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"YEAH BUT I THOUGHT IT WOULD STILL MAKE A SQUARE":
A STUDY OF FOURTH-GRADERS' DISAGREEMENT
DURING WHOLE-GROUP MATHEMATICS DISCUSSION

presented by

Carol J. Crumbaugh

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1!” W14

”YEAH BUT I THOUGHT IT WOULD STILL MAKE A SQUARE”:
A STUDY OF FOURTH-GRADERS’ DISAGREEMENT
DURING WHOLE-GROUP MATHEMATICS DISCUSSION
By

Carol]. Crumbaugh

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment Of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Counseling, Educational Psychology, and Special Education

1998

ABSTRACT
"YEAH BUT I THOUGHT IT WOULD STILL MAKE A SQUARE”:
A STUDY OF BOURTHGRADERS' DISAGREEMENT
DURING WHOLE-GROUP MATHEMATICS DISCUSSION
By

Carol]. Crumbaugh

The purpose of this exploratory study was to examine student
disagreement during whole-group discussion in one fourth grade mathematics
class.

Data collection drew on qualitative method. The focal data source for this
study Of fourth-graders’ disagreements were transcripts from each Of the ten
lessons which comprised the data set. Additional data sources included
students’ math journals, two interviews with the teacher, and two interviews
each with seven randomly-selected students. The data were triangulated and
analyzed to determine (a) what disagreement was in this fourth grade classroom,
(b) what students needed to learn to be involved in disagreements, (c) the role Of
this teacher and these students during disagreements, and (d) the relationship
between disagreements and mathematical tasks.

The findings of this study revealed several discussion-related patterns.
Discussion within which disagreement was situated unfolded in an ebb and flow

to reveal a discourse pattern that this teacher established for whole—group

discussion. In this pattern, mathematical ideas spiraled forward as participants
jointly constructed meaning. The function Of disagreement shed light on its
mathematical significance. Whereas some disagreements functioned to point out
a surface flaw in a representation, others functioned to force children tO get at the
underpinnings Of another’s position, or to hold up two ideas for public scrutiny.
The relationship between purpose, function, and mathematical significance Of
the disagreements led to the examination Of mathematical tasks that fueled
thinking.

Two competencies, social and academic, constituted mathematical
disagreement. Both were necessary for children to mathematically disagree. The
mathematical significance of the disagreements was connected to the teacher’s
role. Disagreement varied in its influence on the discourse. When the
mathematical and interactional aspects Of students’ disagreements were tightly
woven, the mathematical disagreement appeared to contribute to the discussion
and propel it forward. By contrast, when the disagreement was laden by the
interactional aspect, the disagreement created a gap in the discussion. This study
contributes to emerging research on the role Of discourse in mathematics
teaching and learning and suggests that children need to learn more than

mathematics to be involved in mathematical disagreement.

Copyright by
CAROL JEAN CRUMBAUGH
1998

ACKNOWLEDGEMENTS

It is difficult to find the words to adequately express my appreciation to
everyone who helped in the completion of this dissertation.

TO my committee I extend my gratitude for steadfast support and
encouragement through the dissertation process. AS my dissertation director
and mentor, Deborah Loewenberg Ball fostered and sustained my growth as a
researcher and writer. Deborah is an exemplary role model, and I am grateful
for her wisdom and thoughtfulness. Ralph Putnam, my guidance committee
director, generously gave Of his time to meet, read draft documents, and
generally provide support along the way. Thank you. Penelope Peterson,
Sandra Wilcox, and Helen Featherstone nudged my thinking and writing with
insightful questions.

My cohort and colleagues, Troy Mariage, Lauren Pfeiffer, Garnet Hauger,
and Lynn Brice steadfastly provided support beyond measure. Our many
conversations continue to influence my thinking about research, learning, and
teaching. Kara Suzuka, Tammy Lantz, and Cathy Siebert lent special support in
the final stages Of writing. Thank you.

This study would not be complete without heartfelt thanks to Katherine
Vandenberg. Her decision to study her mathematics teaching so early in her

professional development is indicative of her dedication to students and to

teaching. In addition, I salute Pamela Schram and all members of Mathematics
Study Group for all they do on behalf Of children.

TO my family and home-town friends, I embrace you for your love and

support from beginning to end.

ii

TABLE OF CONTENTS

 

 

 

 

 

LIST OF TABLES - -- - - , - vi
LIST OF FIGURES _- , , , - - - - -- _ ....... vii
CHAPTER ONE:
INTRODUCTION TO THE STUDY -- ...... -- -- - - _ -_ - - -1
CHAPTER TWO
LITERATURE REVIEW ........... - - _ ...... ........ - --6
Theoretical Framework ................................................................................................ 6
Analytical Framework ............................................................................................... 10
Participant Structure ....................................................................................... 11
Classroom Discourse .................................................................................................. 14
Mathematical Discourse ............................................................................................ 21
Curriculum ................................................................................................................. 25
Argument and Disagreement .................................................................................... 27
Summary and Conclusion ......................................................................................... 32
CHAPTER THREE
METHOD - - - ........... - ............. ............ - 34
Site, Setting, and Participants .................................................................................... 38
Method ............................................................................................................. 39
Analysis ....................................................................................................................... 46
Use of Hypermedia in Analysis ..................................................................... 49

iii

CHAPTER FOUR

 

 

 

INSIDE DISAGREEMENT _ 55
Lesson Format ............................................................................................................ 55
Discourse Pattern ....................................................................................................... 57
Teacher Moves ............................................................................................................ 61
Asking Questions ............................................................................................ 62
Connecting Ideas and Getting Children to Say More .................................. 62
Behavior and Task Reminders ....................................................................... 62
Typology Of Discussion ............................................................................................. 63
Repetitive Discussions .................................................................................... 64
Discussions With Different Ideas, Without Disagreement ......................... 65
Discussions With Different Ideas, Teacher-Mediated Disagreement ......... 67
Student-to—Student Disagreement ................................................................. 68
What is Disagreement? .............................................................................................. 70
Categories of Disagreement ...................................................................................... 74
CHAPTER FIVE
SINGLE DISAGREEMENT - - - -- - 77
Correct, Revise, or Mediate ....................................................................................... 81
Defend ......................................................................................................................... 85
Kim’s Diamond ............................................................................................... 97
Call for Clarification, Question Another’s Assumptions ...................................... 102
Disagreement Over the Location Of 1 / 2 on the Number Line .................. 114
Monica’s Square: Five Pieces or Four, and Are They Equal? ................... 117
Summary ................................................................................................................... 121
CHAPTER SIX
EPISODES OF DISAGREEMENT..-- - - A- - - 128
May 3 ......................................................................................................................... 126
What is the Disagreement About? ............................................................... 140
The Teacher’s Role ........................................................................................ 146
May 7 ......................................................................................................................... 153
Math on May 7 .............................................................................................. 154
What is the Disagreement About? ............................................................... 168

iv

CHAPTER SEVEN

 

 

 

 

 

 

CONCLUSIONS AND IMPLICATIONS -- -- - ........ 178
Summary and Discussion ........................................................................................ 178
Theory and Practice .................................................................................................. 185

Theory ............................................................................................................ 185

Practice ........................................................................................................... 188
Limitations ................................................................................................................ 189
Questions for Future Study ..................................................................................... 190
APPENDIX A - ..... - 194
APPENDIX B - - - - 196
APPENDIX C 198
APPENDIX D-_ _ _- . - _ _ _ ............................ - . 201
APPENDIX E - _ ...... 203
BIBLIOGRAPHY - - - - - -- 216

 

LIST OF TABLES

Table 1: Purposes Of Disagreement Grouped by Function .................................... 80
Table 2: Participant Rights and Duties .................................................................. 165
Table 3: Student Pronoun Use on May 3 and May 7 ............................................ 174

vi

LIST OF FIGURES

Figure 1: Michaela's Way to Make Half on May 7 .................................................. 61
Figure 2: Ali’s Halves on May 4 ............................................................................... 67
Figure 3: Ionah's Way to Show One-half on May 4 ................................................ 82
Figure 4: Hannah’s Representation for a Fraction on May 3 ................................. 83
Figure 5: Rob's Geoboard Of 1 / 2 and 2/ 4 on May 7 ............................................... 85
Figure 6: Halves Referred to by Shawna on May 7 ................................................ 87
Figure 7: Moira Continues to Outline ”Squares” on May 7 ................................... 88
Figure 8: A Way to Use Partial Squares to Prove Half. .......................................... 88
Figure 9: Overhead Geoboard Referred to by Rob to Indicate Fourths, May 7 91
Figure 10: Monte's Drawing Of Groups on May 3. ................................................. 95
Figure 11: Kim's Diamond on May 11 ..................................................................... 97
Figure 12: Jody’s Attempt to Combine Triangles to Make a Square on May12. 101
Figure 13: Hannah's Drawing of 1 / 2 on May 4 .................................................... 108
Figure 14: Hannah's TV (Drawn from Side View) on May 10 ............................. 111
Figure 15: Monica's Conjecture on May 10 ........................................................... 117
Figure 16: Monica’s Drawing of 3/5 on May 10 ................................................... 118
Figure 17: Monte’s Drawing of Groups on May 3 ................................................ 127
Figure 18: Hannah's Representation for a Fraction on May 3 ............................. 129
Figure 19: Michaela's Way to Make Half on May 7 ............................................... 155
Figure 20: Antonia's and Braden's Ways to Make Half on May 7 ....................... 158

vii

CHAPTER ONE
INTRODUCTION TO THE STUDY

Recent reform documents call for changes in the teaching and learning Of
mathematics (National Council Of Teachers of Mathematics [NCTM], 1989, 1991;
National Research Council [NRC], 1989). Written from a constructivist
perspective, these documents assert that what students learn and how they learn
it is personal and influenced by the learning environment provided by teachers.
Different from the traditional view Of teaching as telling, these reforms seek to
increase student involvement in mathematical activity and discourse. Reformers
believe that this heightened involvement will enhance students' mathematical
power, defined as,

The ability to explore, conjecture, and reason logically; to solve non-

routine problems' to communicate about and through mathematics; and to

connect ideas within mathematics and between mathematics and other
intellectual activity. Mathematical power also involves the development

Of personal self-confidence and a disposition tO seek, evaluate, and use

quantitative and spatial information in solving problems and in making

decisions. Students' flexibility, perseverance, interest, curiosity, and
inventiveness also affect the realization Of mathematical power (NC'I'M,

1991, p. 1).

These documents advocate increased student understandings Of
mathematics and view discourse - ”ways Of representing, thinking, talking, and
agreeing and disagreeing” - as a powerful process through which to develop
and enhance students' mathematical power. In my study, mathematical
discourse reveals itself primarily through whole-group discussion. In discussion,

these students are provided the tools (physical as well as intellectual) with which

to represent their ideas. However, what it actually means for teachers to

promote discourse with the goal Of increased student mathematical power, is an
area Of study in its infancy.

Whereas all classrooms have discourse, the discourse described in reform
documents requires different roles and responsibilities Of teachers and students.
NO longer are teachers to tell while students passively receive knowledge. The
advocated discourse increases students' role in the construction Of their
understandings and draws upon disputes over ideas as ways to refine and
develop ideas. Controversies Of the kind that the Standards (1991) seem to
promote are not common in most classrooms; to the contrary, student consensus
and acceptance are more the rule. My study focuses on an element Of
mathematical discourse, disagreement.

As cited above, discourse encompasses ways Of representing, thinking,
talking, and agreeing and disagreeing. While each construct (i.e., representing,
thinking, talking, agreeing, disagreeing) brings to mind particular classroom
images, it is the notion Of disagreeing on which I want to focus. Why is the study
Of disagreeing, or disagreement important to pursue? First, there appears to be
little research emphasis on disagreement in general, what it is or what goes on
during mathematical disagreement. Whereas disagreement may be simply
defined as those moments when two or more people do not agree or have the
same idea, disagreement may also be thought Of as dynamic and complex.

Second, little is understood about the ways in which students’
disagreements arise and are managed. Rather than encouraging lesson-centered
disagreement, many teachers may wish to avoid disagreements as deterrents to
the forward motion of a predetermined lesson plan. In addition, those moments

when disagreements arise may be unnerving to teachers as times when they are

unsure about whether to manage the mathematics, the social nature Of who is
disagreeing and in what ways, or both. For example, are some disagreements
centered around mathematics and others more personal? In order for
disagreements among students to take place, teachers must relinquish some Of
the previously-held control over discourse and lesson direction, to allow
students' ideas and questions to have some influence over where the
mathematics leads and to provide students the discourse space within which to
explore and own their thinking and mathematical understandings. What is
learned and how it is learned may become more student-centered when
disagreements are permitted and nurtured.

Third, when examined, disagreements may be revealed as a rich source
for understanding the complexities Of mathematics teaching and learning. AS
alluded in the Standards (1991), disagreement embedded within discourse may
be a window for teachers into students’ mathematical learning, at the same time
force children to articulate and justify their reasoning for themselves and for
others. A give and take may result as different ideas make their way to the floor.
In the process, children may also learn respect for each others’ thinking. ”It
seems possible that in our attempts to say things clearly to other people we
progressively learn to build their viewpoints into our own, and thus to see our
knowledge (or perceptions) as hypothetical and Open to change” (Barnes, 1992).
In other words, if an audience Of peers holds children accountable for their
thinking, explanation and articulation spring tO the forefront as important
benefits of disagreement.

Finally, the very existence Of student disagreement in an elementary

mathematics classroom is interesting and important to pursue because it takes

place in a subject commonly perceived tO be consensual. Our culture carries the
belief that in school mathematics, at least, there is one known, correct answer to
each problem posed. Therefore, the notion that disagreement is planned for is
counterintuitive-some would say counterproductive—in a consensual discipline
such as mathematics.

In addition, these disagreements exist in a discourse space created by one
fourth-grade teacher, Katherine Vandenberg, a space within which students are
encouraged to share ideas, clarify, justify, explain, defend, and disagree, through
speech, journal writing, and use Of manipulatives, whether in whole-group
discussion, small-group discussion, individual contemplation, or one-tO-0ne
interaction with her or with one another.

This study is about student-to-student disagreement in a fourth-grade
class. This study Of disagreement may also be thought Of as one of argument, in
the broadest sense. For within argument, which ”refers broadly and neutrally to
reasoning in order to make decisions or test the truth Of claims” (Perkins, p. 155),
is subsumed such things as explore, clarify, justify, explain, defend, and
disagree-characteristics of mathematical power advocated by reform
documents. My exploratory study of student disagreement begins to examine
closely the issues Cited above. Little is known about a) disagreement in general;
b) how individual disagreements during discourse arise and how they are
managed; c) the potential richness Of teaching and learning that exists during
student disagreement; d) the roles Of teachers and students during
disagreements, and; e) the existence of disagreement in a subject commonly

perceived to have known, agreed-upon answers.

This study discusses how during disagreements students are forced to
clarify and defend the reasoning behind their solutions, and through this process
they gain at least some of the increased mathematical power which reform
documents call for. When student disagreement arises during whole-group
discussion, it is imperative that educators (and reformers) have some sense Of
this complex discourse process in which students are expected to gain
mathematical power, and to recognize what students need to learn in order to be
involved disagreement. This study looked closely at a Specific component Of
discourse, student disagreement, for the purpose Of understanding what it takes

to be involved in it in this classroom.

CHAPTER TWO
LITERATURE REVIEW
As I wrote in Chapter One, little is known about disagreement during

whole-group discussion — about disagreement in general, how disagreements
arise and are managed, the potential richness Of teaching and learning that exists
during student disagreement, the roles Of teachers and students during
disagreements, and the existence Of disagreement in a subject commonly
perceived to have known, agreed-upon answers. In this chapter I outline my
rationale for the sociolinguistic study Of disagreement in mathematics
discussions. I follow with a review Of relevant literature.

In the first part Of the chapter I discuss my theoretical framework Of
sociocultural learning theory as the basis of my study. Next I discuss my study’s
analytical framework which is grounded in sociolinguistics. In the following
three sections Of the chapter I discuss classroom discourse, mathematical
discourse, and mathematics (respectively) and highlight the need for research in
these areas. These sections follow the others to underscore the need for studies
Of mathematical discourse and specifically, student disagreement. I close the
chapter with a discussion of the literature on argument and disagreement.

Theoretical Framework

To examine student disagreement, this study is grounded in sociocultural
learning theory, particularly the work Of Vygotsky (1962, 1978) which
emphasizes the importance Of language in learning. In this study Of student
disagreement, children’s use Of language to negotiate meaning is central. For
Vygotsky (1978), language was the most important psychological tool or sign.

As a sign, language mediates learning. That is, it is capable Of transforming

mental functioning. According to this theory, there are three basic themes
regarding learning and language (W ertsch, 1985). The first theme is based on the
premise that in order to understand development Of children researchers must
focus on the process Of development. That is, ”the study Of behavior is not an
auxiliary aspect of theoretical study, but rather forms its very base" (Vygotsky,
1978, p. 65). In this view, behavior is a process and not a product, and
appropriate method takes this into consideration.

The second theme Of Vygotsky’s learning theory is the premise that
learning first occurs in the social domain. In other words, meaning originates
between individuals. ”Every function in the child’s cultural development
appears twice: first, on the social level, and later, on the individual level: first
between people (inter-psychological), and then inside the child (intrapsychological)”
(1978, p. 57). That is, through speech in the social or external domain, the child
transforms and internalizes what is learned. What the child knew previously is
internally reconstructed. In Vygotsky’s view (1978), internalization is not a
unidirectional transfer process. TO the contrary, he viewed it as a series of
transformations:

(a) An Operation that initially represents an external activity is

reconstructed and begins to occur internally.

(b) An interpersonal process is transformed into an intrapersonal one.

(c) The transformation of an interpersonal process into an

intrapersonal one is the result Of a long series Of developmental
events (p. 56-57).
This series of transformations moves back and forth, from the external to

the internal, from the interpersonal to the intrapersonal. Gradually, as

transformation continues, internalization is the result.

Inherent in sociocultural learning theory is the premise that individuals
use talk and action (i.e., behavior) in focal and dynamic ways. In the case Of
mathematical discourse, the implication is that discussion contributes to student
learning, a notion highlighted in recent reform documents (National Council Of
Teachers Of Mathematics [NCTM], 1989, 1991 ; National Research Council [NRC],
1989).

The third theme Of Vygotsky’s learning theory is the claim that to
understand thinking (”mental processes”), we must understand the language
used to mediate it. In his view, language (”signs”) mediates mental processes.

In other words, through language use learning takes place.

In addition to the above themes, Vygotsky (1978) claimed that learning
was guided by a more knowledgeable other through a zone Of proximal (i.e.,
potential) development. He wrote that the zone of proximal development ”is 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” (p. 86). In practical terms, the zone Of proximal development is the
distance between what a child knows about, for example, fractions, and what a
child can learn through discourse about fractions. In the instance Of whole-group
discussion, a teacher creates different zones Of proximal development which lead
to mathematical understandings. That is, ”learning awakens a variety Of internal
developmental processes that are able tO Operate only when the child is
interacting with people in his environment and in cooperation with his peers.

Once these processes are internalized, they become part Of the child’s

independent developmental achievement” (p. 90). From this perspective,
experiences on the social level activate internal processes.

Together, these Vygotskian themes underng this study of fourth-
graders’ disagreement, because to understand what disagreement is and how
children are involved warrants close examination Of language use and
specifically, disagreement. When public disagreement occurs during whole-
group discussion, the discussion as orchestrated by the teacher creates different
zones of proximal development through which children’s learning occurs. In
addition, when children disagree, they push the outer limits of these zones,
forcing one another to explain and clarify what is said and why.

A sociocultural study of language use includes a study of the dynamic use
Of speech as communication. And Vygotsky was concerned that the
interrelationship between communicative and intellectual functions Of speech be
recognized:

The initial function of speech is the communicative function. Speech is

first and foremost a means Of social interaction, a means of

pronouncement and understanding. This function Of speech, which is
usually analyzed in terms of isolated units, has been separated from the
intellectual function Of speech. Both functions are ascribed to speech as if
they were parallel to or independent Of one another. Speech, as it were,
combines within itself both the function Of social interaction and the
function of thinking, but what the relationship is between these two
functions, what brings about the presence of the two functions of speech,
how they develop, and how they are structurally intertwined are
questions that have remained uninvestigated (Vygotsky, 1934, p. 88, cited

in Wertsch, 1985, p. 94).

Vygotsky as cited in Wertsch (1985) wrote that the two functions of speech
- communicative (social) and intellectual (thinking) — are commonly treated as

separate (parallel or independent) entities. In addition, he said this

shortchanges the dynamic character Of speech. In his view, the focus Of speech

studies should be the relationship between the two functions Of speech. Let me
provide an example to illustrate Vygotsky’s statements about the inherent
relationship between the two functions Of speech (communicative and
intellectual). A parallel example Of research on disagreement in elementary
mathematics would be an investigation of the relationship between disagreement
(e.g., what it is, how it arises, how it is managed) and its function in the
discussion (e. g., to converge mathematical thinking, to diverge mathematical
thinking).

What is the appropriate method to conduct studies of the relationship
between the intellectual and social functions of speech (or Of the relationship
between disagreement and its function in the discussion)? In the next section I
discuss the analytical framework Of this study. In addition I continue to build
the rationale for my study Of student disagreement.

Analytical Framework

Separately and in combination, I used sociocultural learning theory and
the method of sociolinguistics - both Of which emphasize the central role Of
language in making meaning - tO shed light on disagreement in the setting Of a
fourth-grade mathematics class. For my analytical framework I drew on the
method Of sociolinguistics, particularly the techniques Of discourse analysis.
Gumperz (1994) wrote that sociolinguistics is a method to investigate the
language use of particular human groups (p. 9). Similarly, Erickson (1986)
defined the method as grounded in the language use Of participants, and the

”immediate and local meaning Of actions, as defined from the actor’s point Of

10

view” (p. 119). As a method, sociolinguistics focuses on the relationship between
speakers and hearers (as well as between writers and readers).

Within the analytical framework Of sociolinguistics, discourse analysis is a
cluster Of techniques to analyze language use. These techniques are used to
investigate language use from the participants’ perspective. For example,
researchers Of classroom discourse strive to understand the perspective Of
teachers and students while they draw on the techniques Of discourse analysis to
investigate its various aspects, such as: speech structure, speech event, speech
acts, intonation, conversational analysis, topic, questions, discourse markers, or
participants’ use of pronouns. The goal is to use an interpretive lens to look for
patterns and relationships which emerge from the data to account for
participants’ perspective.

About research on language use in classrooms, Gee, Michaels, and
O’ Connor (1992) wrote that it is logical to begin with discourse in order to use it
as evidence Of a social and academic processes. Mehan (1979) would term these
processes as competencies — academic and social. A potential social and
academic process to investigate is disagreement in mathematics discussions.

TO participate in discussion and disagreement, children need to learn how
to be involved in it. Below I discuss participant structure, participation norms,
turn-taking patterns, and floor - all Of which must be fostered by the teacher in

order for the children in this study to be able to disagree.

Participant Structure

TO be involved in mathematical disagreement during discussions,

children must learn what to say and when. That is, they must appropriate the

11

participant structures Of the setting. Defined by Erickson and Shultz (1981) as
”differing configurations of concerted action,” participant structures inherently
include norms Of participation. These norms are continually negotiated by
participants. In short, participation norms are ”the rights and Obligations Of
participants with respect to who can say what, when, and tO whom" (Cazden,
1986, p. 437). In the fourth grade setting of mathematics discussions, the rights
and Obligations Of the children included knowing if they could speak about their
fractions ideas, when it was appropriate, and who they should address. TO
disagree, this knowledge is essential while children negotiated meaning-in-
action (Erickson, 1986). And to negotiate meaning-in-action, children needed to
appropriate participation norms, or patterns Of teacher-student interaction.
Teachers do not necessarily Openly teach participation norms, nor teach all Of
them. Some may be indexed (Mehan, 1979), that is, learned about implicitly
through teacher statements, such as, ”When someone is up here sharing their
ideas and we’re calling out or talking, that shows them we’re not interested in
their idea. That’ s not acceptable. Right now you’re supposed to be listening”
(Katherine Vandenberg, April 28).

As stated in the previous chapter, discourse Of the type advocated in
reform documents requires different roles and responsibilities of teachers and
students. With increased involvement in discourse as students directly address
one another, classroom discourse becomes conversation-like. In part it becomes
conversation-like because student-initiated turn taking becomes a participation
norm. According tO Sacks, Schegloff, and Jefferson (1978), turns may be initiated
in any Of three ways. First, an individual may self-select a turn to talk. By

speaking up during or near a juncture or pause in the talk, the person who

12

wishes to speak gains the floor. Second, the next speaker may be chosen by the
current speaker. For example, when I direct a question to a student, I might
conclude my turn with the question, 'What do you think, Amanda?" thereby
indicating specifically who I expect to speak next. Finally, the current speaker
may continue when the other speaker does not initiate a turn during or following
a pause after the speaking turn. Violation Of these three turn-initiation
procedures may result in simultaneous speech in the form Of interruption or
back-channeling (Murray, 1985).1

Once a speaking turn is initiated, according to Sacks, Schegloff, and
Jefferson (1978), the speaker is said to have speaker's rights until he or she
concludes the speaking turn. Whether for a one-syllable utterance (e.g., "Oh?",
"I-Imm") on the one hand, or a lengthy diatribe on the other, once initiated, a
speaker has the conversational right to complete his or her turn. When a
speaker's rights are violated some form Of repair is necessary for the conversation
to proceed. A common repair mechanism during simultaneous speech is when
one speaker stops talking so that the other may continue.

When a speaker has the floor, a turn may not go on endlessly.
Completion rights hinge on "how long 3 / he has been speaking, how Often s/ he
has spoken, the number Of 'points' made in a speaking turn, and the special
rights of some speakers to speak about some topics" (Murray, 1985, p. 31).

In summary, conversations are means by which individuals—adults and

children - communicate. Sacks, Schegloff, and Jefferson (1978) asserted that

 

lWhereas interruption as simultaneous speech is typically interpreted as a violation of a
speaker's rights (Coulthard, 1988; West 6: Zimmerman, 1983; West & Garcia, 1988),
back-Channeling is usually considered a hearer's cooperative, supportive comments to the
current speaker.

13

conversations are context-free speech events which involve interaction between
at least two people who typically acknowledge and honor rights to speak during
turns at talk. However, their assertion that conversations are context free has
been challenged (see for example, West & Garcia, 1988; West 8: Zimmerman,
1983). In addition, their assertion that turn taking is orderly, discussed below,
has been challenged (Edelsky, 1993).

Teachers and students in classrooms also follow rules for structuring
discourse. In what ways is classroom discourse similar to or different from
conversations in other settings? We know that school lessons involve discourse,
and that classroom discourse, similar to conversation, involves the interaction
between at least two people (e. g., teacher and student, student and student).
What other characteristics are shared by conversation and classroom discourse?

Classroom Discourse

Cazden (1986) described spoken language as "the medium by which much
teaching takes place and in which students demonstrate to teachers much of
what they know" (p. 432). As situated language used in a particular setting,
classroom discourse during teacher-led lessons in this country is most
characterized by the basic interactional pattern between teachers and students,
the Initiation-Response—Feedback (IRF) sequencez. Within the three-part IRF
sequence teachers initiate (I) questions, receive responses from students (R), and
follow up those responses (F). Directed by the teacher, the IRF pattern is the most

distinguishable feature that separates classroom discourse from conversation.

 

2Pimm (1987) also refers to this pattern as IRF; however, Cazden(1988) and Mehan (1979)
refer to this pattern as IRE (teacher initiation-student response-teacher evaluation). Whether
termed IRF or IRE, these authors refer to the same three-part interactional pattern between
teachers and students.

14

"All analyses of teacher-led classroom discourse find examples of this pattern,
and anyone hearing it recognizes it as classroom talk and not just informal
conversation" (Cazden, 1988, p. 30).

Whereas the IRF sequence may be used by teachers to control classroom
talk, it may be used alternatively in discussions (Wells, 1993). This three-part
sequence is not necessarily a three-turn sequence. In other words, when the goal
is increased involvement of children in discussion (as opposed to limited
involvement) a teacher may initiate the sequence with a question and seek
multiple responses through further questions. Or, she may seek multiple
responses by connecting the responses of several children. During the feedback
or follow-up part of the sequence, a teacher may use it to take ”an opportunity to
extend the student’s answer, to draw out its significance, or to make connections
with other parts of the students’ total experience during the unit” (p. 30).
Possibly, a teacher would use the follow-up for revoicing (Michaels 8: O’ Connor,
1993). During improvisational discourse, revoicing as a discourse move used by
teachers functions to (a) provide the teacher an opportunity to draw inference
from a student contribution, (b) provide the responding student the right to
validate that inference, and (c) position the responding student and his or her
contribution in relationship to others’ contributions.

When the IRF sequence breaks out of the three-turn sequencing pattern,
classroom discourse takes on new potential. Classroom discourse then has the
potential to become more give-and-take and conversation-like. It is not the same
as conversation, however, this breakout pattern allows for self-selection by

students of turns to get ideas on the floor.

15

In the end, classroom talk is focused on subject matter, for the (Often
debated) goal of schooling in the United States is to educate children in the. core
subject areas of mathematics, science, language arts, and social sciences. Since
the focus of school lessons is teaching and learning of subject matter, it is crucial
that students and teachers work toward bringing subject-specific knowledge into
the public arena. However, in order to demonstrate their subject-matter
knowledge during IRF sequences or at any other time during school lessons,
students must also learn what to say, when and how to say it, and to whom. In
other words, an overall classroom competence is required.

[C]lassroom competence is not limited to academic matters. . .classroom

competence involves matters of form as well as content. TO be successful

in the classroom, students not only must know the content of academic
subjects, they must learn the appropriate form in which to cast their
academic knowledge. That is, competent membership in the classroom
community involves employing interactional skills and abilities in the
display of academic knowledge. They must know with whom, when, and
where they can speak and act, and they must provide the speech and
behavior that are appropriate for a given classroom situation (Mehan,

1979, p. 133).

Mehan's (1979) notion of classroom competence consists of two critical
elements: academic competence and social competence. Academic or subject-
matter learning is the basic function of schooling and is what distinguishes
education from other settings; knowledge of social skills and language use is
required for students to succeed as competent members Of the classroom.
Competence in one area without accompanying competence in the other results
in students' failure to gain the floor for the purpose Of making contributions to
the flow of the lesson. That is, in order for students to get their academic

knowledge on the floor, they must have the social know-how to gain the floor.

Having gained the floor, their voices can be heard and knowledge made visible

l6

and public. Successful participation in lessons then appears to include both
aspects Of classroom competence: public display of subject-matter knowledge
and understanding Of the social means by which one's voice may be heard.

The above discussion of academic and social competence resonates with
the basis of sociocultural learning theory - the use of language to negotiate
meaning (Vygotsky, 1978). The two competencies also resonate with the earlier
quotation from Wertsch (1985). In that quotation he called for studies of the
relationship between the communicative and intellectual functions of speech. In
addition, the competencies are subsumed by sociolinguistics as a method to
investigate language use of participants in discourse (Erickson, 1986; Gumperz,
1994).

As stated above, in order to share one's academic knowledge, students
must first gain the floor. "Having a turn to talk is the minimal requisite for
influencing the course of a lesson. This involves getting the floor. However,
students cannot just talk any time" (Mehan, 1979, p. 140). A significant part of
students' classroom competence is knowing when an appropriate juncture, or "a
break (silence) between turns of talk" (Mehan, p. 140) emerges. It is at these
junctures that students tend to attempt to gain the floor.

At this point it is appropriate to ask how students gain the floor in order
to make their contributions to the classroom talk. Mehan (1979) included the
following as means by which students may gain the floor for a turn at talk:

1. Wait for a direct nomination to speak from the teacher. In this turn-
allocation procedure, the teacher calls on specific students to speak. The student
called upon therefore has the right to speak and complete his or her turn. For

example, consider the following:

17

KV: A half split? Austin, can you add to that some? Did you think it was
both?

Austin: Yea.
(Transcript, May 3).

During her speaking turn the teacher restated and attributed an idea
suggested in the previous turn by a student, and nominated Austin to provide
further comment. Cooperating with this school-related format, Austin waited
until he was called on to speak.

2. Resmnd to the teacher's invitation to bid. Frequently during lessons
teachers invite students to bid through the use of particular phrases or questions.
Whereas bidding typically takes the form of hand raising (which is more
acceptable), it occasionally includes student callouts (which tend to be perceived
as disruptive and less acceptable). For example, students may make a statement
that is followed by the teacher's question, "Nobody has an idea about 200 + 10?”
(Transcript, May 3) or "What do you guys think?" (Transcript, May 3). In the act
of asking these general questions wherein no particular student is addressed the
teacher opens the invitation for students to bid for the floor.

Students may also initiate speaking turns by bidding directly for the floor.
The disruptive student callouts mentioned above belong in this category. At
times these call outs are picked up (i.e., uptake, or an immediate and direct
response [Coulthard, 1988]) by the teacher or other students and the discussion
will continue. Occasionally such call outs are unacknowledged. However, there
are times when a speaking turn is underway and invitations to bid for the floor
are not Opened up by teachers. During these times, students may spontaneously

raise their hands (sometimes accompanied by vocal "Oh, Oh!" or "Ooh, ooh!") and

18

begin to call out a comment while the teacher or a fellow classmate is speaking,
thereby indicating that they have something to add to what has been said and
are willing to talk. However, in these student-initiated bids for the floor, basic
turn-taking rules are generally followed. That is, students tend to allow their
fellow classmates the opportunity and the right to complete their turn. Students
may border on interruption when they noisily raise their hands, lean forward,
and start to call out in hopes Of being called on. When they do not honor
speakers' rights they risk public recognition that their actions are disruptive to
the flow of the lesson, as in the following exchange:

KV: That’ s not appropriate to call out. Shhh. Boys and girls stop and

listen. When someone’s up here [at the overhead] sharing their ideas and

we’re calling out or talking to our table mates that shows them that we’re
not interested in their ideas. That’s not acceptable. Right now you’re
supposed to be taking notes on these ideas that we’re generating and
listening. I see almost half Of Table 2 hasn't written anything yet

(Transcript, April 28).

As she spoke, Mrs. Vandenberg appeared to honor students’ desire to
speak at the same time She stated her interest in the attempted contribution of the
child at the overhead. However, attempts in this situation to gain the floor were
determined to be impolite at a time when Mrs. Vandenberg considered it critical
to listen to others for the purpose of the ensuing discussion.

To summarize this section on classroom discourse, several characteristics
of classroom communication emerge. First, discourse in American classrooms is
typically led by the teacher who controls speaking turns for the most part.
Second, a common feature of classroom discourse is the IRF, or teacher initiation-

student response-teacher feedback, pattern of interaction. During these

sequences, teachers ask a question, seek a response from a student, and follow

19

the student's response with a brief (sometimes evaluative) comment. Third, in
order to share the requisite subject-matter knowledge during lessons, students
are required to be have two types of competence: academic and social.
Knowledge of what to say fails without requisite knowledge of who to say it to,
when, and in what way.

And, students are required to know when and how to get ideas on the
floor. What is ”floor,” and how do children get ideas on it? Typically referred to
and often discussed in studies of classroom discourse (see for example, Cazden,
1988; Mehan, 1979; Shultz, Florio, 8: Erickson, 1982), the concept of floor has
recently changed. Until recently, floor was considered to be ill-defined, at times
”defined as a speaker, a turn, and control over part of a conversation” (Edelsky,
1993, p. 205). This definition was confusing and not helpful in explaining the
meaning participants make of talk. Taking meaning-in-action of participants into
consideration, Edelsky defined floor as ”the acknowledged what’s-going-on within a
psychological time/space” (p. 209). Edelsky argued that floor is mutually
negotiated by participants in talk, and is not the discrete, orderly phenomenon
‘ discussed by others (Sacks, Schegloff, 8: Jefferson, 1978). Her collaborative
definition of floor differs from previous competitive models and captures the
dynamic and interactive nature of floor.

Edelsky (1993) also differentiated between floor and turn, previously and
loosely defined as one-at-a-time, orderly talk. By contrast, Edelsky wrote that
turn is not necessarily orderly. Nor is it necessarily discrete because frequently
in conversation speakers overlap one another so that the end Of a turn is
determined to be when a researcher decided, not necessarily when the speaker

did. Edelsky defined turn as ”on-record ’speaking’ (which may include nonverbal

20

activities) behind which lies an intention to convey a message that is both referential and
functional” (p. 207). Edelsky’s definition of floor and turn allow for the
simultaneous examination of turns and ideas on the floor. In other words,
researchers may examine those instances where turns may come and go, but
ideas remain on the floor through several turns.

When classroom discourse becomes give-and-take and conversation-like
and teachers foster norms Of participation which allow for children to talk
directly to one another, it is reasonable to assume that some of students’ talk will
be exploratory (Barnes, 1992). Exploratory speech is what the term implies,
speech whereby children test ideas and assumptions. In essence, children may
use exploratory speech to rehearse aloud their thinking, ”to rearrange their
thoughts during improvised talk” (p. 108).

What does it mean for classroom discourse to be student-centered when it
is grounded in subject matter? What challenges do teachers of mathematics face
when they have as their goal increased student involvement in discourse while
also fostering the norms Of participation discussed above? In the next section I
discuss recent research on teaching in the spirit of reform.

Mathematical Discourse

As discussed in Chapter One, reform documents favor discourse with
increased student involvement. ”Not commonly viewed as a discursive subject”
(Pimm, 1987, p. 47), mathematics advocated by authors of reform stands in
contrast to traditional school mathematics. Research on the complexities of
teaching in the spirit of reform sheds light on different roles and responsibilities

of teachers and students. Reform-oriented research details some of the

21

challenges faced by teachers and students when they experience the kind of
discourse envisioned by reformers.

Research on teaching in the spirit of the Stanggrgg (NCTM, 1989, 1991) has
highlighted the challenges in doing so (Ball, 1990, 1991, 1993; Ball 8: Wilson, 1996;
Lampert, 1986, 1987; Lampert, Rittenhouse, 8: Crumbaugh, 1996). Drawing on
their practice to engage in research on teaching and learning, the authors
simultaneously focused on teaching, subject matter, learning, and community.

In Ball’s words ”My ears and eyes must search the world around us, the
discipline of mathematics, and the world of the child with both mathematical
and child filters” (Ball, 1990, p. 27).

In addition, practical and moral dilemmas abound in this kind of teaching.
For example, posing the question, What’s all this talk about discourse?, Ball
(1991) underscored the point that the Standards (NCTM, 1989, 1991) cannot be
prescriptive. To the contrary, she wrote that the Standards are a guide, a set of
tools for use by teachers to construct a conception of good teaching. In addition,
the Standards are a springboard for productive conversations about teaching. To
Ball, ”the discourse of a classroom is formed by students and the teacher and the
tools with which they work” (p. 44). In this sense, discourse is situation specific,
hinging on the actions of participants. This formulation is highlighted by the
individuality by which discourse is orchestrated. In other words, there are no
rules that teachers may follow in order to conduct mathematical discussions.
Each teacher must construct an understanding of the subject matter Of
mathematics - and an understanding of the M. And, practice is laden
with decisions - often dilemmas -- about what to do and when. For example,

teachers are constantly considering what questions to ask and how to ask them,

22

what mathematical aspect of students’ contributions to focus on, what an
appropriate representation is, and what knowledge they have of students and
the diversity they represent (Ball 8: Wilson, 1996). And, for teachers, this means
thinking about teaching and learning as relational. Although Ball (1991) does not
explicitly address the notion of relationships among learners, the concept of
relational learning is implied in the following:

The classroom environment, or culture, that the students and teacher

construct affects the discourse in some important ways. The environment

shapes how safe students feel, whether and how they respect one another

and themselves, and the extent to which serious engagement in
mathematical thinking is the norm (p. 45).

New kinds of discursive relationships among students must be fostered so
that Children learn how to speak to one another about their mathematical ideas.
The goal is for heightened involvement in mathematical discourse to enhance
students' mathematical power (as proposed by the Standards [NCTM, 1989,
1991]). Burbules (1993) used the term ”dialogical relation” to describe a relation
between people in the context of discussion, a relation ”to ’carry away’ its
participants, to ’catch them up’ in an interaction that takes on a force and
direction of its own, often leading them beyond any intended goal to new and
unexpected insights” (p. 20).

Lampert and colleagues (1986, 1989, 1992; Lampert et. a1, 1996) also draw
on the teacher's perspective to research mathematical teaching and learning.
Similar to Ball (1990, 1991, 1993) Lampert’s research on mathematics teaching
and learning raises questions about the changing roles of teachers and students
in mathematical discourse of the kind advocated in reform documents.

Lampert’s research also captures the complex demands placed on teachers to

23

draw on mathematical knowledge and pedagogy in conjunction with the
accompanying need to constantly listen to and make sense of students’
developing mathematical ideas.

And, these authors wrote, heightened involvement Of children in
mathematical discourse sometimes brought disagreement. When children
disagreed, additional uncertainties emerged which could affect a child’s feeling
of competence (Ball 8: Wilson, 1996) or discomfort (Lampert, Rittenhouse, 8:
Crumbaugh, 1996).

In essence, the research by Ball (1990, 1991, 1993, 1996) and Lampert (1986,
1989, 1992, 1996) captures a cognitive perception or disposition (Bereiter, 1997)
toward teaching and learning which is at the core of teaching in the spirit of the
m (1989, 1991). As Lampert (1986) wrote, ”What seems most important
is that teachers and students together are disposed toward a particular way of
viewing and doing mathematics in the classroom” (p. 329). In a sense these
researchers draw attention to the fact that in this kind of teaching there is more to
mathematical discourse than mathematics.

While ”managing the intermental” (O’Connor, 1996), or
interpsychological domain of learning, these researchers point to areas of focus
for teachers who wish to teach in the spirit of reform. Regarding the role of the
teacher, it would seem that there is more required than subject matter
knowledge. Knowledge of children and how to manage discussions also figure

into the teaching equation.

24

Curriculum
Teachers who wish to align their practice with current reform need
curriculum resources. One resource for teaching fractions is fling Frgg'ogs
and was developed and piloted in California (Corwin, Russell, 8: Tierney, 1990).
The authors developed this resource around critical ideas in fractions:

‘A fraction can be used to describe all kinds of situations, including parts
of one thing, parts of a group of things, and rates.

‘Different fractions can express the same relationships.

*Estimation of the size of a fraction or the results Of an Operation with
fractions is vital to understanding fractions.

*In part-whole situations, equal fractions Of a whole are the same size;
when dividing something into halves or into fourths, the unit is divided
into equal pieces. However, those pieces need not look the same.

”The comparative order of fractions is absolute, but in part-whole
situations their size is relative to the size of their units.

'To understand a part-whole model, you have to keep in mind the whole,
even when the whole disappears and only the fraction is being used.

*In part-whole situations, fractions which are larger than unit fractions
(fractions with 1 in the numerator) are constructed by repeating the unit
fraction (e.g., 2/ 3 is 1/ 3 and 1/3).

*When fractions are used to express rates, the same rate describes a series
of ratios. (p. 24)

Citing the Curriculum and Evaluation Standards (NCTM, 1989), Corwin
et. al (1990) wrote that NCTM saw "five major goals for students as:

"learning to value mathematics

'becoming confident in one’s own ability

*becoming a mathematical problem solver

*learning to communicate mathematically

*learning to reason mathematically (p. 5).

The authors went on to discuss teaching for understanding. In their view
children should be able to construct mathematical models, teachers should not
tell, children (like mathematicians) should be able to talk and write about their

ideas, and frustration from children is to be expected (Corwin et. al, 1990, p. 6-7).

25

About the connection between the frustration Of mathematicians and the
frustration of students, they wrote,

Approaches [of adult mathematicians] to mathematical problems are

tried, explored, rejected, talked about, compared, and changed. The same

should be true of our students - they need to try things, to come up
against walls, and to try in a different way. . . It is hard to allow our
students to struggle with their frustration. . . Helping your students
make sense of their own methods, helping them express their confusion,
and helping them learn from each other’s approaches will give them much

more than telling them will do (p. 8).

Although Corwin et. al (1990) point to student frustration, they do not
provide teachers insight into what to do when it arises. Nor do they detail what
to do when children disagree. Corwin et. al provide teachers with the rationale
for teaching fractions through activities which are aligned with the vision of
reform. This rationale is discussed up front, apparently to lay the foundation for
the open-ended tasks that follow. The rationale and tasks in in Fra ' are
different from many curriculum resources in elementary mathematics. There is
no script Of what the teacher should say, followed by what students can be
expected to say. Nor is there a list of problems given at the end of a module (e.g.,
geometry, rates, mixed numbers). By contrast, Corwin et. al describe and discuss
activities with an eye on the mathematics (i.e., fractions content) in the activities.

Timely questions about mathematical discourse linger: What is
disagreement? What do teachers do when it arises? How do teachers manage

disagreement? I next discuss studies of argument and studies of disagreement in

mathematics classrooms.

26

Argument and Disagreement

As I wrote in the last chapter, this study of disagreement may also be
thought of in the broadest sense as one of argument. For within argument,
which "refers broadly and neutrally to reasoning in order to make decisions or
test the truth of claims" (Perkins, p. 155), is subsumed such things as explore,
clarify, justify, explain, defend, and disagree - characteristics of mathematical
power advocated by the authors Of reform documents. In the adult world, an
argument is a chain of reasoning on which decisions are made or the truth of
claims are tested (Perkins, 1986). Toulmin (1958) wrote that argument is logic and
has an identifiable structure:

Argument is like an organism. It has both a gross, anatomical structure

and a finer, as-it-were physiological one. When set out explicitly in all its

detail, it may occupy a number of printed pages or take perhaps a quarter

of an hour to deliver; and within this time or space one can distinguish the

main phases marking the progress of the argument from the initial
statement of an unsettled problem to the final presentation of a conclusion

(p. 94).

In his book, The Uses of Argument, Toulmin (1958) primarily discussed
argument as a final product (as Opposed to a process). For example, he detailed
the many components of an argument and their inter-relatedness. These
components of argument include assertion, claim, warrants, backing, data,
qualifier, and so on. Purportedly, he detailed the components of argument to
provide a context in which scholars could concentrate on the finer points of
argument. This concentration, he believed, would nudge logicians toward a
tighter fit between the finer points of argument and their ”gross, anatomical

structure.” Although readers may infer that it is people who make arguments,

27

person-to-person interaction over the development of arguments is essentially
absent from Toulmin’s description.

Perkins (1986) described argument as design — ”a structure shaped to a
purpose” (p. 156). This structure can be as simple as the interaction between
children, as in the following:

Why? Because.

Perkins (1986) went on to say that the purpose of argument is to specify
”why?" Perkins also implied a critical aspect of argument in connection with
this study: the messiness Of argument-in-action. To compare the formal logic of
mathematics with informal arguments (i.e., akin to disagreement between
children) Perkins wrote:

In the formal logic Of mathematics. . . the truth Of the premises is a

matter Of assumption, but in informal argument about real-world matters,

the truth of premises is bound to be in question from time to time. . .

[I]nformal arguments are messy and uncertain in ways that formal

arguments are not (p. 163-164).

Above, Perkins (1986) highlighted the exploratory direction in which
disagreement may take children. And, he positioned the idea of argument-in-
action in the real world, a world which includes children using language to
explore mathematical ideas.

What does it mean for children to argue or disagree (the child version of
argue) in mathematics discussions? Wilkinson and Martino (1993) examined
students’ disagreements in one small group during mathematics. In their work
they based their definition of disagreement episodes on the work of Lindow,

Wilkinson, and Peterson (1984). In the view of Wilkinson and Martino,

disagreement episodes were ”the interaction that followed a verbal assertion of

28

not agreeing with an answer or a step in arriving at an answer to a mathematics
problem” (p. 144). Using a coding system, the authors reviewed transcripts and
video tapes to identify and isolate disagreements. In addition, they defined
disagreement as a three-part process: (1) antecedent position, (2) disagreement,
and (3) resolution moves. Resolution moves included consensus and lack of
consensus. Consensus was indicated by verbal acceptance of a previous
statement. Lack Of consensus was indicated either verbally or nonverbally.

Disagreement was coded as ended ”when consensus was reached or when
the interaction shifted to a new topic” (p. 145). The coded data was triangulated
with additional sources, including results from the mathematics portion of a
standardized test and the results of a sociometric interview. Results showed that
two of the four children in the study acted as catalysts in disagreement. That is,
they ”freely initiated and participated in disagreements as well as group
discussions” (p. 159). Both students also received high sociometric scores from
their classmates. They were perceived as leaders, and their answers and
solutions were attended to by other members of the small group.

In addition, Wilkinson and Martino (1993) found that one particular
problem (i.e., the cube problem) produced more and better mathematical
communication. The cube problem was designed by a research team led by
Karen Fuson (1988) and by Robert Davis and Carolyn Maher and staff. The task
required these first graders to examine several groupings of blocks - a single
small block, a stack of 4 blocks, a wall Of blocks that was 4 x 4 x 1, and a 4 x 4
cube — to respond to the question, How many of the small blocks will we need to make
a big block? Make a drawing of your answer. The authors found that the cube

problem produced more and better responses for three reasons. First, there were

29

less language demands on the children than with other problems. Second, the
children were required to build a physical object (a familiar activity to first
graders). Third, the time taken to build the model in order to answer the
question may have allowed for increased reflection time. No other problem
stimulated the amount or extent of discussion that resulted from the cube
problem.

The work of Cobb and Yackel ( 1994) took a different tack than that of
Wilkinson and Martino (1993). Cobb and Yackel studied the development by
second graders of mathematical argumentation. Yackel and Cobb (1994)
grounded their study in current reform with its renewed emphasis on
mathematical reasoning. In addition, their study drew on the work of
mathematicians (i.e., Davis and Hersh, Lakatos, Balacheff) who have examined
and written about the nature and purpose of proof.

The study had a dual basis. First, teachers with whom they worked
fostered an inquiry tradition which included social norms such as the
development by students Of meaningful solutions, explaining, listening, asking
questions, challenging explanations, justifying interpretations when challenged,
and attempting to reach consensus on answers and solution methods. Second,
these teachers fostered sociomathematical norms, or normative understandings -
a shared understanding Of what to say, when, and to whom (p. 6). Cobb and
Yackel wrote that sociomathematical norms are specific to children’s
mathematical activity and regulate mathematical argumentation.

Yackel and Cobb (1994) found that when they analyzed arguments several
functions of arguments emerged. For example, some arguments functioned to

force children to specify how they interpreted a problem or a solution method.

30

Other arguments functioned to convince others of an error (e. g., computation).
Still other arguments functioned to nudge children’s thinking so that they made
generalizations about thinking strategies.

Lampert, Rittenhouse, and Crumbaugh (1996) discussed the challenges
faced by students in disagreement. In their description of discussion in a fifth-
grade classroom (whole group and small group), a more detailed perspective of
frustration than described above by Corwin et. a1 is revealed:

Experience of academic argument as an amicable mode of interaction in

our culture is rare. . . The school classroom is a place where friends are

made and lost, where identity is developed, where pride and shame and
caring and hurting happen to kids. What they learn from social
interaction cannot be described simply in terms of the mathematics

covered by hashing out logical conflicts between various approaches to a

problem. Mustering evidence to prove that an assertion is right or wrong

is not a decontextualized learning activity. In the classroom, mathematical
argument is done with and to the same people one plays with, eats lunch

with, lives next door to, or has a crush on (Lampert, Rittenhouse, 8:

Crumbaugh, 1996, p. 758, 759).

The work by Lampert et. al (1996) goes to the heart of the challenge faced
by teachers when disagreements arise: How do teachers manage the
mathematics, the discourse, and disagreement while fostering participation
norms intended to support increased mathematical power? And, how do they
do all this so that children feel competent and respected (Ball 8: Wilson, 1996)?

These are timely questions, questions as efforts to reform mathematics
teaching and learning increase. They are also timely questions because they call
to mind the usefulness and power in using sociocultural learning theory and
sociolinguistics to increase understanding of the complexities of mathematical

discourse from the child’s perspective. In the words Of Lampert et. al (1996),

Talking about communication in conjunction with understanding is a
radical idea. Understanding used to be thought of as a function Of

31

individual minds, and teaching and learning as transactions between the

teacher and individual learners, even when there are 30 of them in the

room at one time. But school people are beginning to take into account
the social construction of knowledge, the relations between thought and

language, and the importance of collaboration to real problem solving (p.

759-760).

Lampert et. a1 (1996) recognized that studies Of individual understanding
risk denying the social construction of students’ understandings. Studies which
examine the relationship between social and intellectual aspects of mathematical
disagreement are warranted. From a sociocultural learning perspective, studies
of language use in learning shed light on students’ mathematical sense making.
And sociolinguistics provides the analytical tools with which to get inside the
meaning children make of mathematics content.

Summary and Conclusion

This study stands at the intersection Of sociocultural theory,
sociolinguistics, and mathematical discourse in the context of reform — all of
which appreciate language use. This is an exploratory and interpretive
examination of student disagreements during whole-group discussion in a
fourth-grade classroom. Recent studies of mathematical discourse have as their
basis an appreciation of its complexity, as discussed above. In part, classroom
discourse is so complex because it is unique to each setting. That is to say,
classroom discourse (and disagreement) is constitutive of participants (teacher
and student) and subject matter. It is also constitutive Of the meaning the
participants construct in situ. It stands to reason, then, that the study of student

disagreement would incorporate sociocultural learning theory as a theoretical

framework and sociolinguistics as the methodological orientation. It is through

32

this dual lens that my study of students' disagreements in a fourth-grade

mathematics class developed.

33

CHAPTER THREE
METHOD
With this chapter I share the ways in which this study evolved through

my work with Katherine Vandenberg, an elementary teacher in a Professional
Development School (PDS). In addition, I describe method and analyses in
conjunction with this qualitative study Of student disagreement in a fourth-grade
mathematics classroom. I spent a year in Mrs. Vandenberg's classroom as part of
a larger study, and from this experience questions developed which led to my
study Of student disagreement during mathematics. I first review the work of
the larger study. I then describe the evolution of this research, describing the
site, participants, data modes of Observation, field notes, interviews of the
teacher and randomly selected students, uses of video and audio tape, and the
information derived from spontaneous interactions with Mrs. Vandenberg and
these students.

In May, 1992, I began an assistantship at Edwards Elementary School, a
PDS in a suburban community. My assistantship involved work with
elementary teachers dedicated to thinking about and transforming their teaching
of mathematics. In this collaborative research project, these teachers, Dr. Pamela
Schram, a university faculty member, another graduate student, and I
constituted the Mathematics Study Group Project, wherein we met to discuss,
reflect on, and research the complexity of mathematics teaching and learning.
My research interest in discourse during elementary mathematics classes
mapped well onto the interest in discourse of these teachers. An August, 1992,
interview with two Of the teachers, Pat Smith and Corrine Fox, confirmed that,

indeed, this would be a good match.

In September of that year, I participated in the second weekly meeting Of
Edwards Elementary School's Math Study Group. As a PDS project group, we
set about reiterating our anticipated goals. The teachers, Corrine Fox, Pat Smith,
Felice Wilson, and Katherine Vandenberg discussed and elaborated their plans
for the school year. As each teacher shared her vision, our work for the year
began to unfold. For example, Corrine, a special education teacher, described
her interest in thinking over the year around the theme, What's the point? In
other words, Corrine wanted to reflect on, research, and write about why
allowing students to discuss meaningful mathematical tasks was important to
her. Pat, a first-grade teacher, discussed her interest in thinking about how
students' contributions to the discourse facilitated ongoing assessment. Felice
confirmed her interest, similar to Pat's, in thinking more and writing about
assessment and the ways in which she thought about it. Finally, Mrs.
Vandenberg spoke about her interest in the relationship between discourse and
task in her classroom. More specifically, Mrs. Vandenberg expressed her desire
to write about the ways in which the task affects the discourse. At that moment,
I knew I was in partnership with a group of educators unique in that they
investigated particular questions about their practice - questions arising from
their experiences. Mrs. Vandenberg's focus, especially, seemed to parallel my
interest in unpacking what goes on during mathematics classes when students
are allowed and encouraged to discuss their ideas.

As my work with the Math Study Group moved forward, Mrs.
Vandenberg and I found ourselves sharing a common interest in students'

contributions to the discourse, both spoken and written. Whereas Mrs.

35

Vandenberg asked herself how she could get everyone involved in discussions, I
began asking, What was going on in these discussions, and who was involved?

As I began observations in late September, sitting in the rocking chair at
the rear, right corner of the room, I noted that there were more girls than boys in
the class, 17 girls and 9 boys. One girl was Hispanic, and the rest Of the students
were white.3 I learned that three children received Special Education services,
and noted that three left during math time for Chapter I sessions.

SO that I could gain some understanding Of Mrs. Vandenberg's teaching
background and perspectives on learning, we targeted an hour to meet when
possible during lunch or Math Study Group's scheduled weekly times. During
these Fall sessions, I developed an increased appreciation of Mrs. Vandenberg's
dedication to reflection on her own teaching with the goal of increasing students'
mathematical understandings while enhancing her teaching of the subject. We
discussed math classes I observed and my subsequent field notes, individual
students who were and were not involved in the discussions, mathematical
content and tasks, readings, university courses, and the complexity of teaching
this way. One particular class from October 1 fascinated us both and prompted
ongoing conversation for the rest of the school year.

On this day in October, the topic was Odd and even numbers. Students
were to discuss numbers they generated and tested the day before, with the goal
of deciding if their numbers were Odd, even, or unknown (undecided). Mrs.
Vandenberg and I Observed during the previous class discussion that two

students, Tiffany and Deidra, disagreed about whether 70 was Odd or even.

 

3In January, another white girl joined the class; in April, an African American girl moved
into the district and also joined the class.

36

Deidra said 70 was even and demonstrated that it divided into two numbers that
were the same, 35 and 35. Tiffany argued that 70 was odd, because when you
add 35 and 35, 5 and 5 is 10; carry the 1. One plus 3 plus 3 is seven. Seven's an
Odd number so seventy is odd. In spite of subsequent and lengthy class
discussion, Tiffany maintained an unorthodox stance that was to vex her for
weeks: her belief that 70 was an odd number and that nobody else believed her
but she thought 70 was an Odd number. So determined was Tiffany to convince
others of her belief, she asked Mrs. Vandenberg, me, her parents, and classmates
if we thought 70 was odd.

Mrs. Vandenberg and I discussed this exchange and found ourselves
focusing on the phenomenon of student disagreement and interested in using
video tape as a means to capture teaching and learning. Thinking about this
lesson raised questions for me especially. I asked: How do students disagree?
What did Tiffany do to let her ideas be known? Why wasn't Tiffany‘ convinced
that 70 was even? In the end, Mrs. Vandenberg and I wondered about the larger
question: What is disagreement in this math class? In pursuit of data around
these questions, I began video taping each lesson I observed, on the condition
that as soon as it finished, I turned the tape over to Mrs. Vandenberg for her
review. In the beginning, this was my primary reason for video taping: for Mrs.
Vandenberg's use. As time passed, however, we both reviewed the video and it
generated further questions and discussions about teaching and learning of

mathematics.

 

‘Unfortunately, in early November, Tiffany moved to another community.

37

Site, Setting, and Participants

This study took place in an elementary school on the eastern fringes of
Hart, a suburban community south of a large, Midwestern university. Edwards
Elementary School is part of the second largest school district in the area and is
located 10 miles from the state capital. Hart was originally a farming
community, but is rapidly becoming a suburb of the capital. It is primarily a
residential community with little industry and few small businesses. A
predominantly white community of approximately 11,750, Hart has a diverse
socioeconomic population ranging from working class to middle income. At the
time of the study, Edwards's student population of nearly 500 reflected the
socioeconomic diversity of the Hart community. It also reflected the diversity of
contemporary family forms: 54% Of its students lived with other than their
biological parents. The student population included a large percentage who are
considered at risk, as well as children who are average to above average
students. Of Edwards' students, 18% received free or reduced lunch, and 6%
received Special Education services.

When I joined the Math Study Group, Edwards was in its third year as a
Professional Development School. It was in this school that Mrs. Vandenberg
was in her second year Of teaching. Whereas during the 1991-92 school year she
taught second grade at Edwards, in 1992-93 she moved to fourth grade. In this,
her first year as a fourth grade teacher, I met her in her room after school of the
first full day. As members of Math Study Group, our work together began. Mrs.
Vandenberg did not use the district’s fourth-grade math text, in part because her

goal for the school year was to examine the relationship between math task and

discourse. Instead, she opted to draw from multiple resources, primarily
Marilyn Burns publications and TERC resources. For the geometry portion of
the Fractions unit, she used Seeing Fragtiom (Corwin, Russell, and Tierney,
1990). This book was developed by TERC for the California Department of
Education. In addition, she regularly conferred about resources, tasks, and
specific lessons with Math Study Group colleagues, especially Pamela Schram,
Project Director (who also taught in the room once a week).

In addition to Mrs. Vandenberg, 25 of 26 children in her class were also
participants in this study.5 Consent was not received for one of the boys.
Therefore, although he was in two brief disagreements, his involvement is not
discussed beyond the overall count of students in disagreement.

M

In this qualitative study, I examined the ways in which students in a
regular fourth-grade class disagree during mathematics. I strove to make sense
of student disagreement by drawing on existing Mathematics Study Group
(MSG) Project data collected during 1992-1993. To increase understanding of
these disagreements, a focal question and several sub-questions were developed
to focus the study and analysis. The focal research question was:

D. What is disagreement in this fourth-grade class?
Sub-questions included:
E. What do these students need to learn to be involved in
disagreements?

F. What are the roles of this teacher and these students during

 

5 Because this study was part of the larger Mathematics Study Group Project, parental
and teacher consent for participation were received for that project.

39

disagreements?
G. What is the relationship between disagreements and mathematical
tasks?

Drawing on the method of participant Observation (Erickson, 1986) for the
larger MSG study, I gathered data from September, 1992, to July, 1993, from a
fourth-grade class taught by Mrs. Vandenberg, the teacher assigned to this class.
In Mrs. Vandenberg's classroom I Observed 36 mathematics discussions and
made video and audio tapes of 35 lessons (see Appendix A). Initially, we did not
plan to video tape; however, after my first day Of observation, I asked Mrs.
Vandenberg if she would like to have video of what the children were saying. I
suggested that I would tape lessons, and immediately remove the tape from the
camera for her to take home that day. She agreed, and soon she asked me to
watch these videos, also, so that I could reflect on what the children were saying.
Initially, I recorded video for Mrs. Vandenberg, and wrote field notes for the
MSG Project.

My first goal was to help the children get comfortable with me, so Mrs.
Vandenberg and I agreed that she would introduce me to the class. We also
agreed that she would tell them that I was going to be in the room each week
during math because I was interested in their math ideas. At the beginning of
MSG data collection, Mrs. Vandenberg introduced me to the students and told
the children that we were interested in how boys and girls learned math, and
that I would be in the room each week to tape their math discussions. On each
observation day, I interacted informally with the fourth graders before and after
math time. In addition, when the children worked in small groups, I circulated

around the room to observe their interactions and math journal writings.

4O

When Mrs. Vandenberg agreed to video taping, I asked her if she would
take a few minutes on my first recording day to introduce the camera, which she
did. On that day when I first set up the video recorder, I also had the children
wave to the camera and say hello. On subsequent days, the kids Often
volunteered to help me set up or take down the camera, or asked me to join their
group. Once, when Mrs. Vandenberg was called to the door, I continued class
discussion for a few minutes until she returned (which, in the moment, indicated
to us both how integrated I was into the setting). Throughout data collection, the
children appeared comfortable in my presence, often sharing their journal entries
with me and sometimes asking me questions (e.g., DO YOU think 70 is an odd or
even number?)

As part of the larger MSG Project data, Pamela Schram, Project Director,
and I decided that before and after the fractions unit I would interview a group
Of randomly selected students, in order to assess their learning Of fractions.
Seven students were interviewed, about their fractions learning, and about their
ideas on disagreement. I also spontaneously interviewed four additional
students. We (members of MSG) intended these interviews to draw out kids'
developing math thinking about fractions concepts. In the pre-fractions
interview, I also asked open-ended questions about disagreement. For the
purposes Of my dissertation study, then, the data included existing MSG Project
video, field notes, and student journals. To triangulate the data, I examined
student interviews, specifically the section on disagreement. I also conducted
two formal interviews with Mrs. Vandenberg. Although we attempted to
schedule these interviews during the fractions unit, we were limited by time

constraints and end-of-year school demands. The focus of these interviews was

41

two-fold: to gain a sense of Mrs. Vandenberg’s professional development as a
teacher of elementary mathematics and to learn about her beliefs about
disagreement during mathematics discussions. In addition, Mrs. Vandenberg
and I typically met briefly at the end of each math lesson to discuss mathematical
tasks, issues raised by students, and student participation in discussions.

Early on, when observing in Mrs. Vandenberg's room for the MSG study, I
found myself in a researcher dilemma: how to take more or less extensive field
notes while running the video camera. Because I Observed alone, there was no
one to assist with one or the other. When I recorded field notes, I had to avert
my eyes from watching classroom activity to the pages of notes. In my field
notes I attempted to write as much detail as possible. For example, on October 1,
a day early in data collection when I was learning children's names, at the

beginning of math time I recorded the following details:

12:30

(Sts return to the room.)

12:35

(I sit to write after the camera is introduced.)

KV: Boys and girls, I have something to tell you. Shhh. Right now the
video has been running. . . Say hi to Mr. Vandenberg.

sts: Hi Mr. Vandenberg

KV: I really like the way Rob is ready to go. Can anybody tell, yesterday
Tonya's and Moira's conjecture? Think back to what we were testing.
(pause-no response from students) Think back to what Moira and Tonya
proposed in class with the 10 numbers. What were we trying to find out.
(pause) James, do you remember?

James: NO.
Micah: I have a headache.
KV: You have a headache? I'm sorry to hear that Micah.

42

(Micah says something.)

KV: Even odd numbers. (to all) How do you test numbers to find out if
they're even or odd? Suzanne?

(Suzanne responds.)

KV: Suzanne said we used Unifix cubes or blocks and we divided them
up and you were to record in your journal what you got. As a whole
group I would like to give you the Opportunity to share what your results
were. I have 2 charts. (moves to chalkboard in front of room, where 3
large posters are taped up.)

Steve, would you predict what this (indicates the chart labeled, even.)
chart is for?

(Steve explains that it is for even numbers.)

KV: What do you think will be in these 2 parts (indicates the columns she
has drawn.) What these are for.

(pause. NO student response)

{Do they not understand the directions; or are they not yet attentive?)
Think back to yesterday and what you did to test it.

(DO they wonder what the it is?}

(KV calls on a female st to explain. This female responds and explains.
KV seeks clarification and understanding from students regarding what
they should do with the chart. She calls on a male student; then Kim.
Both students explain what they think they are to do.)

I also have an unknown sheet. What would go there?

Male st in back speaks quietly. KV called on the 2 students in front of
room: You don't know if it's even or odd.)

(KV then calls on a student in the back, who I think is the same one who
spoke up. He says it means I'm not sure.)

KV: Questions about what the chart is about? {Again, no response. Are
students unsure Of directions? Need more connection with yesterday's
work dividing up numbers? Were they unsure Of yesterday's task? I
wonder. . .}

(Students are instructed to record their numbers publicly and share.)
KV: Who would like to share?
(4 females and 2 males raise their hands.)

Shawna? Shawna's going to tell us about the number she tested, the
groups, and record it on the chart she thinks it fits into. {Were you

43

repeating instructions because you sensed some confusion on students
part, or because you sensed this just was a day of sluggish thinking?
Another reason?)

DO you have a number in mind, Shawna?
(Shawna goes to chart.)

Shawna: I tell first, right?

KV: What was your number?

Shawna: I tested 80. I thought of 8, took away 4. . .(continues to explain
her thinking about dividing up 80. Writes 40 + 40 = 80 in the first column
on the even chart. This column was labeled, Number tested.)

KV: She wrote 40 + 40 = 80. What was the number she tested?

sts: 80.

KV: And when she divided into groups what did she get?

sts: 40 and 40.

KV: Could we also write this (points to the 80) in the first column?

In my field notes, I listened closely, to record as much as I could of Mrs.
Vandenberg's questions and her revoicing Of kids' comments, and children's
contributions, whether about the math topic on the floor or something else (such
as Micah's headache).

The fact that I could neither take adequate field notes nor record useful
video if I continued to try to do both made me uneasy because I believe that
much is learned from attending to the verbal and nonverbal communication
during classroom discourse. And it was not acceptable to me to fluctuate
between intense Observing and recording, fearing that while recording I would
miss important goings-on. So much of classroom discourse is generative,
building on what took place before, that I puzzled over an alternative that
captured as much as possible, while recognizing that no method could get it all.

Because I felt that the in situ Observing-recording-Observing cycle had the

44

somewhat limited potential to only get pieces of learning activity and lacked the
continuity I wanted, I chose to capitalize on video technology to enhance my
researcher eyes and mind. If something happened to catch my attention, I
moved the camera toward it. Whenever I swept the room with my eyes,
checking as a teacher does for things like students' level of attention to the task
and activity or movement (e.g., a head lain down, a hand raised, a giggle, a walk
to the pencil sharpener, a poke at a table mate, a grab for a math notebook), I also
visually swept the room with the camera. In addition, as much as possible given
the demands of my graduate assistantship and studies, I wrote field notes
following observations. In this way, I believe I recorded much Of the complex
and emergent nature Of the discourse. In the end, I found the camera work
valuable in subsequent analyses.

My analyses Of student disagreement drew from data collected during 10
lessons from April 28 to May 19, during a unit on fractions. Data collection
proceeded with audio and video recordings of whole- and small-group
discussion. In addition, data was collected from student mathematics journals,
student and teacher interviews, Observation, and field notes. A1513 and mm
recordings of whole- and small-group discussions provided a view and voice of
participants. Student journals gave me an additional discursive window into
student involvement in disagreements. Interviews allowed extended one-to-one
interaction with participants which frequently led to insight into their
perspectives of involvement in discussion and disagreement. My W
and figlg goteg rounded out my data collection as a participant-observer, as I

A documented the general setting in which mathematical disagreements emerged.

45

Analysis

This was an exploratory study of disagreement during mathematics
classes in one setting, and drew on sociolinguistic data analysis using the
techniques of discourse analysis. The lessons analyzed are from a unit on
fractions which began on April 28 and ended on June 11, 1993, with data from
April 28 and May 19. Specifically, data is drawn from the following Observation
dates: April 28, and May 3, 5, 6, 7, 10, 11, 12, 13, and 19. It should be pointed out
that analyses are primarily of whole-group discussion, because the majority of
discussions were in whole. To triangulate analyses, I also examined student
math journals and interview data. I collected and photocopied math journals
during the last week of school. I completed all interviews between April 22 and
June 9.

The purpose of analysis was to locate and examine disagreement. To
address the research questions, my analyses began in Observation, for out of each
observation, a pattern developed. First, before I entered the classroom for each
observation, I had to consider what had transpired up to that point in
combination with what Mrs. Vandenberg wanted me to focus on. In this sense,
the ongoing evolution of classroom discourse and student’s involvement were
always under analysis. Second, after each observation Mrs. Vandenberg and I
usually discussed the math, expanding our sense of detail of students’ thinking.
Third, over time, multiple waves Of analysis followed, based on the work of
Shultz, Florio, and Erickson (1982). These analyses involved repeated viewing of
mathematics lessons supported by field notes, student interviews, teacher

interviews, and students' work recorded in their mathematics journals. First, I

46

searched for lessons with a high level of disagreement using indicators of student
involvement such as increased pace, volume, and overlapping of student speech.
By contrast, low-disagreement analyses examined pauses in speech and speaking
turns, and slower pace, quiet voice, and lack of overlapping student speech.
Second, I examined the mathematics in and around the disagreement; that is,
which fractions tasks did or did not promote discourse and by association,
disagreement? Additionally, I looked for norms of participation in disagreement
to determine whether Mrs. Vandenberg prioritized consensus.

This analysis helped me unpack the notion of disagreement, when and
how it arose, the roles of the participants, the ways in which students used talk
and writing during disagreements, and what students needed to learn to be
involved in disagreements.

To that end, analyses began in the setting during observation. As
disagreement emerged in a class session (and when time permitted), Mrs.
Vandenberg and I would confer while I took down the video camera, for the
most part to begin thinking about why disagreement happened. The majority of
my analyses, however, took place later, primarily through repeated viewing of
video tape.

To locate disagreement in the data, I viewed all lessons non-stop from
beginning to end, in order to develop a disagreement rating (see Appendix B).
This rating - either Sigh, medium or lgwd but especially focusing on high and

 

low - led me to those lessons most likely to have disagreement. First, I searched
for lessons with a high level of disagreement using indicators of student
involvement such as increased pace, volume, and overlapping speech. By

contrast, low-disagreement analyses examined pauses within and between and

47

speaking turns, slower pace Of speaking, quiet voice, lack of overlapping speech,
and silence.

During this wave of analysis, two lessons initially stood out due to the
body language of the children: May 3 and May 7. On these two days, a Monday
and Friday, the children seemed more physically active than on other days.
Whether at the overhead or at their tables, their movement caught my eye. In
order to examine these disagreements more closely and determine what the
activity was about, I completely transcribed the video tapes from May 3 and May
7. These transcripts included as much detail as possible about the children’s use
of nonverbal communication, so that I could draw on them for analyses. On
further examination, I located in each lesson (May 3 and May 7) an episode of
disagreement (the focus Of Chapter Six). Furthermore, to determine the extent of
additional disagreement, I transcribed the remaining eight lessons, although not
at the level of detail as the transcriptions of May 3 and 7. Further analyses
revealed an additional 25 shorter disagreements, which ranged from two to nine
turns (the focus of Chapter Five). In total, I located 25 single disagreements and
2 episodes of disagreement.

Once a disagreement was located, I noted what preceded and followed it,
including what Mrs. Vandenberg said or did immediately before and after
disagreement. I then isolated the disagreement to study it rnicroanalytically.
Then I replaced the disagreement to examine in its immediate context. After I
had reconsidered both the disagreement and its context as a unit - how it arose,
what happened during disagreement, and what followed - I repeated the
process with consecutive disagreements. This process was repeated innumerable

times. In addition, I looked across disagreements to detect patterns among them.

48

For example, when I located a brief disagreement early in math time on May 3, I
extracted it, along with its immediately surrounding talk and noted that it was
connected to the overhead question, What is a fraction? Then, I described it as

follows:

When whole-group discussion was underway, Monte said what a fraction
was ’A fraction is, where you have a big number and you can't solve it so
you split it up into groups so it's easier to solve. . . A fraction is a piece of
the problem.’ While he talked he wrote for his example, ’200 + 10’ and
drew several ’groups’ Of dots to represent his idea. Soon, Mrs.
Vandenberg called on Hannah, who said ’200 + 10 is not a fraction,’ and
began to explain that a fraction is less than one. During the course of her
explanation, the following two-turn disagreement between Hannah and
Monte emerged:

KV: [200 + 10] Is NOT a fraction. But what you showed us is a fraction.
Hannah: It's not a, it's NOT a fraction, because-

Monte: But it CAN be a fraction.

Hannaln because a FRACTION is less than ONE. TEN and TWO
HUNDRED are not less than one. HERE is another kind Of fraction.
There are other kinds of fractions. THIS would be, one-fourth, if you
shaded in one.

KV: OK.

Hannah: And,

KV: How did you know it would be one-fourth?

Whereas Hannah was explaining her position on what a fraction was,
Monte spoke up to re-state his different idea.

In addition, I used a hypermedia data base to analyze data. My goal was
to determine if patterns emerging from analyses existed in another setting.
Use of Hypermedia in Analysis
One layer of analysis perhaps unique to this study was my use of the
M.A.T.H. Project's (Mathematics and Teaching Through Hypermedia, Deborah
Ball and Magdalene Lampert, Co-Principal Investigators) data base to validate
teacher moves that emerged during earlier analysis. This analysis took place

during my involvement with a study group for teacher educators, Teacher

49

Educator Study Group (TESG).‘ Before TESG began, I decided that I would use
my investigation there to play out dissertation questions and hunches. Drawn to
students’ disagreements, I was intrigued when at our first meeting, Ruth Heaton,
the group’s leader, played a June 1990, discussion (and disagreement) around

4/ 4 and 5/5. When I Observed this video I initiated an analysis of disagreement
on June 7 (and others in the data base) and those from my dissertation work
Here, on June 7, the third graders appeared to be disagreeing with one another,
addressing one another's ideas, and, I think, using their own words to try to
express in my mind important fractions thinking.

For example, I remember at the October 5 session Of TESG Observing
video and recording in my journal notes about the June 7 disagreement. In this
disagreement, Lin argued that when comparing a cookie cut into fourths and a
cookie cut into fifths, the shape of the cookie matters; that the shape of the
cookies should be the same. In essence, Lin it's not the pieces or how much you
shade in, it's that the cookies are the same shape. Mathematically, her argument
was about the necessity of equivalence of shapes when comparing wholes under
comparison. By contrast, Christina, who argued that 5/5 is more, appeared to
have a surface view of the situation, by taking the stance that because Sis more
than 4, 5/5 is more than 4 / 4. After all, she said, there are more pieces in 5/5.

She did not seem to see the idea that Lin tried to articulate and illustrate.

 

6 The Teacher Educator Study Group was initiated in September, 1994, and met
approximately two times per month to discuss issues of teacher education. Using the M.A.T.H.
Project's hypermedia environments as a shared context, we regularly discussed mathematics
teaching and learning in a third-grade classroom, and the investigations into mathematics
teaching and learning by university preservice teachers. In addition, each TESG member
developed an investigation Of third-graders' and preservice teachers' learning. We also regularly
discussed these investigations (which we created in the form of an electronic notebook).

50

So, on October 5 at TESG I began by looking for disagreements early in the
year, to see if there was evidence of how the teacher established the norms of
participation in disagreement. I did not find a place where Ball, the third grade
teacher, specifically said something akin to, ”When people disagree with one
another you are to do X, Y, and Z.” I did find implicit ways in which Ball was
nurturing the norms of disagreement. For example, I noted that at the beginning
of the year on September 11, when students disagreed over whether Christina
was 9 or 10 based on the year or year and date Of her birth, the talk appeared to
go through Ball. That is, the children addressed their teacher rather than one
another. Also, their manner of speech seemed more tentative, less assertive, as
they spoke in quiet voices of what they were thinking. It was almost as though
they may have been formulating their thoughts as they came from their mouths,
as though an inward search for sense and whether the sense they were making
mapped onto what they already knew. Sense-making was the term I recorded in
my journal for what the children seemed to be doing as they participated in this
particular disagreement.

These two things, the fact that the students appeared to be disagreeing
with one another and that the level of mathematics seemed more sophisticated at
the end Of the year, seemed to set end-Of-year disagreement apart from what
happened at the beginning of the year. Again, as I did in my analyses of the
presence and absence of disagreement in my dissertation data, I started by
looking where disagreement was and then where it was not, to better understand
the notion Of the concept and its context.

Over the next several weeks Of TESG, I continued thinking about the

differences between disagreement on September 11 and on June 7. I already had

51

analytic notes asking whether Mrs. Vandenberg appeared to fuel, sustain, or
extinguish disagreement and was calling them teacher moves. This was
interesting to me to pursue because it compared the orchestration of
disagreement by a novice, second-year teacher (Katherine Vandenberg) and
Deborah Ball’s management Of it. Ball is a highly recognized teacher-researcher
who chaired the Mathematics Teaching Working Group for the National Council
of Teachers Of Mathematics, to develop the Professional Standgrgg fpr Tgaghing
Mgthgmap’cs (1991). In other words, I began to ask if what Mrs. Vandenberg
does in disagreement in the data may not be the isolated and idiosyncratic
doings of one novice teacher. I asked whether what I found were recognizable
teacher moves used by others who teach in the spirit of current mathematical
reform.

The final step in the multi-layered data analysis was to re-embed the
disagreement in the larger context of what followed on May 3, as well as
subsequent math lessons. I noted that the idea of 200 + 10 is a fraction re-
surfaced later on May 3. When Joanne brought it up she volunteered that she
was trying to make it into a fraction, by drawing a circle with 20 triangles. (She
said, It’ s supposed to be 200 but I don’t know how to draw it.) Although
Joanne’s circular representation Of 200 + 10, drawing 200 triangles with 10
colored in, was incorrect, it did indicate that she continued to think about
whether and how it was a fraction. This example of Joanne’s re-emerging
thinking, evidenced through her representation, was indicative of these students
use of representations. Representations surfaced frequently as the children

worked with their mathematical ideas during the fractions unit.

52

The location of such a micro-disagreement required attention to detail in
order to isolate it. But it also demanded that I re-insert it in context so that I
could connect it to the question on the floor. In addition, I attempted to connect
disagreements over time, to determine if particular fractions concepts were more
problematic than others for these fourth graders. Multi-layered analyses
revealed that mathematical significance Of disagreement was connected to the
task on the floor. As I will discuss in Chapter Five, some disagreements served
different functions in the discourse. For example, some disagreements
functioned to get at the underpinnings Of another’s assumptions and at the
reasoning behind their position. Other disagreements functioned to publicly
hold up two ideas for scrutiny. Still others functioned to pinpoint a surface flaw
in a representation.

In part, my rationale for studying disagreement was to analyze it from an
end-of-year unit, because at that point in the academic year student
disagreement was expected to be evident as a discourse move for which this
teacher had planned for most of the year and, therefore, familiar to students.
Second, bounding the study within the fractions unit allowed me to look closely
at disagreement when content and lesson structure were similar while tasks
differed. Third, what disagreement was and the norms for involvement in it
should have been relatively stable during the time frame under analysis. It was
reasonable to assume that at the end of the school year, these norms were
familiar to the students.

As my study unfolded, I became acutely aware of its complexity, because
to my knowledge, there are no others like it from which I might draw or which I

could duplicate. Therefore, to inform analysis, I drew on two areas of literature:

53

sociolinguistics, specifically the techniques of discourse analysis, and
mathematical representations. Sociolinguistics informed the many layers Of
analyses regarding discourse, disagreement and students' involvement in it. For
these analyses, I drew on the work of Erickson (1986); Shultz, Florio, and
Erickson (1982); Sacks, Schegloff, and Jefferson (1978); Gee, Michaels, and
O’Connor, 1992); Cazden (1986; 1988); Edelsky (1993); O’Connor (1996); Hymes
(1972); Tannen (1991) and); (Burbules, 1993). In my analysis of student
disagreement during school mathematics I drew on the work of Lampert,
Rittenhouse, and Crumbaugh (1996); Wilkinson and Martino (1993), and; Yackel
and Cobb (1994). For analyses Of students’ mathematical representations, I drew
on multiple writings by Ball (for example, 1990, 1996) and Lampert (for example,
1987).

I anticipated learning from this analysis at least three things: a) a better
understanding of what disagreement was, when it arose, how it was managed,
and the role of participants; b) a sense of what students needed to learn to be
involved in disagreement, and; c) the ways in which the existence of
disagreement during mathematics classes enhanced (or didn’t) students'

mathematical power. As analysis proceeded, I learned much more.

54

CHAPTER FOUR
INSIDE DISAGREEMENT
This chapter examines what Mrs. Vandenberg did to establish a discourse

space in which her students can grapple with questions about fractions and,
through discussion, challenge one another’ 5 thinking. Far from a generic entity,
discussion within which disagreement was situated unfolded in an ebb and flow,
with segments of one type or another. In this section I describe discussion
patterns which emerged from data analyses. These patterns include two lesson
formats generally followed by Mrs. Vandenberg, the discourse pattern of whole-
group discussion, a typology of discussion, and, finally, categories Of
disagreement. As I discuss below, Mrs. Vandenberg fostered a whole-group
discussion pattern different from the traditional IRF (Sinclair and Coulthard,
197X), Initiation-Response-Feedback, where she revoiced (O’Connor, 1996)
children’s comments and deflected them back into the discussion. When
disagreement surfaced in whole-group discussions, she usually followed this
same pattern. However, as described below, when extended disagreement
occurred, this was not the case. I begin with a description of the two lesson
formats.
Lesson Format

Mrs. Vandenberg tended to follow one of two formats for math time,
which took place daily immediately following lunch recess and lasted for about
one hour (12:30-1:30 pm.) In the first format, She began with an W
W through which she oriented the students to the math focus for the day.
In this segment, she typically provided an overview of the math problems,

frequently in the form of questions, which she projected at the front of the room

55

from
usual.
their
their

jour

gar
dis
d3
du

V2

qll
Va

from an overhead projector (see Appendix C). During the second mpg,

 

usually lasting 20-30 minutes, students (a) recorded the overhead problems in
their math journals, (b) talked in small groups about the problems, and, (c) wrote
their answers. Typically, when the children worked in small groups and with
journals, Mrs. Vandenberg circulated around the room, interacting with
individual children and small groups. She asked questions about their thinking
and answers they recorded, reminded them of the task at hand, and generally
gauged emergent thinking which she frequently drew on during whole-group
discussion. In the Eli—IQ segment, Mrs. Vandenberg tended to lead whole-group
discussion for about 20-30 minutes. Occasionally, she created journal questions
during whole group, momentarily having the children reflect on and then further
discuss an. issue raised that day. In the Logan segment, she concluded class and
discussion either by reflecting and summarizing that day's discussion or by
generating a journal question, which children recorded and spent a few minutes
writing about.

I also frequently observed a second format for math time, in which Mrs.
Vandenberg started out with orienting comments, then moved immediately into
whole-group discussion because the lesson was an extension or carryover from
the previous day. Typically, she concluded whole group and presented new
questions for the remainder of math time. In the case of the latter, once Mrs.
Vandenberg posed new questions, the lesson structure looped back to resemble

the first format.

56

Discourse Pattern

Within these two lesson formats, whole-group discussion was
characterized by a distinguishable discourse pattern that differed from the
traditional IRF sequence of teacher-led discussion (see Appendix D). Over the
course of a lesson, Mrs. Vandenberg’s pattern resulted in multiple points for
children to get their ideas on the floor. As more and more ideas came to the
floor, the potential also existed for student-to-student disagreement to occur and
re-occur at different points across a lesson. In addition, she fostered a dialogical
relation between people (Burbules, 1993) in the context of discussion, a relation
”to ’carry away’ its participants, to ’catch them up’ in an interaction that takes on
a force and direction Of its own, Often leading them beyond any intended goal to
new and unexpected insights” (p. 20).

Following framing comments, during which she restated or posed the
questions or problems for the day at the overhead projector, Mrs. Vandenberg
generally Opened whole-group discussion by seeking comments from the

children, with questions or comments such as:

Who would like to be the first to share their idea about...
What did you guys come up with?

I'd like to hear some of your ideas about...

Also, she reminded them about behavior or the task. For example, she would

say something like:

Open your journals and also take notes.

You should be writing down

57

When someone is up here sharing their ideas and we're calling out or

talking, that shows them we're not interested in their idea. That's not

acceptable. Right now you're supposed to be
These insertions served to index (Mehan, 1979) a rule about respect for others -
respect for their ideas and about how to talk and listen to each other. When
combined, orientation to the lesson, overhead questions for the day, and
questions and comments to start getting ideas on the floor ultimately launched
the overall process of getting children’s turns and ideas into the discussion.
Therefore, when combined, orientation to the lesson, overhead questions for the
day, and questions and comments to start getting ideas on the floor ultimately
launched the overall process of getting children’s turns and ideas into the
discussion.

After discussion was underway, Mrs. Vandenberg frequently repeated
what children said and then said or asked things which appeared to seek
clarification. As wave after wave Of student responses came to the floor, she

revoiced (O’Connor 8: Michaels, 1993) them and then said things such as:

Can you draw (or write) that?
Tell us about your picture.
How do you know?

What do you mean, ?

Mrs. Vandenberg frequently repeated what a child said back to the group.
But sometimes, to the individual explaining an idea, she tended to ask questions
which invited even more ideas (and potentially disagreement). As more and

more of the children’s thinking went through her, whether intentionally or not,

58

she gave ideas credence through the way in which she either repeated or
synthesized them:

Why does that make sense?

Do you guys agree?

What do you guys think?

....Monte responded to the question, 'What is a fraction?', by writing this

DIVISION7 problem. Any responses?

To further stir the stew of different ideas she brought to the public forum
of whole-group discussion, Mrs. Vandenberg made connecting comments among
children’s ideas, noted her confusion about (apparently) conflicting statements,

or asked questions. To do the first, connect ideas, she would say,

Hannah's last comment was that that was FOUR-fourths. And Ali said it
was ONE, because it was one whole..

(to Micah, about Antonia's halves on the geoboard) Good. One had seven
and a half, one had eight and a half. 50 MIKE showed us how if we just
switched that half we could have eight and eight. SO it looks like
ANTONIA thought about this too. Braden, what did YOU want to say?
To accomplish the second, she might say, ”I'm confused about
something,” then go on to restate two different ideas the children had previously
talked about, either that day or another day. For example, on May 13 during the
discussion of the placement of 1 / 2 on the number line, Jody insisted that 1/ 2
wouldn't be between zero and one - it would be somewhere else. When Steve
followed by saying 1 / 2 would be between 2 and 3 on the number line, Mrs.
Vandenberg said, ”I'm confused about something," and went on to explain that

the day before they had discussed and written in their journals that 1/ 2 + 1/ 2 =

59

1; but that day (May 13) they were saying 1/ 2 was between 1 and 2, or between 2
and 3. This seemed to hold more than one idea up for scrutiny. To get more
ideas on the floor, Mrs. Vandenberg frequently asked questions.

Within this discourse pattern with ideas coming to the floor from the
beginning of whole-group discussion to its conclusion, the potential for
disagreement existed almost from the start. Disagreements did not always occur,
but inherent in this discourse pattern - with more and more ideas coming to the
floor - children had the Opportunity to disagree. In fact, disagreement bubbled
up as early as 2 minutes into one lesson and as late as 51 minutes into another.
When disagreement-like talk began to surface, Mrs. Vandenberg often played an
active role in orchestrating and mediating the interaction. She inserted questions
and comments as the disagreement emerged, so that the turn-taking pattern
became teacher-student-teacher-student. For example, early in math time on
May 7, Michaela shared her way to make halves on the geoboard, an overhead
question for May 5 (see Appendix C). Because her design raised two
controversial ideas, disagreement began to emerge. First, as shown in Figure 1,
Michaela’s shapes did not use all Of the geoboard space, nor the center peg.
Referring to the number of pegs within each shape, she described what she did

to make them equal.

Michaela: Then there’s, 12 in there. And then here’s another 12 and
you’ve got one nail left over. SO I took THAT nail and I made up nails
with rubber BANDS to make it EQUALLY. . .

 

7 Words in call capital letters indicate speaker emphasis. Commas within speaker turn
indicate brief pauses. Vertical dots indicate missing transcript.

60

Second, Michaela said she needed to use the center peg, so she put a lone
band over it, while she also laid other bands down the uncovered portion of the
board. After she said she was ”pretending those are nails,” a teacher-student-
teacher-student pattern of teacher mediated disagreement emerged. Tomika

expressed concern, which Mrs. Vandenberg repeated and asked about:

Tomika: I don’t THINK so.
KV: You don’t think so? Why don’t you think so?

Tomika: Well when she put those two rubber bands in, like, like there.
’Cause you can’t do it.

KV: Tomika doesn’t understand or agree with the rubber bands in the
middle part.

Michaela: Well if there IS a nail you make it EQUALLY. SO I made a part
with rubber bands.
By taking up the children’s dissenting remarks in this way, the disagreement
went through Mrs. Vandenberg, rather than directly to Michaela. However, at
times the children would directly uptake one another to speak for themselves,

resulting in short bursts of student-student disagreement.

 

 

 

 

 

 

 

Figure 1: Michaela's Way to Make Half on May 7

Teacher Moves
As described above, Mrs. Vandenberg generally followed a pattern during

whole-group discussion. Wthin this pattern, she made particular moves to

61

nudge discussion along. These moves included: a) asking questions, b)
connecting ideas, c) getting children to say more, and, c) inserting reminders
about task and behavior.
Asking mestiops
When Mrs. Vandenberg orchestrated the discourse by asking

questions, common questions she used included

What do you guys think about that?
What did you come up with?

Can you Show us that?

How did you know?

What do you mean by _?

DO you agree?

nnectin Ideas an Gettin hi1 ren t a Mor

Second, Mrs. Vandenberg frequently connected kids' ideas, for example,
”Hannah said 4 / 4. Ali said one.” or ”I'm confused about something.” Then she
went on to compare two different things about the same issue, like whether 1 / 2
was less than 1. Or, she would make a specific connection such as, ”That
reminds me of what ALI showed us. DO you remember how ALI showed us her
way of dividing up circles?” Mrs. Vandenberg also said things to get kids to say
more, such as ”Tell us about _,” or ”Explain what you mean by _.”

Behavior and Task Reminderg

She also inserted behavior and task-related reminders as each day's

discourse unfolded, as in the following example from April 28:

I'd like you to Open your journal and also take notes about what is being
shared now. That's not appropriate to call out. Shhh. Boys and girls stop

62

and listen. When someone's up here sharing their ideas and we're calling
out or talking to our table mates that shows them that we're not interested
in their ideas. That's not acceptable. Right now you're supposed to be
taking notes on these ideas we're generating and listening. . . Shhh. It's
really hard to hear people's ideas. You all have such wonderful ideas it's
disappointing when we can't hear some people. Especially some Of you
who have been waiting a while to share your idea.
Insertions such as this one revealed that Mrs. Vandenberg did not take for
granted that the children knew how to be involved in discussions. Instead, this
was an indication of the work she did throughout the year to nurture discussion
norms, affording the children some direct instruction about how to be involved.
Because of these insertions, the majority of disagreements were over the math
content, not interpersonal conflict, or something like shooting bands across the
room (the reason for a lone non-math disagreement). When combined, Mrs.
Vandenberg’s questions, connections, and reminders helped to keep the children
focused on the mathematics while they explored their developing ideas and how
to reason about them.

However, the discourse pattern and Mrs. Vandenberg’s teacher moves
within it as discussed above did not result in one type of discussion. To the
contrary, analyses revealed a typology of discussion across the data.

Typology of Discussion

These discussions took on various nuances, and analyses of them led me
to an examination of disagreement. Indeed, four types of discussion emerged
from the data: repetitive discussions; discussions with different ideas, without

disagreement; discussions with different ideas, with teacher-mediated

disagreement, and; student-to—student disagreement.

63

Ii—

 

Repgtitive Disgpgsippg

Some discussions were flat and difficult to attend to. I sat, awaiting the
breakout of a disagreement, eager to witness how it would begin and what it
would be about. But agreement, not disagreement, marked these discussions.
With the same or similar ideas repeated by students, these repetitive discussions
did not appear to go anywhere. They retraced similar notions, and if discussions
were represented as lines, these would be flat. A small-group discussion on
April 28 illuminates the notion of repetitive discussion. On this, the first day of
the fractions unit, Mrs. Vandenberg followed the first lesson format, beginning
with orienting comments, then questions for the day on the overhead projector:

What is a fraction?

When have you used fractions?

How are fractions represented?

Draw and label pictures.

She read the questions to the class and instructed students to discuss and
brainstorm the questions in small group with their table mates. They were to
draw and label pictures as a group on a large white sheet of paper.

After about five minutes of math time the camera was moved to capture
the first small group's interaction. This group of four consisted Of three girls and
one boy: Connie, Moira, Suzanne, and Monte. A microphone in the middle of
the table captured their talk. They soon were at work on the first question: What
is a fraction? Nearly 6 minutes into their work, Moira suggested that a fraction is
when you ”cut up into pieces.” Monte quickly added, ”it's when you split up a
problem and make it easier,” which he soon repeated. Connie turned to the

camera to say ”we're talking about fractions today,” while Suzanne said nothing

about the question. Her contributions were in the form of talking to and giggling
with Moira. With the large sheet in front of her, Moira spoke aloud as she
recorded, ”A fraction is when you split something up”, to which the others said,
”OK.”

The original ideas shared by Moira and Monte, then, were that a fraction
is when you cut something up or split up a problem and make it easier, which
Moira recorded as ”A fraction is when you split something up.” To them,
cutting or splitting something up was associated with fractions. These
contributions were basically the same and did not differ or provoke
disagreement. They were recorded on the group paper and the group turned to
the second question, When have you used fractions .7, as the camera moved on to the
second small group.

Discussions With Different Ideas. Without Disagreement

On occasion, Mrs. Vandenberg asked for students' ideas about a
mathematics issue or tOpic under consideration, and she got them. However,
students did not disagree with one another; nor was she able to inspire a
disagreement. The different ideas hung in the air without reaction or response. I
labeled these discussions with diflerent ideas, without disagreement. Frequently, I
wondered if the children did not notice that different ideas were coming to the
floor. Mrs. Vandenberg would say something to highlight these differences and
still there would be no student reaction. For example, on May 4, the children
worked for the first time with geoboards on the following task, recorded at the

overhead:

65

How many ways can you show halves (1 / 2)?
Practice on geoboards. Record on paper.
Write one explanation in journal, explaining how

you can prove your idea shows a half.

When Mrs. Vandenberg introduced the task, she had a few children go to
the overhead to begin showing their ways to make halves. When Hannah cut
her drawing“ into fourths, Mrs. Vandenberg asked how she could prove she had
a half. Hannah’s proof was that on one side she had 2 pieces, and ”Four take
away two is two. ’Cause two plus two is four. It’s a half." Others, like Moira,
who made a diagonal cut, said ”They’re the same size so each kid would get it
equally." And, Jody called out that a line down the middle of the board ”has the
same amount Of spots." Later, when Ali showed her different idea (as she said,
her ”odd one”) about halves (see Figure 2, Ali’s Halves on May 4), Mrs.
Vandenberg tried to no avail to fuel disagreement. At the overhead, Ali counted
pegs to prove she had halves, and said she had 9 pegs on each side. Apparently

noting that these halves were not equal, Mrs. Vandenberg asked,

KV: If that was a brownie and you had cut that for me and a friend, I
might say that that’ s not fair because I didn’t get the same size piece. Ali’s
counting pegs. DO you guys agree? DO you guys agree that there’s 9-that
that shows equal halfs?
In spite of these strategic questions to direct the children’s attention to the
discrepancy in Ali’s reasoning, between counting pegs to get halves when she

actually did not end up with equal halves, the children were unresponsive.

 

8 Note that Hannah did not use a geoboard to represent her example. Instead, she drew a
square, calling it a ”brownie."

66

 

 

 

 

 

Figure 2: Ali’s Halves on May 4

In discussion segments like the one above, when Mrs. Vandenberg
persisted, even asking if anyone disagreed, students remained unresponsive. It
is not possible to say whether, in spite of Mrs. Vandenberg's efforts, they did not
recognize the differences for themselves, were out of ideas, or were not
interested. Or, as in the case above, the children may simply not have had
enough time so early in the lesson to think of additional ways to make halves.
Whatever the reasons, the result was that students did take up Ali’s idea to
disagree.

Discussions With Different Ideas, Teacher-Mediated Diaagpagment

At times the discussions became discussions with diflerent ideas, with teacher-
mediated disagreement, for example when Tomika’s disagreement on May 7 went
through Mrs. Vandenberg, or when disagreement over Michaela’s unique way to
use extra bands and ”make up nails” to divide the board in half. I found these
segments of disagreement to be characterized in several ways, first by the fact
that Mrs. Vandenberg mediated them. In a predominantly teacher-student-
teacher-student pattern, kids' comments went through Mrs. Vandenberg. She
sought student responses, they talked to her, she restated the idea, another
student answered, and so on. Second, the students disagreed for the most part

with each other's ideas, rather than the person - a participation norm endorsed

67

by Mrs. Vandenberg. In other words, when Mrs. Vandenberg mediated, the
disagreement tended to elevate the mathematical aspects of the discussion.
When children began to veer from a focus on ideas or toward inappropriate
interactions, Mrs. Vandenberg often stepped in with a management comment, a
third characteristic Of teacher-mediated disagreement.

The following exchange from April 28 demonstrates Mrs. Vandenberg's
involvement through teacher-student turn-taking, her conversational work to get
students to focus on ideas, and her move to manage unacceptable behavior.
After small-group time to work on overhead questions, she began whole-group

discussion about the question, What is a fraction?

KV: By now you’ve generated ideas about what you think fractions are.
I’d like to hear some of those different ideas about what you came up
with. Let’ 3 start with the first question: What is a fraction? What did you
come up with?

St: Stop kicking me! Who’s kicking me?

KV: I’d like you to Open your journal and also take notes about what is

being shared now. I think that’ 11 help you focus.
This immediate interjection to remind students about both behavior and journal
writing quickly turned attention to ideas that came to the floor.

Student-t9_-Student Disaggeement

The fourth discussion type, which is the focus of this study, is student-to-
student disagreement. The predominant characteristic of these disagreement-
based discussions was the idea that the children were disagreeing with one
another. That is to say, there was a mutually engaging aspect of communication
with one student taking up what another said, and disagreeing with it. These

disagreement-based discussion segments, similar to teacher-mediated

68

I_ Vi

disagreement, appeared generative to the overall discussion by drawing in
multiple ideas over which the children interacted. These discussion segments
differed from those where different ideas were on the floor but not acted upon,
and differed from those where disagreement filtered through Mrs. Vandenberg.
Mrs. Vandenberg and I discussed disagreements, Often right after math

time when students were preparing for Writing Workshop. We informally
attempted to tease apart the disagreements to try to figure out why some struck
“8 as good, or constructive, ones. Intrigued by their mathematical content or the
twists disagreements brought to the discussion, or who was involved in them
and in what way, we found the disagreements strategic in our efforts to
understand students' mathematical learnings. Once during the fractions unit the
disagreement took a negative turn. This disagreement was unlike any other in
the data. As I will discuss in Chapter Six, the mathematics under discussion was
overshadowed by the negative interactions of some children. Here, students'
different ideas arose and were disagreed with. However, the difference is that
the mathematical aspect Of the disagreement was not elevated; the interactional
aspect was. If one could imagine a balance scale on which one side held the
weight of the mathematics in the disagreement and the other the interactional,
the scales in that instance would be tipped toward the latter.

To summarize, the lessons format, whole-group discourse pattern, and
discussion types led me to a closer examination of the discursive phenomenon of
disagreement. In the next section, I define disagreement in this setting and
compare it to the more generic notion of argument. I begin with a definition of

disagreement.

69

What is Disagreement?

If discussions took on differing characteristics, some of which resulted in
disagreement, what then was disagreement in this setting? To Mrs. Vandenberg,
in part it was about having and sharing different ideas, as evidenced in both her
questions of the day (at the overhead) and particular questions she asked
children. However, there was a difference here between sharing ideas and
disagreement. When the sharing of ideas was going on, multiple students
offered various answers or strategies and Mrs. Vandenberg orchestrated
discourse with questions and comments. And when disagreement happened,
one or more students would take up the comment Of another, to distinguish their
idea. For example, on May 3, Mrs. Vandenberg solicited students ideas about the
discussion question, What is a fraction? Responses included:

200 divided by 10 is a fraction. (Monte)

A fraction is kind Of like division. (Joanne)

A fraction is less than 1. (Hannah)

Although similar, the children apparently perceived them to be different
because they talked to their teacher, not a classmate, about them. They did not
directly uptake one another's contributions; instead, when called on, they
directed their comments to Mrs. Vandenberg. Without immediate uptake to link
the previous student's idea, it was not disagreement for the purposes of this
study. In these examples, students' ideas went through the teacher who repeated
them, in essence deflecting them back to the children. They appeared to be
addressed to her, she took them up, and rephrased them. For the purposes of

this study, disagreement is defined as those instances where (a) a student had an

70

idea, (b) a second student had a different idea, and (c) the second student
directed it back to the first. This is when students negotiated the floor for
themselves and assumed responsibility both for getting a turn and their idea
onto the floor. They initiated their own turn to disagree with a particular person
around a previously-stated idea. With 26 active fourth graders and Mrs.
Vandenberg daily engaged in the construction Of mathematical meaning and
social norms, disagreement was a mutually engaging undertaking, where one
student contributed an idea to the discussion and others reacted and responded
to it. A statement was made and locked onto by another.

Interestingly, disagreement parameters were not openly stated by the

teacher until late in the hour on May 7:

KV: Let's remember that we're disagreeing with each other's ideas, not the
peOple. Okay? To raise your hand to someone else or, to, yell at someone
else is not appropriate.

With this brief and straightforward reminder to her students, Mrs.
Vandenberg appeared to bring to focus the participation structure (i.e., the
mutual rights and obligations of students, Erickson, 1996) during disagreement.
That is,

Speaker's right: to disagree Others' duty: to listen
Speaker's duty: disagree with ideas Others' right: to expect they will not
be personally attacked

Here, the norms of participation in disagreement appeared vivid: It was
appropriate to disagree with ideas rather than people, and inappropriate to raise

a hand or to yell at someone. Mrs. Vandenberg made a distinction between

71

.sagreeing with ideas and disagreeing with classmates, separating the

Iathematical and interactional aspects of disagreement.

Throughout the data set, Mrs. Vandenberg appeared to hold no
xpectation that students would come to consensus, because she never
pecifically stated that. Rather, the sharing of ideas often appeared to be the

;Oal. For example, on countless occasions Mrs. Vandenberg would say

something akin to the following:

KV: . . . a FEW groups didn't get to SHARE one of their ideas yet.
Monte, would you like to share what YOU came up with for what is a

fraction?
This sharing of ideas seemed important to her desire to develop tasks which
were conducive to discussion, a desire she expressed during a September 26
meeting, when she said the question she wanted to focus on for the year’ 5 Math
Study Group work was, What is the relationship between the task and discussion? In
other words, to her the quality of discussion depended on the mathematical tasks
she developed. And, an important aspect of discussion was sharing ideas.

Curious about Mrs. Vandenberg’s frequent use of ”share,” I asked her the
origin of the language of share, sharing, and share your idea. After a pause, she
said that it may have come from when she taught second grade, when she
thought ”Show and Tell” did not capture the interaction for which she hoped.
She changed to ”share” then because she thought "Show" and ”showing” doesn't
have the social piece. Sharing is more like welcoming feedback or input (field
notes, February 21, 1994). When I asked if, when she used ”share” and
”sharing,” she was thinking mathematically, socially, or both, she answered,

”Socially. Thinking aloud with the group; for example, a question, a

72

representation, or a different idea” (field notes, February 21, 1994). Through this
and other responses, I gained insight into the priority Mrs. Vandenberg placed
on getting children to talk and interact around their mathematical ideas about
fractions content.

Mrs. Vandenberg also elaborated on the importance of disagreement
during discussions. During a July interview, I asked her why she thought

disagreement was important

KV: Well I think the disagreements part is important because I think,
WHEN disagreements occur I think that, that FORCES learners to,
ARTTCULATE what they're thinking about, and to, to JUSTIFY, how, how
they SOLVE something or why they THOUGHT about something in a
particular way. If there's NOT disagreements and things are just
ACCEPTED as, ’OK’, you know and you move ON, then I don't really
think we're forcing them to stretch their thinking and to consider different
alternatives and, um. And then I think a big part Of disagreement, ALSO
is accepting one another's ideas and, seeing the value, in ah, in different
ways Of thinking. Which, which I think is something that takes a lot of
WORK, in the classroom. Um, you know just, just getting children to, to
listen to one another and want to RESPOND to each other and not always
to the teacher. (July 2, 1993)

Although she did not intend disagreement to be her focus, it did become an
interest over time, as she worked to incorporate discussion into math lessons
during this, her first year of teaching fourth grade.

This sense Of interaction through sharing ideas, of children talking with
each other, emerged again and again during discussion. To this teacher and
these fourth graders, disagreement was a more immediate, face-to-face process
than the more formal notion of argument (Toulmin, 1958). Rather than a final-
draft, refined and structured means to present or refute a position, disagreement

to these kids occurred in the moment through exploratory talk (Barnes, 1992),

sometimes as the ideas seemed to come to them in the moment. And the longer

73

 

the disagreement, the more personal it became, as I will discuss in Chapter Six.
In this setting, there was more to disagreement than math, because in
disagreement, the children took opportunities to get both a turn and an idea on
the floor.
Categories of Disagreement

My review Of the data resulted in identification of two types of
disagreement, single disagreement and episodes of disagreement. Although
virtually all disagreement involved fractions content, single disagreement and
episodes of disagreement differed in several ways. First they differed in length.
In the shorter disagreements, which I labeled single disagreement, the number of
turns ranged from two (May 3) to nine (May 12). These disagreements were
bounded by either questions or comments from Mrs. Vandenberg. For example,
on May 13, part Of whole-group discussion was about the question, Where is 1/2
on the number line? Mrs. Vandenberg called on Antonia to show where she put

1 / 2. Antonia first made a mark between zero and 1, saying it was ”in the middle

of zero and one”:

KV: What made you decide that?

Antonia: I don’t know. Or else it could be here. (marked between 0 and -
1)
Jody: But that’s a NEGATIVE number you’re going by!

KV: Wait a minute! A couple Of people came up to ask me things. But I
prefer that you raise your hand. Antonia, why did you put it between

zero and 1?

In this exchange, Mrs. Vandenberg’s questions required Antonia to
explain her thinking, which she seemed unable to do. But as soon as she marked

he spot halfway between zero and negative one, Jody called out her Opposition

74

to the idea. However, after a management reminder, Mrs. Vandenberg stayed
with Antonia, bounding this disagreement (which will be further discussed in
Chapter Five) with her question. This bounding helped keep the focus on the

math under discussion.

The second characteristic of single disagreement was the uptake by the
children of Mrs. Vandenberg’s questions and comments. In the above example,
although Antonia did not answer Mrs. Vandenberg, Jody continued to protest,
this time to her teacher. In this way the children’s attention was directed to the
idea on the floor: the location of 1 / 2 on the number line.

To summarize, single disagreements were identifiable first by length:

They varied between 2 and 9 turns. Second, following these short bursts of
student-tO-student disagreement, Mrs. Vandenberg successfully re-negotiated
her way to the floor and continued to orchestrate the discussion. Third, the
children responded to Mrs. Vandenberg and take up her questions or comments
to answer her. In this cycle - teacher question, student-to-student disagreement,
teacher question - this teacher and these fourth graders mutually negotiated the
floor.

However, there were longer and consecutive waves of disagreement
during which Mrs. Vandenberg receded from the talk and the children appeared
tO take over. In these more extended instances, or episodes of disagreement, the
:hildren either did not hear or ignored Mrs. Vandenberg, and she had to work at
'enegotiating the floor. It was as though the disagreement took on a life of its

Iwn (Burbules, 1993), and whereas the majority of shorter, or single,

lisagreements were over the math content on the floor, in episodes the focus on

1ath seemed to fade. Rather than the math in the disagreement being

75

highlighted, in the episodes, the interactions between the children were what
stood out.

It is important to point out that there is an instance of single disagreement
on May 12 (discussed in Chapter Five) that at first glance may appear to be an
episode. However, the distinguishing feature in this 9-turn disagreement is the
fact that when Mrs. Vandenberg attempted to regain the floor, she succeeded.
As I will point out in Chapter Six, this was not the case during episodes. There,
for some time the children did not respond to Mrs. Vandenberg’s questions,

instead continuing to disagree.

76

CHAPTER FIVE
SINGLE DISAGREEMENT

The disagreements discussed in this chapter concerned fractions content,
Ich as part-whole relationships, equivalence, number and size of pieces Of the
'hOle, whether to count pegs or squares on the geoboard, and the location of a
raction on the number line. As defined in the previous chapter, single
lisagreement is a brief uptake of one student by another, to get a different but
'elated point on the floor. Of the identified 25 disagreements of this type, 24
were mathematical by nature. That is, these disagreements brought to the floor
the children’s developing fractions ideas. I learned that, as these children teased
apart this mathematics content, they relied on representations to explore their
developing knowledge. And, I found that an examination of disagreements
provided a window into students’ emergent thinking about fractions.
Representations were a catalyst for most of the disagreements, because
they helped the children focus on either the mathematical idea being
represented or how a classmate was thinking about and representing that
mathematical idea. Mrs. Vandenberg started whole-group discussion, brought
ideas to the floor and representations with them, and disagreement erupted.
Sometimes, the disagreement was about an underlying mathematical idea
depicted by the representation, and sometimes the disagreement was about the
construction of the representation itself.
Two equally important competencies were at work when these fourth
graders disagreed: academic competence and social competence (Mehan, 1979).
These fourth graders had to be involved in the math content, or academically

competent, in order to know about what to disagree. And, they had to be

socially competent to know how to disagree. One competency was not sufficient.
With one and not the other - academic or social competence — they did not
successfully negotiate the floor with a turn and an idea. For example, if a student
knew her fractions content but not the appropriate juncture or way to disagree,

she likely would not successfully get her disagreement to the public forum of
whole-group discussion. Inversely, if he knew the appropriate juncture at which

to negotiate the floor and get a different idea to the public forum of whole—group
discussion, if he did not know the math, he may not succeed. When a child

could demonstrate both, she was more likely to succeed.

There are three ways to parse these disagreements: by fractions content,

by purpose, and by function. The examination of content revealed that these
fourth graders disagreed over that content which we would anticipate in a unit

on fractions, such as equivalence, part-whole relationships, area, and the location
of a fraction on the number line. The disagreements also differed in purpose -
they could, for example, correct, defend, or question assumptions of another.
Further, some purposes of disagreement were similar, such as disagreement to
correct, to revise, and to mediate (as indicated in Table 1). Together, these
purposes Of disagreement — to correct, to revise, or to mediate — served a function
at a general level in the discourse. Disagreements of these purposes functioned
0 either pinpointed a flaw in what a child said about a representation or
linpointed a flaw in a representation as created. As shown in Table 1, the 24
)ntent-related disagreements fell into six categories of purpose which are the

ganizing frame to this chapter: Correct, Revise, Mediate, Defend, Question

Iother’s Assumptions, and Call for Clarification. These categories of purpose

1 be consolidated to reveal the functions of disagreements. The functions of

78

 

these disagreements in turn identified the mathematical significance of the

disagreements. To demonstrate the purposes, functions, and mathematical

significance of the disagreements, I include in each section examples of children’s

disagreements. I begin with the first category of purpose, Disagreement to

Correct, Revise, or Mediate.

79

Table 1: Purposes Of Disagreement Grouped by Function

 

 

 

 

 

 

9Purpose Function Representations
Correct Another Pinpoint a flaw in what is Joanne’s circle drawing
(n=7) said about a of 200 + 10
representation Circle representation
with overhead question,
Is this 1/2, 2/4, or both?
Why?
Rob’s geoboard of 1/ 2
and 2 / 4
Monica’s drawing of
3/5 (2x)
Jonah’s geoboard
Drawing of 1 / 2
Revise Pinpoint a flaw in a Joanne s r sctangle
(n=2) representation as created drawmg
Fraction bars
Mediate Link 2 different ideas Circle representation
(n=1) with overhead question,
Is this 1/2, 2/4, or both?
Why?
Defend Hold up 2 ideas for 200 + 10 is not/ can be a
(n=6) scrutiny fraction
Monica’s drawing of 3/5
Kim’s diamond
Rob’s geoboard of 1 / 2
and 2/ 4
Question Get at underpinnings of Hannah’s 2 / 4 drawing
Assumptions another’s position to represent 1/ 2
(n=7) Hannah’s TV
Number Line
Call for Clarification Get at reasoning behind Kim’s Diamond
(n=1) another’s position

 

 

 

8O

 

Correct, Revise, or Mediate
Nine disagreements had as their purpose either to revise or correct
someone else. Essentially, these disagreements were to correct some aspect of a
representation, either to correct a discrepancy between what one said and what
one created as a representation or to correct the representation as created. In
addition, one disagreement’s purpose was to mediate two opposing ideas.
Disagreements to correct or revise functioned to pinpoint a flaw in a
representation or what was said about a representation. These disagreements
focused on a specific point. For example, on May 3, Joanne drew one thing and
said another, resulting in a brief disagreement. Joanne attempted to represent
Monte's idea, ”200 + 10 is a fraction,” as a circle. This notion about fractions
came up in response to the overhead questions for math time (Day 2 of the
Fractions Unit) on May 3 (and which were carried over from April 28): What is a
fraction? When have you used fractions? How are fractions represented? Draw and
label pictures. Early in that lesson, Mrs. Vandenberg started whole-group
discussion by asking Monte, ”Monte, what did you think a fraction was? How
did you describe it?” Monte answered ”A fraction is, where you have a big
number and you can't solve it so you split it up into groups so it's easier to solve
. . A fraction is a piece of the problem." He went on to say that ”200 + 10 is a
fraction. ” Monte recorded circled groups Of 10 dots as his representation for 200
o- 10, and later in the hour when Monte's idea again surfaced, Joanne worked at

:reating a circle to represent it. She worked to divide up her circle, in her word,

nto ”triangles," so that her drawing looked like a pie cut into many pieces:

81

KV: So we're assuming that's 200 and you're pretending to color in 10 Of

them?
Joanne: Yes. I'm putting 10 triangles down and colored 5. Five goes into
10. Ten goes into 200.

st: you only have 9.
Joanne: Nuh uh. 1, 2, 3, 4, 5, 6, 7, 8, 9. (colors in one more "triangle")

KV: Nobody has an idea about this 200 divided by 10? I think we should

come back to that.

In the process of working through her picture, Joanne said she didn't
know how to make 200 pieces, so we had to pretend, and that she colored in 10.
When a classmate said she colored only 9, she initially disagreed, but then
colored in a tenth ”triangle." Her action seemed to indicate her agreement. She
assumed she'd colored 10 triangles, but learned she had not. The mismatch
between what she colored and what she said did not alter the correctness of what
she was trying to represent.

A second example of correcting another’s representation occurred on

May 4 , the first day the children worked with geoboards, when Jody corrected

Jonah's representation of one-half Of the geoboard.

 

‘*l

1 O O O i

.__._.l

 

1 O O O 1

 

 

A A A

 

 

Figure 3: Jonah's Way to Show One-half on May 4

'o begin Class, Mrs. Vandenberg recorded these directions on the overhead:

How many ways can you Show halves (1 / 2)?

Practice on geoboards. Record on paper.

82

Write one explanation in journal, explaining how you can prove your idea

shows a half.

More specifically, she passed out geoboards and bands as well as dot
paper. The children practiced making halves on the board, re-created shapes on
the paper, then explained in their journals. Jonah was first to volunteer to show
a way at the overhead, and when his horizontal line didn't connect all the dots

(representing geoboard pegs), Jody corrected him:

KV: Who can come up here and show me one way I could divide this,

assuming this was a geoboard, what's one way I could divide this up into
halfs? Jonah?

Jonah: You could take a rubber band and kind of strap it around like this.
Jody: ALL the way down [points across the geoboard drawing].

Jonah: There aren't enough dots.
Jody: Yeah there is. She's made them on the sides.

KV: I see, Jonah was saying there wasn't another dot at the end. I drew a

square around the dots SO it looked like it was closed in square. That's a

good observation, Jody.

When Jody said there were enough dots, Jonah extended his line across
the overhead board. Jonah assumed this geoboard had 9 pegs and drew his line
accordingly. He appeared to know how to create one-half with a horizontal line.

Similar to the example above about Joanne's tenth triangle, Jonah's action seemed

to indicate his agreement with Jody.

Figure 4: Hannah’s Representation for a Fraction on May 3

The disagreement whose purpose was to mediate occurred on May 3 over
a circle representation introduced by Hannah. Under discussion was an
overhead question that Mrs. Vandenberg spontaneously created to conclude an
episode of disagreement (the episode is discussed in detail in the next chapter).

At the overhead projector, Mrs. Vandenberg pulled together different ideas on
the floor to re-create Hannah’s circle and color in the two sections on the left, as
Hannah did. Then she instructed students to write it in their journals and
respond to the following questions: Is this 1/2, 2/4, or both? Why?

When whole-group discussion resumed and Austin said, "It’s both,"
Carmen disagreed to say it was one-half. Although her explanation was
virtually inaudible, we do know that she disagreed and that she thought it was
one-half. As though wishing to alleviate a perceived distance between their
answers and potential social distance between his classmates, Jonah appeared to
mediate the disagreement when he called out, "I think they’re both the same but
they’re written a different way." With this move, he linked two different ideas
recently placed on the floor by two classmates.

These disagreements to revise or correct - although focused on
mathematical representations - pointed out minor glitches rather than getting at
deeper issues Of sense-making. These disagreements converged thinking over a
technicality. At first glance, they give the impression that students were
mutually creating representations. However to correct another about a line not
quite across the geoboard (as on Jonah's geoboard), although important, was not

he same as disagreeing to defend an idea, as in the next category.

Defend

On the surface disagreements to defend an idea about fractions frequently
appeared to be more adversarial than those focused on correcting, revising, and
mediating. Actually, they served a useful function: to sustain two ideas on the
floor, side by side, for scrutiny. Although prefaced in several instances by ”yeah
but” statements, these disagreements did not place students’ mathematical ideas
in polar opposition. Instead, they showed them to be willing to negotiate the
floor for a turn and an idea equally important to them. Several representations

generated disagreement which held up two ideas at once, including Rob’s

geoboard.

Rob's Way to Shpw 112 and 214

 

 

 

Figure 5: Rob's Geoboard Of 1/ 2 and 2/ 4 on May 7

One disagreement to defend occurred on May 7. The overhead questions

during class were:

Finding ways to show 1/ 2 and 2 / 4 on the geoboard.

Record it on dot paper. Prove one of your ways.

This was the third day of using geoboards, and on each day Mrs. Vandenberg
asked at least one question about making one-half. But this was the first time

since they began working with geoboards that she asked them to think about

85

 

—half and two-fourths at the same time. After the children worked in small
ups and wrote on dot paper and in journals, Mrs. Vandenberg resumed
)le-group discussion. By this time it was late in the lesson, and when she
(ed for someone they had not heard from in a while she called on Rob. Rob's
mesentation of 1 / 2 and 2 / 4 generated three brief disagreements, the first,
own below, about the size of two squares adjacent to his Off-center diagonal.
ee Figure 5 for Rob's geoboard of 1/ 2 and 2 / 4) Rob began to explain his way to
nake 1 / 2 and 2 / 4 by counting pegs on the 2 / 4 side, saying he had 4 above and
below the piece in the middle; and half on the other side (left of the diagonal).
When Mrs. Vandenberg asked him what the piece in the middle was, Rob said he
didn't know, "I just put it there so it'd be 2 / 4 and a half. ” When she asked the
class what they thought Of Rob's way, one boy said he thought it looked equal.
Investigating further, Mrs. Vandenberg asked Rob what he meant, ”They looked
equal." His answer: ”because it's equal things.” Perhaps to learn more about
their understanding Of ”equal,” Mrs. Vandenberg asked if both halves or sides
were equal, and asked Moira to count the squares. Rob took away the band that
made the parallelogram9, and Moira counted and said "They're even. They both
have six.” Mrs. Vandenberg asked individual children, who said they counted
seven and eight and that they have to put together half squares, too. When she
asked Moira to check Off each square as she counted it, Moira marked them and
said, ”Six and a half. " Shawna said it would be six if whole squares were
counted, ”But if you take this one and that one (two partial squares to the left of

the off-center diagonal) that makes a WHOLE.”

 

9This off-center diagonal as a single cut on the geoboard is an example of one-half,
suggested by the authors of gaing Fragipng (p. 30) for teachers to have students prove. The

86

 

 

 

 

 

Figure 6: Halves Referred to by Shawna on May 7

A few classmates thought otherwise:

KV: SO what do we CALL that?

sts: It would be MORE than a whole. 8!

Shawna: It's a little off but it's (gestures to the partial squares) still a half.
Joanne: A little MORE than a half.

KV: Is the other side a little more too?

Mathematically, these children appeared to have different assumptions
about the precision required to make one-half of a square, which was directly
affected by Rob's Off-center diagonal. To Shawna's way of thinking, apparently
half squares that were ”a little Off," when combined, still constituted a whole
square. Proportion, while relevant, did not have to be exact. However, to
disagreeing students, a half had to be a half. Rob's diagonal cut the squares so
they were more than a half, and together made more that a whole.

Based on Shawna's previous Observation that "that one and that one” put
together would be one whole, Mrs. Vandenberg's question, ”What do we call
that?” seemed to be about the whole Shawna constructed. A few students

appeared to disagree over what the partial squares constituted, saying, "It would

 

example in fling Fragtipns cuts in the opposite direction from Rob’s.

87

 

be more than a whole!" Their use of ”it" created confusion. The children
appeared to use the pronoun, it, as a referent for different things. Whereas these
disagreeing students used ”it” in a plural sense to refer to the reconstituted parts,
Shawna pointed to the partial squares individually and referred to each as ”it."
”It's a little Off but it's still a half.” Also, it is interesting that earlier Shawn
referred to these shapes as ”this one and that one,” but here she labeled one as
"half,” which Joanne disagreed with (when she said "A little MORE than a

half. "). Rob’s Off-center resulted in confusion for these children about what

constituted a square and half a square.

 

 

 

 

 

Figure 7: Moira Continues to Outline ”Squares” on May 7

If the children had realized they could examine these partial squares in a
different way, they may have seen a pattern and, therefore, a connection,
between the halves on Rob's board. This connection could have in turn led them
to additional understanding about how to prove the left side was indeed a half

(see Figure 8, below).

 

 

 

Figure 8: A Way to Use Partial Squares to Prove Half.

To encourage children's developing notions of proof, the authors of the
fractions curriculum Mrs. Vandenberg used to teach geoboards offered
suggestions to teachers. Specifically, when introducing halves with the
geoboard, they suggested that teachers ”[Elncourage students to develop and
articulate their strategies for proving their halves are equal and provide as many
materials as you can to help students develop their own proofs” (Corwin,
Russell, 8: Tierney, 1990, p. 16). Teachers were instructed to look for and discuss
fractions which did and did not have the same shape and that they help children
think about equal pieces.

What about the sense Of proof during the disagreements over Rob's
geoboard? What strategies emerged? Mrs. Vandenberg appeared to draw
children' attention to proof by asking whether halves were equal. For example,
when Rob showed his design with 1/ 2 and 2/ 4, she immediately asked, ”How is

that 1/ 2 and 2 / 4?" Subsequently, she asked questions such as,

How do you prove that that's a half over there?

What do you guys think about Ray's way? Lots of you, almost ALL of
you I saw found ways to show this. What do you think of RAY's way?

This is equal to this? Let's take a look at just the halfs first. When Ray
divided this in half, were these two sides equal?

How can you tell?

Do you think if we cut that out that would help us see the two halves?

In addition, Mrs. Vandenberg frequently revoiced (O’Connor 8: Michaels, 1993)
children' contributions during the disagreement. She echoed and amplified their

ideas with these revoicings, saying or asking things like,

89

SO you found HALF on one side and you had 4 pegs and another 4 pegs.

What do you mean, 'it looks equal'?

They both have 6? Can you go point to them?

Amy says you need to put together the two halfs.

This was helpful. When Hannah was counting the squares, remember

when she did this, so we could see?

Regarding Mrs. Vandenberg's role, her initial question about Rob's
representation brought out disagreement. While Moira outlined all partial
squares along the diagonal, Mrs. Vandenberg’s second question, ”Is the other
side a little more, too," drew the children' attention to the right side of the
diagonal, to think about the partial squares there, too. When Moira was done,
they all counted the squares on each side of the diagonal: ”Six and two halfs
make seven.” This question was interesting because the children said "yes,”
even though the squares pointed to on the other side were not the same size.
Perhaps the children thought it all balanced out, basing their thinking on what
they saw; or, maybe they didn't see a difference.

Additional disagreement over Rob's geoboard representation was
generally about the two pieces in the 2 / 4 on his board. Whereas, in the
disagreement above, the children worked to figure out halves in the absence of
Rob’s parallelogram, in the disagreements below, he recreated this unorthodox
way of making 2 / 4 on the right side of his Off-center diagonal. In the first,
described below, although he pointed to each of the fourths (in this case, the
section Of the board above the parallelogram and the section below the
parallelogram) on the board and counted squares, Ali disagreed, apparently

thinking that the partial squares were thirds, not halves:

90

KV: SO Rob, how did you make it into FOURTI-IS? You've got your two
halves [of the board] and we thought those were equal. Then what did
you do?

Rob: I took one rubber band, and made it go down. And then end up
with 1, 2, half; 1, 2, half. 2 and a half EACH.

Ali: 1, 2, and a THIRD

KV: Ali's saying that's a THIRD?

Ali: That's TWO pieces, not one.

Rob: One right here, one right here (points to two parts)

KV: One right HERE. OK. This is a way I saw lots of people talking

about it. Here's a HALF (points to the left). Now I need to make 2/ 4 over

here (points to the right of the diagonal). DO you agree? DO you think

this is 2 / 4 over here, Rob?

In the disagreement, Ali appeared to be saying that the pieces Rob pointed
to were thirds of a square, not halves. And a look at the overhead geoboard on
which Rob counted squares sheds light on this disagreement (see Figure 9,

shown below).

 

 

 

 

 

Figure 9: Overhead Geoboard Referred to by Rob to Indicate Fourths, May 7

When Rob counted squares in his ”fourths," he said, "one, two, half,” and
pointed to the two empty squares and one partially empty square. To him, these
segments Of the board constituted a fourth of the board. However, in his noting
and counting of these segments, he omitted the partial squares outlined moments
before by Moira. In other words, he did not take into account the segments of

the board the children had just debated — the partial squares referred to by

91

Shawna and outlined by Moira. And Ali noticed his omission. She appeared to
be telling him that he left out a part of the half square he had pointed to, to say,
"'That's TWO pieces, not one.” Her meaning is not clear because the ”that" is not
specific, although it is reasonable that she meant Rob counted only one partial
square and there actually were two parts to that square. Rob seemed to assume
that the partial pieces were halves, as indicated when he said, "One, two, half,”
but was Ali talking about them, or something else? When Rob followed up and
said, "One right here, he appeared to agree with Ali's Observation that there were
two pieces. The issue was not resolved, because Mrs. Vandenberg repeated
Rob's "one right here,” and moved on.

What did these students need to think about to be involved in these
disagreements? These brief disagreements originated from Rob's representation
and had the students thinking about two issues. First, they needed to look
closely at the squares on the geoboard and the way in which they were cut up by
his off-center diagonal line. Second, they had to reflect on the question about
making 1/ 2 and 2 / 4 on the board, and whether Rob's design did, in fact, do this.
And, with a design like this which is not a cut across the diagonal, a careful look
by these children was necessary. Their disagreements raised these relevant
issues about squares and space on the geoboard. When Shawna said, ”It's still a
half,” was she implying that because it's not a whole square it shouldn't be
counted? After all, she went to the overhead and said, ”Six if you do that", then
went on to talk about the partial squares making a whole. She didn't originally
say the partial squares were halves - not until this speaking turn When Mrs.
Vandenberg asked, ”Is the other side a little more too?”, she appeared to be
asking if the partial squares on the other side Of the diagonal would be a little

92

 

more, too, because they were talking about those squares being half. Also, while
Shawna and then Joanne spoke their turns, Moira connected kitty-corner partial
squares on the other side of the diagonal. She did that while Mrs. Vandenberg
asked the question about the other side.

As they did in other disagreements, the children took up each other’s
comments to initiate a turn to disagree. To get their differing ideas on the floor,
they spoke for themselves to make the disagreeing moves. Because of the way in
which they took a turn to disagree with a classmate in order to get their idea on
the floor, this disagreement looked conversation-like and simultaneously
challenged the children and Mrs. Vandenberg with new roles and responsibilities
as discourse users.

DiSaggeement Without a Visual Representation: Is 200 + 19 a Fraction?

A second example from the category, Disagreement to Defend, came from
May 3, the second day of fractions, when whole-group discussion was an
extension of April 28. As described in Chapter 4, math time on May 3 followed
Format 2, where the lesson began with Mrs. Vandenberg's orienting comments
and whole-group discussion followed. After the children returned from lunch
recess and settled into their chairs, math began. Mrs. Vandenberg planned to
continue discussion from April 28 when she had the children work in small
groups to respond to three exploratory questions, which she projected at the
overhead: What is a fraction? When have you used fractions? How are fractions
represented? Draw and label pictures. Whereas on April 28, the children worked in
small groups recording their answers on large sheets Of paper for the majority Of

class time, on May 3, Mrs. Vandenberg planned to begin with whole-group

93

discussion. Following one boy's comment about a fight earlier in the day, Mrs.
Vandenberg immediately oriented the children to math time:

KV: I'm gonna remind you right now there's no calling out. OK? If you
choose to continue calling out you're gonna have to go down to the office.
OK. Um, we haven't had math in a while on WEDNESDAY”, we started
talking about FRACTIONS. And, THURSDAY was our field grip and
FRIDAY we only had a half day. So we're gonna pick up where we left off
there when we were talking about the three questions on the overhead.
Jessica put your book away. Question Number One was 'What is a
fraction?’ Question Number Two, 'When have you used them?‘ And
Question Number Three, 'How are they represented?‘ On Wednesday we
got to hear Bart and, Micah's idea. We also heard from, Tomika, and, '

T: Ali and Monica.

T: Just a reminder. The REST of us, will be taking notes, keeping track of
what DIFFERENT tables' ideas were. Remember how we started doing
that on Wednesday?
In a few turns at the start, she overviewed their previous lesson, the questions
then and now, said they would be taking notes in their journals, then began
whole-group discussion while focusing on the first question, "What is a
fraction?”
Mrs. Vandenberg first called on Monte, asking, ”What did you think a

fraction was? How did you describe it?" He responded with words, and then

wrote a number sentence and drew a picture at the overhead. ”A fraction is,

 

10All capital letters denote speaker emphasis.

94

 

where you have a big number and you can't solve it so you split it up into groups
so it's easier to solve. . . A fraction is a piece of the problem", he said. While he

talked he wrote for his example, ”200 + 10" and drew several of the following

”groups” (see Figure 10):

 

 

 

 

Figure 10: Monte's Drawing of Groups on May 3.

Others then aired their ideas when called on by Mrs. Vandenberg. For
example, one girl wondered aloud, 200 + 10 is a fraction?", then said she thought
”division and fractions are related. . . Like you, you can make that [200 + 10]
into a fraction. " Another girl, Hannah, who would be a prominent figure in the
discussion that day, said to Mrs. Vandenberg that 200 + 10 was not a fraction. ”1
don't think it would be a fraction because 200 + 10, if it was a fraction, it would
be um, less than like zero. You would like, you would take—you would have to
take pieces out. . . So it's less than one." Apparently rnisspeaking, Hannah
revised her statement during this turn. She first said a fraction would be less
than zero, then less than one. She seemed to be saying that a fraction is part of
one whole. Hannah's remaining comments on May 3 were consistent with this
position. Subsequently, Hannah said, ”I'll show you what a fraction is." and

went to the overhead to begin what would be an extensive explanation using

11Vertical dots indicate sections of transcript not in the text.

95

 

circular representations. First, Hannah drew a circle divided in half and said,
”There is a fraction. One-half.”

When Hannah went on to say why 200 + 10 is not a fraction, Monte, still
near the overhead, momentarily defended his position that it can be a fraction.
They dueled verbally, point-counterpoint; 200+ 10 is not a fraction—it could be a
fraction. Neither gave a reason for their position, nor did Mrs. Vandenberg seek

one.

KV: [200 + 10] Is NOT a fraction. But what you showed us is a fraction.
Hannah: It's not a, it's NOT a fraction, because-
Monte: But it CAN be a fraction.

Hannah: because a FRACTION is less than ONE. TEN and TWO
HUNDRED are not less than one. HERE is another kind of fraction.
There are other kinds of fractions. THIS would be, one-fourth, if you
shaded in one.

KV: OK.
Hannah: And,
KV: How did you know it would be one-fourth?

Mrs. Vandenberg repeated what Hannah said, then made a clarifying
statement, ”But what you showed us is a fraction." When Hannah said 200 + 10
is not a fraction, Monte disagreed, to say it can be. Hannah continued her
explanation, and Mrs. Vandenberg probed her thinking. This momentary
disagreement demonstrated that these children were attempting to get at the core
of the question on the floor, What is a fraction .7, and in the process of exploring the
meaning of a fraction, brought to the fore two seemingly contradictory big ideas:

that a fraction is a part of a whole, and that a fraction is part of a large number.

96

 

Kim’S Diamong
A third disagreement to defend was Kim’s diamond. The cluster of single
disagreements over this representation provided a window into students’
thinking about what they got when they combined two triangles, and whether

the shaded portion on the diamond was one-twelfth or one-eighth.

 

Figure 11: Kim's Diamond on May 11

This disagreement from May 12 raised the fractions notion of equivalence.
This time the disagreements were prompted by a representation Kim initially
created on the geoboard for a partnered task Mrs. Vandenberg prepared the
previous day. Using a handout printed by Mrs. Vandenberg, on May 11 the
children were to do the following on the top half of the paper: Draw a picture
which represents your fraction. Explain in your journal. Then, on the bottom half of
the paper, she wrote further instructions: What fraction does the picture above
represent? Why? How did you solve this? In other words, the first child was to
think up a fraction and draw a picture to represent it — without writing down or
telling what the fraction was. (Also, the student wrote his or her name.) Then
the second child was to name the fraction from the picture, and on the bottom
half of the handout, write his or her name and explain how he or she solved it.
On the handout, Kim wrote her name and on the back of the paper drew a
diamond shape with one square colored in (see Figure 11, Kirn’s Diamond on

May 11). Following Mrs. Vandenberg’s verbal instructions, Kim also drew the

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picture in her journal and explained it. In her journal, Kim drew the diamond on
the assignment sheet as well as in her journal. Under the journal drawing she
wrote, "My fraction shows 1 / 8 because there is one colered [sic] in and there is 8
all together” (journal entry, May 11). Her partner, Jody, examined Kim's shape,

signed her name, and wrote the following explanation on the handout,

the owser [sic-answer] is 1/ 12 there is 12 things and 1 colored in so it is
1 / 8

1/12 it is a fraction because there is 12 things and 1 colord [sic] so it is a

fraction” (handout, May 11).
A bit difficult to decipher without punctuation and with misspelled words, Jody
appeared to say that there were two answers to this fraction, one-twelfth and
one-eighth, which she talked about when she was invited by Mrs. Vandenberg
the next day to the overhead to share the shape she and Kim worked on.

To begin math time on May 12, Mrs. Vandenberg said they would be
sharing yesterday's pictures. Jody volunteered and went to the overhead with a

prepared transparency on which she had drawn Kim's diamond and written,

there are two ansers to this problem 1 / 12 1/ 8

Jody said, ”There were two ways I thought to solve this problem.” She counted
12 "things" said one was colored in, ”so I thought it would be one-twelfth. But
when I showed Mrs. Vandenberg my answer she said that [pointed to the top
corner of the diamond] equals one." Then, Jody counted pieces again, saying
there were eight without counting with a one-to-one correspondence, number to

piece counted. Subsequently, Mrs. Vandenberg observed that, ”A couple people

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say they don't get it,” and asked Hannah to re-state what Jody said. In her own
words below, Hannah answered,

Jody thought at first there were 4 squares and 8 triangles and that's 12.
And she thought it was 12 and 1 colored in, so it was one-twelfth. But
then she said Mrs. Vandenberg showed her that each triangle, if you put 2
triangles together that would make one square. So, she counted 8, and 1
was colored in so it would be one-eighth.

Elaborating what she said earlier about 2 triangles and one square, Jody
above said when she put them together, what they equal was "just one-half.”

Perhaps conscious of students' confusion about what Jody had been saying, Mrs.
Vandenberg said,

Jody, I think some people are having difficulty understanding what you
meant by 'putting them together.' (goes to the overhead and writes
questions above Kim's diamond) I want each of you in your journal, you
might want to draw a picture to help you. But answer this. Pretend Jody
was YOUR partner, or Kim, and you got this [the diamond]
representation. What fraction does this picture represent?

For the next 5 minutes, the children wrote in their journals while Mrs.
Vandenberg moved from table to table, asking them questions about their
responses. When she resumed whole-group discussion, several students said

they wrote one-twelfth as the fraction in Kim's picture. When she asked Jody

what she thought, Jody answered,

Jody: One colored in of 12. And my other way was one-eighth. 1, 2, 3, 4,
5, 6, 7, 8. (points to and counts each corner as one)

KV: What made you decide to think about it TT-IAT way?

Jody: One of these (points to top corner) is a whole square, and those two
make a HALF.

As Mrs. Vandenberg worked toward gleaning Jody's thinking about her
answers, Jody seemed to move from one-twelfth to one—eighth without firm
convictions about either. Her dubious understanding of what fraction Kim's
shape represented prompted several disagreements between her and Braden.
The first disagreement between Braden and Jody raised the issue of whether two
triangles in her rendition of Kim's drawing made a square. Braden said they

were bigger, but Jody thought they still made a square:

KV: One more time, Braden.

Braden: If you put THESE two triangles together into one square they'd
make a bigger square than one Of these, because if you put say just one of
these triangles, take this triangle (draws a triangle beside Kim's diamond)

Jody: Yeah but I thought it would still make a SQUARE so it didn't
matter.

Braden: Yeah if you put a triangle like this. (He draws one triangle)
KV: I wonder if it would be helpful if we did this on the geoboard, if

somebody did that one on the geoboard for us. How about Kim.

Although Braden said putting two triangles from Jody's drawing together
"would make a bigger square than one of these [a square]", to Jody it didn't seem
to matter if what they made was bigger than a square in the diamond - two
triangles would still make a square, presumably on or off the diamond. Then,
when Braden appeared to start drawing his triangles, Mrs. Vandenberg
suggested that putting the design on a geoboard might be helpful. However,
while she was looking for a volunteer to use the geoboard, Jody tried
unsuccessfully to put two triangles together to form a square (see Figure 12,

Jody’s Attempt to Combine Triangles to Make a Square on May 12).

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a

Figure 12: Jody’s Attempt to Combine Triangles to Make a Square on May12

KV: How about Kim.
Jody: THAT wouldn't make a square ANYWAY.

Braden: Yes it would. Those two and those two. (he draws 2 right
triangles so that they make a square)

KV (to Braden): SO you're saying those two points in the square could be

put together as a triangle?

Whereas Braden formed a square from his triangles, Jody struggled to get
it to work in order to support her claim that two triangles would make a square,
so it didn't matter how big they were. At the end Of the above disagreement,
Mrs. Vandenberg asked Braden a question to clarify what he said, but Braden
either didn't hear her or ignored the question, talking next instead to Jody.

Braden and Jody continued to debate how to make a square from two triangles.

Braden (to Jody): But you see THAT would be bigger.
Jody: You need something like this (again she draws 2 triangles, one

under the other. I don't get it. Those don't look like SQUARE.

Again, Jody attempted to create a square from two triangles, and again
she faltered. Still, she held steadfast to the notion that they should make a
square, and by doing so, continued to hold her idea up beside Braden’s.

Disagreements to defend an idea functioned to sustain two ideas, side by
side. In the moment they did not seem as collaborative as disagreements to

correct, revise, or mediate. Neither did they seem competitive. The children

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appeared to make this move to sustain a second idea, to be weighed by others on
its merits.

Thus far, I have discussed disagreements which functioned to pinpoint a
flaw connected to a representation (Correct, Revise, Mediate) and disagreements
which functioned to hold up two ideas simultaneously (Defend). The
mathematical significance of the first category of purpose drew children’s
attention to a surface characteristic about a representation. The mathematical
significance of the second category was to get these children to look more deeply
into competing ideas on the floor. The third category of disagreement discussed
below, Question Another’s Assumptions or Call for Clarification, forced the
fourth graders to look even more deeply at the mathematical content about
which they disagreed.

Call for Clarification, Question Another’s Assumptions

This category functioned to get at the mathematical underpinnings of
another’s position. For example, disagreement over Kim's diamond did not end
when, as discussed above, Kim struggled to get two triangles to make a square.
To the contrary, the disagreement continued and functioned to get at the core of
Jody’ s reasoning about how much of the whole was represented by the shaded
portion of Kim’s Diamond. Braden shifted the disagreement to be about the
question on the floor: What does the shaded portion of Kim’s Diamond represent? TO
demonstrate Braden’s topic shift, I pick up the disagreement where it left Off,

above:

Braden (to Jody): But you see THAT would be bigger.

Jody: You need something like this (again she draws 2 triangles, one
under the other. I don't get it. Those don't look like SQUARE.

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Braden: When you wrote 1/ 12 down here, I disagree with that. Are you
counting these triangles?

Jody: 1/12. Ijust counted the triangles as ONE.

Braden: like 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12? (counts all pieces, both
triangles and squares)

Jody: Yeah.

Braden: Well that would only be 1 / 12 if they were ALL equal GROUPS.
One of these (points to a triangle) is SMALLER than THAT (points to a
square). It's half size.

Jody: I know but THESE 2 together equals a half. But when I showed
Mrs. Vandenberg she said why don't you just count THESE 2 together as
one whole one.

Braden: But you mean like these 2 (triangles) is one whole one? I get 1, 2,
3, 4, 5, 6, 7, 8.

KV: Who can explain what the PURPOSE was for, how many people got

1/12? Raise your hand if you got 1/ 12 (5 sts raise hands) When you got

1/ 12 you counted this as 12 pieces?
Abruptly, in the middle of disagreeing over how much area two triangles
covered, Braden shifted the topic to disagree about what fraction of the diamond
was colored in (the question Mrs. Vandenberg posed at the beginning Of class).
When Braden said, ”THAT would be bigger,” he pointed to two triangles,
maintaining his earlier stance that two triangles would be bigger than a square.
Jody again attempted to make a square of two triangles, perhaps as Mrs.
Vandenberg showed her the day before; however, for the second time she
couldn't get it to work. When Braden shifted topics back to the fraction
represented by Kim's diamond, he and Jody were quite specific about what they
counted. When Braden replicated Jody's counting and came up with 12 pieces,
Jody agreed that was how she counted. Then Braden raised a new point, that it
could only be one-twelfth if all 12 pieces were equal. In addition, he appeared to

change his mind about two triangles making more than a square, because he said

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they were smaller than a square; a triangle was half size, meaning half of a
square. At that point, Jody said that two triangles equaled a half, even though
she simultaneously restated Mrs. Vandenberg's suggestion to "just count these
two together as one whole one.”

At the end of these disagreements which were prompted by Kim’s
Diamond, Mrs. Vandenberg changed her question mid-sentence, seeking a vote
on how many students got one-twelfth as the fraction on the diamond. Five
children raised hands to vote, and several also said "yeah” in response to her
final question, ”When you got one-twelfth you counted this as 12 pieces?"
Perhaps because some still thought the shape represented one-twelfth, Mrs.
Vandenberg showed Kim's diamond shape, which Kim re-created on a geoboard
(versus a drawing). When asked, many children then said they could see that
two triangles are the same size as one square, and that they thought the fraction
that the diamond showed was one-eighth. In the end, Jody asked Kim, ”So what
was your answer REALLY, Kim?”, to which Kim answered, ”one-eighth," and
demonstrated how she added ”one-halfs" and wholes to get 8 total squares in
her picture. Further, she added, her dad helped her, that ”he told her to write it
down and add it. I tried it and he helped me. . . I knew how many halves to
add because if you add 2 halves it would make a square.” Underscoring her
thinking, she wrote in her journal on May 12, ”It repersents [sic] 1 / 8 becaus [sic]
there is 8 squares 8: 1 is colered [sic] in so it's 1/8." Also, within her diamond
drawing, Kim recorded the fraction each piece represented, 1 / 2 or 1, and wrote,
referring to the triangles in the drawing, ”1 / 2 + 1 / 2 is 1 hole [sic] because all you

have to do is draw a

104

N

picture to make it" (journal entry, May 12).

Mathematically, at the heart of these disagreements were Jody's false
assumptions about Specific areas of this diamond. flat, she based her definition
of a fraction on the number Of pieces in the shape without regard to their size.
Her answer to the handout question, What fraction does the picture above represent?,
was one-twelfth. In addition, she supported this answer with her written
reasoning, ”the answer is 1 / 12 there is 12 things and 1 colored in so it is 1/ 12 it is
a fraction because there is 12 things and 1 colord [sic]." However, her side
conversation when Mrs. Vandenberg suggested putting two triangles together to
count as one whole square prompted her to take this second idea (and different
from her own about ”12 things” in the diamond) to whole-group. Jody shared
Mrs. Vandenberg’s suggestion with the class (and Braden) when she put up the
prepared transparency of Kim's drawing and her own thinking, ”there are two
ansers [sic] to this problem," - 1/12 and 1 / 8. And, on the right edge of the
handout she worked on with Kim she wrote, ”theres [sic] two was thig [sic]
about it 1/ 12 or 1/8.” (That is, ”There's two ways to think about it: 1/ 12 or
1/8.”) In addition, in her journal she struggled with ways to put together two
triangles to get a square, because there she drew a similar picture (this one of
side-by-side triangles) to what she made at the overhead, above. Finally, not to
be overlooked, Mrs. Vandenberg's suggestion, while well intended, may have
done the opposite - either caused Jody to falter in her attempt to grasp how the

two triangles could be combined, or kept her from looking closely enough at the

105

two answers she chose to be able to reason through which one made most sense.
She did not appear to realize, either on the handout or during discussion, that to
form a square the triangles had to be right triangles, and her repeated efforts to
make it work failed. 25% she appeared to think that two triangles in her
borrowed drawing made a square on or Off the picture, so the fact that they were
bigger than a square, as Braden pointed out, seemed irrelevant to her. But if she
maintained this stance, Braden's reason for disagreeing would completely escape
her. Where he questioned whether two triangles of the shape would be the same
size as one of the squares, Jody took the triangle-to-square comparison out of
context. M Jody appeared confused overall about what, exactly, two
triangles constituted. Her first position was that a triangle was one of 12 pieces,
so should be counted as such, regardless of size. Then, to her it was of no
apparent consequence whether a square made from them was a part of this
particular shape or not. Presumably, her teacher told her that two put together
made a square, so Jody interpreted that to be any square, on or off this shape.
Finally, whether she misspoke or said what she meant, in her last speaking turn,
above, she said two triangles were half, apparently half a square. In spite of the
issues Braden raised and Mrs. Vandenberg's suggestion about combining
triangles, Jody seemed not convinced that the colored area on Kim's diamond
was one-eighth.

A second important mathematical issue is the distinction these children
needed to make between a fraction as a number Of discrete Objects versus a
fraction as an area. Repeatedly, Jody worked at combining two triangles to make
a square, and repeatedly her attempts faltered, even though her teacher

suggested she do so. In addition, the shaded portion of Kim’s diamond forced

106

these fourth graders to shift their thinking back and forth, from what portion was
shaded, to the relationship between that square and the diamond as a whole.

A third important mathematical issue is relevant to these disagreements
over Kim's diamond - the question of what constitutes the whole. Whereas
under other circumstances, the geoboard may be used to represent the whole, in
this instance it was not. Instead, it was used as a tool to create a different whole,
the diamond. Therefore, a connection between what they did earlier in the
fractions unit with the geoboard may have been lost.

While the disagreements between Braden and Jody over Kim's diamond
unfolded, none of the other children vied for the floor. While these two
disagreed, what they disagreed about was quite clear. They frequently drew
triangles and squares, pointed to them, and referred while pointing to "these”
and ”that.” What was not clear to Jody was how to make a square out of two
triangles. Apparently in an effort to help Jody "see” that, when combined, the
two triangles in a corner of the diamond would make the equivalent of one of the
diamond's squares, here, Jody either could not remember or did not know for
herself how to put the triangles together. Repeatedly, she stacked one on the
other and, when seeing they did not make the square she anticipated, she
retracted her earlier assertion about putting them together. If she knew or could
remember that in order for the combination to work, they had to be a particular
kind of triangle (right), her confidence in what her teacher told her would not
have been shaken.

Additional disagreements arose which questioned assumptions held by

others, including Hannah’s drawing of two—fourths to represent one-half. In

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these disagreements, there were no disputes over the size of pieces, but how
many, and why, as discussed below.

Hannah' 2 4t how Half

 

 

 

 

 

 

Figure 13: Hannah's Drawing of 1 / 2 on May 4

Here, leading up to the disagreement was a child's decision to create a
drawing, rather than respond with a geoboard representation (which was what
the question on the floor intended). On May 4 (the third day Of data collection
and the second day they worked with geoboards), Mrs. Vandenberg wrote on the
overhead and asked, How many ways can you show halves (1/2)? Children were to
practice on geoboards, record their ways on dot paper, and write one
explanation in their journal, explaining how they could prove their idea showed
a half. Jonah showed his horizontal division Of the geoboard; then Mrs.
Vandenberg asked Hannah to prove his way was a half. Rather than using a
geoboard or the projected drawing of a geoboard to explain her proof, Hannah
put up a new transparency to create a drawing (see Figure 13, Hannah’s Drawing

of1/2 on May 4):

KV: Hannah, how can you prove what Jonah showed us is a half?

Hannah: Say you have a square...a brownie..And you have 4 people-
yourself and 3 friends. SO you've gotta split it like this.

Jody: That's more than a HALF.
Jonah: Yeah.

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KV: Why do you say that's more than one-half, Jody?
Jody: Because it has four pieces.

While she drew, Hannah appeared in the moment to create a story to
accompany it. Her exploratory talk (Barnes, 1992) about the fraction and the
story unfolded in the moment. She referred to her square as a brownie that she
would split with friends. However, Jody, and then Jonah, quickly spoke up to
express disagreement. Subsequently encouraging Jody to say more, Mrs.
Vandenberg asked her about her idea. After Jody answered, ”Because it has four
pieces”, Mrs. Vandenberg suggested they listen to Hannah, which they did. In
her explanation, Hannah went on with her story, saying two of her friends went
home, so they each got two, ”OK see how I colored that in (pointing to the half
colored in) That's one half. . . One half. Four, take away two is two. Cause two
plus two is four. It's a half. "

Asking Hannah about how she proved her idea, Mrs. Vandenberg then
asked, ”Are you saying that you could prove that was a half by seeing how much
of the brownie you got? Anna, is that a way?” When Hannah answered, "Yes,"
Mrs. Vandenberg revoiced (O’Connor 8: Michaels, 1993) Hannah's proof, and
then returned to the geoboard and the first question of the day, "Can you think
of any other ways to show half on the geoboard?”

This and other disagreements revealed an important aspect Of
disagreement in this classroom. Through her question asking and willingness to
have the children speak for themselves, Mrs. Vandenberg and the children
mutually re-negotiated turns and ideas on the floor (Edelsky, 1993). She tended

to ask questions which brought out more and more of the children' thinking.

109

And, whereas she Often was actively involved in the discussion through a
teacher-student-teaCher-student turn-taking pattern, she also momentarily
stepped back, so that children spoke up, student-tO-student and idea-to-idea, to
disagree for themselves. Second (and characteristic of disagreement in this
category), Hannah may have been more comfortable using her own drawing
than the prepared geoboard picture on the overhead. Rather than responding to
the overhead question with use of the geoboard as a representation of one-half,
Hannah returned to a drawing to demonstrate her thinking. After all, on the
previous two days of the fractions unit, it was drawings children used to flesh
out ideas in response to the teacher's questions. Perhaps Hannah did this
because she was less comfortable with the geoboard, or perhaps because at this
beginning point in geoboard use, she preferred her own picture to explain her
conception of one-half. Here, as she did before (and would do again), Hannah
appeared to simultaneously create a story and a drawing, both of which used a
different context and model from Jonah's geoboard half. Third, these children
appeared to hold different notions of what constituted a half. Whereas Hannah
first drew a horizontal line across her "brownie," similar to the way Jonah earlier
divided a picture of a geoboard in half, she then added a second, vertical one,
prompting disagreement from Jody and Jonah. Undergirding Hannah's
representation seemed to be the assumption that one-half and two-fourths are
equivalent. But to Jody and Jonah, Hannah created something more than a half,
for they seemed to think in rigid terms: that one-half is one of two pieces. Also,
they apparently compared the number Of pieces and saw the two pieces in two-
fourths as more than the one piece on one-half. They seemed to be thinking in

terms of number of pieces, not that the area on the right side of Hannah's square

110

was the same, whether one large piece or two equal-sized pieces. That is, Jody
and Jonah may not have seen the equivalence that Hannah did. Also, they could
have been tripped up by the way in which Hannah changed representations,
from use of the geoboard (which the overhead question had them explore), to her
hand-drawn square, or brownie. However, the question about equivalence
raised here by Jody was explained when Hannah continued a story about having
3 friends, saying 2 Of them went home, so the two friends each had one-half of
the brownie. Hannah had a clear sense of what the whole was in her square-
brownie-and-friends scenario. While she spoke, she colored in the two squares
on the right. Mrs. Vandenberg asked, "Are you saying that you could prove that
was a half by seeing how much of the brownie you got? Hannah is that a way?”
Hannah answered, ”OK, see how I colored that in? That's one-half.” Then, Mrs.
Vandenberg redirected the focus and discussion back to the use of the geoboard
to think about the overhead question, above.

Hannah stirred further disagreement in the category, Question Another’ 5

Assumptions, with another drawing, this time of a television (”TV") to represent

lfifi

— _ _ _ — — '

\{

Figure 14: Hannah’s TV (Drawn from Side View) on May 10

one-half, discussed below.

 

 

 

 

Hannah's drawing of one-half on May 10 was the object of two
disagreements which functioned to get at the mathematical underpinnings of

what can represent one-half. The premise of these disagreements was the

111

assumptions undergirding the positions of each participant. The overhead
questions were from their homework: Where do you findfractions? How do you
represent fractions: 1/2, 2/4, 7/14, 4/4? To discuss the first question, Mrs.
Vandenberg asked, ”Hannah, did you find any fractions in your house?” When
Hannah said she didn't have time to do her homework, Mrs. Vandenberg asked

where she might look now:

KV: Where might you look now, that you've heard some of our ideas?
Hannah: At our TV.

KV: At your TV. Why at your TV?

Hannah: 'Cause I could divide my TV in half.

st: NO.

Tomika: How you gonna watch it?

KV: Explain what you mean by that, Hannah.

When Hannah proposed the notion of looking at her television to divide it
in half, two students questioned it. Tomika's practical response went to the core
of the usefulness of a television cut in half. Then, when Hannah drew a

television and explained where she would cut it, children again disagreed.

KV: OK how is that like a fraction?

Hannah: You can cut it in half like this (draws a horizontal line through
it)
Tomika: But then it won't work.

Hannah: It doesn't have to work...

KV: SO you're telling us you can divide anything in half, Hannah?

Again, Tomika's practical response, "But then it won't work," seemed to

indicate that to her, whether or not a television would work was the issue, not

112

whether it could be used as an example Of the idea that anything could be
divided in half, as Mrs. Vandenberg indicated. To Tomika, it appeared that what
mattered was the functionality Of the television. Tomika’s perspective, the
mathematics in Hannah’s drawing was irrelevant. There was no mathematics in
it. Her concern seemed based on the function of the television. By contrast,
Hannah appeared to have more Of a mathematical basis for her TV
representation: anything can be cut in half to demonstrate a fraction.

I included these examples of disagreement to demonstrate that hand-
drawn pictures appeared to be comfortable representations for these fourth
graders, more familiar to them than the geoboard, as I will later show. In the
everyday world of children, there are brownies or pizzas or cakes to be cut.
There's little doubt what the "whole” is, whether one food item or another. After
all, these children created and frequently defined the whole. Not only did
children have a sense of the whole in these situations, they had ways to ensure
the pieces were equal. After all, in real life, ask one person to cut a cake knowing
another gets to choose the first piece. In such an instance, children think about
cutting equal pieces to be fair. But Jody's protest appeared to show that it is also
reasonable from a child's perspective to question whether it's important to think
about the size of a piece or the number of pieces.

Also, hand-drawn representations demonstrated children' ownership of
their ideas. In the drawings, we saw glimpses of how they used real life to
connect to the less familiar domain Of fractions. Even one-half, considered to be
the generic referent by children for a fraction, surfaced in the data again and
again to be a complicated concept for these fourth graders. And in the case of

Hannah's TV, we had a window into a real-world application of one-half (doubly

113

situated in Hannah's world by Mrs. Vandenberg, in her pre-disagreement
question, ”Hannah, did you find any fractions in your house?"), taken to a level
so abstract that Tomika could not grasp it. To the contrary, Tomika took

Hannah's representation completely out of mathematics.

Disaggeement Over the Location of 1(2 on the Number Ling

Disagreement to question another’s assumptions also occurred over the
number line, the most abstract representation used during the fractions unit. The
location of one-half on the number line was especially problematic for Jody. A
cluster of three disagreements erupted on May 13 when the question to begin
class was, Where is IE on the number line? This was these fourth graders' first
classroom experience with fractions on the numbers line, and it appeared to be a
difficult concept for them. To Jody it appeared difficult, because three times she
questioned others' placement of one-half on the number line. First, when
Antonia put a mark mid-way between 0 and 1, and 0 and -1, Jody expressed

concern that 1 / 2 could be in the domain of negative numbers:

KV: What made you decide that?

Antonia: I don't know. Or else it could be here (between 0 and -1)
Jody: But that's a NEGATIVE number you're going by.

KV: Antonia, why did you put it between zero and 1?

Soon after, when Mrs. Vandenberg drew a new number line marked with
numbers 0 to 4, She asked Micah where he would put 1 / 2. When he placed it

between 2 and 3, Jody again raised her doubts:

KV: Where would 1 / 2 be on THAT number line? Micah?
Micah: (responds nonverbally by putting a mark between 2 and 3)

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Jody: Idon't think so.
KV: Why did you put it there, Micah?

The third time Jody disagreed was when Joanne placed 1 /2 in the same

places as Antonia had, between 0 and 1, and between 0 and -1.

KV: . . Joanne, what do you think?

Joanne: Here or maybe right here (she puts 2 marks; one is left of zero)
Jody: I didn't think we were going to the NEGATIVE numbers!

KV: What WOULD it be in between negative 1 and negative 2?

Here, Jody again appeared to disagree with the idea that one-half might
be with negative numbers. Jody initially seemed to Operate under the premise
that one-half was a positive fraction and should therefore be with positive
numbers. In this regard her assumption about 1 / 2 seemed mathematically
sound, and mapped onto the first part of Antonia's and Joanne's assumptions.
To Antonia and Joanne's way of thinking, it also appeared that 1/ 2 could be
between zero and negative 1, perhaps because 1 / 2 was midway between zero
and 1. In this way they appeared to assume that 1/ 2 could be either a positive or
negative number. They seemed to sort of peek into the domain (and placement
of 1/ 2) Of negative numbers. But Jody seemed almost adamant that 1/ 2 did not
belong with negative numbers, and technically she was right. After all, Mrs.
Vandenberg did not ask about locating negative one-half. Yet, it was not just the
placement of 1 / 2 with negative numbers she objected to. She disagreed with
Micah, also, who had put it between 2 and 3. When Mrs. Vandenberg asked
where Jody thought it should be, she said, "I think it's between the 1 and the 2

because you've got the 1, and in between the 1 and the 2 ....you make a half. SO

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you've got one HALF.” In her explanation, Jody seemed to be saying that one-
half belonged between 1 and 2 because there is a 1 and a 2 in ”1/2.” This stance
helps explain why she objected to putting one-half with negative numbers, but is
less helpful when trying to understand why she objected to putting it between 2
and 3. She was comfortable having 1 / 2 between 1 and 2, which are whole
numbers, but not between 2 and 3. The logical explanation appears to be that she
thought this fraction should be between 1 and 2 because it contained these digits.
Or, perhaps Jody thought that if the mark was placed between 2 and 3, that
would be 2/ 3.

The most abstract representation these children encountered, the number
line as a conceptual tool to understand fractions appeared to raise more and
different issues about fractions for these children. They were prompted to think
more about one-half, but in a much different way. Not only did Jody ponder the
notion Of a negative fraction it appeared that for the first time, she examined
whether there was a relationship between how it is written and where it goes on
a number line.

As pointed out, above, Kim’s diamond prompted disagreements of more
than one purpose. Kim’s diamond resulted in disagreements to defend and a
disagreement to call for clarification. Monica’s drawing of three-fifths, discussed
below, also prompted disagreements of more than one purpose. In Monica’s
drawing, these disagreements were to correct (one to correct the discrepancy
between what she drew and what she said and one to correct the representation)
and disagreement to defend an idea. In turn, these different purposes functioned

differently in the discourse, as I discuss below.

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Moni ' uare: Five Pieces r F an Ar The E 1?

Pieces of the whole in Monica's drawing of three-fifths emerged as the
focus of three disagreements on May 10, the first two to correct and the third to
defend. Her drawing grew out Of a discussion of her conjecture about 2/ 4 (see
Figure 15, Monica’s Conjecture on May 10). May 10 was the same day Hannah
drew a television early in the lesson and said she could cut it in half, to represent
the fraction, 1 / 2. On that day after Mrs. Vandenberg got a listing of where the
children found fractions at home, several gave examples of one-half (e.g.,
Hannah's television). Then Mrs. Vandenberg asked how they represented 1/ 2,
then 2/ 4. After Moira and Connie described their drawing of 2 / 4 (a butterfly
with 4 wings, two of which were colored in), she said she wanted Monica to
share her conjecture about 2 / 4, which Monica did. "What I think the 2 means is
how many are gone or colored in. And the 4 is how many there used to be.”

While she spoke, Monica wrote and drew the following:

Q --->how many are gone or colored in

Q -->how many there used to be
Figure 15: Monica's Conjecture on May 10

TO illustrate her conjecture, Monica used a blue pen to draw a square
divided into four pieces. She colored in the right side with the same blue pen
and said it was two—fourths. Mrs. Vandenberg then asked Monica about another
fraction, 3/ 5. "If I just told you three-fifths, could you draw that?” Monica
nodded yes, then modified her earlier square by adding an additional blue line:

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Figure 16: Monica’s Drawing of 3/ 5 on May 10

However, perhaps because Monica wrote in this last dividing line in the same
blue pen with which she'd shaded the right side (making the diagonal hard to
see), some of the children seemed somewhat alarmed that she called it 3/ 5, and

called out a correction:

KV: How did you know that would be three-fifths?

Monica: 'Cause the 3 is how many are colored in and the 5 is how many
there used to be

St: That's FOUR!
Austin: That's two-FOURTHS.

Monica: or how many pieces.

Above, Austin and a second classmate momentarily disagreed about
Monica's 3 / 5 drawing, calling out to her that she had four, and not five, pieces.
This brief disagreement bubbled up, and the discussion continued. Mrs.
Vandenberg asked the dissenters, ”Why are you guys saying that's two-fourths?”

to which one student replied, ”It isn't.” Monica continued:

Monica: Well, let me write it like this. (draws an identical but unshaded
3/5)

Monte: There's 4 boxes.

Austin: Not in the one up THERE!
St: Five!

KV: She did. You just can't see it.

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After Monica drew the second 3/ 5 square, Austin appeared to change his
mind about what she drew, saying, ”Not in the one up THERE!". He seemed to
speak for her to correct Monte when Monte took the position that the drawing
had four boxes. As with the disagreement just before, Mrs. Vandenberg prefaced
this one with a question, and then bounded it with a follow-up comment.

In addition to the above disagreements about the number of pieces or
boxes in Monica's drawing, the children also raised the issue of whether the
pieces should be equal. The discussion continued below, and the issue of equal
pieces arose. Here, Joanne held a second idea up for all and seemed to say that

the idea of 3/5 could be equal, but not the way Monica drew it:

KV: I think that's the point they're trying to make. It's not equal.
Joanne: THAT way it isn't. THAT way it is.

KV: Do you agree with that, Monica?

Monica: It's not. It's not equal.

Joanne: YEAH it is. Just not that way.

KV: Is it important that they BE equal?

Mathematically, what were these disagreements about? In the first two
examples, it seemed to be about 5, or 4, pieces, perhaps an issue of not seeing the
fifth piece because Monica made and colored with the same blue pen. It was
difficult to make the distinction between her original 2 / 4 drawing, and this one
of 3/ 5, where she added a diagonal line through the box in the upper right
corner. Maybe to the disagreeing students, the pieces simply were not visible. In

the third disagreement, Joanne first seemed to say that the way Monica drew 3/ 5

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created unequal pieces. But her second statement in the exchange above, "'IHAT
way it is [equal]" raised a question in my mind. What was the "that” she
referred to? The representations of 3/ 5 projected on the overhead screen were
identical, except one was colored in (the adapted 2 / 4 square that was blue) and
one was not. And, when Monica agreed that her pieces were not equal, Joanne
used the unclear referent, it, for a second time. She may have held an image or
idea in her mind of a way to make equal pieces of 3/5, but we do not see it in the
data, for the discussion moved on. Here, she seemed to say that it is possible to
represent 3 / 5 with equal pieces; it's just that Monica did not represent it with her
drawing of a square.

Discursively, the disagreements over Monica’s drawing functioned to
pinpoint a flaw in the representation and to sustain two differing (yet unclear)
ideas on the floor. The disagreement led to a brief discussion (brief because it
was the end of math time) of number of pieces versus pieces of equal size.

As alluded to earlier in this chapter, some representations may have been
more familiar than others to these children. That is, when they moved from
drawings to the geoboard, the sense of the whole apparently was not clear. The
whole on the geoboard seemed not defined, either for or by them. And, whereas
students’ drawings brought with them inherent issues of part-whole
relationships (equivalence, equal pieces), geoboard representations raised their
own set of problems, such as what constituted the whole, whether they should
count pegs or squares, and on whose authority. After all, with drawings, the
children themselves defined the whole by virtue of what they drew. Further, the
number line as a representation seemed to deny these fourth graders familiar

ground, territory they could create and tell stories about, either with pictures or,

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to some extent, the geoboard. There were no parameters, no limits with the
number line. And as discussed above, the location of one-half on the number
line appeared to undermine the very sense Jody previously held of this fraction.
When the representations became more and more abstract, these children had
less steady ground on which to conceptually tread.

Summary

Where do these categories of purpose and function take us in our
understanding of the mathematical and interactional aspects of disagreement?
The functions of disagreement point back to the math content in them. In other
words, disagreements which functioned to get at the underpinnings of another's
position included those over the number line, Hannah’s representation Of 2 / 4 to
show 1/ 2, and Hannah’s TV. These three representations forced the children to
look most deeply at fractions content. likewise, disagreements which
functioned to publicly hold up two ideas and which were prompted by
representations such as Kim’s Diamond and Rob’s Geoboard, spurred the side-
by-side examination of differing ideas.

What did these children need to do to be involved in disagreement during
math? As Mrs. Vandenberg and these fourth graders took on the more active
role in discourse advocated in reform documents, what happened? Naturally,
they needed to wrestle with big ideas about fractions, and demonstrate their
academic competence (Mehan, 1979) which they appeared to do. Clearly,
representations were prominent in the disagreements. Mrs. Vandenberg's
questions and children' subsequent representations took participants into a

variety of means by which to explore their mathematical thinking about

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important fractions concepts, and included drawings, geoboards, and the
number line.

What makes a good representation from a fourth grader's perspective?
These children drew on their experiences with circles, squares, diamonds, and
even televisions, to incorporate drawings into their developing math
understandings. For these fourth graders, representations seemed to be visual
aids for students, almost as though they were a communicative supplement to
words. And, as discussed in this chapter, some representations prompted
disagreement. Some may argue that there's a lesson here, about the
representations created by these fourth graders and described in this chapter.
For example, Rob's design for 1 / 2 and 2 / 4 could be questioned as unusual.
However, Mrs. Vandenberg's initial intent with geoboards was to draw out
student ideas as they explored the board in conjunction with her overhead
questions, as suggested by designers of the curriculum resource she used
(California Board of Education, 1990). In the end, Rob's design forced such a
close look at the board that the children had to think about issues such as
whether counting pegs or squares made more sense, and what made up
something as small as one square of the geoboard, even though a focus on what
constituted the whole sometimes seemed to be lost. And to Hannah's way of
thinking, a television in her home, when drawn and bisected, seemed to be an
adequate representation of one-half.

But these children learned more than math content. In addition to math,
they were learning how to disagree, as evidenced by their ability to do so. This
was a social competence (Mehan, 1979), an integral part of mathematical

discourse. In the disagreements discussed in this chapter, the children most

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often referred to math ideas, in part because Mrs. Vandenberg actively
questioned them about their ideas and thinking. A few times when they used
pronouns, such as ”it” and ”that," the referent was not stipulated or defined. For
example, on May 7 when Ali claimed "That's TWO pieces, not one,” it was not
clear what the ”that” was. We can speculate that she meant partial squares,
because Rob again pointed them out, "One right here, one right here." However,
most times, these fourth graders were specific about what they meant (i.e.,
Hannah's television, a negative number, four pieces, etc.).

In the setting of fourth-grade mathematics where this novice teacher
worked to foster the type of mathematical she envisioned to be aligned with
current reform, she and the children bumped into strategic aspects of fractions
learning. Part-whole relationships confronted them through their
representations in most disagreements. Pictures and geoboard shapes led them
to examine halves and fourths, equivalence, and equal pieces. And the location
Of one-half on the number line perplexed Jody as she struggled with her
understanding of one-half.

In disagreement, the children’s world met the math world in two ways.
They worked to learn how to disagree, and how be involved in mathematical
discourse Of the type envisioned by the StandardS (1991). They tried on the
language of mathematics, sometimes using their own words as they struggled
for the conventional math words. In addition, they strove to appropriate the
norms for participation in discussion and disagreement which Mrs. Vandenberg
fostered. All in all, important elements emerged through these children'
disagreement over fractions content. At times, Mrs. Vandenberg momentarily

receded from the discussion so that the children negotiated their own turns to get

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an idea on the floor to disagree. Teacher and students mutually negotiated the
floor (Edelsky, 1993) to explore fractions concepts. The students received
guidance from their teacher as they navigated their way through the domain of
fractions, an area of mathematics considered complex by curriculum developers
(Corwin, Russell, and Tierney, 1990). Her question asking and comments
spiraled through their contributions to the discussion, as did children'
contributions. During whole-group discussion, Mrs. Vandenberg appeared to
support student contributions through questions and connections, nudging the
discourse along. And, when children disagreed, the disagreements discussed in
this chapter were usually framed by her comments. In addition, her involvement
outside disagreement through frequent reminders about tasks and behavior
appeared to shape the discourse so that the fourth graders focused during
disagreement on math. However, as I discuss in the next chapter, this was not

always the case.

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CHAPTER SIX
EPISODES OF DISAGREEMENT

The previous chapter focused on the mathematical nature of single
disagreement and demonstrated that these children were so involved in the
discussion and with their ideas that they negotiated their turns and ideas onto
the floor, resulting in disagreements over math content. This was not the case
with episodes Of disagreement. Rather than the math, it was the interactions
between the students which stood out in episodes, where they struggled with
their social competence (Mehan, 1979). In the episodes, Mrs. Vandenberg
inadvertently receded from the discourse, to answer the door or attend to a
student at her side with a question. And, coincidentally, disagreement erupted
at nearly the same moment she was detoured from her role as discourse
conductor. But when she receded, the children had longer strings of talk as they
acted to get their ideas on the floor. And when they did, they did not sustain the
participation norms Mrs. Vandenberg worked to establish. However, the
children were so involved in the disagreement that they went on without their
teacher anyway, to the extent that Mrs. Vandenberg struggled to renegotiate the
floor to manage the disagreement.

In this chapter, I first describe the math lessons on May 3 and May 7, in
which episodes of disagreement occurred. I then discuss the episodes,
highlighting the mathematical and interactional aspects of each. And, I include a
discussion of the mathematics task as well as the mathematical thinking revealed
by these students, and the ways in which they talked and responded to each
other. Second, I discuss what these disagreements seemed to be about; third, the

ways in which these episodes affected the ensuing discussion; and finally, the

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role of this teacher and these students in the disagreements. I follow this with a
similar discussion of the episode of disagreement on May 7. I will show that
when Mrs. Vandenberg receded from these discussions, the children continued,
and the norms of participation she fostered took a back seat to norms the
children may have experienced elsewhere.
May 3

As described in Chapter 4, math time on May 3 followed Format 2, where
the lesson began with Mrs. Vandenberg's orienting comments and whole-group
discussion followed. After the children returned from lunch recess and settled
into their chairs, the math lesson began. Mrs. Vandenberg planned to continue
the discussion from April 28 when she had the children work in small groups to
respond to three exploratory questions, which she projected at the overhead:
What is a fraction? When have you used fractions? How are fractions represented?
Draw and label pictures. Whereas on April 28, the children worked in small
groups recording their answers on large sheets of paper for the majority of class
time, on May 3, Mrs. Vandenberg planned to resume discussion where it left Off
on April 28, by beginning with whole-group discussion. Mathematically, the
discussion to come reflected these fourth graders' thinking about fractions, in
part-whole relationships and as part of a set. Interactionally, the episode of
disagreement revealed that there was more to disagreement over math concepts
than the math. The world of these children outside of school, and perhaps their
more natural ways of interacting, came to the fore as discussion of the questions
of the day unfolded. While the children disagreed in the absence Of their teacher,

the mathematics and norms of disagreement collided.

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Math time began when, following one boy's comment about a fight earlier
in the day, Mrs. Vandenberg immediately oriented the children to the task at
hand:

KV: I'm gonna remind you right now there's no calling out. OK? If you
choose to continue calling out you're gonna have to go down to the Office.
OK. Um, we haven't had math in a while on WEDNESDAY", we started
talking about FRACTIONS. And, THURSDAY was our field trip and
FRIDAY we only had a half day. SO we're gonna pick up where we left off
there when we were talking about the three questions on the overhead.
Jessica put your book away. Question Number One was 'What is a
fraction?‘ Question Number Two, 'When have you used them?‘ And
Question Number Three, 'How are they represented?‘ On Wednesday we
got to hear Bart and, Micah's idea. We also heard from, Tomika, and,

KV: Ali and Monica.

KV: Just a reminder. The REST of us, will be taking notes, keeping track
of what DIFFERENT tables' ideas were. Remember how we started doing

that on Wednesday?

Figure 17: Monte’s Drawing Of Groups on May 3

 

 

 

 

In a few turns at the start, she overviewed their previous lesson, the

questions then and now, said they would be taking notes in their journals, then

 

1"All capital letters denote speaker emphasis.

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began whole-group discussion while focusing on the first question, ”What is a
fraction?” Mrs. Vandenberg first called on Monte, asking, "What did you think a
fraction was? How did you describe it?” He responded with words, and then
wrote a number sentence and drew a picture at the overhead. "A fraction is,
where you have a big number and you can't solve it so you split it up into groups
so it's easier to solve. . . A fraction is a piece of the problem,” he said. While he
talked he wrote for his example, ”200 + 10" and drew several ”groups.”

Others then aired their ideas when called on by Mrs. Vandenberg. For
example, one girl wondered aloud, "200 + 10 is a fraction?", then said she
thought ”division and fractions are related. . . Like you, you can make that [200
+ 10] into a fraction. ” Another girl, Hannah, who would be a prominent figure in
the discussion that day, said to Mrs. Vandenberg that 200 + 10 was not a fraction.
”I don't think it would be a fraction because 200 + 10, if it was a fraction, it would
be um, less than like zero. You would like, you would take—you would have to
take pieces out. . . SO it's less than one.” Apparently misspeaking, Hannah
revised her statement during this turn. She first said a fraction would be less
than zero, then less than one. She seemed to be saying that a fraction is part of
one whole. Hannah's remaining comments on May 3 were consistent with this
position. Subsequently, Hannah said, ”I'll show you what a fraction is." and
went to the overhead to begin what would be an extensive explanation using
circular representations. First, Hannah drew a circle divided in half and said,
”There is a fraction. One-half.”

So, to this point in the lesson on May 3, the discussion about "What is a
fraction?” included the idea that a fraction is a big number you split into groups,

like 200 + 10 (Monte); division and fractions are related (Joanne); and that a

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fraction (for example, one-half) is less than one (Hannah). As the children
responded to Mrs. Vandenberg's questions, they seemed to be talking more to
her than to each other.

When Hannah went on to say why 200 + 10 is not a fraction, Monte, still
near the overhead, momentarily defended his position that it can be a fraction
(described in Chapter 5). They dueled verbally, point-counterpoint; 200 + 10 is
not a fraction—it could be a fraction. Neither gave a reason for their position,
nor did Mrs. Vandenberg seek one. Not responding to Monte, Hannah drew

another circle, this time divided into fourths.

 

Figure 18: Hannah's Representation for a Fraction on May 3

When Mrs. Vandenberg asked her how her circles were like 200 divided
by 10, Hannah restated her idea and basically talked about 200 and 10 as whole

numbers, then continued by shading in the fourths:

Hannah: The reason, OK. Two hundred divided by 10 is not a fraction. .
because. . a fraction is less than one. Ten and 200 are not less than one.
Here is another kind of fraction (another circle drawn, divided in four
parts). There are other kinds of fractions. This would be, one-fourth, if
you shaded in one.

KV: OK.
Hannah: And,
KV: How did you know that would be one-fourth.

Hannah: It would be two-fourths if you shaded in, two, three-fourths and
four-fourths.

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Quietly, these fourth graders Observed Hannah while she drew the circle,
this time divided into four sections. Speaking with more emphasis and apparent
confidence (and volume) than when she first went to the front, her images of a
fraction (the first, a circle divided in half; the second circle divided into fourths)
were projected for all to see. Piece by piece, Hannah colored in and labeled each
section, talking as she colored them. ”One-fourth. . . two-fourths. . . three-
fourths. . . four-fourths, " she said. From across the room, another child, Ali,
soon joined in the talk about fourths.

As Hannah's explanation of a fraction unfolded, the others watched, wrote
in their journals, and began to disagree. Ali reacted to Hannah's talk about
fraction parts, saying, ”And one. Or you could say one," at the same time

Hannah filled in the last piece and said, "Four-fourths.”

Hannah: It would be TWO-fourths if you shaded in, two, THREE-fourths
and FOUR-fourths.

Ali: And ONE. Or you can say ONE.

Hannah's "four-fourths” and Ali's ”one" hung in tandem in the air. Appearing
to capitalize on an appropriate juncture in Hannah's turn, Ali put the idea on the
floor that four-fourths could also be one. The discussion of fourths continued,

first with a question from Mrs. Vandenberg:

KV: How come you say one instead of four-fourths, Ali?

Ali: ’Cause it’s, all colored in. It’s one, group.

Ali, like Hannah, pursued the idea of a fraction as pieces that are colored in;

however, Ali appeared to put the pieces together to think of the figure as a

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whole. Mrs. Vandenberg then spoke to Hannah about the way she colored in her

pieces.

Hannah: It would be two-fourths if you shaded in, two, three-fourths and
four-fourths.

Ali: And one. Or you can say one.
KV: How come you say one instead of four-fourths, Ali.
Ali: 'Cause it's, all colored in it's one, group.

KV: Hannah it was helpful to me the way you went through there and

you colored a piece at a time, one-fourth, two-Did you guys see that?
TWO-fourths, THREE-fourths.

Hannah: In other words if you just put it in half, and you colored in half,
it would be one,

(She records '2 / 2' beside her circle.)
Ali: Two.

Again, Ali and Hannah's turns tumbled over each other, Hannah saying
”one;” Ali saying ”two” when Hannah paused in the midst of saying ”one-half"
to describe the half she'd colored in. Rather than completing that last part of the
fraction's name, Hannah paused, and Ali stepped in, apparently to finish
Hannah's sentence, to become ”one-two.” For the second time, Ali conveyed her
involvement in Hannah's explanation and disagreed, and at a juncture in
Hannah's turn, gave her own interpretation of it. The first time Ali spoke of four-
fourths as one. Here she said one-half would be "one-two,” and again spoke up
at a juncture in Hannah's talk when she recorded, ”2/2”, as though this numeric
writing of the whole finished her sentence. But not finishing this sentence
allowed someone else to do so, which Ali did, as though she was thinking ”And
when you color the other side, you get two, which makes would be one whole.”

Again, Hannah did not take up Ali's call out, instead completing her own

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thought, this time with words. As Mrs. Vandenberg stepped aside to talk to an
adult at the door, in essence she receded from the interaction, which the children

sustained:

Hannah: One-HALF. It would be one-half.

Ali: One-two is one-half.

Jody: But you would STILL split up, two hundred and,
Hannah: And you color the OTHER side would be

Ali: ONE.

Hannah: TWO, halves.

Ali: OR you can just put ONE.

Hannah: NO! Why would you put ONE?

Taking the view that one-two is one-half, Ali appeared to be giving a
second name to the same thing Hannah colored in (one-half); and seemed to be
re-stating her previous position that when the whole is colored in (in this case,
two halves), ”you can just put one." Ali drew on her own children language
presumably to help support her developing understanding of the language of
math, here taking her invention, ”one-two,” to link with the conventional, one-
half.

Whereas it seemed to me that the children were talking generally as a
group, in part because Mrs. Vandenberg asked questions and connected
students’ ideas, when Hannah called to Ali, ”NO! Why would you put one?”, the
interaction became more focused between Hannah and Ali, erupting in a
stronger disagreement over what to call this representation. Also, these students'

interaction took a turn, evidenced by prosodic changes of voice. Ali's previous

call outs were quieter, spoken in somewhat of a soft voice. But here, she turned

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up the volume and left space between each word, ”All - filled - in.” They looked
directly at each other, each taking up the other's comment. Whereas before Ali
reacted to Hannah's comments resulting in two-turn disagreement, now the
disagreement between these two girls became more entrenched, cycling through
more turns examples. With emphasis and clarity, Ali responded to Hannah,

explaining (a second time) why she would put one (versus two, for two halves):
Ali: 'Cause it's L11 filled i_n_.

KV: OK I missed that last ques-
Hannah: NO!

 

Whereas before Ali's contributions about one, and one-two as one-half did
not appear to be acknowledged by Hannah, this "a_l fil_le_d i_n drew a reaction.
Interrupting her teacher, Hannah declared ”NO!” to Ali's comment that she
would put in one, apparently because the circle was all filled in. Mrs.

Vandenberg continued, her question followed by Hannah's comments to Ali:

KV: Hannah's last comment, was that that was FOUR-fourths. And All
said it was ONE, because it was one whole-if we were thinking about that
like a COOKIE, and you were getting FOUR-fourths, would you actually
be getting one whole cookie?

Hannah: Ali Ali I want you to think. This PROBABLY could be one-
second, but then what would, THIS be?

Ali and Sts: One-HALF.

As she had done before, Hannah did not appear to acknowledge Mrs.
Vandenberg's question, instead sticking with her line Of thinking and drawing
more circular representations as she elaborated. More images emerged on the

overhead screen, images that Hannah pointed to and emphasized again while

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she simultaneously launched a series of question-answer exchanges. Also,
Hannah's terms changed, and rather than appropriating Ali's "one-two” (for one-
half), said ”one-second.” Interestingly, when Hannah interchanged these
referents, she subsequently seemed to disagree with the class, here over the use
of "one-half,” the term for the half-shaded circle she began with Her voice was

loud, her words emphatic, and her comments direct.

Hannah- One HALF. That's the same as one—SECOND. There's one-half,
there's one-SECOND. WHAT is the difference between those two?

Ali: Nothing.
Shawna: Nothing she's just said that you can do it either way.

KV: Oh so she's saying that it also would make sense for it to be called
one-SECOND?

At this point, Hannah's question-answer protocol began to take on the
characteristics of interactional sequences described by Cazden (1986), found in
most American classrooms: initiation-response-feedback (IRF), except it wasn't
Mrs. Vandenberg but Hannah who took on the traditional teacher role to control

the 3-part sequence, and the contents of the sequence appeared as:

(I) Hannah: ...This PROBABLY could be one-second, but then what
would THIS be?

(R) Ali and Sts: One-HALF.

(F) Hannah: One HALF. That's the same as one-SECOND. There's
one-half, there's one-half.

(I) WHAT is the difference between those two?
(R) Ali: Nothing
(R) Shawna: Nothing, she's just said that you can do it either way.

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To complete the IRF sequence, Mrs. Vandenberg stepped in to say,

KV: Oh so she's saying that it also would make sense for it to be called

one-SECOND. . .

In the exchange above, perhaps Hannah consolidated the ideas of one-two
and one-second, used one-second, and launched into a position that one-half and
one-second are the same, then asked about their difference. Possibly because it is
what she proposed before, Ali is supported by Shawna, when she responded to
Hannah, ”Nothing.” After Shawna explained what Ali meant, Mrs. Vandenberg

finished the IRF sequence underway, revoicing the talk about one-second.

KV: Oh so she's saying that it also would make sense for it to be called
one-SECOND. . . Since we talked about one-THIRD or one-FOURTH?
You're right. That is two ways to say it and and another way that we've
gotten used to saying it is one-HALF.

Mrs. Vandenberg seemed to acknowledge Hannah's shift from one-two to
one-second, repeating them, then inserting the conventional term, one-half, to
connect it with Hannah's use of one-second. Hannah continued with more

description of one-second, one-fourth, and two-fourths, again addressing

questions to Ali - and getting responses:

Hannah: You COULD call it, one-SECOND. But the REAL one-second is
THIS. Now, er-

(I) Now, what would you call THIS? Right here?
(R) Austin: I call it a HALF.
(R) Ali: A quarter, or a fourth.
KV: A quarter or a fourth?
(F) Hannah: One-FOURTH
KV: DO you agree with that Kim?

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(I) Hannah: What would you call THAT?
(R) Ali: Two quarters or TWO-fourths.
(R) Austin: I call it a HALF.

KV: Wait! I'm curious here. I heard two different ANSWERS. Um, ALI
said it's two-fourths, and AUSTIN said HALF.

More questions, from Hannah and responses from classmates started up
further IRF sequences with Hannah at the helm. Without Mrs. Vandenberg to
foster collaborative norms of participation, Hannah reverted to a school-related
interactional sequence with which she was familiar: Initiation-Response-
Feedback. Taking on a teacher role, a role different from Mrs. Vandenberg’s but
familiar to many school children, Hannah asked questions which required one-
word answers, and to which she gave immediate feedback. In spite Of Mrs.
Vandenberg's attempts to re-negotiate the floor (and the disagreement), Hannah
either did not hear or ignored her, and instead continued talking to Ali. While
Hannah remained at the overhead with Monte nearby and Ali continued, a few
students attempted to enter the fray. However, Hannah continued, below, while
looking out toward Ali. Meanwhile, Mrs. Vandenberg attempted to point out
two different answers she heard. But, again, Hannah did not take up her
teacher’s comment, as more pieces were pointed at and another question asked
by Hannah of Ali. Then, when Mrs. Vandenberg attempted to stop the

discussion for journal writing, Hannah, then Ali, persisted.

St: Same thing.

Hannah: It IS the same thing. Now what would you call THAT? (points
to filled-in 4 / 4)

KV: I want you to stop for a minute and respond to that in your journal.
One second. We'll come right back to this. Can I write down your idea?

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Hannah: FOUR-fourths, not ONE-fourth. If it was ONE-fourth it would
be one.

KV: I want you to respond to this in your journal. Here's the picture
Hannah drew you. OK she had divided this into four pieces, and then she
colored, TWO. OK. And I want you to respond to this question: 'Is this
1/ 2, 2 / 4, or both? Why?’ Austin said one-half. Ali said-Hannah said it's
TWO-FOURTHS. A couple of you said it's both. Tell me what YOU think
and respond to why. And draw it-write the question in your journal
please.

Ali: But it has, FOUR pieces.
KV: Does that question make sense? SO Ali you are starting to explain it

NOW. You think because it has four pieces, you colored in two? You

think it's both? Write the question please. Get started Rob.
In these simultaneous conversations, Mrs. Vandenberg worked to get the
children focused on a journal question about 1/ 2 and 2 / 4, while Hannah asked
Ali another question, then answered it herself. What was Hannah saying? What
was she thinking? The answer to these questions is not clear; however, at least
two explanations are possible. First, she may have reacted to Ali’s earlier focus
on "one” (meaning the whole circle). That is, early on when Hannah said, ”NO!
Why would you put ON E?", she did not listen to Ali’s answer, ”Because it’s fl
mag i_a.” If that were the case, Hannah may have thought at that time that Ali
focused on one of the halves, not one whole. Therefore, in this most recent
exchange, her statement about ”FOUR-fourths, not ONE-fourth...” (note the
emphasis on one in one-fourth) meant that, from her perspective, Ali (earlier and
now) dwelt on one piece, not one whole. Second, and more plausible given
Hannah’s nature, is the possibility that she attempted to control the
disagreement by turning or shifting its focus, a common discourse move among

adults and frequently discussed in conversational analysis literature on male-

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female communication13 (see, for example, West 8: Garcia, 1988). And, not to be
overlooked is that when the disagreement erupted, Mrs. Vandenberg was at the
door, essentially removed from the discussion for the moment. Although Mrs.
Vandenberg fostered a different kind of discourse, Hannah may have reverted to
discourse norms from previous grades. It is notable in these segments with
Hannah that Mrs. Vandenberg worked to achieve different participation norms
in the spirit of the kind of discourse advocated in reform documents. In these
different norms where she could get ideas on the floor and children to reason
about them collaboratively, it turned out that while she was the door, Hannah’s
moves in the disagreement appeared to include two adult-like attempts to
control the talk. She directed the disagreement through several IRF sequences
and seemed to take tighter control by making a powerful discourse move:
shifting topics mid-disagreement.

To end the disagreement, Mrs. Vandenberg consolidated the final
comments of children into an overhead question for their journals. When she did
this she re-created Hannah’s representation of fourths and revoiced (O’Connor 8:

Michaels, 1993) Austin’s observation, "I call it a half":

KV: I want you to respond to this in your journal. Here's the picture
Hannah drew you. OK she had divided this into four pieces, and then she
colored, TWO. OK. And I want you to respond to this question: 'Is this
1/ 2, 2/ 4, or both? Why?’ Austin said one-half. Ali said-Hannah said it's
TWO-FOURTHS. A couple of you said it's both. Tell me what YOU think
and respond to why. And draw it-write the question in your journal
please.

 

'3 Interestingly, this literature focuses on the ways in which males control conversations
(i.e., topic shift, interruptions). But in the instances with Hannah discussed in this study, Hannah
makes these conversational moves (Tannen, 1990) frequently associated with males in literature
on language and gender.

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She concluded the disagreement by bringing everyone's attention to "different
answers,” one-half and two-fourths, she heard and giving directions for journal
writing. With Hannah's and Austin's ideas, she generated a question to which
the class was to respond. Subsequently, students recorded and discussed the
questions at their desks, then wrote responses in their journals. Although the
disagreement ended, thinking about one-half and two-fourths did not.
Following journal writing and discussion in small group at tables, whole group
resumed. I discussed subsequent disagreement on May 3 in Chapter Five.

To sumr‘narize, on May 3 Mrs. Vandenberg appeared to orchestrate
discussion as well as the potential for disagreement. She brought more and more
ideas and speakers to the floor, asked questions, and connected children’s ideas.
However, when called to the door her involvement in the discussion
inadvertently faded, and those norms she fostered did, too. The children
reverted to their own ways of interacting, ways which may have been more
familiar to them than the norms Mrs. Vandenberg orchestrated. But rather than
deter these children, they did not hesitate in her absence, instead assuming the
floor with their own turns and ideas, to the extent that when Mrs. Vandenberg
attempted to re-negotiate her way to the floor, the children overrode her. And,
in the case Of Hannah, some conversational moves (T annen, 1990) which
controlled the disagreement emerged.

By posing an open-ended math task to begin math time on May 3, Mrs.
Vandenberg fostered a discourse space in which disagreement could happen.
The discussion leading up to the disagreement seemed to position disagreement
to occur. For example, Hannah's original thinking that 200 + 10 is not a fraction

differed from Monte's. However, at the time Hannah was talking directly to Mrs.

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Vandenberg, not Monte, so her thinking about Monte's idea went through the
teacher. Ali's idea that you could call four-fourths, one, leaned into
disagreement. Yet, when Mrs. Vandenberg asked Ali why she would say one
instead Of four-fourths, the question and Ali's comments seemed to be between
the two of them, teacher and student. The difference between the pre-episode
airing of different ideas and the initiation of the disagreement was the moment
when Ali spoke to Hannah and their interaction was directed to one another.
Previously, with the exception of the somewhat isolated two-tum disagreement
between Hannah and Monte ("200 + 10 is not a fraction,” "But it could be a
fraction. ”) ideas, although different, were either addressed to Mrs. Vandenberg,
or the children said them without uptake from others.
What ia the Disaggeement Aboht?

The math task on May 3 was discussion about three exploratory questions.
The discussion of the question, ”What is a fraction?", stirred interesting
mathematical thinking about part-whole relationships as well as the notion that a
fraction could be a large number split into groups. Monte's notion that a fraction
is a big number that is split into groups (e. g., 200 + 10) was described and
discussed briefly; however, the majority of the discussion and its subsequent
disagreement revealed the part-whole thinking of some of these fourth graders.
What evolved as points of contention were what to call parts of a fraction. For
example, before the episode, Hannah and Ali differed about four-fourths or one;
two halves or one; then one-half or one-two. During the disagreement, the girls
raised issues about whether to call the whole ”one" or ”two halves,” then,

Hannah questioned the difference between ”one-secon ” or ”one-half."

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Mathematically, the disagreements within the episode were about what to
call Hannah's representations. First, Hannah and Ali disagreed over what to call
colored-in portions of a circle (e. g., four-fourths versus one; one versus two
halves). Then, Hannah argued with Ali and others that "one-second” (which Ali
called "one-two") was "one-half” and seemed to quiz her classmates about the
difference between the two. When Hannah turned Ali’s term for one-half ("one-
two”) into ”one-second," Hannah argued that one-half was the same as one-
second, which was Ali’s position. Whereas the episode was preceded by
Hannah's explanation and drawing (and Ali's comments about ”one”) of what, to
her, a fraction was ("less than one”), the disagreement seemed to zero in on three
issues: a) whether to call two halves, one; b) whether to call one-half, one-two
(subsequently changed by Hannah to become one-second); and c) the difference
between one-second and one-half. In a roundabout way, Hannah ended up
where she began, arguing about one-half. However, by this time, she controlled
the debate not by logic, but through controlling the talk.

Tha Interactions of the Children

When these children disagreed with each other, they interacted with one
another in interesting ways. They took it upon themselves to carry on
disagreeing talk over a span Of time that involved several speaking turns, with
and without their teacher. First, they spoke up for themselves to say publicly
what they disagreed about. They also went about saying it in their own ways.
This willingness to speak, disagree, and do so publicly seemed to indicate their
involvement in the discourse and attachment to their ideas. Through tone of
voice, use Of pronouns, and questions, Hannah, Ali, and others seemed to further

evidence this attachment. For example, as discussed earlier, Hannah's tone of

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voice changed to become louder and more emphatic. Ali also talked louder
during disagreement, and when she said the circles were ”all m i_h", she
separated these words to indicate the emphasis she gave them. m the
children’s use of pronouns shifted, from a sense of collaboration to that of
separateness. They initially seemed to use the pronoun, ”you,” in a plural or
collective sense, as seen in the following pre-disagreement examples where the

children are talking to Mrs. Vandenberg:

Monte: A fraction is, where you have a big number and you can't solve it
so you split it up into groups so it's easier to solve.

Joanne: . . . division and fractions are related. . . Like you, you can
make that [200 + 10] into a fraction

Hannah: And you color the OTHER side would be
Ali: OR you can just put ONE.

Later, when Hannah initiated a disagreement with Ali, her use of "you” became
more direct. When Hannah said, ”NO! Why would you put ONE?" she looked
specifically at Ali, essentially intoning, "Why would you, Ali, put ONE?" With
this move Hannah seemed to diminish Ali’s contention that ”one” could be a
viable alternative. Also, Hannah appeared to separate Ali's math idea from her
own. Her use of ”you” became more personal. Asking, "Why would you put
ONE?” Hannah subsequently talked directly to Ali while also referring to Ali's
idea:

Hannah: Ali Ali I want you to think. This PROBABLY could be one-
second, but then what would, THIS be.

Hannah: One HALF. That's the same as one-SECOND. There's one-half,
there's one-SECOND. WHAT is the difference between those two?

Hannah: You COULD call it, one-SECOND. But the REAL one-second is
THIS. Now, er- Now, what would you call THIS? Right here?

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Hannah: What would you call THAT?

Hannah seemed to single out Ali when she used "you” in her comments and
questions, as though to lesson the status of her contributions. ma, when
Hannah stood at the overhead and questioned Ali, at first glance she appeared to
be getting Ali to clarify her reasons for calling these fractions what she did.
However, the emphasis on the words in the moment seemed to me to serve a
different purpose: to perhaps show that Ali was wrong.

The children used other pronouns as well, such as ”I” and ”we." For
example, Hannah said about 200 + 10, "I don't think it would be a fraction,” and
”I'll show you.” Austin twice called out in response to a question from Hannah,
”1 call it a half. " In these instances and others, when they used pronouns, they
still appeared to talk about math; Hannah about the notion of 200 + 10 as a
fraction and Austin about a particular circular representation of Hannah's. On
May 3, the children’s uses of pronouns, as they appeared in the transcript, can be

listed as follows:

you can't solve it (Monte)

You can make that into a fraction (Joanne)

I don't think it would be a fraction (Hannah)
You would have to take pieces out (Hannah)
I'll show you (Hannah)

if you shaded one in (Hannah)

if you shaded in (Hannah)

Or you could say 'one.‘ (Ali)

PQNF‘WPS’N!‘

if you just cut it in half (Hannah)
10. if you colored in half (Hannah)

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11. But you would still split up (Jody)
12. And you color the other side (Hannah)
13. Or you can just put one (Ali)

14. Why would you put one (Hannah)
15. Ali I want you to think (Hannah)
16. she's just said (Shawna)

17. you can do it (Shawna)

18. You could call it (Hannah)

19. what would you call (Hannah)

20. I call it (Austin)

21. What would you call (Hannah)

22. I call it (Austin)

23. What would you call (Hannah)

Not until #14 in this list did pronoun use seem to shift from a collective sense to
a sense I and you. In addition, their use of negatives was rare, as evidenced in
the first instance from Monte, and the third from Hannah.

A fourth way students showed belief in or attachment to their ideas

 

concerns the use Of disagreement to appear to be right. At the time Hannah sort
of twisted Ali's ”one-two” and argued that one-second and one-half are the
same, this argument seemed Odd to me. I wondered, ”Isn't she arguing with
students although they actually agree with her?” Analyses showed that they
did, indeed, agree with her. Hannah's argumentative stance perplexed me. She
seemed to be jockeying for the position of ”the one who is right,” and therefore,
has to make it appear as if others are wrong. A good way to accomplish this is to
make it look as if they are disagreeing with her, or that she is disagreeing with

them.

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The fact that the children made connections to each other's referents
indicated their involvement and interest in this discussion and subsequent
disagreement. In large part, their referents were clear. For example, early in the
discussion, Monte said, "A fraction is where you have a big number and you
can't solve it so you split it up into groups so it's easier to SOLVE." Here, he
responded to the first question of the day which Mrs. Vandenberg asked him
about, ”What is a fraction?” Soon, Joanne too, connected her thinking to Monte's
when she said, ”You can make that [referring to Monte's number sentence, 200 +
10] into a fraction. " This pattern continued before and during the disagreement.

To begin the disagreement with Ali, Hannah used both a negative and a
singular (Ali)” you” to say, ”NO! Why would you put ONE?” She spoke
directly to Ali and she specified the math-related aspect to which Ali referred
(i.e., coloring in two halves is the same as one whole). And soon after, when
Hannah said to Ali, ”1 want you to think,” she again referred specifically to Ali
and continued to talk about a specific referent, her drawn circle, the object of
what she wanted Ali to think about.

In other exchanges during the disagreement, Hannah said, ”What would
you call THIS?" and ”You COULD call it. . .”, and again, she referred to
something specific when she pointed to her circles, projected from the overhead.
In essence, she asked Ali what she would call the drawn representation, then
said what this circle could be called. These child connections of the specific
referents seemed to mirror Mrs. Vandenberg's weaving Of ideas into the lesson
and further evidenced their belief in an attachment to their ideas. However,
more prominent is how Hannah used language to control the floor in a

competitive way much different from the collaborative floor Mrs. Vandenberg

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fostered. Hannah got louder and used pronouns which separated classmates
and their ideas. She also questioned Ali in a face-off of ideas, ironic because she
negotiated the floor by getting a turn and initially contributing an idea (e.g., a
fraction is less than one) to the discussion of the overhead question, What is a
fraction? But she fell short Of succeeding because she did not know how to hold
the floor. Her controlling actions backfired when she ended up essentially
disagreeing with herself. Without Mrs. Vandenberg to manage the
disagreement, Hannah’s own way of interacting interfered with the math content
she attempted to get on the floor.
The Teacher’s Rolg

Mrs. Vandenberg took on strategic responsibilities in her role as discourse
conductor: she seemed to allow the children to speak for themselves; she
connected students' mathematical ideas across the lesson, and; she attempted to
fuel disagreement. Early in the May 3 math lesson, Mrs. Vandenberg bore the
responsibility for orchestrating the discourse. However, once disagreement was
underway, the children came to the fore, as if to guide it themselves. Whereas
Mrs. Vandenberg called on individual children, sought and connected their
mathematical ideas, and asked questions, during disagreement the children
frequently spoke for themselves, sometimes appearing to either not hear or
disregard Mrs. Vandenberg.

Second, integral to the forward momentum of this discussion and
disagreement was Mrs. Vandenberg’s teacher role as discourse conductor
through which she connected children’s math ideas. For example, in her

orienting comments, Mrs. Vandenberg spoke initially at a general level,

146

connecting the May 3 lesson with the previous one on April 28 when she said,
"we're gonna pick up where we left Off there when we were talking about the
three questions on the overhead." She then talked about the three focal
questions and said which children shared their ideas. As new thinking emerged,
she then connected specific comments. For example, when Monte first proposed
that a fraction is a big number you split up and created his picture of "groups,"
she said, "OK so you're dividing 200 by 10 so you're making groups of ten. How

many groups would you make total?" Their interaction continued:

KV: A fraction is a number, that you split up? IS THAT how you
explained it?.

Monte: Yup. What you can do, to make this problem easier is you can
split it up into groups of 10. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. There's one group. . .

KV: OK so you're dividing 200 by 10 so you're making groups of ten.
How many groups would you make total?

Monte: Twenty.
KV: Twenty groups. And how is that, related to fractions.

Monte: Well if you're splitting a number up into groups to, tO make it
easier.

KV: How is that DIFFERENT than dividing.

KV: You showed us a DIVISION problem 200 + 10, and then you said
you'd go on to make 20 groups of 10?

Monte: Yeah.
KV: And you're saying that's a fraction?

147

Monte: Yeah 'cause a fraction is a PIECE of the problem.

KV: A PIECE of the problem. What do you guys think about that. Monte
responded to the question, 'What is a fraction?', by writing this DIVISION
problem. Any responses?

KV: Comments, questions for him?

Mrs. Vandenberg also asked and then connected Braden's and Joanne's ideas

about Monte's idea that 200 + 10 is a fraction:

KV: What do you think about MONTE'S idea?
Braden: From what I HEARD—

KV: DO you AGREE with it or—

Braden: Well—what I HEARD I don't really think so.
KV: Would you tell him again, Braden.

Braden: I RECORDED the first part of what he said.

KV: HE showed us a division problem, 200 + 10, and he said THAT
would be a fraction.

Braden: I don' agree with him because um,

KV: You AGREE with him?

Braden: NO I DON'T agree with him.

KV: You DON'T agree with him.

Braden: Because a HUNDRED divided by 10, would be 10.
KV: OK.

Joanne: Two hundred divided by 10 would be a fraction?
KV: Braden says he doesn't agree. Joanne you're saying you AGREE.

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Joanne: In a WAY.
KV: Why 'in a way.’

As the dialogue continued, Mrs. Vandenberg continued to connect each
child's idea with what was said before, including the next to speak: Hannah.

KV: Hannah?

Hannah: I might be able to solve 200 + 10. . .

KV: You think with some color tiles you could solve that problem. Is that
a fraction?

Hannah: I don't think it would be a fraction because 200 + 10, if it was a
fraction—er not a fraction—if it was a fraction,. . it would be um, less
than like ZERO you would like, you would take—you would have to take
pieces OUT.

KV: Hannah you're saying—

Hannah: So it's less than ONE. like THAT‘S what a fraction—here I'll
show you what a

KV: You're saying a fraction is less than ZERO?
After more comments, Mrs. Vandenberg connected Hannah's ideas with Ali's:

Hannah: HERE is a fraction. THERE is a fraction. ONE-HALF.

KV: OK. That reminds me of what ALI showed us. Do you remember
how ALI showed us her way of dividing up circles?

Joanne: Yes see,

KV: And how is THAT like 20 + 10 Hannah. Or TWO HUNDRED
divided by 10 excuse me.

Hannah: I keep making my two sideways. OK The REASON, OK. Two
hundred divided by 10 is not a fraction.

KV: Is NOT a fraction. But what YOU showed us IS a fraction.

149

The third way in which Mrs. Vandenberg evidenced her role as discourse
conductor was her teacher moves which opened the door for disagreement.
Whereas she sometimes succeeded in stirring disagreement, there were also
times when the children did not take the bait. As discussed above, Mrs.
Vandenberg often made connections across student ideas, and those linked ideas
were not necessarily identical or similar. Also, after she invited a child to share
his or her idea about the question or problem of the day and that child had done
so, Mrs. Vandenberg would ask questions such as, "What do you think, Kim?",
"Braden, what do you think about that?", ”What do you guys think?", ”Do you
agree?", or ”Do you disagree?” These questions acted as stimulants for more
students' ideas, thereby potentially getting more ideas on the floor.

To summarize, this teacher's role in orchestrating the discussion and
disagreement served to position disagreement in several ways. First, as in the
disagreements discussed in Chapter Five, Mrs. Vandenberg frequently allowed
the children to initiate their own turns to disagree with each other. Here, when
she receded, the children also initiated their own speaking turns and continued
the disagreement in her absence, to the point that she had difficulty negotiating
her way back to the floor. Second, she frequently made connections across
mathematical ideas from the children about the question, ”What is a fraction?"
When she connected and revoiced (O’Connor 8: Michaels, 1993), she placed ideas
on the floor for consideration by all. She sometimes identified particular children
and their association with an idea. She reached back into previous dialogue and
nudged discussion forward through these links. Third, Mrs. Vandenberg Opened
the door for disagreement by asking different students’ questions about what

they thought about others' ideas, and whether they agreed. Over time, as more

150

ideas surfaced, she seemed to Work the discussion toward consideration of the
multiple ideas on the floor, so that a result on May 3 was a disagreement which
involved several speakers and ideas, and lasted several minutes. Eventually,
some of the children spoke up for themselves, locked onto other ideas, and
disagreed.
pr Does ThiS EpiSOde of Disaggeemant Affect the Di§cussion?

On May 3, up to the point when Mrs. Vandenberg generated the overhead
question in the midst Of the disagreement, the children aired several ideas.
Before the disagreement when they responded to the beginning question, ”What

is a fraction?”, the main issues raised by students were:

Monte: A big number you split into groups, with 200 + 10 and circled
groups of dots as his representation.

Hannah: A fraction is less than one, with a circle divided first in half and
then fourths as her representation.

Ali: Four-fourths is the same as one, because it's one group. .
Ali: One-two is the same as one-half, because it's all colored in.

Ali: One is the same as two halves.

Students assumed responsibility for sustaining discourse in disagreement.
In addition, there is another way in which this episode of disagreement affected
the discussion: it provided impetus for more discussion. In the midst of the
May 3 episode of disagreement, Mrs. Vandenberg created a discourse-generated
journal question at the overhead for the children to record and respond to. After
writing, she resumed the discussion, which was a direct result of the
disagreement. She developed an immediate connection to certain students'

publicly-shared ideas: Hannah's about two-fourths, and Austin's about one-half.

151

In this instance with the journal writing, she seemed to sort of put the
disagreement on a discursive burner, to simmer it and then use it to further the
resulting discussion. She appeared to capitalize on the moment near the end of
the disagreement when Hannah asked a question about a specific question:
"What is the difference between those two [her representations of one-half and
two-fourthsl?” The integrity of children's contributions during the disagreement
was sustained by this teacher's spontaneously-created journal questions which
she posed from the overhead.

To summarize, Mrs. Vandenberg started May 3 with a general question,
”What is a fraction?” As a result of discussion, different ideas emerged,
eventually resulting in an episode of disagreement. During the disagreement,
Mrs. Vandenberg created, in flight, an overhead question directly connected to
one asked by Hannah. When she did this, the discussion around the
representations for one-half and two-fourths stopped, but children's thinking did
not. The overhead question appeared to sustain thinking and fuel further
discussion about two specific fractions. In other words, Mrs. Vandenberg
initiated the discussion with an exploratory question, orchestrated discussion
through which children talked about their different ideas, navigated the
disagreement, and concluded it with an overhead question about specific, child-
generated math ideas. In spite of Hannah's use of pronouns, question-asking,
interchange of one-two to one-second, and argument about the difference
between them, the math-related aspects of this episode seemed to prevail. Mrs.
Vandenberg's mid-disagreement overhead question appeared to sustain the
prevalent math ideas, which she then used to launch further discussion about

two specific and disagreement-related fractions, one-half and two-fourths.

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Generally, the disagreement on May 3 appeared to be generative, because
it promoted further discussion about whether two fractions, one-half and two-
fourths, were the same. The notion that these two fractions might be the same
originated in the disagreement and were suggested by participants. And,
discussion about them continued after the episode of disagreement. Different
ideas emerged early on in the discussion, followed by Hannah's initiation of
disagreement with Ali. Mrs. Vandenberg sustained the disagreement with the
journal question, then nurtured further thinking about the specific fractions, one-
half and two-fourths, by orchestrating more post-disagreement discussion.
However, the children’s ideas after the episode were very similar, resulting in a
discussion rather like a flat line, until the end of math time.

May 7

Because the content of math time on May 7 is rooted in previous lessons, I
begin with a brief review of what happened before: May 4, the first day that
students used geoboards, and May 5, when they raised the notion of counting
pegs and squares to find a half on the geoboard.

On May 4 (Day 3 of my study) Mrs. Vandenberg introduced geoboards as
another means to have students explore fractions. Drawing on the teaching
resource, Sgeing Fractions (Corwin, Russell, and Tierney, 1990), her plan was for
students to explore the space on the board. Therefore, as suggested in m
Fractions she introduced a question about making halves. At the overhead she
projected the questions for the day and told the children they were to work with
both geoboards and dot paper (a sheet with 12 reproductions of a geoboard). On

the boards, she said, they were to practice, and then record their ways on the

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paper. They were to also write in their journals, and later would discuss their
halves.

On May 5 (Day 4), students again worked with geoboards to continue
thinking about half Of the board. Whereas on May 4 the children generally
worked with their boards, paper, and journals, on May 5, more class time was
dedicated to whole-group discussion. At the overhead, Mrs. Vandenberg wrote
the questions for the day: Sharing ways to make a half. From yesterday, with
geoboards. Which are congruent? Which are equal? On this second day of geoboard
use, the final two questions, ”Which are congruent? Which are equal?", were
new to the children. Much Of the ensuing discussion honed in on using squares

on the board to make half, although some children thought they should count

pegs.

Math on May 7
Friday, May 7 (Day 5 of this study), was significant because two

simultaneous disagreements occurred (see Appendix E). What interested me
again (as on May 3) was the math content, the ways in which students disagreed
about it, and the fact that what stood out was students’ interactions rather than
the math about which they disagreed. This episode of disagreement differed
from any other during the study in its nature and effect on the discussion
because Mrs. Vandenberg had to step in to stop the disagreement. When she
did, she stated the norms for participation in disagreement:

"the children were to disagree with ideas and not people;

*the children were not to raise a hand to or yell at one another.

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In this section, focusing on disagreements of May 7, I continue the format
of first describing the lesson and its disagreement, again pointing out
mathematical and interactional aspects during the disagreement. Next, I
describe the focus of the disagreements, and the ways in which they affect the
discussion. Finally, I describe the role of this teacher and the students in the
disagreements.

The questions and discussion that day were a carryover from the May 5
introduction of congruent and equal. After a brief comment about a rough
recess, Mrs. Vandenberg began the lesson by telling the children that two girls,
Michaela and Antonia, had ideas to share. First Michaela, then Antonia, both
soft-spoken girls, displayed and described their boards showing halves. At
several points when each girl spoke, Mrs. Vandenberg asked, ”Talk louder,
please,” and said, ”I can't hear you.” Whereas their representations differed,
both said they counted pegs (Michaela said ”nails.”) to make halves on their

board.

 

 

 

Figure 19: Michaela's Way to Make Half on May 7

Several times during Michaela's explanation, children raised the issue of equal

pieces. For example,

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Michaela: Then there's, twelve in there. And then here's another 12 and
you've got one nail left over. So I took that" nail and I made up nails with
rubber bands to make it equally.

Tomika questioned Michaela's use Of extra bands to "make up nails,” and asked

about putting bands on existing pegs (versus laying bands where there were no

P985).
T: Tomika doesn't understand or agree with the rubber bands in the

middle part.

Michaela: Well if there IS a nail you make it EQUALLY. SO I made a part
with rubber bands.

Tomika: But, but all these right HERE, I don't I think it, just, I think you're
supposed to like make it equally, equally, ee-qual-ly.

St: Equally!
Tomika: Yeah. Equally with the pegs that are ON there.

Tomika: And PLUS, the pegs that are right there and are the same as like,
on the other geoboards.

T: The difference here is that, my directions were to divide up the
geoboard evenly. SO the way, MICHAELA divided it up evenly she found
she needed to ADD pegs.

Like Tomika, others continued to wonder about Michaela's use of the
geoboard. While Michaela continued to explain, Mrs. Vandenberg stepped to the
overhead, pointed to Michaela's representation, and asked, the group, ”IS THIS
piece equal to THIS piece?” Some children called out, "Yes," but Tomika noted,

”But there's one more LEFT!” Mrs. Vandenberg noted their shape and said,

 

14I continue to use all capitals to indicate a speaker's emphasis on particular words.

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T: They're even the same SHAPE. What do we call it when they're the
same SHAPE?

Sts: Congruent!
T: Congruent. Good

She used Michaela's shapes to mention congruence and seemed to begin a
transition for the next girl to go to the overhead and talk about her way to make
halves. Whereas Michaela talked about counting pegs as the criteria for figuring
out if she made halves on the board, others, like Tomika, Joanne, and Jody raised
questions about the lone peg in the middle of the board left out Of Michaela's
shoe shapes. Noting this, Mrs. Vandenberg said, "She made TWO equal pieces,
which was part of our direction. Divide your geoboard into half or into two
equal pieces. But she did it in a way that left this out. So she was trying to find a
way to, to justify that, right?" Throughout, Mrs. Vandenberg asked Michaela
about her board, touching on issues of counting pegs, and notions raised in the
questions of the day: congruence and equal. Saying she thought Michaela's
"was a different way to think about the board," she then invited Antonia to share
her idea.

Antonia, like Michaela, talked about counting pegs, saying ”Miss Schram15
and me were talking about Ali's idea'6.” When she paused to recreate Ali's idea
on the overhead geoboard, Mrs. Vandenberg called on Braden, soon to be a
participant in a disagreement. He went to the overhead, and to demonstrate his

way, started to rearrange Antonia's board. Braden stumbled a bit in his

explanation, beginning and stopping, looking confused at the shape he created

 

15Antonia referred to Pamela Schram, the Mathematics Study Group Project's Principal
Investigator, who weekly visited this classroom.

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and telling Mrs. Vandenberg, ”That's not what I did.” He counted the pegs to
the left and right of the jagged line on the overhead geoboard. Appearing to note
his confusion, Mrs. Vandenberg suggested he show the class the geoboard he

was working on, which he did. Braden's shape looked like Antonia's.

 

 

Figure 20: Antonia's and Braden's Ways to Make Half on May 7

While Braden held his geoboard up to the class, Joanne apparently
disagreed with his method of proving half: counting pegs:

Joanne: Wait a minute though! That’ s counting the pegs not the squares.
Suzanne: But they’re not the same size.

Moira: Yeah if you put that ONE thing that’ 5 going like a (in the air with

her finger, she makes a horizontal line)

When Joanne called out her disagreement for the first time in this lesson
the issue of counting squares as a method to figure out halves came up.
Discussed at length on May 5, I was struck by the fact that it came to the floor
this late on May 7. Soon the discourse branched Off into two heated
disagreements, which occurred at the same time Mrs. Vandenberg receded from

the discussion to talk privately with Shawna. The first disagreement, near the

 

1""Ali's idea" refers to a previous discussion where Ali created a shape like Antonia's
(above), recorded it on dot paper, cut it out, and said the parts of the shape were not congruent.

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camera and between Tomika and Braden, was about whether Braden said to not

count the pegs.

Tomika: But you said up THERE (pointing to the overhead where Braden
had been standing) not counting the pegs

Braden: I mean not counting the pegs in the middle.
Tomika: But you didn’t SAY in the middle. You're talking about right

THERE. You’re talking about right THERE in the middle.

Tomika initiated disagreement with Braden and emphasized her
resistance to an earlier statement of his, that said he would not count the pegs.
To clarify, he said, ”I mean not counting the pegs in the middle.” However,
Tomika persisted, and got very specific about what he did and did not say. ”But
you didn't SAY in the middle. You're talking about right THERE. You're talking
about right THERE in the middle." I heard a definite undertone of noise while
many students called out overlapping comments, adding to my confusion about
what was happening to the discussion about counting pegs on geoboards.
Meanwhile, a second disagreement brewed across the room.

Like water about to boil, Joanne’s voice began to rise and Hannah’s
animated movement stood out. The two girls, sitting on the far side of the room
close to where Mrs. Vandenberg stood, appeared to be talking. I recognized
Joanne's voice and saw that Hannah talked, too, but could not tell what they
were saying. Joanne said something about ”pegs” and ”equal,” but in spite of
my best efforts, I could not figure out any more. Hannah responded, leaning in
toward Joanne, who sat kiddy-corner across the cluster of desks.

I was about to learn through the children’s talk of their ideas about

whether they should count pegs or squares to come up with half of the board,

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and how vested in their ideas some children could be. Whereas earlier the first
girls, Michaela and Antonia, touched mostly on the notion of counting pegs,
eventually the idea of counting squares on the board emerged as integral to the
focal disagreement. As the disagreement continued, Mrs. Vandenberg attempted

unsuccessfully to re-negotiate her way to the floor.

KV: I hear a little bit of disagreement. Sh.

Tomika: Up there you said not counting the pegs. You said not counting
the pegs between the back and over THERE.

Hannah: No you don’t! It’ s not a rule about that. No you don’t!

Rather than taking up Mrs. Vandenberg’s comment, several students
continued to talk, among them Hannah, Joanne, Tomika, and Braden. Tomika
continued her disagreement with Braden. At the same time, Hannah became
more vocal about the interaction in which she was involved. ”No you don't. It's
not a rule about that. No you don't,” Hannah declared about something I
couldn't yet detect, her voice getting louder and making this second
disagreement evident. Perhaps seeking to understand what was going on,

perhaps to rein in the discussion, Mrs. Vandenberg asked,

KV: Tell me what you're DISAGREEING about. Right now Ijust hear
comments

Hannah: Right now we’re disagreeing because,
KV: Hannah.

Hannah: (continuing) Joanne and Ali keep saying that a, a geoboard you
HAVE to count the squares not the pegs.

Sts and Joanne: No. Nuh uh.

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Without hesitation Hannah turned to face her teacher and spoke for the
girls, essentially saying, ”Right now we're disagreeing because, Joanne and Ali
keep saying that a, a geoboard you HAVE to count the squares not the pegs."
The girls in question quickly said ”no" and ”nuh uh.” Still at the table and facing
Mrs. Vandenberg, Hannah turned back around to face Joanne and Ali across her
table to say, ”But you don't HAVE to. If you're not-if you don't WANT to.”
Responding so quickly that their speaking overlapped Hannah and Joanne

interacted with rapid-fire turns:

Joanne: I never said you HAVE to.

Hannah: Uh huh! I said I said to you, you don't HAVE to and you goes
'yes you do.‘

Joanne: Not necessarily, but basically yeah you do.
Hannah: Uh uh. There's not a rule or a LAW!
Ali: That’s how you do fractions.

Hannah seemed to take the firm stance that counting squares versus pegs
is not cast in stone in the form of a rule or law. Her verbal and nonverbal cues
indicated that she held to this conviction. She spoke more loudly, got up out of
her chair, and began to walk toward the supply bins to get a geoboard,
continuing to voice her disagreement as she moved across the room. After a
comment from Ali which seemed to support Joanne's idea, Mrs. Vandenberg
attempted to bring Steve into the disagreement, by calling on him. However,
Hannah continued speaking, and her classmates began to lend support to Ali
and Joanne to become a coalition of speakers (Florio & DeTar, 1995). Because
one comment piled on another, I found I had to work hard to glean what they

said. And in the midst of this I said-you said debate Ali spoke up, saying to

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Hannah across the desk, ”That's how you do fractions.” Mrs. Vandenberg called
on Steve, who walked from the back of the room to the overhead projector at the
front. But before Steve could say anything, Hannah continued, and Mrs.
Vandenberg was again distracted, this time by Michaela, who silently beckoned
her to the front of the room. Facing Ali as she stood up and pushed in her chair,
Hannah walked across the room to get a geoboard from a storage bin, and

exclaimed,

Hannah: So. We’re not DOING the fractions!
KV: Sh.

Sts: Yes we are!

Hannah: I said what if we AREN’T!

St: But we ARE!

Monica: But we ARE, Hannah. That’s fractions.
Hannah expressed a second view, that this geoboard work of dividing the

board into halves was not fractions. Her classmates disagreed, saying that this is
fractions. As Mrs. Vandenberg quickly surveyed the room and shushed the rest
of the students, what began as a three-way disagreement between Hannah,
Joanne, and Ali seemed to shift, becoming a disagreement between Hannah and
the class. ”Yes we are [doing the fractions]!” called several students. ”I said
what if we AREN’T!” responded Hannah. There she stood at the rear corner of
the room, facing the entire class.

”But we ARE!" I heard someone say, unable to determine who because
this disagreement both erupted and played out so quickly. ”But we ARE,

Hannah. That's fractions" said a quiet voice from the front of the room, Monica

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as it turned out, Monica who rarely spoke up during discussions. Loudly,
Hannah continued,

Hannah: People can count pegs on this if they want! It's not a LAW that

you—that you CAN ”1‘!”

She leaned toward her classmates and stepped a little closer to Monte who
sat at a table near where she stood. Hannah took a geoboard from the table,
turned, held it up and leaned forward as she looked toward Joanne and Ali while

she emphatically spoke in a louder voice. But, the debate wasn't over:

Joanne: Hannah you don't have to get so burned UP!
Monte: Yeah!

When Joanne said she ”doesn't have to get so burned up,” Hannah did
something unusual, for this student or any other in this classroom during my
nearly 50 observations - she lifted the geoboard over her head in a threatening
fashion, glared at Monte, leaned toward him, and acted as though she was about
to hit him it. Monte and Michaela reacted immediately, with ”Roar!” and
”Whoa!” With this uncomfortable violation of participation norms for
disagreement, Mrs. Vandenberg worked to explicate them. She called Hannah to
where she stood to the right of the overhead projector. In this moment, the
norms of participation were violated in a way found at no other time. Somewhat
indirectly, Mrs. Vandenberg focused on the violation, not directing her
comments specifically to Hannah, but to the entire class. In fact, when she called
Hannah to her Mrs. Vandenberg put her arm around Hannah's shoulder as she
addressed the class. I watched as Hannah briefly smiled when Mrs. Vandenberg

talked to all about the inappropriateness of raising one's hand or yelling at

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someone else. But the fracas continued. ”You have to count the squares” Ali
called. Presumably to nip any physical harm in the bud, Mrs. Vandenberg

stopped the disagreement.
KV: Time out. Time 931;. TIME OUT.

Slowly and more loudly each time, Mrs. Vandenberg spoke and gestured
for everyone’s attention. Her raised arm as she said this was a common signal
for students' attention and quiet was the result. When she lowered her arm, I
didn't hear a sound. Some students quickly resumed talking, and while Mrs.
Vandenberg put her arm around Hannah's shoulder, she spoke finally and

emphatically, to state the norms for participation in disagreement which

KV: TIME OUT! Right now—what does time out mean?” (A child kept
talking while she was saying ”Time out.”)

Moira: It means ’be quiet, please!’
After Moira’s whisper-like definition, Mrs. Vandenberg explained,

KV: It means be a listener. Right now I heard SO many good ideas but I
had a hard time following ANY of them. I also saw one person get a little
bit upset. Let's remember that we're disagreeing with each other's IDEAS,
not the PEOPLE. OK? To raise your HAND to someone else or, to, YELL
at someone else is not appropriate. OK? I called on Steve because Steve,
looked like he had something important to say and he was so
PATIENTLY raising his hand.”

Stepping in during the disagreement and stopping it, Mrs. Vandenberg made
clear the norms for participation in disagreement:
*the children were to disagree with ideas and not people;

*the children were not to raise a hand to or yell at one another.

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Thus re-installing the disagreement norms, she appeared to imply the following
rights and duties of participants:

Table 2: Participant Rights and Duties

 

Speaker 1 Speaker 2
Right: To say idea Duty: To listen to idea
Duty: To listen to reasoning about Right. To disagree with idea
why disagreeing
Right: To clarifyc,1 articulate, defend Duty: To listen to articulation of idea
1 ea

 

 

 

 

Because these norms were violated by the inappropriateness of
” [RAISING] a hand to someone or, [YELLING] at someone else,” Mrs.
Vandenberg intervened to repair the interaction. Then, she redirected the
discussion to what Steve had to say, which was to count the squares. Pointing to
Antonia's geoboard shape (which was still projected from the overhead), Steve
said ”Um what I did was, um I counted all this. Well I changed Joanne's idea a
little bit. I um, I thought I-I counted, I counted squares in this." He pointed to
and counted out loud each square to the right, and then the left, of Antonia's line.
Mrs. Vandenberg then targeted the disagreement-related issues when she said,
”So the disagreement here is whether we count the pegs or squares.”

To review, similar to May 3, the math task initially under discussion on
May 7 was exploratory in nature. Mrs. Vandenberg planned for this task so that
the children had geoboards as an additional representation with which to think
about fractions. Mrs. Vandenberg drew on a resource new to her when she
developed this visual spatial exploration for the children. During students'

descriptions of how they made half, the predominant notion they talked about

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was counting pegs. When disagreements erupted, both included debate over
counting pegs. Whereas Tomika questioned Braden about pegs he said he didn't
count, Hannah raised the issue of whether a person had to count squares and not
pegs, and if there was a law or rule saying one had to do so. Interestingly, on
May Sit was Hannah who asserted that to make half on the geoboard, she
should count squares. Yet here, on May 7, she argued in favor of an additional
idea, as well as the notion that there was not a rule or law saying that they cannot
count pegs. Hannah opened the possibility that she meant that 1/ 2 can be
represented by counting squares or pegs, that there are ways of representing 1/ 2
either way.

However, different from May 3, May 7 began without detail regarding
math questions for the day. Whereas Mrs. Vandenberg initiated math time on
May 3 with specific statements about the questions they would be discussing,
who and what they discussed on April 28, directions about journal writing, and
management, on May 7 she started the lesson differently. Mrs. Vandenberg
invited Michaela and then Antonia to share ”an idea about the work that we
have been doing with the geoboards and fractions," which they did. Michaela
explained the way she counted pegs and worked with extra bands, and Antonia
talked about and projected the shape she and Miss Schram connected to Ali's,
also focusing on pegs as the way to think about making half on the geoboard.
When Braden explained his shape, two disagreements erupted at about the same
time, one between he and Tomika over which pegs he did not count, and the
other initially between Hannah, Joanne, and Ali about counting pegs or squares.
The latter became the focus of everyone's attention and ended with Mrs.

Vandenberg's ”Time outs.”

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From its beginnings when two girls talked about their ways to make half
on geoboards to the moment when the teacher stopped the disagreement about
whether there was a law about counting pegs or squares, May 7 revealed the
students’ thinking about the task of making halves, which for the most part,
consisted of talk about counting pegs, but also about counting squares.
However, in addition to their thinking about halves on the geoboard, the
disagreement prompted me to take notice of the ways in which these children
interacted over each other's thinking.

Similar to May 3, on May 7 Mrs. Vandenberg again stopped the episode of
disagreement. However, the reason she stopped it was not math-related but due
to negative and inappropriate interactions between children. Instead of creating
questions to further the discussion (as on May 3), to stop this episode she re-

stated the norms of disagreement:

T: Let's remember that we're disagreeing with each other's IDEAS, not the
PEOPLE. OK? To raise your HAND to someone else or, to, YELL at
someone else is not appropriate. OK?

Then, she concluded these comments by saying the behavior-related
reason she called on Steve: ”I called on Steve because Steve, looked like he had
something important to say and he was so PATIENTLY raising his hand.” These
post-disagreement norms that Mrs. Vandenberg explicated leaned toward
focusing on a classmate's idea in disagreement, rather than attacking them as a
person, physically or by yelling at them. She also implied that in order to do this,
these children must be polite. She seemed to be getting at disagreeing with each

other's ideas, not the people; but also disagreeing in respectful, civil ways.

167

What is the Disagreement About?

Whereas the disagreements discussed in Chapter Five were about
fractions content, the content in this disagreement was lost to the negative
interactions between children. Yes, they evidenced their involvement in the
disagreement and in their ideas, but the very act of disagreeing quickly overrode
the math content. Lack of social competence prevented academic competence
from playing its role. Mathematically, the disagreement on May 7 between
Tomika and Braden seemed to be over what Braden said not to count. Tomika's
position was that Braden said not to count pegs to make his halves, then said to
count them. Noting this, Braden revised and said he meant not counting
particular (i.e., middle) pegs. The longer and more prominent disagreement
between Hannah, Joanne, and Ali appeared on first glance to be over counting
pegs or squares, whether Joanne said ”you have to,” and if there was a ”rule or
law.” However, the underlying position which Hannah took was that 1 / 2 can be
represented either by counting pegs or by counting squares. And yet, beyond
the mathematics in the disagreement were the pronounced interactions between
children, and the features of these interactions that served to distance their
competing ideas.

The Interactions of the thldren

Although these fourth graders evidenced their involvement in the
disagreement on May 7, it took a negative turn which resulted in the need for
this teacher to step in for interactional and not math-related reasons. Discomfort
in disagreement (Lampert, Rittenhouse, & Crumbaugh, 1993) for these children

and their teacher (and me) evidenced itself as an unanticipated factor. Similar to

168

May 3, Mrs. Vandenberg again stopped the disagreement. However, her reasons
for doing so on May 7 were starkly different. Close examination of student
interactions shed light on why she may have needed to do this, as well as why
the episode took on a negative tone. During the episode, the children talked
louder, used ambiguous pronouns, and shifted to a use of negatives, such as
”didn’t," ”don’t,” and ”aren’t.” And, to support her cause, Hannah called in an
outside expert. However, rather than convince others of her stand on the issue of
counting pegs or squares, Hannah ended up distancing herself from her
classmates.

First, the children spoke more loudly and emphatically during
disagreement than they did on May 3. For example, Tomika turned up the
volume and emphasized particular words when she disagreed with Braden

about the pegs he said not to count (”the ones in the middle”).

Tomika: But you said up THERE not counting the pegs.
Braden: I mean not counting the pegs in the middle.

Tomika: But you didn't SAY in the middle. You're talking about right
THERE. You're talking about right THERE in the middle.

Similarly, Hannah, Joanne, and Ali talked loudly and emphatically, as though in

an attempt to add meaning to their words:

Hannah: Right now we're disagreeing because. . Joanne and Ali keep
saying that a, a geoboard you HAVE to count the squares not the pegs.

Hannah: But you don't HAVE to. If you're not—if you don't WANT to.

169

Joanne: I never said you HAVE to.

Hannah: Uh huh! I said I said to you you don't HAVE to and you goes
'yes you do.‘

Joanne: Not necessarily, but basically yeah you do.
Hannah: Uh uh. There's not a rule or a LAW!

Ali: That's how you do fractions.

T: Steve.

Hannah: So, we're not DOING the fractions!

T: Shh.

Sts: Yes we are!

Hannah: I said what if we AREN'T!

St: But we ARE!

Monica: But we are, Hannah. That's fractions.

Hannah: People can count pegs on this if they want! It's not a LAW that
you—that you CAN 'T!

Joanne: Hannah you don't have to get so burned UP!

In addition to an amplified use of voice in disagreement, these students

increasingly used pronouns. However, unlike May 3 when they to used "you”

either to refer to a particular classmate or to the group generally, on May 7, I

frequently questioned to whom they were talking when they used ”you,"

especially problematic during whole-group discussion with 26 children in the

room. Were they talking to someone in particular, or not? Their generalized use

of ”you” appeared to function to say, when any person does this kind of

problem, then they have to use this method. For example, when Hannah initially

said, ”Joanne and Ali keep saying that a, a geoboard you HAVE to count the

squares not the pegs,” She was looking at Mrs. Vandenberg when she said this

about the squares. Paradoxically, it would seem that Joanne and Ali just told

170

Hannah about counting squares, but here Hannah said ”you.” By contrast, when
Joanne responded, ”. . . but basically yeah you do," she was looking at,
therefore presumably talking to, Hannah. For the rest of the disagreement,
Hannah continued to look toward Joanne and Ali, apparently to underscore who
she talked to. But I noticed something else unique to Hannah's address as she
developed her case for counting pegs: she shifted from the use of ”you” to "we,"

and others followed suit:

Hannah: We're not DOING the fractions.
T: Shh.

Sts: Yes we are!

Hannah: I said what if we AREN'T!

St: But we ARE!

Monica: But we are, Hannah. That's fractions.

Perhaps Hannah’s use of ”we" was her effort to re-establish a sense of
community after she failed with her I-versus-you stance. The sense of
community purported by ”we" would go farther to convince others of her
position on rules about counting squares (or, more precisely, that there is no rule
that they had to, nor a rule that they couldn’t count pegs). Others indicated their
involvement when they called out comments to Hannah and also used "we."
They, too, seemed interested in the notions of squares, pegs, and fractions. Then,
as though building her case from the personal (you) to the larger ”we,” Hannah
seemed to bring an outside authority into this classroom: rules or laws behind
what goes on in math. And bringing in the outside authority allowed Hannah to

call on ”them” to bolster her case against her classmates. Overall, their pronoun

171

use unfolded in the following order, showing how the children spoke in a you
said/ I said fashion, then shifted to we are / we aren’t, to the more general ”we"
and ”people," and concluded with the very personal, "Hannah, you don’t have
to get so burned up:”

You said (T omika)

I meant (Braden)

You M say (T omika)

You're talking about (T omika)

No you M1 (Hannah)

Right now we're disagreeing about (Hannah)

Joanne and Ali keep saying (Hannah)

You have to (Hannah)

But you Mt have to. . if you w want to (Hannah)

PQNQWPQN!‘

10. Im said (Joanne)

11. I said (Hannah)

12. to you (Hannah, to Joanne)

13. basically yeah you do (Joanne)
14. That's how you do fractions (Ali)

15. We're not doing fractions (Hannah)
16. Yes we are (students)

17. I said (Hannah)

18. what if we geluHannah)

19. But we are (student)

20. But we are (Monica)

21. People can (Hannah)

22. Hannah you don't have to (Joanne)

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The students’ use of ”you," ”I,” ”we,” and ”people can,” evolved into an
overall adversarial stance in the episode, especially when combined with the
multiple instances of negatives. Tomika, Hannah, and Joanne inserted words
such as ”don't,” ”don't have to," and "I never said,” as though to differentiate
their thinking from someone else's. The I-you, I never said-I said, we're not-we
are, you have to-you don't have to position-taking of the children created a
formula to breach communication about math, in fact to result in talking past,
rather than to, one another. In addition, it elevated not only the personal nature
of the disagreement, it also distanced Hannah interactionally from her
classmates. This differed from the children’s interactions on May 3 during the
episode of disagreement, when fractions content was at the fore (although not to
the extent as during single disagreements, as discussed in Chapter Five),
amplifying the difference between the episodes on May 3 and May 7. On May 3,
the children also spoke loudly, but not to the extent of May 7. And they also
used pronouns, but on May 3 the pronouns had fairly direct referents to make
clear to what or whom the children referred. What especially distinguished the
May 7 episode from the episode of May 3 were the negatives used by these
children, Hannah’s reference to rules or laws and the very personal, and
negative, tone which the disagreement took on. The following table compares
children's pronoun choice and use of negatives on these two days and highlights

this difference:

173

Table 3: Student Pronoun Use on May 3 and May 7

 

 

 

May 3 Episode May 7 Episode
1. you ga_n't solve it (Monte) 1. You said (Tomika)
2. You can make that into a fraction 2. I meant (Braden)
(Joanne) 3. You didn't say (Tomika)
3. I don't think it would be a fraction 4. You're talking about (T omika)
(Hannah) 5. No you don't (Hannah)
4. You would have to take pieces out 6 Right now we're disagreeing about

(Hannah)

5. I'll show you (Hannah)

6. if you shaded one in (Hannah)

7. if you shaded in (Hannah)

8. Or you could say 'one.’ (Ali)

9. if you just put it in half (Hannah)
10. if you colored in half (Hannah)
11. But you would still split up (Jody)
12. And you color the other side
(Hannah)

13. Or you can just put one (Ali)

14. Why would you put one (Hannah)
15. Ali I want you to think (Hannah)
16. she's just said (Shawna)

17. you can do it (Shawna)

18. You could call it (Hannah)

19. what would you call (Hannah)
20. Icall it (Austin)

21. What would you call (Hannah)
22. Icall it (Austin)

23. What would you call (Hannah)

(Hannah)

7. Joanne and Ali keep saying
(Hannah)

8. You have to (Hannah)

9. But you don't have to. . if you don't
want to (Hannah)

10. I never said (Joanne)

11. I said (Hannah)

12. to you (Hannah, to Joanne)

13. basically yeah you do (Joanne)

14. That's how you do fractions (Ali)
15. We're not doing fractions
(Hannah)

16. Yes we are (students)

17. I said (Hannah)

18. what if we aggrit (Hannah)

19. But we are (student)

20. But we are (Monica)

21. People can (Hannah)

22. Hannah you 51931 have to (Joanne)

 

 

174

 

The third indication from these students of the episode's negative tone
was their lack of connection to each other's ideas. By contrast to their
interactions on May 3, their ideas on May 7 faded so that the children seemed to
step up for a face-off, to separate, rather than connect, thinking, and to compete
rather than collaborate over math ideas. In this episode, the civility Mrs.
Vandenberg fostered faded along with the math focus.

The Role of the Teacher

In some ways, Mrs. Vandenberg's interactions with the children to begin
the lessons were similar on May 7 and May 3. For example, like the children,
during the episode of disagreement on May 7, Mrs. Vandenberg also talked
louder and with emphasis. She also orchestrated the discussion by connecting

students’ ideas, as in the following examples:

T: Oh. This is what happened when we worked with ALI'S idea.
Remember? Every TIME people would change it we ended up back
where we started with the halfs down the middle. What do you guys
think of Michaela's idea?

T: JOANNE had a good point. she said, 'but MICHAELA you had gaps
in the middle.’ She said, 'And it wasn't evenly.’
And, in addition to connecting ideas during the discussion, Mrs. Vandenberg
also laid the discursive foundation for disagreement through questions and

comments:

T: OK. We counting pegs or squares inside?

T: So Michaela when YOU divided up the rectangle, in order to have 2
equal parts you had to get RID of a nail. Is that, is THAT what you're
saying?

T: Is THIS piece equal to THIS piece?

175

However, on May 7, Mrs. Vandenberg did not insert comments and questions as
frequently as she did on May 3. Leading up to May 3's episode, Mrs.
Vandenberg's speaking tums were to ask questions and make connections across
children's contributions. However, prior to the episode on May 7, her
connections and questions to orchestrate the discussion occurred less often and
were less specific.

Also in contrast to May 3, before and during the episode on May 7, Mrs.
Vandenberg was forced to spend considerable energy on management. For
example, while Michaela and Antonia were at the overhead describing their
geoboard designs, Mrs. Vandenberg frequently had to shush the rest of the
children and remind them to listen to the girls. Different from the details she laid
out on May 3, math time on May 7 seemed not as strategically orchestrated. Lost
in the episode were the norms of disagreement Mrs. Vandenberg fostered all
year, and the indexed (Mehan, 1979) classroom rule that children were to
disagree with civility and respect.

On May 7, the second disagreement (first with Joanne, then with others)
seemed weighted down by the personal and negative tone which overtook it.
This episode seemed to take the discussion into a downward spiral. Rather than
stopping the disagreement (but not children's thinking) to sustain thinking about
one or more math ideas from the children, and potentially further disagreement,
on May 7, Mrs. Vandenberg extinguished disagreement to also end
inappropriate behavior. She reminded students that disagreement was supposed
to be about ideas, that they were to be listeners, and that it was not appropriate
to raise a hand to someone else or to yell at them. In addition, when she

subsequently called on Steve, she specifically stated the behavior-related reason

176

why she called on him: ”he looked like he had something important to say and
he was so patiently raising his hand."

The episode of disagreement on May 7 stands in contrast to other
disagreements discussed in this study. During this episode, the children
frequently competed for the floor, and once they negotiated their way onto it,
competed to sustain an idea. Negotiation of meaning became less pressing than
who and what idea had the floor. The children were certainly involved, but
more so in conflict than in disagreement of the sort Mrs. Vandenberg fostered.
The previously-unified dialogical relation (Burbules, 1993) and fractions content
so prevalent in the disagreements discussed in Chapter Five and in the episode
on May 3 were overrun by competition and distance, which Mrs. Vandenberg

repaired when she stopped the disagreement.

177

CHAPTER SEVEN
CONCLUSIONS AND IMPLICATIONS

Summary and Discussion

This exploratory study was an examination of student-to-student address
in disagreement during whole group discussion, those moments when these
fourth graders attempted to independently maintain the norms of mathematical
disagreement fostered by Mrs. Vandenberg. In these instances, these children
independently engaged in a subset of the mathematical discourse similar to what
mathematicians do: they disagreed. This study is timely in that it looked at
mathematical discourse advocated by reform documents (National Council of
Teachers of Mathematics, 1989, 1991 ; National Research Council, 1989), to gain
increased understanding of its complexity. Focusing on the sub-question, What
do these students need to learn to be involved in disagreement, it reveals what
these children needed to learn how to participate in mathematical discourse, and
more specifically, mathematical disagreement, with its norms different than in
traditional math classrooms. And in so doing, there were two competencies the
students had to learn and demonstrate which contributed to the nature of the
disagreements: academic competence (knowledge of fractions) and social
competence (how to respectfully and appropriately disagree). However, there
was more they needed to learn. These children also needed to learn to use these
competencies in synchrony to mathematically disagree. Mathematically, they
had to sort out the legitimacy of assertions, and as they occurred, weigh them in
the face of the assumptions, stated reasoning and justification behind them. And

when they mathematically disagreed, they were to do it respectfully, to give

178

evidence for their position while saving face (Lampert, Rittenhouse 8r
Crumbaugh, 1996).

Whereas, in the traditional math lesson the teacher controls the discourse
and does most of the talking, in lessons such as these where the teacher planned
an open-ended task with discussion, it is envisioned that teachers will involve
students in discourse. There are different expectations of and demands on
teachers and students. For teachers, this adds a layer of complexity to classroom
discussion and disagreement which may be unnerving, because not only is the
task open-ended, so is the discussion. This raises new questions for teachers,
who may be unsure about whether to manage the mathematics, the social nature
of who is disagreeing and in what ways, or both. And, it raises an additional
issue for teachers. Because teachers tend to be insecure both with the subject
matter of mathematics as well as the teaching of it, they may naively assume that
there is at least one aspect of the new mathematical discourse with which they
can succeed: discussion. After all, we "discuss” all the time. However,
participating in discussion and orchestrating it are very different, as this study
revealed. This study shows that managing the interactional aspect of these
disagreements may be as demanding as managing the mathematical aspect,
shedding light on the fact that the combination of both aspects, mathematical and
interactional, results in something greater than its parts, generative mathematical
disagreement.

Focusing on a second sub-question, What is the role of this teacher and
these students during disagreements, this study also underscores the strategic
role of this teacher in the successful management of disagreement during

discourse of this nature. As this teacher fostered norms of mathematical

179

discourse, these fourth graders developed academic and social competencies, to
learn to mathematically disagree about fractions. This is demanding and
complex teacher work, and in the words of O’Connor (1996), ”is not for the faint
hearted” (p. 496). This study reveals in Chapter Five that when Mrs.
Vandenberg momentarily let out the tether, the children negotiated the floor to
directly and publicly disagree with one another and could do so with academic
and social competence. However, this study also shows how, in episodes
(especially on May 7), the interactions of the children overwhelmed their
mathematical, or academic, competence. Without their teacher to manage these
longer disagreements, students were unable to continue the collaborative norms
of disagreement she fostered. In episodes of disagreement, the fractions content
on the floor collided with individuals’ faltering abilities to express them with one
another, and the disagreement as an educative discourse phenomenon
disintegrated. The disagreeing students clearly demonstrated their commitment
to their ideas and to the discourse when they continued to disagree without Mrs.
Vandenberg. However, without the equally important social competence in the
disagreement, academic competence was lost in the fray.

This study differs from other studies of disagreement because it examined
disagreement during whole-group discussion. There was much discursive
responsibility shouldered by Mrs. Vandenberg to orchestrate discussion with 26
fourth graders in the room participating in whole group. From the beginning to
its conclusion, she drew children and their ideas to the public discourse, asked
questions, connected ideas, and inserted task- and behavior-related reminders
while navigating the fractions terrain. Studies of disagreement in small groups,

either without the involvement of the teacher or with limited involvement, do

180

not place on the shoulders of teachers these moment-to-moment demands with
the entire class.

In this setting, disagreement varied in its influence on the discourse.
When the mathematical and interactional aspects of student disagreements were
tightly woven, the mathematical disagreement contributed to the discussion and
propelled it forward. And, when the disagreement was laden by the
interactional aspect, the disagreement created a gap in the discussion, and in the
episode of disagreement on May 7, forced its shutdown. On that day, in spite of
Mrs. Vandenberg’s established norms for respectful disagreement, the autonomy
of disagreeing children got in the way of educative disagreement. Their way of
disagreeing subordinated the math content. Rather than construction of joint
meaning, they engaged in the destruction of joint meaning (0’ Connor, 1996).

At times (for example, on May 3 during the episode when the children
debated whether ”1 /2” was ”one-half,” ”one-two,” or ”one-secon ”) when
disagreement began, the children seemed to be negotiating meaning about
fractions. Although at first glance these children appeared to be disagreeing
about what to name particular fractions, there was additional meaning behind
their statements. I believe they conceptually explored what a fraction was (i.e.,
part of a large number, part of one whole) and in doing so, created drawings and
labels to clarify this thinking. In the episodes, the talk was more from the world
of these children, more theirs than the mathematical discourse Mrs. Vandenberg
fostered. The world of the child met the world of mathematics. That is, their
language and ways of interacting took precedence over what they were learning
in this fourth grade. Let me illustrate with two examples, first of these children’s

language and second, of their ways of interacting. Regarding the language of

181

these children, on May 3 while struggling with what to name the numeric
fraction, 1 / 2, their invented terms, ”one-two” and "one-second," bumped into
the conventional mathematical term, one-half (inserted in the episode by Mrs.
Vandenberg). In the words of Vygotsky (1978), the moment seemed to be a time
when the children attempted to create schooling’s scientific concept (”one-half”)
on their own. In the moment, the children seemed to let go of what Vygotsky
termed spontaneous, non-school concepts of childhood (”one-two” and "one-
second”). Second, regarding their ways of interacting, on May 7 when Mrs.
Vandenberg receded, the fourth graders continued to disagree when she stepped
aside to confer privately with Shawna. In this disagreement, there was no one to
manage the desired norms of respectful disagreement, and the autonomy of the
disagreeing children shone through. Rather than disagreeing about the merits of
counting pegs or squares on the geoboard, the children disagreed about who said
what and to whom.

The relationship between the purpose, function, and mathematical import
of the disagreements led to the examination of mathematical tasks which fueled
mathematical thinking. For example, the partnered task on May 11" which led
to the creation and discussion on May 12 of Kim’s Diamond prompted extended
and strategic reasoning. For much of discussion on May 12, Kim’s Diamond held
the floor. These children teased apart several fractions ideas: equal area (do two
diamonds equal one square) and, in relationship to the whole represented by the

diamond, what did the shaded square represent and how do you know?

 

'7 You will recall that this partnered task was on a handout, on which the children were
to think of a fraction and do the following: Draw a picture that represents your fiaction. Explain in
your journal. Then, on the bottom half of the paper, Mrs. Vandenberg wrote: What fraction does the
above picture represent? Why? How did you solve this?

182

Together, the children nudged one another’s thinking, to determine answers to
these questions. And listeners also appeared involved in this and most
disagreements, as evidenced by their spontaneous call outs to join in the
disagreement; and as evidenced by their attentiveness (e. g., looking at the
representations magnified from the overhead, looking at the speakers). Kim's
Diamond as an example of an offshoot of this partnered task demonstrates that
mathematical disagreements can serve an important function in the discourse: to
get at the underpinnings of another’s assumptions and at the reasoning behind
their position. This example stands in contrast to the episode of May 3 when
several ideas stood in competition on the floor, or on May 4, when Jonah’s
geoboard drawing of one-half was corrected by Jody.

As discussed earlier, ways of getting math ideas on the floor during
episodes of disagreement and ways of talking to each other were negotiated by
the children. When Mrs. Vandenberg receded during episodes, they, rather their
teacher, appeared to be managing it (however unsuccessfully). The children
seemed to act in situ to define the content and direction of this segment of the
discourse.

This study shows that disagreement provided a window into these
children’s developing fractions knowledge. Whereas the single disagreements
were wrapped more tightly around fractions content due to Mrs. Vandenberg’s
management of them, the episode of disagreement on May 3 teetered away from
content, in part because she inadvertently receded from her role as discourse
conductor. And, this study demonstrates that when this teacher receded, for
children disagreement (e.g., in the episode of May 7) can be competitive,
frustrating and uncomfortable (Lampert, Rittenhouse, & Crumbaugh, 1996).

183

However, what is gained from examination of this difficult disagreement, in
combination with that gained from the examination of them all, is that
disagreement can be respectful and collaborative. After all, competition in the
classroom as a positive motivator is good only for the winners.

Analyses of disagreement in Chapter Six demonstrate that, if the authors
of reform documents (and teacher educators who teach in the spirit and context
of reform) have as their goal for students to own their ideas and increase
mathematical power through discourse, a caution is in order. We cannot assume
that the process by which these ideas are owned is easily achievable or a positive
experience. This is a process that requires previously under-recognized teacher
knowledge and skill, so that the educative value and complexity of this discourse
is appreciated.

Regarding the way in which fractions content and social interactions were
separated for examination in this study, of course, the act of disagreeing over an
aspect of fractions content is unified and simultaneous. In real time during a
discussion, there is not one or the other, the content or the process at work. Each
constitutes the other. However, when examined separately for the purpose of
this study, the two competencies, mathematical (i.e., academic) and interactional
(i.e., social), provide new insights into the importance of both in order for
children to mathematically disagree with respect and civility while saving face.

I do not claim that learning takes place only during disagreement. My
claim is that in disagreement during discussion, teachers are afforded a window
into students’ thinking, a useful opportunity if teachers wish to foster the

mathematical activity and learning advocated by the authors of reform

184

documents. Disagreement is a moment at which competing (or, colliding) ideas
have the potential to force children to explain, defend, and justify ideas.

Disagreement is also a moment which may be seized by teachers and
capitalized upon to increase student involvement in discussion. This would
assist teachers in their efforts to enhance students’ mathematical power, as called
for in reform documents (NCTM, 1989, 1991 ; NRC, 1989). As such, disagreement
is a learning nexus from which teachers can learn in the moment what students
think and why. Learning does not require disagreement, but mathematical
disagreement is a public, visible, and spontaneous site in which to gauge student
learning of fractions content and learning of the requisite social competence by
which content is negotiated. And, teacher reflection on the relationship between
task and disagreement can lead to a generative use of disagreement.

The findings of this study, discussed above, draw attention to some of
richness of disagreement in mathematical discourse. In addition, it emphasizes
challenges to teachers and students when disagreements arise during
mathematics discussions. I now turn to implications for theory and practice.

Theory and Practice

Theory
With heightened interest in mathematics teaching and learning that is

aligned with reform documents (NCTM, 1989, 1991 ; NRC, 1989) this study of
disagreement during discussions is timely. It is timely because mathematical
discourse of the type advocated by authors of reform is complex with much yet
to be learned about the new roles and responsibilities of teachers and students.

Studies of student disagreement and argument, although emerging, are rare (see,

185

for example, Cobb & Yackel, 1994; Lampert, Rittenhouse, & Crumbaugh, 1996;
Wilkinson 8: Martino, 1993).

When combined with current interest in sociocultural learning theory and
its basic premise that language use mediates learning, this study reveals that
research on the ways in which children and teachers negotiate meaning through
subject matter discourse is worthwhile. The process by which meaning is
negotiated is situation specific and not to be prescribed. The advocated
discourse increases students' role in the construction of their understandings and
draws upon disputes over ideas as ways to refine and develop ideas. However,
studies which simultaneously draw on language-based theory and use method
grounded in the language use of participants in order to examine discourse-
based mathematics teaching, and disagreement in particular, are rare.
Controversies of the kind that the Standards (1991) seem to promote and were
examined in this study are not common in most classrooms; to the contrary,
student consensus and acceptance are more the rule. More work which draws in
tandem on language and meaning-based perspectives is called for to better
understand student disagreement during mathematics discussions.

In this study I have attempted to show the interrelationship between
sociocultural learning theory, techniques of discourse analysis from the method
of sociolinguistics, and the study of students’ disagreement during fourth-grade
mathematics. This study underscores the complexities of teaching and learning
in the context of reform. The results of my study reveal that there is much for
teachers and students to attend to while negotiating meaning during
mathematics discussions. While subject matter is central, my study shows that

there is more to mathematical disagreement than mathematics.

186

The analyses reveal that these children needed to learn how to participate
in mathematical discourse and more specifically, mathematical disagreement,
with its different norms than in traditional math classrooms. And, in learning
how to participate in this discourse, there were interdependent competencies
students had to learn which contributed to the nature of the disagreements:
academic competence (knowledge of fractions) and social competence (how to
respectfully and appropriately disagree). And in addition (and more
importantly), these fourth graders needed to learn to use these competencies in
synchrony to mathematically disagree.

The analyses also show that the focus of disagreements hinges on the role
of the teacher in the successful management of disagreement. There are norms to
be fostered to promote a dialogical relation (Burbules, 1993), norms which lay the
foundation for the social construction of subject-matter knowledge. These norms
are also central in the formation of academic and social competencies required to
learn to mathematically disagree over subject-matter content.

The analyses show that disagreement varied in its function in the
discourse. When the mathematical and interactional aspects of students’
disagreements were tightly woven, the mathematical disagreement appeared to
contribute to the discussion and propel it forward. And, when the disagreement
was laden by the interactional aspect, the disagreement created a gap in the
discussion, and in one instance of disagreement, forced its shutdown.

The relationship between the purpose, function, and mathematical
significance of the disagreements led to the examination of mathematical tasks
which initially fueled thinking. Mathematical disagreements can serve an

important function in the discourse: that is, to get at the underpinnings of

187

another’s assumptions and at the reasoning behind their position. This stands in
contrast to other disagreements in the data which pointed out a flaw in a drawn
representation; or, in one episode, when several ideas stood in competition on
the floor.

This study shows that disagreement provided a window through which to
spontaneously assess children’s developing fractions knowledge. Whereas some
disagreements were focused on fractions content due to Mrs. Vandenberg"
management of it, one episode of disagreement veered away from content when
Mrs. Vandenberg inadvertently receded from her role as discourse conductor.

What encompasses these findings is the focus on situated language use
and sense-making of participants. Recent research cuts across sociocultural
learning theory, sociolinguistics, and mathematical discourse (see for example,
Lampert, Rittenhouse, 8: Crumbaugh, 1996; O’ Connor, 1996). However,
additional studies are warranted. Individually and in combination, sociocultural
learning theory and the method of sociolinguistics were used to shed light on
what disagreement was in this setting, in the subject matter of fourth-grade
mathematics.

132191132

For educators, this study is useful, because it provides an examination of
how disagreements arise, are managed, and frequently, capitalized upon.
Through this lens, educators may gain insight into ways in which to plan tasks
for disagreement, recognize one as it develops, and use it to gauge learners’
thinking in the moment, and to sustain a discursive course toward learning.

From it, my goal is for educators to recognize such signals as when referents

188

begin to become unclear, when negatives begin to seep into the discourse, and
when I said-you said talk verges on personal rather than mathematical.
Increased understanding of disagreement will assist teacher educators’ efforts to
heighten preservice teachers’ understanding of classroom discourse, so that they
may, to the extent possible, prevent uncomfortable situations. Preservice
teachers require support in their efforts to learn to foster discourse about math
ideas, to become users of respectful discourse and disagreement.

Limitations

This study examined whole-group discussion and disagreement only. A
comparison of student involvement in small- and whole-group discussion would
have added insight into children’s interactions, to determine if they disagreed in
the same way in both forums. The study would be stronger with greater
understanding about who disagreed during small group to determine if norms in
small group differed in the absence of the teacher. Without Mrs. Vandenberg,
would their small-group disagreement be similar to the episodes?

Analyses of additional types of data also would have strengthened this
study. Predominantly, whole-group discussion was examined. Although
listening and writing (journal, picture drawing) were a part of this study, They
were not extensively analyzed. This study of student-to-student disagreement
required an examination of discussion, and was supported by students’ journal
writing.

This study did not set out to examine issues of gender during
disagreement. Many studies exist of adult male-female interaction during
informal conversation (see for example, Tannen, 1990, 1993; West & Garcia, 1988;

West & Zimmerman, 1983; Murray, 1985; Maltz 8: Borker, 1982). Studies also

189

show the ways in which teachers interact in American classrooms more often
with boys than girls (e. g., Brophy, 1985). With recent reform efforts calling for
increased classroom discourse as a powerful process through which to develop
and enhance students’ mathematical power, studies of boy-girl interaction
during classroom discourse is warranted. Such studies during disagreement,
moments which can (as shown in this study) be uncomfortable and frustrating,
are especially timely. Although gender-related studies of classroom discourse
and disagreement are an area of potential, they were not the goal here.

When examined, disagreements may be revealed as a rich source for
understanding the complexities of mathematics teaching and learning. As
alluded to in the Standards (1991), disagreement embedded within discourse
may be a window for teachers into students’ mathematical learning and at the
same time force children to articulate and justify their reasoning, for themselves
and for others. In addition, a give and take may result as different ideas make
their way to the floor. In the process, children may also learn respect for each
others’ thinking.

Questions for Future Study

When student disagreement arises during whole-group discussion, it is
imperative that educators (and reformers) have some sense of this complex
discourse process in which students are expected to gain mathematical power,
and to recognize what students need to learn in order to be involved
disagreement. This study looked closely at a specific component of discourse,
student disagreement, for the purpose of understanding what it takes to be

involved in it in this classroom.

190

Although this study generated findings toward a better understanding of
the territory of mathematical disagreement, it also raised questions about
disagreement and tasks which foster mathematical disagreement:

‘What does it take to create a classroom where children disagree with

respect?

*How do we develop a way to distinguish educative disagreements from

those that are less mathematically worthwhile?

"What does it take to have mathematically worthwhile disagreements?

*How do we create a context that increases the likelihood that

disagreement has mathematical significance and is productive of student

learning?

These questions extend my intent to increase understanding of students’
disagreement during mathematics discussions. Where the goal is changes in the
teaching and learning of mathematics in order to increase students’ mathematical
power, studies developed to address these questions are necessary. Such studies
are necessary to expand researcher knowledge of what it actually means for
teachers to promote discourse with the goal of increased mathematical power.

And perhaps more importantly, studies are warranted to expand researcher

understanding of what it means for students to be involved in disagreement.

191

APPENDICES

192

APPENDIX A

, 193

APPENDIX A

Data Collected, 1992-1993

 

 

 

Field Notes Video Tape: Video fipe: AudiofiTape Student
Math Class Fractions Unit Interviews
September 25 October 1 April 28 Austin Hannah
October 1 October 6 May 3 April 13 December 3
October 8 October 8 May 7
October 13 October 13 May 10 Braden Monica
October 27 November 5 May 11 Monica February 23
November 17 November 24 May 12 Monte
December 1 November 17 May 13 April 26 Jody
January 26 November 18 May 19 March 2
February 2 November 25 Elizabeth
December 1 Discussion Antonia Fractions,
December 3 about May 3 pre- and post-
December 8 discussions: 'n rvi :
January 12 June 9 Braden
January 26 Monica
January 28 Teacher Monte
February 2 Interviews: Elizabeth
February 9 July 2 Rob
February 23 July 13 Connie
March 2 Antonia
March 15
March 16
March 24
April 23
April 26

 

 

 

 

 

194

 

APPENDIX B

195

APPENDIX B
Analyses of Discourse

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APPENDIX C

197

APPENDIX C
Fractions Unit Overhead Questions

April 28. Qgesg'ons to begin glass: What is a fraction? When have you
used fractions? How are fractions represented? Draw and label pictures.
Questions during class: None. Qtestiens at the end ef gess: Do both of these
show half? Why or why not? Show other ways to represent 1/ 2.

May 3. mesfiogs te begin class: Same as April 28. MW
fiss: Is this 1/ 2, 2/ 4, or both? Why? WW: None.

May 4. mestions to begin class: How many ways can you show halves
(1 / 2)? Practice on geoboards. Record on paper. Write one explanation in
journal, explaining how you can prove your idea shows a half. W
m: None. Questions at the end ef class: Here’s Ali’s 2 pieces. How can we

tell if these are equal amounts?

May 5. mestions to begin glass: Sharing ways to make a half. From
yesterday, with geoboards. Which are congruent? Which are equal? Megs
during class: Respond to this in your journal: Is this [Ali’s way] a way to divide
the geoboard in half? Why or why not? mestiogs at the end ef glass: None.

May 7. Qeestions to begin class: None. @estiogs during Qess: Finding
ways to show 1 / 2 and 2 / 4 on the geoboard. Record it on dot paper. Prove one
of your ways. Questions at the end of class: None.

May 10. glenestions to begin class: Sharing responses from May 7

[homework]. Where do you find fractions? How do you represent fractions
(1 / 2, 2/ 4, 7/ 14, 4 / 4)? Questions during glass: Does a fraction have to be less
than 1? Why? Questions et the end of class: Draw a representation for 3/5. Do

the pieces have to be equal? Why?

198

May 11. gflestions to mgin glass: Draw a picture that represents a
fraction and then explain it in your journal. Questions dering eless: None.
Quesg'ees at the end of glass: Is 8/ 4 the same as 4/ 8?

May 12. Questiens to begin class: What fraction does this picture [Kim’s
diamond shape with one square colored in] represent? Why? W
gees: What is 1/2 + 1/2? Why? Draw a picture of this. WW
eggs: Prove that 1/ 2 is the same as 2/ 4.

May 13. Questions to mgt'n eless: Where is 1/ 2 on the number line?
Qdestjefi during class: Is 1/ 2 less than one? Why? WW
dess: Is 1/ 4 greater than or smaller than 1 / 2? Where is it on the number line?

May 19. mestions to begin class: Put in order from smallest to largest:

6/ 6, 3 / 6, 1 / 2. Draw a representation of each. Explain your solution. Qdesdegs
dddng glass: None. Questions at the end of eless: None.

199

APPENDIX D

200

APPENDIX D

Discourse Pattern

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201

APPENDIX E

202

APPENDIX E
Transcript of Episodes of Disagreement, May 3 and May 7

May 3

KV And how is THAT like 20 divided by 10, Hannah? Or TWO
HUNDRED divided by 10, excuse me?

Hannah I keep making my two sideways. OK. The REASON, OK. Two

hundred divided by 10 is not a fraction. Hannah erases, then

writes.

KV Is NOT a fraction. But what YOU showed us IS a fraction.

Hannah It's NOT a-it's NOT a fraction because-

09:51

Monte It CAN be a fraction Monte remains all this time at the OH.

Hannah Because a FRACTION is less than one. TEN and TWO
HUNDRED are not less than one. HERE is another kind of
fraction. There are other kinds of fractions. THIS would be,
one-fourth, if you shaded in one.
Hannah draws a circular representation, divided into four sections.
She points to one of the sections and records '1/4'.

10:16

KV OK.

Hannah And,

203

Hannah

10:36

Hannah

Hannah

Jody

Hannah

How did you know that would be one—fourth.

It would be TWO-fourths if you shaded in, two, THREE-fourths
and FOUR-fourths. Hannah shades each additional section as she
speaks.

And ONE. Or you can say ONE.
Ali says 'one' when Hannah says 'FOLIR-fourths'.
How come you say one instead of four-fourths Ali.

'Cause it's, all colored in it's one, group.

Hannah it was helpful to me the way you went through there
and you colored a piece at a time, one-fourth, two-Did you guys
see that? TWO-fourths, THREE-fourths.

In other words if you just put it in HALF, and you colored in
HALF, it would be one,

Two. Ali says one-two; Hannah says one-half.

One-HALF. It would be one-half.

Hannah now has filled in each of the four sections, and drawn a
new circle with a horizontal line across it:

One-two is one-half.

But you would STILL split up, two hundred and,

And you color the OTHER side would be

204

Hannah

Hannah

Hannah

11:40

Hannah

'Other side' = top portion of the circle.
Rather than finishing her sentence, Hannah records '2/‘2.’
ONE.
TWO, halves.
K V is speaking to someone; Monte colors over a section already
filled in.
OR you can just put ONE.
NO! Why would you put ONE?
'Cause it's ALL FILLED IN.
all filled in - each word spoken separately; speech is not fluid.
OK I MISSED that last ques-
NO!
Hannah continues to draw circular representations of fourths.

Hannah's last comment, was that that was FOUR-fourths. And
Ali said it was ONE, because it was one whole-if we were
thinking about that like a COOKIE, and you were getting
FOUR-fourths, would you actually be getting one whole

cookie?

(calling out) Ali, Ali I want you to think. This PROBABLY

could be one second, but then what would, THIS be.

205

Ali and Sts

Hannah

Shawna

Joanne

Hannah

12:34

this = 1/2 'What would this be? ' = Hannah draws while she asks this
question.

One-HALF.

One HALF. That's the same as one—SECOND. T'here's one-half,

there's one-SECOND. WHAT is the difference between those

two?

Hannah points to what is written on the OH: 1/2 1/2
Nothing.

Nothing she's just said that you can do it either way.

Oh so she's saying that it also would make sense for it to be
called

one-SECOND?
My BROTHER told me-

Since we talked about one-THIRD or one-FOURTH? You're
right. That is two ways to say it and and another way that

we've gotten used to saying it is one-HALF.
I know what the answer is.

Joanne what do YOU want to say?

You COULD call it, you COULD call it-

Hannah looks toward Joanne/Shawna/Ali and speaks loudly.

206

Hannah

Austin

Hannah

Hannah

Dallas

13:13

St

Hannah

You COULD call it, one-SECOND. But the REAL one-second is
THIS. Now, er-Now, what would you call THIS? Right here?

Joanne goes to the OH, joining Hannah.
Hannah draws a circle and shades in the top half.
Joanne reaches under the projector to get a new transparency.

Hannah erases. She redraws a new circle, divides it in fourths, and

shades in the upper right section, pointing to it/‘this '.
I call it a HALF.

A quarter, or a fourth.

A quarter or a fourth?

One-FOURTH. Hannah records ’onefourth' as she speaks. She

continues recording, filling in a second portion of the drawing.
Do you agree with that Kim?

What would you call THAT?

Two quarters or TWO-fourths.

I call it a HALF.

Wait! I'm curious here. I heard two different ANSWERS. Um,
ALI said it's two-fourths, and AUSTIN said HALF.
Same thing.

It IS the same thing. Now what would you call THAT.

207

13:35

Hannah begins to draw, out of camera range.

I want you to stop for a minute and respond to that in your
journal. One second. We'll come right back to this. Can I write
down your idea?

FOUR-fourths, not ONE-fourth. If it was ONE-fourth it would

be one.
I want you to respond to this in your journal. Here's the picture

Hannah drew you. OK she had divided this into four pieces,
and then she colored, TWO. OK. And I want you to respond to
this question: Is this 1 / 2, 2 / 4, or both? Why?

Austin said one-half. Ali said—Hannah said it's TWO-
FOURTHS. A couple of you said it's both. Tell me what YOU
think and respond to why. And draw it-write the question in

your journal please.
But it has, FOUR pieces.
Does that question make sense? So Ali you are starting to

explain it NOW. You think because it has four pieces, you
colored in two? You think it's both? Write the question please.

Get started Rob.

208

KV
Braden
St
13:25

Joanne

Suzanne

Moira

13:34

Tomika

Braden

Tomika

13:48

May 7
Equal on both sides?

Yeah.

But that's counting the PEGS.

WAIT a minute though! That's counting the pegs not the
squares.

But they're not the same size.

Yeah if you put that ONE thing that's going like a

Moira makes a horizontal line in the air with her finger.

Steve raises his hand.

But you said up THERE [points to OH] not counting the pegs.
Hannah leans forward when she responds to Joanne

I mean not counting the pegs in the middle.

Suzanne leaves her desk to get a geoboard from the back of the room.
But you didn't SAY in the middle. You're talking about right
THERE. You're talking about right THERE not in the middle.

Sean goes for a geoboard.

209

Tomika

Hannah

Hannah

14:00

Hannah

Hannah

Sts & Joanne

I hear a little bit of disagreement. Sh.
KVraises her arm, signaling for students ' attention.
Her speech overlaps with Tomika's next turn.

[points toward the OH] Up there you said not counting the
pegs. You said not counting the pegs between the back and
over THERE.

No you don't It's not a rule about that. No you don't!
Hannah '5 voice is louder. She leans in once again to interact with
Joanne. Ali says something to Hannah.

Also talking are Micah/Monte; Tamika/Braden.

That's what we do with fractions.

SO!

Tell me what you're DISAGREEING about. Right now Ijust
hear comments.

Right now we're disagreeing because,

Hannah turns toward K V as she speaks.

Hannah.

(continuing) Joanne and Ali keep saying that a, a geoboard you

HAVE to count the squares not the pegs.

No. Nuh uh.

210

Hannah

Hannah

Joanne

Hannah

14:36

Hannah

But you don't HAVE to. If you're not-if you don't WANT to

Hannah turns back toward Joanne. As Joanne and Hannah interact
their speech overlaps. Joanne/Ali/Hannah continue talking, even

though K Vcalls on Steve, who then goes to the OH.
I never said you HAVE to.

Uh huh! I said I said to you, you don't HAVE to and you goes

yes you do.’

Tamika leaves her table and joins K V near the door.
Not necessarily, but basically yeah you do.

Uh uh. There's not a rule or a LAW!

That's how you do fractions.

A student undertone of comments increases in volume.

Steve.

(facing Ali as she stands up and pushes in her chair) So, we’re

not DOING the fractions!

Hannah appears to ignore the fact that KV has called on
Steve/ofi'ered the floor to him, as she continues to argue with
Joanne and Ali. Steve walks to OH projector. Tamika

returns to her desk. Hannah gets a geoboard from the round table

in the back of the room.

211

Sts

Hannah

St

Monica

Hannah

Joanne:

Monte

Sh.

Now, the lone exchange is between Hannah and, first Joanne/Ali;
but soon is between Hannah and class.

Yes we are!

I said what if we AREN'T!

Hannah turns to face Joanne/Ali as she speaks.

But we ARE!

But we ARE Hannah. That's fractions.

Pointing to the OH screen. KV returns to the OH.

People can count pegs on this if they want! It's not a LAW that
you don't have-that you CAN 'T.

Hannah takes a geoboard from the table, turns, holds it up and

leans forward, toward Joanne/Ali while she emphatically speaks in a

louder voice.

Hannah you don't have to get so burned UP!

Joanne ’5 speech overlaps with Hannah ’5.

Yeah!

Monte makes a face as he speaks.

Hannah lifts the geoboard over her head as though she might hit

Monte with it. He sits about one foot in front of her. Hannah

212

Michaela

E:

Moira

turns around and replaces the geoboard at the round table.
Whoa!

Jessica watched wide-eyed as Hannah swung the geoboard.
Come here Hannah.

You have to count the squares.

Time out. Time out. Time out. (pause)

TIME OUT.

Right now—what does time out mean? (pause)

KV raises her arm and makes a fist, again signaling for students'

attention. This time, K Vspeaks more slowly and emphatically; also,

she lowers her arm. The room is totally quiet for an instant.
Some students resume talking. KV puts her arm around
Hannah '5 shoulder. Braden, Steve, and Antonia are at the OH.
It means be quiet please!

Moira whispers this; I can hear her because she sits near me and
the camera.

It means be a listener. Right now I heard 80 many good
ideas but I had a hard time following ANY of them. I also

saw one person get a little bit upset. Let's remember that we're
disagreeing with each other's IDEAS and not the PEOPLE. OK?

To raise your HAND to someone else or, to, YELL at someone

213

else is not appropriate. OK? I called on Steve because Steve,
looked like he had something important to say and he was so
PATIENTLY raising his hand.

Hannah smiles when KV says "I also saw one person get a little bit

upset. " KV points to Steve. when she says "I called on Steve
because... " The room is quiet when KV finishes saying all this.

214

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