tasks: Bringing the Out-of-School World into School. Bringing two

separate practices together can create conflicts that need to be resolved while completing
a task. Practices are characterized by sets of rules that govern working within those
practices. If two practices and, therefore, two sets of rules clash when brought
together, participants must decide which set of rules to use or negotiate a new set of
rules, drawing on the merging practices.

One tenet of the recent calls for reform in mathematics education (National Council
of Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 1991;
National Research Council, 1989; National Research Council, 1990) holds that students
should understand out-of-school uses of mathematics. This emphasis is shown in this
quote from Reshaping School Mathematics:

Students need to experience mathematical ideas in the context in which they

naturally ariseufrom simple counting and measurement to applications in

business and science. . . Appealing applications should be drawn from the world in
which the child lives, from community events, or from other parts of the
curriculum. . . . The primary goal of instruction should be for students to Ieam to
use mathematical tools in contexts that mirror their use in actual situations.

(National Research Council, 1990, p. 38)

The different situations called for in this document represent different socially-
defined practices that may include rules other than those traditionally taught in school
math classes. To complete tasks that reﬂect out-of school practices, students, their
parents, and classmates must determine the set of rules they will use to govern the task
as they work together. As a result, bringing out-of-school practices into school creates
a situation which includes tremendous potential for conﬂict among practices. Observing
students and their coworkers as they work together on school math tasks that include
out-of-school practices can illuminate the mutual influence of various practices on the
completion of a task.

Evolving Practices: Reforms in Mathematics Education. Conﬂicts between practices

can also result from changes within a practice. Thoughts and actions that are accepted

procedures at one time may fall into disfavor as the practice evolves. Although change is

1 4
an aspect of growth in every practice, it is likely to meet resistance when it is first
introduced (for a description of this phenomenon in the sciences, see Kuhn (1962)).
The resulting conflict represents two colliding practices in much the same way as
bringing together two separate practices did. Each of the practices represents different
sets of rules and the differences between the rules must be reconciled in some way when
the practices are brought together.

Recent calls for reform in mathematics education have suggested that teachers drop
their emphasis on arithmetic computation and develop what has been referred to as
“mathematical pedagogy" (Ball, 1988). Ball describes the goal of mathematical
pedagogy this way:

Rooted in mathematics itself, the goal of mathematical pedagogy is to help students

develop mathematical power and to become active participants in mathematics as a

system of human thought. To do this, pupils must learn to make sense of and use

concepts and procedures that others have invented--the body of accumulated
knowledge in the disciplineubut they also must have experience doing
mathematics, developing and pursuing mathematical hunches themselves, inventing
mathematics, and learning to make mathematical arguments for their ideas (see

(Romberg, 1988). Good mathematics teaching according to this view, should

eventually result in meaningful understandings am mathematics: what it means to

“do” mathematics and how one establishes the validity of answers, for instance.
(Ball, 1988. pp. 3-4)

Richards (1991) has referred to the sort of classroom activity described by Ball
as “inquiry math,” and has contrasted it with the “school math” associated with
traditional classroom instruction focusing on arithmetic computation in the elementary
grades. School math and inquiry math constitute different socially defineduand
potentially conflicting-practices (Cobb et al., 1993). Each practice has its own set of
rules that guide participation in the practice. Whereas in school math arriving at a
correct answer using an established algorithm may signify success, in inquiry math the
emphasis is equally placed on creation and justiﬁcation of new or different solutions and
the construction of mathematical knowledge.

As happened with the “new math” of the 19603, inquiry math is likely to conflict

with parents’ previous school math experience. In reform classrooms, students are

1 5
likely to receive instruction in inquiry math while receiving instruction in school math

at home. As a result, students, parents, and classmates will need to reconcile the
differences between the colliding practices as they complete their school math tasks.
Moving among Practices

The instruction students receive in scth and at home represents different
practices that may be governed by different rules. This possibility increases in the
wake of the recent reforms in mathematics education. As mentioned above, parents’
experience in school math may have focused on arithmetic computation that has
characterized elementary mathematics. The emphasis on assessing situations
mathematically, flexibly applying mathematical tools, and justifying solutions may only
be part of students’ mathematics instruction in school. Students receive instruction
while participating in both practices. Trailing the tasks as they moved from school to
home and back can illuminate the influence different practices had on students’
completion of the tasks.
Summary

Each of these ways of bringing practices together are included in this study.
Although working on school math tasks at home and in school will always inﬂuence how
students complete those tasks, the changes in mathematics education provide an exciting
opportunity to investigate the mutual influence of various practices. As a result,
watching students work at home and in school on tasks that bring together various
socially deﬁned practices can contribute both to a understanding of the impact of
changing mathematics instruction and to the development of situated learning theory.

Overview of the Dissertation

Chapter 2

In Chapter 2 I look more closely at the research that informed this study.
Sociocultural theories of learning begin with the premise that learning is a

fundamentally social process. ”Drawing on the work of Vygotsky, it is argued that all

1 6
higher psychological functions appear first interpsychologically (between people) and

then intrapsychologically (within people). The gradual progression from
interpsychological plane to intrapsychological plane is characterized by two processes:
the semiotic negotiation of tasks and the direction of development as defined by the
practice within which people are working. These two processes provide the framework
for Chapter 2. I begin by reviewing research on negotiation in the zone of proximal
development (ZPD) (Vygotsky, 1978). Although much of this research has focused on
expert-novice dyads in a laboratory setting, some recent work has looked at classroom
interaction and negotiation of tasks among peers.

The review of research looking at the direction of development begins with a look at
research in situated cognition that led researchers to conclude that what and how we
Ieam is determined by the practices in which we participate. Each of the studies
reviewed compared the mathematical systems in situ and the mathematics usually taught
in western schools. Although these studies provided the foundation for a host of
illuminating research on human learning, they failed to account for participation in
various practices at the same time. Looking at the relationship between students’
participation in school and at home provides an opportunity to look at students
participating in more than one practice at the same time.

Although little research has been conducted on the mutual influence of interactions
in the home and school, many researchers have looked at the relationship between
activities at home and success or failure at scth and at homework as a instructional
tool. In this section I review research mat focused on the relationship between the
scth and home. This section is divided into two subsections. First, I review literature
on in-home activities that influence children's development of literacy skills and the
discontinuity in interactional norms between the home and the school. Second, I review

the research on homework. this section is also divided into two subsections. The ﬁrst

1 7
focuses on homework’s effects on student achievement, the second on what homework

sessions look like in different homes.
Chapter 3

In Chapter 3 I discuss the methodology and methods used in this study. I begin this
chapter by revisiting some of the theoretical issues associated with research on Ieaming
in practice and developing a conceptual framework within which I deﬁne the term
“practice“ and outline a methodology for investigating mutually constitutive practices.

I begin the methods section by introducing Ms. Smith, the teacher in whose
classroom this study was conducted, and mathematics instruction in her class. The
introduction includes a brief history of my work in Ms. Smith's classroom and her
vision of teaching in the spirit of the recent reforms in mathematics education.
Following Ms. Smith’s introduction, I introduce the students who participated in the
study and their families. These introductions include the makeup of each family, the
students’ past experience in school, their parents’ school math experiences, and how the
families organize homework sessions at home.

After introducing the students and their families, I describe the tasks Ms. Smith and
I presented to the class. To describe the tasks, I use a framework that includes the
instructional, mathematical, and research reasons for the tasks. It was important that
the tasks fit with ongoing classroom instruction, addressed some of the concerns of
mathematics educators, and addressed some of the theoretical issues described earlier.
How the tasks accomplished these things is addressed in these descriptions.

Following the description of the tasks, I describe the data collection path and the
technology used in each setting to record and document the activities. This section also
includes the response rate of students and their families. Finally, I discuss the methods I

used to analyze the data.

1 8
The next three chapters present the findings of this study. Each of the chapters

focuses on one of the ways of bringing potentially conﬂicting practices together discussed
above.
Chapter 4

Chapter 4 focuses on how students, their parents, and their classmates resolved
dilemmas created when two conflicting practices--school math and grocery shopping--
were brought together. Although the task was designed to see which set of rules the
students and their parents would choose, it became clear that more than the practices
represented in the tasks and how math was done in Ms. Smith’s classroom influenced how
the students and their parents approached and completed the task. Parents’ and students’
experience in mathematics classes and in other out-of-school practices greatly
inﬂuenced how they answered the questions.
Chapter 5

Chapter 5 focuses on continuous and discontinuous mathematical practices. Tasks
two and three reflect the two types of mathematics instruction described by Richards
(1991). Task two included a series of word problems similar to those found in
traditional elementary school math texts.2 This task represented what Richards called
“school math.” The questions posed in Task Three presented were consistent with the
recent calls for reform in mathematics education and represented what Richards called
“inquiry math.” In Task Three students were asked to determine the number of three-
or four-digit numbers possible using a given set of numerals and to explore the value of
each digit in the numbers they found. The questions posed in Task Two were believed to
be continuous with students’ parents’ school math experience, whereas the questions
asked in Task Three were thought likely to be discontinuous with the parents’ school '-

math experience.

 

2For a more complete description of problems presented in elementary mathematics
textbooks, see (Remillard, 1990)

1 9
Parent-student interaction changed depending on the task on which they were

working. While working on tasks that were consistent with parents’ and students’ school
mathematics experience, they quickly agreed on the arithmetic operation to use and how
it should be computed. The parents’ role consisted of monitoring the students’ work and
correcting them when they made mistakes. While working on tasks that were
inconsistent with their school mathematics experience, parents and students roles
changed. Parents, rather than remaining in charge of the conversation, deferred to the
students and their understanding of what was required in the task. AS the students
worked their parents watched and listened intently, learning what was expected in the
tasks. As they Ieamed more and their confidence grew, they contributed more to the
students' homework until, during their conversations for the second part Task 3,
parents’ contributions began to resemble their contributions on earlier tasks.
Chapter 6

Chapter 6 looks across conversations to see how ideas develop as students worked on
the tasks both at home and in school. Looking again at the conversations from Tasks 1 and
3 points out two things. Students do not internalize what they learn in conversation and
use the intact knowledge in subsequent conversations. Rather, with the help of
conventional ways of documenting their work, students reconstruct their answers when
they return to school. Although their may reﬂect the conversations they had at home,
they do not mimic them. Students take part of what they discussed in conversations at
home back to school and reinterpret what they have written down. When the
documentation is lost, students have tremendous difficulty reconstructing their answers
without the help of their parents.

Task 4 supports the assertion that students do not internalize everything they
discuss in conversation for use later on. Task 4 included four questions similar to those

in the ﬁrst three tasks with the belief that parents and students would use the things

2 0
they discussed earlier to help them answer the questions in this task. Neither parents

nor students, however, referred back to the earlier tasks.

The second assertion supported by the conversations from Task 1 and 3 holds that
Ieaming is a gradual process. Students and their parents developed ways of thinking and
acting that allowed them to complete the tasks on which they were working. In
subsequent conversations, the students and their parents changed the ways of thinking
and acting to take into account new data. As their actions changed, so did the students
understanding of the mathematics in the situations.

Chapter 7

In Chapter 7, l revisit the ﬁndings from the previous three chapters and discuss the
implications of this study for classroom instruction and the development of situated
learning theories. Many of the findings of this study support the work of other
researchers working in a sociocultural tradition. As a result, this study may beg the
question “So what?” But pointing out the tremendous number of influences on students'
thinking in school mathematics classes can never be done too often. Even in its
repetition, this study offers refinements or extensions of previously conducted research
on elementary students’ mathematics Ieaming and the influence of various socially-
deﬁned practices on what and how people learn. Those refinements and extensions are

discussed in the final chapter.

 

CHAPTER2
A REVIEW OF THE LTTERATURE

At the end of “The lnternalization of Higher Psychological Functions,“ an essay from
his seminal book Mind in Society, Vygotsky (1978) wrote: “The internalization of
socially rooted and historically developed activities is the distinguishing feature of
human psychology, the basis of the qualitative leap from animal to human psychology. As
yet, the barest outline of this process is known” (p. 57).

Although Vygotsky believed the process of internalizing activities of this sort was
not well known, he had gone a long way in explaining this complex phenomenon. The
process of internalization,‘ according to Vygotsky, is an extended process through which
children fundamentally change the way they think and behave. Very young children's
earliest attempts to perform higher psychological processes such as memorizing items
on a list begin as an unaided, eidetic internal activity. These methods prove difﬁcult and
often unsuccessful. To assist children in learning to memorize things, an adult, or more
knowledgeable peer in some cases, mediates sociohistorically determined methods of
recording the items to be memorized. They might, for example, show them how to notch
a stick, tie a string around their fingers, or write a list. Eventually the child begins to
perform these activities on his or her own and reconstructs them internally; that is, the
child internalizes them. The interaction between the child and the more knowledgeable

other casts the foundation on which Vygotsky's theory of human development rests:

 

1There is an ongoing debate among psychologists about me internalization of concepts.
The term “internalization“ suggests to some that people accept the ideas and ways of
thinking of a culture without, in some way, tailoring those ideas for their own use.
Taken to an extreme, this would lead to the mere reproduction of a culture from one
generation to the next. Those who oppose the term “internalization“ suggest instead that
people “appropriate“ knowledge in a way that makes sense to them. This
conceptualization of this process allows for manipulating and changing culturally
accepted ideas and allows the person to take ownership of his or her own thoughts.
Although this second conceptualization more adequately describes the process of coming
.to know something, I have chosen to use the term “internalization“ here to be consistent
with Vygotsky’s writings.

21

2 2
Every function in the child's cultural development appears twice: ﬁrst, on the social

level, and later on the individual level; first between people (interpsychological),

and then inside the child (intrapsychological). This applies equally to voluntary

attention, to logical memory, and to the formation of concepts. All the higher
functions originate as actual relations between human individuals. (Vygotsky,

1978, p. 57)

The transformation of interpersonal actions into intrapersonal abilities occurs
gradually. During their first attempt to write a list, children may need to be shown
every step. While writing subsequent lists they will gradually complete more of the
task without assistance, finally completing the list on their own. Because
internalization is a gradual process, Vygotsky argued that two levels of development
needed to be understood-the level at which a person can function by his or herself and
the level at which they can function with the help of another person. These two
developmental levels became the boundaries of Vygotsky’s construct of the “zone of
proximal development“ (ZPD). Vygotsky defined the ZPD as “the distance between the
actual developmental level as determined by independent problem solving and the level of
potential development as determined through problem solving under adult guidance or in
collaboration with more capable peers.” (Vygotsky, 1978, p. 86).

Learning within the zone of proximal development is evidenced by the Ieamer’s
movement-both cognitive and behavioraI--toward the socioculturally determined ways
of accomplishing tasks. Adults, or more knowledgeable others in Vygotsky’s words, have
the responsibility of facilitating this change. This conception of Ieaming begs two
questions: How do more knowledgeable others facilitate change in learners' thoughts and
actions? And, what determines what is to be learned, or the direction of development?

As Vygotsky argued, movement within the ZPD is a fundamentally social process and
is negotiated in conversation. The direction of development, it can be argued, is
determined by the accepted methods of completing the task--or by the practice within
which the participants are engaged. In the next sections I describe research that has

explored these two components of learning within the zone of proximal development and

other work that has contributed to this project.

 

 

 

2 3
Negotiation within me ZPD

Vygotsky's deﬁnition of the zone of proximal development suggests the ZPD is an
individual trait. People have two different levels of functioning--an actual level and a
potential level that remain constant no matter where they are or with whom they
interact. This definition of the zone of proximal development as an individual trait,
however, is somewhat shortsighted. Wertsch (Wertsch, 1984) tells the stories of two
studentsuone in ﬁfth grade and one in second grade--who work with the same more
knowledgeable other. In both of these stories, the learner is able to complete a complex
division problem. Even though both students were able to complete the problem with
assistance from the more knowledgeable person, Wertsch suggests they do not have the
same level of potential development. Rather, he has argued that the zone of proximal
development, rather than being a trait of an individual, is constructed through the
interaction of two or more people. And, to understand the zone as a product of interaction
we need to understand three mechanisms of the zone of proximal development: (a)
situation definitions and their corresponding action patterns; (b) the intersubjective
situation definition on which the participants agree; and, (c) the semiotic moves used to
negotiate the intersubjective situation definition.

People construct intrapsychological situation definitions and bring them to bear in
conversation. Although these situation definitions include the spatial and temporal
settings in which the two people are working, they also include much more. They include
the participant’s representation of the problem and objects in the problem and cannot be
separated from the actions in which the people take part; that is, their action patterns.
To understand and analyze the zone of proximal development, then, we need an explicit
account of each person's action pattern.

Wertsch and his colleagues (1984; Wertsch, 1979) have looked at the action
patterns of adults and children as they replicate a model of a barnyard scene. The

children in these studies often choose a piece to put in the model without looking at the

2 4
original. Once they have chosen a piece, they place it in the model without baking to

match the original. This action pattern could be written: (a) choose a piece; and, (b)
place the piece in the model. Adults working with these children, however, often chose
the next piece while looking at the original and then placed the piece in the model in its
appropriate place. This action pattern could be written: (a) Check the original to
identity the next piece needed; (b) Find the piece identified in the first step; and, (c)
Place the piece in its appropriate place in the model.

Movement from one action pattemto the other marks a cognitive change. In most
models of the zone of proximal development there is an assumption that the adult holds to
a representation of the situation that is appropriate to their social and cultural
background and is better or more advanced than the child’s. As a result, the cognitive
change in these models occurs as the child's action pattern comes to resemble the action
pattern of the adult. This change cannot be considered a quantitative shift or addition to
the pattern. In the action patterns mentioned above, moving from the child’s action
pattern to the adult’s action pattern would take more then adding an extra step. Indeed, it
would require a qualitative shift in the child’s representation of the situation. The steps
in the child’s action pattern would need to be redefined. Making this kind of change
requires the child give up his or her previous situation deﬁnition and replace it with the
new definition-some instantiation of the adult’s situation definition. Wertsch
(Wertsch, 1984) summarizes his discussion this way:

Thus, in order to understand the way in which an individual deﬁnes a situation, we

have seen that two interrelated issues are involved: the representation of objects

and the representation of action patterns for operating on those objects.

Furthermore, we have seen that a defining property of the zone of proximal

development is that the participants involved in collaborative problem solving have

different situation definitions. Finally, we have seen that we cannot account for
growth in the zone of proximal development solely in terms of quantitative

increments to an existing situation definition. Rather, we must recognize that a

fundamental characteristic of such growth is what one might term situation

redefinitionusomething that involves giving up a previous situation deﬁnition in
favor of a qualitatively new one. (p. )

2 5
Although the situation deﬁnitions of each participant, and changes in those

definitions, help us see growth within the ZPD, there is a third situation definition that
must be accounted for in adult-child interactions. Along the way to the child’s acceptance
of the adults situation deﬁnition, they work together to complete an agreed upon, or
intersubjective, task. The agreed upon task constitutes a third situation definition that
initially may mirror the child’s and gradually becomes a combination of the child's and
the adults situation definitions. As a result, adults often work with a situation deﬁnition
that is different from the one they would use if they were working independently.

Adults, even though they may work on a situation that reflects both their deﬁnition and
the child's, do not change their “ideal“ definition. The only lasting change is on the part
of the child.

Intersubject'wity hinges on using language in ways both participants understand
within a situation in which they agree to work. Luria (1976), in a study conducted in
the early 19303, compared subjects who had no schooling with those who had several
years of formal instruction. The subjects with no schooling were unable to categorize
items in ways the researchers thought were correct. Instead of using theoretical
classifications, they grouped the objects on the basis on concrete situations or practical
uses. In this study, Luria presented the subjects with pictures of four items--a
hammer, a hatchet, a saw, and a loguand asked them which of the four items did not
belong to the group. Although the researcher and the subjects were able to communicate,
they often understood the task in very different ways; that is, they did not have an
intersubjective situation definition. One 39 year old, nonliterate subject had this
conversation with the researcher. '

1 Subject: They're all alike. I think all of them have to be here. See, if
you’re going to saw, you need a saw, and if you have to split
something you need a hatchet. So they're all needed here.

2 Researcher: Which of these things could you call by one word?

3 Subject: How's that? If you call all three of them a “hammer,“ that
won't be right either.

26

4 Researcher: But one fellow picked three things--the hammer, saw and
hatchet--and said that they were alike.

5 Subject: Probably he's got a lot of firewood, but if we’ll be left without
firewood, we won't be able to do anything.

6 Researcher: True, but a hammer, a saw, and a hatchet are all tools.

7 Subject: Yes, but even if we have tools, we still need wood-otherwise,

we can’t build anything.
In this interaction the subject and the researcher interpreted the situation in different
ways. The researcher brought with him a situation deﬁnition and a set of responses he
deemed correct. The responses resulted from a classiﬁcation system based on what
Wertsch and Minnick (Wertsch, 1985; 1990) refer to as “sign type-~sign type“
relations that exist independent of any speciﬁc situation. In these relations a saw,
hatchet, or hammer are always tools, but a log is never a tool. The subject, in lines six
and seven, rejected the use of a general term for the things in the set. Instead, he
grouped the items in terms of a situation in which he might use themucutting firewood.
In this situation, unless the person has “a lot of ﬁrewood,“ he or she would need all four
items. So, although the researcher and the subject were able to communicate--that is,
the words they used signified the same objects--they did not agree on the “sense“ of the
situation. lntersubjectivity goes beyond sharing word meanings and includes sharing a
situation definition.

Adults and children negotiate the intersubjective Situation definition. To do this,
they offer “bids“ to each other that may or may not be accepted. According to Wertsch,
adults are more conscious of their bids than are the children. They use different bids to
shape the way the children view the situation. These bids constitute the third mechanism
within the zone of proximal development: semiotic mediation.

While working with children, adults often overtly direct the children to do a certain
thing. For example, if an adult and child were working on the modeling task described

. above, the adult may tell the child which piece to choose and where to put it in the model.

2 7
In this case, the child would not need to understand anything about the task, only that by

following the adults instruction they can complete the task. In a different scenario, the
adult might ask the child which piece they need next. Depending on the child’s response,
the adult may revert back to a more overt directive to get the child to move closer to the
adults situation deﬁnition or change their temporary situation definition to more
closely reﬂect the child’s. Finally, the adult might refer to the model to gently nudge the
child closer to their own situation deﬁnition. In each of these cases, however, the child
can either accept or refuse the adults bid and continue using their own situation
definition to complete the task.

Wertsch’s (1984; 1985) model of semiotic negotiation within the zone of proximal
development attempted to describe the scaffolding process in greater detail. It focused on
what the adult does to entice the child to see the situation as they do and to mimic their
actions as they work through it. The changes in the child’s actions during the
conversation represents a transformation from “other-regulated“ activity to “self-
regulated" activity [Wertsch, 1985 #643. The adult, Wertsch argued, uses a set of
semiotic “moves“ (Wertsch, 1984) to direct the child's actions toward the desired
endpoint. The moves grow increasingly directive as the adult assesses the child’s needs.

Wertsch’s early work--and scaffolding in general--while being criticized for
portraying adult-child interaction as being determined solely by the adult, has spawned
a host of research on learning within the ZPD (Elbers, 1991; Elbers, Maier, Hoekstra,
& Hoogsteder, 1992; Griffin & Cole, 1984). The child in early models, the critics
allege, is represented as a passive being who is willing to obey any adult wish. The end
product of the conversation is the child working alone, using the strategies displayed by
the adult in the conversation, to solve the problem. In this model, there is no room for
the creation of new knowledge by the child or altering what is presented by the adult.

Elbers and his colleagues (1991; 1992) have argued that children are not passive

recipients, but, instead, actively participate in the interactions they have with adults.

2 8
They help to define and complete tasks and contribute to what Elbers and his colleagues

refer to as the “mode of conversation.” The mode of conversation is a “contract" that
the participants negotiate. It includes the “norms, mutual obligations and expectations“
(Elbers et al., 1992, p. 106) that guide the interaction. The researchers describe four
modes that characterize conversations: task oriented, instructional, playful, or
affective. How the participants interact depends on how they perceive the conversation.
To illustrate this point the authors point to a study in which researchers compared the
interaction between young children and their mothers and the young children and their
teachers (Olthof, Goudena, & Groenendaal, 1989). The researchers found that children
interacting with teachers took much less initiative then they did in their interactions
with their mothers. The authors contend that the students expected their conversations
with teachers to include more direct instruction; that is, they expected to be told and
shown what to do and to copy the demonstration. As a result, they did little until they
were told what to do.

Even in the school-like interactions where the children said little, they actively
negotiated their role in the conversation. Taking less initiative was a conscious choice
based on their expectations for conversations with teachers. This led Elbers and his
colleagues to suggest that the observable change in the child's actions represents a shift
from “joint-regulation“ to “ self-regulation" (Elbers et al., 1992) rather than the
“other-regulation“ to “self-regulation“ shift posited by earlier conceptions of learning
within the ZPD.

Sociocultural Inﬂuence on the Direction of Development

As we have seen, new ideas and psychological processes begin to develop through
social interaction. Fundamental to this conception of Ieaming is that adults know more
than children and are able to mediate a given body of knowledge to them. As the children
or adult learners continue to interact, the newcomer’s ways of thinking and acting begin

to resemble those of the more experienced person. But, where do the more experienced

 

2 9
person’s thoughts and actions come from? One response to this question is that they are

the product of participating in various socially-defined practices. Practices and the
tools and symbols associated with them shape the way participants think and behave
while engaged in the practice. As a result, thoughts about important questions and
intelligent ways of answering them are situated in the practices and the practices,
themselves, determine the direction of development. This deterministic conception of
practices has led to a growing body of research on “situated cognition.”

In situated cognition studies involving mathematics, participants undertook and
solved complicated tasks involving mathematical calculation. The methods they used to
complete the calculations, however, were sometimes different from the mathematics
normally taught in school. Perhaps some of the best known of this research is Sylvia
Scribner's (1984) work with diary workers, Jean Lave's (1988; Lave et al., 1984)
work with grocery shoppers, and Nunes, Schliemann, and Carraher’s (Carraher et al.,
1985;. Carraher, Carraher, & Schliemann, 1987; 1993) work with Brazilian street
vendors.

In Scribner’s (1984) dairy research, the workers constructed a mathematical
system that allowed them to fill orders and load trucks without resorting to written
computation. Each day the “preloaders' were given “load-out“ order forms that listed
the products and amounts the delivery drivers had ordered. The load-out order forms
were produced by a computer that “cases out” each order by converting units into cases.
If the number of units ordered did not equal a full case, the form listed the number of
cases and an additional amount of units. If the extra number of units was less than half of
the case, the additional amount was listed “+ x,“ and represented the number of units
that needed to be added to the full cases. If the additional amount was more than half a
case, the number of cases was increased by one and the additional amount was expressed
as “- x”-the number of units that needed to be removed from the extra case. For

example, because half gallons of milk came nine to a case, 21 half gallons would be

 

 

30
represented “2+3” in the half-gallon column and 49 pints, which came in cases of 32,

would be written “2-15.”

Scribner found that the workers used the “least-physicaI-effort solution” to each
of the problems. For example, in the diary quarts of milk came sixteen to a case. If a
preloader received an order for 1-6, he or she would need to end up with one full case
and another with 10 quarts of milk in it (16-6=10). If the preloader had the option of
using a full case and removing 6 quarts from a second case (the literal strategy) or
using a case with 2 quarts already in it and adding 8, the literal strategy requires less
physical energy as only six units needed to be moved as opposed to eight in the
alternative strategy. If, in another situation, the partial case had 8 quarts and only 2
quarts needed to be added, filling the order as 8+2 would be the least-physical-effort
solution saving four moves.

The dairy workers were able to do ﬂiese problems very quickly. They did not count
the number of cartons in the partial cases. Rather they shortcut the arithmetic and did
it merely by looking at the case. One preloader explained how he filled an order for 1-8
quarts of milk (half a case) this way:

I walked over and l visualized. I knew the case I was looking at had ten out of it, and

I only wanted eight, so I just added two to it. I was throwing my self off, counting

the units. I don’t never count when I’m making the order. I do it visual, a visual

thing, you know. (Scribner, 1984, p. 26) '

Lave and her colleague’s (1988; Lave et al., 1984) grocery shoppers constructed
mathematical systems in which they accounted for factors other than savings or mere
numbers of items needed. The product or brand that represented the best value was often
one that met requirements other than its cost. Smaller, more expensive packages were
chosen because they fit into available kitchen storage. Smaller packages that cost more
per unit, but less in total cost, were chosen to stay within a monthly budget. Large

quantities of perishable items, even though they cost less, were not purchased if there

was a chance of them spoiling before being eaten. In general, arithmetic calculation of

3 1
cost was used only as a last resort—when the shoppers had narrowed their choice to a

couple brand names and had no preference among them.

When the shoppers did use arithmetic calculations, they, on average, did 2.5
calculations. Although the early computations were often not correct, the shoppers' ﬁnal
calculation and decision concerning the best value was correct an astonishing 98% of the
time. In doing these calculations, the shoppers nearly always simplified the
computation. For example, one shopper compared the cost of different sized boxes of
spaghetti: a four pound box cost $1.98 and a two pound box $1.12. The shopper divided
$2.00 by four getting $.50 and $1.20 by two getting $.60 to compare the cost. On the
basis of these calculations, the shopper determined that the four pound box was the
better deal (Lave et al., 1984, pp. 84-88).

In Brazil, Nunes, Schliemann, and Carraher (Carraher et al., 1985; Carraher et
al., 1987; 1993) found that young street vendors computed the price of their produce in
ways quite disparate from those taught in western classrooms. For example, one street
vendor (M.) computed the price of ten coconuts this way:

Customer: How much is one coconut?

M: Thirty-five.

Customer: I'd like ten, how much is that?

M: [pause] Three will be one hundred and five; with three more, that will
be two hundred and ten. [Pause] I need four more. That is . . . [pause]
three hundred and ﬁfteen . . .I think it is three hundred and ﬁfty.

(Nunes et al., 1993, pp. 18-19)
Nunes et al hypothesized that the computations these children did in the streets was
beyond their knowledge of formal algorithms. As a result, they would have difﬁculty
using school routines to solve the problems. To test this hypothesis they administered
“informal“ and “formal“ tests to the street vendors. The informal tests were conducted
in the streets. Participant observers asked the vendors a series of questions in which
they needed to compute a price. The example given above comes from one of the informal

tests. The formal tests included two sections. In the first section the researchers wrote

 

 

3 2
out similar questions in formal notation. Most of these questions contained the exact

content asked in the informal test. There were some instances, however, where the
researchers asked the inverse question (e.g., 350 + 10 rather than 35 x 10) or moved
a decimal point to Slightly change the problem. In the second section of the formal test,
the researchers embedded the questions in story problems similar to the situations the
vendors found in the streets.

As was expected, the street vendors computed the prices with remarkable accuracy
in the streets and had poorer results on the formal tests. Across the ﬁve subjects, the
interviewers asked 63 informal test questions and the vendors answered 61 of them
correctly (96.8%). The vendors answered 45 of 61 questions correctly in the formal
test word problems (73.7%) and 14 of the 38 questions asked in formal notation
(36.8%).

In each of these examples the knowledge necessary to be successful is embedded in
the practice or activity in which the person is participating. To understand what a
person knows and how that knowledge influences their actions, it seems necessary to
watch the person participate in the practice or activity; psychological investigations of
knowing and learning need to be situated. Because of this association between practices
and the cognitive skills necessary to participate in them, researchers have suggested
“the practices themselves need to become objects of cognitive analysis.“ (Scribner,

1984, p. 14)]. 2

 

2Mathematics educators also have shown a great interest in the mathematics used in out-
of-school settings-~but for different reasons. Whereas situated learning theorists
focused on mathematical systems used in the course of performing other tasks and how
those systems differed from the mathematics taught in school, mathematics educator’s
investigated mathematics in out-of-school situations to develop an elementary school
curriculum that would benefit students throughout their lives. Through surveys of
businessmen (Wilson, 1911/1922); analyses of cookbooks, factory payrolls, retail
sales advertisements, and general hardware catalogs (Mitchell, 1918); analyses of
various occupations (business, insurance, plastering, painting, masonry, salesmanship,
and banking) (Wise, 1919; Woody, 1922); analyses of the mathematics contained in
newspapers and journals (Adams, 1924); and analyses of freshman college courses
(Gallaway, 1923; William, 1921); mathematics educators developed an elementary

3 3
As in the practices investigated by these researchers, the direction of development

in classrooms is determined by the structure of classroom practice. Vygotsky (1986;
1994a; 1994b) wrote that in schools, the direction of development is largely
determined by academic disciplines and student learning is a product of connecting the
concepts students brought with them to class and the concepts accepted within the
various disciplines. Building on Piaget's distinction between spontaneous and non-
spontaneous concepts, he suggested there are two types of concepts: spontaneous or
everyday concepts and scientific concepts which represent the academic disciplines. Van
der Veer and Valsiner (1991) have summarized Vygotsky's discussion of these two types
of concepts this way:
By spontaneous concepts he meant concepts that are acquired by the child outside of
the context of explicit instruction. In themselves these concepts are mostly taken
from adults, but they never have been introduced to the child in a systematic fashion
and no attempts have been made to connect then with other related concepts. Because
Vygotsky explicitly acknowledged the role of adults in the formation of these so-
called spontaneous concepts he preferred to call them “everyday“ concepts, thus
avoiding the idea that they had been spontaneously invented by the child. . . . By
“scientiﬁc“ concepts Vygotsky meant concepts that had been explicitly introduced by
a teacher at school. Ideally such concepts would cover the essential aspects of an
area of knowledge and would be presented as a system of interrelated ideas.“ (p, )
In 1935, Shif (van der Veer & Valsiner, 1991), a student of Vygotsky's, conducted
a study in which she investigated the development of scientific concepts in second- and
fourth-grade classrooms where students were studying the history of the communist
movement in the Soviet Union. Drawing on studies conducted by Piaget, Shif designed
statements she believed illuminated the child's development of both everyday and
scientiﬁc concepts. These statements, as had Piaget’s, all ended with the conjunctions
because or although. In the study the students were asked to complete each sentence.

Piaget‘s questions had all dealt with what Vygotsky termed everyday concepts and were

similar to “The boy fell off his bicycle because . . .“ or “The boy fell off his bicycle,

 

school curriculum that included basic arithmetic operations that were found to be
prominent in these practices.

3 4
although . . . .“ Shif included questions similar to those used by Piaget as well as those

she believed represented scientiﬁc concepts. Among Shif's scientiﬁc questions were
“The police shot the revolutionaries because . . .“ and “There are still workers who
believe in God, although . . .“ (van der Veer & Valsiner, 1991, p. 271).

In the study, the students were asked to complete four different types of statements.
The statements can be thought of as a two by two matrix with everyday and scientific
across the top and because and although“ down the side with each of the four cells
representing a different statement. The statements were either scientiﬁc/because,
scientific/although, everyday/because, or everyday/although.

Shit found that the second-grade students responded significantly better to scientific
questions ending in “because.“ The discrepancy among the scores, She argued, was due to
the classroom instruction. In class, the students are required to answer questions about
the scientiﬁc concepts being presented. These questions, represented specific causal
relations among the ideas being taught. As a result, they developed what Shif and
Vygotsky have referred to as a “conscious realization“ of the relationship among ideas;
that is, an explicit understanding of how the Ideas are related to each other. In everyday
activities, children are not explicitly instructed in the relationship among ideas and are
not often able to articulate them. This lack of a conscious realization of everyday
concepts, Shif argued, led students to respond incorrectly more often to the everyday
questions.

To explain the discrepancy between the second-grade students' scores on the because
and although statements, Shif again turned to the classroom instruction. The
relationships among ideas taught in the classroom, she argued, could best be
characterized as causal. This relationship is best represented by the because
conjunction. Students' poorer showing on the although questions could be attributed, she
argued, to their lack of exposure to the term although and to the fact that the although

. conjunction was more difficult for young children to grasp.

3 5
Although the second grade students responded correctly to most of the

scientiﬁc/because questions and were able to provide an explanation of their response,
their answers were often echoes of what they were told in class and they were unable to
provide concrete examples of the answers they gave. Shif argued that these “schematic“
responses were evidence that the students did not truly understand the answers they
were giving, but had memorized the correct response from their classroom instruction.

The fourth-grade students displayed an entirely different pattern of responses. The
fourth-grade students answered a high percentage of each of the four types of questions.
Shif interpreted this ﬁnding as evidence that students had mastered the causal way of
thinking emphasized in their classroom instruction. She also pointed out that the
students’ answers had lost the stereotypical character displayed in the second-grade
students’ responses. The increase in correct answers to the everyday questions, Shif
argued, was evidence that explicit classroom instruction leads to certain ways of
thinking, that those ways of thinking will gradually spread to other areas and raise the
child’s thinking to new levels, and that, eventually, children meld their everyday
concepts with the scientific concepts presented in formal instruction.

Classroom instruction, in this example, creates a zone of proximal development that
is bounded on one side by the child's everyday concepts and the other by the scientific
concepts being introduced in school. Students' everyday concepts shape the situation
definitions they bring to class and the formal ways of thinking represent the situation
definitions of the teacher. Through extended negotiation, they gradually bring together
these sometimes disparate conceptions of the same task or concept until they reach a
common knowledge or lntersubjectivity (Au, 1990; Edwards & Mercer, 1987;
Panofsky, John Steiner, & Blackwell, 1990).

Mathematics educators have suggested their classrooms function much like those
described by Shif (Bishop, 1991). Students in mathematics classes, they argue, learn

mathematics in ways specific to the classroom in which they are participating. In this

36
way, mathematics classes form their own culture, which includes a set of beliefs and

values that shape how and what mathematics is taught and Ieamed (Bishop, 1991). The
beliefs and values include informal theories of the origins of mathematical knowledge,
the goals of mathematics, and the object of mathematical study. Based on different sets of
beliefs, teachers might choose to emphasize computation, problem solving, logic,

inquiry, or other aspects of mathematics. Schoenfeld described the relationship between
beliefs and instruction this way:

Goals for mathematics instruction depend on one’s conceptualization of what

mathematics is and what it means to understand mathematics. . . . At one end of the

spectrum, mathematics is seen as a body of facts and procedures dealing with
quantities, magnitudes, and forms, and the relationships among them; knowing
mathematics is seen as having mastered these facts and procedures. At the other end
of the spectrum, mathematics is conceptualized as the “science of patterns,” an

(almost) empirical discipline closely akin to the sciences in its emphasis on

pattem-seeking on the basis of empirical evidence. (Schoenfeld, 1992, pp. 334-

335)

Based on these arguments, it is believed that changing teachers’ beliefs about
mathematics, and consequently their instructional goals, will change students’
conceptions and knowledge of mathematics.

The deterministic conception of practices and classrooms ignores the fact that
different people contribute different things to the practices in which they participate.
As a result, the practices that shape the participants’ thinking and behavior are, in
turn, shaped by the participants; to understand a person’s participation, we must
understand his or her contribution to the practice. Both mathematics educators and
situated Ieaming theorists have rejected the deterministic conception of contexts or
practices (Cobb et al., 1993; Lave, 1993) and have begun looking at the relationship
between individuals and the social structure that surrounds them. One way of
conceptualizing this relationship is that people bring with them their experiences in

other socially defined practices. These experiences constitute their contribution to the

immediate practice. To understand a person’s participation in a practice, it is necessary

37
to look beyond the immediate practice to the other practices in which a person
participates. This influence has recently become the focus of research.

Saxe (1990a; 1990b), while investigating the mathematics used by candy sellers
on the streets in Racife, Brazil, sought to explain the influence of school mathematics on
the sellers’ conception and computation of mathematical problems that emerged in their
practice. At the same time, he sought to explain the influence of selling candy on the
sellers’ performance on school-math tasks.

To investigate the inﬂuence of school mathematics classes on the sellers in situ
computations, Saxe compared the pricing and selling strategies used by sellers with
different amounts of scth experience. Through a series of observations and
interviews, he found that candy sellers with more school experience used different-
although no more efficient or mathematically sound--strategies to price the candy in
practice. Saxe characterized the strategic differences this way:

The first grader conceptualizes the mathematical relation with respect to the selling

convention-~3 bars for Cr$1000--a convention linked to the retail transaction

itself. The ﬁrst grader’s solution has a direct mapping on the actual operations of
exchange in that each count of three by Cr$1000 represents a transaction. In
contrast, the schooled child conceptualizes the mathematical relation with reference
to an intermediate value that is not a part of any aspect of the actual transaction
itself--the wholesale price per unit, a value that is accessible by the use of the
standard division or repeated application of a multiplication algorithm. Using this
approach, the schooled seller marks up the derived wholesale price and performs

another computation to translate the mark-up into the street convention. Such a

solution strategy is both distant from the seller’s anticipated transactions and it is

also powerful; the solution strategy is one that can be used across many types of

problems regardless of the particular selling convention. (Saxe, 1990b, pp. 225-

2 2 6 )

Candy sellers with more school experience were also better at recognizing written
numerals, such as those listed on price posters in the wholesale markets. There were no
differences, however, among the groups in their ability to recognize and differentiate
between different currency denominations.

To investigate the influence of selling candy on the sellers’ approach to school math

problems, Saxe presented a series of arithmetic problems to second- and third-grade

sellers and nonsellers. The candy sellers computed more problems correctly than did

3 8
nonsellers and used more ﬂexible strategies to complete the computation. The candy

sellers more often used what Saxe referred to as “regrouping strategies.“ In these
strategies, the sellers broke numbers up into like parts and performed the operation on
those parts. For example, to add 28 + 26, the sellers might have added 20 + 20 = 40
and 8 + 6 = 14, then added 40 +14 = 54.

Beach (1990; 1990) followed high scth students as they moved from school to the
workplace in Nepal. At the time of this study, Nepal had just instituted a public
education system which included mathematics classes in which western algorithms were
taught. The introduction of western mathematics into the village resulted in a growing
contempt for traditional mathematical systems. The high school studentsuand
sometimes the shopkeepers themselves--referred to the mathematics used in local shops
as “guesstimate' or “dumb math.“ In spite of this belief, shopkeepers continued to use
the older mathematical systems in their businesses. The high school graduates Beach
followed from school to work were part of the first class to graduate from the new Nepali
scth system and the first to take western mathematics into the shops.

Following graduation, students were placed in apprenticeships in local shops. As
they began their new jobs the students tried implementing the mathematics they Ieamed
in school. To determine the price of different wares, the recent graduates wrote out the
prices and quantities in traditional western algorithms. This process proved too time
consuming and cumbersome. In its place, shopkeepers instructed the apprentices in the
ways they computed prices in the shop. These procedures were done mentally and
emphasized speed over accuracy. Although the shop procedures were not as accurate as
the traditional algorithms, they estimated measurements and costs accurately enough to
guarantee proﬁts in the shop. In the end, the students combined the two mathematical
systems to construct a system that incorporated both their school mathematics and

traditional shopkeeping.

3 9
Both Saxe’s and Beach’s studies point out the inﬂuence participating in one practice

has on participation in subsequent practices. In each case participants drew on their
experiences in one practice--schooling--to assist them in completing tasks another
practiceushopkeeping and selling candy. In each case the skills learned in one practice
did not remain intact as they moved from one practice to the next. Rather, the
apprentice’s skills were modiﬁed in some way to accommodate the exigencies of the new
practice.

But at the same time these studies point out this inﬂuence, they conceptualized the
inﬂuence as unidirectional-what was learned in one practice inﬂuenced what was done
in a subsequent practice. In Saxe’s study, for instance, the sellers left school before
beginning to sell the candy. As a result, Saxe observed sellers in the streets, but did not
observe students in schools; participation in schools and how sellers might approach
school-like problems was simulated in interviews. In Beach’s study, the students
ﬁnished high school before entering the shops. Neither study looked at people engaged in
two practices simultaneously, yet we are never engaged in only one practice. To fully
understand the mutual influence of practices it is necessary to observe a person’s
participation in two related practices.

Students in elementary mathematics classes work on their school math tasks both in
scth and at home and, as a result, these tasks provide an opportunity to explore the
mutual inﬂuence of various practices. The conversations students have in each of these
settings constitute different socially defined practices-each of which would have its
own set of beliefs and values about mathematics and homework. Mathematics may, for
example, be understood as a set of computational rules or as a system of inquiry.
Homework may, for example, be interpreted as a way for parents to monitor what their
children are doing in school, or it may be interpreted as an ongoing assessment of student
ability, or it may represent an infringement on parents’ time and energy. Each of these

conceptions of mathematics and homework may inﬂuence how students complete their

4 0
school math tasks. And, because students work on the same task in the two settings, I

believed it would be possible to see the inﬂuence the two practices have on students’
completion of their school-math tasks.

The home-scth relationship and mathematics homework has been the focus of a
tremendous amount of research. I turn now to a discussion of that work.

The Home-School Relationship

Educators have long been concerned that underprivileged schoolchildren fail at a
greater rate than other students. Much of the research devoted to explaining this
disparity has focused on the activities or characteristics of students’ homes and
environments that lead to either scth success or failure. The research focused on
school success has looked at the characteristics of students' homes that reliably predict
high levels of school achievement and have resulted in prescriptions for improving the
environments of “culturally deprived“ children. Because middle-class students
performed better in school, it was believed that the characteristics of middle-class
homes would provide a window onto school success. Programs such as Head Start, which
were designed to provide at-risk students with the experiences necessary to succeed in
school, grew out of this research tradition.

The research seeking an explanation of school failure has focused on the notion that
schools do not adequately meet the needs of their students. In general, this research
suggests that students “achieve“ school failure by actively choosing to “not learn“ in
order to save face or preserve their own cultural traditions (Kohl, 1984; McDermott,
1987). The need to save face, these researchers argue, comes about as a result of
schools’ ignoring students' cultural backgrounds and the ways they Ieam things at home.
Much of this research has focused on discontinuities in the ways language is used in
students' homes and their schools. Researchers focusing on discontinuities between the
home and school have suggested that, If teachers were more knowledgeable about how

their students interact in their homes, they would better understand their participation

4 1
in the classroom. What are often interpreted as behavioral problems could instead be

understood as culturally appropriate behavior. This understanding would lead to
increased communication and higher student achievement.

In this section I review both bodies of literature. All of the research reported in
this section focuses on the development of literacy skills. I found no research looking at
the characteristics of students' homes that predict success in school math or research
that suggested a discontinuity between the mathematics used in the home and that taught
in school. Although he research focuses on literacy, there is much to be gained for an
investigation of mathematics learning. If different ways of interacting in school and at
home exist and result in difﬁculty in school, it is likely that different ways of thinking
about mathematics, especially in the wake of the recent reforms in mathematics
education, do exist at home and in school and inﬂuence what and how students learn in
school math classes.

The Inﬂuence of the Home on the Development of Literacy Skills

Catherine Snow (Snow, Barnes, Chandler, Goodman, 8. Hemphill, 1991) and her
colleagues at Harvard undertook a massive study in which they investigated the inﬂuence
of students' home activities on their scores on literacy measures. Snow and her
colleagues reviewed much of the research addressing these issues and, from it,
constructed three models of home inﬂuence on students' development of literacy skills.
The study they undertook was an attempt to test the adequacy of the three models in
explaining students' scores on four different literacy outcomes.

The study, conducted over two years, included 31 families with children in 11
classrooms representing four schools.3 Snow and her colleagues collected data that
illuminated three areas: Literacy activities in students’ homes, literacy instruction in

school, and a measure of students’ literacy.

 

3The number of classrooms grew to 25 in the second year of the study as students were
not kept together as they moved from one grade to the next.

4 2
In the homes, Snow and her colleagues conducted interviews with family members.

Although an attempt was made to interview all family members, the interviews often
included only the mother and the children. The interviews focused on seven content
areas: (a) basic demographics; (b) educational level of the parents and their literacy
practices; (c) children's educational histories and their literacy practices; (d) parents’
view of their children’s school and teachers and their relationship with the school; (e)
children's educational aspirations and parents’ educational expectations for their
children; (f) the ﬁnancial and emotional stress family members were experiencing;
and, (g) the family's television viewing habits and other leisure time activities.

Students in each family were asked to fill out a “time allocation diary“ twice each
year of the study-once during the school year and once in the summer. The diaries
included a log in which students documented what they did, where they did it, and with
whom they did it each 15 minutes from 6:00 AM to 11:00 PM. Each diary also included
a seasonal list of activities in which the students may have participated; the school year
diary included a checklist of activities students may have done while in school and the
summer diary included a similar list of out-of-school activities. Students were asked to
check off all of the activities in which they participated each day.

The researchers also observed each family for 30 minutes while they worked on a
“homework-like“ task. During the task the students, along with one or both of their
parents, completed one page of the time allocation diary. The researchers used Bale’s
(1979) System of Multiple Level Observation of Groups (SYMLOG) to rate the
participants in these conversations on three structural dimensions of group
interactions--negative/positive, dominant/submissive, and task oriented/emotionally
expressive. Although the researchers found these interviews illuminating, they focused
mainly on the data collected in the interviews conducted with the families to test the

three models of home inﬂuences on students' literacy skills.

4 3
In the schools, the researchers looked at school records to document each student’s

history, determine if the students had, at any time, required special assistance from the
school, or had other things in their histories (e.g., health problems or absenteeism) that
may have contributed to their achievement level. The researchers collected, among
other things, students' grades and standardized test scores, the number of times the
students had changed schools, and whether they had ever repeated a grade.

Snow and her colleagues also administered questionnaires to teachers. The
questionnaires focused on two broad areas: their teaching methods and the thoughts on the
focal students. The questionnaire asked them about their classroom activities,
instructional emphases, contact with parents, focal students’ strengths and weaknesses,
and their expectations for students. Finally, each classroom was observed for a total of
three hours. The three hours were broken down into three one-hour observations of
reading instruction, other teacher-led instruction, and one hour of less structured time.

The researchers used three measures to derive four literacy scores for each student.
First, the researchers administered the Diagnostic Assessment of Reading and Teaching
Strategies (DARTS). Although they administered the entire battery to remain consistent
with the test administration instructions, they used the scores from only two sections of
the test. They used the students’ scores on the word recognition and reading
comprehension sections of the test. Second, they collected two writing samples from
each student each year of the study. Each year the students were asked to write a
narrative describing a picture of an elderly woman holding a package of three tomatoes
and an expository paper about a person whom they admired. Their writing quality was
measured by the number of words in the sample. Length scores, the authors claimed
“correlated highly with holistic scoring results, with measures of syntactic complexity,
and with measures of the quality of content.“ (Snow et al., 1991, p. 57). Third, to

measure the students’ vocabulary skills, they administered the vocabulary subtest of the

4 4
WISC-R. As with the DARTS test, the entire WISC-R was administered to remain

consistent with the administration directions.

From the data collected from these activities the researchers developed rating scales
and formed summary variables that were grouped together in different ways to test three
models of home inﬂuence that represent the abundant literature on the home-school
relationship. Each of these models was based on an explanation of student literacy
success or failure commonly found in the literature. The three are models are: (a) the
family as educator; (b) the resilient family; and, (c) the parent-school relationship.

The family as Educator. This model is based on the argument that families that serve
as educating agents better facilitate their children’s success in school (Bloom, 1976;
Bradley & Caldwell, 1984; lverson 8: Walberg, 1982; Walberg & Marjoribanks, 1976;
Ware & Garber, 1972). Although the titles are similar, this notion represents a
fundamental difference from the research conducted at the Elbenwood Center for
Research on the Family as Educator. Elbenwood researchers assume that education is a
fundamental social process that infuses all family activities. As such, all families are
educating agents. The “family as educator“ model presented by Snow and her colleagues,
however, compares those families that educate their children and those that do not.

To construct this model of home influence, the researchers drew on previous work
in which researchers attempted to identify educational characteristics of families and
households that predicted scth success. They constructed a model that includes ﬁve
components. First, the literacy environment of the home was measured by two summary
variables: the observer's ratings of parental literacy and the observer’s ratings of the
provision of literacy. The second component of the model assessed the amount of direct
teaching done by the parents. This component comprised two variables, how frequently
parents helped with homework and the positive/negative dimension of the SYMLOG rating
scale of parent-child interactions during ﬁfe homework-like task. The third component

included whether and how often parents created opportunities for their children to learn.

45

This component was measured by three variables, the number of outings with adults in a
typical week, whether parents restricted television watching time, and whether the
content of television was monitored by the parents. The fourth component measured the
level of parental education. Finally, the fifth component measured the educational
expectations parents had for their children.

The Resilient Family. The resilient family model of home inﬂuence is based on the
notion that different families experience different amounts of stress, and the amount of
stress a family experiences has an effect on the children’s development of literacy skills
(Belle, 1982). At the base of this argument is the belief that working-class families
tolerate greater amounts of financial stress than do middle-class families. Often this
stress is compounded by marital discord, the absence of one marital partner, alcoholism
or drug abuse, or other factors that contribute to a stressful household. Many families
overcome these obstacles to provide an environment which supports and encourages
literacy development. The characteristics of the families able to overcome such
hardship constitute the resilient family model.

To test the resilient family model, Snow and her colleagues developed three
measures. The first of these measures focused on the organization of family activities.
The variables in this component include whether parents monitored what their children
watched on television, whether they controlled the amount of time allowed for television
viewing, and an overall organization score. The overall organizaﬁon score was a
composite score that included “the presence of rules for behavior, some predictability in
scheduling of daily events, reliability of family members in meeting responsibilities,
punctuality, physical neatness, and cleanliness of the house.“ (Snow et al., 1991, p.
92). The second component of this model measured the emotional climate of the ‘-
household. The component included the children’s perspectives on their relationships
with their parents, how punitive or nurturing their parents were and the frequency of

opportunities for having fun with either their parents or with other children. This

4 6
component includes things like the number of outings with parents, the amount of time

parents spend with their children, parent-child relationships, and a punishment scale.
Finally, the model included a measure of family stress. This component included the
family’s income, any life changes that may have influenced the children's literacy
development and a summary rating of the stress on the family and the individual family
members.

Parent-School Partnership. The parent-school partnership model sought to test the
argument that parents’ consistent involvement in their children's education will lead to
increased student achievement (Epstein, 1983; Epstein, 1986a; Epstein, 1986b;
Epstein, 1988; Marjoribanks, 1979; Walberg, 1984). This model included measures
of the parents' formal involvement with their children’s education. Parents who were
formally involved often were members of the parent teacher organization, served as a
volunteer aid, or went along as chaperones on school ﬁeld trips. Another variable
included in this model was the amount of help parents gave their children with their
homework. Parents who helped their children with their homework exemplified this
model in two ways. First, they supported the kind of learning that goes on in school.
Second, they accepted the responsibility for their children completing their school work.
Related to parents’ helping with their children’s homework is the nature of parent-child
interaction during the homework-like task. Parents have the power to make homework
a positive or negative experience. Students whose parents earned a positive rating on the
SYMLOG scale had higher literacy scores. The parent-scth partnership model also
included the number of contacts parents had with teachers and students’ tardiness
records.

Each of the three models explained most of the variance on one or more of the four
literacy scores. The family as educator model explained 45 percent of the variance in
children's word recognition skills and 60 of the variance in the children’s vocabulary

scores. Individual variables in the family as educator model that accurately predicted

47
student literacy scores include: (a) the home literacy environment; (b) mother’s

educational level; and, (c) mother's educational expectations for their children. The
resilient family model accounted for 43 percent of the variance in students’ writing
scores. The variables most closely associated with high writing scores included: (a)
parent-child relationship during the homework-like task; (b) observer ratings of
organization in the home; (c) presence of television viewing rules; and, (d) the number
of activities listed in the time allocation diaries. The parent-school partnership model
explained “respectable amounts“ (Snow et al., 1991, p. 118) of variance on each of the
four literacy measures (21 percent on reading comprehension, 32 percent on word
recognition and writing, and 38 percent on vocabulary). Of the variables included in
this model, parents' formal involvement accounted for the majority of the variance as it
correlated highly with each of the four literacy scores.

Summary. The research conducted by Snow and her colleagues, and those that came
before them, is valuable in pointing out that many factors contribute to a students'
development of literacy skills. Different familial activities contribute in different ways
to a child’s success in school. Taken to perhaps a ﬂippant extreme, problems with
differential school achievement could be solved if we could make sure all families of
school age children have characteristics similar to those identified with scth success;
that is, we can change the rate of school success by changing households. At the same
time, however, findings that associate more formal education for parents, higher
educational expectations, families that emphasize literacy activities, homes where the
rules are similar to those in school, a wide variety of educational activities, and parental
involvement in school functions with increased literacy scores seem to suggest that
schools are serving the needs of only a small segment of the population. This ﬁnding is
supported by many other reports of life in schools (Powell, Farrar, & Cohen, 1985;
Sizer, 1992). Is it possible to change schools in order to accommodate the ways of

thinking, knowing, and doing that exist in other segments of our population? What

4 8
happens to the other students whose lives don't match up well with their classrooms?

These questions are addressed by the researchers whose work is presented in the next
section.
The Discontinuity between Home and School

As I mentioned above, the research reviewed in this section has focused on the
discontinuity between language use at home and in school. Much of this work has been
conducted by anthropologists and sociolinguists using a form of participant observation
as their methodology. ‘

In perhaps the best known of these studies, Shirley Brice Heath (1982; 1983)
spent several years in two communities in the Piedmont Carolinas. Heath spent ﬁve
years working with teachers and students in classrooms and in their homes to better
understand the interaction patterns of the region. She found that the questions teachers
asked in school were very similar to the interaction in their homes. The teachers’
questions were characterized by two things. First, teachers often asked questions to
which they already knew the answers. Second, the questions were often
“about things being about themselves” (Heath, 1982, p. 105), as one student described
them. These questions asked students for information about the labels, attributes, or
characteristics of objects or events removed from their naturally occurring contexts.
As did the homework sessions investigated by the researchers with the Elbenwood center
(McDermott, Goldman, & Varenne, 1984; Varenne, Hamid-Buglione, McDermott, 8-
Morison, 1982), these questions required students to display their knowledge publicly.
Both of these patterns were in conflict with the interaction patterns in students homes.

In their homes, Trackton children learned to interact with adults in speciﬁc ways.
They were not considered “information-givers” (Heath, 1982, p. 119) nor were they
expected to answer questions to which the adults already knew the answer. The questions
Trackton adults asked were about whole activities or objects. They compared events or

objects, asked about the uses for an object, explored causal relationships, or started

4 9
stories. In each of these situations it was wrong to ask questions to which the questioner

already knew the answers.

The discontinuities between how interaction patterns in school and at home led
students to remove themselves from classroom conversations. Their removal often was
interpreted as failure or poor performance in school. Similar relationships between
interactions at home and in school have been documented by other researchers.

Susan Urmston Philips (Philips, 1993) spent two academic years and the summer
in between on the Warm Springs Indian Reservation in cenUal Oregon. Philips had two
items on her agenda during this time. First, she sought to document culturally
distinctive uses of the English language, the primary language of all tribal members.
Second, students' test scores at the Warm Springs high scth were traditionally lower
than their anglo counterparts in nearby schools. Although the difference in scores
existed across all academic subjects, the difference was greatest on the verbal sections of
the tests. Philips wanted to explore the notion that the cultural differences in language
use may contribute to the difficulty students were having in school.

Philips found that Warm Springs students learned socially appropriate ways of
Ieaming and interacting that were very different than what was expected in school. She
found that children in Warm Springs families Ieamed through what she termed the
“visual channel.“ Children often watched their elders while they cooked, chopped wood,
sewed, or engaged in other tribal activities. The elders offered very little instruction
during these observations. The children would replicate what they watched until they
believed they did it correctly. Then they would display their newly developed skill by
doing whatever it was they had learned.

Verbalizing was left to the tribal elders who were believed to be much wiser as a
product of their age. At Tribal administration meetings, only the elders would discuss
important Tribal decisions. The younger members of the tribe were expected to

contribute physically. They would carry out the mandates of the tribal elders. Also at

5 0
the Tribal meetings, members would move their chairs out of rows to the perimeter of

the room. This enabled them to see the people with whom they were talking and stopped
anyone from sitting behind them.

Most of the conversations the Warm springs Indians had were centered around a
speciﬁc activity or task. In these conversations they used much less exaggerated actions
than did anglos. They did not, for example, maintain a gaze into either the speaker's or
listener's eyes and they seldom nodded in agreement or offered “Uh-huhs“ or “Um-
hmms“ to show they were paying attention as anglos often do. They used fewer changes in
volume or inflection to reestablish their Iisteners’ attention.

In school, Warm Springs students exhibiting behaviors consistent with Philips’
description of tribal conversations were often scolded or punished in some other way. In
school they were expected to show competency and understanding by offering answers to
teachers’ questions. This was in direct conﬂict with the teaching and learning they
experienced before coming to school. These students were accustomed to listening to the
elders speak and carrying out what they said and did at tribal meetings. They were also
accustomed to Ieaming through observation and demonstrating their competence by
performing the task. Offering their ideas was in direct conﬂict with the ways of
interacting common in tribal meetings and in their previous Ieaming experiences.
Students were often scolded for not paying attention because they would not maintain a
steady gaze at the teacher or nod their head as a sign of understanding. These gestures
were culturally determined ways of conveying attention that were not part of the Warm
Springs culture.

The discontinuity between children’s uses of language in and out of scth was also
the focus of a study conducted by Sara Michaels (1981; 1986). As part of a larger
ethnographic study of communication in the home and school, Michaels focused on the
narrative structures students used during “sharing time“ in a first-grade classroom.

Sharing time, sometimes referred to as “show and tell,“ was a time in class when during

51
which students could tell the class something they thought was important. The teacher

was actively involved in the students’ stories, maintaining the ﬂoor for them and
interjecting questions and comments throughout the story.

Michaels found that the students’ narratives fit into one of two categories: (a) topic
centered; or, (b) topic associating. In topic centered narratives, students developed the
story or theme through a linear presentation of information that described on one event
or object. In topic associating narratives, students presented a sequence of personally
associated personal anecdotes. At ﬁrst glance, these narratives appeared to have no
beginning, middle, or end and, as a result, no point. After more in-depth analysis,
however, the collections of anecdotes in these narratives were found to be related to one
event. Thus, students employing a topic associating narrative structure developed their
themes through anecdotal association rather than linear description.

Michaels described the class in which she conducted this study as comprising “half
white and half black children“ (Michaels, 1981, p. 426). The different narrative
styles were aligned with the different ethnic groups in the classroom. Topic centered
narratives best characterized the stories told by white students and topic associating
narratives best describe African-American students’, particularly girls’, stories. The
teacher--an excellent teacher Michaels carefully points outuwas skillful at helping
the white students craft their stories, but had difficulty working with students who told
topic-associated stories. While working with the white students she quickly picked up
the topic of the story and helped the students expand it through a series of well-timed
questions and comments. As a result, the teacher and student engaged in a synchronized
conversation that allowed the student to maintain the focus or direction of his or her
story.

With the black students, however, the teacher had difficulty discerning the topic of
the story and asked questions that seemed ill-timed and threw the students off course.

The difficulties led the teacher to institute a set of rules governing the conversations.

5 2
These rules included choosing an important topic to talk about and only talking about one

thing-~things on which the teacher believed the black students needed to work. The
different narrative structures and misinterpretation of prosodic clues led to gross
misunderstanding about the children and their abilities.

In another study, researchers at the Kamehameha Early Education Program sought
to explain the disparity between the reading scores of children of Hawaiian ancestry and
those of majority ethnic groups. When early attempts based on increasing student
motivation and questioning students’ general intellect failed, these researchers began to
contemplate other reasons for the disparity. The hypothesis they posited suggested that
there were differences between the ways students Ieamed out of school and what they
were asked to do in school and that the differences might impede the students' Ieaming.

In response to their hypothesis, a team of teachers, psychologists, anthropologists,
and linguists worked together to develop a culturally congruent reading program. The
reading program includes elements of “Talk Story,“ a Hawaiian story telling tradition.
Interactions in talk story are characterized by “mutual participation.“ The teacher,
rather than explaining the story to her students or merely quizzing them on the story,
works together with the students to reconstruct the story; that is, they “co-narrate“ the
story. Co-narratlon is the joint narration of a story by two or more people that is
common to Hawaiian story telling traditions. In the KEEP program, the teacher and
students co-narrate the story, adding comments whenever they believe it is appropriate.
According to western standards, the children in the KEEP program talk out of turn and
aggressively butt into the conversation. But, allowing them to co-narrate stories like
this has greatly improved their reading scores.

In each of these studies, students' classroom instruction was vastly different from
the ways in which they spoke and Ieamed at home. At home they talked, listened, and
conveyed attention in ways that were interpreted as misbehaving or as evidence as a

deficit in their classrooms. In the instances where these differences were reconciled,

5 3
students' achievement increased. Although, there is no evidence that discontinuities

similar to these exist in mathematics, the research conducted by these researchers does
provide insight into a study of home and school influences on students' completion of
school math tasks. As in other classes, students in school math classes will use the
interactional patterns they have grown accustomed to at home. As a result, it is
important to look for these differences while listening to students' interactions. Second,
the existence of discontinuities like these suggest that there may be differences in the
conceptions of mathematics used in school and at home. This possibility increases as
teachers begin to teach math in the spirit of the current reforms in mathematics
education. As mathematics instruction gradually changes, parents' conceptions of mam .
may grow more and more different from those expressed in schools and, eventually, they
may conflict. It is equally important to keep an eye open for conceptual differences
between the home and the school.
Homework

Over the years attitudes toward homework have oscillated. In the early twentieth
century, homework was an integral part of schooling. Two wide-spread beliefs may
account for this emphasis. First, people imagined the mind as a large muscle. Taking
part in certain intellectual activities exercised the muscle, increased its strength, and
disciplined the mind (Brink, 1937; Cooper, 1989). One of these activities--
memorizing names, dates, and factsuwas easily accomplished in the home. As a result,
teachers frequently sent memorization tasks home.

The second belief may have arisen in response to the first one. Around the turn of
the century, Thorndike and Woodworth (1901) rejected the notion that the brain is a
muscle and that academic practices such as memorizing dates and facts exercise it. In its
place, they suggested that knowledge comprised a series of relationships between a
person and the environment that was represented by a collection of bonds between

environmental stimuli and appropriate responses. Once a person established a bond, he

5 4
or she would exhibit the same response each time they came across the stimulus.

Educators needed to provide their students with tasks that would strengthen the
appropriate stimulus-response bonds. Students could strengthen their stimulus
response bonds by practicing the appropriate responses as part of their homework.

The sentiment surrounding homework changed often during the first three quarters
of the century. At times educators were calling for reforms that are being echoed today
(Cooper, 1989). In these calls, educators asked teachers to emphasize problem solving
rather than the drill and practice advocated by earlier theorists. And, like the calls we
hear today, educators questioned the usefulness of homework. Indeed, the life adjustment
movement suggested that homework intruded on students' rights to pursue other out-of-
school interests (LaConte, 1981). At other times educators believed teachers needed to
assign more homework. For example, after the Russians launched Sputnik Americans
became concerned that schools were not preparing students for the future. In response,
educators looked to homework as a way of accelerating students’ acquisition of subject
matter knowledge.

In the 1960s, students' “home environments” led to another controversy around
homework (Austin, 1979; Cooper, 1989). In homes where academic work was valued,
students were expected to complete their homework and perform well. This expectation,
educators argued, had mixed effects. On one hand, working hard on their homework
assignments could develop students’ self-discipline and good attitudes toward school. On
the other hand, the emphasis on performing well might get parents to move beyond the
role of tutor or assistant and complete the students’ work or students may copy from
other students. In other homes, it was believed noneducated parents could not help their
children learn the material they needed to know.

Although the sentiment toward homework has changed over time, the research on
homework has focused on the same small group of questions. Reviewers (Austin, 1979;

Coulter, 1979; Keith, 1987) have identified three major paradigms of research on

5 5
homework. The first of these paradigms investigated the effects of homework on

students' school achievement. COOper (1989) has further refined this category to
include subcategories of homework versus no treatment, homework versus in-class
supervised study, and time spent on homework assignments. In all of these studies
students' achievement was measured either by their scores on standardized achievement
tests or on teacher constructed tests. The major findings of this group of studies tended
to support the view mat regularly assigned homework increased student achievement.
Somewhat surprisingly, some researchers in this paradigm drew other conclusions
(unsupported according to Coulter (1979)) and suggested the abolition of homework.

For example, in 1937, DiNapoli (1937) investigated the influence homework had
on fifth- and seventh-grade students in nine New York elementary schools. The design
consisted of two conditions: compulsory homework and no compulsory homework. Fifth-
grade students in the compulsory group were given up to one hours worth of homework
each evening; seventh-grade students were given up to one and a half hours. Students in
the no compulsory homework condition were not required to do any homework although
many chose to. DiNapoli found the compulsory group of fifth-grade students scored
higher on achievement tests, but the differences between the conditions in seventh grade
were insignificant. From these findings DiNapoli concluded homework was not a good
idea.

In another study, Crawford and Carmichael (Crawford & Carmichael, 1937) spent
six years investigating the effects of homework on students' school achievement. The
researchers in this study, rather than developing an experimental situation,
investigated the change during a district wide policy shift in the El Segundo, California
schools. During the first three years of this study students were assigned mandatory
homework. The second three years no homework was assigned. Each year the
researchers administered the Stanford achievement test to fifth-, sixth-, seventh-, and

eighth-grade students. Crawford and Carmichael found that the SAT scores were better

5 6
the ﬁrst three years than they were the second three. This difference was attributed to

compulsory homework.

Maertens and Johnson (1972) conducted a study in which the parents of 400
fourth-, fifth-, and sixth-grade students were trained to provide feedback on arithmetic
assignments. The study included three conditions. In the first condition teachers
assigned no homework and did not allow students to complete unfinished work at home. In
the second condition teachers assigned homework daily and parents were instructed to
provide feedback immediately after each problem. In the third condition, students were
assigned homework and parents provided feedback but only when the student had
completed the entire assignment. The researchers found no difference between
conditions two and three, but students in those two conditions scored better than students
in condition one. From these findings the researchers concluded that homework
structured to include parental involvement and feedback enhances student achievement.

More recently, Gray and Allison (1971) looked at the relationship between
homework and mathematics achievement and found the differences between the homework
and nonhomework treatments to be insignificant. In a series of interviews conducted
after the study, these researchers found that no student understood the fractional
concepts and computation being taught. From this study the researchers concluded we
need to be careful attributing success or failure to homework.

In the late sixties and early seventies there was a shift in the focus of research in
this group to looking at the effects of certain kinds of homework on certain types of
students. This research compared students in homework conditions to students in
supervised study halls and compared the effects of homework on high and low achieving
students. Doane (1972), for example, found a significant relationship between
homework and high achievers and an insignificant relationship between homework and

low achievers. Ten Brink (1967) found that higher achieving students thrived under

5 7
traditional homework conditions, but not under supervised study. The results were

reversed for low achieving students.

In all of the studies included in the first group, homework was conceptualized
quantitatively (i.e., amount of time spent working, whether or not is was assigned). In
general, the researchers concluded that homework increases students achievement and
ought to be an integral part of classroom instruction. In the second group of studies, the
question shifted from whether or not homework should be assigned to how homework
assignments ought to be structured and introduced in the classroom. Researchers
working in this paradigm investigated how homework should be organized to accomplish
different tasks such as introducing new topics or reviewing topics already covered.

One group of researchers (Butcher, 1975; Laing, 1970; Urwiller, 1971) looking
at these questions compared the effects of distributed or spiral assignments and massed
or traditional homework assignments. In the distributed assignment condition students
were assigned sets of exercises that included things taught that day as well as exercises
reviewing topics covered in past lessons. In the massed or traditional condition students
were assigned exercises pertaining only to the topic discussed on that day. Although none
of the finding of this research were statistically significant, the results tended to
support the spiral or distributed homework condition.

Another group of researchers (Friesen, 1975; Peterson, 1971) looked at the
effects of exploratory homework assignments on students' achievement. These
researchers concluded exploratory homework is a good introduction to new material as it
provides an intuitive base for instruction.

The third group of researchers looked at feedback and grading strategies teachers
might use when homework assignments are brought back to school. In general, they
found that when teachers determined students' grades by from number of correct
answers, rather than only checking that the homework was completed, led to increased

performance.

5 8
In sum, homework has been a research topic for quite some time. The research done

has focused on three different questions. The first question, should homework be
assigned, has generated the largest literature. Conclusions from this work, along with
the life adjustment movement and concern over students' home environments led to a
large amount of controversy over homework. Although the data suggested homework was
a good practice, researchers often claimed the beneﬁts were small enough to warrant the
abolition of homework (Coulter, 1979). Later research accepted homework as a
worthwhile practice and focused on the structure, uses, and feedback on the work
students did. In each of these traditions homework per 89 was not investigated; that is,
no one looked at what homework looked like in different homes or explored the nature of
the influences it had on students‘ completion of the tasksnonly that it did or did not
influence student achievement. There is, however, one group who has looked at
homework in different homes. This work out of the Elbenwood Center for Research on
the Family as Educator is the subject of the next section.
Families as Educators

To discuss the research on families as educators, I draw heavily on the work of Hope
Jensen Leichter (1979; 1985) and her colleagues at the Elbenwood Center for Research
on Families as Educators. Outside of the work done at this center, research on families
as educators is sparse. The research conducted at the Elbenwood Center, however,
provides an excellent view of educational practices within families. As most researchers
investigating the relationship between schools and homes have done, researchers at the
Elbenwood Center have focused on the development of literacy skills. To investigate this
inﬂuence they have developed three strands of research: (a) the influence of family
activities on the development of literacy skills; (b) families' routines surrounding
television viewing; and, (c) adolescents moving through, engaging in, and making sense

of educative experiences throughout their social networks.

5 9
The work at the Elbenwood Center has led to a set of characteristics of familial life

and education. The characteristics, rather than simplifying the study of families as
educators, describe the difficulties inherent to such an agenda of study. Perhaps the
most telling of these characteristics is that family life consists of simultaneous streams
of activity within which are embedded other activities. Family members, for example,
may be preparing dinner, answering homework questions, watching television, and
cleaning a scraped knee, all at the same time. These embedded activities make the onset
and conclusion of action sequences difﬁcult, if not impossible, to identify. As a result,
educative patterns are virtually inseparable from other activities.

Interaction among adults and their children change throughout the life cycle of the
family. These changing relationships also render educative patterns difficult to predict.
Whereas the changing life cycle of the family makes some things difficult, the families
share a history throughout the cycle that inﬂuences the ways in which they
communicate. This history often leads to the development of a shorthand family
members use to communicate with each other. Written communication often becomes
incomprehensible to others when written in this shorthand and verbal communication is
sometimes only accessible to the initiated.

These difﬁculties in documenting educative situations in families became more
apparent as a group of researchers from the Elbenwood center began to look at homework
sessions in different households (McDermott et al., 1984; Varenne et al., 1982). This
group of researchers spent time with 12 families while they participated in activities
that contributed to their children's development of literacy skills necessary for school
success. The researchers documented ongoing household activities including housework,
doing and checking homework, watching television, conversations, meals, and family
outings. They also interviewed both nuclear and extended family members, neighbors,
and teachers and other school personnel. The interviews focused on the family histories,

past and current literacy practices, each family's history of contact with the school,

6 0
other family members’ school literacy successes, and family members’ perceptions of

the functions of literacy. After the researchers had established a rapport with the
families, they videotaped one event. In each family, the researchers tried to record a
formally deﬁned (by the family) homework session. Although some families refused and
suggested instead that they record a meal or other family event, the tapes of the
homework sessions provided an insight into homework in different students' homes.
Most education arises from the children's participation in activiﬁes deeply
embedded in the “flow of everyday goals and possibilities” (Varenne et al., 1982, p.
11). Homework, however, was an exception. Unlike other educational activities in the
home, homework was a special event. The families had well speciﬁed routines for how,
when, and where the work would be done. For Joe Kinney, for instance, homework
included two separate, but related routines. First, Joe went to his grandmother's house
immediately after school where he sat at the kitchen table and worked. His grandmother
often helped him with his homework and, at times, did some of it for him. The second
routine began when Joe’s mother picked him up after work. After picking him up, Joe
and his mother went home where Joe finished his homework at the dining room table.
Joe and his mother looked over what needed to be done and his mother devised a strategy
for completing the work. Once the strategy was determined, Joe worked alone on his
homework while his mother prepared dinner and took care of Joe's younger sister.
Most of the recorded homework sessions were divided into two subsections where
one of the participants worked while the other waited for his or her turn. In the Kinney
family, Joe's mother went first, setting up things for Joe to do. While she did this, Joe
actively waited to do what whatever his mother set up. While his mother determined
what to do, Joe attended closely so when his turn came he could begin immediately. In
another home, the Farrells, the two subsections were reversed. Sheila, the student, did
her homework alone in the first subsession. In the second subsession. Sheila's mother

checked her work and asked questions to determine how Sheila arrived at the answers on

61
the paper. Although the work was completed by different people at different times, the

work represents a collaboration between the students and their family members. The
collaboration, rather than being on the actual homework task, comes in the coordination
of the different subsessions of the homework session.

Regardless of the sequence of activity in the different households, the families knew
how to act “school like” and the interactions between students and family members
closely resembled the interaction teachers and students have in school. The “canonical
form" (Varenne et al., 1982, p. 104) of these conversations had the adult asking
questions to which the student supplied an answer. The questions/answer exchanges
were similar to the classroom exchanges identiﬁed by other educational researchers
(Cazden, 1988; Edwards & Mercer, 1987; Mehan, 1979) where teachers ask students a
question to which they already know the answer, the student responds, and the teacher
evaluates what the students said. The Elbenwood center researchers identiﬁed two types
of questions: (a) eliciting questions where the students displayed their knowledge of a
certain domain and (b) elaborating questions where the student was asked to explain how
they arrived at a certain answer.

Most of the families in which the children were successful in school put great effort
into the homework sessions. Their goal for these sessions may not have been for the
student to learn something, however, but for them to take correct answers back to
school. Through the students' work and their question and answer sessions with family
members, the families could ensure correct answers to the homework. This emphasis,
the researchers argued, comes from the spotlight placed on students and their family by
homework. Teachers and schools not only grade students, but families. As a result, if a
family member feels competent, they are unlikely to let their children take wrong
answers back to school. As a result, the homework sessions were not instructional
sessions, but instead are opportunities for students to display what they have learned in

school or other places. The researchers summarized their argument this way:

62

Homework is organized as a school knowledge display scene for purposes of
evaluation. There is no definite suggestion that in homework children Ieam. At best

they display a knowledge that they have acquired elsewhere and “elsewhenf In no

sense can we say that our children learn through their families by doing homework.

(Varenne et al., 1982, p. 101)

The work of the Elbenwood center was the ﬁrst to look at what homework was and
meant to different families. At the same time that it illuminated the practice of
homework within the different families, it also pointed out that homework creates a link
or conduit between the home and school. Through the conduit, school work comes home,
and families become responsible for the completion and quality of the homework when it
returns to school. The social relationship developed between families and schools shapes
how families do homeworkuwhat their goals are--and the value they place on it. This
relationship, however, is not a direct link between the school and home, but is nested
within the other relationships that influence how families operate. Job demands, sibling
needs, and everyday household chores, among other things, influence how families
organize homework sessions.

Summary

The research presented in this chapter presents a portrait in which all learning
results from interactions with more knowledgeable people. By mimicking the more-
knowledge-other’s actions, we internalize their situation definitions in ways that allow
us to use them in subsequent interactions. The direction of development (e. 9., what is
learned) is determined, or at least shaped, by the socially-defined practices within
which the conversations take place.

Elementary school mathematics and homework are socially-defined practices that
shape how students think about elementary school mathematics. But the relationship
between the home and school is complex. Whereas teachers may send mathematics
assignments home to supplement the work they have done in class, parents may see

homework as serving different purposes. They may see homework as a way of keeping

parents informed of classroom activities, or they may see it as an imposition on their

6 3
time, or they may see it as a way of monitoring parental obligations. As a result, the

goal of homework sessions in students' homes may not be to help students learn
mathematics, but to send them back to school with correct answers in order for the
family to save face. How homework sessions are organized, the goals families have for
the sessions, and the roles children and their parents play in them must be taken into
account in an investigation of the inﬂuence of homework on students' completion of

elementary school mathematics tasks.

CHAPTER 3
lI/EII-IODQOGYANDIIEIHODS

The title of this chapter, Methodology and Methods, reﬂects a distinction between the
two sections that make it up. The Oxford English Dictionary deﬁnes methodology as “the
branch of knowledge that deals with method and its application in a particular ﬁeld.“
Following from this definition, the section on methodology addresses questions about the
phenomenon being investigated and what might characterize an appropriate way of
investigating it. Stephen Toulmin (1982) has argued that there is nothing about a given
subject matter that dictates the methods that should be used to investigate it. Rather, the
methods need to be consistent with the researchers' conception of the subject they are
investigating. In the first section, I lay out my conception of learning in practice and the
mutually constitutive nature of participating in multiple practices. The discussion
focuses on the unit of analysis appropriate for an investigation of learning in practice.
Following that I discuss the characteristics of a method that might allow researchers to
investigate the mutual inﬂuence of participating in various practices.

Method, according to the Oxford Dictionary, is “a mode or procedure; a (defined or
systematic) way of doing a thing, especially in accordance with a particular theory or as
associated with a particular person.“ In the methods sections I describe the speciﬁc data
collection and analysis activities—the methodsuused in this study and how they
represent the characteristics of the methodology developed in the first section.
Speciﬁcally, I introduce the people who took part in this study, the neighborhoods in
which they lived, and the classroom in which the study was conducted. I discuss the tasks
that were used in the study and how the classroom teacher and I developed them. Finally,
I discuss the analytic framework I used to begin analyzing the data.

Methodology
Recent advances in psychological investigation have included the notion that

psychologists need to adopt a unit of analysis that is larger then, but inclusive of the

64

65
individual (Cole, 1990; Rogoff, 1990). A unit of analysis of this type departs from

psychology's traditional focus on the individual to include people doing things. A basic
tenet of this new approach is that what and how people know is inseparable from the
things they do and where they do them. The search for a more encompassing unit of
analysis has led researchers to look at Ieaming in context, in practice, as an event, in
activities, and in other settings that appear to inﬂuence a person's thinking.

The current emphasis on situated learning or Ieaming in context was foreshadowed
in writing of John Dewey (1938/1991). Dewey suggested that the things we do are
always connected to the other things we have done; they are part of an experienced world.
He referred to the relationship among the things we do as situations, which he deﬁned
this way:

What is designated by the word “situation“ is not a single object or event or set of

objects or events. For we never experience nor form judgments about objects and

events in isolation, but only in connection with a contextual whole. This latter is

what is called a “situation.“ (Dewey, 1938/1991, p. 72)

Dewey goes on to suggest that psychological investigations have ignored the situated
nature of Ieaming and knowing and, instead, have focused on isolated skills or events.
Hewmm:

Psychological treatment takes a singular object or event for the subject-matter of

its analysis. In actual experience, there is never any such isolated singular object

or event; an object or event is always a special part, phase, or aspect of an

environing experienced world—a situation. (Dewey, 1938/1991, p. 72)

Dewey’s call for a new unit of analysis that takes into account the situations people
encounter while carrying out a definite purpose (Dewey, 1895/1964) is echoed in the
work of many contemporary psychologists and other social scientists (Laboratory of
Comparative Human Cognition, 1983; Lave & Wenger, 1991; Leontiev, 1981; Rogoff,

1990; Scribner, 1984; Wertsch, 1981). Consistent to all the calls for change in

6 6
psychology is the notion that investigations of learning and knowing need to include the

things people do; that people learn new things in order to fulﬁll a goal or for some other
purpose and that these things are completed within an expansive, elaborate setting. .

Researchers searching for a unit of analysis similar to Dewey’s situations use many
different terms to express the situated nature of learning. Along with other terms, these
researchers often talk about practice, activity, or context. In choosing a term or phrase
to describe the things people do, I wanted to accomplish two things. First, and foremost,
I wanted the term to adequately describe the idea that people do many different things in
many different places that inﬂuence what and how they know and Ieam. At the same
time, I wanted a term that was consistent with the way other educators and psychologists
have thought about these issues. The second part was quite difficult as researchers seem
to use the terms practice and context in different and sometimes confusing ways.

I chose the term practice rather than context for one reason. Whereas contexts are
everoexpanding networks of connections unique to a particular time and place, socially
deﬁned practices have their own histories which allows them to be discussed independent
of the participants. I can talk, for instance, about the practice of farming without
talking about a specific farmer. I can talk about how the practice varies based on
geographical settings—the differences between dry-land farming in the Western United
States and the taming in the lush soils of the midwestem states, for example. Or, we
can talk about the history of farming and how technological advancements have changed
the practice.

Practice, as I use it here, refers to the routine activities in which people
participate. Practices comprise both a socially determined structure of activity and the
meaning participants attach to the activities. Practices can be as large as the practice 'of
medicine, law, or farming. Or, they can be a small as reading x-rays, writing a brief,

or selling grain. Each of these practices is shaped by a fuzzy set of rules or guidelines

6 7
and expectations that set out certain elements of the practice. The rules and expectations

surrounding a practice form the socially constructed structure of the practice.

Within the structure of a practice, people participate in different ways. How a
person participates in a given practice is influenced by their knowledge, skill, and the
technology to which they have access (Scribner & Cole, 1981). Writers who use word
processors, for example, may write very differently than those who use pen and paper,
as would an experienced writer as compared with a novice.

To understand a person's participation in practice, it is necessary to understand
both the socially constructed structure of the practice and the meaning the participant
attaches to different parts of the practice. But how does participating in two or more
socially-constructed practices at the same time inﬂuence what and how we know and
learn? This question is somewhat problematic for two reasons.

First, although the question asks about the mutually constitutive nature of
participation, it seems to present a situation in conflict with our everyday experience--
we do not usually participate in different practices at the same time. If a person is a
Methodist, a plumber, a softball player, and a father, he seldom goes to church, ﬁxes a
leaky faucet, ﬁelds a ground ball, and disciplines his child simultaneously. Yet, how he
plays softballnwhether he plays one or four nights a week, for instanceumight be
inﬂuenced by his being a father. A person with kids might choose to play less frequently
so that more time could be spent with family. The inﬂuence might go the other way as
well; playing softball might become a part of his interaction with his kids. They might
play catch, hit “fly balls," or the father might provide more formal instruction on how
to play the game. In this way the influence is two way: being a father inﬂuences how and
when he plays softball and playing softball inﬂuences how he interacts with his kids. So,
while we participate in many different activities each of which influences how we
participate in other activities, we do not participate in all those activities at the same

time. Rather, we move from one practice to another at different times. But, as we move

6 8
among practices, we don't leave behind the other things we do. It might be best to say one

practice dominates the others at different times, or even that the practice changes in
response to the setting in which the person ﬁnds himself.

The second problem arising from this question is, given the definition of mutually
constitutive practices, we can never ﬁnd the origin of “ways of thinking" or ideas.
Rather, everything we Ieam or do is inﬂuenced by the other things we have Ieamed or
have done. In Chapter One, I suggested that researchers have traditionally
conceptualized the relationship among practices as unidirectional. For this
conceptualization to be useful, the researchers needed to identify the origin of ideas or
knowledge in one practice, for example, the practice of school mathematics, and see how
that knowledge inﬂuenced (i.e., whether or not it was used or how it changed) a person’s
participation in another practice, filling dairy orders, for example. Locating the origin
of knowledge is problematic in two ways. First, if we believe that collateral
participation in socially constructed practices influences how we participate in different
practices, then whether or not the strategies used to fill the dairy orders are identical to
the traditional mathematical algorithms taught in school, the mathematics the dairy
workers Ieamed in school inﬂuenced the strategies they constructed in the dairy.
Second, in most studies comparing in- and out-of-school math, the relationship between
practices has been investigated by documenting the mathematics used in various
situations and then comparing the workers performance in situ to their performance on
a school-like math test. The people in these studies are, however, no longer in school.
The studies look at what these people have remembered years after their school days and
compare that knowledge to the mathematical strategies they constructed in the
workplace. To truly investigate the inﬂuence of participating in one socially-
constructed practice on another, we need to look at practices in which people are

participating at roughly the same time; that is, at collateral practices.

6 9
A methodology for investigating collateral participation needs to take into

consideration three things. First, the methodology needs to reflect the everyday
experience of moving among various practices; investigating collateral participation
needs to follow or trace a person moving among various practices. Second, the
methodology needs to focus on something other than the origin of a person's knowledge;
investigating collateral participation needs to focus on inﬂuences rather than origins.
Third, it needs to look at people participating in two or more related practices during the
same time period. Finally, I would like to add one more necessary component: a study of
collateral participation requires we look at related practices; the practices must include
a shared domain of knowledge. For instance, all of the practices must be mathematical
practices or include a mathematical component.

To look at the two-way inﬂuence described above and to meet the criteria laid out in
the previous paragraph it is necessary to ﬁnd something that crosses boundaries among
practices; something that people do while participating in different settings or practices.

The things people do-the tasks they undertake-often begin and end in the same
practice. They may, however, move among practices for short periods of time. For
example, a person who sets out to complete a household landscaping project might take
the project from her home to the hardware store, to the nursery, and back home before
finishing the project. While the initial plans for the project may have been discussed
and drawn at home and the project is completed at home, the interactions the woman has
at the hardware store and the nursery will change the project. She might find out about
a new tool at the hardware store, or, at the nursery, she might Ieam that the plants she
chose won’t do well in the shade around her house, so she needs to choose shrubs better
suited to her environment. In either case, the ﬁnished project will have been inﬂuenced
by the people with whom she talked. By looking closely at the interactions people have

while participating in various practices and understanding how the project or task

70
changes over time we can begin to understand how participating in different activities

inﬂuences what a person does.

In this study I looked closely at the practices of scth mathematics and homework.
The tasks were school-math homework assignments. Like the landscaping example I
presented above, homework begins and ends in the same practice--school math-~but
crosses boundaries and is discussed and worked on at home. The tasks for this project
started in the classroom, went to students' homes, and returned to the classroom where
students talked in small groups and then in whole class discussions about the tasks. I
documented interactions around the tasks through observation, videotaping, and
audiotaping in the classroom, and by audiotaping interactions around homework in the
students’ homes. By following the tasks across the different settings we can see changes
in the tasks and infer the inﬂuence the different conversations had on the students'
completion of the tasks.

Understanding how participating in the two practices inﬂuenced students'
completion of the tasks required understanding the structure of the practices and the
meaning the students, their parents, and teacher attached to them. School math in this
classroom was dominated by small-group and whole-class discussions. Homework in
each of the students' homes also included conversations between the students and a family
member. In the next section I report the speciﬁc data collection methods used to
document students' conversations in school and in their homes. Along the way, I will
introduce the school, classroom, teacher, students, and families that took part in this
study.

Methods
The Classroom and the Teacher
The research I report here was conducted in Alice Smith's1 fifth-grade classroom.

My collaboration with Ms. Smith began as part of a study conducted by researchers with

 

1The teacher's and the students' names used in this dissertation are pseudonyms.

7 1
the Center for Learning and Teaching of Elementary Subjects at Michigan State

University. In 1989, researchers with the Elementary Subjects Center were interested
in collaborating closely with a small number of teachers and supporting them in making
meaningful changes in their teaching. Much of this research was conducted In
professional development schools affiliated with the Partnership for the New American
Education housed at Michigan State University. Based on the model presented by the
Holmes Group (1990), professional development schools were intended to provide
teachers with opportunities to work with each other and in collaboration with university
researchers to examine and make changes in their teaching as well as play a larger role
in the administration of their school. Teachers in these schools were given release time
to meet and reﬂect with university researchers and there was a general expectation at
the schools that teachers would enter into collaborative research projects with
university researchers. Smith was one of four teachers who responded to a request we
made to teachers to form a working group to think about mathematics teaching and
learning and changes they might make in their teaching. As a part of our collaboration,
we studied the difficulties involved in changing interactional norms in the classroom and
attending to students' mathematical thinking (Perry, 1981; Wertsch, 1984).

In the fall of 1992, the upper elementary level teachers decided to departmentalize
the intermediate grades. Smith was scheduled to teach four mathematics classes--two
fourtlvgrade classes and two fifth-grade classes. Smith wanted to teach these classes in
a way that reflected the spirit of current reforms in mathematics education. (National
Center for Research on Teacher Education, 1989; National Commission on Excellence in
Education, 1983; National Council of Teachers of Mathematics, 1989; National Council
of Teachers of Mathematics, 1991; National Research Council, 1989). To help her
reach this goal, she attended summer seminars on mathematics education and joined a

group of teachers and university personnel who were reading and talking about

72
alternatives to traditional mathematics instruction. Each of these things contributed to

the pedagogy Ms. Smith used in her classroom.

Ms. Smith presented her students with many different types of tasks. The previous
Spring, the school district had adopted a new mathematics textbook. Smith was
interested in using the text as much as possible. As a result, Ms. Smith's students were
presented with tasks that sometimes represented more traditional school-math work and
others that reflected the problem-solving emphasis of the recent reforms. In each case,
students were expected to complete the tasks and justin what they had done to complete
them. Often, new tasks were introduced as homework. Students were expected to take
the task home to work on and return to school with either a justiﬁcation of their solution
or a list of questions their classmates and teacher could help them answer.

In the classroom, students worked in “math groups“ made up of three or four
students. In these groups, the students would discuss the tasks and their solutions or
questions. The ultimate goal of these groups was to construct a consensus solution to the
task and a justiﬁcation for the solution. While the students worked, Smith wandered
from group to group “putting out fires,“ entertaining questions and rewarding groups
that remained on task.

When most of the groups were ﬁnished with the task, Smith usually brought the
class together to discuss their solutions. These discussions began with one group
presenting their solution to the task and often ended in exciting conversations about
different mathematical ideas-conversations which sometimes lasted for days. During
these conversations, students were encouraged to write the different solutions in their
math notebooks.

In short, although Smith was ultimately concerned with answer to the task, she
stressed understanding why the solutions her students used arrived at the correct

answer and that there may be many solutions to the same task. In her teaching, she

73
worked diligently to make her students' thinking public through conversation and

writing.
The Students and their Families

The students and their families that took part in this study represented the social
make-up of the classroom, the school, and the neighborhood. The students lived in one of
two neighborhoods. One feature that divided the two neighborhoods was whether students
walked to school or were bused from a “downtown“ neighborhood. The students who
walked to school came from the neighborhood immediately surrounding the school. This
neighborhood comprised mainly Caucasian working-class families. Although Ms. Smith
described virtually all of her class as having “working-class backgrounds“ and “blue-
collar home environments,“ the neighborhood surrounding the school was the more
attractive of the two. Two of the six students in this project came from this
neighborhood.

The downtown neighborhood comprised mostly minority families and was considered
something less than attractive. Ms. Smith described it as an “unprotected neighborhood
with old cars in the driveways and lawns.“ In the middle of this undesirable
neighborhood there was an area called “Capital Commons,“ a subsidized housing
community where rent costs were based on the family's income. Ms. Smith described
this area as a protected island in the middle of an otherwise dangerous neighborhood. The
people living in Capital Commons, Ms. Smith told me, came from middle-class
backgrounds and had middle-class values. One student who participated in this project
lived in Capital Commons, the other three students came from the downtown
neighborhood surrounding Capital Commons.

Of the six students who took part in this study, three were boys, three girls; three
were African American and three were Caucasian; four of the students came from single
parent families, two from two parent families. Each family was supplied with an

audiotape recorder, tapes, and instructions on how to use the recorders to tape

7 4
homework conversations. At the conclusion of the project, the tape recorders were given

to the students for their participation.

Although the students share many characteristics (e.g., they are all in fifth grade
and have the same teacher) they differ in many ways as well. Each student represents
different social and ethnic backgrounds. They bring with them both academic and out-
of-school histories that influence their completion of the tasks. Among a number of
other things, there are three things that stand out as particularly important to this
study. These characteristics include parents' experiences with math in school and in
their “everyday lives,“ the students' success or failure in scth math, and the normal
homework routine in each student's home. In this section I look at the characteristics of
the students and their families that might influence their completion of the school math
tasks. These characteristics are summarized in table 3.1.

Kathy

Kathy, according to Ms. Smith, was and always had been a good math student. She
was popular among a group of students Ms. Smith called “potential leaders.“ These
students were characterized by their middle-class backgrounds and values and involved
parents. All of the students in this group participated in the “Math-a-Rama“ program-
-a program in which students from schools throughout the district competed in
mathematical contests which emphasized computational speed and accuracy.

Kathy lived in Capital Commons with her father and younger brother. Kathy's
father was in school at a nearby university, majoring in chemistry. At the time of this
project he was applying to medical schools across the country. To put himself through
college he worked as a medical chemist in a local laboratory. He told me he used
mathematics constantly in his work.

Well I work as a medical chemist and we use, ah, all the way from extremely

complex polynomials to figure things out to, you now, just simple ratio problems.

Oh, the concentration of an unknown is equal to the, ah, the percent concentration of

unknown, you know, divided by the, ah, the concentration of a standard. Different

things like that, you know, just very simple things all the way to very complex.
~ And then, everyday in class, ah, I'm taking a class right now where it's a, um,

75

physiologic, ah, study of physiological problems with a hand-held computer and so
we're taking complex equations and, and manipulating them in a computer and using
them to calculate cerebral blood flow and um, ah, metabolic rates and different
things like that, so, it’s a real common thing.

Table 3.1

Characteristics of Student Participants and their parents

 

 

Academic Household Parents'
Student Neighborhood Etihnicity history makeup background
Karen Surrounding Caucasian High average Father, Auto worker,
Brother Geometry ll
Kathy Capital Caucasian Leader, Father, Medical
Math-a-Rama Brother chemist, pre-
med student
Pete Surrounding Caucasian Emotionally Mother, Aunt, Grandmother
impaired, Grandfather, helped with
temper Grandmother, homework
tantrums two siblings
Ronnie Downtown African- Goodstudent, Mother Goodmath
American college student, school
material, math had
Math-a-Rama charged
Shaundra Downtown African- Goodlangiage Mother, Father
American arts student, Father interested in
difficulty in engineering and
math math.
Tony Downtown African- Not motivated, Mother, Mother
American athletically Step-father, confident in
oriented Brother, school math.

Step-brother

Although he had taken many math courses, Kathy's father was not completely

comfortable with what he called the “new math.“ His discomfort came from the “creative

part of math.“ He recounted math classes where he was given a textbook with pages of

computational problems. Now, he suggested, students are asked to develop their own

problems and formulas. The expectation to create unique problems and formulas was

part of his university math classes as well. In those classes, he felt somewhat

7 6
disadvantaged in relation to recent high school graduates. They had experience doing

these sorts of things; he did not.

His discomfort did not result in scorn for the creative part of math. Rather, he
thought it was important and he was disappointed it was not part of his mathematics
experience. It did, however, make him a bit uncomfortable when helping Kathy with her
math. Ms. Smith sometimes asked the class to write problems they thought their
classmates could solve. Kathy's father told me he was sometimes uncomfortable helping
her with this part of her homework. He said, “In the story writing aspect she might be a
little bit better than I am already just because I wasn't exposed to it as early.“

Studying or doing homework was a family activity in Kathy's home. Each person had
his or her own work station. Kathy's father sat at the dining room table, Kathy sat at her
desk and Kathy's brother, Bobby, sat at a smaller table nearby. On most nights they
listened to music while working. They took turns choosing the evening's music. Because
they all had work to finish, there were rules for when Kathy or her brother could
interrupt their father. He needed, he told me, to maintain a train of thought to complete
his homework and couldn't always be interrupted. Kathy and Bobby were asked to go on
to other problems and keep any questions they had until their father was at a place in his
work where he could take a break. When he was between problems or came to a break in
his work, Kathy's father asked Kathy and her brother if they needed any help.

Tony

Tony lived downtown with his mother and stepfather and many children. One
brother had recently been released from prison on an educational program. Ms. Smith
characterized another brother as being “pretty temperamental.“ Tony's stepfather also
had two children from previous marriages. His son, Tony's stepbrother, was in the
other fifth-grade class in this school. His daughter was in Ms. Smith's class at the

beginning of the year, but soon moved out of state to live with her mother.

7 7
Ms. Smith described Tony as “not motivated and athletically oriented.“ Tony had not

been academically successful in his years in this school. “He can surprise you, though,“
Ms. Smith told me. Tony read at a third grade level and was weak in math, but he “knows
his facts.“ Tony, Ms. Smith told me, was not very attentive in class, although he would
ask some questions. He had a slight hearing problem that may have influenced his being
perceived as inattentive.

Tony's mother worked wim him on his homework. She told me she was comfortable
helping him. she made it clear that she stressed school work in their home. Tony, she
said, came home from school and sat at the kitchen table to do his work. They worked
together while she prepared dinner. Throughout the homework sessions, she was
available to answer questions. During our conversation, she asked if it was okay if she
raised her voice during the recorded conversations.

Karen

Karen lived with her father and brother in the neighborhood surrounding the school.
Karen's mother left many years before. It was a difﬁcult experience for Karen and her
brother. Ms. Smith told me of a “heart wrenching“ story Karen wrote about her mother
leaving and that her brother had been labeled “explosive“ during that time. In spite of
this turmoil, both Karen and her brother had adjusted very well. Karen made the ﬁrst
quarter honor roll as did her brother in middle school.

Karen had spent most of her academic career at this school. She did, however,
transfer to another school for a short time. While at this other school her teacher
thought Karen had academic problems and referred her to the special education program.
Her father became very upset and pulled her out of the school and brought her back to
this school. Ms. Smith told me that in this school population, Karen, although she has
'some weaknesses,“ is a strong average student. Indeed, in Ms. Smith's classroom she is
an above average student and receives mostly 'Bs' on her report card. Ms. Smith

suggested some of this success might be attributed to her “tremendous level of

7 8
motivation.“ Ms. Smith used some examples to illustrate Karen's abilities and

weaknesses. She told me Karen did ﬁne on her weekly spelling lists, but had problems
with common words in writing; she did well in verbal performance, but her punctuation
was weak. Ms. Smith characterized Karen as a “pretty good math student.“ She is, Smith
told me, sometimes confused by mathematical concepts but she perseveres until she
understands. Karen contributed substantially to nearly every class discussion.

Karen's father told me he had gone through Geometry II in high school. He told me
students were expected to do more these days than when he was in high school. His son
was in an accelerated algebra class in seventh grade. This surprised Karen's father
because he remembered algebra as a ninth-grade subject. He told me he still uses some
of the things he Ieamed in math. As part of his job at a local auto assembly plant, he
needs to calculate how many man-hours and parts are needed for jobs described on
assignment sheets. He told me:

I have to take care of parts and make sure the parts are going to be here so as far as

multiplying and dividing by days, you know, and how many we're going to use a day,

that kind of thing. I use it a couple times a week. . . I make sure we have enough, you
know, to ﬁnish the week.

There was a breakfast bar in the kitchen at Karen's house where they sat to do the
homework. Her father described this area as having “stools, good lighting, and a hard
surface“ all of which made it a good place to work. The kitchen was adjacent to the living
room where the television was usually on during homework time. Karen usually worked
alone at the breakfast bar asking questions only when she struggled with her work. In
one conversation Karen's father told me he usually helps with her spelling, but Karen
doesn't need much help with her math homework. Karen's father told me he was
comfortable helping Karen with her math homework, but that Karen often did not need

help.

7 9
Pete

Pete had a history of violent temper tantrums. Ms. Smith told me he had been in the
principal's office nearly every day the previous year. He was placed in special education
classes and had been labeled emotionally impaired. Pete often refused to participate in
class. When asked questions or to demonstrate something at the overhead projector, Pete
would often refuse to talk or get out of his desk. His participation in class increased the
year of this project. Although he would still refuse to participate at times, he
contributed substantially to both small group and whole class discussions.

Along with his refusal to'participate in class, Pete was generally not engaged in
school subjects. Ms. Smith, however, pointed to two things that made her believe Pete
was interested in what she called “material with substance.“ The class had recently
taken a trip to a local museum. Pete took the trip very seriously, preparing questions
for museum guides and Ieaming much about state history before going. During the trip,
he was quite engaged, asking the questions he had prepared and follow-up questions to the
responses given by museum guides. The second example Ms. Smith used to support the
assertion that Pete was interested in substantive material came in class. Ms. Smith
often reads to her class just after lunch. In conjunction with a unit on American Indians,
the class read a book entitled 'l Sailed on the Mayflower“ by Roger Pilkington. One day,
Ms. Smith told the class they could read more after lunch. Some class members objected;
they wanted her to read from a book they had been reading after lunch. Pete argued quite
strongly for reading the “Mayflower“ book.

Pete lived in the neighborhood surrounding the school with his mother, his mother's
sister, his mother's parents, and two younger siblings. Because his mother sometimes
felt uncomfortable helping Pete with his math homework, he often worked with one of
the other adults in his household. Elementary math, she told me, is much more advanced
than it was when she went to school. When I asked her if the math she learned in scth

was helpful when she worked with Pete she told me:

8 0
At the beginning it was, now it’s, doesn't really do any good because I don't know

what he is talking about. . . Whatever it is that he’s doing right now. I've just, I've

given up because I, the first time he brought it home I said 'take it into papa.’

(laughs) I don't even know what you're dealing with there.

Pete's mother thought some of the math she Ieamed in school was useful in her
everyday life. She used an example of buying meat in the grocery store to show how she
needed to calculate prices and the total cost of groceries to stay within a budget.

Doing homework was a daily activity in Pete's house. He and the person helping him
would usually sit at the dining room table. In the background Pete's grandfather would
watch television and listen to the conversation at the table, adding insight as needed. As
it turned out Pete worked primarily with his grandmother.

Shaundra

Shaundra came to this school in the middle of the previous year. Her mother
reported that Shaundra “went on hold“ when they moved to this area and started having
lapses in her school work. Ms. Smith told me the lapses, while sometimes still a
problem, were much fewer and farther between than they had been the previous year.
She told me a story of Shaundra missing an assembly because of unfinished work. While
Shaundra was sitting doing her work she looked up and said, “If you think I'm not doing
well this year, you should have seen me last year.“ This year Shaundra was completing
her work and participating in class discussions.

Ms. Smith told me she was quite impressed with Shaundra's ability to make
connections and draw sophisticated conclusions in language arts. This ability, however,
was not as apparent in math; math was difﬁcult for Shaundra Some of the difﬁculty may
have come from her father's interest in mathematics. He had wanted to be an engineer"
and really enjoyed math. As a result, he spent a lot of time working with Shaundra on
her math. Ms. Smith, trying to teach in a way that reﬂected the current calls for reform

in math education, emphasized mathematical reasoning and problem solving in her class.

8 1
Shaundra would often say her father was telling her to do it a different way. Ms. Smith

thought the different explanations may have led to Shaundra's confusion.
Ronnie

Although Ronnie was quiet and didn't often volunteer his thoughts, he was one of Ms.
Smith's best students. She referred to him as “college material“ and described him as
“Usually quite calm, very self-controlled, diligent with his work, performs well.“ But,
like Shaundra, Ms. Smith thought “this kind of math“ was difﬁcult for Ronnie; that he
would be more comfortable if math were taught in a more traditional fashion stressing
computation rather than reasoning. Like Kathy, Ronnie was quite successful in the
Math-a-Rama program where students compete in computational speed and accuracy.

Ronnie, his mother, and brother moved to the area just before the school year
started. His mother had enrolled in a court reporting program at a local community
college. Ronnie's mother told Ms. Smith they had moved frequently before they moved
here. As a result, Ronnie had not completed a school year in one school. Because Ronnie
liked this school so much, she told Ms. Smith she would make a concerted effort to keep
him in this school.

Ronnie's mother had been a good math student. Math came easily to her and, as a
result, she became the math expert in her family-even her older sisters went to her
for help. she told me of times in her math classes where she would finish an assignment
“like that“ (snapped her fingers) and then just sit and “waste time.” Even with the
mathematical history, Ronnie's mother worried that she had forgotten nearly all the
math she had learned and was a little concerned about the help she could provide Ronnie.
She told me a story of one assignment where Ronnie got “everything wrong.“

I helped him one time and he got everything wrong, okay, and it was my fault. He

was very upset with me and told me he would never ask me to help him with his

homework. . . . It had to do with fractions and l was telling him to reduce them to the

8 2
lowest terms the wrong way. . . . So, after that, it’s like he doesn’t want my help

unless he needs it, then he’ll ask.

She went on to say that scth math had changed so much that there were things she didn’t
know about.

I'm embarrassed to say this, but, he brought, he had his homework one day, I didn’t

understand it at all. . . . I don't remember what it was, it had something to do with,

ah, I don’t know if it was rounding off or something you asked me. No it wasn't
rounding off cause I know that. It was something and it had to do with some kind of
problem that I'd never seen before, you know and I was like ‘I don't understand this’
you know and he’s like ‘some on mom’ and I say ‘I don’t’ you know and I think that's
kinda, I need brushing up on the, the more up-to-date math--how they do it, you
know, the way they do their problems and solve them so that I could know because it
makes it easier for both of us. If I don’t know, it's embarrassing for me to tell him

I don’t know, I don’t understand it.

Her feeling that she had forgotten the math she Ieamed in school and that different
things were being asked of elementary math students that she knew nothing about
contributed to Ronnie’s mother’s concern for her ability to help Ronnie with his
homework.

In spite of the one bad experience and Ronnie’s mother's concerns about her
knowledge of mathematics, Ronnie and his mother worked together on his homework.
Ronnie's mother was a self proclaimed “TV. freak” and that Ronnie worked better in
front of the television. She described their homework sessions this way:

He'll get his books and he’ll come sit with them on the floor and I’ll be on the couch;

I’m a T. V. freak. It seems like he’ll work better when the T. V.’s on. . . . He does his

work, ya know, and then if he has a problem he can just come up on the couch and

tell “Well, I don't understand this or that" you know. That's basically how it goes.

83
The Tasks

The tasks used in this project were designed by the teacher, Ms. Smith, and myself.
Three things were taken into consideration in designing the tasks: instructional
considerations, mathematical considerations, and research considerations. Instructional
considerations included how comfortable Ms. Smith felt about the content and how the
task ﬁt with her ongoing classroom instruction. At the beginning of the year, the school
district in which this project was conducted adopted a new mathematics textbook.
Although teachers were not required to use the text, Ms. Smith wanted to incorporate the
text into her instruction in some way. Some of the tasks we sent home during this
project began as lessons from the new textbook. None, however, came directly from the
textbook. Each task, in some way, grew out of a previous activity or led into a new
mathematical area into which Ms. Smith wanted to move.

Once the topic of the tasks was decided, we looked more closely at the mathematical
ideas we wanted students to explore. Three of the four tasks included an introductory
paragraph followed by four or five questions about the paragraph. The questions
following the introductory paragraph were designed to address speciﬁc mathematical
ideas we deemed important. These ideas ranged from making sure both quotative and
partitive models of division were represented in the ﬁrst task to including zero as a
choice when working with combinations and permutations in the third task. So, although
the topic for the task usually came from Ms. Smith’s classroom instruction, the specific
questions asked were chosen based on different mathematical considerations.

Research and theoretical considerations also led us to use certain tasks and
questions. For instance, recent psychological theories have suggested adults scaffold
(Piaget, 1966; Stipek, 1988) children as they work together on tasks. In an attempt to
illuminate this phenomenon, the tasks often included some repetition of ideas. It was
believed if parents referred back to previous problems or provided less support or

assistance in subsequent questions or tasks they were "tearing down or building up the

8 4
scaffolding.“ Another research consideration concerned the mathematical content of the

tasks. In the initial interviews with parents, they told of math classes they had in
school. In general, the classes they described emphasized a computational understanding
of mathematics. Calls for reform in mathematics education are suggesting students Ieam
to assess situations mathematically, flexibly apply mathematical tools and justify their
solutions using accepted canons of evidence (National Council of Teachers of Mathematics,
1989; National Research Council, 1990). Tasks were chosen that reflected one of these
disparate perspectives to see if the mediation in or out of school changed in relation to
the content being discussed.

Instructional, mathematical, and research considerations influenced each task in
different ways. In the next section I look more closely at each of the tasks used in this
project. What may already be clear is that these considerations are inseparable. The
instructional considerations include mathematical thought, mathematical considerations
involve both instructional and research/theoretical considerations, and
research/theoretical considerations also include both instructional and mathematical
considerations. For the purposes of analysis, however, it is necessary to distinguish
among the three considerations.

Task 1

Instructional Considerations. Just before we started this project, Ms. Smith gave
each student in her class a shopping guide from a local grocery store. Along with the
guide, she gave them a series of math problems2 that could be solved using the
information provided on the guide. Students computed amounts for various items based
on their advertised prices. The problems Ms. Smith assigned focused, for the most part,

on multiplication and division. We decided the ﬁrst task should reﬂect both shopping, a

 

2The term problem is used here in its traditional sense; that is, as a computational
exercise or a short story problem. This usage is consistent with the teacher’s use of the
term.

“real-life” situation,

had.

As a result, we designed a task in which Willie, a student like the students in Ms.
Smith’s class, agreed with two friends to take turns packing lunches. Willie was going to
pack lunches for the first week. Below the introductory paragraph in which Willie and

his task are introduced, there were four questions the students were asked to solve to

85

help Willie pack the lunches. The assignment sheet is shown in ﬁgure 3.1.

 

 

1.

1.8%.le

Willie and two of his friends decided they would take turns packing lunch for each
other. They talked about what the lunches should include and decided on a
sandwich, a piece of fruit, and a dessert for each person. Willie volunteered to
pack lunch for the ﬁrst week. When he checked his refrigerator he found he
needed some food in order to pack lunches for the week. So, Willie went to the
market. When he got there he found buying groceries was a bit more confusing
than he thought and he is asking for your help. Please help him solve the
following problems.

Willie believed that all sandwiches should have tomatoes on them. The
supermarket advertised tomatoes at 3 for $1.00. Willie decided he only
needed two tomatoes for the week.

0 How much would the tomatoes cost?

Willie decided he wanted to put apples in the lunches two days in the ﬁrst
week. The store sold ﬁve apples for $2.00.

0 How many apples will Willie need?

0 How much will Willie pay for the apples?

Twinkees are Willie’s favorite dessert. He wanted to put them in as many
lunches as possible. At the store Willie found that Twinkees come in
packages of 12 and cost $4.56.
0 How many days would one package of Twinkees last?
0 How much would it cost to put Twinkees in the lunches for one

day?

Willie and his friends decided they would split the cost of the lunches each
week.

0 How much did Willie spend at the supermarket?

- How much did it cost each person?

 

earlier, division can be used to answer the questions we chose for this task. Division can

Figure 3.1: Task 1 Assignment Sheet

Mathematical Considerations. Like the shopping guide problems Ms. Smith assigned

and multiplication and division like the shopping guide activity

 

8 6
be thought of in two ways. First, in a partitive model of division the divisor represents

the number of sets among which the dividend is divided. Partitive division answers the
question: “How many items will be in each group if I divide XX into X groups?“ An
example of partitive division would be dealing a deck of cards (52) to four people and
counting the number of cards each person received (13). Second, in a quotative model of
division the divisor represents the number of items in each set and answers the question
“How many groups of X are there in XX?” An example of quotative division would be
taking the same deck of cards and, if you play a game where each player gets four cards,
how many people can play at one time? Both partitive and quotative interpretations of
division were represented in the questions for this task.

Questions one, two, and four each can be answered using a partitive model of
division. In question number one the cost of three tomatoes ($1.00) is divided into
three groups with the quotient representing the amount of money in each group (i.e., the
cost per tomato). One step in a possible solution for question number two would be to
divide the total cost of five apples by five to get the amount of a single apple. Like
question one, the total cost of the apples is divided into five groups with the quotient
representing the amount of money in each group. In question four, students were asked
to calculate the total cost of groceries Willie bought and ﬁgure out how much each person
should pay. The total cost is divided among three people with the quotient representing
the cost each person needs to pay. Question three can also be answered using division,
but a quotative model of division. In this question students are asked how many days a
box of twelve Twinkees would last if Willie put them in each lunch. Since there are
three lunches being packed each day. the question asks students how many groups of
three Twinkees are included in a box of twelve.

Research Consideraa'ons. One of the questions we were interested in during this
project was whether the practice within which people were participating inﬂuenced the

way they solved problems. In this task, if the task is interpreted as a “real life“

8 7
problem, the practice of grocery shopping could constrain the solutions students

construct. Because the numerals represent money in this task, solutions like thirty-
three and one third cent are not appropriate. Thus, when working on the first question,
students needed to negotiate a remainder and round to the nearest cent. Students who
divided one dollar by three came up with either three groups of thirty-three cents with
one cent left over or they wanted to add two cents so that they could have three groups of
thirty-four cents. If, however, the problem was interpreted as a school math task and
the numbers were taken out of the practice of grocery shopping, answers such as thirty-
three and one third might be acceptable.

A second research consideration arose from the notion of scaffolding mentioned
above. Three of the questions following the introductory paragraph could be solved using
a partitive model of division. These questions were written this way to see if parents or
other family members would refer to previous problems while helping the students with
their math homework.

Task 2

Instructional Considerations. When we began discussing task number two, Ms.
Smith suggested we use a set of questions she had already constructed. She wanted to use
these questions to get students thinking about place value and base ten. The assignment
would provide a segue from the material covered in task one to a project in the new
textbook in which students would rewrite numbers in expanded notation. The assignment
sheet is shown in ﬁgure 32:3

Mathematical Considerations. As I mentioned in the section on instructional
considerations, Ms. Smith wanted to get her students thinking about place value and base
ten. Because of this, she wrote questions which her students could solve by multiplying
or dividing by multiples of ten. Questions 1, 3, 4, 5, and 6 fit this model. Question ﬁve

is the only question students might use division to solve. In this question they might

 

3The sheet was actually hand written by Ms. Smith.

8 8
divide 1,420 by 100 to get the number of books in a stack. Computing this problem,

however, requires students to manipulate a remainder. Students solving this problem
without taking into account the fact that books cannot be divided up into pieces of books to
even out the stacks might answer the question saying there would be 14 remainder 20
books in each stack or 14.2 books per stack. An answer for the question taking into
account the nature of books might be eighty stacks of fourteen books and twenty stacks of

fifteen books.

 

Math

1. Keisha had 142 books. Her mom told her she’d give her $10 for
every book she read. How much money would she receive if she read
all of her books?

2. If she received $3 for each book, how much would she receive?

3. If, over three years, she received 100 times as many books as she
had in problem one, how many books would she have?

4. If she received 1000 times as many books, how many would she
have?

5. If Keisha had 1420 books and she put them in 100 stacks, how many
books would be in each stack?

6. How .many groups of 10 equal 100?
H u u H 10 N 1 .000?
H H H H H H 10,000?
If H u H H u 100,000?

 

 

 

Figure 3.2: Task 2 Assignment Sheet

Research Considerations. The research considerations in choosing this task are
twofold. First, it was important to me that the classroom teacher feel a part of this
project This importance was manifested in a couple of ways. As I mentioned before, the
tasks needed to reﬂect ongoing classroom instruction. I did not want to construct tasks
that would require Ms. Smith to teach something other than what she would teach if I
were not in the room. To accomplish this, is was important for her to be involved in the
construction of the tasks. She constructed this task with specific reasons in mind--

reasons that were consistent with her ongoing classroom instruction. Second, the

8 9
interviews with parents suggested they were accustomed to school math in which they

were asked to compute problems like those included in this task. Other tasks on which
students and their family members would work would ask them to think about
mathematics in a different way. These questions provided a “baseline“ with which to
compare other conversations.

Task 3

lnstrucﬁonal Considerations. As I mentioned above, Ms. Smith wanted her students
to start thinking about place value. In the district’s newly adopted textbook, Ms. Smith
found an activity in which students were given a set of three notecards, each with a digit
written upon it. In the activity the students were asked to arrange the cards to show as
many three digit numbers as possible and write each number they found in expanded
notation. Ms. Smith thought this activity would give the students an opportunity to think
about place value. We used the activity as a starting point and designed a task in which
students were asked to arrange the cards and write each number in expanded notation, as
it was described in the textbook. We expanded the activity to include two other things we
thought were important. The students were asked to determine when they had found all
the possible numbers, and, in the end, determine a way to predict how many numbers
could be found from a set of digits.

This task is the only one that went home twice. The assignment sheets are shown in
figure 3.3.

Mathematical Consideraﬁons. Among the major emphases of the recent reforms in
mathematics education (National Commission on Excellence in Education, 1983; National
Council of Teachers of Mathematics, 1989; National Research Council, 1989) are the
students' understanding of number systems and number theory and their ability to
reason through various mathematical problems. We hoped this task would require

students to do those things.

9 0
Math

 

1. Compile a list of three digit numbers that can be made using the digits
3, 4, and 7.

A. How many numbers did you come up with?
B. How do you know when you’ve found all the numbers?

2. List all the different numbers in order from smallest to largest
A. Tell why you ordered the numbers as you did.

3. Write each of the numbers you find in expanded notation.

 

 

‘ A Explain what each digit means in each number.

 

Math

1. Write down as many four digit numbers as you can using the digits 0,
2, 5, and 9.

A. How many numbers did you find?
B. How do you know when you’ve found all the numbers?
2. List all the different numbers in order from smallest to largest.
A. Tall why you ordered the numbers as you did.
3. Write each of the numbers you find in expanded notation.
A. Explain what each digit means in each number.
4. Now that you have worked with both three and four digit

combinations, can you think of a way to predict the number of
combinaﬁons you can make with a certain number of digits?

 

 

 

Figure 3.3: Task 3 Assignment Sheet
The task asks students to explore two different mathematical areas. First, students
were asked to arrange the mree digits in as many ways as possible, which provided them
with an opportunity to think about permutations and combinations. Second, students
explored place value by ordering the numbers and writing them in expanded notation.
The task also provided two opportunities for students to reason mathematically. First,

in traditional mathematics classes, students have completed their problems or

9 1
assignments when there are no more problems to compute and they know they are right

or wrong when the teacher tells them so. Yet, in mathematics there is no one available to
tell you when you are finished or whether your solution is adequate. Developing the
ability to assess the reasonableness of a solution is one of the goals of mathematics
education. After students were asked to record the three digit numbers they found, they
were asked to explain how they knew they had included all the possible permutations;
that is, how they knew their solution was adequate. The second part of this task was
given to students after they had discussed their answers to part one. Part two was
designed to give students an opportunity to gather similar data with a different number
of digits and make sense of the entire corpus of data. This part of the task differed in two
important ways. First, zero was included in the four digits. A four digit number with
zero in the thousands place (0295) represents the same value as the three digit number
without the zero (295). Students , we believed, might question whether zero could be
placed in the thousands place in a four digit number. Second, part two culminated in
with a question about a general method for estimating the number of permutations that
could be made with a given number of digits.

Research Considerations. As with the other tasks, the different considerations
overlapped for third task. This task was designed to address two research questions.
First, the content of this task, although consistent with the recent calls for reform, was
perhaps at odds with parents’ mathematics experiences. The task was, in part, designed
to see if the content of the task changed the conversations between parents and children.
Second, this task, like me ﬁrst task, was designed to see if parents or students would
refer back to earlier problems or tasks during the conversations.

Task 4

Instructional Considerations. Task number four was the last assignment before the

Winter Holidays. Because of approaching holidays, Ms. Smith wanted to “wrap things

up.“ The task included something from each of the three previous tasks. It began with a

9 2
story about Nick and his family’s holiday celebration. Following the story was a series

of questions about the festivities. Each of the questions reﬂected, in some way, one of the

first three tasks.

 

Math

Nick and his family always celebrate the holidays in the same way. They
gather together at Nick's house and exchange a few gifts. After that they
sing holiday songs and eat cookies and fruit. Nick decided this year he was
going to give his brother and sister some presents. He would like you to
help him with his Holiday plans.

1. Nick has two sisters and two brothers. He wanted to give each of them
a pair of socks, so he bought four pairs--a red pair, a blue pair, a
green pair, and a purple pair. Nick couldn't decide what color to give
each person. How many different ways can Nick arrange the socks for
his brothers and sisters?

2 . Nick was also in charge of choosing the songs they were going to sing.
He decided, based on what they had done in the past, they would sing
for two hours. Each song, he thought, would take about ten minutes.
How many songs did Nick need to choose?

3. As was their tradition, Nick wanted to buy his brothers and sisters
some fruit. He thought he would get each of them an apple and an
orange. At the market apples cost 3 for 99¢ and oranges cost 2 for
60¢. How much would Nick need to pay for the fruit?

4. Nick also wanted to give each of brothers and sisters some holiday
cookies. He bought 20 cookies. How many cookies would each person

‘ get?

 

 

 

Figure 3.4: Task 4 Assignment Sheet

Mathematical Considerations. Students were asked in the first question for the
number of permutations Nick could make with four pairs of socks. This first question is
similar to what the class did in task number three, but asks only for the number of
permutations and not the permutations themselves. The second, third, and fourth
questions reﬂect tasks number one and two. In the second question students were asked
to help Nick decide how many songs he would need to choose. This question represents a
quotative model of division; that is, it asks 'How many groups of ten minutes are there in
two hours?“ The third question asks the students to compute the cost of fruit at a certain

price. This problem reflects the connection to a “real life“ situation from task number

9 3
one. Finally, question number four asks students to compute a partitive division

problem.

Research Considerations. The major research consideration for this task was to see
if students, their parents, or anyone else would make connections among the problems
they were working on for this task and those they had worked on earlier.

The Data Collection Path

The data collection path is shown in figure 3.1. The path started in scth where Ms.
Smith introduced each task during large group discussions. Each of the introductions, as
was all large group interaction in the classroom, was videotaped or audiotaped.

Collecting the data for this project depended heavily on the cooperation and
assistance of the volunteer families. These people all agreed to record the conversations
they had with their children about their math homework. If they chose not to record
these tapes, no data could have been collected for the in-home portion of this project.
Four tasks were sent home as part of this project. The overall return rate for the tapes
recorded in students’ homes was 79 percent (19 of 24 tapes). For the first task, all six
families returned the tapes. On the second task, which was recorded the Tuesday before
Thanksgiving, three of six tapes were returned. All six families returned tapes for the
third task which was the only task that lasted more than one evening. For this task,
students and their family members discussed related tasks on two consecutive evenings.
These conversations were recorded on opposite sides of the same tape. All families
recorded conversations around both parts of the task. The fourth task resulted in four of

six tapes. The tapes returned by each student is documented in table 3.2.

94

         
     
   

    
 

 

 
   
 

     
 

  
  

  
       

 
     
              
          
  
 

 
 

Students’
Whole class homes Math groups Whole class

Karen and

two others

\ Kathy and

Kathy , 9 three
/ others

:atrhen, Shaundra Satin,
‘ a y, Shaundra and three a y,
Pete, \3 Pete,

, others .
Ronnie, Ronnie,
Shaundra, Shaundra,
Tony and Tony and
their Pete their
classmates classmates

 
 

 

 
 

Pete, Ronnie,
Tony, and
Jason

   

l Ronnie
N
Tony

Figure 3.5: The Data Collection Path

 

  

 

After the students worked on the tasks at home, they brought them back to the
classroom. The normal routine in the classroom included a short time during which
students met and talked in small groups. The six students who participated in this
project were members of four different small groups. Tony, Pete, and Ronnie, were in
the same group. The other three students, Shaundra, Kathy, and Karen were members 'of
different groups. The small groups including the student participants were either video
or audio taped. Each day, one group was video taped and three others were audiotaped.
Which group was videotaped depended in part on which students were in class on a given

9 5
day, and which groups had been video taped in the recent past. An effort was made to

video tape each group the same number of times. After students met in small groups,
they would come back together as a large group to discuss their solutions to the tasks.
Representatives from different groups presented the solution their group agreed upon
and other students would comment on the solution. Usually as a result of these
discussions, students would deem solutions either acceptable or unacceptable. These
discussions were all videotaped.

Table 3.2

Returned tapes

 

Students

Task Karen Kathy Pete Ronnie Shaundra Tony

1 Notape x x x x x
2 x Notape x Notape x Notape
3 x x x x x x
4 x Notape Notape x x x

In both settings, at home and in school, students were encouraged to record all their
work on paper. They were asked to leave all marks on their papers so that their work
could be followed while listening to the conversations they had while working on the
tasks. Students' written work was collected and photocopied following each task.

Methods of Analysis

The analysis followed the data collection path. At each stop along the path, I asked
different questions to better understand the interactions at home or in school and how the
interactions influenced the students’ work. At the same time I looked at individual
conversations, I looked for strands of inﬂuence on the students' work; that is, was there
evidence that their interactions in one setting inﬂuenced their completed task. The

researcher task is an analytic one. Varenne, Hamid-Buglione, and McDermott, and

9 6
Morison describe it this way: The researcher's task "consists in guiding an audience

through the artifact to highlight selected properties. In the process, it is not only
permissible, but actually required, that the analysis substitute theoretically meaningful
words for those of the informants." (Varenne et al., 1982, p. 63). Different theoretical
perspectives contributed to the data analysis in this study. In what follows I explain in
more detail the analysis at each stop along the data collection path.

Ms. Smith’s introductions to the tasks were analyzed for their contribution to the
students' completion of the tasks. Ms. Smith told me she was concerned that the students
might not answer the same questions if they did not clarify things in the classroom
before taking the tasks home. As I looked at the introductions, I tried to understand how
Ms. Smith worked to alleviate ambiguity in the tasks.

Wertsch’s model of negotiation in the zone of proximal development served as a
starting point from which to analyze the interactions students had at home. As mentioned
in Chapter Two, Wertsch’s model includes three components: (a) the situation
definitions participants bring to the conversations; (b) the intersubjective situation
definition; and, (c) the semiotic moves the participants use to gain intersubjectivity.
For each of the conversations I looked for evidence of the participants’ situation
deﬁnitions. The situation definitions participants brought to bear in the conversations
were analyzed in terms of the socioculturally defined practices they represented.
Practices, it was believed, could inﬂuence what students’ work in two ways. First,
deﬁning the problem in terms of different practices might lead to using different rules
to guide the completion of the task. Second, the participants may bring discontinuous
conceptions of school math to the conversations. I looked for evidence of either of these
influences in the conversations.

I also looked for evidence of an intersubjective situation definition in the
conversations and how the participants negotiated intersubjectivity. Wertsch’s model

included three semiotic moves adults use in conversation to negotiate intersubjectivity.

9 7
These moves include (a) directly telling the child what to do; (b) asking the child to

determine the next move and responding to their suggestion in one of two ways: directing
them to a more appropriate move or changing the way they deﬁned the situation to more
closely match the child’s; or, (c) referring to a completed example for the child to
model. I looked at the conversations for evidence of each of these moves. Although
Wertsch developed his model to explain adult-child interactions, the model also served
as a framework for analyzing the interactions students had with their classmates.

Learning in the Wertsch’s model is evidenced by the child “taking on“ the adult’s
situation deﬁnition during the interaction. Learning from a situated cognition
perspective (Lave, 1993) includes using what has been done in one practice to aid
participation in other practices. To investigate the inﬂuence of students’ interactions in
one setting on their interactions in other settings and on their completion of the tasks, I
looked for evidence of both deﬁnitions of learning.

Whereas using the theoretical perspectives as a starting point for the data analysis
in this study helped me understand the conversations l was listened to and the ongoing
classroom instruction, it also provided an opportunity to look closely at the theories and
their adequacy in describing the data. The analysis led to a series of questions about the
theories themselves. Does Wertsch’s model of negotiation within the zone of proximal
development adequately describe the conversations recorded in this study? what is
lntersubjectivity and what does it mean to gain lntersubjectivity? Does the content
inﬂuence which of the semiotic moves family members used in the conversations? How
do the answers to these questions refine or expand the theoretical perspectives used to

analyze the data?

CHAPTER4
BRINGING THE OUT -OF-SCHOOL WORLD INTO SCHOOL

Situated learning theorists have suggested that socially-defined practices shape the
way people think and learn. Candy sellers, grocery shoppers, and weight watchers all
developed mathematical systems other than the formal mathematics taught in schools
that allowed them to accomplish goals situated in their particular practices. So strong
was the belief that practices shaped cognition that theorists argued the practices
themselves, rather than the individuals who make up the practices, should become the
objects of psychological investigation (Scribner, 1984). But, people participate in
more than one practice at the same time. What happens when two or more practices with
conﬂicting conventions convene? How do people resolve the conflict?

One way to explore the inﬂuence of various practices is to embed potentially
conflicting practices in the tasks presented to different people. In this chapter, I look at
the results of embedding an out-of-school practice, grocery shopping, in a elementary
school mathematics task. Embedding out-of-school practices in the tasks was meant to
illuminate issues of interest to both mathematics educators and educational psychologists
interested In situated learning. Situated learning theorists have argued that the
practices in which we participate shape our thoughts and behavior. Following from this,
it is likely that students, when presented with tasks representing an out-of-school
practice, would work within the rules of that practice. The influence of out-of-school
practices on systems of mathematics is also emphasized in the recent calls for reform in
mathematics education (National Council of Teachers of Mathematics, 1989; National
Research Council, 1989; National Research Council, 1990). The reform documents
have argued that elementary school students, and other mathematics students, should
recognize the situated nature of mathematics. Simple calculation rules traditionally
taught in mathematics classes are not sufficient in all situations. Rather, different

- levels of precision, different computational conventions, and other slight differences

98

9 9
change the mathematical procedures used in different practices. Bringing two practices

with somewhat different mathematical rules together, we hoped, would create a conflict
that the students and their parents would need to resolve as they worked on the task. In
an attempt to design a task that was consistent with the reforms in mathematics
education and issues in situated learning, we brought the out-of-school world into the
classroom by creating a school-math task about grocery shopping.
Task 1

Task 1 began with a story in which the students were asked to help Willie make
some decisions and compute the prices of different items during a grocery-shopping
trip. The assignment sheet is shown in figure 4.1. The practices of school math and of
grocery shopping have different sets of rules that govern computation (Lave et al.,
1984). Whereas traditional mathematics instruction has emphasized computational
accuracy and stringent rules that govern computation, grocery shoppers often loosely
estimate the cost of products to monitor how much money they are spending, to choose
among different brands of the same product, or to make other decisions associated with
grocery shopping. Because Task 1 was a school math task about grocery shopping it was
likely that the two practices would conflict. The conversations the students had with
their parents pointed out that more than these two practices influenced how they
approached and answered the questions.
Question 1: The Rules of Practice

Students in elementary mathematics classrooms are taught to round off numbers to a
certain significant place by following a specific set of rules. James and James (1976)
define these rules this way:

When the ﬁrst digit dropped is less than 5, the preceding digit is not changed; when

the first digit is greater than 5, or 5 and some succeeding digit is not zero, the

preceding digit is increased by 1; when the ﬁrst digit dropped is 5, and all

succeeding digits are zero, the commonly accepted rule (computer’s rule) is to

make the preceding digit even, i.e., add 1 to it if it is odd, and leave it alone if it is
already even.

1 0 0
Task #1

 

Willie and two of his friends decided they would take turns packing lunch for each
other. They talked about what the lunches should include and decided on a
sandwich, a piece of fruit, and a dessert for each person. Willie volunteered to
pack lunch for the ﬁrst week. When he checked his refrigerator he found he
needed some food in order to pack lunches for the week. So, Willie went to the
market. When he got there he found buying groceries was a bit more confusing
than he thought and he is asking for your help. Please help him solve the
following problems.

1. Willie believed that all sandwiches should have tomatoes on them. The
supermarket advertised tomatoes at 3 for $1.00. Willie decided he only
needed two tomatoes for the week.

0 How much would the tomatoes cost?

2. Willie decided he wanted to put apples in the lunches two days in the ﬁrst
week. The store sold ﬁve apples for $2.00.
0 How many apples will Willie need?
0 How much will Willie pay for the apples?

3. Twinkees are Willie's favorite dessert. He wanted to put them in as many
lunches as possible. At the store Willie found that Twinkees come in
packages of 12 and cost $4.56.

0 How many days would one package of Twinkees last?
- How much would it cost to put Twinkees in the lunches for one
day?

4. Willie and his friends decided they would split the cost of the lunches each
week.
0 How much did Willie spend at the supermarket?
0 How much did it cost each pgrson?

 

 

 

Figure 4.1: Task 1 Assignment Sheet
Grocery shoppers, in contrast, use a different set of rules: when confronted with prices
including a fraction of a cent, grocery shoppers round the price up to the nearest cent.
Using this rule accomplishes two things. First, grocery stores, as do other stores, round

prices up at the checkout counter.1 Mimicking the procedure used by the store will

 

1During one presentation of this study, an audience member who had spent 20 years in
the grocery business questioned my conception of grocery shopping. With the advent of
scanners, he told me, grocery stores no longer always round up to the nearest cent.
Rather, it depends on the order groceries go through the check out lane. If three cans of
soup were sold for one dollar, they would cost thirty three cents a piece if they went
over the scanner at different times. He cited public relations and legal reasons for this
pricing: If the store rounded up and a customer bought three cans of soup priced at three
for a dollar and had them scanned at different times, the cost would come out to $1.02.

. The two cent difference between the advertised price and the actual cost of the soup could
anger customers or even lead to law suits. To avoid these problems, grocery stores

1 01
provide a more accurate estimate of the total cost of the items in the shopper’s cart.

Second, many grocery shoppers have budgets which constrain their spending. Rounding
prices up helps insure they remain within their budgets.

Question 1 was designed to bring together these two sets of rules and see which set
the students and their parents--and classmates--would choose to follow. The question
asked the students to determine the price of two tomatoes if three tomatoes cost one
dollar. If the students and their parents used the rules of scth math, the tomatoes
would cost 33 cents each and 66 cents for two. If they used the rules of grocery
shopping, the tomatoes would cost 34 cents each and 68 cents for two. The answers each
student arrived at for Question 1 are summarized in table 4.1.

Five of the six students (83%) answered Question 1, saying two tomatoes cost 66
cents. Although only Kathy and her father listed the cost of the tomatoes at 68 cents, the
choice between sets of rules was apparent in three of the ﬁve conversations recorded."Z
In each conversation different decisions were made about how to answer the questions
that reﬂect different practices and, perhaps, other inﬂuences on how students and their

parents complete school math tasks.

 

round the price down to the nearest cent. So, three cans of soup scattered throughout
your groceries would cost 99 cents. If, however, you sent two cans of soup through the
check out at the same time, the price would be added together (33 1/3+33 1/3=66
2/3) and rounded up to 67 cents (see Shaundra’s father’s explanation in the next
conversation) and a third can coming through later would cost thirty three cents for a
total cost of one dollar. Finally three cans going through at one time would cost one
dollar. Being a grocer, though, is different from being a grocery shopper. Although, my
new understanding of the grocery business will undoubtedly change the way I buy
groceries, Ms. Smith and I used the understanding of grocery shopping described in the
text to design Task One. And, it was Kathy's father's conception of grocery shopping, not
what grocers do, that was brought to bear in their conversation. As inauthentic as these
conceptions might be, they did influence the students’ completion of the task. Newman,
Griffin and Cole (1989, p. 15) have agreed that “Not all change is for the better. In
many cases, the interactions [between teachers and students] can have detrimental
effects or result in changes that can not be assessed as better or worse.” Perhaps we
participated in such an interaction.

2Karen and her father did not record their conversation. Karen's answers were taken
from her written work.

102
Table 4.1

Solution Table for Task 1, Question 1

 

 

Student Cost of the tomatoes Rules
Karen $1.00 + 3 : .33 School math
.33 + .33 = .66
Kathy $1.00 +3: .34 Grocery shopping
.34 x 2: .68
Pete “'00 + 3 = 333— School math
.33 + .33 : .66
Ronnie $1.00 +3: .33 School math
.33 x 2: .66
Shaundra $1.00 +3: .33 r. 1 School math
.33 x 2: .66
Tony $1.00 +3: .33 r. 1, School math
.33+.33=.66

ln Kathy and her father’s conversation, Kathy had trouble answering the question
within the rules of school math. When she arrived at a quotient with a remainder, she
was unsure of how to interpret it. Her father, using rules consistent with grocery

shopping, directed her to an answer.

1 F Okay, so in the ﬁrst question, um, it says, that Willie thought they had to
have tomatoes. Okay, he thought you had to have tomatoes in a sandwich and
that they were three for a dollar. And, ah, so Willie decided he only needed
two for the week. So, if they were three for a dollar and he only had to buy
two, how much would they cost?

2 I43 I, I don't know because do, thirty-three cents it would be three times and
it would go into ninety-nine cents though. It wouldn’t go into a dollar. So
you have to go up one, thirty-four times three is a, is a dollar two. So it
would have to be thirty-four cents remainder two. But, but, but the st,
but

3 F But now hold it. How much is each tomato?

II

103

4 K3 thirty-three cents or thirty-four.

5 F And your saying thirty-four cents, right? So, and he wants to buy tWO, so
how much is it going to cost him?

6 IG Wait, oh thirty-three cents, sixty-six.
7 F Yeah, but you said thirty-four cents, right? You said you had to go up one.

8 I6 Oh, oh yeah, thirty-four. then it would have to be sixty-eight. A dollar
sixty eight, no, just sixty-eight cents.

9 F Right, that’s the answer to your first question, right?
1 0 IG Yup.

Prior to this conversation Kathy had done some preliminary computation. When her
father asked her how much the tomatoes would cost she responded with two possibilities
for the price of one tomato. She was a bit confused, though, because one price-33
cents--when multiplied by three equals 99 cents and the other, 34 cents, when
multiplied equals $1.02. In her thinking at this point, neither of these seems right
because the multiplication she used to check her division should come out to exactly 100
cents and neither of the answers do that.

Kathy seemed to solve the problem without considering the conventions of the
practice in which the task is situated. Instead, she dealt only with the numbers she
extracted from the assignment sheet and without considering the situation neitheruor
both-responses made sense. Looking at the problem this way provided no guidance in
ﬁguring out which way to round the quoﬁent and, as a result, she found herself
confronted with a dilemma.

Her father, however, saw the task as grocery shopping. Although his deﬁnition of
the situation was not explicitly stated, he did and said some things that suggested his
definition was consistent with the practice of grocery shopping. In line twelve, after

Kathy told him the tomatoes could cost either 33 or 34 cents apiece, he, by asking a

pointed question, guided her to agree that each tomato costs 34 cents. Furthermore, in

104

the conversation Kathy's option of 34 remainder two3 was stripped of the remainder;
that is, the extra two cents were no longer an important consideration--a move that can
be interpreted as being consistent with the practice of grocery shopping. Once they
decided that 34 cents was the correct answer, Kathy added that amount to itself and she
and her dad agreed, after a bit of confusion over the computation, that two tomatoes cost
68 cents.

Although Kathy and her father’s conversation was the only one in which the rules of
grocery shopping determined the correct answer, the choice between two sets of rules--
and consequently two possible answers--arose in two other conversations. In
Shaundra’s house there appeared to be no question about which set of rules to follow
while answering the question, but a dilemma emerged as they recorded their answer.

After Shaundra read the introduction to the first task, her father read the ﬁrst
question and wondered aloud about how to answer it. The question asked how much Willie
would pay for two tomatoes if the store sold three tomatoes for one dollar. His wondering
led him to ask how much one tomato cost.

1 D Okay, lets take a look here. Problem number one. Okay, Willie believed
that all sandwiches should have tomatoes on them. The supermarket
advertised tomatoes at three for a dollar. So, Willie decided he needed
two tomatoes for the week.

2 3) Problem.

3 D Uh-huh, that’s a problem. He needs only two tomatoes but they're three
for a dollar. So how much would he pay for the two tomatoes that he
bought?

4 S) Um, I see now.

5 D So, if there’s three tomatoes for a dollar, then how much would one
tomato cost?

6 8) Um,

 

3Kathy’s used the term “remainder" nontraditionally in this example. Rather than
signifying the left over portion of the dividend, Kathy used the term to represent the

~ number of cents over one dollar that results from multiplying 34 x 3. Multiplying 34

cents by three results in a product of $1.02.

105

7 D That’d be three divided into one dollar.
8 82 Three divided into one dollar.
9 D Right.

Shaundra did the division narrating each step as she went along. 'Three goes into
one no times. Three goes into ten three times. Three times three, nine . . . . " When she
paused, even for the shortest of times, her father joined in asking her questions about
what to do next or telling her what to do. Together Shaundra and her father worked
through the division and came up with an answer of 33 remainder one. Although her
father pointed out the remainder and made sure Shaundra had it written down, they did
not discuss it or what to do about it. Instead, they dropped it and determined the cost of
one tomato to be 33 cents with no mention of the remainderuuntil they had answered
the question. After they had agreed that Willie would have paid 66 cents for the tomatoes
and Shaundra had written her answer down on her paper, her father said:

The thing I was wondering is that, you know, when you go to a store a lot of times,

well, this won’t be rounded off. A lot of times they round it off to the next number.

Which it might not be sixty-six cents it might be sixty-seven. Because they're

gonna make sure that they get the advantage with that penny that’s left over. They

always charge you that extra penny. Whether you want to pay it or not. So, that
way they never lose any money.

In Shaundra and her father’s conversaﬁon, they used rules that are consistent with
those taught in traditional elementary mathematics classrooms. But, when they
ﬁnished, Shaundra's father wondered aloud about an alternative answer that was

consistent with the rules of grocery shopping.4 Although he recognized the choice

 

4Shaundra’s father’s conceptualized the rules of grocery shopping slightly different than
the conceptualization used to develop this task. Rather than rounding the price of each
tomato to the nearest cent, Shaundra's father rounded the price of two tomatoes to the
nearest cent. This conception is similar to the rules explained to me by the audience
member experienced in grocery shopping.

1 0 6
between the two sets of rulesuand the resulting difference in answers--they did not

change their answer. What they, or he. believed was expected in scth held more
authority than did the situation in which the task was set. Tony’s mother also pointed out
different possible answers, but chose the answer consistent with school math in their
conversation.

Tony began the conversation by reading the assignment sheet aloud. As he read, he
stumbled over words such as “refrigerator” and “confusing“ causing his mother to
become upset; both words were included in Tony’s spelling list and, she suggested, he
should recognize them. Despite their initial difﬁculty with the assignment, Tony and his
mother worked their way through Question 1.

Tony and his mother decided that the cost of one tomato could be determined by
dividing one dollar by three. They worked through the algorithm together: Tony narrated
the steps as he worked and his mother questioned him about the reason for each step.
When they finished the division, they arrived at an answer of 33 with a “1” at the
bottom of the division bracket. Tony’s mother asked what they should do about me “1.”

1 M This is dollars and cents, remember, so is you going to round one off or
you going to keep on going?

2 TW Keep on going to cents. (pause) No, round it off.

3 M So, how much would each tomato cost? How much would it cost for two
tomatoes?

Uh, each would cost thirty-four cents?

Apiece.

It couldn’t actually cost 33 cent, I mean 34 cent apiece then that would
make it be more than a dollar. If you add thirty-four up three times

TW

5 M Apiece? Or, together?
TW
M

8 TW Let me think, yeah. It’s 33 cents.
Recognizing his difficulty interpreting the remainder, Tony’s mother suggested they
“put the decimal point behind the three“ and “try adding a zero on.” This strategy

reflects another heuristic or rule often taught in mathematics classrooms: carrying a

107

division problem out to more places may result in an quotient with no remainder. Tony
added the decimal point and a zero and carried out the division. Again he arrived at a
quotient with a remainder which his mother helped him interpret.

9 M So in order to round it off, you can take the one and leave it as the
remainder, but we would say they cost thirty three and a half cents
apiece, right?

1 0 Um-hmm, we would get (pause)
1 1 For two. Which you would get if. . . Now you gotta do what, add?
1 2 Um-hmm

1 3 And, what are ya gonna add? (pause)
15 All right then, s6, what did you come out with?

16

TW
M
TW
M
14 TW Thirty-three (pause) two times
M
TW That'd be sixty-six cents.
M

1 7 All right then. Actually sixty-seven cents but you’re not that far

advanced.

Tony's mother, as did Shaundra's father, recognized that different solutions were
possible for this question. One possible solution included dropping or ignoring the
remainder and adding 33 + 33 to arrive at a cost of 66 cents for two tomatoes. The
other solution involved rounding 33 r. 1 to 33.5 and adding 33.5 + 33.5 for a cost of 67
cents. Although it is not clear that she was following the rules of grocery shopping5 the
two possible solutions reﬂect the different sets of rules common to grocery shopping and
school math. But, the decision about which set of rules to follow was not inﬂuenced as
much by the setting in which the question was situated or originated as it was by Tony’s

mother’s beliefs about his ability. She saw the ﬁrst solution (33 + 33 = 66) as a

 

5Tony's mother may be using a “buggy” set of school math rules in which she interprets
any remainder as one half. At the beginning of her explanation, she used the same rules
Kathy did in the conversation presented above. She ruled out 34 cents as the answer
because 34 times three equals $1.02. That rule was attributed to school math classes.

It may be that she, as was Kathy, was searching for a school math rule that ﬁt the
situation.

1 0 8
developmentally prior solution to the “more advanced” solution of 33.5 + 33.5 : 67

cents.

In the conversations Kathy, Shaundra, and Tony had with their parents, they
appeared to choose the practice to which to attach the questions. Depending on their
choice, different sets of rules governed how the question was answered. The choice of
which rules to follow was determined by different things in different conversations. In
one conversation presented here, the choice was determined by the practice in which the
task was set--rules consistent with the practice of grocery shopping were used to
determine an appropriate answer. In the second conversation, the set of rules was
determined by the practice in which the task originated and where the students’ answers
would be assessednrules consistent with the practice of school math were used to guide
their work. In the final conversation, the rules reflected the parent's determination of
the student's ability level. In this conversation the rules used to shape their answer
changed as the parent better understood the student’s ability.

The conversations about Task 1, Question 1, point out that things other than the
practices embedded in the tasks inﬂuenced how students and their parents answered the
questions. The inﬂuence of other practices became evident in the conversations the
students and their parents’ had as they worked on the rest of Task 1. In their
conversations about Question 2, the assumptions the students and their parents made
while answering the questions shaped their answers and may have masked the
mathematical thought in the conversations.

Question 2: Situation Definitions

Question 2 was similar to the Question 1. In both questions the students needed to
determine the cost of one item and, using that cost, determine the cost of a group of
items. Whereas in the ﬁrst question they were asked to determine the cost of a smaller
number of items than were advertised, in Question 2 they were asked to determine the

cost of more items than were included in the advertised cost. The similarity between the

1 0 9
questions was intended to see if parents would refer to the earlier question while

assisting their children in answering Question 2. Referring back to a completed model
or past experience is consistent with instruction within the zone of proximal
development (Newman et al., 1989; Wertsch, 1984). None of the parents referred to
the previous question.

The ﬁrst part of Question 2 asked the students to determine the number of apples
Willie would need if he wanted to pack them in the lunches on two days. Five of the six
students who participated in this study agreed that Willie would need six apples. How
they arrived at this answer varied slightly but, in essence, they took into account the
number of days Willie wanted to pack apples in the lunches (2) and how many lunches
were being packed each day (3). The strategies the students used are summarized in
table 4.2.

The second part of Question 2 asked the students to determine the cost of six apples if
the store sold five apples for two dollars. Two similar strategies emerged among the ﬁve
students who agreed that Willie needed six apples to pack the lunches. In each of the
strategies, the students and their parents first determined the cost of one apple by
dividing two dollars by ﬁve arriving at a cost of 40 cents. To determine the cost of the
six apples Willie needed, the students either multiplied .40 x 6 or added .40 to $2.00
(the cost of ﬁve apples) to arrive at $2.40. Either strategy concluded that six apples
cost $2.40. The strategies students used are summarized in table 4.2.

Although five of the students agreed that Willie needed six apples and they would cost
$2.40, one student (Tony) did not agree with those answers. He answered the questions
saying Willie needed four apples and they would cost ﬁfty cents. Although his answers
are different, there are some similarities between Tony’s solutions and those of the
other students. To understand these similarities we need to look at the conversation Tony

had with his mother as they answered the questions.

1 1 0
Table 4.2

Solution Table for Task 1, Question 2

 

 

Student Number of apples needed Cost of the apples
Karen 6 $2.00 + 5.: .40
.40 x 6 = $2.40
Kathy Intuited 6 $2.00 + 5 = .40
.40 x 6 = $2.40

aid

$2.00+.40 = $2.40

Pete 3 x 2 : 6 $2.00 + 5 : .40
$2.00 + .40 = $2.40
Ronnie Directed to 6 $2.00 + 5 = .40

$2.00-FAQ: $2.40

Shaundra 3 + 3 : 6 $2.00 + 5 = .40
.40 X 6 = $2.40
Tony 2 + 2 = 4 $2.00 + 4 : .50

Tony’s mother read Question 2 aloud: “Now problem number two. Willie decided he
wanted to put apples in the lunches two days in the ﬁrst week. The store sold apples,
sold ﬁve apples for two dollars.” When she had finished reading the question, she
summarized it and helped Tony come up with an answer.

1 M Okay, he only wanted to put apples in the lunches two days in the ﬁrst
week. [Um-hmm] Okay, so the apples are five for two dollars, so what
[are you] going to do with that problem? . . . The key is right here. He
wanted to put apples in me lunches two days in the ﬁrst week.

2 TW That means he needs two apples.

3 M You gotta keep remembering that Willie is making lunch for himself and
his friend. . . two days in the week.

4 TW He gonnaneedfour.

111

5 M He gonna need four, all right.

Tony and his mother determined the number of apples Willie needed by ﬁrst
determining the number of lunches being packed (2) and the number of days Willie was
packing apples in the lunches (2). Using these figures they‘determined Willie needed
four apples. Although the answer Tony and his mother arrived at was different than the
other students’ answers, they used a similar strategy. They, as did the other students,
multiplied the number of apples needed per day by the number of days Willie needed the
apples or added the number of apples used each day together the number of days they
were being packed in the lunches. What differed were the assumptions they used to
define the situation and to determine what would constitute an appropriate answer.
Rather than Willie packing lunches for himself and two friends, they deﬁned the question
(and the entire task) as Willie packing lunches for himself and one friend.

Tony and his mother's answer to the second part of Question 2 is harder to explain.
In the beginning of their conversation, Tony's mother pointed out that five apples sold
for $2.00, but to answer the second part of the question, Tony and his mother divided
$2.00 by four--the number of apples they determined Willie would need--to determine
the cost of one apple. From, this calculation they determined one apple would cost 50
cents. This quotient was the only answer discussed in their conversation and the only
answer recorded on Tony’s assignment sheet. Their answer might reﬂect another
redefinition of the question. They may have redeﬁned the question to read: Willie paid
$2.00 to put apples in the lunches on two days. How many apples would he need? How
much would each apple cost? If these were the questions they answered, their answers
would have been correct.

Although Tony and his mother’s answer might be explained by understanding the
assumptions they used while answering the question, how those assumptions changed is
not explained. Looking more closely at their conversation and the practice of homework

in Tony’s household, provides some insight on how changes like this may come about.

1 1 2
Tony began their work on the second part of Question 2 by writing out the division

bracket. His mother saw what he had written and offered her approval by saying “All
right, talk to me brother.“ As Tony began to do the computation, the loud snap of a mouse
trap being sprung sounded in the background and a voice announced “Well, you got ‘em”
indicating a long sought mouse had just been caught. When she heard this, Tony's mother
excused herself from the table saying “call me when you get ready to do the problem, but
right now, I got to do something.”

Tony continued to compute his answer, narrating each step. His mother monitored
the computation from a distance. Tony quickly suggested that four went into 20 two and
than six times. Believing that Tony had merely guessed, his mother threatened to make
him write his “times tables“ if he did not stop guessing. Tony offered four as the
number of times four went into twenty which elicited the response his mother had
promised. Tony turned his paper over and wrote out the multiplication facts from 4 x 1
to 4 x 5. When he realized 4 x 5 : 20, he offered five as the answer to the second part of
Question 2. His mother agreed and together they completed the computation arriving at
an answer of ﬁfty cents. When they ﬁnished, Tony’s mother instructed him to write his
answers on his paper. Tony wrote: four apples, 50¢.

The subject of their conversation had changed. In the earlier conversation, Tony and
his mother talked about the number of apples Willie needed and other things that shaped
the questions they were answering. But as they began working on the second part of
Question 2, Tony’s mother's attention was diverted away from Tony’s homework and
directed toward other household tasks. Although their conversation continued, it focused
on Tony’s computation and not on whether that computation fit the question they were
answering. The computation Tony did was correct--$2.00 + 4 does equal 50 cent --
and we can judge Tony and his mother's conversation successful if we accept the topic

they chose for their interaction. But Tony's answers were wrong and their conversation

1 13
was unsuccessful if we situate it in the larger picture of Tony’s math assignment; his

answers were determined incorrect when he returned to school.

Homework, itself, represents a socially-defined practice that represents only one of
many streams of activity that occur simultaneously in students' homes (McDermott et
al., 1984; Varenne et al., 1982). At the same time that they assist with homework,
family members fold laundry, make dinner, discipline children, clean house, pay bills,
and participate in other household chores. Each of the practices, except maybe
homework, contributes to the overall operation of the household. Homework may, in
fact, take away from the otherwise smooth operation of the home. As a result, homework
may seem less important than other activities that more directly contribute to the
household. Confronted with a household activity that needs attention, parents may need to
divert attention away from their children’s homework. This appears to be what
happened in Tony and his mother’s conversation. Although she continued their
conversation, her divided attention may have allowed the conversation to change
direction just enough to miss its target. As a result, Tony’s computation, although done
correctly, did not address the question in a way that was accepted in school.

Looking ahead to the conversation Tony and his mother had about Question 3 suggests
that the practice of homework is also characterized by accepted norms of interaction. In
their conversation about Question 2, it became apparent Tony’s mother assumed two
lunches were being packed rather than three. During their discussion of Question 3,
Tony questioned that assumption and received a warning from his mother about the
consequences of doing so.

Tony began their conversation about Question 3 by reading it aloud. The question
asked how many days a package of Twelve Twinkees would last. When he ﬁnished
reading, his mother instructed him to answer the ﬁrst part of the question. “Solve the
ﬁrst one ﬁrst, child. How many days? It says as many days as possible, now, keep that

in mind.” Tony told his mother the Twinkees would last four daysuan answer

I753;- _,_, _

1 14
consistent with the assumption that three lunches were being packed. His mother,

however, disagreed and directed Tony to a different answer. She said, “Now, so there’s
two people, so what would you do?” Tony, recognizing that his mother had defined the
problem differently than he had, objected. “I thought it was Willie and his two friends”
to which his mother responded: “No it’s just Willie and one friend. Willie only got one
friend, it’s two people. If Willie gets another friend I’m gonna go to bed.“ Faced with
the prospect of losing his mother’s assistance, Tony accepted her situation deﬁnition and
never again mentioned his alternative. Tony and his mother completed their work on
Question 3 and the rest of Task 1 assuming two lunches were being packed. This is how
their conversation about Question 3 ended:
1 M Now, what Willie trying to do is put Twinkees in lunch for hisself [sic]
and his friend for as many days as he can. Okay, there's twelve
Twinkees. Now, if two people ate a Twinkee everyday, how many days or
how many times could they have Twinkees per week? (pause) Count by
twos.
2 TW Six
3 M Right!

In Tony’s household, he was expected to accept his mother’s contribution to the
conversation without question. His attempt to question her assumptions resulted in his
being warned about the consequences of continued questioning--his mother would leave
him to work alone. Although Tony’s mother's reaction might be interpreted as
inappropriate or even rude, it may also characterize culturally appropriate norms of
interaction in their household. As in the Warm Springs tribe (Philips, 1993) where
council elders are not to be challenged, it may be that Tony’s role in his household does
not afford him the option of questioning his mother’s position. All of the students who
participated in this study interacted in ways that represented other conversations they
had in their homes. Indeed, Tony's mother, in our first conversation, asked if it was

okay if she raised her voice while they worked on Tony’s homework as that characterized

the conversations they had in the past.

1 1 5
Tony and his mother’s answers to the questions in Task 1 were inﬂuenced by two

forces other than those associated with the practices of school math or grocery shopping.
First, they, or perhaps Tony’s mother, defined the problem in a way that was
inconsistent with the definition expected when Tony returned to school. In their
definition, Willie was packing lunches for two, rather than three people. Although this
assumption caused different values to be used, Tony and his mother used the same
strategies as the other students to determine the number of apples Willie needed. As a
result, Tony’s answers make sense in the context of their conversations although they
were not accepted when he returned to school.

The practice of homework also influenced Tony’s answers. Homework is a complex
practice. It is completed within a tangle of household activities, many of which demand
attention at unpredictable times. As a result, the goals of understanding the mathematics
in the task may subordinated to the goals of other household chores. When Tony’s
mother's attention was diverted to other necessary household activities, the topic of
their conversation changed. Rather than talking about the fit between the computation
and the question they were answering, they began talking only about the computation.
This slight change resulted in an incorrect answer that Tony took back to school. The
practice of homework also includes conversational norms within which students and
their parents agree on deﬁnition of the question they are asking and appropriate ways to
answer it. The interactional norms that characterize these conversations may represent
ways of interacting common to other conversations students have with their parents. As
a result, the practice of homework is different in different students' homes and will
Inﬂuence their completion of school-math tasks in different ways. In Kathy’s home, as
we shall see below, questioning her father's contribution is accepted and even
encouraged.

As the conversations students had about Questions One and Two suggest, many things

contribute to Students’ completion of school math tasks. But, along with the inﬂuence of

1 1 6
the practice of school math, the practice in which the task is situated, and the practice of

homework, students’ previous experience in school math classes-~and, perhaps their
parents’ experience in school math—also inﬂuence how they approach tasks. This
influence is illuminated in the conversations students had as they answered Question 3.
Question 3: Differing Assumptions

Question 3 asked the students to determine the number of days a box of twelve
Twinkees would last and the cost of putting Twinkees into the lunches each day. Taking
the number of Twinkees and determining how many groups of three were in the box
represented a different model of division than was used in Questions One and Two.6

Only Tony’s mother mentioned the difference between the earlier two questions and
this one. The difference, however, was not in the models of division, but in the
question's level of difficulty. During their conversation Tony was having difficulty
decided what operation he needed to use and what values to place in the algorithm. As he
struggled, Tony’s mother interrupted him saying:

This one here is kind of difficult for you then. You did the ﬁrst step. How many days

would a package of Twinkees last? Okay, we determined that two people Is eating out

of it so we determined that it will last six days. Now we got the answer six. Six,

now we got to divide into the cost. How much do Twinkees cost?
Tony and his mother went on to divide $4.56 by 6, the number of days the Twinkees
would last. From their calculation, they determined the daily cost to be 76 cents. As
happened in their conversation about Question 2, Tony and his mother used a strategy
used by other students. But, because of their assumption that two, rather than three,
lunches were being packed, they ended up with incorrect answers.

The other ﬁve students all determined the Twinkees would last four days. In each
case the students and their parents determined the number of days by dividing 12 (the

number of Twinkees in the box) by 3 (the number of Twinkees packed each day). The

students answers to Question 3 are summarized in table 4.3.

 

6See chapter three for a more complete description of the different models of division.

1 1 7
Table 4.3

Solution Table for Task 1, Question 3

 

 

Student Number of days Cost

Karen 12 + 3 : 4 $4.56 + 12 : .38
.38 x 3 = $1.14

Kathy 12 + 3 = 4 $4.56 + 4 = $1.14
Pete 12 + 3 : 4 $4.56 + 12 = .38
.38 x 3 = $1.14

Ronnie 12 + 3 : 4 $4.56 + 12 : .38
.38 x 3 = $1.14

Shaundra 12 + 3 : 4 $4.56 + 4 : $1.14
Tony 6 $4.56 + 6 : .76

Although the students all ended up using the same strategy to determine the number
of days the Twinkees would last, two of them--Ronnie and Shaundra-«began answering
the question in a way that illuminates the influence of their experience in traditional
elementary mathematics classrooms on how they completed the task.

In traditional mathematics classrooms, assignments often include a series of
problems which focus on a particular aspect of computation. To complete these
assignments students must figure out the focal procedure and repeat it each time they
compute a problem. The same format is often used in story-problem assignments:
students must ﬁgure out the strategy for solving the ﬁrst problem and apply that same
strategy to the remainder of the problems in the assignment. Participating in
classrooms where assignments are formatted in this way would lead students to develop
conceptions of school math that reﬂect that structure. Conceptions consistent with this

model may explain Ronnie and Shaundra’s initial attempts to answer the ﬁrst part of

1 1 8
Question 3 and provide a reason for one of the strategies used to answer the second part

of Question 3.

Both Questions One and Two could be answered using a two step strategy. In the ﬁrst
step, students determined the cost of one item (tomatoes or apples) by dividing the
advertised amount by the number of items sold for that amount. After the item cost was
determined, the total cost could be computed by adding the cost for each item needed or by
multiplying the cost by the number of items needed. This strategy could be carried out
by merely pulling the numbers out of the text of the problem and applying the sequence
of operations to those numbers. Most students answered the questions this way: out of 12
answers to the two questions, 11 (92%) reflected this strategy. If students held a
conception of mathematics instruction similar to the one described above, and had
answered both questions using the same strategy, it is likely they would attempt to use
the strategy to answer the remaining questions. Ronnie and Shaundra did just that.

After Ronnie read Question 3 aloud, his mother asked him how he could figure out
the number of days the Twinkees would last. Ronnie suggested the answer could be
determined by dividing “Twelve into four-fifty-six.“ Carrying out this computation
would determine the cost of one item (a Twinkee) and reﬂects the ﬁrst step of the
strategy commonly used in Questions One and Two. The numbers Ronnie used were the
only numbers included in Question 3 on the assignment sheet. Ronnie, it appears, had
taken the numbers from the story and began to apply the sequence of operations that
successfully answered Questions One and Two. '

Ronnie's mother interpreted his response as jumping ahead to the second part of
Question 3 where the cost of putting the Twinkees into the lunches for one day was
computed. She redirected him to another strategy and reserved his strategy for the
second part of Question 3.

1 M No, no, no, no. Not yet. Don't, I’m not, that’s not what I‘m sayin’ first.

Okay, you, you serving lunch for three kids, right? [yeah] So, what's
three to make twelve? How many times does three go into twelve?

2

3

4

RB
M

RB

Three goes into four times.

Okay, so you can get, they can get a Twinkee in their lunch for how many
days?

FouL

In a similar conversation, Shaundra’s father asked her how she would determine the

number of days the Twinkees would last. Shaundra was more explicit about the strategy

she wanted to use.

1

2

9

8)

D
S)
D

B

8'.)

Well, ﬁrst we gotta ﬁnd out how much one cost.
Oh, you do?
Right

Okay, well let’s see. let’s see, I’m trying to ﬁgure. Okay, First off, I
want to do me ﬁrst problem of how many days would one package of
Twinkees last. Okay, we got three people eating Twinkees, right?

Right.

We've got twelve Twinkees. Twelve Twinkees, three people are eating
them, eaﬁng these twelve Twinkees. They could, they going to eat three
Twinkees, three Twinkees are going to be gone for each day.

Right

Right? So, you have to divide that three those three Twinkees into the
twelve Twinkees and you’ll get, and that" tell you how many days the
Twinkees will last. Do you know why? Because you're using three
Twinkees a day. And you have to use, you have to divide the amount of
Twinkees you’re using up.

Right.

At this point, Shaundra’s father used an example to explain the situation to

Shaundra. He told her the Twinkees were like the packages of bologna her parents buy

for her and her sibling’s lunches. Each day, he told her, they use three slices of bologna

and can determine how many days a package of bologna will last by dividing the number

of slices used each day into the number of slices in the package. At the end of the

example, Shaundra began dividing three (the number of Twinkees used each day) into

twelve (the number of Twinkees in a box).

1 2 0
In each of these conversations the students began to answer Question 3 by using the

strategy that worked to answer the first two questions. Their parents, in both cases,
directed them to a more appropriate strategy for answering the ﬁrst part of the
question. The conversations, however, illustrate that the students’ experience in
mathematics classes inﬂuence how they approach school math tasks. Not only are their
answers inﬂuenced by the immediate practices in which they are participating
(homework, grocery shopping, and Ms. Smith’s classroom), but by the practices in
which they have previously participated.

The influence of previous scth math experience may also have contributed to the
choice of strategies students and their parents used to solve the second part of Question 3.
Students and their parents used one of two strategies to answer the second part. The
strategies students and their parents used are summarized in table 4.3. In the ﬁrst of
these strategiesuthe strategy Tony and his mother used--the students and their parents
determined the number of days the Twinkees would last and divided the total cost of the
Twinkees by that number. Three of the six students (including Tony) used this strategy.
The other three students answered Question 3 by determining the cost of one Twinkee
(38 cents) by dividing the total cost of the Twinkees ($4.56) by the number of
Twinkees in a box (12) and multiplying the cost by three (the number of Twinkees
packed each day). This strategy is the same strategy students and their parents used to
answer Questions One and Two. For each question, the dyads determined the cost of one
item and then, by multiplying or adding, determined the cost of the number of items
needed. Using this strategy may reﬂect students’ and parents’ previous school math
experience where sets of questions could be answered using the same strategy.

In sum, students and parents brought to bear their previous school math experience
as they worked on Task 1. This experience may have included a conception of school math

tasks in which all the questions included in the assignment could be answered using the

1 21
same algorithm or strategy. Part of completing the assignment was to ﬁgure out the

strategy and apply it to all the questions.

Math in Ms. Smith’s classroom, however, was different. In Ms. Smith’s classroom
students were encouraged to explore different ways of answering questions. Ms. Smith's
conception of school math also influenced how the students answered the questions in
Task 1. This influence is particularly apparent in the answers the class accepted for
Question 4.

Question 4: Accepted Answers

Question 4 asked the students to pull together what they had done to answer the
previous three questions and determine the total cost of the groceries Willie bought and
the amount each person needed to contribute. Although the students’ answers varied
(there were four different answers), their responses represented two strategies. The
students' answers are summarized in table 4.4.

Table 4.4

Solution table for Task 1, Question 4

 

 

Student Total cost Cost per person
Karen .66+2.40+4.56 : 7.62 $7.62 + 3 = $2.54
Kathy .68+2.40+4.56 = $7.64 $7.64 + 3 = $2.54 r2
Pete .66+2.40+1.14 = 4.20 $4.20 + a = $1.40
Ronnie .66+2.40+4.56 : 7.62 $7.62 + 3 = $2.54

Shaundra .66+2.40+4.56 = 7.62 $7.62 + 3 = $2.54
Tony .66+.50+.76 = $1.92 $1.92 + 2 = .96

In a sense, all six students used a similar strategy to answer Question 4. Each
student extracted amounts from the ﬁrst three questions, added those amounts together,

and divided the sum by the number of lunches being packed. The amounts students chose

122
to add together, however, differed and reflected two different strategies and, perhaps,

different conceptions of school math assignments.

Two of the students (Pete and Tony) added the ﬁnal answers to the ﬁrst three
questions together and divided that amount by the number of lunches they believed were
being packed. Tony’s answers are conspicuous because of the assumptions he and his
mother made while answering the questions. Pete, however, answered Questions One,
Two, and Three in ways that were consistent with the other students’ answers. Taking
the final answer to each of the ﬁrst three questions led Pete to include the cost of putting
Twinkees in the lunches one day rather than the cost of an entire box. As a result, Pete
computed a total cost of $4.20 and an individual cost of $1.40. The strategy Pete and
Tony used may represent a conception of school math assignments where the last story
problem draws together the answers from earlier problems and a arithmetic operation
is applied to produce a ﬁnal answer. Pete’s conversations with his grandmother as they
answered this questions supports this interpretation.

Pete’s grandmother coached him through each of the steps in the strategy described
above. As they began to work on Question 4, she instructed him to look through his
answers to the previous questions. Pete listed their answers to Questions One and Two
and with each answer he received approval from his grandmother; “Okay,“ she said
after Pete wrote down each number. After collecting the answers to the first two
questions, Pete’s grandmother interrupted his work and asked him how he might answer
the question. Pete told her which arithmetic operation he intended to use by responding
“add it.” His grandmother agreed and told him to write down “the cost of your third
day.“ Pete wrote “1.14” on his assignment sheet and his grandmother approved and
asked him for the total cost to which Pete replied “Four twenty.” His grandmother

agreed and Pete carried out the division to determine the cost per person.

1 2 3
The other four students (Karen, Kathy, Ronnie,7 and Shaundra) added the cost of the

items Willie bought at the store and divided that cost by three to determine the cost per
person. Although Kathy used the same strategy as the other three students, she arrived
at a different total cost and cost per person. The difference resulted from the rules she
and her father used to answer Question 1. Kathy and her father used rules consistent
with the practice of grocery shopping and determined the cost of two tomatoes to be 68
cents. In the other three students’ solutions, the tomatoes cost 66 cents. This difference
led the class to accept a list of different appropriate answers.

In class the students worked in their math groups to compare their answers to the
questions. As was common in Ms. Smith’s classroom, each group arrived at a consensus
cost for each item, the total cost of the groceries, and how much each person needed to
pay. When all the groups had reached consensus, each group presented their answers to
class which debated them until appropriate answers for each question were determined.

The class accepted three possible answers to Question 4. The answers the class
accepted were based on different assumptions the students made while completing Task 1.
The answers the class accepted are summarized in table 4. 5. The class agreed that the
apples cost $2.40 and the Twinkees cost $4.56. The difference among the answers .
stemmed from the rules students followed while answering Question 1 and, if the rules of
grocery shopping were followed, what to do with the “r.2.” In the end, the students
decided if they went to a “nice store” that charged 33 cents for each tomato, the total
cost of the groceries was $7.62 and each person needed to pay $2.54 . If the students
went to a “greedy store” they would have paid 34 cents for each tomato and the groceries

would have cost $7.64 which, when divided, left a cost per person of $2.54 r. 2. The

 

7Ronnie began to answer the question using the prices listed in the questions on the
assignment sheet. To determine the total cost, he initially added $1.00 (the price of
three tomatoes), $2.00 (the price of five apples), and $4.56 (the price of a box of
Twinkees). His mother quickly stopped him and redirected him to the strategy presented
here. His initial attempt may also signify a conception of school math assignments
similar to the one used by Pete and Tony. Ronnie and his mother’s conversation about
this question is explored in greater detail in chapter six.

1 24
students determined two ways to interpret the remainder. The ﬁrst, which was attached

to the greedy store solution, was to round the cost up to $2.55. The second way of dealing
with the remainder led to a third solution the class called the “work solution.“ In this
solution Willie’s two friends each paid $2.55 and, because he had done the shopping,
Willie paid $2.54. Neither Tony nor Pete’s answers were included in the list of
appropriate answers.

Table 4.5

Solutions Accepted by Ms. Smith 's Class

 

 

Solution Computation
“Nice“ store .66 + 2.40+ 4.56 : 7.62
7.62+3=2.54

Cost per person: $2.54
“Greedy“ store .68 + 2.40 + 4.56 : 7.64
7.64+3=2.54 r2
Cost per person: $2.55
“Work” solution .68 + 2.40 + 4.56 : 7.64
7.64+3=2.54 r2
Willie's cost: $2.54,
Two friends’ cost: $2.55
The nice store solution used the rules of school math, the greedy store used the rules
of grocery shopping, and the work solution provided a pragmatic way of solving a real
problem. In Ms. Smith’s classroom none of these sets of rules was given primacy.
Instead, Ms. Smith encouraged and expected students to think about different ways to
answer the questions and justify their solutions. Many times the class agreed on a set of
correct answers rather than one correct answer. It may be that Tony’s answers would

have been accepted had he presented a justiﬁcation for his answers in class.

125

Summary

The conversations the students and their parents had as they worked on Task 1
suggest that no one practice determines how students approach and complete school math
tasks. In the conversations described above, the practice embedded in the task, the
practice in which it originated, parents' perception of students' ability, students'
conceptions of school math, the practice of inquiry math in Ms. Smith's classroom, the
intertwined nature of homework in students’ homes, cultural norms of interaction, and
the assumptions made by parents and students as they worked, all inﬂuenced how
students and their parents interpreted and answered the questions.

All of these inﬂuences contributed to a new set of rules the students and their
parents used to guide their conversations. The new set of rules were amalgams of
different practices and beliefs the students and their parents brought to bear in the
conversations. Although these different practices were combined to produce a new set of
rules--a new or amalgamated practice--the different practices did not contribute
equally. Instead, the different practices competed for prominence. In the conversations
the students recorded, school math was weighted more heavily than the other practices
represented in the conversations.

Kathy and her father’s conversation in the next section provides an example of the
different inﬂuences on their completion of school math tasks. In the conversation both
Kathy and her father bring many things to bear that shape their answers to Task 1,
question 3.

The Practice of Math Home work in Kathy's Household

Kathy began the conversation by reading the question aloud.

Number three says Twinkees are Willie’s favorite desert. He wanted to put them in

as many lunches as . . . possible. At the store Willie found that Twinkees come in

packages of twelve and cost four ﬁfty-six. Hold it, he wanted to put them in as many
lunches as possible. (pause) [inaudible] They can have each one in each group.

They can have four in each group. Oh, Ms. Coleman, Ms. Coleman said that there’s
single packs. She said that there is twelve Twinkees there are twelve Twinkees in

the package.

1 2 6
In the middle of reading the question, Kathy paused and drew a diagram on her paper

@0069

Figure 4.2 Kathy’s Diagram

that looked like this:

Through out her instruction, Ms. Smith encouraged her class to draw pictures of the
questions they were trying to answer. Drawing pictures, she believed , helped the
students better understand what they are doing in the tasks. In this case the division
could be done using a picture rather than the division algorithm. Thus, Kathy began the
problem in a way consistent with her school math class. At the same time, she
recognized a discrepancy between her conception of grocery shopping and the one
presented on the assignment sheet-Twinkees don’t really come in packages of one, she
thought, but because Ms. Coleman told her they do, it was okay to use packages of one for
this question. This inconsistency led to a discussion about whether Twinkees come in
packages of twelve in which each Twinkee is wrapped individually and whether or not
this question authentically represented the practice of grocery shopping.

1 D Yeah, you can buy a box of twelve. That’s what they’re talkin’ about.

2 I0 That’s stupid though, because when they have 'em, they have regular
Twinkees though there’s two in each pack.

3 D Right, yeah.

4 IO And there’s twelve packages. Like, okay, you buy a big box . . .

5 D No, there’s not twenty-four

6 IC- Yes there is, Yes there are.

7 D Well I’ve never seen ‘em that way, but anyway it doesn’t matter we have
to do with what they’re giving us here. Okay?

8 IG But, Ms. Smith . . .

9 D So Twinkees are Willie’s favorite desert and he wanted to put them into as
many lunches as possible. At the store Willie found that Twinkees come in
packages of twelve and cost four fifty-six. How many days would one
package of Twinkees last? (pause)

127

10 IG Four

1 1 D Um-hmm, because he’d use three a day.

1 2 IO Three a day?

1 3 D Yes. Where do you think you got four?

14 I6 Oh. Oh,lsee,aday.

1 5 D You’d use three a day divided by twelve is going to give you four days. So,
a package will last four days. Okay, so the first answer is four.

1 6 IO Okay. Okay, I got to label. Six

1 7 D Oh yeah, that’s right. Make sure you put your units on there. Very

common mistake in story problems not to put your units.

1 8 IG Six. How many apples will Willie need? (pause) Four days. Four, How
many days will one package of Twinkees costalast. Four days, days.

1 9 D You probably don’t need to record this all.
Although Kathy's father told her he had seen packages of twelve individually wrapped
Twinkees (his conception of grocery shopping), in the end he invoked the rules of school
math saying “it doesn't matter we have to do with what they’re giving us here“ (line 7).
In the face of Kathy’s protest, her father reread the question and Kathy provided the
number of days the Twinkees would Iast--four. Kathy, although she agreed on the
answer, questioned her father’s eXplanation that Willie would use three Twinkees per
day. Kathy's diagram showed four Twinkees for each person, not four days. In order for
her to understand her father's explanation, she needed to reinterpret her diagram.

After they resolved this dilemma, Kathy began to document their work on her
assignment sheet. In line 16, Kathy mentioned that she needed to “label“ her answers.
Her father agreed and told her to “make sure you put your units on there.“ The term
“units“ was never discussed in Ms. Smith's class. Rather, it appears to be a remnant of
Kathy's father's university math classes. Kathy's father often brought up things from
his classes while they discussed these tasks. During their conversations about Task 1,

Question 2 he questioned the assumptions Kathy was making about the taskuwas Willie

1 2 8
making a lunch for himself and two friends or just his two friends. He told her that the

assumptions she made would inﬂuence how she would answer the questions. Kathy told
him Willie was packing three lunches. Her father told her she “probably was right,“ and
they would assume that through the rest of the questions, but that she always needed to be
aware of the assumptions she was making. Like the term “units,“ the idea that the
students need to make their assumptions explicit was not discussed in Ms. Smith's
classroom.

As Kathy documented their work, her father suggested that she did not need to
“record this all” (line 35). This is the first time the “rules” associated with this
study are explicitly discussed by Kathy and her father. The collection of practices that
inﬂuenced Kathy and her father's completion of this task now included grocery shopping,
Kathy’s conception of school math, her father’s conception of school math, and this study.
Each of these practices contributed to what would count as a correct answer and an ‘
appropriate presentation of the answer.

Both the rules of this study and Kathy's father’s school math experience came up
throughout their conversations. In the next segment of this conversation for example, in
line 23, Kathy’s father reminded her to write everything on her paper; another rule of
this study. Near the end of this conversation, Kathy's father again brought to bear his
School math experience. In line 37 he told Kathy “there's many ways to do it“ and later
he translates four times four into “four squared“ (line 53). Both of these statements

are likely to have come from his school math experience.

2 0 I6 How much would it cost to put Twinkees in the lunch for one day? Okay,
so, four days it would cost four fifty-six.

2 1 D So you can divide four fifty-six by . . .

22 I6 Four.

2 3 D By four, if you would like to do that. No, no, no, no. Remember what he
said. Do all work on the paper you're going to hand in.

24 I6 Okay. Four fifty-six

25
26
27

28

29
30
31

32

33

34
35
86
37

88

:39

40

<1

<2

<3

44

45

603505 050505060

U

129

Do it right here.
Four into four fifty-six.
Um-hmm

No, I don’t think so. Cause it said how much would one day [um-hm], so

one day...

How many days? How many days did you say up here?
Four days.

Four days.

But, wait, it says ‘how much will Willie pay for,’ for wait, okay ‘how
much would it cost to put Twinkees in the lunches for one day.

But we already know what it’s gonna cost to put them in for four days [Tl
for four days]. So, if we divide that number by four that’ll tell ya, tell us
what it is for one, right?

Oh, ‘okay.

Right?

Four

There's many ways to do it.

‘kay. Four goes into four one time, one times four, I need a favor, I need a
better background.

You’re right you do. Use this for now.

‘Kay, so. one into four once is four, zero bring down five, four goes into
five, that’s one, four goes into one, zero, zero bring down your six, four
goes into six . . . It’s not going to cost eleven something. I did something
wrong. It’s divide

No, not eleven something. Oh, have you done division with decimals yet?

I don’t think so.

I don’t think you have either. Well instead make it four hundred and
fifty-six pennies then instead of four dollars and ﬁfty-six cents. I mean I
could show you how to do decimal division but if you haven’t covered it yet
it’s probably just better to learn what you know. Four hundred fifty-six
cents now. Remember, so you’re answer is going to come out in cents.
Okay. (pause)

You should even label that right out here. (pause)

46

47

48

49

50

51
52
53

54

55

56

57

58

59

60

61

62

63

64

65

60

6°60

C360

I'G

D

130

Cents. Okay. Four days, four days goes into that sounds dumb, “four days
goes into . . ." four hundred fifty-six cents.

Yeah, it’s better just to leave that un . . ., you know, you don’t have to
worry about it because it's not going to come out

‘Kay

Actually it should.

Four one, four that’s zero bring down the five, one goes into four four
times, that's one bring down your six, four goes into sixteen . . . Is it four
times four equals sixteen?

Hmm?

Four times four is sixteen isn’t it?

Um-hmm, four squared.

Four goes into sixteen four times. four times four is sixteen, zero.
Exactly one fourteen.

Now wanna hold it. Exactly one hundred and fourteen, right?

YUP

Remember, we have this in cents so it's exactly one hundred and fourteen
cents.

Okay

Which you know is a dollar fourteen but still you have to leave it in the
same unit. You can’t change that.

Okay.
Where’s the graph paper Kathy?

That I gave you? Up in your room. I gave you a whole pile of graph paper
up in your room. There's no graph paper down here.

Well, we don't need to record this, okay? [okay] anyway, go ahead, go on
to the next one.

Okay. Number four. Okay so each one would cost one fourteen. fourteen a
day. Ah, boy, okay. Now . . .

Hold it that’s one fourteen per day.

Throughout the last part of this conversation, Kathy's father brought to bear a rich

knowledge of units. After he and Kathy agreed that they needed to divide $4.56 cents by

1 31
four to determine the cost of putting Twinkees in the lunches for one day, Kathy began to

compute the answer. After working a short time, Kathy announced she came up with
“eleven something,“ but that could not be correct. In line 41, her father asked her if
she had “done division with decimals yet?" When Kathy answered that she had not, her
father told her “its better to Ieam what you know” and suggested they change the units
from dollars to cents. By doing that, Kathy could divide 456 by 4--a division problem
with which she was already familiar. Kathy’s father reminded her often throughout the
rest of their conversation that they had changed the units and she needed to document
that.

In essence, Kathy ﬁrst checked to see if the question represented an authentic
grocery shopping task. If it did not, then the discontinuity could be resolved by invoking
the rules of a different practice. In this case, whether or not the question was authentic
was overridden by the rules of school math where it often does not matter if the
questions are authentic. Documenting their work on the task included drawing a picture
of the solution in a way that was consistent with doing math in Ms. Smith’s classroom,
making sure that the units they used were clearly marked on the paper in a way that
represented Kathy‘s father‘s university math classes, and pufﬁng all documentation on
one sheet for this study. All of these things represent a collection of practices that
contributed to the amalgamated practice of “math homework in Kathy's household.“

Summary and Conclusions

Completing Task 1 in each student’s household represented a unique practice--an
amalgamated practicenthat reflected the participants’ experiences in other socially-
deﬁned practices and different conceptions of the practice of elementary school
mathematics. Although the students all worked on the same tasks and were held to the
same assessment standards when they returned to school, their responses to the

questions in Task 1 fulﬁlled requirements beyond those of Ms. Smith’s class. The

132
requirements of each of the practices brought to bear in the conversations were at least

partially met in the conversations.

Amalgamated practices are likely to be developed in any household where students
and parents work together on school math tasks. In any conversation, students and their
family members are going to bring to bear experiences in various, related practices.
Their attempts to fulfill the requirements of those practices will always lead to the
development of an amalgamated practice. This developmental process is magniﬁed in
instances where the out-of-school world is brought into schools and homes as more
practices are likely to contribute to the amalgamated practice.

The development of amalgamated practices as the students worked on Task 1 both
refutes and supports the contentions of situated Ieaming theorists. On one hand, the
notion of amalgamated practices refutes early contentions that practices determine what
and how people think and do. Although different practices clearly inﬂuence people, they
are not deterministic as suggested by Scribner (1984) and other theorists. As Cobb
(1993) and his colleagues have suggested, practices, while they have their own history,
are made up of the people who participate In them and the experiences they bring to bear
while participating. To understand a practice, and a person's participation in a practice,
it is necessary to understand the experiences the participants bring to the practice as
well as the practice’s own history.

The development of amalgamated practices, on the other hand, supports the more
contemporary contention that Ieaming involves extending what is Ieamed beyond the
immediate situation. In each of these conversations, the students and their parents used
things learned while participating in other practices. What they brought with them to
these conversation reﬂected their own experience in school mathematics classes and in
other socially defined practices. Their participation was not limited to the things in

their immediate surroundings.

1 33
But, just as bringing different practices together in a conversation can lead to

conﬂict, so to can changes within a practice. The conversations presented in this chapter
have foreshadowed the inﬂuence different conceptions of the same practice can have on
students’ work. Recent calls for reform in mathematics education have suggested that
teachers change their instruction to focus on conceptual understanding rather than the
traditional focus on computation. The calls for reform have created a situation in which
different conceptions of elementary school mathematics are likely to collide. In chapter

ﬁve I look at the influence the evolving practice of mathematics education has on

students’ conversations with their parents.

CHAPTER 5
THE INFLUENCE OF EVOLVING PRACTICES ON ST UDENT S’ COMPLETION OF THE TASKS

The conversations presented in Chapter 4, pointed out that the students and their
parents brought to bear various experiences that shaped how they approached and
answered questions. In their conversations they constructed a new set of rules—an
amalgamated practiceuthat drew on and often satisfied the sometimes disparate
experiences the participants brought to bear. To construct the amalgamated practices,
the students and their parents needed to resolve discrepancies among the various
practices. As a result, practices often competed for prominence and, as a result, collided
in ways that illuminated the differences among the practices and how the participants
resolved those differences.

The evolution of a practice can also create situations in which practices, or
conceptions of the same practice, compete. Technological advancements, changing beliefs
and values, and other conditions that demand the practice adapt can all lead to the
evolution of a practice. In tool and die shops, for instance, the development of computer
numeric control (CNC) Iathes have changed what it means to be a machinist (Martin 81
Beach, 1992). No longer can machinists rely on their mechanical knowledge of a metal
lathe and their ability to precisely turn a piece manually. They now must understand the
abstract connections between the computer image of the piece they are fashioning and the
mechanical performance of the machine. As a result, what it means to be a machinist has
changed over time; machinists are now technicians as well as craftsmen.

If two machinists-one a traditional craftsman and the other a newly trained
technician--sat down to solve a problem, they would bring to bear different experiences
that would shape how they defined and solved the problem. The traditional craftsman
might look for the problem in the lathe, whereas the newly trained technician might look

for the problem in the computer. As they worked, their different perspectives might

134

135
compete for prominence as did the various practices brought together in Task 1. This

sort of competition occurs in any evolving practice.

Mathematics education is an evolving practice. Recent changes in beliefs about the
mathematical skills and abilities students should developwhile participating in
mathematics classes (i.e., what it means to do mathematics) have led to different ideas
about mathematics instruction. Reform documents (National Council of Teachers of
Mathematics, 1989; National Council of Teachers of Mathematics, 1991; National
Research Council, 1989; Naﬁonal Research Council, 1990) have argued that teachers
need to reduce their emphasis on isolated computaﬁonal skills and include opportunities
for students to recognize mathematical elements in situations, flexibly apply
appropriate mathematical tools, and engage in mathematical reasoning. Richards
(1991) has labeled mathematics instruction in the spirit of the reforms inquiry math
and traditional school mathematics instruction school math. Inquiry math classrooms,
he argued, emphasize the mathematical skills called for in the reform documents. In
inquiry classrooms, computation, although it still plays an important role in
mathematics education, is not the sole focus of instruction. Students spend time playing
with numbers, debating mathematical deﬁnitions, and exploring various mathematical
snuaﬁons.

Just as the “new math“ in the late 19503 was incompatible with parents’ school
math experience, current calls for reform suggest a conception of mathematics that often
differs from the experience of parents of many school-aged children. Kathy’s father told
me in an early interview that:

The math when l was a kid was, you know, they give you a book and you have a set of

problems, you do those problems. Now I think it’s an awful lot more, urn, where

math is set up to make children, ah, formulate their own problem, a lot of story
problems. I think they start at a much earlier age now then they did when l was in

school. In fact, Ms. Smith and l were talking about this. Ah, when I took Algebra and

136
Trig for the first time in college, when they introduced us to creating our own story

problems, writing our own formulas and stuff like that, I was completely inept at it.

Whereas people who had just come out of high scth were quite a bit better at it. So

I think that it is slowly changing.

The evolution of elementary mathematics instruction can have disconcerting
consequences. Pete’s mother felt the changes were quite intimidating and stopped her
from working with Pete. She told me:

We didn’t start math as early as they do in grade school. . . . It was, I don't know, a

lot different. Nothing was as required as it is now. Like, whatever it is they’re

doing now in math, I have absolutely no idea how to do any of it. So, it’s always
someone else in the house that’s helping him with that. . . . We got as far as
fractions. That’s how far I went in school.

Although some parents, such as Kathy’s father, may embrace the changes in
mathematics instruction, others, like Pete's mother may shy away from being involved
in their children’s mathematics homework. In either case the changes will have an
impact on how students and their parents interact while completing school math tasks.
As occurred with the hypothetical machinists, these different conceptions of the same
practice may compete when parents and their children work together on inquiry math
tasks. This chapter looks at the influence different conceptions of elementary
mathematics had on students’ and their parents’ conversations.

Tasks 2 and 3 presented students with mathematical tasks that reﬂected the two
views of mathematics education presented by Richards. In Task 2 the students were
presented with a series of word problems reminiscent of those found in traditional
elementary mathematics textbooks and in school-math classes. In Task 3 the students
were asked to determine how many three digit numbers could be formed from the digits
3, 4, and 7, how many four digit numbers could be formed from the digits 0, 2, 5, and

9, and, based on their work, how they might determine how many numbers are possible

137
from any set of numerals. Both of these tasks focused on place value, the next topic in

Ms. Smith’s ongoing classroom instruction. In Task 2 the students explored the impact
of multiplying or dividing numbers by multiples of ten. Task 3 asked the students what
digits stood for in different places in the numerals and how they helped determine the
value of numbers.

The conversations students had with their parents as they completed Task 2 were
quite different from those they had while working on Task 3. One explanation of the
differences in the conversations centers around the parents’ familiarity with the tasks
students were asked to complete. When the tasks were continuous with the parents’ past
experience, they were much more willing to offer assistance and guide the students to an
answer. When the tasks contrasted with their experience--that is, when the practices
were discontinuous--parents contributed less to the conversation and deferred to the
students or classroom teacher more often. In the following sections, I look closely at the
conversations students and their parents had while working on Tasks 2 and 3 and the
solutions the students brought with them to school.

Task 2

Task 2 included a series of story problems about a girl named Keisha, some books,
and an agreement she made with her mother to earn a certain amount of money for
reading each book. In the ﬁrst two questions the amount of money Keisha would receive
for reading each book changed and the students were to determine how the change would
effect Keisha’s situation. In Questions 3 and 4 the number of books Keisha had increased
by a multiple of ten. Students were asked to compute the number of books Keisha now
had. In Question 5 Keisha divided her books into 100 stacks. The students were asked to
determine how many books would be in each stack. Question 6 asked the students to

divide various multiples of ten by ten. The assignment sheet is shown in ﬁgure 5.1.

138

 

Math

1. Keisha had 142 books. Her mom told her she'd give her $10 for
every book she read. How much money would she receive if she read
all of her books?

2. If she received $3 for each book, how much would she receive?

3. If, over three years, shereceived 100 times as many books as she
had in problem one, how many books would she have?

4. If she received 1000 times as many books, how many would she
have?

5. If Keisha had 1420 books and she put them in 100 stacks, how many
books would be in each stack?

6. How many groups of 10 equal 100?
it u u u 10 u 1,000?
N u H H H H 10,000?
H H H H H u 100,000?

 

 

 

Figure 5.1: Task 2 Assignment Sheet

Students’ answers to Task 2 are summarized in table 5.1.1 The table includes
students’ work as it appeared on their assignment sheets and was discussed in their
conversations at home. All six students (100%) arrived at the same answers for
Questions 1 and 2. Although Pete’s computaﬁon for Question 1 looks slightly different
than the other students’, it is likely that they all used the same rules to guide their
computation. The rules commonly taught in school math classes hold that writing the
largest number (i.e., the one with the most digits) above the smaller number makes
multiplication easier. The difference in Pete’s computation might be explained by the
inclusion of a the decimal point and two zeros indicating cents in his work. By adding the
decimal point and zeros to ten dollars, the number becomes bigger than 142. Following

the rule, $10.00 should be placed above 142 in the multiplication algorithm.

 

1Three of the students (Karen, Pete, and Shaundra) recorded their conversations for
Task Two. The other answers included in table 5.1 were taken from students’ written
work.

Table 5.1

Solution Table for Task 2

139

 

 

_Q_ues. ff Kaﬁn Kathy Pete
1 . 142 142 $10.00
x10 x 10 x 142
000 000 2000
__:J..&29_ _+1.AZSI. 4000
$1,420.00 $1,420 _ 19999
$1,420.00
2 . 1 142 1
142 X 3 142
x 3 $426 x 3
$426.00 $426.00
3. 142 142 300
£92 2.1% __+_.142
000 000 442 books
0000 0000
_JZ—H 00 M
14,200 14,200
4. 1,000 1,000 1,000
x 142 x142 x 142
2000 2000 2,000
40000 40000 4,000
100000 100000 10000
142,000 142,000 142,000
5. 14 R20 14 r2 142
100_)1420 W20 10W
100 _ 10 .190
420 42 420
__400 _40 .M
020 2 200
M
14 in each stack 0
6a 1 0 1 0 1 0
6 b 1 0 0 1 0 0 1 0 0
6c 1,000 1,000 1,000
6d 10,000 10,000 10,000

 

 

 

Table 5.1 cont.

Solution Table for Task 2

140

 

 

Ques. # Ronnie Shaundra Tony
1. 142 142 142
x 10 x 10 x 10
1,420 000 1.420
_LIAZSI.
$1,420
2. $426 142 142
X 3 X 3
$426 $426
3. 142 142 142
x 100 x 100 x 100
880 000 14,200
0
——142 +104O2o
14’200 14 200
+ 100 ’
14,300
4, 142,000 142 1.000
x 1000 x 142
000 142,000
000
+142
142,000
5. 122 r20 14.20 1420
100)1420 100)1420 ﬂ
_ 10 -100 14.20
22 420
20 .490
20 20
_ 0
20
14 stacks
6a 1 0 1 0 1 0
6b 100 100 10 hundreds
60 1,000 1,000 100 hundreds
6d 10,000 10,000 10,000

1 41
Four of the six students (67%) answered Question 3 saying Keisha now had 14,200

books. All four of these students multiplied 142 by 100 to answer the question and their
written work was virtually identical. Ronnie, although he began his solution in the same
way as the other three students and arrived at 14,200 midway through his computation,
determined that Keisha now had 14,300 books.

Pete’s answer of 442 books can only be explained by looking at the conversations he
had both at home and in school. While Pete worked with his grandmother they had this

conversation.

1 G Number three. If, over three years, she received a hundred times as many
books as she had in problem one, how many books would she have? [pause]
What do you have to do with that problem? Over three years she received a
hundred as many, hundred times as many books as she had in problem one.
How many books would she have? [pause] What do you have to do to this
problem?

2 PC Add it. [pause]

3 G Right. Okay, number four.

After Pete suggested that he add in line 2, he wrote 300 + 142 on his paper and
added the values together arriving at an answer of 442. Although there is no explanation
of where the values Pete added together came from, his grandmother supported his choice
of addition as the appropriate way of answering the question and Pete returned to scth
believing his answer was correct. The origin of the values Pete had added together
became more clear when he discussed the questions in his math group the next day.

When Pete, Tony, Ronnie, and Jason compared their answers to Questions 1 and 2
they found they all answered the questions the same way. That was not true for Question
3; no one in their math group had the same answer. As they compared their answers
they were joined by a special education teacher who was observing students in their

class. The special education teacher asked the group questions about their answers that

illuminated the origin of the values Pete had added and Jason's answer of 56,800 books

as well.

142
What caused the different answers was the students’ different interpretations of

“over three years“ in the question. Entering the numbers into his calculator, Jason
explained his answer this way:

One-four-two times one-zero-zero, one thousand four hundred and two.2 Plus you

need it for over three years, times four equal ﬁve, see . . .

Jason showed the special education teacher his calculator with 56,800 written in the
display.

Jason had interpreted “over three years“ to mean that the number of books
increased one hundred fold each year for four years. As a result, he multiplied Keisha’s
initial 142 books by 100 and then by four to arrive at an answer of 56,800 books.

As Jason entered the numbers into his calculator, Ronnie, who Pete had convinced
that 442 books was the correct answer, explained their solution this way: “Take one-
forty-two, add a hundred to it, two-forty-two. Add another hundred to it, three-forty-
two. Add another hundred to it, four-forty-two.“ Ronnie3 and Pete had interpreted
“over three years“ to mean that Keisha added one hundred books a year for three years
to her total number of books.

To these students all of the information in the question needed to be used in the
solution; their answers needed to reﬂect “over three years“ in some way. As was
pointed out in chapter four, the students’ previous experience in school math classes
may have led them to believe that story problems included no irrelevant information.

The students answered Question 4, which did not include a time span, but asked the
students to determine the numbers of books Keisha would have if she received 1,000
times as many as she had in Question 1, with no difficulty. All six of the students who

participated in this study and Jason, the only member of Pete's math group who did not

 

2Jason misread his calculator here. It read 14,200.

3The paper Ronnie turned in at the end of this task-~the one that determined his grade-
listed the answer to this question as 14,300. Ronnie arrived at that answer by
multiplying 142 by 100 getting 14,200 and then adding 100 arriving at a final answer
of 14,300. That answer is as yet unexplained.

143
record conversations at home, listed 142,000 as the number of books Keisha would

receive. Their ease in answering this Question 4 supports the notion that the time span
in Question 3 confused the students as they answered that question.

Question 5 asked how many books would be in each of 100 piles if Keisha had 1,420
books. Only two students, Ronnie and Kathy, labeled their answers. Ronnie, although his
computation was incorrect, wrote that Keisha would have 14 stacks. Kathy, whose
computation also is incorrect, wrote there would be 14 in each stack. In both of these
answers the student dropped the remainder from the quotient. This may reflect their
understanding that the remainder could not be evenly divided among the piles. The other
four students used no labels and reported answers of 14 r 20 or 14.2. Both answers
were computationally correct, but neither interpreted the remainder in the quotient--
there could not be 14.2 books in each pile.

All the students answered Question 6 the same way. They each listed the answers to
the division problems and showed no computation. The students’ answers were all
correct.

The Conversations around Task 2

In the conversations for Task 2, parents were quite directive. For each of the six
questions, the students used the traditional multiplication and long division algorithms-
-often talking aloud throughout the computation. The parents' role was to guide the
students to the appropriate operation and to monitor the students’ computation. In an
illustrative conversation, Shaundra and her father were trying to determine how much
money Keisha would receive if she was paid 10 dollars for each of the 142 books she

read.

1 8) Keisha had a hundred and forty two books. Her mom told her she’d give her
ten dollars for every book she read. How much money would she receive if
she read all of her books?

2 F Okay, so we have a hundred and forty two books to read. And, if you're
going to give somebody ten dollars for every book, that means you're gonna
do what to it

3 so
4 F
s a:
6 F
7 en
a F
9 a)
10 F
11 s:
12 F
13 so
14 F
15 so
16 F
17 a:

144

You could times it or you can just count by tens up to a hundred and forty
two. Well,

You‘ll also end up . . . It's always better to multiply. That saves you a lot
of time than adding up ten four hundred and forty two times. So . . .

I‘m talking about if you put like ten, twenty, thirty, . . .

Oh. that's the long way. that’s the long way and if you do that, you’ll be
you'll be way behind anybody in the class trying to ﬁnish that. That's why
you have to multiply. Because it cuts it short. You can get through a lot
faster multiplying, that's why you multiply. That's the only way. So, you
multiply a hundred and forty two times what?

Ten.
A hundred forty two times ten. And, what do you get?

Zero times two is zero. Zero times four is zero. Huh-mm [no] You do that
and you gotta search [inaudible] Cause if you go, cause when you do that
you always times it from the bottom number and if you go zero times two is
zero, zero times four is zero, and zero times one is zero.

You always do the first number ﬁrst. It'd be one times two, one times
four, and one times one and then you bring the zero over one. You did it
backwards. Go right ahead and tell me how you're going to do it.

What I'm going to do is, all that is times zero is zero.

All of what is limes zero?

Anything times zero is zero. Cause a hundred and forty two times ten you
get [Okay] zero times two is zero and zero times four is zero and zero
times one is zero. [okay] So, what I'm going to do is I’m gonna go over to
the next number and times it by one.

Okay.

All right, one times two is two, one times four is four, and one times one is
one and my answer is one thousand four hundred and twenty.

One thousand four hundred and twenty is your answer. One thousand four
hundred and twenty what?

Dollars

Shaundra began this conversation by reading the question aloud and attempting to

answer it with no assistance from her father. In line 3, she offered her father a choice

of two different operations. Her father, citing the efﬁciency of multiplication, steered

her away from addition as the correct choice. In lines 4 and 6, he told her multiplication

1 4 5
was always better than addition because it saves time and will help her stay up with the

rest of her class. Once they agreed that multiplication was the correct choice, Shaundra
began the computation. As she worked, her father paid close attention to what she wrote
and asked questions when he was unsure of her actions. Finally, Shaundra's father made
sure she labeled her answer correctly.

Although Shaundra carried out the steps in the computation, her father remained in
control while answering this question. He directed her to a specific operation--one he
believed was the best suited for the questionuand guided her through the computation
making sure she carried it out correctly and labeled her answer in an appropriate way.

The direction Shaundra’s father provided in the conversation was apparent in all of
the conversations the students had while working on Task 2 at home. In the conversation
between Shaundra and her father, Shaundra made choices that did not always ﬁt with her
father’s conception of the problem. When that occurred, he redirected her to a more
appropriate way of answering the question. In other conversaﬁons, the students'
responses ﬁt closely with the parents conception of the questions. Conversations in
which the students and their parents agreed on the what needed to be done were very
short and to the point. In the following conversation, Pete and his grandmother computed
the amount of money Keisha would receive it she read 142 books and was paid three
dollars for each book she read.

1 G Okay, number two. If she received three dollars for each book, how much
would she receive? (pause) I'll read it again, the dogs kind of interrupted
us. If she received three dollars for each book, how much would she
receive? So, how many books did she have?

2 PC A hundred and forty two.

3 G And the number is three dollars so you have to, what do you have to do to
ﬁgure that one out?

4 PC Times it.

5 G Okay. So it‘s three times a hundred and forty two. [pause] Let's see. Um,
three times two is . . .

6 PC Six.

146

7 G Okay. Three times four is . . .

8 PC Twelve

9 G Okay, three times one is . . .
1 0 PC Four.
1 1 G No.
1 2 PC Three.
1 3 G plus one F
1 4 PC Four 0
1 5 G Okay. [pause] So, she would receive, that's right. Four hundred and

twenty-six dollars.

 

Pete’s grandmother began this conversation by reading the question aloud. After
reading the question, she asked Pete a series of questions that led to using the traditional
multiplication algorithm to compute the amount of money Keisha earned. By asking the
questions she gradually talked Pete through the multiplication algorithm as he did the
computation, correcting him when he made mistakes. Finally, she told Pete he was right
and rephrased the answer to include dollars.

Pete and his grandmother's conversation was similar to Shaundra and her father’s
in important ways. Both conversations, and all the conversaﬁons students had with their
parents during Task 2, comprised four sections: (a) read the question aloud; (b) choose
the appropriate Operation; (c) compute the answer; and, (d) present the answer in an
appropriate way. In each of these four sections, except perhaps the ﬁrst, the students
initially worked by themselves. When the parents recognized an error or that the
student was having trouble, they intervened and directed the student to the correct
answer.

The direction shown by the parents in these conversations was also apparent in the
conversations they had while working on Task 1. In order for the parents to direct the

. students as they did in these two tasks, they needed to be familiar with what was being

1 4 7
asked of the students. Tasks 1 and 2 likely represented practices with which the parents

were familiarngrocery shopping and a traditional conception of elementary school
mathematics. Task 3, although it too represented the practice of school math, included
things the parents were less likely to have encountered in their school math experience
or used in their daily lives; that is, it represented a different conception of the practice
of elementary school mathematics. As a result, the conversations students had with their
parents were also different from their conversations on the previous tasks.

Task 3

Task 3 was assigned on two consecutive nights. On the first night, the students were
asked to compile a list of three-digit numbers using the digits 3, 4, and 7 and explain
how they knew when they found them all; to write those numbers in order from largest
to smallest and explain why they ordered them as they did; and to write them in expanded
notation and explain what each digit meant in the numbers. On the second night, they
repeated the assignment, but with four digits-0, 2, 5, and 9, and a fourth question that
asked the students if they could predict how many numbers could be generated from a set
of digits. The assignment sheets are shown in ﬁgure 5.2

Task 3 resulted in more incomplete or unanswered questions than did the other
tasks. Although all six students recorded conversations for this task, the structure of
the conversations was different than those for the other tasks and they differed among the
households as well. The structure also changed from the ﬁrst night to the second.

Each night, Karen had a short conversationuwith her father as she read her answers
into the tape recorder. They did not, however, work together to answer the questions.
Although the conversations they recorded often seemed to be cursory inspecﬁons of
Karen’s completed work, Karen recounted their conversations in great detail as she
worked in her math group in school. Karen, for instance, taught the class an alternative
way of writing numbers in expanded notation that she claimed her father taught her.

- That conversation was not recorded.

148

 

Math

1. Compile a list of three digit numbers that can be made using the digits
3, 4, and 7.

A. How many numbers did you come up with?
B. How do you know when you've found all the numbers?

2. List all the different numbers in order from smallest to largest
A. Tall why you ordered the numbers as you did.

3. Write each of the numbers you find in expanded notation.

 

 

A. Explain what each tigit means in each number.

 

 

Math

1. Write down as many four digit numbers as you can using the digits 0,
2, 5, and 9.

A. How many numbers did you find?
B. How do you know when you’ve found all the numbers?
2. List all the different numbers in order from smallest to largest.
A. Tall why you ordered the numbers as you did.
3. Write each of the numbers you find in expanded notation.
A. Explain what each digit means in each number.
4. Now that you have worked with both three and four digit

combinations, can you think of a way to predict the number of
combinations you can make with a certain number of digits?

 

 

 

Figure 5.2: Task 3 Assignment Sheets
Kathy worked with her father both nights. The ﬁrst night they worked together
through all three questions. But in spite of her father's presence, Kathy completed the
task alone after having the directions clariﬁed or the task prepared in some other way.
The second evening, Kathy began working with her father, but, after they reconstructed

the algorithm that predicted how many numbers were possible,4 Kathy's father left her

 

4Kathy and her father’s conversations are analyzed more closely in chapter 6.

1 4 9
to complete the task by herself. As Kathy expanded the numbers in her list, her father

watched a television show about Malcom X and commented on his discontent with the
federal government.

Pete was ill the first night, so he and his grandmother completed both parts of Task
3 the second night. They worked together on the ﬁrst two questions of Task 3a, but when
the third question asked the students to write the numbers in expanded notation, Pete’s
grandmother suggested he leave his paper blank and ask Ms. Smith when he returned to
school.5 Their conversation for the second part was similar. Pete and his grandmother
answered the questions together. When they were to write the numbers in expanded
notation, Pete’s grandmother read the question and said: “Write each of the numbers you
ﬁnd in expanded notation and we don’t know what that means.” Pete did not include an
answer to that question and they went on to answer the last question together.

Ronnie and his mother also set out to complete both parts of Task 3 the second night,
but they worked on the assignments In reverse order. After completing the second part
of Task 3, Ronnie suggested they begin working on the ﬁrst part. His mother, however,
told him it was too late and that he needed to get some sleep. Ronnie never did complete
the ﬁrst part of the task.

Shaundra and her mother also worked on both parts of Task 3 on the second evening.
Shaundra's mother, as were many of the parents, was uncertain about the content of the
task. Rather than leaving Shaundra to work alone, however, Shaundra’s mother worked
closely with her as she completed the ﬁrst part. She eagerly watched Shaundra as she
compiled her list of numbers, wrote them in ascending order, and wrote them in
expanded notation. What she learned from the ﬁrst part of Task 3, she used to provide
direction on part two. But, when Shaundra began “expanding the numbers“ on the
second part, her mother left her to work alone. Before she left, she instructed Shaundra

to write out four numbers and a note for Ms. Smith saying that 24 was too many

 

5Pete never asked Ms. Smith about expanded notation.

150

numbers to expand. Shaundra did not complete the second part of Task 3. As she worked

the tape recorder shut off and Shaundra came back and recorded a message to me on the

tape: “I won’t get to the last problems. Sorry.”

Tony and his mother began working on Task 3 together, but when the assignment

asked them to write the numbers in expanded notation, Tony’s mother shut the tape

recorder off and the remainder of their conversation was not recorded. The second night,

Tony read his incomplete answers into the tape recorder alone.

Table 5.2

Solution Table for Task 3a

 

 

# Karen Kathy Pete
1a 6 6 6
1b KM: How do lknow? KG: I've used all the G: you have two that start
numbers. with three, two that start
F: You used your threes, with four, and two that
fours, and sevens. F: Every number has start with seven.
occupied every position.
2a 347, 374, 437, 374, 347, 374, 437, 374, 347, 374, 437, 374,
734, 743 734, 743 734, 743
2b F: you put the smallest Looked at the Hundreds, G: you wrote it down
ﬁrst because that‘s the then tens, then ones. because this number came
easiest. before this number?
3 Explained how she did one No response G: told Pete to get help
and said she did the same when he went back to
for the rest of the school.
numbers.
3a Hundreds, tens, and ones. KG: I did three equals Ones, tens, hundreds

three hundred, the four
equals forty, and the
seven equals seven. I did
that to all of 'em.

Table 5.2 (cont.)

Solution Table for Task 3a

151

 

 

It Ronnie Shaundra Tony
1a 6 6 6
1b No response SQ: I've used all the RB: I used them all and l
numbers because I’ve can't find no kinda way.
mixed all the numbers up
as many ways that are
possible
2a No response 347, 374, 437, 374, 347, 374, 437, 374,
734, 743 734, 743
2b No response from lowest to highest No response
3 No response Constructed number No response
sentences that included
equaﬁons for each digit.
(E.g., 347 could be
written as :
(3x1)+(7-3)+(6+1)
4 No response the three stands in the one No response

hundreds place and it
would become three
hundred and the seven
stands in the tens place it
becomes seventy now go to
the four . . .the ones place
And, in that place stands
the four, which is just
plain four.

In general, the students worked alone more the second night than the ﬁrst. The

parents' decreased role in these conversations might be explained by the repetition of

the two parts of Task 3. Many of the ﬁrst evening’s answers could be expanded for the

second evening's questions. The parents may have believed the students did not need the

amount of assistance they needed the ﬁrst night. An alternative explanation suggests the

parents' uncertainty with the content led them to withdraw from the conversations.

Many of the parents stopped contributing when the students began writing numbers in

1 5 2
expanded notation the second night. The parents often explicitly stated that they did not

know how to write numbers in expanded notation and told the students to work alone.
Despite working alone, the students all completed Task 3.

The students’ responses for Task 3 are shown in tables 5.2 and 5.3. Students’
responses were derived from their conversations and their written work. The students'
written work for Task 3, more than any other task, failed to match what was said in the
conversations. At times the conversations included important aspects of the students’
answers that were not written down. As a result, the students or their parents are
sometimes quoted in the tables. When quotes are used, the students’ initials or the
initial of their parent is given to credit the speaker.

Task 3 A

Question 1. On the first evening of Task 3, all of the students and their parents
found six three-digit numbers. The students’ explanations of how they knew they were
finished all focused--at least initially--on the idea that they had used all the digits. In
three conversations the parents expanded the explanation to describe a more systematic
approach to their search for the numbers. Kathy's father included the number of
positions a digit could occupy in their explanation. Throughout their conversation,
Kathy's father tried to determine how many numbers they should be able to ﬁnd.
Although he was not completely convinced, at the end of the conversation, he concluded
the number could be determined by adding the number of digits available to ﬁll each
position in the three digit numbers (3+2+1:6). Pete’s grandmother recognized that
each digit appeared in the hundreds place in two of the numbers in their list. She
expanded Pete's explanation to include multiplying the number of digits available by how
many numbers begin with each digit (3x2:6). Shaundra’s mother also recognized that
each number appeared in the hundreds place twice. She told Shaundra about her
discovery this way:

But you know what else I Ieamed? Okay, in each one of the numbers, like three
forty seven, three seventy four, you can only mix them up twice on each one. See

1 5 3
the sevens, you have seven forty three and seven thirty four. And the fours, you

have four thirty seven and four seventy three. So, there’s only certain ways you

can turn it around.

Each of these extensions inﬂuenced how the students and parents approached and
answered the questions the second night of Task 3. In the conversation between Kathy and
her father, for instance, Kathy’s father continued his search for the way to predict how
many numbers were possible. In Shaundra and her moﬁrer’s conversation, Shaundra’s
mother drew on the patterns she saw in Task 3 A to devise a strategy that would make
Shaundra’s work more efﬁcient.

Question 2. All five of the students who answered this question (100%) ordered the
numerals in the same way. Four of the five (80%) justified their order saying they
ordered the list from “lowest to highest.” Although Ms. Smith had talked about place
value in class before the students took Task 3 home, none of the students explicitly
mentioned place value as a reason for ordering the numbers.

Question 3. Question 3 asked the students to write the numbers they found in
expanded notation. Sensing the students might be unfamiliar with this topic, Ms. Smith
had introduced expanded notation and asked the students how they might write their
numbers. The students who were familiar with expanded notation went to the board and
demonstrated how the numbers could be written. The class agreed that 48,265 should be
written 40,000+8,000+200+60+5 as the numbers in this representation reﬂect the
value of each digit in the original number. Despite this discussion, ﬁre students and
their parents found this question difficult as they worked at home: ﬁve of the six parents
were uncertain about how to answer this question.

Four of ﬁre ﬁve parents (80%) who were unsure of expanded notation asked ﬁre
student how it was done. The ﬁfﬁr parent, Tony’s moﬁrer, shutoff ﬁre tape recorder and

they did not record the rest of their conversation. In ﬁrree of the four recorded

conversations, the students showed their parents how to write the numbers in expanded

154

notation.6 Once the student did one or two, ﬁre parents became more willing to help ﬁrem
ﬁnish writing the numbers.

All of the students who recorded conversations answered ﬁre last question same way.
They all used place value labels to explain what ﬁre different digits meant. They did not,
however, make any connection between ﬁre expanded notation and the last question.

Task 3b

For ﬁre second part of Task 3, ﬁre students repeated ﬁre activity ﬁrey had done ﬁre
previous evening, but with four digits (0, 2, 5, and 9). At ﬁre and of the second part of
Task 3, ﬁre students were asked how ﬁrey might predict how many numbers could be
derived from a given set of digits. The students responses to Task 3 B are summarized in
table 5.3. ﬁre lists of numbers the students compiled and ﬁre lists of the numbers in
order from smallest to largest are not included in the table to preserve room.

Question 1. The students all compiled different lists of numbers ﬁre second evening.»
Even ﬁrough they did not agree on how many numbers were possible, all but one student,
Kaﬁry, held fast to ﬁre notion ﬁrat ﬁre best way to know ﬁrey had found all ﬁre possible
numbers was to “use all the numbers.“ Kaﬁry and her faﬁrer, in contrast, determined
that ﬁrere should be twenty-four numbers based on an algoriﬁrm Kathy’s faﬁrer
remembered from his school-math experience. After a lengthy discussion during which
they sought out a way to explain the twenty-four numbers Kaﬁry found, Kaﬁry's father
remembered ﬁrat the number of possible permutations could be determined by
multiplying the number of positions available for each digit (4x3x2x1:24).

Question 2. The students’ lists of numbers in order from smallest to largest varied
depending on the numbers included in their original lists. All of ﬁrem, however, listed
the numbers in ﬁre proper order. Only Ronnie mentioned place value while explaining

why he ordered ﬁre numbers as he did. Other than Tony, who said “because if I didn’t I’d

 

6ln the fourﬁr conversation, Pete's grandmother asked him if he understood expanded
. notation. Pete told her he did not and she suggested he talk to his teacher about it when he
returned to school. Their conversation ended there.

155
get ﬁre wrong answer,” all ﬁre students (67%) used the same answer ﬁrey used ﬁre

previous evening.

Quesﬁon 3. All of the students wrote the numbers out in expanded notation using the
meﬁrod Ms. Smith introduced in class or ﬁre meﬁrod Karen taught the class earlier in
school. Karen’s faﬁrer, the only parent who was completely sure of how to answer ﬁris
question, taught her a different way to write expanded notation that Karen brought back
and presented to ﬁre class. The class agreed ﬁrst ﬁre meﬁrod worked and accepted it as an
altemaﬁve to the meﬁrod on which ﬁrey had ﬁrey agreed. Using Karen’s father’s
notation, 347 would be written (3x100)+(4x10)+(7x1).

Table 5.3
Solution Table for Task 3b

 

 

# Karen Kathy Pete

1a 1 6 2 4 2 5

1b KM: Idid ﬁre ﬁrousands, Multiplying PC: Ichecked over it. I
ﬁre hundreds, the tens, 4x3x2x1:24 helped checked all ﬁre
and ﬁre ones. ﬁrem determine how many combinations.

numbers they should find.

2b KM: Smallest to ﬁre Looked at ﬁre ﬁrousands, PC: smallest to largest

largest. hundreds, tens, ﬁren ones.

3a ﬁrousands, hundreds, KG: It means it has ﬁre Ones, tens, hundreds,

tens, and ones ﬁrousands, hundreds, the ﬁrousands
ones and tens.

4 You can make sixteen. KG: If you have ﬁrree, you Multiply the number of
can make six. If you have combinations possible for
four, you can make each number by the
twenty-four. number of digits you

have. E.g., for ﬁrree
digits multiply two times
three, for four digits
multiply 4 times six.

156
Table 5.3 (cont.)

Solution chart for Task 3b

 

 

# Ronnie Shaundra Tony
1a 1 8 2 3 1 2
1b M: 4x4:16 SQ: I know I found ﬁre TW: by switching ﬁrem,
numbers because I by scrambling them up,
RB: Ichecked all of them. worked with every and looking.
I checked to see if I could numeral possible.

make any more numbers.

2b RB: I used my place SQ: to ﬁnd how many TW: Because if I didn't I
value. different possible ways I wouldn't get ﬁre answer.
could use ﬁre numbers

3a Ones, tens, hundreds, and No response You take two ﬁrousand,
ﬁrousands. plus zero plus ﬁfty plus
nine and get two ﬁrousand
fifty nine.
4 M: I would ﬁrink ﬁrat you No response No response

would take the number
and, you know like, four
digits times the same
amount.

Four of ﬁre ﬁve students who answered Question 3a listed ﬁre place value of each
number when describing what each digit meant in ﬁre numbers. Tony read an example of
ﬁre number sentences he had written down. Although he did not expliciﬁy mention place
value in his answer, he did read ﬁre places correctly in his number sentences.

Question 4. Of ﬁre four students who answered Question 4, two only reported how
many numbers they found each evening. Alﬁrough Kaﬁry and her father agreed that the
algoriﬁrm determined how many numbers were possible, Kaﬁry was one of ﬁre students
who reported how many numbers she had found while completing the two parts of Task 3.
The oﬁrer two, Ronnie and Pete, answered Question 4 saying ﬁrat ﬁre number could be

calculated in some way. Ronnie and his mother suggested ﬁrey could predict four-digit

1 5 7
numbers by multiplying 4 x 4:16. Pete’s grandmother summarized their answers

saying ﬁrey could multiply the number of combinations possible for each number by the
numbers of digits to determine how many were possible. Pete and his grandmother’s
solution is described in more detail below.

The conversations about Task 3

In contrast to ﬁre conversations ﬁre students and their parents had for Task 2, ﬁre
parents were much less directive as they worked on ﬁre ﬁrst part of Task 3. In the
conversations for Task 2, ﬁre parents were able to recognize difﬁculties the students
were having and direct them to a more appropriate strategy for answering ﬁre questions.
The students seldom questioned ﬁre parents’ strategies and ﬁre parents only corrected the
students when ﬁrey were mistaken. The adults clearly directed ﬁre conversations.

In Task '3a, however, the parents were often uncertain about ﬁre tasks and more
often asked for and accepted students’ explanations--even when ﬁreir explanaﬁons were
wrong. The parents’ uncertainty about the tasks led to two reactions. In two households,
ﬁre parents left the students to answer ﬁre questions on their own or ended the
conversation and shut off the tape recorder. In the oﬁrer ﬁrree households, ﬁre students
and ﬁreir parents worked togeﬁrer to answer the questions.7 When ﬁris occurred, the
students, raﬁrer than ﬁreir parents, led the conversations. In extreme situations they
switched roles: ﬁre student, based on his or her conception of what was expected upon

returning to school, became the more knowledgeable oﬁrer and ﬁre parent became ﬁre

 

7Only five conversations were recorded for Task 3a. Ronnie did not complete ﬁrst part of
ﬁre task. In ﬁreir conversation for the second part of Task 3, however, Ronnie and his
moﬁrer worked togeﬁrer to compile the lists and answer the questions. At ﬁre and of ﬁreir
conversation, Ronnie’s mother contributed to Ronnie’s understanding in ways that typify
the conversations in other students' homes. Karen's recorded conversations with her
father did not exemplify the conversations in other students’ homes. Karen did,
however, recount unrecorded conversations wiﬁr her father as she worked in her maﬁr
group in school. Many of ﬁre ideas she recalled from those conversations inﬂuenced her
group’s and ﬁre entire class’ ﬁrinking about this task. These conversations suggest that
the conversations in which the parents contributed to the students’ thinking, even when
parents were uncertain about the content, may be even more prevalent ﬁran ﬁre data
presented here suggest.

1 5 8
Ieamer. The students explained to ﬁreir parents what expanded notation was and

demonstrated how expanded numbers could be written and wrote them down.

After the students finished ﬁreir instruction, ﬁrey worked alone. As they worked,
ﬁreir parents watched closely to ﬁgure out what the students were doing and began to
contribute when they ﬁrought they had figured it out. When the students ﬁnished, ﬁreir
parents often reviewed ﬁreir work and elaborated ﬁre things the students said or wrote.
The parents' elaborations did not seem to reﬂect ﬁreir knowledge of maﬁrematics or
ability to solve maﬁrematical problems as much as it did a disposition to become engaged
in their children’s work, even when ﬁre work deviated from their conception of
elementary mathematics content and instruction. The parents in these conversations,
like ﬁre students, were trying to ﬁgure out the answers as they worked. Their
observations led ﬁrem to ask different questions and offer insight the second evening they
were unable to offer at first.

The parents who were actively engaged in ﬁre ﬁrst part of Task 3 were able to
contribute to ﬁre conversations about Task 3b in one of two ways. Some parents helped
ﬁre students use ﬁre lists of numbers they generated for each part the task, and patterns
within ﬁrose lists, to answer Question 4 on ﬁre second part of Task 3. Question 4 asked
ﬁre students how ﬁrey might predict how many numbers would be possible wiﬁr a given
set of digits. This contribution is shown in ﬁre conversations Pete and his grandmoﬁrer
had as ﬁrey worked on the two parts of Task 3.

After Pete compiled his list of numbers for Task 3a, his grandmother read ﬁre next
part of ﬁre question. The question asked how they knew ﬁrey had found all the numbers.

G How do you know when you found all ﬁre numbers? (pause) How many
combinations of each number do you have.

PC two?

G So you have two that start wiﬁr ﬁrree, two ﬁrat start with four, and two
ﬁrst start wiﬁr seven.

159

Pete’s grand moﬁrer had recognized a pattern in ﬁre list. The pattern was ﬁrst ﬁrere

were two combinations for each number; that is, ﬁrst each number was in the hundreds

place twice. Pete wrote ﬁris down as ﬁre reason he knew he had found all ﬁre numbers.

Pete compiled his list for the second part of Task 3. When he had ﬁnished, his

grandmoﬁrer recognized a similar pattern in Pete’s list: each number was in the

ﬁrousands place six times. The two psttems Pete’s grandmoﬁrer found served as the data

from which they derived a way of predicting how many numbers would be possible.

They had ﬁris exchange:

G

Okay, it says, now that you've worked wiﬁr both three and four digit
combinations, can you ﬁrink of a way to predict ﬁre number of
combinations you can make wiﬁr a certain number of digits? With your
ﬁrree digit ones you had, how many combinations of each number did you
have?

What?

With your three digit numbersnright there on that page--how many
combinations did you have? Of each number?

Two.

Okay, so you know that you can get two combinations out of each, each three
digit, out of each ﬁrree, each of the ﬁrree digits. So you have . . .?

Two, four, six.

That, okay, so you know you can get six combinations out of a ﬁrree digit.
so, on your four digit one here, how many combinations did you get on each
number?

Six.

Okay and there's four numbers. And you came up wiﬁr how many
altogether? What did you tell me?

Twenty-four. -
Okay. So, six times four is what?

Twenty-four.

160

G Um-kay. So, you have six combinations, six, yeah, six combinations in
each number and six times four numbers is twenty-four. So, if you had,
um, as long as you know how many combinations you have, like two times
ﬁrree is six, six combinations. Six times four is twenty-four. So, do you
understand ﬁrst you can mul, you simply multiply that number out? If you
knew your multiplication numbers better, it'd be easier wouldn’t it? We
have to work on those. Um-kay, ﬁrst completes your math for today.

Pete’s grandmoﬁrer was onto someﬁring. Her meﬁrod of determining how many
numbers were possible would work in all situations, provided you could determine “how
many combinations“ were possible for each number. Given an opportunity, they may
have gone on to discover ﬁrst they could use factorial multiplication to determine how
many combinations were possible and extend ﬁreir response to include multiplying
togeﬁrer ﬁre number of digits available while ﬁlling each place in ﬁre numbers (i.e.,
4x3x2x1:24).

But, Pete and his grandmoﬁrer's search for the answer may be more important than
the answer itself. Students not only Ieam what is done in interactions with adults, but
how it is done as well (Newman et al., 1989). Pete's grandmother insightfully
recognized ﬁre patterns in each of ﬁre lists she and Pete compiled. By juxtaposing the
two patterns she was able to generate a rule ﬁrst predicted how many numbers were
possible wiﬁr a certain set of digits. Pete’s grandmother's contributions included many
of the characteristics called for in the reform documents in mathematics education.
Togeﬁrer she and Pete gathered data and made maﬁrematically sound decisions about how
to interpret ﬁre data. In essence, she and Pete were exploring msﬁrematics as they
worked on ﬁre tasks.

The same ﬁring can be said about Shaundra and her moﬁrer as ﬁrey worked on Task 3.
As ﬁrey worked togeﬁrer to compile the lists, Shaundra’s moﬁrer, as did Pete’s
grandmoﬁrer, searched for patterns and struggled to make sense of ﬁre content of ﬁre
tasks. Shaundra's moﬁrer’s role in these conversations characterizes the second way in

which parents contributed to ﬁreir children’s work on Task 3: Parents gradually became

more willing to suggest strategies ﬁrst would assist ﬁre students. The parents' initial

1 61
uncertainty and reliance on ﬁre students’ conception of what was expected in school and

the parents’ gradual contribution to the task is evident in these two conversations
between Shaundra and her moﬁrer. In the first conversation Shaundra’s moﬁrer was
uncertain about expanded notation as ﬁrey worked on Task 3a and so she asked Shaundra if
she knew what it was.8 Shaundra assured her mother she did and began writing out ﬁre
numbers. As she watched Shaundra write numbers in expanded notation Shaundra's
mother worked to Ieam ﬁre rules wiﬁrin which ﬁre Shaundra operated. Once she
understood ﬁre rules, she was more willing to contribute to the conversation.

1 M What’s expanded notation? Do you know?

 

2 S) Yedr
3 M Tell me. E
4 8) It’s like, um, you do like Karen did, her dad told her ﬁrst um, that like
ﬁrree hundred forty seven you could write it like ﬁris: three times one and
ﬁren say plus, um,
5 M Ch
6 83 four and then, um,
7 M Probably minus
8 $1 Um,
9 M Well, I see you got ﬁrree times one notation
10 82 is three
11 M ﬁren times, or plus four so that’s the problem right ﬁrere wouldn’t it? I
mean that equals ﬁrree, four and seven.
12 8) No, like three hundred forty seven.
1 3 M Okay, go ahead and do it cause you understand it.
14 8) And, plus, um, eight minus one.
15 M Okay good, ah write each of ﬁre numbers you find

 

aShaunda recorded her conversation for the ﬁrst part of Task Three a day later than the
other students. the day after Task 3a was assigned, Karen taught her method of writing

. numbers in expanded notation. In this conversation Shaundra refers to Karen’s method
of writing numbers in expanded notation.

1 6 2
As she talked, Shaundra wrote out a maﬁrematical expression. The expression,

(3x1)+4+(8-1), included smaller expressions that, when simplified, would produce
the digits in the number with which she started. In this example, 3x1:3, 4:4, and 8-
1:7. Put in succession, ﬁrose numbers read 347, the number Shaundra was writing out
in expanded notation. As Shaundra worked, her moﬁrer struggled to make sense of what
she was doing. In line 11 she believed she understood: 3x1 is 3, plus 4 is 7, and since
those are the digits in the original number, they are ﬁnished. Shaundra, however,
disagreed and continued to write out ﬁre number sentence. Her moﬁrer accepted her
error and told Shaundra to continue on because “you understand it.“

Shaundra’s mother continued to watch and, at the end of ﬁrelr conversation,
contributed to one of Shaundra's number sentences. The last number Shaundra worked
on was 374. As she worked she and her moﬁrer had this conversation.

1 6 8) Problem of number is three hundred and seventy four. So, I'll just go
ﬁrree and eight minus one and ﬁre oﬁrer number was

1 7 M How about four? Can it be anything like four minus zero?

1 8 82 Okay.

1 9 M It has to equal up to four, right? That’d be something different.

2 0 32 Minus zero that'd be four so that make ﬁrree hundred forty eight, what
happened?

21 M No,

2 2 8) Seven

23 M Right. Three hundred seventy four.

Shaundra’s moﬁrer figured out ﬁre rules wiﬁrin which Shaundra was working. In line 17
she contributed an expression, 4-0, that fit Shaundra’s rules and added to her number
sentence. Shaundra accepted her moﬁrer’s contribution and, after a slight confusion,
declared herself ﬁnished. Her moﬁrer, now understanding ﬁre rules, also pronounced the

expression “right“ and repeated the number to signify their conclusion.

1 6 3
Shaundra's conception of expanded notation was wrong. Even ﬁrough what Shaundra

had written down resembled Karen’s method of expanding numbers, her expressions did
not represent either of ﬁre ways agreed on by Ms. Smiﬁr and her class. Shaundra's
moﬁrer, however, did not know ﬁrst. And, being unsure of expanded notation herself, she
accepted Shaundra’s explanation and worked hard to understand it as Shaundra worked.

During ﬁris conversation, Shaundra's mother learned someﬁring new, albeit
something incorrect, in order to help her daughter while she worked on her homework.
What began as an assignment ﬁrst may have been discontinuous wiﬁr her past experience
was now part of her experience. When she and Shaundra worked on the second part of
Task 3, the ﬁrings they had done and Ieamed while working on the ﬁrst part informed
their decisions and actions. Shaundra’s moﬁrer, srrned with newly constructed
maﬁrematical knowledge, contributed more to ﬁre conversations. This inﬂuence can be
seen in the conversation Shaundra and her mother had while compiling ﬁreir list of
four-digit numbers.

Whereas Shaundra’s moﬁrer helped Shaundra compile the list in ﬁre ﬁrst part, she
offered no strategy that might assist her in doing so. When she had ﬁnished, Shaundra’s
moﬁrer recognized a pattern in the numbers and told Shaundra about it. In ﬁreir second
conversation, Shaundra’s mother, recognizing that Shaundra was going to have trouble
keeping track of the numbers she found, offered a strategy that would assist her. The
strategy she offered was consistent with ﬁre pattern she recognized in ﬁre earlier

conversation. She told her:

I'm going make this, try and make ﬁris easier for you, okay? Why don’t you try to
work with all your nines first, like how many ways you can turn nine hundred, you
got nine, two, ﬁve, oh. Then you can do nine ﬁve two oh. . . . You know what I mean?
Do all ﬁre nines ﬁrst and then do all ﬁre ways you can do five and all ﬁre ways you
can do two. Okay, you know what I mean?

1 6 4
After Shaundra’s mother offered this strategy, they easily compiled a list of twenty-two

numbers.9 Shaundra’s moﬁrer contributed to the search by finding numbers Shaundra
had not yet seen and by offering bits of advice as ﬁrey worked.
They began their search by looking for “the nines“--numbers with the digit 9 in

the thousands place. As ﬁrey started looking, Shaundra guessed ﬁrey would ﬁnd only

three and quickly compiled a list of three numbers that began wiﬁr the digit 9. When she
showed the list to her moﬁrer, her moﬁrer remarked, “ls ﬁrst ﬁre only ways you can do
it? Nine two ﬁve oh, Nine ﬁve oh two, Nine oh two ﬁve. How about Nine two oh ﬁve? ls
ﬁrst a different way?“ Shaundra agreed ﬁrst it was a different way and added 9,205 to
their list. With their amended list, ﬁrey assumed they had found all the possible
numbers and went on to find “the twos.“ Shaundra now predicted they would ﬁnd four
“two numbers“ and quickly compiled a list of four numbers wiﬁr the digit 2 in ﬁre

ﬁrousands place. Her mother, looking over Shaundra's work, saw anoﬁrer number ﬁrst

was not in ﬁreir list.

1 M I see one you haven’t got.

2 83 Oh yeah! I forgot. Um, yup, ﬁve oh nine.
3 M Yeah.
4 8'.) Now ﬁrere were one, two, ﬁrree, four, five.
5 M So, you got ﬁve out of there, you probably should of got five out of the
oﬁrer ones.
6 8'.) Well, I don't know. I ﬁrink Ms. Smith said ﬁrst they don’t all have to be
even.
7 M Oh, okay. Well still if you got ﬁre same numbers. Just make sure you ain’t

got two of the same. Two nine five oh, Two nine oh five, Two oh nine ﬁve,
You got two oh ﬁve nine, okay that’s it. Okay.
82 And, now, we’re going on.

9 M What about, then I see one more ﬁren. If you flip ﬁrose two around it’d be
two ﬁve nine zero. ls ﬁrst anoﬁrer way of doing it? You ain't got that one.

 

9The other two numbers in ﬁreir list were found when they listed the numbers in order
from smallest to largest and were added later in ﬁreir conversation.

165

10 8) Wait, two, nope. Two ﬁve nine oh. Now we go on. So we found six for ﬁrst
one. And ﬁrst was just ﬁre maximum for ﬁre other one.

Shaundra’s mother was more directive in this conversation than she was in part one
of Task 3. She had a clear conception of how to answer this question, and when Shaundra
began doing ﬁrings ﬁrst did not match ﬁrst conception, she quickly directed her to a
strategy ﬁrst would help her answer ﬁre question. As Shaundra began using ﬁre strategy,
her moﬁrer monitored her work. She provided hints about missing numbers and even
contributed numbers that Shaundra had missed.

Shaundra and her mother went on to look for the “fives” and the “ohs.” In spite of
their new belief ﬁrst each digit would be in the ﬁrousands place of six different numbers,
ﬁrey did not go back to complete the list of nines. Raﬁrer, ﬁrey moved ahead to ﬁnd ﬁre
six numbers that began with ﬁve and ﬁre six that began wiﬁr the digit 0. Wiﬁr each
successive search, they reﬁned ﬁreir strategy. In the end, they no longer searched for
fives or ohs, but for “five ohs“ or “oh fives.“ This change represented their
understanding that each of ﬁrese new categories would comprise two numbers and if ﬁrey
found one, ﬁrey could “flip“ the last two digits to ﬁnd the other one. Using ﬁris strategy,
Shaundra and her mother easily compiled the list.

Shaundra and her moﬁrer’s search for the numbers in ﬁre ﬁrst part of Task 3 began
as a random gaﬁrering of numbers. The different permutations could be intuited easily
and “I used ﬁrem all“ was perhaps the best explanation available for knowing when they
were ﬁnished. But, as they worked over the two parts of Task 3, Shaundra and her
mother constructed an understanding of the situation that allowed ﬁrem to efficiently
compile the lists of three and four digit numbers. On first glance, Shaundra's assertion
ﬁrst she knew she was done because she “worked wiﬁr every numeral possible” does not
explain how she knew. But if “worked with every numeral“ is rewritten to say “found
all six numbers ﬁrst began with each of ﬁre digits presented by recognizing that, wiﬁrin

each group of six numbers, each of ﬁre other ﬁrree digits will be in ﬁre hundreds place in

1 6 6
two different numbers, and, if I found one of ﬁrose numbers, I could ﬁnd ﬁre oﬁrer by

ﬂipping ﬁre numbers in ﬁre tens and ones places,“ it becomes clear ﬁrst Shaundra and
her moﬁrer, although they may have begun the activity with little understanding of what
was being asked of ﬁrem, Ieamed how to approach and answer ﬁrese quesﬁons.

Shaundra’s moﬁrer, however, never overcame her uncertainty about expanded
notation. When she and Shaundra started writing numbers in expanded notation,
Shaundra's mother told her, “Okay, now I’m going to leave that up to you,“ and she left
ﬁre table to take care of some oﬁrer household needs. She continued to talk wiﬁr Shaundra
from a distance, but their conversation focused on how many of ﬁre 24 numbers
Shaundra needed to write out in expanded notation. In ﬁre end, ﬁrey decided Shaundra
should write out four numbers and a note to Ms. Smiﬁr explaining ﬁrst 24 was too many
to write out.

Summary and Conclusions

The ultimate goal of any homework conversation is to send ﬁre students back to
school with correct answers (Varenne et al., 1982). This goal is difficult, if not
impossible, to meet when parents do not understand ﬁre content of the assignments.
Parents need to know what a correct answer is in order to guide their children to one.
When boﬁr the students and ﬁreir parents bring similar conceptions of elementary school
maﬁrematics to a conversation, there is litﬁe or nothing to negotiate. The process of
determining what information is required to answer ﬁre question or what combination of
operations to use to arrive at an answer is obvious to the parﬁcipants--especially ﬁre
parents.

The conversations recorded for Task 2 are examples of homework conversations in
which parents knew what a correct answer was. It is likely that the story problems
presented as part of ﬁrst task were consistent wiﬁr their elementary school mathematics
experiences and, consequently, their understanding of mathematics. In ﬁrose

conversations ﬁre parents quickly determined how to arrive at the correct answer and

1 6 7
efﬁciently guided their children to that answer. When ﬁreir children’s actions deviated

from ﬁre course ﬁrey had chosen, they quickly redirected ﬁrem to more appropriate
actions. The assignments were completed quickly and ﬁre students answered most of the
questions in the same way as ﬁreir classmates.

When parents do not know what a correct answer is, they have two options. They
can stop participating in the conversations and let ﬁre students work alone or they can
turn to ﬁre child to ﬁnd out what a correct answer is and then use ﬁrst description, right
or wrong, to guide the students while working on subsequent questions or assignments.
In Task 3 many of the parents did not know what a correct answer was. Alﬁrough some
parents stopped participating in ﬁre conversations, most parents sat wiﬁr the students as
ﬁrey worked on ﬁre first part of Task 3 and sought to Ieam what the students were doing.
When they understood ﬁre task, the parents' participaﬁon more closely resembled ﬁreir
contributions to the conversations they had for ﬁre earlier tasks.

The change in parents’ participation was apparent boﬁr within and across the
conversations. As parents watched ﬁreir children working, ﬁrey interjected comments
whenever ﬁrey believed ﬁrey understood the rules. These small contributions tested
ﬁreir hypotheses about what a correct answer was and how it could be arrived at. If ﬁre
students accepted their contributions, ﬁrey were on the right track; if the students
rejected their contributions, ﬁrey needed to reﬁrink what ﬁre students were doing and
retest their new understandings. By ﬁre and of ﬁre conversations, the parents and
students agreed on ﬁre task and how ﬁre questions could be answered. When ﬁre students
and parents began working on the second part of Task 3, parents entered ﬁre conversation
wiﬁr a more thorough understanding of what the task entailed. Using ﬁreir newly
constructed understanding they were more directive and their contributions more
closely resembled ﬁrose in the earlier conversations.

The conversations for Task 3 point out that parents, or other adults, do not always

have a better, or more sophisticated, conception of what is required in elementary school

1 6 8
maﬁrematics tasks. Children can be teachers and parents can be learners and, at least in

these conversations, parents are willing learners. They paid close attention to what
their children were doing and Ieamed everything they could in order to assist them later
on.

The change in focus and content of elementary school mathematics instruction to
someﬁring inconsistent with their experience did not stop the parents from participating
in these conversations. It just changed how they contributed. In Tasks 1 and 2, which
were set within practices familiar to ﬁre parents, they stayed in command of the
students’ actions, correcting them as they worked. In Task 3, they became “co-
explorers“ of the situation who modeled maﬁrematical inquiry in ways ﬁrst are
consistent with ﬁre calls for reform in maﬁremstics education. Alﬁrough ﬁrey did not
always answer the questions correcﬁy, the parents and students gathered data, searched
for patterns, devised explanations of ﬁre patterns they found, and used what ﬁrey learned
in one conversation to inform their work in subsequent conversations--all of which are
dispositions maﬁrematics educators hope their students develop.

In Chapter 4 I argued ﬁrst ﬁre conversations students had at home greaﬁy inﬂuenced
the work ﬁrey brought back to ﬁre classroom. In their conversations with ﬁreir parents
ﬁrey were exposed to many different practices and conceptions of the same practice ﬁrst
shaped ﬁre way they deﬁned ﬁre tasks and inﬂuenced ﬁre answers ﬁrey brought back to
school. The conversations presented here point out the school's inﬂuence on what
students do at home. The tasks presented to students have a great impact on how parents
interact with their children about their homework. But the inﬂuence of the school on
ﬁre home and of ﬁre home on ﬁre school occurs at ﬁre same time; that is, ﬁrey are mutual
influences. The mutual inﬂuence of the school and home is explored in greater depth in

Chapter 6.

Cl-IAPTER6
TOHOMEAND BACK

In the beginning of Life on ﬁre Mississippi, Mark Twain told ﬁre physical and
“historical“ history of the Mississippi River. He presented the historical history as
four “epochs“ in the river’s existence. He wrote:

Let us drop ﬁre Mississippi’s physical history, and say a word about its historical

history-80 to speak. We can glance at its slumbrous ﬁrst epoch in a couple of

short chapters; at its second and wider-awake epoch in a couple more; at its flushest
and widest-awake epoch in a good many succeeding chapters; and ﬁren talk about its
comparatively tranquil present in what shall remain of the book.
In each epoch, people who came to see or lived near ﬁre river came to know it in new
ways-ways that fit wiﬁr the ﬁrings ﬁrey needed to do or included new technologies (e.g.,
the riverboat) that allowed them to do new things. Throughout its history, the river
changed both in its physical structure and in the meaning given it by its visitors and
residents.

In ﬁre ﬁrird epoch, ﬁre one he refers to as ﬁre “widest-awake“ epoch, Twain
chronicled his own Ieaming of the river. Twain used ﬁre same structure to describe his
own learning. Much like ﬁre changes in ﬁre ways ﬁre explorers and residents knew the
river, his knowledge of the river underwent qualitative changes with different “epochs“
of learning. At the beginning of his apprenticeship on the river, Mr. Bixby, Twain's
mentor, instructed him to keep a notebook of the river's features. As a result, he ﬁrst
Ieamed the river, he recounted, as a list of the physical characteristics-~ﬁre towns,
sandbars, points, and bends. Although he could recite the list for all but “ten miles of
river in every ﬁfty,“ Mr. Bixby made sure he did not become complacent and pointed out
that such a list was inadequate for captaining a riverboat. To better prepare himself for
his future position, Twain, with the help of Mr. Bixby, learned the river as a “dark

hallway.“ Knowing ﬁre river in this way allowed him to “feel” his way in the dark

169

1 7 0
constanﬁy knowing where he was and what to expect ahead. This way of knowing the

river also proved inadequate, as the river changed constantly. That constant change
forced Twain to Ieam ﬁre river in yet anoﬁrer way. In ﬁre and, he Ieamed to read ﬁre
river as if it were an ever-changing novel. The novel, he wrote, was new each time he
read it and each new novel demanded a close reading. It was ﬁris ﬁnal way of knowing the
river that allowed him to become a highly competent riverboat captain.

Twain goes on to say ﬁrst he not only changed ﬁre way he ﬁrought about the river, but
he gave up ﬁre ways he knew the river before becoming a riverboat captain. He wrote:

Now when I had mastered ﬁre language of ﬁris water and had come to know every

trifling feature that bordered ﬁre great river as familiarly as I knew the letters of

ﬁre alphabet, I had made a valuable acquisition. But I had lost something, too. All
ﬁre grace, ﬁre beauty, ﬁre poetry had gone out of ﬁre majestic river! (Twain,

1883/1984, p. 95)

Twain went on to describe a beautiful sunset he remembered from his past and what ﬁre
patterns of water, ﬁre color of ﬁre sky, and floating objects meant to him ﬁren and what
they meant now. Whereas in ﬁre past ﬁrey were ﬁre “glories and charms“ of ﬁre river,
they now were predictors of tomorrow’s weather, the rise and fall of the river’s water,
and the changing physical structure of ﬁre river. To fulﬁll his goal of becoming a
riverboat captain, it was necessary for Twain to give up old ways of knowing the river
and take on new ways ﬁrst made his task easier.

Twain’s knowledge of the river developed gradually and involved both ﬁre accrual of
new knowledge and qualitative transformations in how he knew ﬁre river. He could not
forget the list of towns, points, and sandbars he memorized earlier, but as he gained
experience on the river, they took on new meanings--meanings ﬁrst were necessary for
him to accomplish ﬁre ﬁrings he set out for himself.

Children Ieam maﬁrematics and oﬁrer school subjects in much ﬁre same way. They

do not merely know or not know how to compute different maﬁrematical operations, solve

1 71
msﬁrematical problems, or understand maﬁrematical concepts. They develop ﬁrese

abilities over time. Their learning, like Twain’s, involves both the accrual of new
knowledge and qualitative transformations in ﬁre way ﬁrey assess maﬁrematical
situations, use maﬁrematical tools, and make sense of ﬁreir ﬁndings.

The gradual development of concepts and procedures was important as well to
Vygotsky (1978). He argued that children, through interaction with more
knowledgeable adults or peers, gradually develop ﬁre ability to participate in socially
deﬁned and historically determined activities. In their initial attempts, children
require a large amount of assistance to complete the task. Over time ﬁre child assumes
more responsibility for the finished product and the adult provides less assistance.
Eventually, the child reconstructs the activities internally and completes ﬁrem
independenﬁy.

Throughout this process ﬁre child’s ability increases and the gap between ﬁre aspects
of ﬁre task he or she can do alone and ﬁrose ﬁrey can do wiﬁr assistance shrinks. This gap
represents ﬁre zone of proximal development. Evidence of Ieaming within ﬁre zone of
proximal, and within a specific conversation, comes from changes in what children do--
their actions and thoughts--while working wiﬁr adults or more experienced peers to
solve a problem. The activity is initially displayed in the actions of ﬁre more
experienced oﬁrer in the conversation; that is, in ﬁre physical manifestation of ﬁreir
situation deﬁnitions. The child observes, imitates, or obeys ﬁre adult’s actions or
directions until ﬁrey can complete ﬁre same tasks wiﬁr no help. Wertsch (1984)
referred to the things people do in conversations as their “action patterns.“ If a child
begins to mimic the more experienced participant’s action pattern, they have Ieamed.

But, as Lave has pointed out, Ieaming includes “extending what one knows beyond
the immediate situation“ (1993, p. 13). So, alﬁrough ﬁre changes students made in

their action patterns during ﬁreir conversations at home may be evidence of Ieaming, it

172
might be beneﬁcial to watch the student in subsequent interactions around ﬁre same tasks

to see how they extend what ﬁrey did in those conversations to other conversations.

Looking beyond the immediate situation points out ﬁrst people do not merely
internalize ﬁre things they are exposed to. Raﬁrer, Ieaming, as Vygotsky may have
intended, is a active, gradual process in which learners personalize the things they are
exposed to. Drawing on the work of Leont'ev (1981), Newman, Griffin, and Cole
(1989) have argued that children, rather than internalizing knowledge, “appropriate“
it in ways ﬁrst aid ﬁrem in doing ﬁrings necessary in their immediate surroundings.
Whereas internalization suggests learners add intact pieces of knowledge to existing
cognitive structures, ﬁre notion of appropriation suggests that children might use part of
what is being taught or change what is being taught to fit ﬁreir own purposes. Newman,
Griffin and Cole summarized ﬁris argument saying: “The child has only to come to an
understanding that is adequate for using ﬁre culturally elaborated object in ﬁre novel life
circumstances he encounters“ (1989, p. 63). In ﬁre case of elementary school
maﬁrematics, the “culturally elaborated objects“ are the concepts, procedures, and
other mathematical tools. Students may take part of a concept or maﬁrematical tool and
use it to achieve some and only to ﬁnd ﬁrst ﬁrey may need to refine the “crude tools“ ﬁrey
have constructed to work in anoﬁrer instance.

In this chapter, I looked beyond ﬁre situation in two ways. In the first section, I
discuss two cases ﬁrst illustrate ﬁre development and extension of mathematical ideas
across conversations. These cases were drawn from Tasks 1 and 3 ﬁrst were discussed in
more detail in the two previous chapters. In ﬁre ﬁrst case, “It’s Two Something,“
Ronnie and his moﬁrer answered a question at home only to have Ronnie return to scth
where he failed to fully reconstruct the answer. Alﬁrough he could not recount the work
he and his moﬁrer did, it is still clear that ﬁreir conversation inﬂuenced Ronnie’s
interaction in school. This case also points out ﬁre role culturally deﬁned tools play in

recollecting past actions. In ﬁre second case, “Reconstructing a tool,“ Kathy and her

1 73
father ﬁgured out a way of predicting how many three- and four-digit numerals can be

made from a given set of digits. Although ﬁreir initial “tool” worked to determine the
number of three-digit numerals, they found it did not work for four-digit numerals.
They needed to, and did, reﬁne ﬁre tool to work in oﬁrer situations and, at ﬁre same time,
still work in the initial situation. Kathy brought the tool they constructed back to the
classroom and convinced her classmates and teacher that it did, indeed, predict ﬁre
number of numerals that can be made with a given number of digits—but not wiﬁrout
some resistance from her classmates. The cases begin in ﬁre students’ homes where they
began working on ﬁre tasks. As a result, each case revisits many of ﬁre issues discussed
in chapters four and ﬁve. Boﬁr cases also look beyond the immediate conversation to help
explain ﬁre movement seen wiﬁrin ﬁre conversations in students' homes.

In ﬁre second section, I look at ﬁre students' conversations as ﬁrey completed Task 4.
Task 4 was made up of questions ﬁrst included content similar to that contained in ﬁre
ﬁrst three tasks. Although the students had successfully completed ﬁre previous ﬁrree
tasks, neither ﬁre students nor the parents referred back to ﬁre earlier tasks as they
worked on task 4. Although this may be explained by ﬁre students’ ability to complete
many of the questions wiﬁrout assistance, it also suggests, as does “It’s Two Something,”
that what is Ieamed in one setting, may not be available in its entirety to ﬁre student in
subsequent conversations.

It’s Two Something
AtHome

Ronnie and his moﬁrer sat down to work on his homework. As usual, they sat in
front of the television, Ronnie on ﬁre floor, his mother on the couch behind him
(interview, 11/12/92). This night they were working on Task 1 which asked the
students to assist Willie, who was packing lunches for two friends, on a trip to ﬁre
grocery store. Ronnie and his moﬁrer discussed each of ﬁre questions as ﬁrey worked on

them. For the ﬁrst two questions, ﬁrey agreed on ﬁre unit price of ﬁre different items

1 74
and ﬁre number of each item Willie would need to pack ﬁre lunches. When ﬁrey had

figured out ﬁrese values, Ronnie mulﬁplied ﬁre two numbers to find the total price and
his moﬁrer checked his computation. The ﬁrird question involved a different model of
division.1 In spite of this difference, ﬁre interaction between Ronnie and his mother
remained largely ﬁre same. As they discussed each of ﬁre ﬁrst ﬁrree questions, Ronnie’s
moﬁrer repeated ﬁre question as it was written on ﬁre assignment sheet and let Ronnie
begin answering it. When she recognized Ronnie was having trouble, she would
interrupt his work and direct him to an appropriate answer to the question. When ﬁrey
got to ﬁre fourﬁr question they had ﬁris conversation:
1 RB (Reading the question from ﬁre assignment sheet.) Willie and his friends
decided ﬁrey would split ﬁre cost of ﬁre lunches each week. How much did
Willie spend at ﬁre supermarket? How much did it cost each person?
2 M Um-hmm

3 RB One dollar (Ronnie began writing on ﬁre back of his assignment sheet.)

4 M What do you, do you add it up on ﬁrst side of the paper? No, no, no, no, wait -
a minute ﬁrough. Where you getting four ﬁfty-six from? (pause)

RB Four fifty-six cause there’s twelve Twinkees in the problem
M Oh, you goin‘ back this way instead of down. [yeah] Okay. (pause)
7 RB [inaudible]

M What else do you got down ﬁrere?

9 RB Seven fifty-six. Willie spent. (pause) I already know that it's two fifty-
two.

10 M What’s two fifty-two

11 RB The answer.

 

1Questions one and two involved a parﬁﬁve model of division and question three a
quotaﬁve model. A partitive model of division answers questions like “If you twelve
items into three groups, how many items will be in each group?“ A quotaﬁve model
answers ﬁre question “ How many groups of ﬁrree are there in twelve?‘ For a more
- complete description of the division models see chapter three.

12

13

14

15
16
17
18
19
20
21

22

23
24
25
26
27
28
29
30
31
32

33

3

35333338

423232323293

175

No, no, no, no. See it says how much did Willie spend at ﬁre supermarket?
Okay up here he needed two tomatoes which were sixty-six cents. Okay
ﬁren he needed six apples but only ﬁve of them was for two dollars so he

had, you had to put how much more wiﬁr it?

Two forty

Right. 80 sixty-six plus two forty. Okay ﬁren down here he bought, he had
to get the Twinkees, right? [Um-hmm] which was four ﬁfty-six. And
ﬁren, then, then you add ﬁrose up. So, you’d be adding up the sixty-six

cents ﬁrst he spent [uh-huh] and ﬁre two . . .

The two forty

Right. and ﬁren ﬁre . . .

The four fifty-six.

Right.

So, I can do that here. Four fifty-six, two forty . . .

With ﬁre sixty-six cents he paid for the tomatoes. (pause)

[inaudible] Seven sixty two.

Let me see. (pause) carry ﬁre one, six plus tan is sixteen, put down ﬁre
six carry the one, yup seven sixty two. [80, ﬁrree] That's how much he
spent, no, no, no that’s how much he spent at ﬁre grocery store for his

week. For his . . . (pause)

Okay

Okay, you gotta make sure you write ﬁrst down for ﬁre answer ﬁrough.

I don't need to.

Yes you do.

Uh-uh, she knows it’s on the back.

Oh, okay. And how much did it cost each person?
Now I gotta average. Three goes into seven sixty-two

Um-hmm. (pause) You must use ﬁrese up here. (pause)
That’s what that'll be.

What?

Two fifty-two, four.

176

34 M Two dollars and fifty-four cents.
Ronnie began ﬁre conversation by posing ﬁre question as it was stated on ﬁre

assignment sheet and began answering it on his own. He started by writing down ﬁre cost
of ﬁre Twinkees in Question 3. In line 4 his mother stopped him and asked him “Where
you getting four fifty-six from?” Alﬁrough Ronnie was only going in a different
direction (i.e., listing the values from Question 3, Question 2, and ﬁren Question 1
raﬁrer ﬁran the reverse order), Ronnie’s moﬁrer ﬁrought he was doing someﬁring
different from ﬁre procedure she thought was appropriate. When she understood what he
was doing, she recognized it as an alternative way of answering ﬁre question and changed
her situation definition to match his.

Ronnie then listed amounts from the first three questionsunot Willie‘s cost, but

ﬁre prices listed in the questions on ﬁre assignment sheet and added ﬁrem togeﬁrer

arriving at a total cost of “seven fifty-six“ ($4.56+2.00+1.00:7.56). Ronnie,

apparenﬁy having done some work before ﬁre conversation, announced ﬁrst he already
knew ﬁre cost per person was $2.52. As he wrote down his answer, his mother (line

12) recognized that he had listed amounts that were inconsistent wiﬁr her situation
definition and became more directive. She stopped him and told him he needed to use
different amountsuthe amounts on which they had agreed earlier in their conversation-
-and ﬁrst he needed to add ﬁrose amounts to ﬁnd out how much Willie spent at ﬁre grocery
store. After her explanation, Ronnie and his moﬁrer said ﬁre amounts together as he
wrote ﬁrem down. Ronnie added ﬁre amounts and arrived at a sum of “seven sixty-two.”

His moﬁrer checked his computation by talking aloud ﬁrrough the addition algoriﬁrm and

explained ﬁrst the number represented the total cost of ﬁre shopping trip.

In lines 24-28, it became clear that Ronnie’s and his moﬁrer’s situation definitions
included an appropriate presentation of ﬁre answer as well as ﬁre answer itself. Ronnie
did all the calculations on the back side of his paper. His moﬁrer, perhaps drawing on

her conception of school maﬁr, told him to write ﬁre sum on the front of his paper.

1 77
Ronnie declined, saying he could write it there because his teacher would know where to

ﬁnd It. His mother accepted his explanationuagain changing her situation definition to
match his-and ﬁrey went on to ﬁre next part of ﬁre question: How much would each

person need to pay? Ronnie announced he needed to “average,“ divided ﬁre total cost by
three--ﬁre number of people eating ﬁre lunchesnand ended up with a cost of $2.54 per
person.

To ascertain Ronnie’s zone of proximal development, we need to determine what he
was able to do on his own and what he was able to do wiﬁr his moﬁrer’s assistance. Ronnie
knew that he needed to collect amounts from the ﬁrst three questions, add ﬁrem togeﬁrer,
and “average“ them. Ronnie also knew how to complete all of ﬁre computation necessary
to answer the question. This informaﬁon, however, was not enough for him to answer
ﬁre question correctly. To accomplish ﬁrst, his moﬁrer needed to assist him.

Ronnie’s situation definition differed only slightly from his mother's. She too
believed ﬁrey needed to gather amounts from each of the previous ﬁrree quesﬁons, add
ﬁrem togeﬁrer, and divide by ﬁrree. But Ronnie and his moﬁrer may not have agreed on
which amounts needed to be collected and ﬁre reason ﬁre sum needed to be divided by
ﬁrree. Wiﬁr his mother’s help, Ronnie was able to answer the question in an appropriate
way. Ronnie’s moﬁrer was able to assist him in determining which amounts to gather
and, perhaps, understanding why they needed to divide ﬁre sum by ﬁrree. And, in ﬁre and,
when Ronnie and his mother began reciting ﬁre values in unison, it appeared that Ronnie
had changed his situation deﬁnition to more closely resemble his moﬁrer’s.

The shift in Ronnie’s situation deﬁnition suggests ﬁrst he Ieamednor
appropriated--his mother's strategy for answering the question. The shift represents
the qualitative change in his actions that signifies Ieaming in the zone of proximal ‘

development. But, even though Ronnie appeared to change his situation deﬁnition, there

are still pieces of Ronnie’s solution that are difficult to understanduunless we look at

what he did in his classroom before this conversation. Further, ﬁre appearance ﬁrst

178
Ronnie has learned how to solve this problem is brought into question if we look at what

he did in his classroom after ﬁris conversation.
There are many ways in which this conversation reﬂects maﬁrematics in Ronnie's

classroom. The most obvious connection is ﬁrst Ronnie and his mother are talking about
maﬁr homework. They probably would not have had ﬁris conversation if ﬁre task had not
been assigned as homeworknand perhaps they would not have had ﬁris conversation if

ﬁrey were not participating in this study. Second, Ronnie's ideas about where to record
the different pieces of his solution reﬂect the way ﬁrey “do math“ in his classroom. This
conception is different from ﬁre conception his moﬁrer has, but when he invokes ﬁre
authority of the classroom (line 27: “She knows it’s on the back“), Ronnie's mother
changes her deﬁnition to match Ronnie’s conception of math in Ms. Smiﬁr’s classroom.
Finally, Ronnie announced ﬁrst he needed to average. Taken alone, Ronnie's
announcement might not make sense; he was not ﬁnding an average cost. But, a few
weeks prior to this task, ﬁre class completed a short unit on averages. During ﬁrst unit
ﬁrey Ieamed an algorithm for computing averages. The algoriﬁrm was someﬁring like:
“Add up all the different values, divide ﬁre sum by ﬁre number of values you added
togeﬁrer. The quotient will be ﬁre average.“ In a sense, Ronnie used ﬁris algorithm in
ﬁris task. Alﬁrough the task was not an average problem, he did add up ﬁrree values and
divided ﬁre sum by ﬁrreeuﬁre number of values he added together. So, the algoriﬁrm he

used is consistent wiﬁr ﬁre one he Ieamed to compute averages and ﬁre connection he made

seems plausible.

In School
On most days students met in small groups to discuss ﬁre tasks before talking about

ﬁrem as a class. When Ronnie returned to ﬁre classroom, he met wiﬁr two oﬁrer students,
Jason and Pete, to discuss Question 4. Tony, who was also a member of ﬁris math group,
was absent. As they began ﬁreir discussion, Jason and Pete were trying to see who could

get the largest number in ﬁre display of ﬁreir calculators. To increase ﬁre size of ﬁre

1 7 9
number, they entered an addition problem and ﬁren repeatedly and quickly pushed the

“+“ button with the eraser end of their pencils. In the midst of ﬁris competition, Ronnie

paraphrased Question 4 from ﬁre assignment sheet and ﬁrey had ﬁris exchange:

Now Willie and his friends, how much did they spend at ﬁre supermarket
altogether?

2 PC How much did ﬁrey spend altogeﬁrer?
Check ﬁrst out y’all (Shows his groupmates a very large number in the

IFIB

3 JS
display of his calculator). Get ﬁrst much.
4 RB (Working on the assignment) One dollar, two dollars, four ﬁfty-six
5 PC Don’t push it (ﬁre addition button on Jason’s calculator) anymore and I’ll

beat it. (pause)
Jason stopped pushing ﬁre addition button, tossed his calculator onto Pete's desk, and

turned to face Ronnie's desk. He watched Ronnie work on Question 4. When Ronnie
ﬁnished, he announced that the total cost of the groceries was “seven fifty-six“ (the
erroneous total he had gotten before his mother corrected him at home) and he looked to
Pete and Jason for conﬁrmaﬁon. Jason was looking at his paper and Pete was still
punching the addition button on his calculator. Ronnie, in an angry voice, repeated his
answer to Pete who, wiﬁrout stopping or looking at Ronnie’s work, nodded and repeated
his answer, “seven ﬁfty-six,“ a different total than ﬁre one written on Pete's
assignment sheet. Jason, however, ﬁrought Ronnie's answer was wrong and turned to
face his own desk and checked his ﬁgures.

As Ronnie and Jason worked on Question 4, Pete picked up the microphone, ﬁrrust it
at Jason saying “I’ll smack you.“ Pete’s action captured ﬁre Interest of a classmate who
asked what ﬁre microphone was. Pete told the classmate “it’ll record you.“ Jason then
turned to ﬁre microphone and said some nonsense words into it. Ronnie, who was still
working on the question, turned to boﬁr Pete and Jason and reprimanded ﬁrem for not
working on their assignment. In response to Ronnie's reprimand, Jason ﬁrreatened to

leave the group and turned to his own desk. Ronnie turned to Pete. Jason, however, soon

 

1 8 0
returned to ﬁre conversation and he and Ronnie discussed ﬁre questions as Pets continued

to push ﬁre buttons on his calculator.
6 RB Three goes into seven . . .
7 JS No,it’ssupposedtobe...

8 RB how many times?

9 PC (still pushing the addition button on his calculator) Three goes into
seven?

1 0 RB Two times, right?

You’re supposed to see how much it costs for everyﬁring and it comes out to

11 J3
five sixty-four,2 right?

12 PC What numbers do you have to push to spell ‘hell?’

1 3 J3 And you have to times3 it by ﬁrree people because ﬁrree people are
splitﬁng ﬁre cost. So ﬁrst equals . . . it ends up for me, it ends up for me a

dollar eighty-eight for each person. A dollar eighty-eight for each person.

14 RB That’s not right. It’s two something.
15 J3 A dollar eighty-eight for each person. I'll even do it. (pause)
Alﬁrough Ronnie was having trouble reconstructing ﬁre answer he and his moﬁrer
agreed on in ﬁreir conversation, he was sure that Jason’s answer, “a dollar eighty-
eight,“ was not correct.4 Ronnie remembered ﬁrst the answer he and his moﬁrer agreed
on was “two something.“ Jason however, still believed his answer was right. To

support it, he began punching ﬁre numbers he had used to determine ﬁre cost of the

 

2Jason arrived at his answer of $1.88 by adding $2.04 (34 cents-wthe cost of one
tomato--by 6-the number of tomatoes he believed Willie needed, he ﬁren added $2.40

(the cost of six apples)+$1.20 (the cost of putting Twinkees in the lunches for one day)
and then dividing the sum ($5.64) by ﬁrree (the number of people sharing the cost) to

arrive at a cost per person of $1.88.
3Although Jason used ﬁre term “times” to describe what he had done, he divided ﬁre total

cost ($5.64) by three.

4Jason, Tony, and Pete had all used ﬁre same strategy to answer Question Four. They all
took the answer to ﬁre last part of questions one, two, and ﬁrree, added them together, and
divided the sum by three to get ﬁre cost per person. As a result, ﬁrey all took ﬁre cost of
putting Twinkees into ﬁre lunches each day rather ﬁran the cost for the week. This
strategy might represent their understanding of how to do word problems in school math.
Their strategy might include a provision for sets of problems where the last question, in
some way, summarizes the earlier questions. In ﬁrese instances, you take ﬁre answers to

the earlier questions and manipulate ﬁrem in some way.

181

groceries Into his calculator. When he ﬁnished, he showed Ronnie and Pete the display

and said “See, a dollar eighty-eight.“ Ronnie still was not convinced and asked Jason,

“Where did you get five sixty-four from?” to which Jason replied “From everything“

and grabbed his paper to show him. As ﬁre conversation continued, Ronnie tried to show

Jason that the values he (Ronnie) added togeﬁrer resulted in a total cost of $7.56. Jason

initially balked at Ronnie’s explanation, ﬁren reluctanﬁy added the values.

16

17
18

19

20

21
22

23

24
25
26
27

28
29
3O
31
32

33

RB

JS
FIB

JS
FC

RB
J8

J8
RB
JS
RB

JS

RB
J3

J8

That was three for one dollar, ﬁrey paid for that. Then ﬁrey pay right here,
four fifty-six. Then, oh
I got my answer right, that’s final.

Look and two dollars. Two dollars plus one dollar plus four fifty-six is not
no five something.

Five point sixty four divided by ﬁrree equals a dollar eighty-eight.
(Looking at Jason’s calculator) Ah, dude, you only got eighteen hundred
dollars and ninety-five.

Look Pete, I'm gonna add it up in front of your face, look.

(reading ﬁre number in the display on his calculator) Eighteen hundred
ninety-five.

Look y’all. One dollar, two dollars, and four ﬁfty-six is seven fifty-six.
Tell him it's right y'all. Tell him it’s right.

That's wrong.

Tell him it’s right. You gotta add ﬁrst one dollar . . .

Let me see this. I’ll add it all up.

ﬁrst two dollars and ﬁrst four fifty-six. And ﬁrst comes up to seven ﬁfty-
six. Ain’t that right?

(Beginning to add ﬁre costs together again) One point zero zero plus

Yes, that's right.

Yeah, he’s comin’ up with ﬁve something.
two point zero zero plus

Do it on your calculator

four point fifty-six equals seven fifty-six. Seven fifty-six.

182

 

34 RB That’s what I had, no. You so dumb.
35 JS You so dumb. (pause)

36 PC Seven ﬁfty-six, that’s the answer.

37 J8 Two fifty-two.
Much like ﬁre families during homework sessions, Ronnie and his groupmates were

involved in competing activities. Pete, alﬁrough he occasionally contributed to the
discussion of ﬁre homework, was equally concerned wiﬁr ﬁre race for ﬁre largest number
on the calculator and oﬁrer calculator tricks. He reduced his role in conversation to

conﬁrming ﬁre results of ﬁre other members while Ronnie and Jason discussed and

 

argued out a solution to Question 4. His conﬁrmations, however, seldom involved more

‘r

ﬁran a cursory nod or verbal agreement. Alﬁrough it appears ﬁrst Pete was not prepared
for this assignment, he and his grandmoﬁrer successfully answered ﬁre questions at home

two nights before ﬁris conversation. In many instances, ﬁre work Pete and his

grandmother did while answering ﬁre questions may have helped Jason and Ronnie

successfully answer ﬁre questions.
Throughout ﬁre conversation, Ronnie used ﬁre situation definition he used when he
started working wiﬁr his mother. He did not use ﬁre solution on which ﬁrey finally
agreed. He wrote down ﬁre amounts listed in ﬁre questions on ﬁre assignment sheet:
$1.00, $2.00, and $4.56 (lines 16, 18, and 23). When Jason suggested a different
solution, Ronnie, in line 14, said ﬁre solution made no sense, ﬁrst it needed to be “two
something,“ but he could not completely reconstruct the strategy he and his moﬁrer used
two nights before. In line 16, Ronnie talked about the amounts Willie paid for the
different items on his list. This, too, was part of ﬁre discussion Ronnie and his mother
had at home. Each of ﬁrese ﬁrings suggests ﬁrst Ronnie took someﬁring away from ﬁre

conversation he had wiﬁr his moﬁrer. He knew ﬁrst the answer could be determined by

collecting ﬁre amount Willie spent for each item, adding ﬁre amounts togeﬁrer, and

1 8 3
dividing by three. He also knew ﬁre answer should be “two someﬁring.“ But, while he

appropriated pieces of ﬁreir solution, he did not internalize it entirely.
Later in class, Ronnie was called on to explain his group’s answer. Ronnie again had

trouble reconstructing their solution. Ms. Smith asked him if he had explained
everyﬁring he had done to answer the question. Ronnie told her he had not and explained

ﬁrst he had lost his homework paper and had no written record of his work at home. The

only thing he had written down was ﬁre computation he had done wiﬁr his group and his

record of ﬁrst was sketchy.
If Ronnie had his paper it is likely ﬁrst he could have reconstructed the solution on

which he and his mother agreed. Vygotsky (1978) suggested ﬁrst ﬁre development of

higher psychological processes was mediated by more knowledgeable oﬁrers who

introduce culturally defined tools or activities designed to facilitate ﬁre process. When

developing memory, for instance, adults may teach children to notch a stick or write a

list. The children can use this method until they develop ﬁre capacity to remember

wiﬁrout the tool. In Ronnie's case, his wriﬁen record may have served the same

purpose. It may have allowed him to recall processes ﬁrst were not yet fully developed.

Alﬁrough Ronnie's difficulty reconstructing the strategy in school might suggest he did

not change his ﬁrinking in a way that more closely reﬂects that of his moﬁrer, anoﬁrer

plausible interpretation focuses on the importance of ﬁre tools or mnemonic devices

Ronnie used to help him ﬁrink about ﬁre question. Had he held on to his homework, ﬁre
notes and calculations he had written down may have helped him reconstruct ﬁre solution
he and his mother agreed to. But wiﬁrout ﬁrem he needed more assistance ﬁran was ‘
available from his groupmates to answer the question.

The class went on to accept ﬁrree possible answers to Question 4. The answers ﬁre

class accepted were based on different assumptions the students made while answering
all the questions in Task 1. If students went to a “nice store” ﬁrst charged 33 cents for

each tomato, each person would need to pay $2.54 (.66+2.40+4.56:$7.62.

 

‘ “ll

 

 

1 84
7.62+3=$2.54). If the students went to a “greedy store“ ﬁrey would have paid 34 cents

for each tomato for a total cost per person of $2.55 (.68+2.40+4.56:7.64.
7.64+3=$2.54 r2 rounded to $2.55). The third acceptable answer was a variation on
the greedy store solution that they termed the “work solution.“ In this solution Willie's
two friends each paid $2.55 and, because he had done ﬁre shopping, Willie paid $2.54.
The answer Ronnie and his maﬁr group had constructed was not included in the list of

acceptable answers. The answers Ronnie and his moﬁrer agreed to—the nice store

solution--was.
Summary
In their description of the zone of proximal development, Newman, Grifﬁn, and Cole

(1989) wrote:
The concept refers to an interactive system within which people work on a problem

which at least one of them could not, alone, work on effectively. . . .
Meﬁrodologicslly, cognitive change can be observed as children pass ﬁrrough or work

within the zone. (Newman et al., 1989, p. 61)
This description ﬁts ﬁre conversation between Ronnie and his mother as they worked on
Task 1, Question 4. Togeﬁrer ﬁrey worked to answer a question ﬁrst Ronnie could not
answer on his own. As ﬁrey worked togeﬁrer, Ronnie changed his action pattern to more
closely resemble his moﬁrer's. Ronnie’s revised action pattern provided evidence of
cognitive change within ﬁre zone of proximal development.

But Ronnie and his mother’s interaction was not that simple. Some important
aspects of the conversation are overlooked in this short description. In the conversation,
the role of the more knowledgeable oﬁrer shifted. At the same time Ronnie’s mother
helped him answer the question, Ronnie helped his mother understand what it meant to’do

math in his classroom. As in ﬁre conversations reported in chapters four and five, this
trade off resulted in an amalgamated practice ﬁrst represented ﬁre conceptions of school

math and other practices ﬁrey brought to ﬁre conversation. And, again as in ﬁre previous

1 8 5
chapters, ﬁre influence was not evenly distributed across practices. Rather, because ﬁre

task began and ended in school, what was required for an appropriate presentation in
Ronnie's classroom carried more weight than ﬁre other practices.

Although Ronnie revised his solution to more closely resemble his moﬁrer’s, it still
reflected his experience in Ms. Smiﬁr’s classroom. Near the end of their conversation
Ronnie announced ﬁrst he needed to “average“ ﬁre costs. Had we listened only to ﬁre
conversation between Ronnie and his moﬁrer, we may have questioned his use of that
term—perhaps going so far as to say he did not understand the concept of averaging. The
recognition ﬁrst he used a algorithm that had been labeled “ﬁre averaging algorithm“ in
class makes his choice of words reasonable.

Looking across the conversations points out the gradual appropriation of ideas and
activities posited by Vygotsky (1978) and other sociocultural theorists (Newman et al.,
1989). Although we can argue that we were able to observe cognitive change in Ronnie's
conversation wiﬁr his mother, the change was not lasting. Ronnie’s difﬁculty recounting
ﬁre solution when he retumed to school suggested that he did not completely understand
how to answer the question. Yet, his interaction wiﬁr his maﬁr group did include ideas
about which he and his mother talked. What he appropriated from ﬁre conversation with
his mother was a parﬁal understanding ﬁrst may be filled out in subsequent
conversations. Ronnie’s difﬁculty in scth also highlights ﬁre importance of the
cultural tools available to document conversations. During ﬁre conversation with his
mother, Ronnie wrote down what he and his moﬁrer agreed upon and even debated what
his documentation of the solution ought to look like. But, wiﬁrout his written record,
Ronnie was unable to reconstruct ﬁre solution in school. Perhaps if he had his paper he
could have.

Newman, Griffin, and Cole (1989) continue their discussion of the zone of proximal
development saying, “When cognitive change occurs not only what is carried out among

participants, but how they carry it out appears again as an independent psychological

1 8 6
function that can be attributed to ﬁre novice.“ (Newman et al., 1989, p. 61). Students

not only Ieam maﬁrematical ideas in ﬁreir conversations, but how to do math as well.
What ﬁrey Ieamed about doing maﬁr in their conversations at home influenced ﬁreir
participation in school. This phenomenon is clear in ﬁre second case, Reconstructing a
Tool. In the case, Kaﬁry and her faﬁrer worked togeﬁrer to ﬁnd a way to predict how
many numbers can be formed from a given set of digits. Over two nights’ discussion,
they arrived at an appropriate way to predict how many numbers were possible.
Although this skill is valuable in itself, participating in the process of constructing it
may have been even more valuable for Kaﬁry. The case also highlights ﬁre gradual
development of maﬁrematical tools and ideas.
Reconstructing a Tool

AtHome

Kaﬁry and her father worked together at ﬁre dining room table. Normally, Kathy,
her brother Bobby, and her father all did ﬁreir homework together. This evening,
however, Bobby watched a television program about the “old west” while Kaﬁry and her
faﬁrer worked on Task 3. Alﬁrough Kaﬁry and her father's answer to ﬁre question of how
ﬁrey knew ﬁrey had all ﬁre numbers was similar to the answers of the oﬁrer students,
their conversation was quite different. Kaﬁry and her faﬁrer were the only people who
tried to determine a way of predicting how many numbers were possible while working
on ﬁre ﬁrst part of Task 3. The other students generated the list of numbers and ﬁren
tried to explain how they generated the list.

Kathy read ﬁre ﬁrst question aloud and began answering it. The question asked the
students to list as many ﬁrree-digit numbers as possible using the digits 3, 4, and 7.
Kaﬁry found six numbers that she could write using ﬁre digits. She showed ﬁre list to her
faﬁrer and he his initial response was ﬁrst ﬁrere should be more numbers. They had this

conversation:

11511611511

10

11

5

IG
F

187

It should be more ﬁran six.

What, it can't be.

There are.

No, daddy, I've used all ﬁre numbers.

Um-hmm

You have to use all three of ﬁrem at one ﬁme.

I know. Three should be able to take up every position, right? So here
it's in ﬁris position, that position, and ﬁris position. Four should be able
to be in every position. So, here it's in second position [347], last
position [374], first position [437], first position [473].

let's see, right here.

Yeah, but if you look at ﬁris, three should be able to be in ﬁrree positions,
four should be able to be in four positions, and seven should be able to be
in ﬁrree positions. That's nine total numbers that you should be able to
come up wiﬁr.

Four shouldn't be able to be in four different posiﬁons.

Three different positions.

Alﬁrough Kathy found only six three-digit numbers, her father believed more

numbers were possible. His reasoning about how many ﬁrree-digit numbers were

possible focused on ﬁre number of positions each digit could ﬁll. Because each digit could

be in ﬁrree positions, and ﬁrere were three digits to use, nine numbers were possible

(3+3+3=9L

Keeping ﬁre idea ﬁrst nine numbers were possible in mind, Kathy and her faﬁrer

looked for three more numbers to complete their list of all possible numbers. They

looked at ﬁre list to make sure each digit had occupied each position an equal number of

, times. When they could not find any more numbers Kathy’s faﬁrer somewhat reluctanﬁy

changed ﬁre way he predicted ﬁre number possible.

12

F

Well see you might be right, I can’t remember how this works, but, the
number in ﬁre first position can have ﬁrree positions, the number in ﬁre
second can have two, and ﬁre number in ﬁre last position can have one.
That’s a total of six.

188

1 3 IG See, ﬁrst’s what I got.

14 F Yeah, I know, but I still keep ﬁrinking there’s more combinations.

1 5 IC- that's seven ﬁrirty four. I've got six. So the answer would be . . .

1 6 F Okay, go ahead and go, go with your answer because I can’t come up with
any other.

1 7 I6 Okay, we'll just say, we'll just say six numbers.

1 8 F Remember we‘re supposed to leave all ﬁris work on here.

1 9 IG I know. Okay, for 'B’ it says ’How do you know when you‘ve found all ﬁre
numbers?’ All ﬁre letters are like, gone.

20 F All the letters?
21 I6 I mean all the numbers are gone.
2 2 F Every number has occupied every position.

Kaﬁry and her faﬁrer’s strategy changed in this part of their conversation. Rather
than predicting how many numbers were possible and ﬁren trying to ﬁnd ﬁrst many
numbers, ﬁrey accepted the list of six numbers Kaﬁry had written down and tried to
ﬁgure out a way of predicting that number. Their second attempt at predicting how many
numbers were possible still focused on ﬁre number of digits and the positions each digit
could fill, but took into account the fact ﬁrst once a position is filled it is no longer
available to the oﬁrer digits. As a result, the first digit can be placed in three positions,
ﬁre second in two positions and ﬁre ﬁrird in the leftover posiﬁon. Even ﬁrough Kathy’s
father toyed with this meﬁrod of predicting how many numbers were possible, Kathy and
her faﬁrer were not completely convinced ﬁrst this strategy was correct. Instead, Kaﬁry
wrote down “all the numbers are gone“ as the reason for knowing they had found all the
numbers.

In School

The next day Kaﬁry took her answers back to school. As on most days, Ms. Smiﬁr's

class met in maﬁr groups to discuss their homework. Kaﬁry met with Elaine and Helen.

Their discussion was short, all agreeing there were six possible numbers, how ﬁrose

1 8 9
numbers should be ordered from smallest to largest, and how they could determine the

relative size of the numbers. That day each math group produced a chart showing ﬁre
numbers ﬁrst could be made with ﬁre ﬁrree digits and ﬁre answers to the oﬁrer questions.
Kaﬁry and her-group added an answer to the bottom of ﬁreir chart which said “We never
have anything to talk about because we always agree.“ Kaﬁry and her faﬁrer’s discussion
about how to determine how many numbers were possible was not discussed in ﬁreir
maﬁr group, only the answers on the assignment sheets. Near ﬁre and of class, Ms. Smiﬁr

passed out ﬁre second part of Task 3 and instructed ﬁre students to take it home and

complete it overnight.
Back at Home

As ﬁrey had ﬁre previous evening, Kaﬁry and her faﬁrer worked at ﬁre dining room
table. In ﬁre background, Bobby watched a television show about Malcom X. Throughout
the conversation, Kaﬁry’s, and sometimes her father’s attention, shifted back and forth
between ﬁre television show and Kathy’s homework.

Question 1 asked ﬁre students to generate as many four-digit numbers as possible
using the digits 0, 2, 5, and 9. Kaﬁry had written a list of numbers before they started
recording. she started ﬁre tape by reading the list aloud. Near the end she asked her
faﬁrer if she could use zero in ﬁre ﬁrousands place. He told she could, but would not say
“zero ﬁroussnds“ when she read ﬁre number. Kathy completed the list and announced she
had 24 numbers. Her announcement took her faﬁrer by surprise.

1 I6 I have twenty-four.

F Twenty-four. numbers?

(ION

IG Yes, ﬁrst I made out of ﬁrose. I have two, two oh ﬁve nine. two ﬁve oh
nine. Two oh nine ﬁve. Two ﬁve oh nine. Two five . . .

Yeah, ﬁrat's ﬁre same number. . .
nine oh.

I can predict ﬁrst ﬁrere’ll be . . .

\IOrU'I-h
ﬁ'nﬁ'"

Two nine oh ﬁve, Two nine ﬁve oh. See

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

“"6115 6

E5

190

Fourteen. There should be fourteen numbers.

One, two, ﬁrree four, ﬁve, six, seven, eight, nine, ten, eleven, twelve,
thirteen, fourteen, ﬁfteen, sixteen, seventeen, eighteen, nineteen,
twenty, twenty-one, twenty-two, twenty-three, twenty-four. Twenty-
four, not fourteen. Cause I used ﬁre zero.

No, shouldn’t matter.

Watch, for each one it should have at least six.

Well how many numbers did you come up with for ﬁrree? Remember
when you had ﬁrree?

Three?

Yeah, when you had ﬁrree, how many number combinations did you come
up wiﬁr? You came up wiﬁr six didn’t you?

Yeah.

Okay. Now think about ﬁrst. You had three numbers. Okay, lets just call
it A, B, and C.

I know, but now I have . . .

And you come up with six combinations. Right?

Um-hmm [yes]

So, we could either say ﬁrst ﬁrat’s two times as many as ﬁrese or you could
look at it ﬁrst A here can be in three different positions, right, it can be in
this position, this position, or this position.

Um-hmm

But if it‘s in this position, ﬁren these two can only occupy two positions.
And if ﬁris one is in this position and ﬁris one's in ﬁris position, this one
can only occupy one position, all right? So, we take and we add ﬁrem up.
Three plus two plus one and what number does ﬁrst give us?

(pause) Six.

Right. And we came up with six combinations, right? Now let's look at
this number. We have four of ’em: A, B, C, and D, all right? Now we have
four different possibilities. Let’s use the same rule we came up wiﬁr up
here. This one can be in four positions [Um-hmm]; this one can be in
three positions; ﬁris one can be in two positions; ﬁris one can be in one
position. How many does that give us?

Ten.

Oh, you're right.

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

5116116115

611611611

5115116116116“

191

seven, eight, nine ten . . .

How did I come up wiﬁr fourteen? There should only be ten combinations.
But I can get that many, Daddy! I can't help it.

Are you sure you don't have any repeats like you did ﬁrere above ﬁrough?
Two, ah, you listen. Two oh ﬁve nine . . .

No, l have to look at them, I can‘t just listen.

Two oh nine five. Two ﬁve oh nine. Two five nine oh. Two nine oh ﬁve.
Two nine ﬁve oh. Two five, no, Five two oh nine. Five two nine oh. Five
nine oh two. Five nine two oh. Five oh two nine. Five oh nine two. Oh
Two hundred ninety-five. Two hundred fifty-nine. Five hundred ninety
two. Five hundred two, twenty nine. Nine hundred twenty ﬁve. Nine
hundred fifty two. Okay, Nine two oh five. Nine two five oh. Nine ﬁve oh
two. Nine ﬁve two oh.

You're right.

Nine oh two ﬁve.

All right, we'll have to rethink this then.

Nine oh ﬁve two.

So, maybe,

This can be in four posiﬁons, that can be in four, that can be in four, and
that can be in four. One two, ﬁrat'd be eight and . . .

That’s four to ﬁre fourth.

What?

Four to the fourth. It’s four times four times four times four.
No.

Oh, you're adding them togeﬁrer.

Yeah. That's four

plus four plus four plus . . .

Sixteen. I don’t know how I got ﬁrese.

Four times four is sixteen.

but they‘re all logic right?

192

5 0 F Yeah, but how many, how many do we have?

51 IO Twenty-four.

52 F Twenty-four, lets see how we can come up with twenty-four out of this
then. Oh, ohll Ah-halll I know what it is. Look at ﬁris. We can come up
with that. This is not plus, this is times. Look three times two is six,
times one is six. Now watch. Four times ﬁrree times two times one. Four
times three is twelve [twelve], times two is twenty-four, times one is
twenty-four. Dah, da-da-dall

Kaﬁry’s faﬁrer’s initial reaction to her announcement rests on ﬁre strategy ﬁrey used
ﬁre previous night to predict how many numbers they could make. Although they were
not completely convinced of ﬁre strategy ﬁre previous night, Kaﬁry’s faﬁrer used it here
to predict how many numbers were possible. The strategy ﬁrey used was to add the
number of positions in which a digit could be placed. For ﬁrree digits the strategy was to
add 3+2+1 for a total of six numbers. For four digits ﬁre strategy was to add 4+3+2+1.
Kathy's faﬁrer’s initial thought that there were 14 possible numbers was likely caused
by a computational error.5

When ﬁrey determined that Kaﬁry’s list included 24 different numbers, they
realized they needed to reﬁrink their strategy (line 36). Kaﬁry began ﬁris process by
trying out the strategy her father first used the night before (lines 39-48). The ﬁrst
digit, she suggested, could be in four positions, the second could be in four positions, and
ﬁre ﬁrird and forﬁr digits could also be in four positions. Adding according to ﬁris
strategy resulted in a prediction of 16 numbers. But they had 24 numbers. Kaﬁry’s
fsﬁrer told her ﬁrey needed to ﬁgure out how ﬁrey could get 24. Almost immediately, he
said, “this is not plus, this is times“ (line 52). To support this idea he pointed out ﬁrst
the strategy worked for both three-digit numbers (3x2x1:6) and four-digit numbers
(4x3x2x1:24). Kathy's father celebrated their solution by singing a trumpeter’s

announcement “Dah, da-da-da.“

 

5Although ﬁrere is no evidence of his computation, it may be that he added
4+4+3+2+1=14.

1 93
The strategy Kathy and her faﬁrer constructed to predict how many numbers were

possible with a certain number of digits developed over two conversations. Over the
course of its development the strategy took on three identities, or-to use Twain’s
termnwent ﬁrrough three epochs. Each of the epochs represented a relationship
between the number of digits and the positions in which the digit could placed. At ﬁrst
Kaﬁry and her faﬁrer believed ﬁrst each digit could be in ﬁre same number of posiﬁons.
To predict how many numbers were possible, ﬁrey multiplied the number of digits by the
number of positions. When the lists of numbers did not match their predictions, Kaﬁry
and her father changed their strategy. They started to think ﬁrst ﬁre number of posiﬁons
in which each digit could be placed depended on its position in ﬁre number. In a three-
digit number, the ﬁrst digit could be in three positions, ﬁre second in two positions, and
ﬁre ﬁrird in one position. By adding ﬁrese values together, they could predict six possible
numbers. This strategy worked for ﬁrree-digit numbers, but when ﬁrey used it to
predict how many four-digit numbers could be made, ﬁrey found ﬁre strategy was
inadequate. In its final epoch, ﬁre strategy was to multiply the number of positions in
which each digit could be placed togeﬁrer. For four-digit numbers they multiplied
4x3x2x1 for a total of 24 possible numbers. This strategy was supported by the fact
that it worked for both three and four-digit numbers.
Back in School

The following day Kaﬁry did not go to school but the events in class form important
background for understanding what happened when she did return to school. Elaine was
ﬁre only member of Ksﬁry’s math group in school that day. Ms. Smiﬁr asked her to join
Mark, Karen, and Joan’s group to discuss ﬁreir answers. As Elaine moved her desk over
to the group, she asked Karen how many numbers she found. Karen told her she found
sixteen. Elaine said she found fifteen. Karen began comparing the lists to ﬁnd ﬁre
number Elaine missed. As Joan entered ﬁre group, Karen asked her how many numbers

she found. Joan told her “eighteen.“ Dumbfounded ﬁrst someone had found more

1 9 4
numbers ﬁran she, Karen grabbed Joan's paper and began looking at ﬁre numbers. She

and Elaine compared ﬁre lists. Although Karen found numbers on Joan’s list that were
not on her or Elaine’s lists, she was not ready to change what she had written down.

Joan’s list did not include numbers that began with zero. She believed that numbers
could not start with zero-0259 was ﬁre same as 259 and did not count as a four-digit
number. Karen was even more concerned ﬁrst Joan had eighteen numbers and had not
used numbers with zero in ﬁre thousands place. She argued ﬁrst her own list was correct
because “I have my dad at my house recording me every time we get something like
this.“ “Big deal!“ Joan responded and Karen elaborated, saying her faﬁrer was 36 years
old. Neiﬁrer being convinced of ﬁre oﬁrer person’s position, Joan and Karen began a
discussion of wheﬁrer zero was a digit that lasted ﬁre entire math period. At the end of
ﬁre day noﬁring had been resolved. Karen and Joan each held to their own beliefs about
zero and their own lists of four-digit numbers.

When Kaﬁry returned to school ﬁre following day, she joined Karen, Joan, and Fred's
maﬁr group. Mark and Elaine were absent. As she entered her new group, she was met
wiﬁr cheers. Ksﬁry was very influential in class. Her ideas often were the focus of class
discussions. Karen’s reaction reﬂects Kathy’s position in class.

Fred was not feeling well. The group started ﬁreir conversation by advising Frank
ﬁrst he should go home. He could not, he told ﬁrem, because his moﬁrer was Christmas
shopping and could not come to pick him up. They tried to convince him to go see ﬁre
school nurse, but Frank refused. Instead, he laid his head on his desk while the other
group members discussed ﬁre assignment. Karen, Kaﬁry, and Joan began by comparing
how many numbers ﬁrey found. (KM denotes Karen; KG denotes Kaﬁry; JS denotes Joan)

1 I6 How many numbers did you get?

2 KM Sixteen.

3 I6 You got sixteen, I got twenty-four.
4

JS I got eighteen

195

5 KM What ﬁre heck, where's your paper?

Karen, Kathy, and Joan started comparing ﬁre lists of numbers ﬁrey generated ﬁre
previous night. But, rather ﬁran focusing on the numbers included in each list, they
focused on how many numbers should be in the lists. Their conversation, as did many of
the conversations in class, focused on supporting and justifying the answers ﬁrey
brought back to class. Kathy started by telling how she and her faﬁrer determined how

many numbers were possible:

See, watch, okay. There's four numbers. Then you can put zero in four places, you

could put two in ﬁrree places, you can put ﬁve in two places, and you can put nine in

one. Now four times three is twelve, times-what is that number?-times two is

twenty-four, times one is twenty-four. So you should have twenty-four.
Karen disagreed wiﬁr Kaﬁry saying only sixteen numbers were possible. She supported
her contention in ﬁre two ways she had ﬁre day before. As on the previous day, these
arguments did not convince her group. At ﬁrst she told ﬁre group “I get tape recorded“
believing that being part of ﬁris study added credence to her work. By recording her
conversations wiﬁr her father, her answers gained strength and allowed her to be more
forceful when presenting her answers. Kaﬁry also recorded her conversations and told
Karen so. In response, Karen looked for anoﬁrer way to substantiate her list.

Karen's second attempt to support her answer came when she told ﬁre group ﬁrst her
faﬁrer had showed her how to predict how many numbers were possible. She told her
groupmates, “My dad told me if ﬁrere‘s four digits you should have four here, four here,
four here, four here.“ Karen explained ﬁris argument more clearly later in ﬁre
conversation saying “There should be, [for] zero there should be four of them, you got
ﬁrst right. Two should be four numbers, ﬁve should be four numbers, and nine should
be, have four numbers.“ Karen was arguing ﬁrst ﬁrere should be four numbers

beginning with 0, four beginning with 2, four beginning wiﬁr 5 and four numbers

beginning wiﬁr 9.

1 9 6
Karen had constructed ﬁris strategy with her father earlier in the week. The

strategy was similar to the first strategy Kathy and her faﬁrer used. Kathy and her
faﬁrer changed ﬁre strategy when ﬁre predicted number did not match ﬁre numbers Kathy
found. In Karen’s conversation with her faﬁrer, ﬁre strategy matched ﬁre list of numbers
she had written down; ﬁrst is, Karen’s list included 16 numbers and the strategy
predicted sixteen possible numbers. Given ﬁris match, ﬁrere was no reason to believe
ﬁre strategy did not work. And, to give up ﬁre strategy wiﬁrout some resistance meant
giving up what she and her father agreed was the right answer.

Varenne et al. (1982) have argued ﬁrst, in areas in which parents feel competent,
they work very hard to send ﬁreir children to school wiﬁr correct answers on ﬁreir
homework. Parents work wiﬁr their children because they believe, Varenne et al. argue,
that teachers assess families as well as ﬁreir students. Parents who allow ﬁreir children
to come to school wiﬁr incorrect answers are held in lower esteem ﬁren ﬁrose parents
whose children come with their homework done correcﬁy. Students, it appears, also
work to save face for their families. They apparently believe ﬁrst their parents would
not allow ﬁrem to take incorrect answers back to school. As a result, students often used
their parents as support for ﬁreir answers. Kathy used her father’s instruction as
support later in the conversation when she said, “I understand it and maybe you won’t
and I don't know how to explain it. My dad just showed me ﬁrst way and it works.“

Joan disagreed wiﬁr boﬁr Kaﬁry and Karen. She had eighteen numbers and pointed at
Karen's paper while telling her there should be six there, six there, six there and six
there.“6 In the and, Karen, Kaﬁry, and Joan were no closer to agreeing on how many
numbers were possible. Their conversation ended this way:

6 KM Well, all I know is that I got sixteen numbers and I think it’s right.

7 IC-t I think twenty-four is right.

 

6ﬁrese numbers add up to 24 numbers. Joan did not have 24 numbers in her list at ﬁris
point.

8.18

197

I got ﬁre same ﬁring as you [Kaﬁry] but I didn't use the zeros.

Joan’s comment ﬁrst she didn't use ﬁre zeros brought back ﬁre unresolved issue

from the previous day: Could zero be used in ﬁre ﬁrousands place in the numbers?

Alﬁrough Karen and Kathy agreed ﬁrst it could, Joan did not. The group turned to this

issue before returning to the original topic.

Kathy raised the question of zero in the ﬁrousands place with her father two nights

before. During their conversation, Kaﬁry’s father assured her zero could be used, but

ﬁrst she would not say “zero thousands“ when saying ﬁre number. Karen and her faﬁrer

did not discuss wheﬁrer zero could be in the thousands place; ﬁrey included it wiﬁrout

question. In scth Kathy and Karen worked together to persuade Joan ﬁrst zero could be

used in ﬁre ﬁrousands place.

9

10

11

12

13

14

15

16

17

18

19

20

21

I61

JS

KM
JS
KM
JS
KM

JS
KM
JS

KM

I didn’t put it when I did my expanded notation. I didn't, I didn‘t put the
zero, but, um, when I put it down, I had to put it or else people would
complain ’where's the zero?’

Okay. I understand that zero is a digit, but using zero ﬁrousand I don’t
think that’d really sound right because that . . .

Its just zero ﬁrousand two hundred fifty nine.

I know, but if you have no ﬁrousands . . .

It’s just two hundred fifty nine.

Why don’t you just say two hundred ninety ﬁve.

It's just two hundred fifty-nine. You just put ﬁre zero, you just add the
zero.

And you don't say it, you just say ‘two ﬁfty nine.‘ And then, and then if
they ask you a question say 'But I didn't want to say ﬁre zero.’

Yeah
She understand now?
So, I can add ﬁre zeros if I like. All right.

It says four digits, so that‘s how you guys could prove zero is a digit cause
it said four-digit numbers.

That’s what I was trying to tell her the oﬁrer day.

198

2 2 JS I know, but I didn't understand because ﬁrey, zero thousand, I didn’t
understand that. If you have zero ﬁrousand, why not just say two hundred
fifty nine?

Kathy and Karen’s experience in different practices is evident in this short
exchange. Kathy's initial statement ﬁrst she included zero because “people would
complain ‘where’s ﬁre zero'“ (line 9) if she did not reflects her understanding of how
maﬁr is done in her classroom. Kaﬁry’s understanding of math in her classroom is
evident again in line 16 when she offers a way of supporting ﬁre notion ﬁrst zero could be
in ﬁre ﬁrousands place and not be said when reading the numbers. To explain why zero
could be used, Kathy offered her faﬁrer’s explanation: “And you don’t say it, you just say
‘two ﬁfty-nine.” Although Karen did not make direct reference to her conversation
wiﬁr her father, she was steadfast in her belief ﬁrst zero could be in ﬁre thousands place,
just as she was ﬁrst sixteen numbers were possible.

At ﬁre and of ﬁris exchange, Joan was convinced ﬁrst zero could be used in ﬁre
ﬁrousands place and that she could “add ﬁre zeros if I like.“ As Kaﬁry and Karen talked,
Joan wrote out another six numbers all beginning wiﬁr zero. When she finished, her list
had expanded to 24 numbers and included all the numbers on Kaﬁry’s list. Now she and
Kaﬁry agreed ﬁrst there were 24 numbers, but Karen held to her list of sixteen. This
issue proved more difficult to resolve.

23 KM I got sixteen numbers, ﬁrat‘s all I can say.

2 4 IG I did, I only got six. On number four. I mean on the one before ﬁris wiﬁr
three numbers I only got six. That would be right. Three, two and one.

2 5 JS Yeah six for each one.

2 6 IC- Three times two is, you could put ﬁrree in ﬁrree places, you could put two
in two places, you could put one in, let’s say it was seven, say this is a
seven. And you could put, ﬁren you could put ﬁrere in ﬁrree places, Frank
stop, and you can put seven in two places and you can put open in one
place. And, three ﬁmes two is six plus, times one is six. Okay, just

2 7 KM What, what, what ﬁrree, two and one.

2 8 I6 ['To Frank] Get ﬁrst out of here. (Frank ﬁrrusts a piece of paper in front
of Kathy) Three

29

30
31

32

33

34

35

36

37

38

39

40

41

42

43

44

KM

KM

KM
JS
KM

JS
KM
J3

J8
KM
K3

199

Now what, listen to me, Kathy, but, if I remember right, I ﬁrink I had like
twelve numbers

I don’t ﬁrink you could have twelve
Or nine or something like ﬁrst.

Let's just experiment. Say it’s . . . Frank stop it. Get it out of here.
Okay, we could do ﬁrree . . .

I'll put it [ﬁre tape recorder] on pause.

twenty-one. Three one two. That's all you can do right ﬁrere that's all.
And ﬁren you have to go to ﬁre twos. Then it’d be two one ﬁrree. Two ﬁrree
one. Okay so ﬁrat’s one, two, ﬁrree, four. And, then you go to ﬁre ones.

One ﬁrree two. One two three. And, its six numbers and ﬁrat's all you can
make out of it, see? See, one, two, three, four, ﬁve, six.

What I don’t get is . . .
Yeah, but see you're switching the numbers around.

I‘m talking to Kathy just for a minute. Say here‘s the same numbers. But
here, I don’t get what you’re doing.

There's only four digits.

I only get four numbers.

I think there’s more.

I ﬁrink ﬁrere‘s six. Why do you ﬁrink ﬁrere's . . .Okay, let me see what you
have. Two oh . . .Zero two ﬁve nine. Then you have zero two nine ﬁve.
Then you have zero five two nine. But you can also have zero ﬁve nine
two. From zero nine two ﬁve, you can have zero nine ﬁve two.

We had ﬁrst.

We just bring . . .

Fine, I’m not even gonna use ﬁris. You guys can do that. Me and Joan will
make our own numbers.

Frank came alive as they discussed how many numbers were possible. But, raﬁrer

than entering into ﬁre conversation, he started playing with the recording equipment and

ﬁre oﬁrer students’ papers. Kaﬁry, Karen, and Joan’s attention was divided between ﬁre

task and Frank’s playfulness as they tried to answer the question.

200

At the beginning of ﬁris exchange, Kathy tried to show ﬁrst her strategy worked for

both ﬁrree and four-digit numbers. Using an example similar to what she and her father

had done, Kathy predicted that six ﬁrree-digit numbers would be possible. Karen

objected saying she thought she had found more ﬁran six numbers on ﬁre first part of

Task 3. Although she could not remember exactly how many numbers she found, her

second guess-~nine--fits the strategy she was using in this conversation: If a four-digit

number would have 4+4+4+4=16 numbers, then a three-digit number would have

3+3+3=9 numbers.7 With Karen still not convinced, Kathy took her paper to look over

her list of numbers. In the end, her frustraﬁon took over and she suggested ﬁrst she and

Joan _would leave and start their own group. With ﬁre ﬁrreat of her group splitting up,

Karen asked for help. I had the following exchange wiﬁr ﬁre group.

45
46
47
48

49

50

51

52

53

54

55

KM
IG
JR
KM
JS

KM

JR

KG.
J6

J8

Please help us

With four numbers you can make twenty-four combinations can’t you?
What do you think?

I think

We're trying to explain to her how we’re getting six numbers on each one
of ﬁre, like nine ﬁve two zero. We're trying to explain to her how we're
39.an six numbers instead of four and she, we can't ﬁnd a way to explain
You‘re trying to explain it to who?

She ﬁrinks there is only sixteen. l ﬁrink there is twenty-four.

Did you show her you're list?

YES!

And, we tried to explain it.

She doesn’t have half the numbers I have.

 

7Karen and her faﬁrer found six numbers on the ﬁrst part of Task Three. In their
conversation ﬁrey said they knew ﬁrey had found all the numbers when Karen “took the
ﬁrrees and you took the tours and you took ﬁre sevens.“

201

5 6 JR So, you don‘t have all ﬁre numbers she has? {KM nods) So do ﬁre numbers
you don't have work?

5 7 JS Yes, ﬁrey all work

5 8 JR Then you would, would you have to add those numbers to your list?
5 9 KM I don’t know

6 0 JR What do you think?

6 1 KM Yeah.

After our conversation, the group compared and combined ﬁreir lists, eventually
ending wiﬁr a list of 24 numbers. As ﬁrey did, Ms. Smiﬁr called the class back togeﬁrer to
discuss ﬁre questions.

During the whole class discussion, Kaﬁry told ﬁre class her group had found 24
numbers. Kaﬁry's announcement was met with a chorus of surprised reactions. No oﬁrer
group found more ﬁran 16 numbers. Ms. Smiﬁr asked Kaﬁry to read her group’s list and
explain how ﬁrey knew how many numbers were possible. She did, using ﬁre same
explanation she used in her small group. When constructing ﬁre number you have a
choice of four positions for the ﬁrst digit, three positions for ﬁre second digit, two
positions for the third digit, and one position for the final digit (4x3x2x1:24 possible
numbers). After a short discussion, ﬁre class accepted Kathy’s explanation, agreed on
ﬁre list of 24 numbers, and accepted Kaﬁry’s meﬁrod as a way of predicting how many
numbers were possible.

Summary

In ﬁreir conversations at home, Kathy and her father worked together to reconstruct
a strategy her father had used in the past. Their initial strategyuto add ﬁre number of
choices for each digit (3+2+1:6)--worked to predict how many three digit numbers
were possible. When Kaﬁry returned to scth she and her group agreed on ﬁre list of

possible numbers and the answers to ﬁre oﬁrer questions in the task. As a result, ﬁrey

2 0 2
never questioned the strategy Kathy or the oﬁrers used to determine how many numbers

were possible and it is reasonable to say ﬁrst Kaﬁry believed ﬁre strategy worked.

When ﬁrey worked on ﬁre second part of Task 3, Kaﬁry and her faﬁrer were
confronted wiﬁr a situation in which their strategy did not work. When they realized
that Kathy had found more numbers than their initial strategy predicted, Kaﬁry's father
announced they needed to “rethink“ what ﬁrey had done and refine ﬁreir strategy. After a
short time, Kathy’s faﬁrer proclaimed ﬁre correct strategy was to multiply the number
of choices for each position.

As in ﬁre ﬁrst case, Kaﬁry and her faﬁrer’s conversations highlight the gradual
development of mathematical ideas and activities. The strategy they reconstructed went
ﬁrrough a series of changes that support ﬁre contentions of sociocultural theorists. These
theorists argue ﬁrst people appropriate understandings that allow ﬁrem to function in
their immediate situations. Kathy and her father’s initial strategy worked in the
situation in which ﬁrey were working and ﬁrere was no reason for them to look beyond
ﬁrst situation. As ﬁrey worked on ﬁre second part of Task 3, ﬁreir situation changed.
Faced wiﬁr ﬁris change, ﬁrey changed ﬁreir understanding to include both parts of Task 3.

Perhaps even more important ﬁran getting the right answer, Kaﬁry was able to see
her faﬁrer's willingness to conjecture about how to predict the numbers and to change
his conjecture when it was refuted by new evidence. Kaﬁry’s conception of school math,
as well as being influenced by her participation in school, was influenced by her
conversations at home. Modeling inquiry as Kathy's father did may have helped Kaﬁry
participate in school in more sophisticated ways. At the time of this study, Kathy’s
faﬁrer was taking university mathematics classes where students were asked to develop
what he referred to as ﬁre “creative part of math“ in which, he explained, students
develop their own problems and formulas. Kaﬁry’s ability to explain how she and her
faﬁrer determined how many numbers were possible may have come from her extensive

involvement in the conversations with her faﬁrer. Whereas many other students were

2 0 3
conﬁned to a role in which they only computed ﬁre final answer to ﬁre questions, Kaﬁry

and her father “developed their own formulas.“ Her involvement not only allowed her to
contribute her answer to ﬁre class, but to explain what she and her faﬁrer had done at
horns.

Kathy, Karen, and Joan's small group conversation points out that students place a
tremendous amount of faiﬁr in ﬁre ﬁrings ﬁreir parents tell them. As a result they
support ﬁre answer ﬁrey bring back to school by claiming it was their parent’s answer
and Iisﬁng their parent’s attributes. Their faith someﬁmes resulted in group members
threatening to leave their group. Rather ﬁran reﬁrinking or changing ﬁreir answers, the
students would rather leave to preserve ﬁre answer they had written down. As Twain
suggested, Ieaming involves giving up old ways of thinking and accepting new ways.
Alﬁrough the shrdents eventually did change the way ﬁrey answered the questions, ﬁre
give-and-take aspect of learning was often very difficult for the students.

Finally, other practices, such as this study, influenced how students participated in
class. Students who participated in ﬁris study used ﬁreir recorded conversations as
support for their answers.

The gradual learning process is also apparent in students' conversations for
subsequent tasks that involved similar content. Task 4 included four questions similar
to ﬁrose asked in Tasks 1, 2, and 3. Watching students as they answered ﬁrese questions
supports many of the ﬁndings presented above.

Task 4: Making Connections Across Tasks

Looking at students' conversations in different settings as ﬁrey worked on a task
illuminated ﬁre idea that concepts and understanding develop gradually. Participating in
a conversation does not insure the ability to do ﬁre ﬁrings discussed in anoﬁrer setting.
Nor does a solution in one setting insure successful solutions in oﬁrer settings. But, even
ﬁrough students may not be able to use ﬁre ﬁrings discussed in a conversation in their

entirety, they take away pieces that eventually will resemble ﬁre whole activity.

204
Implicit in the belief that learning is gradual is the notion that responsibility for

completing tasks is gradually taken on by ﬁre student while participating in subsequent
tasks involving similar ideas or questions. According to Vygotsky (1978), in
subsequent sessions, learners do more on ﬁreir own whereas more knowledgeable
othersuparents, in this case-contribute less. Parents, it has been suggested
(Wertsch, 1984; Wertsch, Minick, 81 Arns, 1984) refer to previous attempts to point
out things ﬁre learners have already done wiﬁr assistance. The previous attempts serve

as models for the student to follow.

 

Math

Nick and his family always celebrate ﬁre holidays in ﬁre same way. They
gaﬁrer togeﬁrer at Nick’s house and exchange a few gifts. After that ﬁrey
sing holiday songs and eat cookies and fruit. Nick decided ﬁris year he was
going to give his brother and sister some presents. He would like you to
help him wiﬁr his Holiday plans.

1. Nick has two sisters and two broﬁrers. He wanted to give each of them
a pair of socks, so he bought four pairs-a red pair, a blue pair, a
green pair, and a purple pair. Nick couldn’t decide what color to give
each person. How many different ways can Nick arrange ﬁre socks for
his brothers and sisters?

2. Nick was also in charge of choosing ﬁre songs ﬁrey were going to sing.
He decided, based on what they had done in ﬁre past, ﬁrey would sing
for two hours. Each song, he thought, would take about ten minutes.
How many songs did Nick need to choose?

3. As was their tradition, Nick wanted to buy his brothers and sisters
some fruit. He ﬁrought he would get each of them an apple and an
orange. At ﬁre market apples cost 3 for 99¢ and oranges cost 2 for
60¢. How much would Nick need to pay for ﬁre fruit?

4. Nick also wanted to give each of his broﬁrers and sisters some holiday
cookies. He bought 20 cookies. How many cookies would each person

, get?

 

 

 

Figure 6.1: Task 4 Assignment Sheet.
Task 4 was designed to explore parents’ decreasing assistance and to see wheﬁrer
they referred back to previous tasks while working. All of the questions in Task 4 were
similar to those asked in ﬁre three earlier tasks. Question 1 asked ﬁre students to ﬁnd

out how many ways Nick could distribute gifts to his brothers and sisters. This question

205
was similar to questions in Task 3. Questions two and four included division problems

similar to those in Task 2. Question 3 took ﬁre students back to the grocery store as in
Task 1. The assignment sheet is shown in ﬁgure 6.1.

Four of the six students recorded conversations for Task 4 (66%). The students’
answers to Task 4 are summarized in table 6.1. Kathy and Pete are not represented in
the table as they did not record their conversations or turn in ﬁreir written work. Task
4 began the last week of school before winter break and continued after ﬁre break in
January. On ﬁre last day before break, Ms. Smiﬁr asked her students to clean out ﬁreir
desks. Many students ﬁrrew their maﬁr papers away before ﬁrey were collected.

Table 6.1

Solution Table for Task 4

 

 

# Karen Ronnie Shaundra Tony
1 1 6 2 1 6 1 6
2 1 2 1 2 1 2 1 2

3 Apples: $1.32 Apples: $1.32 Apples: $1.32 Apples: $1.32
Oranges: $1.20 Oranges: $1.20 Oranges: $1.20 Oranges: $1.20
Total: $2.62 Total: no answer Total: $2.52 Total: $2.52
4 5 5 5 5
The students answered questions two, three, and four wiﬁr no difficulty. Their
conversations as ﬁrey worked to answer the questions were very reminiscent of ﬁrose
ﬁrey had for Tasks One and Two. The students worked wiﬁr ﬁreir parents to determine
what operation was necessary and ﬁren to compute an answer. In no conversation did ﬁre
parents or students refer back to the questions in ﬁre earlier tasks. This short
conversation between Shaundra and her father illustrates ﬁre conversations ﬁre students

and ﬁreir parents had for ﬁrese questions.

10

11

12

13

B

“15118

206

Um-hmm, Okay. So, now for number two. It says, “Nick also was in
charge of choosing ﬁre songs ﬁrey were going to sing. He decided, based on
what they had done in the past, ﬁrst he would sing for two hours. Each
song, he ﬁrought, would take about ten minutes. How many songs did Nick
need to choose?“ '

So how many minutes do you have in an hour?
Sixty.

Okay, so he’s gonna sing for ten minutes. How many times does ten minutes
go in sixty? Ten divided into sixty?

Um, six times.
Okay, six times. So, ah, ﬁrat’s one hour.
Yedr.

So, you got six takes care of one hour. Six, so ﬁrat’s six songs ﬁrst you can
sing in one hour. But we got two hours. So what do you have to do?

Just, um, like if you take boﬁr of those sixes from, if you divide them, ﬁren
if you add them boﬁr up you get twelve.

No, you don’t divide ﬁrem, you just, you can eiﬁrer multiply ﬁrst six times
two hours, or you can add another six for the second hour, which is, like
you said, it equals, ah,

Ah, twelve.

So, it equals twelve, so what are you saying, you got twelve songs?

Yeah, so he’ll need, ﬁrere’s um, sixty minutes in one hour, so he’ll need

six songs for one hour and ﬁrere’s two hours so ﬁrst would equal twelve and
he would have to ﬁrink of, um, twelve songs.

In ﬁris conversation, Shaundra answered a series of questions that led her to ﬁre answer.

Her father, alﬁrough Shaundra did the computation, structured ﬁre conversation to guide

Shaundra to the right answer.

As was ﬁre case wiﬁr Task 1, none of ﬁre parents or students referred back to

previous questions ﬁrst included similar content. Perhaps the best explanation of this "is

ﬁrst ﬁre students were able to answer these questions wiﬁr little or no assistance.

Therefore, there was no reason for the parents to refer back to ﬁre oﬁrer tasks. In ﬁre

conversations ﬁre students seldom had difﬁculty choosing an operation or computing an

207
answer. When ﬁrey did have trouble, their parents quickly directed ﬁrem to an

appropriate operation or corrected their computation.

Question 1, however, was different. Although Karen, Shaundra, and Tony all said
sixteen combinations of the socks were possible--an incorrect answeruﬁrey arrived at
the answer in different ways. Karen multiplied four times four to get sixteen. Shaundra
grouped the socks in matching colors. For ﬁre “red group“ she included “Red, red; red,
green; red, blue; and red, purple.“ For each color group she constructed four matches.
Togeﬁrer, ﬁre four groups totaled 16 combinations. Tony wrote out this array:

B R G P

R B P G

G P B R

P G R B

Figure 6.2: Tony's Array
Alﬁrough the matrix represents four permutations of the socks, ﬁrere are sixteen items
in ﬁre matrix. Tony counted ﬁre items and not the permutations.

As wiﬁr the other three questions, neiﬁrer the students or ﬁreir parents referred
back to Task 3 while answering Question 1. This is particularly interesting as Shaundra
and Karen had been involved in conversations ﬁrst led to the correct number of
permutations while working on Task 3 and seemed to understand how to generate the lists
of permutations.

Karen and Kaﬁry had worked together on Task 3. Their group's work was presented
in the vignette above. In their group Karen initially held fast to the list of sixteen
numbers she and her faﬁrer had compiled. But, after a ﬁerce argument, she conceded,
wrote down ﬁre six missing numbers, and seemingly accepted Kathy’s method of
determining how many numbers were possible. But, when she answered this question
for Task 4 she did not hesitate. She read the questions, said “sixteen“ wiﬁrout taking a

breath, and went on to the next question. On her paper she wrote “4x4:16.“ She did

208
not write out a list of permutations or in any other way represent ﬁre arrangements, she

merely stated her answer and went on. The meﬁrod she used to determine how many
arrangements were possible is ﬁre same meﬁrod she and her faﬁrer used for Task 3.

Shaundra and her moﬁrer had worked hard to compile a list of 24 four digit numbers
in Task 3. As ﬁrey worked, Shaundra's mother pointed out ﬁre patterns in ﬁre numbers
ﬁrey found. At the end of their conversation, Shaundra used ﬁre patterns to make sure
she had found all ﬁre numbers. But when she and her father started working on Task 4,
Shaundra did not refer to ﬁre patterns she and her mother found ﬁre week before.

Rather, they started looking for ways of matching the socks in pairs. Shaundra’s list
included sixteen different ways of pairing the socks.

As did ﬁre two cases presented above, boﬁr of these conversations portray ﬁre gradual
learning process. both of ﬁre conversaﬁons support ﬁre notion that what appears to be
Ieamed in one setting may not extend beyond that conversation. Boﬁr Shuandra and Karen
had accepted and used an appropriate way of determining the possible permutations with
a group of four items. But, neiﬁrer of ﬁrem used it to answer ﬁris question.

Summary

People Ieam things gradually. Throughout ﬁre process they accrue new knowledge
and reﬁne things ﬁrey already know. As a result, ﬁre ﬁrings we know and do go ﬁrrough a
series of qualitative changes. Our earliest attempts to do ﬁrings are often aided by more
experienced people and ﬁre tools ﬁrst aid us in completing the tasks. During our
interactions with these people and tools, we observe ways of ﬁrinking about and doing
things that help us better understand our situation. Over time, we become more skilled
at the things we have observed and practiced in our interactions. As our abilities
increase, our dependence on ﬁre tools and people around us diminishes. But along wiﬁr
our growing independence comes ﬁre necessity of giving up previous ways of knowing.

The gradual change in a person's ﬁrinking, it has been posited, can be seen as he or

she works wiﬁr other people in problem-solving situations. When ﬁre person changes

2 0 9
what ﬁrey do to more closely resemble ﬁre things ﬁrey’ve heard, seen, and practiced in

the interaction, they have learned. In ﬁre ﬁrst case presented in this chapter, Ronnie
made changes of this sort as he talked with his moﬁrer about Task 1. As they talked,
Ronnie changed his strategy to one his mother suggested would better answer ﬁre
question. As they worked, Ronnie wrote down ﬁris new way of answering the question.
But somewhere between home and school Ronnie lost his paper and, without his written
record, was unable to completely reconstruct ﬁre strategy in school. Instead he reverted
to ﬁre strategy he used at ﬁre beginning of ﬁre conversation wiﬁr his mother.

Alﬁrough it looked as though Ronnie Ieamed how to answer the quesﬁon from his
mother, he had not completely appropriated the strategy. What looked like clear
evidence of Ieaming in their conversation, turned out to be something slightly less.
Ronnie had appropriated parts of the solutionuhe remembered that the answer was
“two something,“ for instance-but had not appropriated the entire solution.

Recent work in situated learning (Lave, 1993) has suggested ﬁrst learning includes
extending what is appropriated beyond ﬁre immediate situation. The tools used within
speciﬁc practices work as an aid in this extension. Had Ronnie not lost his assignment
sheet, he may have been able to reconstruct ﬁre solution. But, wiﬁrout the tools and his
moﬁrer's assistance, he could not.

The second case presented here also portrayed ﬁre gradual Ieaming process. This
portrayal, however, was slighﬁy different. Whereas the first case pointed out the
necessity of looking beyond the immediate situation for evidence of learning, this case
highlights two other aspects of the gradual development of maﬁrematical ideas. First,
Newman, Griffin, and Cole (1989) argued that people appropriate only what is
necessary to make sense of their immediate situation. As ﬁrey encounter new situations,
they change what they think to fit their new circumstances and explain ﬁreir past
experience as well. Kaﬁry and her father worked together to ﬁnd a strategy to predict

how many numbers could be made from a given set of digits. Their efforts provide a

210
clear example of ﬁre process of reﬁning msﬁremaﬁcal ideas and activities to

accommodate new situations.

Second, through participation in conversations, people not only appropriate what
they talk about, but how ﬁrey talk about it as well. By observing her faﬁrer conjecture
about how to predict how many numbers were possible and revise his ﬁrinking when the
list of numbers ﬁrey found did not match their prediction, Kathy may have appropriated
the skills that allowed her to reason through different situations.

Both of ﬁre cases also point out ﬁrst many ﬁrings inﬂuence students’ development of
mathematical ﬁrinking ability. As did the conversations reported in chapters four and
ﬁve, the conversations the students had at home greaﬁy influenced what ﬁrey did in the
classroom. Students placed a great amount of faith in ﬁreir parents’ mathematical
abilities. They believe, if for no other reason than their parents’ age, that they know the
maﬁrematics that is being taught in school. They have, after all, gone ﬁrrough it
ﬁremselves. As a consequence of ﬁreir faith, the interaction in ﬁre maﬁr groups often
began with ﬁre students sticking up for their parent’s answer. If their answer was not
accepted by the group, ﬁre students would threaten to leave raﬁrer than changing ﬁreir
answer.

Parents do not, however, know maﬁrematics in the same way. Kaﬁry’s faﬁrer, for
example, may have had a richeruor perhaps just more contemporary-understanding
of ﬁre mathematical content of the tasks ﬁran did ﬁre oﬁrer parents. Through her
interactions with him, Kathy developed more sophisticated ways of participating in
class. Kathy’s faﬁrer modeled mathematical inquiry in ﬁreir conversations in a way ﬁrst
was not present in ﬁre other conversations. As a result, Kaﬁry brought more ﬁran
answers back to school.

In school, as in their homes, ﬁre students’ math groups were involved in competing
streams of activity. The students were often involved in other classroom activities that

were important to their social standing in class. Whether ﬁreir ideas were accepted by

21 1
ﬁreir group was sometimes as much a social issue as it was a maﬁrematical or academic

one. In ﬁre and, they would turn to a classroom authority--in this case either Ms. Smiﬁr
or me-to resolve conﬂicts. Even than, students would only reluctanﬁy change their

BI'ISWSTS.

CHAPTER7
SUMMARY AND IMPLICATIONS

Students’ conversations at home and in scth have a profound influence on their
mathematics learning. In conversations, students, along wiﬁr their teachers, parents,
and classmates, bring togeﬁrer different conceptions of maﬁrematics and other
experiences ﬁrst shape what and how ﬁrey learn. Students, however, do not easily learn
the things they discuss. Raﬁrer, learning is a complex process filled wiﬁr choices, risks,
and other social pressures that students and their conversational partners must
negotiate as they talk. In the and, students cannot always do or say what appeared so
easily Ieamed in ﬁreir conversations. And yet, ﬁre influence of ﬁreir conversations at
home and in school remains great. In this final chapter, I review the findings of ﬁris
study ﬁrst led to this brief summary and contemplate their contribution to broader
ﬁreories of learning and mathematics education.

Students’ conversations in different places illuminated different aspects of learning
maﬁrematics. In ﬁre following sections l revisit students’ participation in conversations
at home and in school and as ﬁrey move from one conversation to ﬁre next. At ﬁre and of
each section, I discuss ﬁre contributions of ﬁre findings to previous research and theory
in educational psychology and maﬁrematics education.

Following the summary of the ﬁndings, I look back on ﬁre data collection meﬁrods
used in ﬁris study. Alﬁrough ﬁrese methods provided a glimpse of ﬁre inﬂuence students’
conversations at home and in school had on their completion of elementary scth
mathematics tasks, there were many uncontrollable gaps in the “data trail.” In this
section I discuss ﬁre gaps ﬁrst existed and how ﬁrey may have inﬂuenced ﬁre data
constructed for this study and changes in ﬁre methods that might result in a more
complete data trail.

Finally, I explore the Implications of this study in light of recent calls for reform

, in education and mathematics education. Calls for reform in education have identified

212

 

II i|l1 rill.» lllllllrriI.

 

21 3
eight goals (National Education Goals Panel, 1994) that are meant to address and reverse

negative trends in America’s schools. One of ﬁre goals is to increase parents’
involvement in their children's schooling. The ﬁndings of this study can inform
attempts to develop productive home-school partnerships that can help fulfill ﬁris goal
and ease ﬁre implementation of ﬁre recent reforms in maﬁremaﬁcs education. The
current emphasis on home-school partnerships also provides an opportunity to continue
ﬁre study of ﬁre inﬂuence of students’ homes and schools on what and how ﬁrey learn. In
this ﬁnal section I look more closely at the goal of increased parental participaﬁon and
how extensions of ﬁris study might contribute to its fulﬁllment.

Students’ Conversations
Conversations at Home

The conversations students and ﬁreir parents had at home constituted one of many
streams of ongoing activity ﬁrst characterize most households. In all but Shaundra’s
house, the students and ﬁreir parents listened to music or watched television as they
worked on the homework tasks. Parents sometimes cooked dinner, took care of family
pets, or talked wiﬁr other family members as ﬁrey worked wiﬁr their children. Parents
sometimes talked to students about their homework from across ﬁre room or from
adjacent rooms. It was clear ﬁrst homework at certain times was subordinated to oﬁrer
household practices ﬁrst required immediate attention. Parents whose attention was
divided among household tasks were less able to monitor or contribute to ﬁreir children’s
work.

At times, parents’ divided attention caused them to overlook errors in the students’
work. Although ﬁre students in these conversations often used the same sﬁategies as did
oﬁrer students, misunderstandings or confusion over important aspects of the tasks or
values used in computation produced incorrect answers. Given the assumptions and

values ﬁrey chose for their calculaﬁons, ﬁreir work was-done correcﬁy. When they

21 4
returned to school, however, their answers were considered wrong by boﬁr their

teachers and classmates.

The students and their parents interacted in ways that were consistent with normal
conversation in ﬁreir household. Different rules applied in different homes. In one
household, for instance, the student was not allowed to question his mother’s
contribution to the tasks; in anoﬁrer, the student was expected to question her father’s
contribution. The rules wiﬁrin which ﬁre students and ﬁreir parents worked inﬂuenced
the students' participation in school.

In general, ﬁre students began all of ﬁre conversations at home by trying to answer
the questions wiﬁr no assistance from ﬁreir parents. Their parents, however, sat nearby
and intervened at the slightest hint that the students were having difﬁculty answering
the questions. Although ﬁris scenario brieﬂy describes the conversaﬁons in students’
homes, what ﬁre students and ﬁreir parents contributed to ﬁre conversations depended
greaﬁy on how consistent the task was with ﬁreir experience in elementary school
maﬁrematics classes or other out-of-school practices.

Consistent Tasks. Most of ﬁre tasks ﬁre students and ﬁreir parents worked on during
this study were consistent wiﬁr some aspect of ﬁreir past experience. While completing
these tasks, ﬁre students and ﬁreir parents drew on the different practices in which they
participated. When ﬁrese practices were brought togeﬁrer in conversation, ﬁrey often
created conflicts ﬁrst demanded resolution before the students and ﬁreir parents could
answer ﬁre questions.

Tasks ﬁrst were consistent wiﬁr students’ and parents' past school mathematics
experience (Task 2, for Instance), presented litﬁe or noﬁring to resolve. As students
and their parents completed ﬁrese tasks, ﬁrey quickly agreed on what operations were
required and how ﬁrey could be computed. As a result, the parents monitored students’
work, only correcting minor computational errors. Tasks that brought out-of-school

' practices into school draw more practices into the conversation, consequently giving

21 5
rise to more conflicts. These conﬂicts were resolved in one of two ways. Conﬂicts could

be resolved by bringing togeﬁrer aspects of different practices to form a new,
amalgamated practice. Students and parents ﬁren worked within ﬁre rules of ﬁris new
practice, with the answers they constructed meeting ﬁre requirements of each of the
contributing practices. Conflicts could also be resolved by invoking ﬁre rules of a more
powerful or influential practice. In many of the home conversations, for example, the
practice of inquiry math in Ms. Smith’s classroom was weighted most heavily and could
override any other practice brought to bear in ﬁre conversations.

Inconsistent Tasks. When the tasks presented activities that were inconsistent with
ﬁre students’ or parents’ experience, the parents could not provide the sort of guidance
ﬁrey did on ﬁre oﬁrer tasks. Alﬁrough ﬁre conversations around ﬁrese tasks started out as
did those discussed above, when students experienced difficulty, parents initially either
deferred to ﬁre students and allowed ﬁrem to work on ﬁre tasks alone or ended ﬁre
conversation. When they deferred to the students one of two ﬁrings occurred. When ﬁre
students, too, were uncertain about what to do, the parents deferred to Ms. Smiﬁr and ﬁre
students returned to school wiﬁrout completing ﬁre assignments. When the students
believed they knew what to do, ﬁrey worked alone as ﬁreir parents looked on.

The parents were not passive participants in ﬁrese conversations. Raﬁrer, ﬁrey
watched intenﬁy as their children worked to answer the questions. As they watched,
ﬁrey asked questions to check ﬁreir growing understanding of what ﬁre student was doing-
-even when ﬁre student was doing something ﬁrst led to an incorrect answer. When ﬁre
parents believed they understood the task, ﬁrey contributed more to the conversations-
even to wrong answers. When ﬁre parents felt competent, they offered ﬁre students
advice about how to answer ﬁre questions. They contributed strategies ﬁrst would
increase efficiency, algorithms from their scth mathematics experience, and other

ﬁrings that helped students answer and document their answers to the questions.

21 6
In conclusion, the conversations students had at home were enmeshed wiﬁr other

household activities ﬁrst limited participation in the conversations. Even wiﬁr these
limitations, students and parents contributed whatever they believed would aid in
answering ﬁre questions. What ﬁrey were able to contribute depended on ﬁre connecﬁons
ﬁrey made between ﬁre immediate task and various aspects of ﬁreir experience. When
parents were certain about ﬁre mathematics, they directed ﬁre students to correct
answers. When parents were uncertain about ﬁre mathematics asked for in ﬁre tasks,
ﬁrey deferred and attended to ﬁre students’ knowledge of what was discussed in class. As
ﬁrey watched, ﬁrey Ieamed about the mathematics and eventually contributed to the
conversations. These conversations boﬁr supported and refuted contentions of
sociocultural ﬁreorists, maﬁrematics educators, and oﬁrer educational researchers. In
ﬁre next section I review claims of these researchers and how this study contributes to
their ﬁndings.

Contributions to Theory. McDermott, Varenne, and ﬁreir colleagues (McDermott et
al., 1984; Varenne et al. 1982) have suggested that parents may have different goals for
homework sessions ﬁran do teachers. Whereas teachers view homework as practice of
important skills and ideas raised in ﬁreir classrooms, parents strive to send students
back to school with correct answers. The conversations presented here support ﬁris
contention. When parents were certain about what ﬁre students were being asked to do,
ﬁrey directed students to what they knew were correct answers. When parents were
uncertain about how to complete the tasks, some, raﬁrer ﬁran risking an incorrect
answer when ﬁre students retumed to school, suggested students not complete ﬁrst part of
ﬁre task and ask ﬁreir teacher about it when ﬁrey returned to scth ﬁre following day. It
was better, it seems, to go back with no answers ﬁran wiﬁr wrong answers. In other
' households, parents worked hard to understand what it was ﬁre students were being asked
to do. When ﬁrey believed ﬁrey knew what a correct answer was, ﬁrey again became quite

- directive in the conversations. In extreme cases, parents avoided telling their children

21 7
about new mathematical ideas to insure correct answers. Kathy’s father told her, “I

mean I could show you how to do decimal division but, if you haven't covered it yet, it’s
probably just better to learn what you know.“

Wiﬁrin these conversational structures, students and their parents deﬁned ﬁre tasks
on which ﬁrey worked and what constituted an appropriate answer. This process, too,
has been the focus of much work in ﬁre sociocultural tradition and ﬁre conversations
recorded here boﬁr support and extend this previous research.

Wertsch (1984), in his seminal discussion of negotiation within the zone of
proximal development, suggested ﬁrst, in order for someone to learn someﬁring in a
conversation, participants must enter the conversation with different situation
definitions. A person’s situation deﬁnition is manifested in the ﬁrings ﬁrey do as ﬁrey
participate in the conversation (i.e., in their action patterns). Learning in ﬁre
conversations can be evidenced in ﬁre changes in one person’s action pattern to more
closely resemble the actions of oﬁrer participants. The findings of this study support
this contention. While ﬁre students and their parents worked on tasks that were either
inconsistent wiﬁr ﬁreir school maﬁrematics experience or drew on more than their
school maﬁrematics experience (Tasks 1 and 3), ﬁrey came to ﬁre conversations with
different situation deﬁnitions ﬁrst needed to be negotiated as they worked. While the
students and their parents worked on tasks that were consistent with their school
maﬁrematics experiences they brought similar if not identical situation definitions and
there was little or nothing to negotiate (i.e., Task 2). In ﬁrese conversations it is
doubtful that ﬁre studentsuor the parents--leamed anyﬁring they did not know before
entering ﬁre conversation.

In Wertsch’s (1984; 1985) discussion of Ieaming in the zone of proximal
development, he also suggested that adults enter conversations wiﬁr a more appropriate
situation definition ﬁran do children. Through the course of their conversations,

- children take on ﬁre adults’ action patterns and begin to deﬁne the situation as the adult

21 8
does. Adults, however, make only temporary changes in their deﬁnition to accommodate

the child’s immature approach to the task. Elbers and his colleagues (1991; Elbers,
Maier, Hoekstra, 81 Hoogsteder, 1992) have criticized this transition from “other-
regulation to self-regulation“ as ignoring the contributions of ﬁre child to ﬁre
conversations. Instead, they argue, the transformation moves from “joint-regulation to
self-regulation,“ with both participants actively contributing to the formation of the
task. The conversations recorded for ﬁris study support Elbers' claim.

When ﬁre students and their parents brought different situation definitions to the
conversations, ﬁrey negotiated a shared definition of the task ﬁrst drew on ﬁre
experiences each of ﬁrem brought to the conversation. At different times ﬁre students and
ﬁreir parents assumed ﬁre role of ﬁre more knowledgeable other. Students brought
knowledge of how mathematics was done in Ms. Smith's classroom and ﬁreir parents
brought ﬁreir own experiences boﬁr in and out of school ﬁrst contributed to the shared
deﬁnition. In the end, ﬁre student completed ﬁre task in a way ﬁrst reﬂected the different
experiences both participants brought to the conversation.

At ﬁre same time ﬁrst these conversations supported many claims of sociocultural
theorists, they also call into question other findings and provide empirical support for
more contemporary theories of Ieaming. Sociocultural ﬁreorists have been concerned
with the relationship between the individual and the social structures within which they
live, work, and play. At times, ﬁreorists have argued ﬁrst ﬁre practices in which people
participate determine what and how they learn (Scribner, 1984). More recenﬁy,
however, ﬁreorists have argued ﬁrst people move among various practices (Cole, 1990;
Rogoff, 1992; Valsiner, 1993; Wertsch 81 Hickman, 1987) and that each of the
practices in which they participate influences their participation in other practices.

The conversations students had at home suggest ﬁrst no one practice determined what
or how students learned. Raﬁrer, students and ﬁreir parents drew on their experiences

. in many different practices to construct a new set of rules-—a new practice--within

21 9
which they worked. Their experience in any practice was available for ﬁrem to use

while working on ﬁrese tasks and the new practice they constructed included aspects of
oﬁrer practices they believed could contribute to their answers to ﬁre questions
presented in the tasks.

The different practices did not contribute equally. In most households, what the
students and ﬁreir parents believed Ms. Smith expected in her classroom was given more
weight than other practices. The rules of “doing math“ in Ms. Smith’s classroom, they
believed, determined the data ﬁrey had to work with and appropriate presentation of
ﬁreir answers. Although their varied experience may have led ﬁrem to question ﬁre tasks
sent home or what students were doing in school, in ﬁre and, parents insisted ﬁrst ﬁre
students comply with ﬁre expectations in Ms. Smith’s classroom.

This ﬁnding becomes particularly important in light of ﬁre recent calls for reform
in mathematics education. Cobb and his colleagues (Cobb, Pateman, 8 Bednarz, 1993;
Cobb, Wood, & Yackel, 1993) have resisted the contention of sociocultural ﬁreorists that
practices exist independent of the participants in the practice or activity. This study
supports ﬁreir stand. The practices the students and ﬁreir parents brought to bear
reﬂected their experience in school maﬁrematics classes and oﬁrer out-of-school
practices. How ﬁrey did maﬁr homework in students’ homes depended on ﬁreir past
experiences and how they fit homework into other ongoing streams of activity.

At the same time that conversations recorded for ﬁris study support Cobb and his
colleagues, they call into question ﬁre assumption ﬁrst schools provide students’ sole
exposure to mathematics. Maﬁrematics has often served as a site for psychological
research in education. Kilpatrick (1992) has suggested psychology’s interest in
maﬁremstics may have stemmed from

perceptions regarding its important role in the school curriculum; its relative

independence of nonschool influences; its cumulative, hierarchical structure as a

school subject; its abstraction and arbitrariness; and the range of complexity and
difficulty in the learning tasks it can provide.

220
Included in Kilpatrick’s list of reasons is the notion that classrooms provide students’

sole or primary exposure to mathematics. The conversations recorded here, however,
suggest ﬁrst students’ families may have as much or more inﬂuence on the maﬁrematics
students Ieam than do teachers and classmates. When ﬁre parents felt certain about ﬁre
content, they were directive in ﬁreir instruction; students listened to what they said and
mimicked ﬁreir actions. As I will discuss in more detail later, upon returning to school,
ﬁre students often aggressively defended ﬁre ﬁrings ﬁreir parents taught them at home;
they were reluctant to change ﬁreir answersueven when their assumptions, strategies,
or answers were at odds wiﬁr oﬁrer ﬁrings discussed in class. When the content deviated
from parents’ elementary school mathematics experience, parents exhibited some of ﬁre
traits associated wiﬁr ﬁre calls for reform in mathematics education. They explored ﬁre
situation with their children, tried to figure out what was going on, and, when they felt
comfortable, began participating in the conversations in ways that resembled ﬁreir
earlier contributions. What ﬁre students learned in the conversations was not always
consistent wiﬁr what Ms. Smith intended. Their answers, however, sﬁll adequately
answered ﬁre questions, in ﬁreir ﬁrinking, and inﬂuenced what students brought back to
scth more so than did ﬁreir participation in school.

In sum, the conversations in students’ homes presented a portrait of homework as a
complex practice that is constructed from parents’ and students’ experiences boﬁr in and
out of school. What was Ieamed in ﬁre homework sessions was determined as much by
what happened in other classrooms and out-of-school practices as it was by the
immediate practice of elementary school maﬁremstics in the students' present
classroom. When the students completed their homework, ﬁrey brought ﬁre products of
ﬁreir homework sessions back to school where ﬁrey compared ﬁreir answers to those. of
other students. What ﬁrey Ieamed at home and how ﬁrey arrived at ﬁreir answers served

as preparation for their discussions in school.

221

Conversations in School

In many ways, the conversations students had in their math groups at school were
reminiscent of ﬁrose they had at home. Students' conversations about mathematics in
school represented just one strand of ongoing activity. As they worked in ﬁreir maﬁr
groups, ﬁre students engaged in conversations about calculators, about ﬁre equipment
used to video and audio tape ﬁreir conversations, about aﬁrletics, and about other out-of-
school activities. Ms. Smith sometimes interrupted their work to ask about assignments
ﬁrey had not completed or other pieces of classroom business. Daily announcements came
over the intercom telling students about scth activities or calling students to ﬁre ofﬁce.
Each of dress things represented different strands of activity to which students needed to
attend as ﬁrey worked.

The students each brought different experiences to ﬁre conversations that shaped
what ﬁrey believed needed to be done. Their experiences at home heavily inﬂuenced what
they brought to their maﬁr groups in school. But; what they brought to ﬁre
conversaﬁons in school was different from what they brought to the conversations at
home. Because the tasks were completed as homework before ﬁrey were discussed in
school, the students came to school armed with answers, not just ideas about what to do.

The students’ answers became the initial focus of ﬁreir conversations in school, and
having--not gettingnthe right answers was the goal of ﬁre conversations. The
conversations began wiﬁr ﬁre students comparing ﬁreir answers. If they all had the same
answer, they went on to the next question without discussing how they arrived at their
answers or the content of ﬁre question. If they had different answers, their task changed.
Now the goal became selecting ﬁre correct answer from among ﬁrose proposed in ﬁre
conversation. The students’ objective was to defend ﬁreir answers and convince their
group members that they were correct.

The students’ ﬁrst line of defense was to check ﬁreir computation. Students either

wrote out the computation on a new sheet of paper or punched ﬁre numbers into a

2 2 2
calculator to show how ﬁrey had computed their answers. When they completed the

computation, and the result agreed wiﬁr ﬁre answer they had written down, they
presented it to their groupmates as evidence that their answers were correct. When
discrepancies were found in ﬁre computation, the students willingly changed their
answers.

If everybody’s computation was done correcﬁy and they still had different answers,
ﬁre students moved to a second line of defense. Their objective now was to determine who
had ﬁre correct answer. They set out to support their answers and, at ﬁre same time,
discount answers oﬁrer ﬁran those ﬁrey had written down. Just as ﬁre parents wanted to
send their children back to school with right answers, ﬁre students argued vehemenﬁy
for the answers ﬁrey constructed with their parents. Even when confronted wiﬁr
convincing arguments for other answers the students would refuse-net least initially--
to change ﬁre answers ﬁrey had constructed at home. In a sense, ﬁrey chose to “not
learn“ (Kohl, 1991) what other students were telling them in order to preserve their
family’s position.

The students used many things to support their answers. Their parents, by virtue
of ﬁreir age and having done ﬁrese sorts of tasks before, were set up as authorities who
would not send ﬁrern back to school wiﬁr wrong answers. Students told ﬁreir groups ﬁrst
their answers must be correct because their parents had worked wiﬁr ﬁrem the night
before.

The students also used their and oﬁrer students' standing in Ms. Smith’s class to
support the answers they chose. The students accepted answers from students wiﬁr
higher standing in the classroom more readily ﬁran those of lower standing student --
even when ﬁre lower-standing students’ answers were correct. The students who
enjoyed higher standing in the classroom participated more and were more instrumental
in determining ﬁre answers on which ﬁre class agreed. Many factors contributed to a

student’s classroom standing. Students could gain standing for their contributions to

223
class discussions in math class or ﬁrey could gain standing from ﬁreir accomplishments

in oﬁrer academic and non-academic activities. The students also used ﬁreir

participation in ﬁris study as evidence of ﬁreir standing in class. Students who recorded
their conversations sometimes reminded their math-group partners of their
participation in this study as support for their answers. Invoking their participation as
evidence did not always work, however, as other students in ﬁre math groups often also
recorded their conversations.

If the students could not convince ﬁreir groupmates of their answers or agree that
one group member's answer was correct, they sought assistance from a classroom
authorityuMs. Smith or me. They asked which answer was correct and Ms. Smiﬁr or I
talked with ﬁre students to help ﬁrem sort out ﬁreir difﬁculﬁes. The students would
accept ﬁre outcome of ﬁrose conversations as correct answers and ﬁrey changed ﬁreir
answers to reﬂect the conversaﬁons.

As did ﬁre conversations students had at home, these conversations both support and
refute contentions of sociocultural theorists. In the next section, I discuss how these
ﬁndings support, extend, or refute the contentions of various theorists.

Conﬁibutions to Theory. Students’ conversations in school support many of the
contentions of sociocultural theorists and maﬁrematics educators. The conversations in
school, as did those at home, support the contention ﬁrst practices are made up of ﬁre
peeple who participate in ﬁrem (Cobb et al., 1993). Although the rules of maﬁrematics
in Ms. Smith’s classroom included exploring alternative answers and changing what
students believed based on ﬁrings said by oﬁrer students, ﬁrese rules were not always
followed in the small group conversations. Groups saw their task as determining one
correct answer and each student believed his or her answer was correct. What counted
as warrants for answers in the maﬁr group discussions was determined by the students
and, often in the heat of argument, students invoked many ﬁrings oﬁrer ﬁran maﬁrematical

reasoning to support their answers.

224
Vygotsky (1978) argued that more knowledgeable peers assist their coworkers or

classmates in Ieaming new ﬁrings. Wiﬁr ﬁris assistance, less knowledgeable people hone
their skills and abilities as if they were working with adults. Recent classroom
research (Bivens, 1990; Eichinger, 1992; Tudge, 1990), however, suggests that
students do not always assist ﬁreir classmates in such a democratic manner. Raﬁrer,
students resist working with some people, accept the ideas of popular students over ﬁrose
of more marginalized students, and worry more about ﬁreir standing wiﬁr other students
ﬁran with ﬁre teacher. Indeed, good standing wiﬁr the teacher is often avoided. The

, conversations recorded for this study support the contentions of these researchers.
Students may have tried to help oﬁrer students, but ﬁrey would often refuse help.
Answers they accepted had as much to do wiﬁr students' standing in ﬁre class as it did

with ﬁreir mathematical worﬁr. Standing, however, was sometimes granted for
contributions to ﬁre class discussions.

Although ﬁre findings from this study support those of other researchers looking at
small group interaction, Ms. Smith’s class always arrived at mathematically sound
answers to ﬁre question in ﬁreir large group discussions. During these discussions, each
maﬁr group presented ﬁreir answers and ﬁre class was able to ask questions to clarify
what was presented. Often students’ questions forced oﬁrer students to reﬁrink the idea
ﬁrey were presenting. In the end, wiﬁr Ms. Smiﬁr’s help, ﬁre class discussed all of ﬁre
answers presented and determined which of ﬁrem were appropriate. The whole class
negotiation of appropriate answers might provide a fruitful site for future research.

The conversations students had at home and in school show students changing ﬁreir
thoughts and actions in ways that reflect their conversational partners’ situation
deﬁnitions. Alﬁrough ﬁre change was not always easy, in ﬁre and students appeared to
mimic ﬁre action patterns of the people wiﬁr whom they talked. But even though students
changed ﬁre ﬁrings they did and said as they participated in conversations at home and in

school, it became evident as ﬁrey moved among conversations ﬁrst changing what ﬁrey do

225
in one conversation may not be deﬁnitive evidence of ﬁreir Ieaming. In the next section,

I look more closely as students move across conversations.
Looking across Conversations.

What students apparently learn in their conversations at home and in school is not
always available to ﬁrem in subsequent conversations. Students often learned only part
of what was discussed and did not always recognize situations in which ﬁrey can use what
ﬁrey learned in previous conversations.

What ﬁre students had available to them when ﬁrey returned to school depended, in
part, on ﬁreir documentation of their conversation at home. When the students lost
ﬁreir documentation or did not document their conversation, they were less able to
reconstruct ﬁre answers ﬁrey had agreed upon at home. Even when ﬁre students could not
reconstruct ﬁreir entire answer, however, it was clear ﬁrst their conversations at home
inﬂuenced their ﬁrinking. Students brought incomplete understandings of the questions
back to school that included ideas about which answers might be correct. Indeed, when
they heard ﬁreir groupmates’ answers, they often knew which ones were incorrect and
why.

Even with fully documented and reconstructed answers, ﬁre students did not
recognize situations in which they could use what they had discussed in previous
conversations. At home, the students and ﬁreir parents approached the tasks as if ﬁrey
were separate, unconnected activities. While working on Tasks 1, 2, and 4, the students
and ﬁreir parents did not refer back to things they had done in previous tasks.1 This
phenomenon was particularly evident as they worked on Task 4, which was speciﬁcally
designed to explore ﬁre connections students and ﬁreir parents would make to previous

conversations. When ﬁrey did refer to their experience, their references were to rules

 

1There is one notable exception to this pattern. Pete’s grandmother, while working an
Task 2, reminded him of computational rules she had told him ﬁre week before. In their
conversation, she repeated how ﬁrey computed the answer to a question from Task 1 and
told Pete to do ﬁre same ﬁring to answer ﬁre questions on Task 2. She did not, however
refer back to the earlier tasks while completing Task 4. '

2 2 6
of parﬁcipationuwhat needed to be included in answers and how they should be

presented, when to use labels, rules for the same operation in different practices--but
seldom about a conceptual understanding of mathematical ideas.2

Only while ﬁre students and ﬁreir parents worked on Task 3, which explicitly asked
ﬁrem to look back at their previous work, was ﬁre gradual development of ideas visible.
As they worked on ﬁre ﬁrst part of Task 3, students and their parents compiled a list of
three-digit numbers and then explained how ﬁrey compiled the list. As ﬁrey worked,
ﬁrey did not emphasize ﬁre maﬁrematical ideas embedded in the task. Rather, ﬁrey focused
on answering ﬁre questions. As a result, maﬁrematically suspect answers were accepted
when ﬁrey believed ﬁrey had sufﬁciently answered the questions in the first part of Task
3. The second part of Task 3, an extension of ﬁre ﬁrst part, asked ﬁrem to compile a list
of four-digit numbers, review what they had done in the first part, and devise a method
of predicting how many numbers could be made using a given set of digits. As ﬁrey
compiled their lists of four-digit numbers, ﬁre students and parents referred back to
what ﬁrey had done on ﬁre ﬁrst part. They used strategies ﬁrey had devised and revised
their ﬁrinking about how to predict how many numbers were possible. When they
ﬁnished, the families were able to conjecture about a way to predict how many numbers
were possible with a given set of digits.

In contrast to their conversations for the other three tasks, the students and ﬁreir
parents made connections between maﬁrematical ideas ﬁrey had constructed earlier, ﬁre
ﬁrings ﬁrey had done in previous conversations, and ﬁre immediate quesﬁons. As they
worked, ﬁre students and their parents reﬁned their strategies, honed their ﬁrinking
and, in essence, did mathematics. Indeed, many of ﬁreir actions were consistent wiﬁr ﬁre

calls for reform in maﬁrematics education. They did these things, however, only when

 

2Ronnie's declaration ﬁrst he was going to average the cost of the groceries in Task 1 may
be an exception to ﬁris finding. In ﬁrst instance, Ronnie recognized a similarity between
what he was doing in ﬁre immediate task and a maﬁrematicsl idea he had previously
discussed in school. Alﬁrough he declared the usefulness of ﬁrst procedure, he did not
discuss why the notion of averaging was applicable in ﬁrst task.

227
the tasks speciﬁcally required them. Perhaps more ﬁran anyﬁring else, these

conversations suggest ﬁrst making connections between tasks and maﬁrematical ideas is a
skill ﬁrst must be encouraged In maﬁrematics classrooms. Along wiﬁr ﬁre practical
implications of these conversations, they also inform recent theories of Ieaming.
Contributions to Theory. Watching students as they participated in this series of
conversations provides empirical support for--and clarification or extension of-the
claims of sociocultural theorists. One of the major tenets of Vygotsky’s (1978) theory
of development was ﬁrst people Ieam ﬁrings gradually. Over time, and with the
assistance of more experienced oﬁrers, people develop skills and abilities that allow
them to accomplish other, larger socially-defined tasks. Vygotsky (1978), and other
soviet psychologists (Luria, 1978) sought to document ﬁris process by watching the
“microgenetic“ development of cognitive abilities. In Western psychology, ﬁre
microgenetic development of cogniﬁve abilities has been taken to mean ﬁrst Ieaming can
be seen in one observation. Wertsch and Stone (1978) described microgenesis as “the
development of a skill, concept or strategy within the context of a single observational
session“ (p. 8) and Newman, Grifﬁn, and Cole suggested ﬁrst “Meﬁrodologically,
. cognitive development can be observed as children pass ﬁrrough or work wiﬁrin the zone

[of proximal developmentl.“ (p. 61).3

 

3Catan (1989) has argued ﬁrst, although this deﬁnition of microgenesis has been widely
accepted in the west, it represents a “misuse“ of Vygotsky‘s original intent. Vygotsky
used microgenesis to describe a meﬁrodology with which he could explore the cultural
history and ontogenetic development of psychological functions in a miniature,
accelerated form. To document the development of writing in children, for instance,
Luria (1978) used children between 5 and 9 years old because he believed they would
have the same psychologically salient characteristics of preliterate adults. In a series of
short experimental sessions ﬁrrough which, he hypoﬁresized, he could accelerate the
course of development, he presented ﬁre children wiﬁr sets of phrases ﬁrst would be
difficult to remember with no means of recording. Asking ﬁre children to remember
these phrases and sentences, he hypoﬁresized, would require the children to develop a
system of recording ﬁrst would resemble ﬁrst development of similar systems at more
macro levels. Neiﬁrer the research conducted by Wertsch and Stone, Newman, Griffin,
and Cole, nor ﬁre research conducted as part of this project would fall into the category
of microgenetic research thus deﬁned.

2 2 8
This study, however, suggests that what students have apparenﬁy learned in one

conversation may not be available to ﬁrem in subsequent conversations. This ﬁnding
draws into question ﬁre notion ﬁrst learning can be seen in one observational session.
Rather, researchers need to look for evidence of Ieaming in subsequent interactions and
to understand how people use what was discussed in ﬁre earlier conversations to inform
ﬁreir actions. Traditionally, ﬁris issue has been viewed as one of transfer.

Lave (1988) and others (Beach, 1990; Pea, 1988) have suggested ﬁrst to examine
ﬁris issue we need to reconceptualize the transfer problem. The transfer problem arose
from Thorndike’s rejection of the notion ﬁrst learning academic subjects such as Latin
would beneﬁt people in other academic endeavors (Thorndike 81 Woodworth, 1901). In
its place, Thorndike suggested ﬁrst skills would transfer from one setting to another if
boﬁr settings contained common elements. This conception of transfer held that ﬁre
actors wiﬁrin ﬁre settings merely responded to environmental stimuli and suggested ﬁrst,
when taught how to respond to a given stimulus, a person’s response would be consistent
across situations. The common elements ﬁreory of transfer, though still apparent in
some places, has come under fire from many directions.

Perhaps the greatest fervor is around ﬁre notion ﬁrst individual agency plays a role
in the transfer of knowledge from one situation to another. Transfer, it is currenﬁy
argued, does not rest on the knowledge a person has, but on ﬁreir ability to access ﬁre
knowledge at appropriate times. What is transferred is determined by ﬁre actor’s
perceived similarities (Brown, Bransford, Ferrara, 81 Campione, 1983; Pea, 1988)
among situations, raﬁrer than common elements residing in the situations ﬁremselves,
and ﬁreir determination of what is and is not useful as ﬁrey work in unique situations. In
this conception of transfer, the prior knowledge and experience of ﬁre learner and the
continuities (what is the same across situations), discontinuities (what is different),
and transformations (how people change ﬁreir ﬁrinking and actions) across situaﬁons

(Beach, 1990) in which they are working need to be considered.

2 2 9
The conversations students and ﬁreir parents had while working on Task 3 support

ﬁris conception of transfer. The two parts of ﬁre task can be described in terms of their
continuities (boﬁr parts asked ﬁre students to compile lists of numbers given a set of
digits, boﬁr asked students to write ﬁre numbers in order and in expanded notation),
their discontinuities (the second part included four rather than ﬁrree digits), and in how
the students and ﬁreir parents transformed ﬁreir thinking across the conversations. As
they worked on ﬁre ﬁrst part of Task 3, ﬁre students and their parents constructed
incomplete, erroneous, or underdeveloped understandings of permutations ﬁrst were not
always helpful as they worked on ﬁre second part. Their recognition of ﬁre continuities
between ﬁre tasks, however, led ﬁrern to use what they learned while answering ﬁre first
set of questions. When they determined ﬁreir strategies did not work, they transformed
their strategies into someﬁring useful in both situations. Indeed, ﬁre discontinuity
between the two parts of Task 3 may have led students and their parents to a more
complete understanding of permutations.

Students’ and parents’ construction of incomplete or underdeveloped understandings
of permutations as they worked on the ﬁrst part of Task 3 supports oﬁrer claims made
by scoiocultural ﬁreorists. Newman, Griffin, and Cole (1989) have argued that, raﬁrer
than internalizing skills and abilities intact, children appropriate pieces of what is
discussed ﬁrst ﬁrey ﬁnd useful in completing ﬁre task at hand wiﬁrout consideration of
future tasks. When they encounter unique situations, however, ﬁrey may find what they
have appropriated wanting. Faced with ﬁris dilemma, ﬁrey must change ﬁre way ﬁrey
deﬁne ﬁre situation or change ﬁreir understanding of ﬁre ideas ﬁrey are bring to bear in
ﬁreir work. The students and their parents did ﬁris as they worked on ﬁre two parts of
Task 3.

In addition, what students appropriate and have available in subsequent situations is
inﬂuenced by ﬁreir documentation of the conversations in which ﬁrey participate.

(Remember Ronnie’s inability to reconstruct ﬁre work he did wiﬁr his mother when he

 

 

230
left his homework paper at home.) This supports Vygotsky’s (1978) claim ﬁrst higher

psychological processes are mediated by cultural tools and artifacts developed to
facilitate the process.

Although sociocultural theorists agree that researchers must look beyond ﬁre
immediate situation to document ﬁre gradual development of ideas and abilities and ﬁrst
ﬁre inﬂuence of tools designed to aid in ﬁre development of higher psychological processes
need to be documented, there has been little empirical support of ﬁris contention. This
study provides initial empirical evidence of ﬁrese claims and recent conceptions of
transfer. It shows people referring to their documentation of previous conversations

(or ﬁre results of losing ﬁrst documentation) to retrieve ideas ﬁrey began to understand.

 

It also shows them changing ﬁreir thinking when they discover the ﬁrings ﬁrey Ieamed h
while working on the earlier tasks were not helpful.
Summary.

The conversations in which students participate are only one of many concurrent
streams of activity. Both at home and in school, ﬁre streams of activity compete for
students’ attention, sometimes pulling participants’ attention away from the immediate
task. Each of the participants in ﬁre conversations brings varied experiences ﬁrst
contribute to how they deﬁne the conversation and, as a result, what students, ﬁreir
parents, and classmates Ieam by participating in one conversation is heavily inﬂuenced
by the experiences they and their conversational partners bring to bear in the
conversation.

The reasons people choose to invoke different experiences in conversations also
vary. Alﬁrough ﬁre majority of ﬁrings people do and say are done to complete the
immediate task, participants sometimes choose to invoke ﬁrings in order to preserve
oﬁrer aspects of ﬁreir lives. These things may include saving time for other activities or
to save face. Participants accept or deny the contributions of ﬁreir conversational

partners for various reasons. Students’ standing in class, parents’ or teachers'

2 3 1
authority in different settings, and oﬁrer ﬁrings contribute to what students accept as

appropriate solutions.

Finally, alﬁrough people appear to learn someﬁring in one conversation, ﬁrey may
not be able to use what they Ieamed in subsequent conversations. What is available to
students as ﬁrey participate in successive conversations depends on ﬁreir documentation
of ﬁre earlier conversations. Even wiﬁr complete documentation, however, students
often can reconstruct only part of what they did earlier.

Each of ﬁrese findings contributes to a growing literature on human Ieaming and can
contribute to the conversations surrounding ﬁre implementation of recent reforms in
mathematics education. The ﬁndings from ﬁris study support the notion that Ieaming
cannot be seen in one observational session and raise questions about what evidence we
need in order to say a person knows something. Indeed, ﬁre ability to use what is
discussed in one conversaﬁon in subsequent conversations (i.e., extending what is
learned beyond the immediate setting) may better illuminate what a person truly knows.
The connections people make among conversations and ideas, however, are inﬂuenced by
many ﬁrings and may not be a natural characteristic of Ieaming. Rather, ﬁre willingness
to make productive or appropriate connections may need to be taught.

The findings of ﬁris study can also inform the implementation of reforms in
maﬁrematics education. Maﬁrematics educators cannot assume ﬁrst students’ sole
exposure to maﬁrematical ideas comes in school. Parents contribute greaﬁy to students’
maﬁrematical understanding. Alﬁrough ﬁre mutual inﬂuence of ﬁre home and school has
always existed, it is particularly important in the wake of the reforms which include
new deﬁnitions of what it means to know and do maﬁrematics. Parents and students may
have conceptions of school maﬁrematics that conflict with ﬁre spirit of ﬁre reforms.
Parents’ and students’ unwillingness to change ﬁreir conceptions of elementary-school
mathematics might create barriers to students’ msﬁrematical understanding. As a

result, successful implementation of the reforms may include parent education as well

2 3 2
as changes in elementary- and secondary-school classrooms. Parent education and

developing successful home-school partnerships provides a fruitful opportunity for
further research.
Meﬁrodology and Meﬁrods

Alﬁrough prior to ﬁris study ﬁreorists had begun suggesting that what and how people
learn is influenced by their participation in many socially-deﬁned practices, no
research had documented successive conversations to explore how ideas change as people
move from practice to practice. Consequently, no established methods existed to follow
people among practices and to document their developing ideas. The meﬁrods I used in
ﬁris study were developed as an initial attempt to document cognitive change across a
series of conversations. In general, ﬁre data collection meﬁrods were successful.
Listening in on students' conversations at home and watching them parﬁcipste in school
illuminated many of ﬁre influences on students’ ﬁrinking. But even ﬁrough ﬁre meﬁrods
provided a glimpse at ﬁre influence of different conversations on students’ completion of
ﬁre tasks, ﬁrere are aspects of ﬁre meﬁrods that can be improved. In ﬁris section I review
ﬁre methods used to collect ﬁre data and discuss some of the difﬁculﬁes ﬁrst resulted.
Finally, I discuss some changes ﬁrst might lead to a more complete data trail.

The data collection methods I used in this study comprised two components. The first
component focused on documenting classroom activity. To collect ﬁris data, I videotaped
Ms. Smiﬁr’s ongoing classroom instruction and videotaped or audiotaped small group
conversations during class time. Before I began collecting data, I had resigned myself to'
collecting conversations of only one small group each day. As I began ﬁlming and
recording in class, ﬁre students who recorded ﬁreir conversations at home also began
recording ﬁreir conversations in school. Many of them brought their tape recorders and
turned ﬁrem on at ﬁre beginning of math class and left ﬁrem on until the end. Taking a
hint from them, I started placing audio tape recorders on in ﬁre small groups ﬁrst were

not being video taped on a given day. In ﬁre end, I produced a series of tapes, boﬁr audio

2 3 3
and video, ﬁrst documented whole class instruction and each of ﬁre small group

conversations in which the students who took part in this study participated. To
document the students’ written work I collected ﬁreir assignment sheets after each task.

The second data collection component focused on documenting ﬁre conversations
students had wiﬁr ﬁreir parents as ﬁrey worked on school maﬁr tasks at home. To collect
ﬁrese data, I provided each of ﬁre six students wiﬁr an audiotape recorder and tapes. I
asked them to record ﬁre conversaﬁons ﬁrey had wiﬁr ﬁreir parents and write their work
on ﬁre assignment sheets Ms. Smith handed out in class. I instructed ﬁre students to
return the tapes the following morning.

The goal of ﬁrese meﬁrods was to document an uninterrupted trail of ideas moving
from school to home and back to school. The trail, however, was beset wiﬁr gaps. Some
of ﬁre gaps resulted from unrecorded conversations. In ﬁre beginning of the study,
primarily, ﬁre conversations recorded in school documented only part of what the
students discussed. Because of technological limitations (e.g., the number of tape
recorders and microphones available), absent students, and other logistical hurdles,
every conversation in which ﬁre students participated could not be recorded. This caused
a problem as illuminating conversations between students and their parents could not
always be followed up with their conversations in school.

Collecting students’ written work also created a gap of sorts. Part of Ms. Smith’s
conception of teaching in the spirit of the recent reforms in maﬁrematics education
included having students revise ﬁreir answers based on their small group conversations
and ﬁren again during whole class discussions if changes were warranted. Alﬁrough the
students were instructed to write down all of ﬁreir work and asked not to erase earlier
work, they often did erase and write over previous answers. In ﬁre videotapes of small
group interaction, students erased or in some way changed ﬁre answers they had written
down earlier. Yet, on ﬁre assignment sheets collected at ﬁre and of each task, the students

answers often were the same as ﬁrose ﬁrey constructed with their parents at home. This

234
made it difficult, if not impossible to determine what changes were made in the small

group interactions.

Oﬁrer gaps were created in the conversations at home. In each conversation recorded
at home ﬁrere were times when ﬁre tape recorder clicked off and back on. There may be
many reasons for ﬁre recorders being turned off and, at times, hints to the reasons were
given on the tape. For example, Kaﬁry’s faﬁrer told her to shut ﬁre recorder off while he
went to check something as ﬁrey worked on Task Three. Based on where ﬁre tape was
stopped and ﬁreir conversaﬁon boﬁr before and after, it seems reasonable to conclude that
he went to look up someﬁring ﬁrst helped him determine how to predict how many
numbers were possible. Kathy’s father, however, never alluded to where he want or
what he did when ﬁre recorder was off. In another example, Kathy’s father told her she
“didn’t need to record all ﬁris“ as she wrote down her answer to a question on Task One.
Alﬁrougfi he may have been preserving tape, ﬁre gap may also have included some
substantive conversation ﬁrst was not recorded. In yet anoﬁrer example, Tony’s mother
turned off ﬁre recorder when she got “heated up“ with Tony. When ﬁre recorder was
turned back on, she was no longer heated up and they continued to work on the
assignment.

Each of ﬁrese examples points out that parts of ﬁre conversations students had in
ﬁreir homes were not recorded. The conversations students had in school also contained
evidence that the parts of ﬁreir in-home conversations were not recorded. While
discussing Task Three, for example, Karen explained the strategy she and her faﬁrer used
to predict how many numbers were possible. Her father had told her, she recounted,
that for four-digit numbers ﬁrere would be four numbers that began with each digit. So,
to determine how many numbers were possible she needed to multiply four times four
arriving at a prediction of 16 numbers. Although this strategy fit the answer Karen had
written down, it was not discussed in ﬁre conversation she recorded the night before. On

ﬁre tape Karen spoke alone to ﬁre tape recorder. Her response to ﬁre question about

2 3 5
predictions was only “you can make six numbers and you can make sixteen.“ These

numbers--six and sixteen--were how many three- and four-digit numbers Karen and
her father found while working on ﬁre two parts of Task Three.

In many cases an account of what ﬁre student had done at home could be constructed
from listening to ﬁre conversations ﬁrey had in school and at home. But,—the missing
conversations and written work meant providing an incomplete picture of how the
students completed school-maﬁr tasks in ﬁreir homes, and, as a result, an Incomplete
picture of ﬁre influence ﬁre different interactions had on students’ completion of ﬁre
tasks. Participating in a practice means knowing what resources are available and how to
use them. Having mathematical resources like ﬁrose ﬁrst may have been available in
Kaﬁry’s home and using ﬁrem during ﬁre conversations is an important aspect of ﬁre
practice of doing math in a student’s home. Written documentation of students’ ﬁrinking
also helped to piece together the trail of ideas necessary in a study of collateral
participation in various practices.

Oﬁrer gaps were created when as ﬁre adults’ participation in ﬁre recorded
conversations decreased. Over ﬁre course of ﬁre study, the tapes grew progressively
shorter and scantier. In some cases, the parents stopped participating in ﬁre recordings,
leaving ﬁre student to narrate what was already written on their papers. On ﬁrese tapes
it is not clear whether ﬁre parent had worked with the student on ﬁre task or ﬁre student
completed ﬁre task wiﬁr no help. The shorter tapes might be explained, in part, by the
different content in the tasks. The earlier tasks contained questions wiﬁr which the
adults may have been familiar. Task 3, however, contained questions ﬁrst ﬁre parents
may have been uncomfortable answering. Parents may be more willing to engage in
conversation when they feel comfortable, or competent, with the content.

Alﬁrough ﬁre reasons for ﬁre shorter tapes cannot be conclusively determined, ﬁrey
may result from asking parents to record conversations that do not normally occur in

their households. The conversations recorded later in this study were often carried on

236
from a distance. In Tony and his moﬁrer’s conversation for Task Four, for example,

Tony’s mother talked with him from a different room. Her comments are difﬁcult to
hear on ﬁre tape and Tony often asked her to repeat what she said. Throughout the
conversation, Tony's mother was doing other things important to ﬁreir family's life. The
later tapes in which parents talked from a distance and engaged in oﬁrer household
activities may be a better representation of authentic conversations students and ﬁreir
family members have about their school work. Asking the parents to sit down and record
their conversations for this study may have created an artificial conversation--a
conversation that took parents away from other Important household obligations.

Of ﬁre ﬁrings that created gaps in ﬁre data, only a few are controlled by ﬁre
researcher. The ﬁrings beyond ﬁre control of the researcher include parents’ and
students’ decisions to turn ﬁre tape recorder off and on while recording ﬁre
conversations, ﬁre incomplete record of written work, and ﬁre parents decreased
participation in the conversations. Although many of these difﬁculties are unique to this
study, parents’ decreasing participation in the conversations is consistent wiﬁr Varenne
et al.’s (1982) difficulties while videotaping homework sessions.

The controllable aspects of ﬁre study include ﬁre missed conversations in ﬁre
classroom and ﬁre content of ﬁre different tasks. The meﬁrods would be stronger if ﬁre
people who recorded ﬁreir conversations at home worked together in a maﬁr group at
scth and all of ﬁreir conversations in each setting, as well as all ﬁreir contributions to
large group discussions were recorded.

In sum, ﬁre data collection meﬁrods I used in ﬁris study illuminated the influences
different conversations have on what and how people Ieam. In ﬁris way, ﬁrey contributed
to ﬁre development of a meﬁrod to investigate ﬁre mutual inﬂuences of various practices
on Ieaming. The meﬁrods did not, however, provide a complete data trail. Rather, gaps
existed in the data ﬁrst might mask some aspects of ﬁre various practices in which the

students participated. Alﬁrough some of ﬁre gaps could be eliminated by careful selection

2 3 7
of students and placement in ﬁreir maﬁr groups, most of ﬁre gaps are created by

difﬁculties inherent in this type of research. Even with these difficulties, however, the
meﬁrods produced a clear vision of the sociocultural influences on students' completion
of a series of elementary-school mathematics tasks with implicaﬁons for practice and
further study.

Implications for Practice

This study provides a detailed look at ﬁre inﬂuence of students' interactions at home
and in school on ﬁreir maﬁrematical ﬁrinking. Alﬁrough many of the findings support
contentions of oﬁrer ﬁreorists and researchers, never before have they been brought
together to describe the phenomenon of Ieaming across conversations in ﬁris way. As a
result, ﬁre ﬁndings of ﬁris study have implications for both learning theory and
elementary-school maﬁrematics. I discussed ﬁre theoretical implications in ﬁre
previous sections. In this final sections I explore the practical implications in light of
ﬁre calls for reform in education and mathematics education.

Recent reports (National Commission on Excellence in Education, 1983; National
Education Goals Panel, 1992) have suggested that America’s schools are failingo-test
scores are dropping, violence is increasing, and students are not being prepared for
their futures. In response to dress reports a bipartisan coalition of educators (Naﬁonal
Education Goals Panel, 1994) has formulated a set of eight goals meant to address and
reverse ﬁre negative trends in our nation’s schools. One of ﬁre eight goals is to increase
parents’ participation in ﬁreir children’s academic development through home-school
partnerships.

Successful home-school partnerships require more ﬁran parents merely asking
their children if ﬁrey have finished their homework. Indeed, ﬁre recommendations
suggest parents be actively involved in their children’s schoolwork. This is meant to
accomplish two ﬁrings. First, parents, it is assumed, can provide additional assistance

while their children complete their schoolwork. Second, when both teachers and parents

2 3 8
have worked wiﬁr ﬁre student on the same tasks, ﬁrey will have a shared experience about

which ﬁrey can talk. Although ﬁrese suggestions, if followed, may assist parents and
teachers in their work wiﬁr ﬁreir children and students, ﬁrey rely on an overly
simplistic view of students’ homes and ﬁre home-school relationship.

The research presented here provides an initial look at home-school relationships
that might inform the development and maintenance of home-school partnerships.
Parents in ﬁris study were deeply concerned about ﬁreir children’s academic success and
wanted to do whatever ﬁrey could to help ﬁrem wiﬁr ﬁreir schoolwork. Running a
household, however, is a tremendously complex task ﬁrst requires parents to juggle
many different-sized spheres of ongoing activity. Merely asking parents to assume more
responsibility for their children’s schoolwork may be unsuccessful. Likewise, the
failure of parents to fully participate cannot be taken as a rejection of ﬁre program.

In the study reported here, parents' participation in their children’s schoolwork
varied depending on their perceived competence with ﬁre material on which the students
were working and their own schedules. When it ﬁt into ﬁreir schedules, ﬁre parents
worked closely with their children on ﬁreir schoolwork. When ﬁrey felt qualified, they
efﬁciently directed their children to answers ﬁrey believed were correct, but when they
were unsure of what was being asked, they shied away from ﬁre questions, often telling
their children to work alone or, if the student was unsure, to ask the teacher the next
day.

The parents’ reactions to ﬁre content of assignments gains importance in ﬁre wake of
ﬁre calls for reform in maﬁrematics education. As was the case wiﬁr ﬁre “new math“ of
the 1950s, mathematics assignments in reform classrooms are likely to be inconsistent
wiﬁr parents’ maﬁremaﬁcs experiences. Consequently, changing mathematics
instruction may enlarge ﬁre discontinuities that already exist between ﬁre home and

school and may make it more difficult to form home-school partnerships.

239
The study reported here is an initial attempt at understanding ﬁre home-school

relationship and ﬁre influence ﬁre two institutions have on students' mathematics
learning. Using ﬁre findings of ﬁris study and, perhaps, those of subsequent studies, it
may be possible to develop meﬁrods for developing and maintaining productive home-
school partnerships.

In light of the interest in home-school partnerships, there are two logical
extensions of ﬁris research. First, I may continue to study ﬁre relationship between
homes and schools to better understand ﬁre inﬂuence each has on students' schoolwork.
To conduct ﬁris research, I may duplicate the study I have reported here taking into
account ﬁre shortcomings mentioned above to ensure a more complete data trail. An added
component of the study would include an Instructional component for parents
participating in the study. The instructional component would most likely take the form
of a newsletter explaining ﬁre mathematical ideas being discussed in class. The success
or failure of ﬁre home-school partnerships could contribute to the development of
meﬁrods or prescriptions for practicing teachers. A second logical extension would be to
teach an elementary school maﬁrematics class myself and attempt to develop productive
home-school partnerships with my students' families. This research could contribute
methods pitfalls of developing home-school relationships.

Summary and Conclusions

The study reported here, more than anyﬁring else, supports recent advancements in
learning ﬁreory. It provides evidence in support of ﬁre notion that Ieaming is not an
individual process, but, rather, a fundamentally social process in which primacy can
given be given to neither ﬁre individual or ﬁre social structures within which they
function. The study provides empirical support for ﬁre noﬁon ﬁrst people do many things
concurrenﬁy and each of ﬁrose ﬁrings influences how they ﬁrink about maﬁremaﬁcs and,
perhaps, school in general. It points out ﬁrst learning is a gradual process during which

learners make choices about what they are to learn or believe within a constrained set of

 

240
choices offered by the socially-defined practices within which they participate. Finally,

it points out that conflicts can arise when various practices containing conflicting rules
are brought together in conversation.

These ﬁreoretical tenets can be used to describe elementary school students as ﬁrey
ﬁrink about and discuss mathematics at home and in school. Elementary mathematics
students, use and learn maﬁrematics in boﬁr settings. When conflicting conceptions of
mathematics, homework, or school collide in their conversations, ﬁre conflict must be

resolved. The resolution, however, entails many risks. Students may be faced with

- #3,...

choices about whose rules to follow. Following the wrong rules can result in answers
ﬁrst are inconsistent wiﬁr those expected when the students return to school.

The recent calls for reform in mathematics education have increased the chances of

 

‘Tr-

conflict in students’ conversations. The calls for reform have drastically changed what
it means to know and do maﬁremaﬁcs in elementary-school classrooms. These changes
are likely to conflict with parents’ and students' previous maﬁrematics experiences.
Students exposed to different conceptions of maﬁremsﬁcs at home and in school may be
slow to Change ﬁre way they ﬁrink about maﬁremaﬁcs. As a result, teachers will need to
probe students’ answers to understand ﬁre assumptions they used when answering ﬁre
questions and to remain open to alternative interpretations of mathematical situations
and answers. And, successful implementation of ﬁre reforms might need to include ﬁre
development of home-school partnerships in which parents are instructed in the new
conceptions of elementary-school mathematics as well as changes in elementary-school
mathematics classrooms.

At ﬁre same time ﬁrst the implementation of ﬁre reforms can be aided by the findings
of ﬁris study, the reforms continue to provide an opportunity to explore the
relationships among various practices and their influence on Ieaming. The
conversations students parﬁcipate in at home and in school reﬂect changing practices in

which students Ieam mathematics. Understanding how they change and ﬁreir

2 4 1 .
contribution to students’ mathematics learning can contribute to ﬁre growing body of

knowledge about human Ieaming. These two opportunities for furﬁrer study suggest that
maﬁrematics educators and researchers interested in human learning have reasons to

continue working together.

 

IJSTOFREFERENCES

UST OF REFERENCES

Adams, H. W. (1924). The mathematics encountered in general reading of newspapers
and periodicals. Unpublished Master’s thesis. Department of Education, University

of Chicago. Reviewed by F. K. Bobbitt in, Elementary School Journal Volume 25,
No. 2, (PP. 133-143).

Au, K. H. (1990). Changes in a teacher’s views of interactive comprehension

instruction. In L. C. Moll (Ed.), Vygotsky and Education: Instructional implications
and applications of sociohistorical psychology (pp. 271 -286). New York:
Cambridge University Press.

Austin, J. D. (1979). Homework research in mathematics. 79, 115-121.

Bales, R. F., 81 Cohen, S. P. (1979). SYMLOG: A system for the multiple level

observations of groups. New York: Free Press.

Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what
prospective teachers bring to teacher education. Unpublished doctoral dissertaﬁon,
Michigan State University, East Lansing, MI.

Beach, K. (1990). Rural Nepali students reconstrucﬁon of mathematics while moving
from school to work. Paper presented at the annual meeting of ﬁre American

Educational Research Association, Boston.

Beach, K. (1993). Becoming a bartender: The role of external memory cues in a work-

directed educational activity. 7, 191-204.

Beach, K. D. (1990). From school to work: A social and psychological history of maﬁr

in a Napali village. Himalayan Research Bulletin, 18(2/3), 18-23.

242

243
Belle, D. (Ed.). (1982). Lives in stress: Women and depression. Beverly Hills, CA:

Sage.

Bishop, A. J. (1991). Mathematical enculturation. Boston: Kluwer academic
publishers.

Bivens, J. A. (1990, April). Children scaffolding children in the classroom: Can this
metaphor complete/y describe the process of group problem solving? Paper
presented at ﬁre annual meeting of ﬁre American Educational Research Association,
Boston, MA.

Bloom, B. (1976). Human characteristics and scth Ieaming. New York: McGraw
Hill.

Bradley, R. H., 8 Caldwell, B. M. (1984). The relation of infants’ home environments
to achievement test performance in first grade: A follow-up study. Chi/d
development, 55, 803-809.

Brink, W. G. (1937). Direct study activities in secondary schools. New York:
Doubleday.

Bronfenbrenner, U., Kessel, F., Kessen, W., 8 White, S. (1986). Toward a critical
social history of developmental psychology. American Psychologist, 41(11),
121 8-1 229 .

Brown, A. L., Bransford, J. D., Ferrara, R. A., 8 Campione, J. C. (1983). Learning,
remembering, and understanding. In J. H. Flavell, 8 E. M. Markman (Ed.),
Cognitive development, Vol. III, Handbook of child psychology (pp. 77-166). New
York: Wiley.

Butcher, J. E. (1975). Comparison of the effects of distributed and massed problem
assignments on ﬁre homework of grade nine algebra students. Unpublished doctoral

dissertation, Rutgers University, New Brunswick, NJ.

244
Carpenter, T. P., Peterson, P. L., Chiang, C. P., 8 Loef, M. (1988). Using knowledge of

children ’s mathematics thinking in classroom teaching: An experimental study.
Paper presented at ﬁre annual meeting of ﬁre American Educational Research
Association, Paper presented at the annual meeting of ﬁre American Educational
Research Association, New Orleans, LA.

Carraher, T. N., Carraher, D. W., 8 Schliemann, A. D. (1985). Maﬁrematics in the
streets and in schools. British Journal of Developmental Psychology, 3, 21-29.

Carraher, T. N., Carraher, D. W., 8 Schliemann, A. D. (1987). Written and oral
mathematics. Journal for Research in Mathematics Education, 18, 83-97.

Catan, L. (1989). The dynamic display of process: Historical development and
contemporary uses of the microgenetic meﬁrod. 29, 252-263.

Cazden, C. B. (1988). Classroom discourse: The language of teaching and Ieaming.
Portsmouth, NH: Heinemann.

Cobb, P., Pateman, N., 8 Bednarz, N. (1993). Constructivism and activity theory: A
consideration of their similarities and differences as ﬁrey relate to mathematics
education. In H. Mansﬁeld, N. Pateman, 8 N. Bednarz (Ed.), Mathematics for
tomorrow‘s young children: International perspectives on curriculum. Dordrecht,
Netherlands: Kluwer.

Cobb, P. (1986). Contexts, goals, beliefs, and learning mathematics. For ﬁre Ieaming
of mathematics, 6(2), 2-9.

Cobb, P., Wood, T., 8 Yackel, E. (1993). Discourse, mathematical thinking, and
classroom practice. In E. A. Forrnan, N. Minick, 8 C. A. Stone (Ed.), Contexts for
learning (pp. 91-119). New York: Oxford University Press.

Cole, M. (1990). Cultural psychology: A once and future discipline? In J. J. Berrnan

(Ed-)1” :— .2 11'“ n '1 m 201 '2' rot: I g o: .; I: ..

31 (pp. 279-335). Lincoln, Nebraska: University of Nebraska Press.

Cooper, H. M. (1989). Homework. New York: Longman.

 

245
Coulter, F. (1979). Homework: A neglected research area. 5(1), 21-33.

Crawford, C. C., 8 Carmichael, J. A. (1937). The value of home study. Elementary
School Journal, 38, 194-200.

Davis, R. B. (1967). The range of rhetorics, scale, and oﬁrer variables. Journal of
research and development in education, 1, 51 -74.

Dewey, J. (1895/1964). What psychology can do for ﬁre teacher. In R. D.
Archambault (Ed.), John Dewey on education (pp. 195-211). Chicago: University
of Chicago Press.

Dewey, J. (1938/1991). Logic: The ﬁreory of inquiry. Csrbondale, IL: Southern
Illinois University Press.

DiNapoli, P. J. (1937). Homework in the New York City elementary schools . Teachers
College, Columbia University, New York.

Doane, B. S. (1972). The effects of homework and locus of control on arithmetic skills
achievement in fourﬁr grade. (Doctoral dissertation, New York University,
Dissertation Abstracts (International, 73-8160)

Eckert, P. (1989). Jocks and Bumouts: Social categories and identity in the high
school. New York: Teachers College Press.

Edwards, 0., 8 Mercer, N. M. (1987). Common knowledge: The development of
understanding in ﬁre classroom. New York: Methuen.

Eichinger, D. C. (1992). Analyses of middle school students' scientific arguments in
collaborative problem solving contexts. Unpublished Doctoral Dissertation,
Michigan State University, East Lansing, MI

Elbers, E. (1991). The development of competence in its social context. Educational
psychology review, 3(2), 73-94.

Elbers, E., Maier, R., Hoekstra, T., 8 Hoogsteder, M. (1992). lnternalization and

adult-child interaction. Learning and Instruction, 2, 101-118.

246
Epstein, J. L. (1983). Longitudinal effects of person-family-school interactions on

students outcomes. In A. Kerckhoff (Ed.), Research in sociology of education and
socialization, (Vol. 4) ( Greenwich, CT: Jai.

Epstein, J. L. (1986a). Parents' reactions to teacher practices of parent involvement.
Elementary school journal, 86, 277-294.

Epstein, J. L. (1986b). Toward a theory of school and family connections (Report No.
3). Center for Research on Elementary and Middle Schools, The Johns Hopkins
University, Baltimore, MD.

Epstein, J. L. (1988). How do we improve programs for parent involvement?
Educational horizons, 66, 58-59.

Friesen, C. D. (1975). The effect of exploratory and review homework exercises upon
achievement retention and attitude in a first year algebra course. (Doctoral
dissertation, University of Nebraska, Dissertation Abstracts International, 76-
4516)

Fuson, K. C. (1988). Children's counting and concepts of number. New York:
Springer-Verlag.

Gallaway, M. T. T. (1923). Maﬁrematics needed in a freshman course in clothing. In W.
W. Charter (Ed.), Curriculum construction (pp. 241-243). New York:
MacMillian.

Gray, R. F., 8 Allison, D. E. (1971). An experimental study of the relationship of
homework to pupil success in computation with fractions. School Science and
Mathematics, 71, 339-346.

Grifﬁn, P., 8 Cole, M. (1984). Current activity for the future: The zo-ped. In B.
Rogoff, 8 J. V. Wertsch (Ed.), Children ’s Ieaming in the “zone of proximal
development“ (pp. 45-64). San Francisco: Jossey-Bass.

Hadamard, J. (1945/1954). An essay on the psychology of invention in the

mathematical field. New York: Dover.

2 4 7
Heaﬁr, S. B. (1982). Questioning at home and school: A comparative study. In G.

Spindler (Ed.), Doing ﬁre ethnography of schooling: Educational anthropology in
action (pp. 102-131). New York: Holt, Rinehart, 8 Winston.

Heath, S. B. (1983). Ways with words: Language, life, and work in communities and
classrooms. New York: Cambridge University Press.

Holmes Group. (1990). Tomorrow’s schools: Principles for the design of professional
development schools. East Lansing, MI: Auﬁror.

lverson, B. A., 8 Walberg, H. J. (1982). Home environment and school Ieaming: a
quantitative synthesis. Journal of experimental education, 50, 144-151.

James, G., 8 James, R. C. (1976). Mathemaﬁcs dictionary (4ﬁr ed.). New York: Van
Nostrand Reinhold.

Keith, T. Z. (1987). Children and homework. In A. Thomas, 8 J. Grimes (Ed.),
Children 's needs: Psychological perspectives (pp. 275-282). Washington, D. C.:
National Association of School Psychologists.

Kilpatrick, J. (1987). What constructivism might be in mathematics education. In J.
C. Bergeron, N. Herscovics, 8 C. Kieran (Ed.), Proceedings of the 11th
International Conference for the Psychology of Mathematics Education (Vol. 1, pp.
3-27). Montreal: lntemational Group for the Psychology of Maﬁrematics Education.

Kilpatrick, J. (1992). A history of research in mathematics education. In The
handbook of research on maﬁrematics teaching and Ieaming (pp. 3-38). New York:
Macmillian.

Kohl, H. R. (1984). Growing minds: On becoming a teacher. New York: Harper and
Row.

Kohl, H. (1991). I won’t Ieam from you: The role of assent in Ieaming. Minneapolis:

Milkweed editions.

248
Kohlberg, L. (1971). From is to ought: How to commit the naturalistic fallacy and get

away with it in ﬁre study of moral development. In T. Mischel (Ed.), Cognitive
development and epistemology (pp. 151-235). New York: Academic Press: '

Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago: University of
Chicago Press.

Laboratory of Comparative Human Cognition. (1983). Culture and cognitive
development. In W. Kessen (Ed.), History, theory, and methods, Vol 1 of P. H.
Mussen (Ed.) Handbook of child psychology (pp. 295-356). New York: Aldine de
Gruyter.

LsConte. (1981). Homework as a learning experience: What research says to ﬁre
teacher . ERIC document reproduction service No. ED 217 022,

Laing, R. A. (1970). Relative effects of massed and distributed scheduling of topics on
homework assignments of eighﬁr-grade maﬁrematics students. Unpublished doctoral
dissertation, Ohio State University, Columbus, OH

Lampert, M. (1990). When the problem is not the question and the solution is not the
answer. American Educational Research Journal, 27(1), 29-63.

Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday
life. Cambridge: Cambridge University Press.

Lave, J. (1993). The practice of Ieaming. In S. Chaiklin, 8 J. Lave (Ed.),
Understanding practice: Perspectives on activity and context (pp. 3-32). New
York: Cambridge University Press.

Lave, J., Murtaugh, M., 8 de la Rocha, O. (1984). The dialectic of arithmetic in
grocery shopping. In B. Rogoff, 8 J. Lave (Ed.), Everyday cognition: Its
development in social situations (pp. 67-94). Cambridge, MA: Harvard University
Press.

Lave, J., 8 Wenger, E. (1991). Situated learning: Legitimate peripheral participation.

New York: Cambridge University Press.

249
Leichter, H. J. (1979). Families and communities as educators: Some concepts of

relationship. In H. J. Leichter (Ed.), Families and communities as educators (pp.
3-94). New York: Teachers College Press.

Leichter, H. J. (1985). Families as educators. In M. D. Fantlni, 8 R. L. Sinclair (Ed.),
Education in scth and nonschool settings. Eighty fourﬁr yearbook of ﬁre National
Society for the Study of Education (pp. 81-101). Chicago: University of Chicago
Press.

Leinhardt, G., 8 Greeno, J. G. (1986). The cognitive skill of teaching. Journal of
Educational Psychology, 78(2), 75-95.

Leinhardt, G., 8 Smiﬁr, D. A. (1985). Expertise in mathematics instruction: Subject
matter knowledge. Journal of Educational Psychology, 77(3), 247-271.

Leont’ev, A. N. (1981). Problems of ﬁre development of mind. Moscow: Progress
Publishers.

Leont'ev, A. N. (1981). The problem of activity in psychology. In J. V. Wertsch (Ed.),
The concept of activity in soviet psychology (pp. 37-71). Armonk, NY: M. E.
Sharpe, Inc.

Long, R. S., Meltzer, N. S., 8 Hilton, P. J. (1970). Research in mathematics education.
2, 446-468.

Luria, A. R. (1976). Cognitive development: Its cultural and social foundations.
Cambridge, MA: Harvard University Press.

Luria, A. R. (1978). The development of writing in ﬁre child. In M. Cole (Ed.), The
selected writings of A. R. Luria ( New York: M. Sharpe. '

Maertens, N., 8 Johnson, J. (1972). Effect of arithmetic homework upon the attitude
and achievement of fifﬁr and sixﬁr grade pupils. School Science and Mathematics,

7 1, 117-126.
Marjoribanks, K. (1979). Families and their Ieaming environments: An empirical

analysis. London: Rouﬁedge and Kegan Paul.

250
Martin, L. M. W., 8 Beach, K. (1992). Technical and symbolic knowledge in CNC

machining: A study of technical workers of different backgrounds (MDS-146).
National Center for Research in Vocational Education, University of California at
Berkeley, Berkeley, CA,

McDermott, R. P. (1987). The explanation of minority school failure, again.
Anthropology and Education Quarterly (Special issue: Explaining ﬁre school
performance of minority students), 18(4), 361-364.

McDermott, R. P., Goldman, S. V., 8 Varenne, H. (1984). When scth goes home: Some
problems in the organization of homework. Teachers College Record, 85(3), 391-
409.

Mehsn, H. (1979). Learning lessons: Social organization in the classroom. Cambridge,
MA: Harvard University Press.

Michaels, S. (1981). “Sharing time“: Children's narrative styles and differential
access to literacy. Language in society, 10(3), 423-443.

Michaels, S. (1986). Narrative presentations: An oral preparation for literacy with
ﬁrst graders. In J. Cook-Gumperz (Ed.), The social construction of literacy (pp.
94-116). New York: Cambridge University Press.

Mitchell, M. H. (1918). Some social demands on the course of study in ariﬁrmetic. In
N. s. f. t. s. 0. education (Ed.), Seventeenth yearbook of ﬁre National Society for ﬁre
Study of Education, Part I (pp. 7-17). Bloomington, IL: Public School Publishing.

Moll, L. C., 8 Whitmore, K. F. (1993). Vygotsky in classroom practice: Moving from
individual transmission to social transaction. In E. A. Forrnan, N. Minick, 8 C. A.
Stone (Ed.), Contexts for Ieaming (pp. 19-42). New York: Oxford University

Press.

2 5 1
National Center for Research on Teacher Education. (1989). Section B 8 C of

elementary teacher interview: Teaching elementary scth mathematics . Michigan
State University, National Center for Research on Teacher Education, East Lansing,
MI.

National Commission on Excellence in Education. (1983). A nation at risk: The
imperative for educational reform . U. S. Government Printing Ofﬁce, Washington,
DC.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation
standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Maﬁrematics. (1991). Professional standards for
teaching maﬁremaﬁcs. Reston, VA: Auﬁror.

National Education Goals Panel. (1992). The national education goals report: Building a
nation of learners. Washington, D.C.: Author.

National Education Goals Panel. (1994). The national education goals report.
Washington, D.C.: Auﬁror.

National Research Council. (1989). Everyboo)I counts: A report to the nation on the
future of mathematics education . Washington, D.C.: Auﬁror

Naﬁonal Research Council. (1990). Reshaping school maﬁrematics: A philosophy and
framework for curriculum. Washington, D.C.: National Academy Press.

Nesher, P. (1989). Microworlds in maﬁrematical education: A pedagogical realism. In
L. B. Resnick (Ed.), Knowing, Ieaming, and instruction (pp. 187-216). Hillsdale,
NJ: Erlbaum.

Newman, 0., Griffin, P., 8 Cole, M. (1989). The construction zone: Working for
cognitive change in school. New York: Cambridge University Press.

Nunes, T., Schliemann, A. D., 8 Carraher, D. W. (1993). Street maﬁremaﬁcs and

school mathematics. New York: Cambridge University Press.

252
Olﬁrof, T., Goudena, P., 8 Groenendaal, J. (1989). Effectieve commumicatiepatronen

tussen opvoeder en kind: Een Iogitudinaal onderzoek zan de verbs/e ouder-kind en
leerkracht-kind interactie in taakgerichte situaties bij pruters en k/euters (3-6
jaar) (Effective communication“ patterns between educator and child: A longitudinal
study of verbal parent-child and teacher-child interaction in task-oriented
situations with children from 3 to 6 years-of-age). Unpublished Dissertation,
Utrecht (The Neﬁrerlands): University of Utrecht, department of Child Studies,

Panofsky, C. P., John Steiner, V., 8 Blackwell, P. J. (1990). The development of
scientific concepts and discourse. In L. C. Moll (Ed.), Vygotsky and education:
Instructional implications and applications of sociohistorical psychology (pp. 251 -
267). New York: Cambridge University Press.

Pea, R. D. (1988). Putting knowledge to use. In R. S. Nickerson, 8 P. P. Zodhiates
(Ed.), Technology in education: Looking toward 2020 (pp. 169-212). Hillsdale,
NJ: Erlbaum.

Perry, I. (1988). A black student’s reflection on public and private schools. Harvard
Educational Review, 58(3), 332-336.

Perry, W. G. (1981). Cognitive and ethical growth: The making of meaning. In A.
Chickering (Ed.), The modern American college (pp. 76-116). San Francisco:
Jossey-Bass.

Peterson, J. C. (1971). Four organizational patterns for assigning maﬁrematics
homework‘. School Science and Mathematics, 71, 592-596.

Philips, S. U. (1993). The invisible culture. Prospect Heights, IL: Waveland Press.

Piaget, J. (1966). Need and signiﬁcance of cross-cultural studies in genetic
psychology. lntemational Journal of Psychology, 1(1), 3-13.

Powell, A. G., Farrar, E., 8 Cohen, D. K. (1985). The shopping mall high school:

Winners and losers in the educational marketplace. Boston: Houghton Mifﬂin.

2 5 3
Remillard, J. (1990). Analysis of elementary curriculum materials: Perspectives on

problem-solving and its role in Ieaming mathematics. Paper presented at the
annual meeting of ﬁre American Educational Research Association, Boston, MA.

Resnick, L. B. (1987). Education and Ieaming to think. Washington, D. 0.: National
Academy Press.

Richards, J. (1991). Maﬁrematical discussions. In E. von Glasersfeld (Ed.),
Constructivism in mathematics education (pp. 13-52). Dordrecht, Netherlands:
Kluwer.

Rogoff, B. (1990). Apprenticeship in thinking. New York: Oxford University Press.

Rogoff, B. (1992). Children 's guided participation in cultural activity. Invited address
at ﬁre I Conference for Sociocultural Research, Madrid, Spain.

Romberg, T. (1988). A common curriculum for mathematics. In G. Fenstermacher, 8
J. Goodlad (Ed.), Individual differences and ﬁre common curriculum (82nd yearbook
of the National Society for the Study of Education ( Chicago: University of Chicago
Press.

Romberg, T. A. (1982). An emerging paradigm for research on addition and subtraction
skills. In T. P. Carpenter, J. M. Moser, 8 T. A. Romberg (Ed.), Addition and
Subtraction: A cognitive perspective (pp. 1-8). Hlllsdale, NJ: Erlbaum.

Romberg, T. A., 8 Carpenter, T. P. (1986). Research on teaching and Ieaming
mathematics: Two disciplines of scientific inquiry. In M. C. Wittrock (Ed.),
Handbook of research on teaching (3rd ed.) (pp. 850-873). New York: MacMillan.

Saxe, G. B. (1990a). Culture and cognitive development. Hillsdale, NJ: Erlbaum.

Saxe, G. B. (1990b). The interplay between children’s Ieaming in school and out-of-
school contexts. In M. Gardner, J. Greeno, F. Reif, A. H. Schoenfeld, A. diSessa, 8 E.
Stage (Ed.), Toward a scientiﬁc practice of science education (pp. 219-234).

Hillsdale, NJ: Erlbaum.

254
Schoenfeld, A. H. (1983). Beyond the purely cognitive: Belief systems, social

cognitions, and metacognitions as driving forces in intellectual performance.
Cognitive Science, 7, 329-363.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of
research on mathematics teaching and Ieaming (pp. 334-370). New York:
macmillian. .

Scribner, S. (1984). Studying working intelligence. In B. Rogoff, 8 J. Lave (Ed.),
Everyday cognition: Its development in social situations (pp. 9-40). Cambridge,
MA: Harvard University Press.

Scribner, S., 8 Cole, M. (1981). The psychology of literacy. Cambridge, MA: Harvard
University Press.

Shuell, T. J. (1986). Cognitive conceptions of Ieaming. Review of Educational
Research, 56(4), 411-436.

Sizer, T. R. (1992). Horace's compromise. Boston: Houghton Mifflin Company.

Skemp, R. R. (1976). Relaﬁonsl understanding and instrumental understanding.
Mathematics Teaching, 77, 1-7.

Snow, C. E., Bsmes, W. 8., Chandler, J., Goodman, I. F., 8 Hemphill, L. (1991).
Unfu/filled expectations: Home and school influences on literacy. Cambridge, MA:
Harvard University Press.

Stipek, D. J. (1988). Motivaﬁon to Ieam: From practice to theory. Englewood Cliffs,
NJ: Prentice Hall.

Ten Brink, D. P. (1967). Homework: An experimental evaluation of the effect on
achievement in maﬁrematics in grades seven and eight. (Doctoral dissertation,

University of Minnesota, Dissertation Abstracts International, 65-15326)

255
Thorndike, E. L., 8 Woodworth, R. S. (1901). The influence of improvement in one

mental function upon the efficiency of other functions. Psychological Review, 8,
247-261, 384-395, 553-564.

Toulmin, S. (1982). The construal of reality: Criticism in modern and postrnodern
science. Critical Inquiry, 9, 93-111.

Tudge, J. (1990). Vygotsky, ﬁre zone of proximal development, and peer collaboration:
Implications for classroom practice. In L. C. Moll (Ed.), Vygotsky and Education:
Instructional implications and applications of sociohistorical psychology (pp. 155-
172). New York: Cambridge University Press.

Twain, M. (1883/1984). Life on the Mississippi. New York: Viking Penguin Press.

Urwiller, S. L. (1971). A comparative study of achievement, retention, and attitude
toward maﬁremaﬁcs between students using spiral homework assignments and
students using traditional homework assignments in second year algebra. (Doctoral
dissertation, University of Nebraska, Dissertation Abstracts International, 71 -
19521)

Valsiner, J. (1993). Bi-directional cultural transmission and constructive
sociogenesis. In R. Maier, 8 W. de Graf (Ed.), Mechanisms of sociogenesis ( New
York: Springer.

van der Veer, R., 8 Valsiner, J. (1991). Understanding Vygotsky: A quest for
synthesis. Cambridge, MA: Blackwell.

Varenne, H., Hamid-Buglione, V., McDermott, R. P., 8 Morison, A. (1982). “I teach
him everything he Ieams in school“: The acquisition of literacy for Ieaming in
working class families (143). Teachers College, Columbia University, New York.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological
processes. Cambridge: Harvard University Press.

Vygotsky, L. S. (1986). Thought and Language. Cambridge, MA: MIT Press.

2 5 6
Vygotsky, L. S. (1994a). Imagination and creativity of ﬁre adolescent. In R. van der

Veer, 8 J. Valsiner (Ed.), The Vygotsky reader (pp. 266-288). Cambridge, MA:
Blackwell.

Vygotsky, L. S. (1994b). Thinking and concept formation in adolescence. In R. van der
Veer, 8 J. Valsiner (Ed.), The Vygotsky reader (pp. 185-265). Cambridge, MA:
Blackwell.

Walberg, H. J. (1984). Families as partners in educational productivity. Phi delta
kappan, 65, 397-400.

Walberg, H. J., 8 Marjoribanks, K. (1976). Family environments and cognitive
development. Review of educational research, 46, 527-551 .

Ware, W. B., 8 Garber, M. (1972). The home environment as a predictor of school
achievement. Theory into practice, 11(3), 190-195.

Wertsch, J. V. (1979). From social interaction to higher psychological processes: A
clarification and application of Vygotsky's theory. Human Development, 22, 1-22.

Wertsch, J. V. (1981). The concept of activity in soviet psychology: an introduction.
In The concept of activity in soviet psychology (pp. 3-36). Armonk, NY: M. E.
Sharpe, Inc.

Wertsch, J. V. (1984). The zone of proximal development: Some conceptual issues. In
B. Rogoff, 8 J. V. Wertsch (Ed.), Children ’s Ieaming in the “zone of proximal
development“ (pp. 7-18). San Francisco: Jossey-Bass Inc.

Wertsch, J. V. (1985). Vygotsky and the social formation of mind. Cambridge, MA:
Harvard University Press.

Wertsch, J. V. (1991). Voices of the mind. New York: Oxford University Press.

Wertsch, J. V., 8 Hickman, M. (1987). Problem solving in social interaction: A
microgenetic analysis. In M. Hickman (Ed.), Social and functional approaches to

language and thought (pp. 151-171). New York: Cambridge University Press.

257
Wertsch, J. V., Minnick, N., 8 Arns, F. J. (1984). The creation of context in joint

problem-solving. In B. Rogoff, 8 J. Lave (Ed.), Everyday cognition: Its development
in social context (pp. 151-171). Cambridge, MA: Harvard University Press.

Wertsch, J. V., 8 Minnick, N. (1990). Negotiating sense in ﬁre zone of proximal
development. In M. Schwebel, C. A. Maher, 8 N. S. Fagley (Ed.), Promoting
cognitive growth over the life span (pp. 71-88). Hillsdale: Erlbaum.

Wertsch, J. W., 8 Stone, C. A. (1978). Microgenesis as a tool for developmental
analysis. Quarterly Newsletter of the Laboratory for Comparative Human Cognition.
1, 8-10

William, L. W. (1921). The maﬁrematics needed in freshman chemistry. School
science and mathematics, 21(7),

Wilson, G. M. (1911/1922). Connersville course in mathematics. Baltimore:
Warwick and York.

Wise, C. T. (1919). Survey of arithmetical problems arising in various occupations.
Elementary School Journal, 20(October), 118-136.

Woody, C. (1922). Types of ariﬁrmetic needed in certain types of salesmanship.

Elementary School Journal, 22(7), 505-521.