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dissertation entitled

TEACHERS' USE OF A PROBLEM-SOLVING ORIENTED
SIXTH-GRADE MATHEMATICS UNIT: '
TWO CASE STUDIES

presented by

Anthony Dane Rickard

has been accepted towards fulfillment
of the requirements for

Ph.D. degreeinfiugric“ Teaching, & Ed. Pol

Dept. of Teacher Education

Major Professors:

Sandra K. Wilcox Deborah Loevenberg Ball

Date 6/1/93

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TEACHERS' USE OF A PROBLEE-SOLVING ORIENTED
SIXTH-GRADE IATEEIATICS UNIT:
TWO CASE STUDIES

By

Anthony Dane Rickard

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Teacher Education

1993

ABSTRACT

TEACHERS' USE OF A PROBLEM-SOLVING ORIENTED
SIXTH-GRADE MATHEMATICS UNIT:
TWO CASE STUDIES

BY

Anthony Dane Rickard

Problem solving is a central issue in current reform initiatives
in mathematics education. However, while curriculum developers design
problem-solving oriented curricula to help move reforms into K-12
mathematics classrooms, little is known about how teachers actually use
problem-solving oriented mathematics curricula to teach.

This study investigates how two sixth-grade mathematics teachers
used a problem-solving oriented unit on perimeter and area. A four-
dimensional framework is developed and employed to explore how each
teacher's knowledge, views, and beliefs shaped her use of the unit.
Using data collected through interviews, classroom observations,
conversations with teachers and their students, samples of students'
work, teachers' lesson plans, and the unit on perimeter and area, two
case studies are presented to portray how each teacher used the unit in
her classroom.

This study shows that each teacher's use of the unit was
consistent with her underlying views and beliefs, and with some aspects
of the intentions of the curriculum developers who designed the unit.
However, other aspects of the teachers' use of the unit varied from the
intentions of the curriculum developers. This study shows further that
each teacher's use of the unit was shaped by interplay between her own

views, beliefs, and knowledge, and the unit. Therefore, both the

perimeter and area unit and the teachers shaped the teaching which
occurred in their classrooms.

This study suggests that while problem-solving oriented curriculum
can play a role in shaping mathematics teaching, the views, beliefs, and
knowledge of teachers should be addressed in curriculum. This study
also points to issues for future research that are connected to

teachers' use of problem-solving oriented curricula.

This work is dedicated with love and respect to my father,
who has provided unending support throughout all of my endeavors,
and to the memory of my mother.

iv

ACKNOWLEDGEMENTS

A number of people helped make completing this work easier than it
might have otherwise been, and each deserves a note of thanks.

To my father, David Rickard, and my sisters, Tracy and Tiffany:
Thank you for your understanding and support throughout my collegiate
and graduate career.

To my major professors Deborah Loewenberg Ball and Sandra K.
Wilcox: Thank you both for helping me better understand and appreciate
myriad issues, ideas, and dilemmas in mathematics, teaching, and doing
research in mathematics education. Thank you also for your insights,
encouragement, and support throughout work on this dissertation and
throughout my doctoral program.

To my advisory and dissertation committee members William
Fitzgerald, Bruce Mitchell, and Ralph Putnam: Thank you for guidance,
many learning experiences, and valuable advice during my doctoral
program at Michigan State University.

To the Connected Mathematics Project directors -- Glenda Lappan,
Elizabeth Phillips, and William Fitzgerald: Thank you for the
conversations which helped me in conceptualizing this study and for the

opportunity to learn about and develop mathematics curriculum.

TABLE OF CONTENTS

CHAPTER 1
INTRODUCTION ......................................................... 1
The Problem ...................................................... 1
The Purpose of the Study ......................................... 6
The Significance of the Study .................................... 8
Teachers' Conceptions of Problem-Solving Activity .......... 8
Teachers' Use of Problem-Solving Oriented Curricula ........ 9
Instructional Decisions ................................... 10
Problem—Solving Reform .................................... 11
Overview of the Dissertation .................................... 12
CHAPTER 2
REVIEW OF THE LITERATURE .......................................... 15
Research on Problem Solving ..................................... 15
Teaching and Learning Problem Solving ..................... 15
Problem Solving in Classroom Settings ..................... 19

Research on Factors Influencing Teachers'
Use of Curricular Materials ..................................... 25

Constructing a Framework ........................................ 31

Teachers' Views and Beliefs About

Mathematics and Problem Solving ........................... 33

Problem—Solving Activity in Classroom Settings ............ 37

Subject-Matter Knowledge .................................. 41

Teachers' Perceptions and Beliefs About

Student Learning...... ...... .... .......................... 45
Summary..... .................................................... 51

vi

vii

CHAPTER 3
METHODOLOGY ......................................................... 54
Case Study Research ............................................. 54
Overview of Methods ............................................. 56
Teacher Participants ............................................ 58
Conducting the Study ............................................ 59
Interviews: Design and Rationale ......................... 60
Observations: Studying Teacher Participants .............. 64
Analysis .................................................. 65
CHAPTER. 4
PROBLEM SOLVING AND THE INSTRUCITONAL UNIT:
BACKGROUND AND ANALYSIS ........................................... 68
The Connected Mathematics Project ............................... 68
The CMP and Middle School Mathematics Reform .............. 68
The Connected Mathematics Project: Views and
Beliefs About Mathematics and Problem Solving ............. 7O
Problem-Solving Activity in Classroom Settings ............ 75
New Roles for Students ............................... 75
New Roles for Teachers ............................... 76
Subject-Matter Knowledge .................................. 80
Perceptions and Beliefs About Student Learning ............ 81
Covering and Surrounding ........................................ 82
Overview and Description of the Unit ...................... 82
Problems and Problem Solving in
Covering and Surrounding .................................. 87
Looking Ahead: Teaching Covering and Surrounding ............... 90
0 CHAPTER 5 A
THE CASE OP KAREN KNIGHT .......................................... 92
A Profile of Karen Knight ....................................... 92
Karen and the Domains of the Framework .......................... 97

Views and Beliefs About Mathematics
and Problem Solving ....................................... 97

viii

Problem-Solving Activity in Classroom Settings ........... 100
Subject-Matter Knowledge ................................. 102
Perceptions and Beliefs About Student Learning ........... 105
Summary .................................................. 110
Karen's Use of Covering and Surrounding ........................ 111
Summary of Unit Coverage ................................. 112

Sequencing of Investigations and Perceptions
of Problem Solving: The Power of Beliefs ................ 114

The Room Design Problem and Skill
Maintenance .............................................. 119

Different Curriculum, Similar Use of Teacher Materials...128

Karen and Covering and Surrounding:

'I’m Trying to Change' ................................... 135
Summary ........................................................ 139
CHAPTER 6
THE CASE OP BETTY WALKER ......................................... 141
A Profile of Betty Walker ...................................... 141
Betty and the Domains of the Framework ......................... 144
Views and Beliefs About Mathematics
and Problem Solving ...................................... 145
Views and Beliefs About Problem-Solving Activity ......... 147
Subject-Matter Knowledge ................................. 150
Perceptions and Beliefs About Student Learning ........... 153
Summary .................................................. 156
Betty's Use of Covering and Surrounding ........................ 157
Summary of Unit Coverage ................................. 157

Problem Solving: Different Degrees of

Exploring and Experimenting .............................. 159
Use of Teacher Materials: 'I Haven't Ever

Looked at That!‘ ......................................... 169
Same Teaching, Unchallenged Beliefs ...................... 176

Summary ......... . .............................................. 178

CHAPTER 7

ix

KAREN AND BETTY: THE INTERPLAY BETWEEN TEACHER
AND CURRICULUM .................................................... 179

Interplay: Reciprocal Influence Between
Teacher and Curriculum ......................................... 179

Unpacking Karen's and Betty's Uses of Covering and
Surrounding: A Perspective from Curriculum Development ........ 183

Unpacking Covering and Surrounding: Karen's

and Betty's Perspectives ....................................... 187
Views and Beliefs About Problem Solving .................. 188
Problem-Solving Activity in Classroom Settings ........... 189
Subject-Matter Knowledge ................................. 190
Perceptions and Beliefs About Student Learning ........... 191
Commentary ............................................... 192
Synopsis of Findings ........................................... 193
CHAPTER 8
"HAT LIES AHEAD: CHALLENGES AND DILEMMAS FOR
CURRICULUM DEVELOPMENT ........................................... 195
A Dilemma for Curriculum Developers ............................ 197

Karen and Betty: Teachers Who Provide Snapshots of Dilemmas
for Curriculum Developers ...................................... 199

Computation and Skill Maintenance ........................ 200

The Sufficiency of Curriculum Materials for

Supporting Teachers ...................................... 203
Teaching and Learning Subject Matter ..................... 207
Teachers' Perceptions and Beliefs About Learners ......... 214
Final Comments ........................................... 217
A ......................................................... 219
B ......................................................... 223
C ......................................................... 227
D ........................................................ .231
E ......................................................... 244

Table

Table

Table

Table

Table

Table

Table

4.

5.1

5.

5

6.

6.

6.

1:

.3:

LIST OF TABLES

Summary of Covering and Surrounding ....................... 85

Summary of Problems, Follow-Up Questions, and ACE
items assigned by Karen .................................. 113

Karen: Assigned number of exercises compared to total...113

Karen: Percentage of exercises assigned in each covered
investigation ............................................ 114

Betty: Summary of Problems, Follow-Up Questions,
and ACE items assigned ................................... 158

Betty: Assigned number of exercises compared to total...159

Betty: Percentage of exercises assigned in
each covered investigation .............................. 159

Figure 2.1:
Figure 4.1:
Figure 5.1:
Figure 5.2:

Figure 5.3:

LIST OF FIGURES

Geometric representation of squaring a binomial ........... 42
The pentomino problem ..................................... 89
Students' non-standard way of finding perimeter .......... 103
Converting between units for measuring area .............. 103

Matt, Trevor, and Aaron's rectangle for proving

triangle area ............................................ 116
Figure 5.4: Mr. Dull’s room .......................................... 120
Figure 5.5: Shakaya's room ........................................... 123
Figure 5.6: Diagrams of door and window wall sections ................ 130
Figure 5.7: Parallelogram on a grid .................................. 134
Figure 6.1: Student's work on finding perimeter of a rectangle ....... 150
Figure 6.2: Student's conjecture about square units for area ......... 151
Figure 6.3: Mr. Dull's room ........................................ ..160
Figure 6.4: Max’s room ............................................... 162
Figure 6.5: Shaded square and circle diagram ........................ .172
Figure A.Bl:Subject-matter knowledge item 1 from interview ........... 209
Figure A.B2:Subject-matter knowledge item 2 from interview ........... 210
Figure A.BB:Subject-matter knowledge item 3 from interview ........... 211

xi

CHAPTER 1

INTRODUCTION

The Problem

Over the last 10 years, calls for integrating problem solving into
K-12 mathematics have steadily gained momentum. Major initiatives in
mathematics education at both state and national levels have provided
agendas for reforming K-12 mathematics, envisioning problem solving as
perhaps the most central aspect of the curriculum (e.g., California
State Department of Education, 1991; National Council of Supervisors of
Mathematics [NCSM], 1989; National Council of Teachers of Mathematics
[NCTM], 1980, 1989, 1991). Another common feature of these reforms is
their acknowledgement that currently in K-12 mathematics many teachers,
for a plethora of reasons, do not teach mathematics as problem solving,
and therefore students are not learning mathematics as problem solving
(Putnam, Lampert, & Peterson, 1990). Instead, reformers argue, current
K-12 mathematics instruction tends to concentrate on developing
students' computational proficiency and skills in applying algorithms
(NCTM, 1989; Stodolsky, 1988). As a consequence, K-12 students have
little opportunity to develop higher-order skills in mathematics such as
problem solving (Kulm, 1991; NCTM, 1989; Putnam et al., 1990).

Seeking to transform this predominant state of affairs,
substantial changes have been proposed for the K-12 mathematics
curriculum. The NORM (1989) Curriculum and EValuation Standards for
School Mathematics and the Mathematics Framework for California Public
Schools (California State Department of Education, 1991) are likely the
nmet ambitious plans describing problem-solving oriented curriculum

1

recommendations for school mathematics. California's Framework draws
heavily on the NCTM Standards, including adoption of the NCTM position
that 'Problem solving should be the central focus of the mathematics
curriculum' (NCTM, 1989, p. 23).

Echoing the problem-solving emphasis of these curriculum agendas
are new visions for how students should learn, know, and experience
mathematics in school. Reformers highlight the importance of connecting
mathematics in concrete ways to the world around us and learning about
relationships between mathematical concepts and processes (see Steen,
1990). Advocates of problem-solving centered curriculum reform seek to
deemphasize instruction on mechanical and often disconnected algorithms
and computation, and to increase instructional emphasis on important
mathematical concepts (e.g., measurement, number, shape) via problem
solving. Mathematics education reform is calling for sweeping change
throughout the K—12 curriculum to move away from rulebound textbook
learning. The clear message is that a school mathematics curriculum
should help teachers teach mathematics as problem solving and students
learn mathematics as problem solving (Greeno, 1991; Lester & Kroll,
1990; NCSM, 1989; NCTM, 1989, 1991; Putnam et al., 1990).

In response to the problem-solving focus of all recent major
curriculum reform initiatives in K-12 mathematics education, the last 10
years have also seen a flurry of activity in mathematics curriculum
development. For example, the Heed Numbers project is an elementary
mathematics curriculum that centers on students collecting, generating,
organizing, representing, and making sense of data (see Friel, Makros, &
Russell, 1992). The Middle Grades Mathematics Project produced five

detailed units incorporating a problem-solving based learning model

developed by the authors -- the units focus on measurement, spatial
visualization, factors and multiples, probability, and similarity (see
Lappan, 1983; Shroyer & Fitzgerald, 1986). The Camputer Intensive
Algebra project is an algebra curriculum that integrates computers and
computer software into high school algebra (see Fey & Reid, 1991). The
Connected Mathematics Project, recently funded by the National Science
Foundation, seeks to develop a complete middle school mathematics
curriculum by 1996 that emphasizes connections among mathematical
concepts and between mathematics and other disciplinary areas (see
Fitzgerald, Lappan, & Phillips, 1991). All these curricula assume that
problem solving is a central activity in mathematics that K-12 students
should be engaged in when studying mathematics. These and other
problem-solving oriented curricula assume that the student is not a
passive receiver of mathematical facts and procedures. Rather, learners
are active in constructing their own understandings of mathematics
through problem solving in mathematically rich contexts (e.g., Shroyer &
Fitzgerald, 1986).

Problem-solving oriented curricula imply not only new roles for
students but also imply new roles for teachers (Cohen & Ball, 1990;
NCTM, 1991). If teachers are to teach mathematics as problem solving,
instruction cannot be limited to what Jackson (1986) calls the
'transmission model' of teaching -- the teacher tells the students
information and demonstrates procedures, and students show, by doing
what the teacher does, that they have received the information and
procedures. In contrast, teachers who teach mathematics as problem
solving, reformers argue, employ multiple representations of concepts

and relationships, model and engage students in dialogues where

conjectures about problem situations are offered, tested, and revised.
Teachers help students articulate, represent, and modify their own
ideas, and journey with students through a mathematical terrain of
important concepts and connections (Ball, 1990a; Lampert, 1990; NCTM,
1989, 1991; Putnam et al., 1990). Teaching mathematics as problem
solving may also include teaching students specific problem-solving
strategies to solve particular kinds of problems (e.g., Meyer & Sallee,
1983), or teaching students global strategies applicable to varieties of
problems (e.g., Charles & Lester, 1982).

Because problem-solving oriented curricula imply new teaching
practices for teachers, they are often viewed as vehicles for teacher
learning and change. For example, in their National Science Foundation
proposal connected Mathematics, the developers of the Connected

Mathematics Project note that:

In order to help teachers make the kinds of changes in
instructional thinking and planning implied by the goals of
connected Mathematics, the materials developed will take
seriously the need to provide instructional strategies and
organizational help for teachers so that they can develop
new modes and habits of instruction (Fitzgerald et al.,
1991, p. 13).

The NCTM (1991) emphasizes that for a teacher to change his or her
practice of teaching mathematics to a problem-solving orientation,
ongoing effort to implement new practices and analysis of one's own
teaching are required.

It is well-known, however, that what curricular materials imply
for teaching practice or are intended to accomplish in classrooms and
what actually occurs can be quite different. There is substantial
evidence that teachers enact curricular materials in many different

ways. As persons who work in institutions where curricular and other

learning materials are generally imposed by others, teachers tend to
shape curriculum to their own immediate situations and available
resources (Lipsky, 1980; Lortie, 1975; Sarason, 1982). Teachers also
have varying degrees of subject-matter knowledge of mathematics and
pedagogical content knowledge about representing and connecting
mathematics and problem solving to learners (Ball & McDiarmid, 1990;
Wilson, Shulman, & Richert, 1987). Teachers hold different perspectives
and beliefs about mathematics, problem solving, and the role of problem
solving in mathematics and the mathematics curriculum (Rickard, 1991;
Silver, 1985; A. Thompson, 1989; Wilcox, Schram, Lappan, & Lanier,
1991). Teachers can also be constrained or motivated by the context in
which they teach (Wilcox, Lanier, Schram, & Lappan, 1992). For example,
teachers can feel overwhelmed by perceived time constraints,
discouraging them from being open to new ideas about teaching, or be
challenged and motivated to change by the learners they encounter in
their classrooms (Lortie, 1975; Wilcox et al., 1992). All of these
factors -- available resources, subject matter knowledge, pedagogical
content knowledge, different perspectives and beliefs, context --
contribute to how teachers use curricular materials to teach. The
presence of so many factors suggests that it is uncertain how a teacher
will use problem-solving oriented curricula in the classroom.

Despite the uncertainties associated with teachers' use of
problem-solving based math curricula, teachers are still being pushed
from many directions to use these materials. Yet, how teachers use such
materials in the classroom is uncertain, and different materials can
imply different perspectives on problem solving (c.f., Meyer & Sallee,

1983; Shroyer & Fitzgerald, 1986). In the current context of problem-

solving reform in K-12 mathematics, there is an acute need to study
factors that shape how teachers use problem-solving oriented curricular
materials. For while such materials are available, with more
mathematics curriculum development efforts currently underway, it is not
at all clear how teachers actually use these kinds of materials in their
classrooms. Studies seeking to investigate teachers' use of problem-
solving oriented curricular materials can inform the continued push
toward teaching mathematics as problem solving and hold implications for
teaching, teacher education, curriculum development, and educational

policy.
The Purpoee of the Study

The purpose of this study is to investigate how a piece of
problem-solving oriented curriculum is used by teachers in classroom
settings. The study will help to better identify and understand the
issues that need to be considered in trying to conceptualize how
teachers use problem-solving oriented curricula in classrooms. The main
question central to this study is How do teachers use a piece of
‘problen-eolving oriented curriculum in their classrooms?

In this research, I study two sixth-grade teachers, each teaching
a unit developed by the Connected Mathematics Project (CMP). The unit,
Covering and Surrounding (see CMP, 1992a, 1992b), is a geometry unit
that focuses on the measurement concepts of perimeter and area, and the
relationships between these concepts. I include an argument in Chapter
4 justifying the use of the Covering and Surrounding unit in this study.
I argue that the unit is congruent with the NCTM Standards documents
(see NCTM, 1989, 1991) and that it is designed to facilitate teachers'

use of problem solving as a context for instruction in their practice

(see Fitzgerald et al., 1991). Through problem solving, the unit
developers intend to accomplish at least two instructional goals -- to
learn mathematical content (i.e., perimeter and area) and to connect
concepts (i.e., develop and understand relationships between perimeter
and area).

Embedded within the main research question are several research
areas related to teachers' use of problem-solving oriented curricula and
corresponding sub-questions. These research areas and questions have
guided my thinking over the course of the research and have proved
useful in framing and conceptualizing the study. Addressing these
research areas and how they inform the main question of 'How do teachers
use a piece of problem-solving oriented curriculum in their classrooms?‘

is the focus of this study:
. Conceptualizing problem-solving activity in classrooms:

What kinds of issues and challenges do teachers encounter as
they teach Covering and Surrounding?

What does problem-solving activity look like in the
teachers' classrooms and how is it organized?

What do teachers believe their students learn about problem
solving from the unit?

. USe of problem-solving oriented curricula:
How do teachers use problem solving when teaching the unit?

How and to what extent do teachers teach/emphasize problem
solving?

How and to what extent does the unit influence teachers'
teaching of problem solving and mathematical content?

How do teachers' perceptions and beliefs about student
learning influence their use of the Covering and Surrounding
unit?

0 Comparisons between the intended and the enacted curriculum:

How do the curriculum developers intend mathematical
concepts to be taught via problem solving?

How do the intentions of the curriculum developers compare
with how the teachers use Covering and Surrounding in their
classrooms?

- Teacher change and implications for mathematics education:

How and to what extent does teaching the unit cause change
in teachers' practice?

What are the implications of the findings from this study
for teacher education, curriculum, and mathematics education
reform?

The above guiding research areas and sub-questions are intended to
provide a means for thinking broadly about the main research question.
My intent has been to focus on the teachers and their use of Covering
and Surrounding without being blind to factors that influence how
teachers use a piece of problem-solving oriented curriculum in their
classrooms.

Significance of the Study

This study contributes to research on teachers' use of problem-
solving oriented curricula from different perspectives: (a) how
teachers conceptualize problem-solving activity; (b) establishing a
research-based framework with which to examine and conceptualize
teachers' use of problem-solving oriented curricula in classroom
settings; (c) identifying, describing, and using the research-based
framework to analyze teachers' instructional decisions when using a
piece of problem-solving curriculum and examine what they take into
account; (d) using findings that inform (a), (b), and (c) to inform
implementation of education reforms through problem-solving oriented
materials.

There has been little research on how problem solving looks in

classrooms and the factors that shape how problem solving is organized

in mathematics classrooms (Greeno, 1991; Silver, 1985). By examining
teachers' use of a problem—solving oriented unit, this study can unpack
how participating teachers conceptualize problem-solving activity and
how their conceptions influence their teaching. This study does not
provide a means for framing problem solving in all classrooms, as
problem solving can vary significantly between classrooms (see A.
Thompson, 1985). In contrast to the extensive work that has already
been done on conceptualizing problem-solving for students and small
groups (e.g., Schoenfeld, 1985a), this study examines teachers'
conceptions of problem solving and how it interacts with other beliefs
and knowledge to shape their use of a problem-solving oriented unit.

Teachers do not simply enact curricular materials. Just as a chef
modifies a recipe to suit particular tastes, teachers enact curricular
materials in different ways based on their own knowledge, beliefs, and
the learners they teach. Curricula to a teacher, like a recipe to a
chef, is a kind of shorthand for what a lesson might be like. No matter
how detailed a lesson plan or description of an activity, curricula
cannot specify everything a teacher will do. Teaching necessarily
involves making decisions and constructing interpretations (see Jackson,
1986).

In seeking to integrate materials to fit their teaching
circumstances, teachers may use curricula in particular ways because of
availability of time, contextual constraints, negotiations with
students, managing tensions internal to teaching like how students will
learn from the curriculum (Cohen, 1988; Lipsky, 1980; Lortie, 1975;

Sarason, 1982; Wilcox et al., 1991, 1992). I establish a framework

10

based on prior research to conceptualize teachers' use of problem—
solving oriented curricula in classroom settings. The purpose of the
framework is to better identify and understand the factors which shape
teachers' use of problem-solving oriented curricula. The relationships
of the teacher participants' own conceptions about problem solving and
using problem-solving oriented curricula to knowledge and beliefs
defined by the domains of the framework are studied to understand how
they use Covering and Surrounding.

Directly related to how teachers mold curricular materials to
their own practice are the kinds of instructional decisions teachers
make when using the materials. For example, faced with time conflicts
teachers frequently make judgement calls about what portions of the
curriculum are most suitable or inappropriate for their students (e.g.,
Freeman & Porter, 1989; Lortie, 1975; Wilson, 1990). This study
examines teachers' instructional decisions and rationale for making
these decisions while teaching Covering and Surrounding. Drawing on
analysis of the unit and classroom observations and interviews, I
explore aspects of the unit perceived by teachers as especially suitable
or inappropriate,for their teaching situation. I employ the research-
based framework to piece together why teachers make the instructional
decisions they do and what factors shape these decisions. Understanding
teachers' instructional decisions when using problem-solving oriented
curricula holds implications for curriculum development, implementing
problem-solving reforms, and teacher education (see Fitzgerald et al.,
1991; The Holmes Group, 1990; NCTM, 1989, 1991; Shulman, 1987; Wilson et

al., 1987).

11

W

The majority of mathematics teachers, math educators, curriculum
developers, policymakers, teacher educators agree that in our
increasingly diverse, complex, information-based society, being skilled
in problem solving is essential for all citizens (e.g., California State
Department of Education, 1991; The Holmes Group, 1990; NCSM, 1989; NCTM,
1989, 1991). But implementing problem-solving approaches to teaching
mathematics in school is proving to be a difficult challenge. In many
instances, the recognized need to teach mathematics as problem solving
in K-12 mathematics is at odds with persistent and seemingly competing
demands for computational proficiency, skill in using algorithms to get
right answers, and increased performance on standardized assessment
tests (NCTM, 1989; Nicholls et al., 1991). Dilemmas of implementing
problem solving when many K-12 math curricula are still structured
around fragmented behavioral objectives, assessing students' problem-
solving performance and understanding, working within the confines of
limited time, meeting the needs of diverse students, understanding
teachers' beliefs and knowledge of mathematics, are all issues that
challenge making problem solving an integral part of K-12 mathematics
instruction (Ball & McDiarmid, 1990; The Holmes Group, 1990; Lester &
Kroll, 1991; NCTM 1989, 1991; Nicholls et al., 1991; Sarason, 1982;
Wilcox et al., 1991, 1992).

A significant facet of this study is that the above issues
surrounding problem solving are informed by studying how teachers use
the Covering and Surrounding unit. The insights provided by this study
should be useful to teachers, teacher educators, curriculum developers,

policymakers , mathemat ics educators :

12

What dilemmas do teachers face when using problem-solving
oriented materials?

What do teachers need to know about mathematics and content
specific pedagogy (i.e., pedagogical content knowledge) to
teach mathematics as problem solving?

What does the previous question imply for preservice and
inservice teacher education?

How do teachers use a piece of problem-solving oriented
curricula in their classrooms and how does their enactment
compare with the intent of curriculum developers?

How do teachers’ beliefs about mathematics and problem
solving, views and beliefs about student learning, subject-
matter knowledge, and conceptions of problem—solving
activity interact to shape their use of a piece of problem-
solving oriented curricula?

This study seeks to examine these questions and issues in the context of
teachers using problem-solving oriented materials in their classrooms.
I study the nature of teachers' use of a problem-solving based unit and
try to unpack the influences that shape the activity and the rationales
that explain it.

Overview of the Dieeertation

In this chapter, I have established the central question of this
study -- ”How do teachers use a piece of problem-solving oriented
curriculum?’ I have also argued how this study addresses the problem
and can also inform other issues within mathematics education (e.g.,
teaching mathematics as problem solving, teacher education, mathematics
education reforms).

In the next chapter, I survey research on problem solving in
mathematics and research on teachers' use of curricular materials. I
then link these two areas of research by developing a four-dimensional
framework to aid in conceptualizing teachers' use of problem-solving

oriented curricular materials in teaching mathematics.

13

In the third chapter, I detail the methodology employed in this
study. I describe why constructing case studies is appropriate for
addressing the research question and I explain how I selected the
teacher participants. I also discuss the interview instruments and
protocol I designed for this study. I describe how I observed teacher
participants in their classrooms as well as the sources of data and
methods of data collection.

The fourth chapter is an analysis of the Covering and Surrounding
unit. The chapter describes, in terms of problem-solving activity, how
the developers of the unit intend problem solving and mathematics to be
portrayed in the classroom via the problems and tasks provided in the
unit. The chapter unpacks the intentions of the developers of the
Covering and Surrounding unit to enable comparison between the intended
curriculum of the unit developers and the curriculum enacted by the
teacher participants as they teach the unit in their classrooms.

In the fifth and sixth chapters, I present the cases of how
teacher participants Karen Knight and Betty Walker (both teachers' names
are pseudonyms) used the Covering and Surrounding unit in their
classrooms. Each chapter provides a profile of the teacher and her
instructional setting and describes in detail how she used the unit in
her classroom. Using interview data and data collected through
classroom observations and conversation, I employ the four-dimensional
framework developed in the literature review to conceptualize each
teacher participant’s use of the unit. The case studies are intended to
be non-evaluative and portray teachers' use of Covering and Surrounding.

In the last two chapters, I describe how the main research

<question is informed by this study. The study findings are presented in

14

Chapter 7 and are supplemented by arguments in Chapter 8 that address
issues in mathematics education linked to teachers’ use of problem-
solving oriented curricular materials. The two final chapters are

followed by appendices and references.

CHAPTER 2

REVIEW OF THE LITERATURE

Research. on Problem. Solving

I l' i I . E l] i 1 .

Studying the methods of solving problems, we perceive
another face of mathematics...Mathematics presented in the
Euclidean way appears as a systematic, deductive science;
but mathematics in the making appears as an experimental,
inductive science (Polya, 1945, p. vii).

Learning to solve problems is the principal reason for
studying mathematics. Problem solving is the process of
applying previously acquired knowledge to new and unfamiliar
situations...problem solving strategies involve posing
questions, analyzing situations, translating results,
illustrating results, drawing diagrams, and using trial and
error (National Council of Supervisors of Mathematics
[NCSM], 1989, p. 471).

Although problem solving is a central component of recent reform
initiatives in mathematics education (e.g., NCSM, 1989; NCTM, 1989,
1991), mathematics educators have been concerned with problem solving
for over four decades (Kilpatrick, 1985). In his classic work on
problem solving, How to Solve It, Polya (1945) presents one of the
earliest detailed discussions of problem solving in mathematics and the
mathematics curriculum. It was in this work that Polya’s now famous
heuristic for problem solving was first presented: (1) Understanding
the problem; (2) devising a plan; (3) carrying out the plan; (4)
looking back (Polya, 1945). Although Polya had contemporaries who
investigated problem solving processes (e.g., Bloom and Broder, 1950;
Duncker, 1945), flew to Solve It and other works by Polya (e.g., Polya,
1967, 1968) remain some of the earliest and most influential efforts to
formally study problem solving and how problem solving might be taught

and learned in school classrooms.

15

16

Despite the efforts of Polya and others, however, research on
problem solving remained limited for some time. Regarding the status
of research on teaching and learning problem solving available around

1960, for example, Kilpatrick (1985, p. 9) notes that:

[authorities] were suggesting that the best advice the
research literature had to give was that problem solving
should be taught by giving students lots of problems to
solve.

But over the next 10 years, some researchers developed approaches to
teaching problem solving where students were taught to classify problems
and then select and apply a rule or procedure to obtain a solution
(e.g., Dahmus, 1970). For example, in solving word problems elementary
or middle school students would typically be taught to first identify
keywords like 'total' or 'difference'. Students would then be
instructed to classify the problem (e.g., as an addition or subtraction
problem) and perform the corresponding arithmetic operation with the
data given in the problem to obtain the answer. However, such
'algorithmic' approaches to problem solving quickly became
controversial. Some researchers argued that problem solving is more
complex than simply selecting and following recipes and that teaching
students to classify problems by type and then apply a memorized
sequence of problem solving steps can be difficult for students and also
counterproductive (e.g., Brian, 1967; Kilpatrick & Wirszup, 1972).
Researchers began emphasizing that heuristics of problem solving (e.g.,
Polya's four problem solving steps) should be considered as flexible
frameworks and not as iron clad procedures guaranteed to solve any

problem.

17

By the early 1980s, problem solving moved into the limelight of
mathematics education. In the Agenda for Action, NCTM (1980) pushed for
problem solving to become ”the focus of school mathematics" (p. 1).
During the 1980s, findings by cognitive scientists studying how computer
models solve problems suggested that problem solving involves complex
information-processing and metacognitive processes (Schoenfeld, 1987;
Silver, 1987). Goldin (1992), for example, argues that the findings of

cognitive science in the area of problem solving imply that

... problem solving is not one thing. It involves a highly
complex aggregate of internal psychological processes, which
occur to varying degrees and in various combinations (p.
275).

Although some conceptions of problem solving may paint problem solving
as a linear process (e.g., Polya's four problem solving steps), findings
from cognitive science strongly suggest that problem solving includes an
interconnected web of components and processes (Goldin, 1992). For
example, researchers have identified pattern recognition,
representation, memory schemata, and how individuals interact with
technology as interactive components and processes that shape problem
solving behavior (see Kaput, 1985; Schoenfeld, 1987; Silver, 1987,
1989; P. Thompson, 1985).

Despite gains in understanding problem solving processes, however,
various researchers have argued that cognitive models of problem solving
behavior should not be the limit of problem solving instruction.

Problem solving instruction should go beyond teaching students set
problem solving procedures and models (Greeno, 1991). Instead, problem
solving should be pursued in a broader context of mathematical discourse

where problems and solutions are negotiated and revised (see Greeno,

18

1991). In the early 1980s, researchers shifted much of their attention
to studying the learning and teaching of problem solving in the context
of cooperative groups (e.g., Noddings, 1982, 1985; Schoenfeld, 1982).
Much of this research has pointed to the usefulness of cooperative
groups and small-group discussion as effective vehicles for teaching and
learning problem solving.

Also during the early and mid 1980s, a sharper picture of the
components of problem solving performance for individuals and small
groups emerged from research. For example, Schoenfeld (1985a)
constructed a framework to conceptualize problem solving performance in
individuals from his research with college undergraduates. Schoenfeld's
(1985a) framework consists of the following components: resources, or
the mathematical knowledge that an individual has access to in trying to
solve a problem (e.g., formulas, concepts, methods of proof);
heuristics, or the strategies and techniques that can be employed to
work toward a solution of a problem (e.g., working backwards, using
related problems); control, or how resources and heuristics are selected
and implemented, and how these judgements are made (e.g., self-
monitoring and assessment of progress on a problem); belief systems, or
individual conceptions and perceptions about mathematics which influence
problem solving behavior (e.g., beliefs about self and attitudes toward
mathematics) (see Schoenfeld, 1985a). These four components interact
and can influence each other (e.g., heuristical knowledge and resources
influence control), and also help to map out the territories of problem
solving processes (see Schoenfeld, 1985a).

In the current reform climate, the terms 'problem solving' and

'mathematical problem solving' are being used more frequently and freely

(I)

56

19

than ever in the mathematics education community. However, there is
little shared understanding regarding what problem solving is, how
problem solving might be taught, and how it is learned. In particular,
there is little research which focuses on problem solving in the setting
of mathematics classrooms, and surveying the literature reveals that
what problem solving activity in classrooms looks like is poorly
conceptualized (Grouws, 1985; Silver, 1985). Most research on problem
solving has tended to focus on understanding how individual students or
small groups of students, sometimes working from programmed instruction
booklets, attempt to solve specific kinds of problems (e.g., Dahmus,
1970; Noddings, 1982; Schoenfeld, 1982, 1985a). While study of
individual or small group problem solving has provided insight into the
complexity of problem solving (e.g., Schoenfeld, 1985a), future research
needs to examine how mathematics as problem solving is organized and
orchestrated in classroom settings (Greeno, 1991; Grouws, 1985; Silver,
1985).

An ironic characteristic of most prior research aimed at informing
the teaching and learning of problem solving is that the role of the
teacher in classroom instruction is almost never addressed (Grouws,

1985). As Silver (1985) notes:
... the teacher is truly the 'forgotten soul' in research on
the teaching and learning of mathematical problem solving.
In general, the teacher and teacher-related variables have

been systematically controlled (for statistical reasons) or
unconsciously ignored by researchers (p. 262).

Silver (1985) also points out that available research on teaching and
learning mathematical problem solving is limited by a 'lack of good

description of what actually happened in the classroom when problem

20

solving was taught” (p. 248, emphasis in original). Grouws (1985)
echoes Silver's concerns and argues that ”What is needed is careful
scientific inquiry on the teacher's role in the acquisition by students
of problem-solving ability” (p. 301).

Commenting on possible future directions of research on teaching

and learning mathematical problem solving, A. Thompson (1989) notes:

I hope that ten years from now we will be more knowledgeable
about effective ways and techniques of teaching problem
solving and have a better sense of how to go about preparing
teachers in their use. I also hope that by then we will
have had an opportunity to examine curricular materials for
teaching problem solving and how they are used by teachers
in their classrooms (p. 243).

The general lack of research on how problem solving is taught and
learned in classroom settings has emphasized that future inquiry should
consider the role of the teacher, the nature of curricular materials
used to foster problem solving ability in students, how teachers use
these materials, rich description of classroom activity, and the
implications of findings for teacher education and curriculum
development (Shulman, 1985; A. Thompson, 1989, 1992; P. Thompson, 1985).
Researchers investigating problem solving have only recently begun
to grapple with the complexity and methodological dilemmas presented by
classroom settings, like defining classroom problem solving, creating
assessment instruments and criteria for problem solving, and
conceptualizing the roles of students and teachers when engaged in
problem solving activity (Grouws, 1985; Lampert, 1990; Lester and Kroll,
1991; Silver, 1985). For example, the issue of representation (i.e.,
how specific concepts are represented and how mathematics is portrayed
as a discipline) is a key construct in problem solving research (Kaput,

1985; A. Thompson, 1985). Yet, there is little research which analyzes

21

the disciplinary and pedagogical implications and ramifications of
teachers' and students' use of instructional representations in
mathematics (see Ball, 1990a for an analysis of representations in
teaching fractions). Selection and use of representations in problem
solving is also directly related to issues of teacher knowledge.
Current research is exploring what kinds of knowledge teachers draw on
to choose, design, and use representations in their teaching (Ball,
1990a; Greeno, 1991; Kaput, 1985; Wilson et al., 1987).

In examining current research on problem solving, it is apparent
that problem solving in classroom settings is complex and is influenced
by an array of factors. Silver (1985) argues that at least ten factors
significantly affect teaching and learning mathematical problem solving
in the classroom -- these factors include students' beliefs and
teachers' beliefs. Students' beliefs include how they perceive
mathematics (e.g., as relevant only to scientists, as an interesting
field of inquiry open to all and applicable to wide varieties of
situations) and how they perceive themselves in relation to mathematics.
Teacher beliefs include how the teacher perceives mathematics, as these
beliefs will be implicitly, if not explicitly, communicated to students
and will influence learning (Silver, 1985). Teacher beliefs also
include how teachers view themselves as learners of mathematics, their
conceptions of teacher and student roles in the classroom, and their
beliefs about what it means to know mathematics (Wilcox et al., 1991).

Other issues which have emerged from recent inquiry into problem
solving include questions about what a problem solving oriented
mathematics curriculum looks like (Shulman, 1985), what teaching

practice from a problem solving perspective looks like (Grouws, 1985),

22

and how to assess students' problem solving ability and performance
(Lester & Kroll, 1991; NCTM, 1989; Romberg, Zarinnia, & Collis, 1991).
For example, in a problem solving oriented mathematics curriculum, does
any kind of problem necessarily engage students in problem solving?
Would the problem ”Eddie hands the grocery clerk a $20 bill for a $16.38
purchase -- how much change does he get?” belong in a problem solving
oriented mathematics curriculum? This kind of example points to an
emerging question in problem solving instruction and curriculum design:
What kinds of problems involve students in problem solving and help
develop students' problem-solving skills?

Although the question of what kinds of problems belong in problem
solving oriented curricula remains open (see Shulman, 1985), some
researchers have posited that some problems are better than others.
Polya (1968, p. 139), for example, classified problems according to the
following scheme:

1. One rule under your nose

2. Application with some choice

3. Choice of a combination

4. Approaching research level.
The change problem above would be an example of a ”one rule under your
nose” problem. The second kind of problem requires recall of a rule
from among two or more possibilities (e.g., choosing between solving a
quadratic equation with the quadratic formula or by factoring), while
the third entails combining two or more rules/procedures in some
combination (e.g., finding the amount of paint needed to cover a house).
The fourth kind of problem not only requires the application of multiple

rules/procedures, but also involves making judgements and weighing,

23

formulating, and selecting possible paths to a solution (e.g., what is
the best position and synchronization for traffic lights in the downtown
area of your city to best facilitate traffic flow?)1. Kilpatrick (1985)
notes that Polya felt that the ”educational value” of problems increases
from type 1 to type 4. Unfortunately, however, the kinds of problems
researchers know the most about are of the first two types (Kilpatrick,
1985). Other math educators since Polya have tried to refine problem
classification for teaching and learning problem solving (e.g., Charles
& Lester, 1982), but in general the roles of different kinds of problems
in curricula and in teaching and learning problem solving is not well
conceptualized (Kilpatrick, 1985; Kulm, 1991; Silver, 1989; P. Thompson,
1985).

Another factor complicating the roles of different kinds of
problems in curricula is the interplay between curriculum and
assessment. Traditionally, assessment has been separate from
mathematics curricula, serving mainly to decide how well students have
mastered material and to assign grades. For problem-solving oriented
curriculum and instruction, however, assessment can be viewed as an
integral component of curriculum, teaching, and learning. Assessment
can be employed to shape instruction and provide learning experiences
for students (Lester & Kroll, 1991; NCTM, 1989). However, recent
emphases on assessment in problem solving oriented mathematics
instruction (e.g., Fitzgerald et al., 1991; Kulm, 1991; NCTM, 1989,

1991) are complicated by a lack of understanding and consensus as to

 

1Polya’s classification scheme for problem types does not address the relative nature
of problems to the solver. For example, what might be a rule under the nose of an
eighth grader could be very problematic for a second grader. The notion that what
counts as a problem or problem type is relative to the solver will be examined later
in this chapter and revisited in Chapter 4.

24

what problem solving should look like in classrooms and what the roles
and importance of different kinds of problems are in problem solving
oriented math curricula (Greeno, 1991; Grouws, 1985; Lester & Kroll,
1991; Silver, 1985, 1989). Problem solving influences assessment
because what problem solving entails implies what should be assessed in
problem solving (NCTM, 1989). However, assessment influences problem
solving because what assessment reveals about how students understand
problem solving should shape problem solving instruction (Lester &
Kroll, 1991).

Overall, while prior research on mathematical problem solving has
proved useful in better understanding the complexity of problem solving
processes and behavior, few studies have investigated how teachers teach
problem solving in classrooms. Although there is an abundance of
problem solving oriented curricular materials available for teachers,
little is known about how teachers actually use these materials in their
classrooms. Researchers have begun to investigate how teachers'
knowledge, beliefs, and perceptions (e.g., knowledge of mathematics,
beliefs about mathematics and problem solving, perceptions of student
learning) shape their teaching of mathematics and how problem solving is
enacted in the mathematics curriculum (e.g., Grouws & Cramer, 1989).
Clearly evident from the literature on teaching and learning
mathematical problem solving is wide agreement that inquiry needs to be
conducted on conceptualizing how teachers enact problem solving oriented

curricula and teach problem solving in classroom settings.

25

Research on Factors Influencing
Teachers' Use of Curricular: Materials

While there is only sparse understanding of how problem solving
plays out in the classroom, there is no shortage of curricular materials
to help teachers and students engage in problem solving. The diverse
approaches different curricula take toward problem solving is testimony
to the scattered perspectives that exist about problem solving in
classroom settings. For example, some text books treat problem solving
as a discrete topic in the mathematics curriculum (e.g., Mathematics in
Action, 1992) whereas others seek to integrate problem solving
throughout the curriculum (e.g., COmprehensive School Mathematics
Program, 1978). Other materials, intended to supplement the existing
curriculum, pursue problem solving by teaching specific strategies to
solve specific kinds of problems (e.g., Meyer & Sallee, 1983), by
teaching general heuristic strategies for more global applications to
wide varieties of problems (e.g., Charles & Lester, 1982), or weaving
problem solving into learning activities without explicitly teaching
problem solving strategies (e.g., Shroyer & Fitzgerald, 1986).

But while there is an abundance of curricular problem solving
materials available, little is known about what teachers actually do
with these materials in their classrooms. Available research suggests
that teachers vary considerably in how they use curricular materials.
For example, Freeman and Porter (1989) found that teachers in their
study did not determine the mathematical content of lessons from their
texts, yet Remillard (1991a) studied a math teacher who relied heavily
on her text to structure and define the content of her mathematics

teaching. Other studies illustrate differences in how math teachers use

26

curricular materials, articulate beliefs about problem solving, and
teach problem solving (Bockarie, 1980; Graybeal & Stodolsky, 1987; A.
Thompson, 1985). Research has also uncovered discrepancies between how
teachers talk about using curricular materials in their classroom and
how they actually use the materials in practice (e.g., Stake & Easley,
1978).

There is much research to show that teachers vary in how they use
curricular materials, policy guidelines, curriculum objectives in their
classrooms (e.g., Cohen & Ball, 1990; Lipsky, 1980; Lortie, 1975;
Sarason, 1982). Factors contributing to influencing how teachers shape
their practice and use curricular materials include the degree of risk
they are willing to take in instruction, their knowledge and beliefs
about the subject matter, to what extent the teacher perceives the
instructional context to be constraining or empowering, the availability
of time (Floden, Porter, Schmidt, Freeman, & Schwille, 1981; Jackson,
1986; Sarason, 1982; Wilcox et al., 1991, 1992). However, despite this
variability and complexity, policymakers and curriculum developers
interested in moving problem solving reforms (e.g., NCTM, 1989) into
classrooms view curricular materials as a vehicle for teacher learning
and change. Problem solving oriented curricular materials are often
central to implementing problem solving oriented reforms (e.g.,
California State Department of Education, 1991; Cohen & Ball, 1990).
How teachers use curricular materials, and how their practice compares
with the intentions of policymakers and curriculum developers, can be
difficult to interpret and are unlikely to match exactly (Ball, 1990b;

Elmore, 1979; Wiemers, 1990; Wilcox et al., 1992; Wilson, 1990).

27

Additional reasons for why teachers may vary in their use of
curricular materials, or enact a curriculum differently than intended by
its developers, can be tied to differing views about the roles of
curriculum and problem solving. For example, Stanic and Kilpatrick
(1989) argue that students should not only be taught to select and
follow the ”right strategy” to solve a given problem. Stanic and
Kilpatrick (1989) warn that such an approach reduces problem solving to
becoming ”... a skill, a technique, even, paradoxically, an algorithm”
(p. 17). Instead, they argue, problem solving should be taught as
”art”, where the teacher plays a key role in helping students learn to
flexibly and fluently understand and solve problems (Stanic &
Kilpatrick, 1989). This view contrasts sharply with the findings of
Hembree (1992), who concluded in his meta-analysis of 487 studies on

problem solving that:

Heuristics [problem solving strategies] in middle grades 6-8
seemed mildly better than other approaches and gained a
distinctly superior status in high school. A positive
impact on students' [problem solving] performance also
resulted from teachers especially trained in heuristical
methods (p. 242).

This finding points to explicitly teaching selecting and implementing
strategies as a good way to teach problem solving (Hembree, 1992).
Curriculum developers have produced problem-solving oriented curricula
that arguably subscribe to these different perspectives (c.f., Meyer &
Sallee, 1983 [i.e., teaching selection and implementation of
strategies]; Shroyer & Fitzgerald, 1986 [i.e., problem solving is
entwined in doing mathematics]).

Teachers' use of curricular materials can be fairly congruent with

the intent of curriculum developers (e.g., Remillard, 1991a), although

28

this is not necessarily (and perhaps generally not) the case (e.g.,
Wiemers, 1990). Teachers may enact curricular materials differently
than intended by curriculum developers because teachers and
researchers/curriculum developers can harbor different views about the
math curriculum and its use. For example, Prawat, Putnam, and Reineke
(1991) assessed the perspectives on elementary math curriculum of seven
experts (4 researchers, 3 teachers). The researchers in the sample
explicitly addressed curriculum and materials and instruction, whereas
the teachers were less explicit about the curriculum and focused on
developing in students positive attitudes about mathematics (Prawat et
al., 1991). While all of the experts agreed with current, broad problem
solving oriented reforms (e.g., NCTM, 1989), an array of views on what
ideal math curricula should encompass and what are the most important

aspects of classroom teaching practice emerged:
The problem for curriculum developers and teachers is that
underneath a seemingly unified call for major changes in the
way mathematics is taught actually lie a number of
strikingly different assumptions and images of what good
mathematics teaching should look like. We argue that for
teachers to make sense of the advice and calls for change
that bombard them, they need to realize that they may be

based on multiple - and possibly incompatible - assumptions
(Prawat et al., 1991, p. 64).

Not only has research confirmed that teachers often have different
priorities for instruction than curriculum developers (e.g., Lortie,
1975; Prawat et al., 1991; Sarason, 1982), but teachers can also
interpret the goals of a call for reform differently and enact curricula
differently based on these different assumptions (c.f., Ball, 1990b;
Wiemers, 1990; Wilson, 1990).

As well as teachers' use of curricular materials being shaped by

their views on curriculum, and that these views may differ from those of

29

curriculum developers, teachers also vary in their perceptions of
mathematics. For example, a teacher may perceive mathematics as an
arbitrary collection of rules or as a field of creative inquiry subject
to revision and change (Ball & McDiarmid, 1990; A. Thompson, 1992;
Wilcox et al., 1991, 1992). But different views of mathematics can also
exist within the discipline of mathematics and may translate into
different views of teaching and learning mathematics. For example,

Hershkowitz (1990) identifies two instructional views of geometry:

There are two main ”classic” aspects of geometry: viewing
geometry as the science of space and viewing it as a logical
structure, where geometry is the environment in which the
learner can get a feeling for mathematical structure (p.
70).

Depending on how a teacher perceives mathematics both in relation to
herself and as a discipline, different approaches to teaching
mathematics may emerge. For example, geometry perceived as a ”logical
structure” may translate into teaching students definitions, rules, and
two-column proofs -— the ”logical structure” of geometry should be
mastered. Geometry perceived ”as the science of space” may promote
engaging students in investigating the geometric patterns and properties
of objects and phenomena arising in both mathematics and nature (see
Steen, 1990). If curriculum developers and teachers subscribe to
different perspectives on learning mathematics, the intent of curriculum
developers and teachers' interpretation of the curriculum are unlikely
to coincide (Elmore, 1979; Sarason, 1982).

Current research has demonstrated that how teachers perceive
mathematics and themselves in relation to mathematics also influences
how they use curricular materials, teach problem solving, and choose or

design learning experiences for students (A. Thompson, 1985, 1992;

IN

,4.

30

Wilcox et al., 1991, 1992). For example, A. Thompson (1985) studied two
teachers whose teaching practices emphasized different positions on the
importance of problem solving in school mathematics. One of the
teachers, who viewed mathematics as ”fixed and predetermined”,
predominantly taught mathematics by presenting the math content ”in a
clear, logical, and precise manner” (A. Thompson, 1985, p. 286). In
this teacher's classroom, students learned by listening to the teacher
and answering the questions posed by the teacher. In contrast, the
second teacher, who viewed mathematics as ”more a subject of ideas and
mental processes than a subject of facts” generally taught mathematics
by encouraging students to ”guess and conjecture” and by being
”receptive to the students’ suggestions and ideas” (A. Thompson, 1985,
pp. 289-290). While problem solving was an integral part of the latter
classroom, it was scarce in the first, especially if problem solving
includes making conjectures, connecting ideas, and developing processes
more than mastering fixed facts and predetermined procedures. Both of
these teachers taught mathematics in a way congruent with their beliefs
about mathematics and illustrate the great influence teachers' beliefs
have on how they teach mathematics and problem solving.

An implication of findings about teachers' use of curricular
materials is that teachers do not simply enact a curriculum, but
interpret and filter it through their own perspectives, knowledge, and
beliefs (Cohen, 1988; Cohen & Ball, 1990; Lipsky, 1980; Lortie, 1975;
Sarason, 1982). In using curricular materials that seek to integrate
problem solving (in some way) into mathematics instruction, teachers
will also play a role in shaping the curriculum and problem solving in

the classroom. Yet, while both the teacher and the curricular materials

31

play roles in molding how problem solving looks in the classroom, it is
difficult to assess the roles of the curriculum and teacher and how they
interact (see Grouws, 1985; Remillard, 1991a). Further research is
needed into how teachers use problem solving oriented materials in math

classrooms. Remillard (1991b), for example, argues that

It is this question of how teachers use and interpret
alternative curricula which needs further study. We need to
know more about what teachers bring to new and nonstandard
mathematics curricula and how they make sense of and use
them (p. 60).

Researchers are increasingly acknowledging that in the context of
problem solving careful research is needed to better conceptualize the
role of curricular materials in problem solving instruction (P.
Thompson, 1985), the role of the teacher in problem solving instruction
(Grouws, 1985; Silver, 1985), and how teachers and curricula interact in
classroom settings (Ball, 1990b; Ball & Cohen, 1990; Grouws, 1985;
Remillard, 1991a; Wiemers, 1990; Wilson, 1990).
Constructing a Framework

Teaching and learning problem solving in classroom settings is
poorly conceptualized. Despite the proliferation of ”problem solving”
workshops and inservices for practitioners and math educators, our
understandings about problem solving instruction as it occurs among
teachers and students in classrooms is sketchy at best. As A. Thompson

(1989) notes:

Reports of instructional studies in problem solving have
generally lacked good descriptions of what actually happened
in the classroom ... and have often failed to assess the
direct effectiveness of instruction. ... As a result, our
knowledge of desirable instructional practices in problem
solving is mostly of folklore rather than research evidence
(p. 232).

32

Similarly, how teachers use curricular materials in their classrooms,
especially to teach ”mathematics as problem solving” (see NCTM, 1989),
remains an open question. For example, teachers may integrate problem
solving into their mathematics teaching or may neglect it almost
entirely, viewing open-ended problems, for example, as frustrating to
their students (e.g., A. Thompson, 1985). Significant variation is
common among the curriculums teachers enact, even when using the same
curricular materials. For example, Freeman and Porter (1989) found that
teachers may center their mathematics teaching tightly around a textbook
to meet district objectives, while other teachers may use the same text
only as a convenient resource, focusing instead on the mathematical
topics and skills they themselves regard as important. Teachers and
math educators can also exhibit diverse -- and sometimes disparate --
assumptions about the role of the curriculum in mathematics instruction.
For example, Prawat et al. (1991) found that teachers tended to think of
the curriculum ”as something that is given to them by the school
district” (p. 62), whereas university based math educators tended to
view the curriculum as constructed in the classroom by teachers and
students and consisting of ”rich problem situations, developed around
important clusters of mathematical ideas” (p. 63).

However, despite the scattered state of research on problem
solving and teachers’ use of curricular materials, the literature in
these areas and the connections between them can inform the question of
how teachers use problem solving oriented curricula in their classrooms.
Reviewing the literature reveals that teachers' use of problem solving
oriented curricular materials can be influenced and shaped by at least

four issues: (a) Teachers' views and beliefs about mathematics and

33

problem solving; (b) Teachers' views and beliefs about problem solving
activity in classroom settings; (c) Teachers' subject matter knowledge
of mathematics; (d) Teachers' perceptions and beliefs about student
learning.

While these four regions overlap (e.g., if a teacher believes
problem solving is an individual activity, she probably won't use
cooperative groups as a setting for problem solving activity), the
literature indicates that each plays a role in shaping how teachers use
problem solving oriented curricular materials. In the next four
sections, I will survey the literature in each area. Together, the four
regions comprise a framework that will be useful in conceptualizing and
analyzing teachers’ use of the Covering and Surrounding unit.

Regarding views and beliefs about problem solving within the
mathematics education community, Stanic and Kilpatrick (1989) argue

that:

The term problem solving has become a slogan encompassing
different views of what education is, of what schooling is,
of what mathematics is, and of why we should teach
mathematics in general and problem solving in particular (p.
1, emphasis in original).

”Problem solving” and ”mathematical problem solving” are loaded terms
that call forth myriad images about teaching and learning mathematics.
The notion of problem solving in school mathematics is not only open to
wide interpretation by mathematics educators in general, but especially
among teachers and curriculum developers (see A. Thompson, 1989).
Problem solving has become a familiar term to math teachers over the
past 10 years, but how teachers address problem solving in their

classrooms, and what problem solving means to them, varies considerably

34

and may be at odds with the goals of school mathematics reforms (c.f.,
Ball, 1990b; A. Thompson, 1985; Wiemers, 1990; Wilson, 1990).

One of the reasons teachers vary in their views and beliefs about
problem solving is that there are aspects of problem solving which are
difficult to define. Earlier, I noted that all problems may not be
created equal, citing Polya's (1968) scheme for classifying problems as
a way of differentiating between different kinds of problems and his
view that some problems are of more ”educational value” than others.
But another fundamental issue in teaching and learning problem solving
and in curriculum development is trying to better conceptualize ”What is
a problem?” For example, in surveying both past and current research
on teaching problem solving, Kilpatrick (1985) notes that the question
of ”What is a problem?” is an issue that emerges as being at the heart
of teaching problem solving. Polya (1967) asked the question of ”What

is a problem?” and stated:

...to have a problem means: to search consciously for some
action appropriate to attain a clearly conceived but not
immediately attainable aim. To solve a problem means to
find such action...Yet some degree of difficulty belongs to
the very notion of a problem: where there is no difficulty,
there is no problem (p. 117, emphasis in original).

Charles and Lester (1982) answer the same question this way:

A problem is a task for which:

1. The person confronting it wants or needs to find a
solution.

2. The person has no readily available procedure for
finding the solution.

3. The person must make an attempt to find a solution (p.

5, emphasis in original).
Despite these formulations of what a problem is, however, there is still
a lack of consensus as to what problems are and how they should be used
to teach students problem solving (Kilpatrick, 1985). Grouws (1985) and

Shulman (1985) argue that research needs to address the issue of how to

35

identify or construct and use ”good” problems in teaching problem
solving which help meet learning goals for students and inform teaching
practice.

Teachers' views and beliefs about problems are directly related to
their teaching of problem solving and their views and beliefs about
problem solving. In the absence of research findings teachers are
largely left to their own devices, or the textbook, to determine the
kinds of problems with which their students engage. Even given
classification schemes for problems to help determine what are ”good”
and ”not so good” problems (e.g., Polya, 1968), teachers still vary
considerably in the kinds of problems and problem solving activity they
feel their students need to learn about math content and problem solving
(c.f., A. Thompson, 1985; Wiemers, 1990; Wilcox et al., 1991, 1992;
Wilson, 1990).

Predictably, the nature of problems teachers develop and/or
provide for students is connected to their own views and beliefs about
what are ”good” problems for their students (see Putnam, Heaton, Prawat,
& Remillard, 1992). For example, does a teacher view the problem ”What
is the perimeter of a rectangle with length = 8 and width = 18?” and
”What is the perimeter of a rectangle with area 144?” as essentially the
same (e.g., because they both ask about perimeter of a rectangle) or
different (e.g., because the first problem has one answer and the second
has multiple solutions)? Do students have comparatively limited problem
solving opportunity with the first problem than with the more open-ended
second problem? This example attempts to illustrate research findings
which indicate that teachers' views and beliefs about problem solving

are, at least in part, shaped by their beliefs about what kinds of

36

problems with which their students should work (Ball, 1990a; Charles,
1989; Kilpatrick, 1985; Wilcox et al., 1992).

There is considerable research that addresses how teachers'
beliefs about mathematics influences their mathematics teaching (see A.
Thompson, 1992; Wilcox et al., 1991, 1992). Some of this research has
focused on how teachers’ beliefs about mathematics is connected to
problem solving instruction. Teachers who view mathematics as a fixed
set of given rules and procedures to be mastered through practice also
tend to approach problem solving as a set of procedures; teachers who
see mathematics as a domain of creative inquiry tend to address problem
solving as a creative and flexible endeavor inclusive of students' ideas
as well as useful heuristics (e.g., A. Thompson, 1985, 1989). With
respect to student achievement in problem solving, Hembree's (1992)
meta-analysis revealed that ”The better that teachers felt toward
mathematics, the better their students performed with problems” (p.
254). Hembree’s finding corroborates and connects the work of other
researchers who argue that teachers' views and beliefs about mathematics
influence student learning through their teaching (e.g., Grouws &
Cramer, 1989; Silver, 1985) and that an individual's beliefs about
mathematics provide a context for his or her problem solving behavior
and performance (e.g., Schoenfeld, 1985a).

Teachers' views and beliefs about problem solving seem strongly
tied to beliefs about mathematics, including what problems are
appropriate for student learning, and may thus shape students' problem

solving behavior.

3‘.-

37

The scant and scattered state of research on problem solving in
classroom settings makes answering the question ”What does problem-
solving activity in classroom settings look like?” difficult (Grouws,
1985). The research that is available suggests that the kinds of
problem-solving activity teachers involve their students in is shaped by
their underlying views and beliefs about mathematics and problem solving
(e.g., Putnam et al., 1992; A. Thompson, 1985). Rather than being based
on research evidence or documented pedagogical approaches (i.e., careful
descriptions of how problem solving is taught in different classroom
situations and the effects of the instruction on students and teachers —
- see Grouws, 1985 for an argument about the need to document teaching
problem solving in classrooms), problem-solving instruction emerges from
teachers' beliefs and assumptions about mathematics and problem-solving
processes (A. Thompson, 1989). Some researchers, such as Schoenfeld
(1985a, 1985b) and Lampert (1990) offer descriptions of problem-solving
activity in classroom settings that are based on detailed assumptions
about mathematics and problem solving.

Schoenfeld (1985a, 1985b) argues that problem solving is complex
and that successful problem solving necessitates monitoring one's own
progress in solving problems. Problem solving entails being able to
control a problem by making appropriate decisions about selecting and
applying the kinds of resources (i.e., mathematical knowledge) and
heuristics (e.g., problem solving strategies) that will facilitate
solving the problem (Schoenfeld, 1985a). In the classroom, problem
solving entails students developing control skills so that students can

productively select and use heuristics/strategies to implement their

38

mathematical resources. Students must learn how to monitor their
progress on a problem and make decisions about how to allocate their
mathematical resources and select and implement heuristics (Schoenfeld,
1985b). Learning control is a critical facet of successful problem
solving performance. For even if appropriate strategies and heuristics
are available to a student, without good control behaviors (e.g.,
consciously assessing progress, metacognition), a problem solving

attempt may be unsuccessful. Schoenfeld (1985a) stresses that:

...inefficient control behavior can cause individuals to
squander the problem solving resources potentially at their
disposal and thus fail to solve problems that are easily
within their grasp. The issue [of control] is not just the
use of one's heuristic knowledge; it is how all of one's
mathematical knowledge is called into play (p. 114).

With respect to his own teaching practice, Schoenfeld (1985b)
describes how he attempts to teach college undergraduates control. In
some cases, Schoenfeld acts as a ”manager”, where he poses a problem to
the class and the students' role is to make suggestions for its
solution. Schoenfeld (1985b) says that in this instance the teacher's
role ”...is not to judge the suggestions”, but to ”monitor” them,
”...raising questions about the efficacy of suggested steps (both when
they are useful, and not useful)” (p. 373). Under this kind of
”monitoring”, students collectively construct a solution to the problem.
However, Schoenfeld notes that problem solving in the classroom also .
entails students working together in small groups (of three or four)
where the teacher acts as a ”consultant” to the various groups. In this
approach, he gives students a problem to solve in their small group.
Schoenfeld then circulates around the room to each of the groups helping

them productively control the problem by asking questions like ”What

39

(exactly) are you doing?”, ”Why are you doing it?”, and ”How does it
help you?” (see Schoenfeld, 1985b, p. 374). Schoenfeld argues that this
kind of pedagogy employing these kinds of questions helps students
develop ”effective” control which is critical to successful problem
solving performance (Schoenfeld, 1985a, 1985b). Teaching problem
solving in this manner requires the teacher to not only have a rich and
flexible knowledge of mathematics, but also knowledge of how students'
and his or her own mathematical knowledge is employed to work towards
solutions of problems.

In her teaching practice, Lampert (1990) emphasizes that
mathematics is not a collection of fixed facts, rules, and procedures
that must be practiced until mastered. Rather, doing mathematics means
formulating conjectures and solutions within problem situations and then
formulating proofs or counter-examples (i.e., mathematical arguments) to
prove or refute potential proofs and solutions. Problem solving,
therefore, entails crafting a mathematical argument which provides and
justifies a problem solution. Lampert (1990) notes that in the

classroom

...the teacher must make explicit the knowledge she is using
to carry on an argument with them [i.e., the students] about
the legitimacy or usefulness of a solution strategy. She
needs to follow students' arguments as they wander around in
various mathematical terrain and muster evidence as
appropriate to support or challenge their assertions, and
then support students as they attempt to do the same thing
with one another's assertions. As the teacher moves around
in mathematical territory in a flexible manner, she is
modelling an approach to problem solving (p. 41).

In Lampert's view, the teacher needs to know more than the rules for
doing mathematics - she needs to know how to prove mathematical rules

and assertions and how to make this knowledge accessible to students.

40

Problem solving is pursued by the classroom as a community, requiring
the teacher to become a facilitator as students formulate their own
conjectures and arguments supporting or challenging each others'
solutions (Lampert, 1990).

Other research, however, suggests that in many classrooms problem-
solving activity looks much different than Schoenfeld and Lampert
portray, and is often based on different assumptions about mathematics
and problem solving. For example, in discussing four case studies of
how fifth-grade teachers approach teaching mathematics for
understanding, Putnam et al. (1992) argue that teachers often assume
that students need to be told how to do mathematics. This ”transmission
model” of teaching (see Jackson, 1986) for mathematics means the teacher
needs to tell students math facts, demonstrate mathematical procedures,
and then the students need to memorize/master the facts/procedures.
Mathematics is assumed to consist of a fixed body of facts and
procedures, and problem solving means applying the facts and procedures
successfully to get the right answer. While teachers may acknowledge
that understanding is important, they also may argue that understanding
can only come after facts (e.g., how to interpret a fraction) and
procedures (e.g., how to divide fractions) have been mastered.

Regarding how this view of mathematics and problem solving translates
into problem solving activity in the classroom, Putnam et al. (1992)

note:

Problem solving in these classrooms involved applying well-
practiced computational skills in particular situations,
rather than opportunities to figure out what is reasonable
and sensible in these situations. This view of problem
solving is consonant with the hierarchial view of learning
we discussed earlier - that students must learn the basics
before they can understand and apply them.

Cg!

41

Schoenfeld (1985a, 1985b), Lampert (1990), and Putnam et al.
(1992) describe different kinds of classroom activity that, depending on
the underlying assumptions about mathematics and problem solving, could
all be described as ”problem solving”. In trying to conceptualize
problem-solving activity in classroom settings, therefore, it is
critical to understand how mathematics and problem solving are viewed by
the teacher, and that these views and beliefs will play an important
role in shaping ”problem solving” activity (see Greeno, 1991; Grouws,
1985; Shulman, 1985).

Subiect;Matter_Knouledas

Schoenfeld (1985a) argues that knowledge of mathematics (resources
in Schoenfeld's terms) has a direct influence on mathematical problem
solving. While subject matter knowledge interacts with other factors
which together shape problem solving behavior (e.g., selection of
appropriate problem solving strategies), knowledge of mathematics is a
fundamental component of problem solving performance (Schoenfeld,
1985a). Lampert (1990) argues that teachers need a degree of subject
matter knowledge of mathematics when modelling approaches to problem
solving because teachers need to make explicit to students the
mathematical knowledge they draw on. Schoenfeld (1985b) echoes
Lampert's (1990) position, stressing that a teacher should model
appropriate problem solving behavior to students and that this includes
drawing students' attention to the mathematical ”resources” being
brought to bear on the problem.

Further evidence that subject matter knowledge shapes how problem
solving is taught by teachers in their math classrooms comes from

connections among research on problem solving and research on teacher

42

knowledge and teacher education. Shulman (1987), Ball and McDiarmid
(1990), and Wilson et al. (1987) demonstrate that teachers' subject-
matter knowledge influences how they design, select, and use
instructional representations. For example, a teacher may teach his or
her students how to square a binomial by the familiar FOIL method
(multiply the First terms, then the Outside terms, then the Inside
terms, and finally the Last terms):

(a + b)2 = (a + b)(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2
However, the teacher whose subject matter knowledge of algebra includes
connections between algebra, geometry, and mathematical proof could also
employ this representation of squaring a binomial:

Figure 2.1: Geometric representation of squaring a binomial:
° D__|

0 Al A2

 

 

 

 

 

 

 

 

Each side of the square has length a + b, so the area of the
square is (a + b)2. However, from the diagram, the
subdivided regions of the square A1, A2, A3, and A4 have
areas A1 = a2, A2 = ab, A3 = ba, A4 = b2. Since the area of
the square is also equal to A1 + A2 + A3 + A4 = a2 + ab + ba
+ b2 = a2 + 2ab + b2, the area of the square has two
equivalent forms with the result that (a + b)2 = a2 + 2ab +
b2.

The first representation is a rule which does not attempt to convey why
(a + b)2 = a2 + 2ab + b2 but only asserts what to do to square a

binomial -- it is a rule, not a proof or a justification. The second

43

representation tries to show, geometrically, how the binomial square can
be visualized and proved in terms of an object (i.e., by subdividing the
area of a square with edge a + b). Either representation can be used to
show students how to square binomials, but they vary in how they
represent mathematics (i.e., as a fixed set of rules, as interrelated
concepts, respectively). The representations also vary in what they
communicate is important for students to know about mathematics ~—
knowing how to perform rules and procedures versus knowing how to prove
relationships and connect ideas. The accessibility of these
representations to a teacher will be influenced to a significant degree
by his or her subject-matter knowledge of mathematics, including
knowledge of relationships between mathematical ideas and knowledge of
mathematical proof. Silver (1985) and Kaput (1985) link this finding
from teacher knowledge and teacher education research to problem solving
by arguing that representations play a key role in problem solving
processes, problem solving performance, and teaching and learning
problem solving.

While subject-matter knowledge plays a direct role in problem
solving performance and in teaching problem solving, Shulman (1987) has
argued strongly that subject matter knowledge is important in teaching
in general (see also Ball & McDiarmid, 1990). Shulman and his
colleagues (e.g., Wilson et al., 1987) have stressed the need to study
the role of subject matter knowledge in teaching. Research on teaching
has traditionally neglected the specific subject matter being taught,
and future inquiry needs to consider the subject matter and the
knowledge teachers need about the subject matter to teach it well. As

Putnam et a1. (1992) comment:

44

Commonplace is the claim that it is not just a matter of
teachers needing more mathematics courses; they need richer
knowledge about mathematics for teaching.

The above passage adds weight to Charles's (1989) argument that a
critical question to wrestle with in preparing teachers to teach problem
solving is ”How good a problem solver must a teacher be to be an
effective teacher of problem solving” (p. 263)?

In the above passage, (i.e., Putnam et al., 1992, above), what
kind of teacher knowledge is implied when emphasizing a need for ”richer
knowledge about mathematics for teaching”? Researchers have called for
conceptualizing the kind of knowledge teachers need for teaching by
linking subject matter knowledge with knowledge of learners (e.g.,
Shulman, 1987; Wilson et al, 1987). This kind of knowledge, frequently
referred to as pedagogical content knowledge, is knowledge of subject
matter that is powerful for teaching. Wilson (1989) describes that

pedagogical content knowledge

... consists of understandings and beliefs about the range
of alternatives for teaching a particular piece of subject
matter to particular students in particular schools, as well
as knowledge and beliefs about the ways in which students
learn the content in question. This knowledge also enables
teachers to generate instructional representations that are
justifiable on the basis of the discipline itself, on
theories of teaching and learning, on knowledge of the
interests and prior knowledge of students, and on
educational goals and objectives (p. 1).

In teaching problem solving, then, what does a teacher need to know ”to
be an effective teacher of problem solving” (Charles, 1989, p. 263)?
Knowledge of appropriate instructional representations for teaching
problem solving is consistent with Wilson's portrayal of pedagogical
content knowledge and research that has emphasized the importance of
representations in teaching problem solving (e.g., Hembree, 1992;

Silver, 1985). As Kaput (1985) emphasizes, ”Whatever the details, most

45

would agree that some idea of representation seems to be at the heart of
understanding problem-solving processes” (p. 381, emphasis in original).

Issues of subject matter knowledge, pedagogical content knowledge,
and problem solving seem to coincide when considering how teachers
generate and use instructional representations to teach problem solving
(c.f., Kaput, 1985; Schoenfeld, 1985a; Wilson et al., 1987).

Research has also indicated that teachers’ beliefs and perceptions
about student learning interact with other factors which all shape
classroom instruction and the nature of learning experiences afforded to
students (A. Thompson, 1992). For example, classroom instruction is
molded both by teachers' beliefs and perceptions about what is important
for students to know about mathematics and by the context of instruction
(Wilcox et al., 1992). Although teachers may believe that students
should learn mathematics by engaging in structurally rich, problem
solving oriented situations, contextual constraints such as limited
time, large class size, multiple and demanding responsibilities, the
views of other teachers and administrators, can discourage teachers from
enacting instruction congruent with these beliefs (Wilcox et al., 1991,
1992). Kohl (1984) has argued that how teachers perceive their students
(e.g., as cooperative, as disciplinary problems) directly influences how
and what teachers teach.

However, contextual factors that influence teachers' perceptions
and beliefs about student learning extend beyond individual classroom
situations or building policies. Research has demonstrated that factors
such as social class, gender, and race shape perceptions and beliefs

about what particular kinds of students should learn in particular

46

schools (e.g., Oakes, 1985). Anyon (1981) found that the predominant
social class of a school's community corresponded with different
learning opportunities for students. For example, while students in
working class schools had their attention largely focused on lessons in
all subject areas that were limited to memorizing facts and mastering
procedures, students in upper-class (i.e., executive elite) schools
engaged in inquiry based learning experiences (Anyon, 1981). For
students in the working class school, knowledge is told to them by
others and is found in books; for the upper-class students knowledge is
created by people and is subject to change (Anyon, 1981). Teachers
within both of these respective kinds of schools often supported their
students' learning experiences, emphasizing, often in terms of the
predominant social class of the school's students, that they were
teaching their students what they needed to know. Some teachers in the
working-class schools noted that since their students could not do
arithmetic well, they needed considerable practice so that they could
later, for example, calculate correct change. Upper-class teachers
argued that their students needed more open-ended, creative learning
experiences to prepare them to be future leaders in society (Anyon,
1981). This example points to the powerful impact students' class
status can have on how teachers perceive them as learners and determine
what is appropriate for them to learn.

Gender differences in mathematics is an issue which has continued
to demand attention from mathematics educators (see Dessart and Suydam,
1983; Fennema, 1989; Swafford, 1980). At least part of the reason for
the controversy and uncertainty which pervades debates on the role of

gender in mathematics teaching and learning lies in an abundance of

47

inconsistent and often contradictory research findings. For example,
some studies have concluded that there is no evidence of gender
inequality among math students when the number of mathematics courses
taken by both sexes is controlled, whereas other research has shown that
under these conditions differences in achievement (in favor of males)
persists (Dessart & Suydam, 1983). However, with respect to problem
solving, gender differences seem more pronounced. As Fennema (1989)

notes:

Although there has been much research about gender
differences in mathematics and many interventions have been
developed to alleviate the differences over the last 15
years, there is still powerful evidence that males achieve
at higher levels than do females, particularly in tasks of
high cognitive complexity, such as true problem solving (p.
211).

Proposed reasons for gender differences in mathematics vary, and include
arguments that the factors of class and race affect females differently
than males (Hallard & Eisenhart, 1988) and that males tend to be more
confident than females about mathematics (McLeod, 1989). Some
researchers have argued that teachers, often unconsciously, may treat
males and females differently in the classroom (Strauss, 1988). A
prevailing view on gender differences in mathematics is that educators
need to be aware of gender differences in mathematics and provide equal
learning opportunities for students of both sexes (NCTM, 1991; Strauss,
1988).

Not only class and gender, but also race is a portion of the
overall context that shapes teachers' perceptions of their students and
their beliefs about student learning. Academic tracking in schools,
where different students are placed in different subject matter

sequences, is a mechanism where racial differences are often quite

48

visible. For example, in her study of 25 different K-12 schools, Oakes

(1985) notes that:

... it is clear that in our multiracial schools minority
students were found in disproportionately small percentages
in high-track classes and in disproportionately large

percentages in low-track classes. ... this pattern was most
consistently found in schools where minority students were
also poor. ... In academic tracking, then, poor and minority

students are most likely to be placed at the lowest levels
of the schools’ sorting system (p. 67).

That student class, race, and gender are all important contextual
factors in shaping how teachers perceive their learners is evident in
current math education reform initiatives. For example, in the
Professional Standards for Teaching Mathematics, the NCTM (1991)
emphasizes the need to educate ”every child”:

As a professional organization and as individuals within
that organization, the Board of Directors sees the
comprehensive mathematics education of every child as its
most compelling goal. By ”every child” we mean

specifically:

0 students who have been denied access in any way to
educational opportunities as well as those who have
not;

. students who are African American, Hispanic, American

Indian, and other minorities as well as those who are
considered to be a part of the majority;

0 students who are female as well as those who are male;
and

. students who have not been successful in school and in
mathematics as well as those who have been successful
(p. 4).

As research and educational reforms both indicate, factors which
influence teachers' beliefs and perceptions include not only personal
views and perspectives and individual classroom or building situations,
but also encompass the broader societal issues of class, gender, and
race (Anyon, 1981; Bereiter & Scardamalia, 1987; The Holmes Group, 1990;

NCTM, 1989, 1991; Oakes, 1985; Putnam et al., 1992; Strauss, 1988;

49

Wilcox et al., 1991, 1992). All of these factors (i.e., teachers’ own
perceptions and beliefs, class, race, gender) contribute to create an
overall context in which teaching and learning occurs.

But while teachers' beliefs and perceptions may interact with
myriad contextual factors to shape instruction, teachers' beliefs about
what their students need to know and how they should learn it remain a
critical filter through which instruction must pass (Ball, 1990b; Cohen
& Ball, 1990; Silver, 1985; A. Thompson, 1985; Wilson, 1990). One
reason for the resilience and powerful influence of teachers' beliefs is
that they have generally been established over a long 17 year period in
K-16 schooling (Wilcox et al., 1991). As Lortie (1975) argues, however,
learning about teaching through the eyes of a student is inherently
limited because the motivations, beliefs, rationales, and knowledge
behind the teacher behavior a student witnesses are largely hidden and
unknown. Teachers' beliefs being powerfully shaped by limited
observation and then influencing instruction is especially significant
in teaching mathematics. This is because beliefs about the nature of
mathematics can be communicated to students through instruction
(Lampert, 1990) and because the content and structure of students'
learning experiences can lead to confining or empowering learning
opportunities and events (e.g., see Anyon, 1981 and Oakes, 1985 for
examples of how tracking and student class and race interact with
teacher beliefs).

In a synthesis of the research on teachers' beliefs, A. Thompson
(1992) notes that teachers do have views and beliefs about how students
learn and that these beliefs shape how mathematics gets taught in the

classroom. Teachers' beliefs and perceptions about student learning are

50

also connected to their views and beliefs about mathematics and subject
matter knowledge of mathematics (Ball & McDiarmid, 1990; Wilcox et al.,
1991, 1992). For example, in analyzing case studies of how four
different teachers approach teaching fifth-grade mathematics, Putnam et
al. (1992) found that the teachers felt basic skills had to come before
understanding, that learning only occurs over time, and that students
might not be ready for the abstract thinking needed for understanding.
Putnam et a1. (1992) discuss how these beliefs about student learning

shaped the teaching of mathematics in the teachers' classrooms:

All these beliefs about learning -— that basic skills must
be mastered before one can understand, that learning takes
time, and that children might not be developmentally ready
for abstract thought -- support the traditional emphasis on
computational skills and facts, leaving the understanding of
mathematics to be dealt with later. Beliefs about learning
are also related to what is being learned. What teachers
know and believe about mathematics necessarily influences
their beliefs about how students learn it.

Ihutnam et al. (1992) note that in their study the teachers’ beliefs
inbout learning taking time is, at least on the surface, congruent with
<:urrent problem solving oriented reforms (i.e., California State
IDepartment of Education, 1991). California state-level reforms argue
tzhat learning takes time because students need a variety of rich,
Iproblem solving oriented activities to provide multiple opportunities to
llnpack and connect mathematical ideas (California State Department of
IEducation, 1991). Some teachers in the study, however, felt that
Ilearning takes time because students first have to spend time practicing
iand mastering basic facts, skills, and procedures (Putnam et al., 1992).
Codlectively, these beliefs reinforced students learning mathematical

facts, procedures, and algorithms and little about problem solving.

51

It is important to acknowledge that teachers' beliefs and
perceptions about student learning are not the only factors that shape
their teaching of mathematics. For example, teachers' views of what is
important for students to learn and how students learn can be influenced
by the views of other teachers and administrators (e.g., Wilcox et al.,
1992). Standardized assessment and achievement tests at local, state,
and national levels are also powerful in shaping how and what teachers

teach (Nicholls et al., 1991). As Kulm (1991) emphasizes:

Basic computational skills have been the focus for
competency tests. ... Teachers have been legitimately
concerned that if they ”fight the system” and teach higher
order thinking [e.g., problem solving], their students would
suffer on the computationally oriented tests that they are
required to pass. Many educators believe that very little
change will occur in mathematics curriculum and teaching
without a concurrent change in testing, especially in state
and national standardized tests that are used to assess and
compare school-by-school achievement (p. 72).

Overall, research indicates that teachers’ beliefs and perceptions about
student learning, which directly influence classroom teaching‘(e.g., A.
Thompson, 1992), are shaped by a variety of factors, including their own
experiences in school, the context of instruction, the expressed views
and beliefs of others, tests and policies, personal beliefs,
perceptions, and knowledge about learners and subject matter (Ball &
McDiarmid, 1990; Kulm, 1991; Lampert, 1990; Lortie, 1975; Putnam et al.,
1992; Sarason, 1982; Schoenfeld, 1985a, 1985b; Silver, 1985; A.
Thompson, 1985, 1989, 1992; Wilcox et al., 1991, 1992).
Summary

Research indicates that while advances have been made in

conceptualizing problem solving processes and individual problem solving

performance (e.g., Schoenfeld, 1985a), teaching and learning problem

52

solving in classroom settings has not been significantly addressed
(Kilpatrick, 1985; Grouws, 1985; Silver, 1985, 1989). Similarly,
research on how teachers use problem solving oriented mathematics
curricula is scant (A. Thompson, 1989). While an abundance of problem-
solving oriented curricular materials exist, these materials take
different approaches to teaching problem solving, reflecting a wide
array of views about what problem solving is and how it fits into the
school mathematics curriculum (Kilpatrick, 1985; c.f., Meyer & Sallee,
1983; Shroyer & Fitzgerald, 1986; see Stanic & Kilpatrick, 1989).
Little is known about how teachers actually use curricular materials in
their classrooms to shape and guide instruction, despite the importance
such findings would have for teacher education, curriculum development,
and mathematics education reform (Prawat et al., 1991; Remillard,
1991a).

Research seeking to investigate teachers’ use of problem solving
oriented curricular materials is informed by four areas of existing
research: (a) teachers' views and beliefs about mathematics and problem
solving, (b) teachers' views and beliefs about problem solving activity
in classroom settings, (c) teachers’ subject matter knowledge of
mathematics, (d) teachers' perceptions and beliefs about student
learning (including the multiple dimensions of student class, race,
gender as well as teacher beliefs). The importance and influence of
knowledge and beliefs in teaching is well-documented (Ball & McDiarmid,
1990; A. Thompson, 1992), and teachers' knowledge and beliefs interact
with external factors (e.g., limited time, class and race of students,
standardized tests) to shape classroom practice (Kulm, 1992; Lortie,

1975; Putnam et al., 1992; Sarason, 1982; Wilcox et al., 1991, 1992).

53

Therefore, research indicates that the nature of curricular materials,
teachers' knowledge, beliefs, and the context of instruction are all
critical factors to consider in investigating teachers' use of problem-

solving oriented curricular materials.

CHAPTER 3

METHODOLOGY

Case Study Research

In prior research, case studies have been used as a means to study
how teachers use mathematics curricular materials. Although these
studies generally do not focus specifically on how problem-solving
oriented curriculum is enacted by teachers, the studies in this area do
investigate how teachers use curricular materials. For example, Freeman
and Porter (1989) studied four elementary teachers to learn how they
interacted with their math texts to make decisions about the mathematics
content of lessons. Remillard (1991a) crafted a detailed case study
which explores and analyzes how one teacher enacted an elementary
mathematics curriculum that attempts to engage students in problem-
solving situations and discussions about mathematical ideas.

Other researchers have used case methods to study how teachers'
knowledge and beliefs interact with policy reform to shape elementary
mathematics instruction. This line of research has exposed a variety of
similarities, disparities, and tensions between the intentions of state-
level mathematics education reform and how teachers enact the reform-
based elementary math curriculum. For example, Wilson's (1990) case
study analysis demonstrated that teachers’ ability to enact mathematics
curriculum as outlined by reforms (in this case the California
Framework, see California State Department of Education, 1991) can be
constrained by their knowledge of mathematics and their need for ”...
help in learning about new [teaching] methods, help in finding time to

teach for understanding, resources for evaluating such understanding”

54

55

(Wilson, 1990, p. 41). Wiemers's (1990) case study portrays a fifth-
grade mathematics teacher whose beliefs about teaching mathematics are
at odds with the vision of the California state-level mathematics
reforms. The case shows how the teacher filters and transforms the
reform policies through his existing beliefs into an interpretation of
teaching that he views as somewhat compatible with the reforms and his
own beliefs (Wiemers, 1990). Ball's (1990b) case study presents a
teacher whose practice reflects some of the California state-level
reforms, but is inconsistent with these policies in other respects.
Collectively, these three case studies (Ball, 1990b; Wiemers, 1990;
Wilson, 1990) are examples of how case study research methods can be
used to explore relationships between intended and enacted curricula,
teachers' perceptions and understandings about policy and curricula, and
teachers' knowledge and beliefs.

While some case study research has focused on how teachers use
mathematics curricular materials, or explore connections between
intended reforms and curricula and teachers' actual practice, other case
study research has focused explicitly on teachers' knowledge. For
example, Shulman (1987) used case studies to investigate the kinds of
knowledge teachers draw on in their teaching. This line of inquiry has
helped to conceptualize the knowledge that seems specific to teaching
and teachers (e.g., pedagogical content knowledge -- see Wilson et al.,
1987). Research techniques that are used in constructing case studies,
such as careful analysis of protocols and interview transcripts, have
been used to explore individual and small-group problem-solving
performance (e.g., Noddings, 1982; Schoenfeld, 1985a). Such research

has led to further insight into the nature of the knowledge of

56

mathematics and heuristics required for successful mathematical problem
solving (Schoenfeld, 1985a). Other characteristics of case study
research, such as extensive observation, constructing rich descriptions
of settings, are congruent with calls for research on problem solving
that incorporates detailed accounts of teachers' actions and instruction
in classrooms (Grouws, 1985; Silver, 1985, 1989; A. Thompson, 1989).
Overview' of Methods

Two teachers participated in this study. The participating
teachers teach in different school districts whose students, in general,
come from contrasting socioeconomic backgrounds. The two classroom
settings studied contrast along various dimensions: size of the
schools; building/district policies (e.g., tracking): socioeconomic
status and demographics of the school communities. These differences
and contrasts between the classroom settings are desirable for this
study because factors such as tracking and social class of learners have
all been shown to have a significant impact on how teachers teach and on
what students have an opportunity to learn (Anyon, 1981; Bereiter &
Scardamalia, 1987; Lortie, 1975; Oakes, 1985). My choice of research
sites with contrasting characteristics was not intended to provide for
generalizeability of the research findings. Rather, the variance in
characteristics between the sites has allowed me to study two settings
across which a variety of factors and issues are present that can
potentially influence teachers' use of the Covering and Surrounding
unit.

I interviewed each teacher prior to observing their classrooms.

Each teacher interview was conducted in two parts (see Intezyieua;

Deaign_and_BaLignalg section). All interviews were audio taped and

57

transcribed for analysis. I then observed each classroom over a period
of approximately eight weeks. I spent one class period per week for the
first three weeks observing the teachers as they taught and becoming
familiar with their classroom settings prior to Covering and Surrounding
instruction. For the next five weeks I observed each teacher at least
one and usually two classes per week teaching the Covering and
Surrounding unit. During the five weeks of Covering and Surrounding
instruction, I observed one teacher eight class periods and the other
nine class periods. Observational data was collected during classroom
observation via field notes and audio taping. Periodically during the
observations I had conversations with students and the teacher
participants (conversations were audio taped) and collected data from
documents (e.g., teachers' lesson plans, samples of students' written
work). At the conclusion of the unit, I had a conversation with each
teacher participant to reflect on their experience of teaching Covering
and Surrounding. Confining the study to two classes allowed me to
collect rich yet manageable data.

Using the Connected Mathematics Project unit Covering and
Surrounding in this study makes methodological sense. As a contributing
developer of the unit I am familiar with the intentions of the
curriculum developers. Throughout the study, however, I have been
careful not to let my familiarity with the unit blind me to issues that
were not considered or anticipated in its development, but might still
be significant factors in the study. The unit is suitable not only
because it is problem-solving oriented, but because it also focuses on
the challenging middle school mathematics concepts of perimeter and

area. Moreover, the unit was designed assuming that the teacher plays a

58

central role in guiding and orchestrating instruction (see Fitzgerald et
al., 1991; Shroyer & Fitzgerald, 1986). The unit provides contrasts in
how problem solving is used. For example, in some investigations
students engage in problem solving to learn content (e.g., the concepts
of perimeter and area). In other investigations, students explore
problems to learn primarily about mathematical connections and
relationships between concepts (e.g., how the perimeter of rectangles
can change when area is held constant). While the unit's use of problem
solving to teach content and make connections overlaps in each activity,
some investigations emphasize one more than the other. The unit is also
congruent with curricular recommendations of current problem solving
based reform agendas in mathematics education (e.g., NCTM, 1989).
Teacher Participants

One of the first challenges of this study was how to select
teacher participants. Both teacher participants needed to teach sixth-
grade mathematics, be willing to teach the Covering and Surrounding
unit, and discuss with me her use of and views of the unit. I also
required that the teachers be relatively open to new ideas, curricular
materials, and teaching practices in mathematics.

I selected Betty Walker1 from her school district from a pool of
teachers currently affiliated with the Connected Mathematics Project.
While I am not acquainted with any of these teachers, they had all made
tentative commitments to pilot Connected Mathematics Project units
during the 92-93 school year. Betty was interested in mathematics

education reform and was willing to use innovative curricular materials

 

1Betty Walker is a pseudonym.

59

and try new pedagogical practices that could facilitate changes in
practice. She also agreed to provide detailed feedback and engage in
discussions of her use of Connected Mathematics Project materials.

In selecting Karen Knightz, who teaches in another school
district, I contacted the district's mathematics coordinator and
requested recommendations for sixth-grade mathematics teachers who might
be interested in participating in this study. The mathematics
coordinator is familiar with the CMP materials and serves on the
advisory board for the project. After receiving the district
coordinator's recommendations, I contacted the teachers to describe the
study and gauge each of their interest levels in participating. I then
met with the most interested teacher, Karen Knight, to discuss the study
in greater detail, and at this meeting she agreed to participate.

Betty walker and Karen Knight both exhibited the aforementioned
characteristics of openness toward trying new materials and practices in
teaching mathematics, and both were willing to talk about their views on
and use of the Covering and Surrounding unit.

Conducting the Study

I employed fieldwork methods to conduct this dissertation study.
The nature of my inquiry -- trying to understand how teachers use a
problem-solving oriented unit -- is especially suited to the on—site
observation and interview techniques of fieldwork research (see Bogdon &
Biklen, 1982; Hammersley & Atkinson, 1983). In addition to interviews
and observations, I also collected data from relevant documents, such as

the Covering and Surrounding unit, samples of students' work, teachers’

 

2Karen Knight is a pseudonym.

60

lesson plans, information on the schools and their communities, district
curriculum objectives and policies. For example, I have described the
Covering and Surrounding unit and have drawn on the Connected
Mathematics Project NSF proposal Connected Mathematics (Fitzgerald et
al., 1991) to construct a portrait of the intended curriculum (see
Chapter 4). As a researcher, I have triangulated multiple sources of
data (i.e., analysis of the curriculum, interview and observational data
on participants) to understand how the study participants used Covering
and Surrounding. The data is used to construct cases that incorporate
assertions and supporting arguments to explain how each teacher uses the
unit. I use the cases to raise educational issues related to teachers'
use of problem-solving oriented curricular materials.

Following are detailed discussions of the two main phases of data
collection and then analysis of the data: Interviews -- including
design of and rationale for the interview materials; Observation -- how
the teacher participants were studied in their classrooms; Analysis --
the predominant areas that emerged from the data which characterize and
shape the teacher participants' use of COvering and Surrounding, that
are employed to construct case studies and obtain findings.

The first step in data collection was to interview both teacher
participants. The purpose of the interview was to acquire general
background information on the teacher participants (e.g., number of

years teaching, number of years teaching mathematics) and to assess:

(a) Teachers' views and beliefs about mathematics and
problem solving.

(b) Teachers' views and beliefs about problem-solving
activity in classroom settings.

61

(c) Teachers' subject-matter knowledge of mathematics.
(d) Teachers' perceptions and beliefs about student
learning.

These four areas represent regions which the research literature
indicates influence how teachers use curricular materials and how
teachers teach, understand, and perceive mathematics and problem
solving. My intent in the interviews was not to gain information to
predict teacher behavior, but to assess teachers' views and beliefs
within these regions to better gauge issues shaping teachers' use of the
Covering and Surrounding unit over the course of data collection (i.e.,
as teachers teach the unit).

In each part of the interview, I provided teachers with items
(i.e., excerpts from curricular materials or vignettes of classroom
situations) to react to. Problem solving is an issue that is difficult
to discuss in depth directly, without reference to a particular problem,
context, or situation (Grouws, 1985; Kilpatrick, 1985). The excerpts
from curricular materials all deal with the concepts of perimeter and
area in order to be consistent with the focus of the Covering and
Surrounding unit. The vignettes depict teachers and students in
classroom situations. Using interview items oriented around the same
topics as the instructional unit affords more meaningful comparisons
between the interview data and classroom observation data than if the
interview items dealt with concepts different than perimeter and area.
However, to provide for some range of math concepts among the interview
items, not all deal with perimeter and area. For example, a brief

vignette of a sixth-grade mathematics lesson, used to assess teachers'

62

views and beliefs about problem-solving activity in classroom settings,
does not incorporate perimeter or area.

The importance of assessing teachers' knowledge and beliefs by
providing them with authentic classroom materials and situations is
emphasized by Barnes (1987). Barnes (1987) argues that teaching is very
complex and that teachers' knowledge is drawn from a variety of sources
(see also Shulman, 1985; Wilson et al., 1987). Moreover, the specific
context of a classroom situation or the nature of particular curricular
materials has a large influence on how teachers access their knowledge
and how their knowledge and beliefs shape their instructional decisions
(Barnes, 1987; Wilcox et al., 1991, 1992). Asking or providing teachers
with decontextualized questions or situations (e.g., ”What are your
views on teaching mathematics as problem solving?”) often does not yield

much about teachers’ knowledge or beliefs. As Barnes (1987) notes:

If teacher assessment is to be valid, it must ...
approximate the realities of classroom teaching [and] the
assessment ... must allow teachers to create their own
meanings from the information provided (p. 9).

By providing teacher participants with a variety of classroom situations
and curricular materials to discuss and react to, I have attempted to
”approximate the realities of classroom teaching” in the interview
assessment. By asking teacher participants open-ended questions
throughout the interview, I have also tried to allow them room to
construct and articulate their own meanings and knowledge of the
curricular materials and classroom vignettes.

The first part of the interview consisted of some preliminary
background questions and then three sets of items and questions, each

attempting to address the role and use of curricular materials, problem

63

solving, and problem-solving activity in classrooms. In each of these
sections, teacher participants were provided with excerpts from various
curricular materials (i.e., Hake & Saxon, 1985; Math in Action, 1992;
Shroyer & Fitzgerald, 1986), and a description of a problem-solving
lesson in a sixth-grade mathematics classroom (see Appendix A for the
vignette). I expected that using these curricular excerpts and the
vignette would provide data for interesting and rich comparisons between
their professed views and their actual practice (see also A. Thompson,
1985). This part of my conversation with the teacher participants was
structured around open-ended questions so that the teachers would have
freedom and latitude in their responses.

The three items used to assess subject-matter knowledge are
designed to help understand different facets of teachers’ subject-matter
knowledge of perimeter and area. Item 1 is intended to assess teachers'
knowledge of the relationships between perimeter and area as well as
what teachers think counts as proof. Item 2 tries to find out if
teachers think conceptually about area (e.g., as the number of square
units needed to cover a figure). Item 3 asks teachers to make sense of
a student's (hypothetical) response to perimeter problems where correct
answers are obtained in a non-standard way and helped me learn how
flexible the teachers were in their understanding of perimeter (i.e.,
does a teacher think that the only way to find perimeter is 2L + 2W)
(see Appendix B for items 1, 2, and 3 assessing subject-matter
knowledge).

The last section of the interview assesses teachers' beliefs and
perceptions about student learning by asking them to react to a vignette

that describes a sixth-grade mathematics teacher and her interactions

64

with her students (see Appendix C for this vignette). The vignette
describes how the teacher has crafted three different learning
situations for three different groups of students, and how she teaches
mathematics individually to an individual student from each group. The
interviews with both teacher participants were transcribed in their
entirety.

After being interviewed, both Karen Knight and Betty Walker were
observed in their classrooms once per week for three weeks during the
month prior to beginning their instruction of the Covering and
Surrounding unit. I observed Karen Knight and Betty Walker at least one
and usually two classes per week‘for the next five weeks as they taught
the Covering and Surrounding unit. Each class was audio taped and I
wrote detailed field notes during each class. I also, on occasion,
spoke with participating students (i.e., those students who consented to
be audio taped and for whom I had also received consenting parental
permission forms) about their perceptions of the unit. Conversations
with students were also audio taped. I also taped several interactions
between the teacher participants and their students.

On most occasions, I was able to expand my field notes the same
day as I observed the class. Audio tapes of each class session were
reviewed, and selected portions of these tapes were transcribed. I also
periodically initiated conversations with both teacher participants
about specific lessons on the same day of the lesson to get information
on their perceptions while they were still fresh in their minds. In a
variety of instances, I was able to compare teacher and student

perceptions of the same lesson.

65

I frequently collected samples of teachers' lesson plans and
samples of students' written work. Representative samples of students'
writing were collected. At various points during the Covering and
Surrounding unit, I asked the teachers to react to and discuss specific
examples of students' written work with me. I also asked the teacher
participants to talk with me about particular events during classroom
instruction (e.g., students' conjectures, students’ arguments to support
their solutions or conjectures).

All of these strategies for data collection were aimed at
understanding teachers’ use of the Covering and Surrounding unit. My
goals were to collect data that would help me in answering questions
like ”How do teachers use the Covering and Surrounding unit?”, and ”How
and to what extent does the teachers’ teaching address problem solving?”
Answering these questions was guided by how each teacher perceived the
unit, roles of themselves and their students in the classroom, subject-
matter knowledge about mathematics, beliefs about mathematics and
problem solving.

Upon finishing the instruction of the unit, I had a reflection
session with each teacher. This consisted of an extensive conversation
where I tried to assess their overall impressions and reactions to the
experience of teaching the unit. I also explicitly asked both Karen
Knight and Betty walker if they felt that teaching the Covering and
Surrounding unit had motivated any changes in their teaching practice.
Analxaia

My first task in data analysis was to carefully analyze the
background and assumptions and purposes underlying the Covering and

Surrounding unit. I developed this analysis prior to observing the

 

 

66

teacher participants teaching Covering and Surrounding in their
classrooms. The result of this analysis appears as the next chapter.

My purpose in analyzing Covering and Surrounding is to conceptualize the
intentions of the developers of the unit. My analysis indicated that
the unit developers intend Covering and Surrounding to provide a context
for interactive roles for students and teachers in the mathematics
classroom and engage students with a variety of problem-solving oriented
learning experiences. Throughout my analysis of the unit, I draw
heavily on the Covering and Surrounding student (CMP, 1992a) and teacher
(CMP, 1992b) materials, the Connected Mathematics Project NSF proposal
(Fitzgerald et al., 1991), and the Curriculum and Evaluation Standards
for School Mathematics (NCTM, 1989) and the Professional Standards for
Teaching Mathematics (NCTM, 1991). Collectively, these documents
represent the curricular materials teachers and students use in this
study (i.e., CMP, 1992a, 1992b), and the conceptual and ideological
foundations for the development of the Connected Mathematics Project
curricular materials as represented by the Covering and Surrounding unit
(i.e., Fitzgerald et al., 1991; NCTM, 1989, 1991).

After conceptualizing the intentions of the developers of the
Covering and Surrounding unit, I conducted interviews with both teacher
participants and then observed them teaching the unit. Analysis of this
data was ongoing. For example, I compared my observational data on
participants as they taught Covering and Surrounding with their
responses to the interview items and questions. My intent was not to
predict or check the behavior of the teacher participants with respect

to their use of the unit, but was to help alert me to issues occurring

67

(or not occurring) in teachers' discussions with me in the interview
setting and in their actual teaching practice.

I kept a chronological study log as the participants taught the
unit in which I reflected on my observations and conversations with
Karen Knight, Betty Walker, and their students. At the conclusion of
the unit instruction I carefully examined my data from all sources
(i.e., field notes, interviews with the teacher participants prior to
teaching the unit, conversations with the teacher participants while
teaching the unit, reflections after teaching the unit, samples of
teachers' lesson plans, samples of students’ work, conversations with
students, conversations between students and teacher) to document issues
and patterns emerging from the data. I then employed the four-
dimensional framework I developed in the literature review to better
understand and conceptualize teacher participants’ actions and decisions

in developing the final case studies.

CHAPTER 4

PROBLEM SOLVING AND THE INSTRUCTIONAL UNIT:
BACKGROUND AND ANALYSIS

The Connected Mathematics Project1

W

Responding to calls for reform of K-12 mathematics education
(e.g., NCSM, 1989; NCTM, 1989, 1991), the National Science Foundation
recently funded four middle school mathematics curriculum development
projects. The Connected Mathematics Project was funded for $4.8 million
over five years, 1991 through 1996. The principal investigators for the
CMP are Glenda Lappan, William Fitzgerald, and Elizabeth Phillips of
Michigan State University, Susan Friel of the University of North
Carolina at Chapel Hill, and James Fey of the University of Maryland at
College Park. The goal of the project is to develop a complete middle
school curriculum for grades six, seven, and eight that is congruent
with the Curriculum and EValuation Standards for School Mathematics and
the Professional Standards for Teaching Mathematics of the National
Council of Teachers of Mathematics (see NCTM, 1989, 1991). As well as
providing students with a curriculum that delivers engaging, problem—
solving oriented mathematics, the CMP intends to develop a curriculum
that is usable by teachers and will help move teaching and learning
school mathematics ahead into the next decade. The CMP curriculum is
being designed to help middle school mathematics teachers implement
reforms widely advocated in mathematics education (e.g., teaching and

learning mathematics as problem solving, making mathematical

 

1For convenience, I will generally refer to the Connected Mathematics Project as the
CMP.

68

69

connections, teaching all students for understanding) (NCSM, 1989; NCTM,
1989, 1991; Putnam et al., 1990).
The CMP is committed to teaching powerful mathematics to all

middle school students.

The proposed Connected Mathematics curriculum will be
designed to meet the needs of all students with experiences
that are stimulating and challenging to middle school kids
with a variety of interests and aptitudes. We believe that
students of different interests, abilities, and learning
styles can learn with and from each other (Fitzgerald et
al., 1991, p. 6).

Teaching connected mathematics for understanding to all students is
congruent with other initiatives aimed at reforming teaching and teacher
education. For example, The Holmes Group (1990) outlines a reform
agenda aimed at restructuring existing K-12 schools into Professional
Development Schools, or PDSs. PDS sites are schools where school
administrators, teachers, university faculty, the private sector, and
local, state, and federal government work together to improve teaching
and learning for understanding in all subject areas for all students.
The Holmes Group (1990) envisions that ”The Professional Development
School will be a place where everybody's children participate in making
knowledge and meaning -- where each child is a valued member of a
community of learning” (p. 29).

The title connected Mathematics Project reflects the overall
commitment of the project to develop a complete middle school
mathematics curriculum that attempts to make connections among
mathematical topics, to other school subjects, between the elementary
and secondary curricula, and to the real world. As the CMP proposal

states:

70

The working title of this proposed middle school curriculum,
connected Mathematics, has been chosen to suggest the team's
interest in developing students’ knowledge of mathematics
that is rich in connections - connections among the various
topic strands of the subject, connections between the
planned teaching/learning activities and the special
aptitudes and interests of middle school students, and
connections between the preparation developed by elementary
school mathematics and the goals of secondary school
mathematics (Fitzgerald et al., 1991, p. 2).

In contrast to typical curricula that tend to be disjointed, fragmented,
and computation driven (NCTM, 1989), the CMP emphasizes
interdisciplinary connections between mathematics, social studies,
science, and business, and devotes special attention to connections
among important mathematical ideas (e.g., relationships between

perimeter and area).

I): ill] 'E'-!' ”1.:

A central focus of the CMP curriculum is on teaching and learning
mathematics as problem solving. In the CMP curriculum, teachers are
expected to guide students as they explore rich problem solving
situations and contexts. The role of problem solving in the CMP

curriculum is congruent with the position of the NCTM:
Problem solving is the process by which students experience
the power and usefulness of mathematics in the world around
them. It is also a method of inquiry and application...to
provide a consistent context for learning and applying
mathematics. Problem situations can establish a ”need to

know” and foster the motivation for the development of
concepts (NCTM, 1989, p. 75).

Unlike many middle school mathematics texts and curricula, the CMP does
not include problem solving as an isolated topic in school mathematics.
Problem solving is not presented in the CMP curriculum as a few narrow
behavioral objectives (e.g., ”the student will be able to solve a three-
step problem”), nor is problem solving disconnected from mathematical

content by appearing only at the end of exercise sets. Rather, the CMP

71

takes the view that problem solving should be integrated throughout
school mathematics as a central part of what it means to learn and do
mathematics (Lampert, 1990; Lester & Kroll, 1990; NCTM, 1989; Putnam, et
al., 1990; Shroyer & Fitzgerald, 1986).

The key role of problem solving in the CMP curriculum is evident
in the learning goals for students. Fitzgerald et al. (1991) identify

eight process goals:

COUNT - Determine the number of elements in finite data
sets, trees, graphs, networks, permutations, or combinations
by application of mental computation, estimation, counting
principles, calculators and computers, and formal
algorithms.

VISUALIZE - Recognize and describe shape, size, and position
of one-, two-, and three-dimensional objects and their
images under transformations; interpret graphical
representations of functions, relations, and symbolic
expressions.

COMPARE - Describe relations among quantities and shapes
using concepts such as equal, less than, greater than, more
or less likely, congruence, similarity, parallelism,
perpendicularity, symmetry, and rates of growth or change.

MEASURE - Assign numbers as measures of geometric objects,
probabilities of events, and choices in a decision-making
problem. Choose appropriate units or scales and make
approximate or exact measurements by successive
approximation or application of formal rules.

MODEL - Construct, make inferences from, and interpret
concrete, symbolic, graphic, verbal, and algorithmic models
of quantitative, visual, statistical, probabilistic, and
algebraic relations in problem situations. Translate
information from one model form to another.

REASON - Bring to any problem situation the disposition and
ability to observe, experiment, analyze, abstract, induce,
deduce, extend, generalize, relate, manipulate, and prove
interesting and important patterns.

PLAY - Have the disposition and imagination to inquire,
investigate, tinker, dream, conjecture, invent, and
communicate with others about mathematical ideas.

USE TOOLS - Select and use intelligently calculators,
computers, drawing tools, and physical models to represent,
simulate, and manipulate patterns and relations in problem
settings (pp. 5-6).

72

Counting, visualizing, comparing, measuring, modelling, reasoning,
playing, and using tools are all mathematical processes that are used in
problem solving. By developing and connecting these processes in the
context of rich problem situations, the CMP intends to provide students
with opportunities to learn mathematics as problem solving and build
problem solving skills (see Lester & Kroll, 1990; NCTM, 1989;
Schoenfeld, 1985a; Silver, 1987).

To conceptualize mathematics as problem solving, the CMP materials

are organized around five instructional themes:

Qontent_eno_£tooeeee§ -— The curriculum will be organized
around a selected number of important mathematical content
and process goals, each to be studied in depth.

gonneotione —— The curriculum will emphasize significant
connections among various mathematical topics that are
presented and between mathematics and problems in
disciplines that are meaningful to students.

MAthanatisALInxestigations -- Instruction will emphasize
inquiry and discovery of mathematical ideas through
investigation of structurally rich problem situations.

Representations -- Students will grow in their ability to
reason effectively with information represented in graphic
numeric, symbolic, and verbal forms and move flexibly among
these representations.

Ieohnolooy -- Selection of mathematical goals and teaching
approaches will reflect the information processing
capabilities of calculators and computers and the
fundamental changes such tools are making in the ways people
learn mathematics and apply their knowledge to problem
solving tasks (Fitzgerald et al., 1991, p. 3).

These instructional themes are compatible with criteria for teaching and
learning mathematics as problem solving as described by the NCTM in the
Professional Standards for Teaching Mathematics (NCTM, 1991) and in the

CUrriculum and EValuation Standards for School Mathematics (NCTM, 1989):

...students should encounter, develop, and use mathematical
ideas and skills in the context of genuine problems and
situations...they [students] should develop the ability to
use a variety of resources and tools, such as calculators

73

and computers and concrete, pictorial, and metaphysical
models (NCTM, 1991, p. 20)..

...persistent attention to recognizing and drawing
connections among topics will instill in students an
expectation that the ideas they learn are useful in solving
other problems [outside the mathematics class] and exploring
other mathematical ideas... . Curriculum materials can
foster an attitude in students that will encourage them to
look for connections, but teachers must also look for
opportunities to help students make mathematical connections
(NCTM, 1989, p. 85).

The tasks in which students engage must encourage them to
reason about mathematical ideas, to make connections, and to

formulate, grapple with, and solve problems (NCTM, 1991, p.
32).

The vision of mathematics teaching that is advocated by the CMP is not
that teachers ”give” students information. Rather, the CMP curriculum
is intended to be used with teaching that engages ”...students' interest
and intellect while providing opportunities to deepen understanding of
the mathematics being studied...” (Fitzgerald et al., 1991, p. 12).
Since a pedagogy that is committed to teaching mathematics as problem
solving is likely to be new to many teachers (see NCTM, 1991), the CMP

curricular materials are intended to serve as a lever for teacher

change:
In order to help teachers make the kinds of changes in
instructional thinking and planning implied by the goals of
Connected Mathematics, the materials developed will take
seriously the need to provide instructional strategies and
organizational help for teachers so that they can develop

new modes and habits of instruction (Fitzgerald et al.,
1991, p. 13).

The CMP curriculum is intended to help middle school teachers implement
problem-solving oriented mathematics teaching.

Problem solving in the CMP curriculum means engaging with and
exploring mathematically rich problem situations. The developers
describe how students learn mathematics through problem solving

processes:

74

Starting with a problem situation, students are encouraged
to observe patterns, generalize, extend, connect, reflect on
their solutions, evaluate, record, and communicate their
findings verbally and in writing. Concepts and skills
evolve from the explorations of problems (Fitzgerald et al.,
1991, p. 13)

The developers do not view problem solving as a set of strategies (e.g.,
”guess and check”, ”working backwards”, ”draw a diagram”) that should be
presented to, and then practiced by, middle school students. For
example, in the Covering and Surrounding teacher's edition, the

developers note:
We expect that by the end of this unit students will have
gained insight and skill in the following aspects of problem
solving involving measurement:

1) Recognizing situations in which measuring perimeter or
area will produce answers to practical problems.

2) Finding perimeters and areas of regular and irregular
figures by using transparent grids, tiles, or other objects
to cover the figures and string, straight line segments,
rulers, or other objects to surround the figures.

3) Cutting and rearranging parts of figures to see
relationships between kinds of figures, in particular,
parallelograms, triangle and rectangles. Then devising
strategies for finding areas by using the relationships
observed.

4) Observing patterns in data by organizing tables and
graphs to represent the data.

5) Using multiple representations to understand situations,
in particular, physical models, tables, graphs, symbolic and
verbal descriptions (CMP, 1992b, pp. 2-3).

In the CMP curriculum, problem solving is represented, in a global
sense, as the process of learning and doing mathematics. The Covering
and Surrounding problem goals also emphasize that the developers view
certain kinds of tools (e.g., tiles, grids) and processes (e.g.,
”cutting and rearranging parts of figures”) as especially appropriate to
learning about perimeter and area concepts. The Covering and Surrounding

problem-solving goals represent the developers' pedagogical position

75

that, ”It is crucial that we educate our students to think in terms of
the ’process' of doing mathematics as much as they do about the
'product' or solution” (Fitzgerald et al., 1991, p. 16).

E ll _5 J . E I' °! . :1 i ! .

Neu_3olee_fiot_$tndente. With its focus on learning mathematics as
problem solving, the CMP curriculum implies new roles for students.
Students' roles are not oriented around performing numerous and tedious
rote computations. Students learn by engaging in mathematical problem
solving -- investigating, conducting experiments, modelling, reasoning,
measuring, making connections in a variety of problem situations. In
Covering and Surrounding students will study perimeter and area by
taking on the role of an architectural consultant who has to figure out
room designs to meet the specific needs of clients. In this scenario,
students use tiles to measure and create floor plans subject to various
constraints (e.g., a client must have an area of at least 16 square
units and wants lots of windows). Students are expected to discuss
their designs with the class, establish criteria by which to compare the
appropriateness of different designs, and offer explanations for how
they would (or would not) alter their designs given different
constraints and conditions (see CMP, 1992a).

The CMP envisions students as interactive participants in
classrooms. Students are expected to formulate and determine the
validity of conjectures, develop mathematical relationships, discuss
their ideas about the mathematics in question with peers and teachers in
written and oral form (see CMP, 1992a, 1992b; Fitzgerald et al., 1991).

This is congruent with the NCTM's (1991) portrayal of students' role in

76

mathematical discourse as outlined in Standard 3 of the Professional

Standards for Teaching Mathematics:
Standard 3:
Students’ Role in Discourse

The teacher of mathematics should promote classroom
discourse in which students -

. listen to, respond to, and question the teacher and one
another;

0 use a variety of tools to reason, make connections, solve
problems, and communicate;

- initiate problems and questions;
0 make conjectures and present solutions;

. explore examples and counterexamples to investigate a
conjecture;

0 try to convince themselves and one another of the validity
of particular representations, solutions, conjectures, and
answers;

- rely on mathematical evidence and argument to determine
validity (p. 45).

Neu_Rele§_er;Ieachers- But making changes in how students learn
mathematics means helping teachers think about how they teach
mathematics and how they might change and develop their teaching
practices (Charles, 1989; Cohen, 1988; A. Thompson, 1989; Wilcox et al.,
1991, 1992). For students to learn mathematics as problem solving,
teachers must take an interactive rather than a transmitive stance in
their classrooms (e.g., Lampert, 1990; Schoenfeld, 1985b). The
teacher's role is to guide students' learning of mathematics, not tell
students about mathematics. The teacher's role should include helping
students articulate and refine their own ideas, and revise prior
conceptions in light of new findings and insights (Ball, 1990a; Lampert,

1990). Researchers and reformers agree that teaching mathematics should

77

not be oriented around showing and telling students how to do
mathematical procedures (e.g., teaching division by working many long
division problems for students, and having students do the same to
master the algorithm), but should provide frequent opportunities for
students to develop, refine, and revise their understandings of
mathematics as they explore problem situations (Cobb, 1989; NCTM, 1991;
Noddings, 1990; Von Glaserfeld, 1983).

The CMP views the teacher as an interactive instructor who will
encourage and foster mathematical discourse around important
mathematical ideas, concepts, and processes and the connections among
them (Fitzgerald et al., 1991). The NCTM (1991) describes discourse in

the mathematics classroom:

Discourse refers to the ways of representing, thinking,
talking, and agreeing and disagreeing that teachers and
students use to engage in ... tasks [e.g., projects,
questions, problems, applications]. The discourse embeds
fundamental values about knowledge and authority (p. 20).

The developers of the CMP curriculum see the role of the teacher as a
facilitator of discourse and problem solving in the mathematics

classroom:

The image of mathematics teaching to which the team of
Connected Mathematics is committed parallels that put
forward by the NCTM Curriculum and EValuation Standards for
School Mathematics and the NCTM Professional Teaching
Standards. We envision teachers guiding individual, small
group and whole group activities; selecting mathematics
tasks to engage students' interest and intellect while
providing opportunities to deepen understanding of the
mathematics being studied; using tools and teaching students
to use tools such as calculators, computers, and
manipulatives to explore mathematics; and orchestrating
mathematical ideas while deliberately seeking connections to
previous knowledge. This is in sharp contrast to teachers
giving students information (Fitzgerald et al., p. 12,
emphasis in original).

78

The vision the CMP developers have of the teacher’s role in the
curriculum is congruent with the teacher's role in mathematical
classroom discourse as described by the NCTM (1991) in Standard 2 of the

Professional Standards for Teaching Mathematics:
Standard 2:
The Teacher’s Role in Discourse
The teacher of mathematics should orchestrate discourse by -

- posing questions and tasks that elicit, engage, and
challenge each student's thinking;

0 listening carefully to students' ideas;

0 asking students to clarify and justify their ideas orally
and in writing;

0 deciding what to pursue in depth from among the ideas that
students bring up during discussion;

0 deciding when and how to attach mathematical notation and
language to students' ideas;

- deciding when to provide information, when to clarify an
issue, when to model, when to lead, and when to let a
student struggle with a difficulty;

- monitoring students' participation in discussions and

deciding when and how to encourage each student to
participate (p. 35).

In Covering and Surrounding, students are given the task of designing a

room using l-inch tiles (CMP, 1992a, p. 2):

 

Problem 1: Mrs. Hide likes the amount of floor space in Mr. Dull's
design, but she wants more windows and a more interesting shape for her
room. She asks you to help her out.

a) Use 12 square tiles to create a floor plan design for Mrs. Hide.
Remember that a window or a door can go into each section of wall space.
Mrs. Hide tells you that she wants at least 14 sections of wall space,
including windows and doors, in the room you are designing.

b) After you have designed the floor plan for the room make a
drawing to show the location of the door and where each window is.
Write a paragraph to tell Mrs. Hide why your design is better than Mr.
Dull's.

 

 

 

79

The teacher's edition (CMP, 1992b) stresses that this problem is
intended to help students visualize perimeter and area and develop
techniques for finding these measures (e.g., counting tiles to find

area, counting tile edges to find perimeter):
It is important that students get a clear picture of the
relationship between the physical tiles representing squares
of carpet [i.e., area] and the prefabricated wall sections

[i.e., perimeter] that are the same width as the edge of a
carpet tile (p. 23).

The teacher's edition also specifies problem-solving goals that the

designing rooms problem addresses:

0 Developing awareness of the differences between area and
perimeter.

- Developing concept images (i.e., mental images) that help
distinguish between area and perimeter.

- Finding area and perimeter through covering with tiles
and counting edges and numbers of tiles (CMP, 1992b, p. 23,
emphasis in original).

In this problem, as the NCTM (1991) notes:
One aspect of the teacher's role is to provoke students'
reasoning about mathematics. Teachers must do this through
the tasks they provide and the questions they ask. For
example, teachers should regularly follow students'

statements with, ”Why?” or by asking them to explain (p.
35).

In the Covering and Surrounding unit, and in the CMP materials in
general, the intent is for teachers and students to work together to
make sense of mathematical concepts and ideas. The ”transmission model”
(Jackson, 1986) of teaching -- teachers telling students facts and
demonstrating procedures, students absorbing facts and mechanically
recalling/practicing the facts/procedures -- is explicitly discouraged

in the Covering and Surrounding teacher's edition (CMP, 1992b):

The student materials have been written to support
investigative classwork. The expectation is that the
investigation will be launched by the teacher and the class
working together followed by students working in pairs or

80

small groups to make sense of the problem. Then the class
returns to whole class format for a class summary and
discussion of the findings. The best student learning will
occur if students are allowed to do some exploratory work on
their own, to discover strategies, and to share their
findings with their class, not if the teacher launches each
class with an explanation of a rigid procedure to do the
forthcoming problems (emphasis in original, p.5).

The intent in Covering and Surrounding is for the teacher to guide
students' learning as they explore rich mathematical problems. The
intended roles of students and teachers in the CMP curriculum echoes the

position of the NCTM (1991):

Instead of doing virtually all the talking, modeling, and
explaining themselves, teachers must encourage students to
do so. Teachers must do more listening, students more
reasoning (p. 36).

W
The developers of the CMP materials take the position of the NCTM

(1991) that

Knowledge of both the content and discourse of mathematics
is an essential component of teachers' preparation for the
profession. Teachers' comfort with, and confidence in,
their own knowledge of mathematics affects both what they
teach and how they teach it. Their conceptions of
mathematics shape their choice of worthwhile mathematical
tasks, the kinds of learning environments they create, and
the discourse in their classrooms (p. 132).

The developers' position that teachers' subject-matter knowledge shapes
the entire classroom environment for learning mathematics is why the CMP
teacher materials provide suggestions for teaching. While teachers’
subject-matter knowledge of mathematics varies (Ball & McDiarmid, 1990),
the developers assume that the mathematics and teaching pedagogy
required to use the CMP curriculum as intended will be new to many
teachers (Fitzgerald et al., 1991). Therefore, as well as to provide
answers and solution strategies, the CMP teacher materials are designed,

for example, to help the teacher ask students questions to enrich their

81

understanding and to ”help the teacher make much better judgements about
what mathematics students understand and can use” (Fitzgerald et al.,
1991, p. 15).

But no curriculum can provide teachers with teaching suggestions
for every classroom situation. Inherent in teaching is managing the
myriad dilemmas and uncertainties of practice (Ball, 1990a; Cohen, 1988;
Cohen & Ball, 1990; Jackson, 1986). Teachers address dilemmas and
uncertainties arising in practice by making instructional decisions
based on knowledge and beliefs, the subject matter being taught, and the
characteristics of the learners and the instructional context (Jackson,
1986; Lortie, 1975; Wilcox et al., 1991, 1992). Therefore, while the
CMP teacher materials are intended to provide assistance in using the
curriculum, teachers will necessarily need to draw on their own subject-
matter knowledge of mathematics as well as other knowledge and beliefs
(see Chapter 2).

E I' i E J' E E] | 5! i I .

Paralleling calls for reforming teaching and learning in schools
(e.g., The Holmes Group, 1990; NCTM, 1989, 1991), the CMP developers
intend the curriculum to teach powerful mathematics to all students
(Fitzgerald et al., 1991). To accomplish this goal, the developers
craft interesting and appealing instructional situations around topics

which are relevant to all middle school students:

The topics chosen [in the CMP curriculum] provide intriguing
and thought-provoking mathematics for all students, and the
open-ended problem explorations through which we plan to
develop those topics make such a 'mathematics for all'
curriculum feasible (Fitzgerald et al., 1991, p 6).

By embedding important mathematical concepts and processes in rich

problem situations and contexts that are engaging for middle school

82

students, the developers intend to make mathematics interesting and
accessible to all students.

As described earlier in this chapter, the developers believe that
students learn mathematics by investigating rich problem situations.
The view of student learning taken by the developers can be summarized
as going through three stages (see Fitzgerald et al., 1991; Lappan,
1983; Shroyer & Fitzgerald, 1986):

. Students engage with a mathematically rich problem.

0 As students explore the problem they develop ideas about
mathematical concepts and formulate problem-solving
strategies.

0 Students refine their understandings of the mathematics
and problem-solving strategies embedded in the problem by
comparing their ideas with others and reflecting on and
testing their own ideas.

This conception of how students learn mathematics is congruent with the
developers’ view that students are active participants in their own
learning and that the teacher's role is to guide students as they
explore the mathematical terrain of the problem.
Covering and Surrounding

The CMP unit Covering and Surrounding (CMP, 1992a, 1992b) is the
second of three sequenced geometry units that together constitute the
geometry strand of the sixth-grade CMP curriculumz. The unit focuses on
measurement concepts, specifically perimeter and area, and the
relationships between them. As the title of the unit suggests, area is

represented as a measure of the number of the square units needed to

 

2The Covering and Surrounding student edition (CMP, 1992a) and teacher edition (CMP,
1992b) used in this study are both working drafts of the unit materials. The CMP
expects that covering and Surrounding, along with other developed materials, will be
revised before final publication (see Fitzgerald, et.al., 1991).

83

cover a shape; perimeter is a measure of the length or distance needed
to go around or surround a shape. Maxima and minima are subthemes of
the unit. The unit provides problem-solving situations where perimeter
and area, and the relationships between them, are used to investigate
questions of ”What's the biggest?” and ”What's the smallest?”.
Minimizing costs and materials, maximizing area and distances,
formulating arguments and recommendations, and estimating measurements
are some of the different kinds of activities students engage in.

Covering and Surrounding is divided into seven major activities
called Investigations (see CMP, 1992a). Each of the seven
investigations in the unit begins with one or more problem situations.
Labelled Problems, they are intended to engage students with
mathematical concepts and processes. Problems in Covering and
Surrounding are frequently supplemented by Follow-Up Questions. Follow-
up questions are intended to provide additional problem-solving
situations to help students refine their understandings about the major
concepts and processes embedded in the Problem. These opening
activities in each investigation introduce new content and are
structured to include class discussion in which students would share
ideas and conjectures (see CMP, 1992a, 1992b; see Appendix D for a
complete investigation from Covering and Surrounding).

Following the problems and follow-up questions is a section called
”Applications-Connections-Extensions”, or ACE. This section consists of
supplementary problems which are divided into three sub-sections —-
Application problems, Connections problems, and Extension problems. The
teacher’s edition explains the purpose of each kind of problem (CMP,

1992b):

84

Each of the seven investigations in Covering and Surrounding
ends with Applications, Connections, and EXtensions.
Application problems are intended to help students apply
what was just learned in the investigation. Connection
problems link the current investigation's topics to other
mathematical ideas (e.g., to other topics in the same or
another unit). Extension problems are challenging exercises
that require students to apply the investigation concepts in
ways not explicitly covered in the investigation. While the
ACE section of an investigation may be a source of homework
problems, some problems from this section will warrant
classroom time (emphases in original, p. 10).

The table below summarizes the Covering and Surrounding unit, including

sample Problems and ACE items from each Investigation:

Table 4.1:

85

Summary of Covering and Surrounding (CMP, 1992a):

 

Investigation [and
representations
used]

Central
Concept(s) and/or
Process(es)

Sample Problem

Sample ACE Item

 

l - Measuring
and Designing
Rooms [1-inch

Understanding &

”Use 12 square
tiles to make a
floor plan design

”If you walk around
the perimeter of a
rectangle, through

 

square tiles, grid distinguishing for Mrs. Hide. how many degrees
paper, outlines of between area and Mrs. Hide tells you have you turned
rooms that are perimeter. she wants at least when you get back
measured with the 14 sections of wall to where you have
tiles] space” (p. 2). started” (p. 12)?
2 - Areas and Students use a grid Students outline
Perimeters of Developing to estimate the their shoe on grid

Figures with
Irregular Edges
[grid paper and
transparent grids]

techniques to
estimate area and
perimeter.

perimeter and area
of a marsh in 1985
and again in 1990
(p. 13).

paper and find the
area of their

shoeprint (p. 18).

 

3 - Going Around in
Circles [grid
paper, transparent
grids, 3-D models]

Estimating the area
and circumference
of circles;
developing an

understanding of n.

”Find an estimate
for the area of the
circle... that

is smaller than the
actual area” (p.
21).

Students are asked
to locate and
describe a circular
object in their
neighborhood and
describe it and its
use (p. 28).

 

4 - Constant Area,
Varying Perimeters
[1-inch tiles,

Varying the
perimeter of
rectangles while
holding perimeter
constant to study

Students find all
the rectangular
rooms that can be
made with 24 square
tiles and recommend

Students use 20
tiles to build the
rectangles with the
largest and
smallest perimeter

 

graphs, grid paper] relationships one as the best for (p. 34).
between area and office space (p.
perimeter. 31).
Extending ”Hilda made a
understandings ”Working in your rectangle from

S - Constant

Perimeter, Varying
Area [l-inch tiles,
graphs, grid paper]

about relationships
between area and
perimeter by
holding perimeter
of rectangular
figures constant
and varying area.

group, find all
rectangles that you
can make with
square tiles that
have a perimeter of
18 units” (p. 38).

square unit tiles.
It has an area of
16 square units and
a perimeter of 16
units. Draw a
picture of Hilda’s
rectangle” (p. 40).

 

6 - Finding
Relationships:
Connecting
Parallelograms,
Triangles, and
Rectangles [grid
paper, transparent
grids, models]

Using knowledge
about rectangles to
develop ways of
finding area and
perimeter of
triangles &
parallelograms.

”Can you find a way
to make 1 cut
through a
parallelogram so
that the 2 pieces
can always be
reassembled into a
rectangle” (p. 42)?

”If you know that a
rectangle measures
12 cm by 15 cm, how
can you use this
information to find
the area of the

rectangle” (p. 47)?

 

 

7 - The Bee Problem
Revisited
[transparent grids,
pattern blocks, 1—
inch tiles]

 

Extending area and
perimeter concepts
to other shapes and
developing and
verifying
conjectures.

 

”Why do bees make
hives from hexagons
rather than other
tessellating shapes
like triangles or
squares” (p. 54)?

 

”As we increase the
sides of a regular
polygon, what
figure does the
polygon seem to
become” (p. 58)?

 

 

86

The teacher’s edition (CMP, 1992b) provides suggestions for
teaching as well as answers for the problems, follow-up questions, and
ACE items in the student materials. For example, as noted in Table 4.1,
Investigation 6 uses understandings of area and perimeter of rectangles
to develop strategies for finding area and perimeter of triangles and
parallelograms. The teacher’s edition (CMP, 1992b) offers this advice

to teachers:

Try not to tell students area formulas (e.g., Areatrnnmle =
1/2 x base x height) as the investigation is structured to
help students develop these or equivalent algorithms. As
students work through the problems of finding the
relationships between rectangles and parallelograms and
parallelograms and triangles, you might instead ask them
questions about how they are dividing up their
parallelograms or rectangles. For example, when a student
discovers that when she cuts a parallelogram along one of
the diagonals two congruent triangles are formed, you might
ask her how the area of the triangles relates to the area of
the original parallelogram. You might ask her further,
then, to think about how the area of a triangle could be
expressed in terms of the area of the original
parallelogram. These kinds of questions help students
develop their own problem solving skills and understanding
of perimeter and area rather than just learning a rule
(emphasis in original, p. 64).

The teacher edition is meant to help the teacher, not to direct teacher
action (CMP, 1992b; Fitzgerald et al., 1991)3. The developers intend
the teacher materials to ”provide sufficient help that a well-motivated
teacher can teach the materials with little or no inservice” (Fitzgerald
et al., 1991, p. 15). The teacher materials are designed to help the
teacher in her classroom instruction by highlighting ”subtle points of
the mathematics and to help the teacher become more reflective and

curious about student learning” (Fitzgerald et al., 1991, p. 23).

 

3Appendix D provides a complete Investigation from Covering and Surrounding, and
Appendix E provides the corresponding material from the teacher’s edition. Appendices
D and E may be consulted to provide an example of the support the teacher edition
provides to the teacher in teaching Covering and Surrounding.

87

As I argued earlier, teaching and learning problem solving in
classroom settings is not well-defined, nor is it clear what kinds of
problems are most appropriate for helping students learn about
mathematics and problem solving (Charles & Lester, 1982; Grouws, 1985;
Kilpatrick, 1985; Stanic & Kilpatrick, 1989). Yet, as reformers
emphasize, it is important to carefully consider the kinds of problems
students have opportunities to work with, as problems are bridges to

mathematical learning goals that go beyond getting the ”right answer”:

... problems that are used in school instruction should be
means to more ambitious educational goals, rather than being
the goals themselves. We hope that students will be able to
reason effectively about mathematical concepts and
principles and about the events and systems that they
encounter in their lives (Greeno, 1991, p. 75).

Problem solving is one of the ”ambitious educational goals” the CMP
developers have for students. Problems and problem situations in the
CMP curriculum are designed with the intent of helping students learn to
reason and work with mathematical concepts as well as learning about
mathematical concepts (CMP, 1992b; Fitzgerald et al., 1991).

But conceptualizing ”What is a problem?” and ”What is problem
solving?” in the Covering and Surrounding unit is not a clear—cut task,
despite the mathematical content and process goals the CMP curriculum
sets for students (see Fitzgerald et al., 1991). While Polya (1967,
1968) and Charles and Lester (1982), for example, have offered
”definitions” of what a problem is (see Chapter 2) and heuristic
characterizations of problem solving, there is no single means for
defining problems and problem solving in covering and Surrounding. Part

of the difficulty is because Covering and Surrounding arguably includes

88

a variety of different kinds of problems. Consider this classification

scheme utilized by Hembree (1992):

Standard (word or story) problems require the translation
of verbal statements into mathematical operations.
Nonstandard (process or open-search) problems encourage
the use of flexible methods; the solver possesses no routine
procedure for finding an answer. Real-world problems
entail situations where students will need to select and
apply the tools of mathematics at their discretion.

Puzzles depend on luck or guessing or a use of unusual
strategies toward their solution (p. 249, emphases in
original).

The following problems from the Covering and Surrounding unit (CMP,
1992a) seem to fit into this framework:

Standard (word or story) Problem

A rectangle measures 8 cm by 10.5 cm. What is the area? (p.
47).

Nonstandard (process or open-search) Problem

Using 16 tiles, build the floor plan for a rectangular
building with the greatest perimeter. Now rearrange the
tiles to build the floor plan with the smallest perimeter.
Write a sentence to describe each of the rectangles (p. 34).

Real—world Problem

Investigate where measurement is used in the world around
you. To do this, you can choose to do one of the following
assignments:

a) Interview an adult about how he or she uses measurement
on the job. For each kind of measurement that you find, ask
how the measurement is made. What units are used? What
kinds of instruments are used to make the measurements?

b) Look carefully at a newspaper or magazine and find uses
of measurement. You might find examples of measurement in
advertisements or articles. Be sure to record what units
are used to make the measurements and what the measurements
are used for (pp. 19-20).

89

Puzzle Problem

Figure 4.1: The pentomino problem:

 

Problem 1: Can You Make 18?
Construct the following pentomino with your square tiles:

 

 

 

 

 

Find a way to add tiles to this figure to create a figure with a
perimeter of 18 units. When adding tiles to any pentomino, remember
that the tiles must touch edge to edge (p. 36):

 

 

 

It is important to also consider the possibility that some kinds of
problems may not be incorporated into Covering and Surrounding. For
example, Charles and Lester (1982) include ”Drill Exercise” in their
categorization of problems as one kind of problem, which they illustrate

with the example

346
3.2.8. (p. 6) .

There are no ”naked number” drill exercises of the kind above included
in the Covering and Surrounding unit.

The drill exercise also illustrates a semantic difficulty in
trying to determine ”What is a problem?” in curricular materials. While
some educators and teachers might think of a drill exercise as a
legitimate problem, others, such as Greene (1991) suggest that,

depending on the solver, a drill exercise might not be a problem:

mathematicians regularly work on problems that take many
hours -- indeed, if a question can be answered by
mechanically applying a known procedure, mathematicians do
not think it is a problem at all (p. 83).

9O

Greeno’s (1991) remarks address an important issue with respect to
teachers' use of problem-solving oriented materials. The conception of
problem solving a teacher brings to her use of problem-solving oriented
curricula may influence how she perceives problem solving being
addressed by the materials. For example, if problem solving is
conceived as solving drill exercises, then Covering and Surrounding
could be perceived as being devoid of problem solving!

Looking Ahead: Teaching* Covering and Surrounding

The developers of the CMP curriculum intend Covering and
Surrounding to engage students with rich, connected mathematics via
problem solving. However, university based curriculum developers and
practicing teachers can approach mathematics curricula from different
perspectives. For example, curriculum developers may intend curricula
to develop students' understanding of problem solving and mathematical
concepts, while classroom teachers may focus on using the curricula to
make mathematics more enjoyable for their students and to help them
develop positive attitudes about mathematics (Prawat et al., 1991).

This can contribute to teachers emphasizing different content and
processes (e.g., computation instead of problem solving) in their use of
curricula than those intended by the curriculum developers (e.g.,
Wilson, 1990).

The next two chapters present the cases of how two different
teachers used the Covering and Surrounding unit in their classrooms. In
each of the cases, a profile of the teacher participant is provided and
is followed by an assessment of her knowledge, views, and beliefs in
each of domains of the framework developed in Chapter 2. Each teacher's

views, knowledge, and beliefs is then used to conceptualize her use of

91

Covering and Surrounding described in the subsequent section. For each
teacher participant, her use of Covering and Surrounding is unpacked to
address the main research question, ”How do teachers use a problem-

solving oriented piece of curriculum in their classrooms”, and identify

issues connected to the main research question.

CHAPTER 5

THE CASE OF KAREN KNIGHT

A Profile of Karen Knight
With her students watching intently, Karen Knight writes
”DIVISIBILITY LAWS” on the overhead transparency. She then writes the

number 246 and addresses the class:

Karen: Let's review our discussion from yesterday. Is this number
divisible by 2?

Marcusl: Yea!

Karen: And how do you know that this number is divisible by 2?

Marcus: Because each digit can be divided by 2.

Karen: No -- not quite. Think back to what we talked about
yesterday.

Alicia: Because the six can be divided by 2?

Karen: Good! You need to remember this [underlines the 6 in 245]

-- a number is divisible by 2 if the ones digit is divisible
by 2. Now think about some of the other divisibility laws
we used yesterday -— what else is this number [points to
246] divisible by?

Ellen: Three!

Karen: And how do you know that?

Ellen: Because the numbers add up to 12!

Karen: Okay -- but why does that mean that the number is divisible
by 3?

Eric: Because you can divide a number by three if, like, when you

add the digits together that number divides by 3.

Karen: Good! Remember, a number is divisible by three if the sum
of its digits is divisible by 3.

 

1All students’ names are pseudonyms. In selecting pseudonyms, I have attempted to
represent each student’s ethnicity to provide a sense of the diversity of Karen’s

classroom.

92

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Karen continued the lesson by stating new divisibility laws for 5 and
10. ”A number is divisible by 5 if it ends in a 5 or 0,” Karen
declared. She also emphasized that ”A number is divisible by 10 if it
ends in a 0.” Karen wrote both of these divisibility laws on the
overhead and then illustrated how the number 350 is divisible by both 5
and 10 because ”it ends in a zero.”

Having completed examples illustrating the divisibility laws for
2, 3, 5, and 10, Karen put the number 11930 on the overhead. Addressing
her class again she said, ”Copy this number down and test it for
divisibility by 2, 3, 5, and 10. Now, I don't want 'yes’ or 'no'
answers either -- you need to explain why it is or isn't divisible.”
Circulating around the room, Karen surveyed her students as they worked
silently on the task. She returned to the front of the room after
letting the students work for about two—minutes and said, ”Okay, what
did you find out?” Four different students gave the correct responses
for divisibility of 11930 by 2, 3, 5, and 10, explaining their reasoning
by reciting or paraphrasing the appropriate divisibility law. Just
before the bell sounded, however, Trevor raised his hand and was called

on by Karen:

Trevor: I think that you could say 11930 is divisible by 5 because
it can be divided by 10.

Karen: Could you explain that again Trevor?

Trevor: Well, if a number is divided by 10 it has to be divisible by

5 too because fives divide tens.
Karen: Very interesting observation Trevor! Did everyone hear
that? [nods from the class] Since 5 divides 10, a number

that is divisible by 10 has to be divisible by 5 also. Very
good Trevor!

Trevor, obviously pleased at the acknowledgement of his contribution,

smiled just before the bell rang. Karen dismissed her students by rows

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after quickly giving a short assignment from the textbook on testing
numbers for divisibility.

This excerpt from Karen Knight's 10/14/92 lesson on divisibility
laws occurred about three weeks before she began teaching Covering and
Surrounding. As in the other lessons I observed her teach prior to
Covering and Surrounding, Karen kept her students focused on the
mathematics topic at hand and insisted on participation and correct
answers from her class. Karen demands that her students be able to
explain how they arrived at their solutions to exercises and encourages
them to share their ideas about mathematics. An organized teacher and a
strong leader in the classroom, Karen emphasizes practice and
proficiency in her mathematics teaching, assigning her students homework
several times each week. Karen describes herself as focusing on
teaching her students about mathematical procedures, formulas, and how
to use them.

The following week I observed Karen leading her students in an
analysis of the ”Factor Game”, an activity from a teacher resource book
called Factors and Maltiplesz. The activity involved students finding
all of the proper factors of the numbers 2 through 30 (i.e., all the
factors including 1 but excluding the number itself) and computing the
sum of these numbers. Students organized their work on an activity
sheet Karen had duplicated from the resource book. After allowing her
students some time to work on the task in pairs, Karen placed a

transparency of the activity sheet on the overhead that had the answers

 

2Factors and.Multiples (Fitzgerald et. al., 1986) is a unit produced by the Middle
Grades Mathematics Project which was a curriculum development project that some of the
CMP developers were involved in. The unit features a non-standard teaching guide
which provides detailed suggestions for teaching as well as answers. The topics of
the unit include factors, multiples, primes, and composite numbers.

95

filled-in but concealed the answers with a sheet of paper. Karen then
had students volunteer answers (i.e., the proper factors and their sum
for each number 2 through 30), revealing each answer after it was given
to check it. Classtime ran out before the entire sheet was checked and
Karen told the class that they would finish checking the answers the
next day.

After class I asked Karen about how she liked the Factors and
.Multiples unit. She replied, ”I like to use it near the beginning of
the year like this because it's good review of number facts and the kids
like it.” I asked Karen about the suggestions for teaching provided by
the unit and how useful she thought they were. After pausing to think
about this, she said, ”I guess I don't use the teaching suggestions that
much because I've taught the unit before. I mainly use it for answers,
like I did today to let the students check their work.” I then asked
Karen about how useful she found her textbook teacher edition compared

to Factors and Maltiples, and she said that

Well, I usually let the students use the teacher text to
check their answers when we work from the text, or I use it
to check their answers if I'm grading their assignment.

Karen also noted that the textbook teacher edition ”really doesn't have
much in it except answers.”

Karen's lesson on divisibility laws and her comments about her use
of the Factors and Maltiples unit and textbook teacher edition
foreshadow two important features of her practice: (1) Karen's usual
mode of teaching mathematics is to focus on mathematical rules,
procedures, formulas, and how to use them -- Karen describes this as

”abstract teaching”; (2) Karen's use of teacher materials centers on

96

obtaining answers to allow students to check their own work or for Karen
herself to check their work.

A teacher with over 25 years of teaching experience, Karen has
taught first, third, fourth, and sixth grades in a variety of school
settings and has held teaching positions in Alabama, New York,
Louisiana, and Michigan. She has taught a wide range of subjects over
her career, including mathematics, social studies, music, aerobic
dancing, and computer programming. In teaching mathematics, Karen says
that she finds working with numbers, making and using graphs and
diagrams, and using the language of mathematics to be particularly
interesting.

Karen teaches in a large middle school which enrolls about 1,350
students in grades six, seven, and eight. Karen taught Covering and
Surrounding in her Enriched Mathematics class -- about 25% of the sixth-
graders in her school are enrolled in Enriched Mathematics. In most
cases, students are placed in Enriched Mathematics by scoring in the top
25% on standardized achievement tests in mathematics near the end of the
fifth-grade. The 75% of sixth-graders who are not enrolled in Enriched
Mathematics are placed in General Mathematics. According to Karen, the
main difference between sixth-grade General Mathematics and Enriched
Mathematics is that the Enriched sections use a seventh-grade textbook.

The majority of students enrolled in Karen's school are from
working-class socioeconomic backgrounds. For example, some of the
students' parents work in heavy industry (e.g., automobile
manufacturing). About half of the students at Karen's school qualify
for the district's reduced-cost lunch program, and some students live in

areas of their community where low-cost housing is provided and crime

97

and drugs are problems. However, the students in Enriched Mathematics
do not necessarily share the socioeconomic background of most of the
students in the school. Karen commented to me that one of the reasons
students in her Enriched Mathematics class tend to be innovative in
their thinking is because ”Their parents, being professionals, they
teach their kids a little bit more than the regular parent who doesn't
have the knowledge.”
Karen and the Domains of the Framework

During the summer and early fall prior to teaching Covering and
Surrounding, I interviewed Karen to learn about her views, knowledge,
and beliefs within the four domains of the analytic framework developed
in Chapter 2 (see Appendices A, B, and C for interview protocol and
items). In this section, I explore key aspects of Karen's views,
knowledge, and beliefs about mathematics and problem solving, problem-
solving activity in classrooms, subject-matter knowledge, and
perceptions and beliefs about student learning. Karen's views,
knowledge, and beliefs identified in this section will be utilized
throughout this chapter to understand her use of Covering and
Surrounding.

Problem solving is an important topic within the mathematics
curriculum for Karen. Karen believes that problem solving can help her
students in real life. She emphasized this to me when I asked her about

how important problem solving is for middle school students:

AR: How important would you say problem solving is for middle
school kids?

Karen: Oh, very important.

98

AR: Well, some teachers would say that at the middle school
level it's most important to have kids learning their facts,
making sure they can --

Karen: All of life’s a problem! You see, they have to go to lunch
-- that’s a problem and they have to solve it. What if, for
instance, mom only gives you a dollar this morning because
that's all she has? You have a problem to solve to feed
yourself at lunch time! It's a problem.

But although she places heavy emphasis on problem solving being a part
of real life, Karen sticks closely to the textbook when teaching her
students about problem solving. When I asked her about how frequently
she teaches problem solving in her classroom, Karen said that she
teaches problem solving ”At least twice a week, or as it comes up in the
textbook.” Karen went on to elaborate about how problem solving is

integrated into her mathematics teaching:

AR: Ideally, do you think you'd like to spend more time on
problem solving, or are you doing about right?

Karen: I'm not doing it about right. We're going more and more
toward problem solving, but I'm not spending that much time
on it. I use problem solving if it arises in the textbook.
But I want to make sure that they have the steps for solving
problems, but I have yet to do a whole unit on just problem
solving. So problem solving is kind of part of my lessons,
but they're not lessons on problem solving.

Karen believes that she addresses problem solving in her teaching,
but not to the degree that she should. Problem solving, in Karen's
view, is part of her lessons because she does assign exercises from the
text which she feels require her students to do problem solving to
obtain a solution. Karen thinks that problem solving is part of her
lessons because within her usual lessons on other topics students
occasionally do problems that require problem solving. While Karen
implies that she should teach problem solving more, she believes that

her students are learning about ”the steps for solving problems.”

99

Because Karen teaches problem solving when ”it arises in the
textbook”, understanding how Karen’s textbook addresses problem solving
is part of understanding Karen's beliefs about problem solving. Here is
an example of a problem from a section labelled ”Problem Solving .
Applications” in Karen's textbook (Abbott & Wells, 1985) which she said
is an example of the kind of problems she typically assigns her students

to address problem solving:

The diameter of Joanna’s bicycle wheel is 66 centimeters.
How far does the wheel travel in one revolution (p. 345)?

In terms of Hembree's (1992) framework (see Chapter 4), the above
problem is a ”standard (word or story) problem” that requires ”the
translation of verbal statements into mathematical operations” (p. 249).
In the above problem, the student needs to know the circumference
formula C = 2nr and then translate the word statement into C = 2nx:66
centimeters to get the answer.

Although she believes problem solving is important because it can
help her students in real life, word problems of the kind Karen pointed-
out in the textbook are generally not the same as problems in real life.
Translating word problems into formulas is not the same as solving
”real-world” problems ”where students will need to select and apply the
tools of mathematics at their discretion” (Hembree, 1992, p. 249).
Translating a word problem into a formula or operation, for example, is
not the same as a real-world problem because the path to the right
answer may not be so obvious. Therefore, there are differences between
the kind of problem solving embedded in Karen's mathematics teaching and

the problem solving she believes students will need to do in real life.

100

Karen believes that problem solving in the mathematics curriculum
is important because it is relevant to real life. However, her views
about teaching problem solving are centered on the textbook and solving
standard (word or story) problems. While Karen's reasons for why
problem solving is important for her students are connected to solving
real-world problems, her portrayal of problem solving in the classroom
is focused on solving routine word problems. This comparison shows that
how Karen addresses problem solving in her classroom varies from her
beliefs about problem solving. Her use of the textbook seems to be at
the root of this variance. The ”problem solving” sections of Karen’s
textbook are predominantly sets of standard word problems. Moreover,
she believes that problem solving is important for real life, yet also
teaches problem solving as it occurs in the text. The product of the
interplay between Karen's beliefs and the text is that she maintains her
beliefs about problem solving while her textbook-oriented teaching of
problem solving focuses on standard word problems. This interaction
creates differences between Karen's beliefs about problem solving and

how it is enacted in her teaching.

E 1] -fi 1 . E I' 'l . :1 E |'

AR: What kinds of activities would you say your students do when
they are working on problem solving?

Karen: Basically, problems in everyday life. I remember last week
we did one with two boys with summer jobs, and we compared
how much one had made over the other, and then one went
shopping and who had more saved. The guy who had the more
saved was the one who made less money. The guy who made
more money, I guess he felt free to go shopping. So we made
that comparison, but it wasn't a unit, it was just a page in
that book. Its got two pages and one page talks about steps
of problem solving and then they give them problems to
solve. We just solved it and went through there and found
out how you would solve it, what steps you would use,
eventually, before school is out in June, I will be

101

assigning them some problem solving, some more problems to
solve.

Karen describes in the above excerpt that students in her class
are doing problem solving when they are solving word problems in the
text. That Karen conceptualizes problem-solving activity in her
classroom as solving textbook word problems is reinforced by her views
about the challenges of teaching problem solving. Karen said that ”I
find the biggest problem with problem solving is reading and
understanding the language that the textbook is about.” Karen's
description of problem-solving activity as solving word problems from
the textbook is consistent with how she describes herself teaching
problem solving (i.e., teaching problem solving when it arises in the
text).

Karen and I also talked about what students do when they are doing
problem solving in her classroom. Karen described that, in her view,
problem solving involves reading a problem ”from the textbook”, figuring
out how to solve it, getting an answer, and then seeing if the answer
”makes sense”. Karen's description of problem-solving activity is very
similar to Polya’s (1945) well-known problem—solving heuristic:
Understand the problem; formulate a plan; carry-out the plan; look back.
When I asked Karen to talk about how she had come to think about problem
solving in terms of these steps she replied, ”I've seen problem solving
stuff on posters before.” Karen was referring to the popular kind of
posters that display steps for problem solving, Polya's heuristic being
one of the most common (see NCTM, 1980). Karen emphasized that
understanding the problem and then seeing if the answer ”makes sense”
are what she stresses the most when her students are doing problem

solving.

102

Just as Karen's views and beliefs about problem solving are tied
closely to her textbook, her conceptions about problem-solving activity
in classroom settings are also linked to the text. Karen conceptualizes
problem-solving activity as occurring in her classroom when students are
working on word problems from the text. This claim is reinforced by
Karen's citation of the word problem in her textbook in the prior
section as an example of what her students work on when doing problem
solving. Although Karen referred earlier to problem solving being
important in real life, in our discussions about problem-solving
activity in her classroom she only talked about doing word problems from
the text. These descriptions from Karen about what constitutes problem-
solving activity in her classroom reinforce the earlier claim about the
differences between her beliefs about problem solving and how she enacts
problem solving in her classroom: While Karen believes that problem
solving is important because it is useful in real life, problem-solving
activity in her classroom focuses on standard word problems because that
is how the textbook, which she strongly adheres to, addresses problem
solving.

Wedge

Karen is not confident of her own subject-matter knowledge of
perimeter and area. While she is willing to entertain students' non-
standard ideas about perimeter and area even if they are not familiar to
her, she prefers to consult the textbook before taking a position as to
the validity of students' non-standard ideas. For example, this is how
Karen responded to a question in which she is asked to help a colleague
make sense of a student’s non-standard way of finding perimeter (which

is also shown below):

103

Figure 5.1: Student’s non-standard way of finding perimeter.

 

 

 

 

9
P = 2 x (18/9 +18/2): 2 x (36/18 +162/18)
= 2 x198/18 =198/9 = 22

First, I would refer her to the text and we’d look up the
perimeter. We would find out what the definition is for
finding perimeter. I can't say this is wrong until I find
out how do you find perimeter, make sure the child
understands that we’re finding perimeter, and then see if
this makes sense. We would go to the textbook and start
there.

Karen is unwilling to either dismiss, accept, or take some other
position on the student's work until she refers to the textbook.

Karen also expressed a desire to consult the text in describing
how she would respond to a student who had developed a non-standard
method for converting between different square units for measuring area:

Figure 5.2: Converting between units for measuring area.

6 yards

 

8 yards

A = 1/2 x B x H = 1/2 x 6 x 8 = 1/2 x 48 = 24 square yards
3 feet = 1 yard so Area = 24 x 3 = 72 square feet

AR: New, suppose that during your instruction on perimeter and
area one of your students raises his hand and is very
excited. He says that he has figured out how to convert
between different units used to measure area. And he shows
you his solution to this area problem [referring to Figure
5.2 above] on this triangle and that he came up with 24
square yards. Now, he says that since there are 3 feet in a

104

yard he can convert the area from square yards to square
feet How would you respond to this student?

Karen: Converting square yards to square feet?

AR: Yes. He's multiplying the square yards by 3 since there are
3 feet in a yard.

Karen: OK, into square feet Hmm.. (long pause) Well, I don't
know. I could try it, I'd let him try his theory. And I'd
like him to do some experimentation, again, to see if it
makes sense, to see if it’s mathematically correct. ... I'd
ask him which book did he use and let me see it —- if he's
used his text I'd like to see what they've [i.e., the
textbook authors] have done there -- and it might be that if
he’s really excited about this I'd let him share this with
his classmates. I'd make a transparency of this. So this
would be okay.

Karen is uncertain about the validity of the student's faulty
conjecture, but she eventually accepts it as she indicates in the above
transcript. In understanding Karen's subject-matter knowledge, an
important point about the above response is her tentativeness in
accepting the student’s theory and her desire to consult the textbook.

As well as showing how Karen refers to consulting the textbook in
making sense of students' non—standard work, the two excerpts also
demonstrate that she does not cite the textbook as the final authority
for judging students' ideas. In the perimeter question, Karen stresses
that she ”can't say this is wrong” until she investigates what is meant
by perimeter and how the student understands perimeter. In converting
area units above, Karen is careful to allow the student an opportunity
to explain and ”let him try his theory.”

While the textbook clearly plays an important role in Karen's
pedagogical reasoning, it is not the central authority guiding her
teaching. Karen's uncertainty about her own subject-matter knowledge is
manifested in her desire to consult the textbook before making decisions

about the validity of students' ideas. However, she is adventurous

105

enough to entertain students' unfamiliar conjectures and allow them
opportunities to develop their ideas. Karen's strong tendency to
consult the textbook is the result of her uncertainty about her own
subject-matter knowledge of perimeter and area combined with her
willingness to give students' ideas a chance.

Karen believes that middle school students increasingly need to do
”hands-on” activities to understand mathematics. By ”hands-on”
activities Karen means mathematics lessons or tasks in which students
use manipulatives or tools (e.g., tiles, grids, cubes) to ”concretely
understand and work with the concepts.” For Karen, the intent of hands-
on activities is to give students opportunities to physically work with
mathematical concepts so they can understand them.

Prior to teaching the Covering and Surrounding, I asked Karen to
share her impressions of several different pieces of curriculum (e.g.,
Shroyer & Fitzgerald, 1986). Some of the curricula involved activities
where students use square tiles to measure perimeter and area of plane
figures. Reacting to the materials, Karen said ”My usual material
doesn't really include exercises where they use the tiles, and that is
my goal this year -- to get them to do more hands-on, use more
manipulatives.” Karen noted that she is ”not a tile person” but that
since fewer kids are able to think abstractly she is trying to change

her own style of teaching:

I find out that as we go further and further into the future
we have fewer and fewer kids that can think abstractly --
they need more hands-on. So I have to change my own style
of teaching.

106

In subsequent conversations, Karen said that the Covering and
Surrounding unit, with its use of a variety of manipulatives (e.g.,
tiles, transparent grids) is ”really hands-on.” Noting that ”hands-on
is the wave of the future”, Karen said that since her students need
hands-on activities to understand, she must change her teaching to
include more hands-on to meet their needs. Karen views the Covering and
Surrounding unit as something that will provide her students with the
hands-on experiences using manipulatives that they need to understand
perimeter and area.

By ”thinking abstractly”, Karen means being able to understand
concepts, like perimeter and area, by knowing and using formulas. Karen
described to me that her usual mode of teaching is ”teaching
abstractly”3, meaning that she would tell her students what the concepts

were and then teach formulas or procedures:

After I had told them the idea of area and perimeter, then

we'd go into formulas rather than hands-on. I think most

middle school teachers are similar, that we think a lot of

the hands-on is done in elementary school, so we don't have

to do hands-on and we treated them like junior-high students

when they got here. Now we're finding that even some of the

junior-high kids still need hands-on.
Karen believes that sixth-graders need hands-on activities because they
can't think abstractly -- as Karen said, ”They cannot really understand
the concepts of perimeter and area by having me explain it to them, they
need to do hands-on activities to understand.” Karen also noted that

after doing hands-on students ”are more able to think abstractly about

the concepts, like using the formulas correctly.” Karen has assumed in

 

3”Teaching abstractly” and ”think(ing) abstractly” are phrases that Karen used
frequently throughout the study to describe her own teaching. I use these terms with
quotations in the text to emphasize when Karen used the term(s) to describe or explain
aspects of her own practice.

107

the past that sixth-graders don’t need hands-on, and therefore she
hasn't done hands-on in her teaching. Now, however, she feels that her
students need to do hands-on before she can teach abstractly.

Karen's lesson on divisibility laws at the beginning of this
chapter is an illustration of her tendency to emphasize the abstract.
Karen presented her students with the divisibility rules for different
numbers and demonstrated them. She then had her students practice the
rules by finding whether the number 11930 was divisible4 by 2, 3, 5, and
10. While Trevor made a contribution to the class discussion that went
beyond the four divisibility laws Karen had presented, she restated his
conjecture to the class in the form of a rule. None of the rules in
Karen's lesson, with the possible exception of Trevor's explanation of
his own conjecture, were conceptually motivated or concretely
illustrated. Karen taught the divisibility laws ”abstractly” -- she
told her students the rules, demonstrated their use, and then had them
practice using the rules.

With respect to teaching and learning perimeter and area, Karen
wants her students to learn the same mathematics from hands-on
activities that she would ordinarily tell them about in her abstract
teaching. When I asked her about what she would like her students to
know about perimeter and area, Karen said she wants her students to
”understand the basic concepts -- what perimeter and area are and how to
find them and how to use them to solve simple problems.” She described

how her students have difficulty in ”doing the basic concepts, like

 

4As emphasized in Chapters 2 and 4, an important aspect of problem solving is that it
is relative to the solver -- a problem for one individual is not necessarily a problem
for somebody else. While Karen's problem to her students of deciding if 11930 is
divisible by 2, 3, 5, and 10 might be considered a problem for sixth-graders, the way

Karen used it was as a drill exercise, i.e., to practice using divisibility laws.

108

finding the correct areas of shapes” and that she thinks ”hands-on is
going to help them with that.” Karen noted that the textbook doesn't do
hands-on but just has formulas which the students are supposed to learn.
However, Karen said that, ”The kids really can't think abstract like
that, just learning formulas cold, so I need to do hands-on so that they
can understand them.” Karen elaborated on this point, noting that to
understand perimeter and area her students need to know ”when to use
which formula to get the right answer -- the hands-on should help them
know to, like, when to multiply for area and add for perimeter.”

While Karen has decided to teach perimeter and area using a hands-
on approach, her comments suggest that this change in her mode of
teaching has not been accompanied by a change in what she believes
students should know about perimeter and area. Karen had surveyed both
the student and teacher materials for Covering and Surrounding prior to
our first discussion. However, she only cited topics from the textbook
in talking about what she would like her students to know about
perimeter and area. The learning goals Covering and Surrounding posits
for students (see Chapter 4), for example, include understanding
relationships between perimeter and area, developing estimation
strategies, and finding area and perimeter of irregular figures to which
formulas may not apply. However, Karen's expectations of what her
students should know/understand about perimeter and area focus on ”doing
the basic concepts” which are oriented around computation and the kinds
of standard word problems found in the textbook. Using formulas
correctly is at the heart of what Karen wants her students to know about

perimeter and area.

K:

109

Karen also believes that maintaining students' interest in lessons
is critical to successful student learning. For example, in discussing
the quizzes included in the teacher materials for the unit (see CMP,
1992b), Karen talked about how she would decide whether or not to use

them:

I will determine whether I'm going to use them based on
their interest. You have to deal with sixth-graders,
they're kind of flighty on some days, and if they're not
interested in the thing it starts going downhill after the
first two or three assignments. Even though they're [i.e.,
Karen's Enriched Mathematics students for the upcoming 92-93
year] an enriched group, they could be a much slower group
for interest. Again, it all depends on the interest of the
students I have.

Karen believes that students need to be interested in the mathematics
they're doing or the lesson will inevitably start ”going downhill.”
Karen pointed out to me on other occasions that what she would be able
to accomplish in the Covering and Surrounding unit with her students
would all depend on their level of interest. As I explain next, student
interest, for Karen, is a necessary condition for successful student
learning.

Karen noted that Covering and Surrounding would be a piece of
curriculum that would probably maintain students' interest. For
example, Karen commented prior to teaching the unit that the last
investigation, ”The Bee Problem Revisited”, would likely be interesting
to her students -- ”The bee project looks fun and I bet they would like
it, and if they like it they'll be more interested.” However, Karen
also commented that all the work with tiles in the first investigation
might not be as interesting because it ”might get boring after awhile
for them [i.e., her students] and they might just start multiplying.”

Karen's comments provide insight into what counts as student interest

110

for her -- if students are bored with an activity they won't be
interested, whereas if they like an activity they will be interested.

Karen’s conception of student interest focuses on whether students
like the activity involved in a lesson, as opposed to whether or not
they find the mathematics involved in a lesson engaging. The activity
part of a lesson is what students are doing (e.g., cutting out shapes,
using manipulatives, using a calculator, multiplying numbers by hand);
the mathematics component of a lesson is the mathematical concept(s)
embedded within students' activity. For example, Covering and
Surrounding has students use tiles to measure rooms (the activity) to
learn about perimeter and area (the mathematical concepts) in the first
investigation. It is conceivable that in instructional situations
students might find the mathematics interesting (e.g., perimeter and
area concepts and relationships) and find the activity not interesting
(e.g., pulling numbers from rectangle diagrams to find area via rote
multiplication). Karen's attention to student interest centers on the
activity component of lessons and not on the mathematics involved.
Summary

Karen believes problem solving is an important component of middle
school mathematics because it can help her students in real life.
However, Karen teaches problem solving from the textbook, which focuses
on solving standard word problems, not real-world problems. Problem
solving for Karen means solving word problems from the textbook. Karen
believes that problem-solving activity occurs in her classroom when
students are solving textbook word problems and following a linear

process of problem solving steps -- reading and understanding the

111

problem, figuring out a plan, getting the answer, and seeing if the
answer makes sense.

Karen is uncertain about her own subject—matter knowledge about
perimeter and area. She consistently refers to the textbook as a
resource in making sense of students' non-standard work because she is
hesitant about her own knowledge, yet willing to entertain students'
conjectures. Providing students with hands-on learning experiences and
maintaining their interest are two central issues in Karen's perceptions
and beliefs about student learning. However, while Karen wants to
change her mode of teaching perimeter and area from abstract to hands-on
activities, her beliefs about what students should know about perimeter
and area remain focused on learning how to use formulas. Moreover,
Karen's conception of student interest is attentive to whether students
like the activity of a lesson but does not address whether they find the
mathematics engaging.

Karen's Use of Covering and Surrounding

This section begins by providing a summary of Karen's coverage of
Covering and Surrounding. I give a synopsis and brief analysis of the
exercises Karen assigned from each investigation in order to illustrate
how she used the unit. The subsequent sections each describe and
analyze a specific aspect of Karen's use of Covering and Surrounding. I
focus on four key aspects: (1) how Karen's sequencing of investigations
and perceptions of problem solving compare with the intentions of the
curriculum developers, (2) the coexistence of hands-on activities and
skill maintenance while teaching Covering and Surrounding, (3) her use
of teacher materials, and (4) the influence of Covering and Surrounding

on her practice. Throughout these sections, the assessment of Karen's

112

views, knowledge, and beliefs in the four domains of the framework is
employed to unpack and understand her use of Covering and Surrounding.
W

Karen's instructional decisions about sequencing investigations
and perceptions of problem solving in teaching Covering and Surrounding
were shaped predominantly by her own beliefs, not by the teacher
materials. Karen selected and taught investigations in Covering and
Surrounding in a different order than intended by the curriculum
developers because of the power of her preexisting beliefs about student
learning. Moreover, Karen did not believe her students were learning
about problem solving in Covering and Surrounding because her beliefs
about problem solving are different than the problem solving occurring
in the unit.

Karen began teaching Covering and Surrounding in her Enriched
Mathematics class on 11/3/92 and concluded the unit on 12/3/92. After
teaching the first investigation of Covering and Surrounding, Karen
allowed her students to determine the subsequent investigations studied
by majority vote. To portray Karen's use of Covering and Surrounding,
it is helpful to first show which investigations she covered in the unit
and the Problems, Follow-Up Questions, and ACE items she assigned.
Table 5.1 below summarizes the sequence in which Karen taught
investigations and the Problems, Follow-Up Questions, and ACE items she

assigned:

Table 5.1:

Summary of Problems,

assigned by Karen.

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Follow-Up Questions,

and ACE items

 

 

 

 

 

 

 

 

 

Investigation Assigned Assigned Follow- Assigned ACE
Title Problems Up Questions Items

1 - Measuring none - extra
and Designing l, 2, and 3 1 credit

Rooms

6 - Finding Follow-Up

Relationships - Question to

Connecting 1 and 2 Problem 1 and l and 2
Triangles, the Follow-Up

Parallelograms Question to

and Rectangles Problem 2.

7 — The Bee Follow-Up

Problem 1, 2, and 3 Question to l

Revisited Problem 3.

5 - Constant Follow-Up none - extra
Perimeter, 1, 2, and 3 Question to credit
‘Vagying Area Problem 1.

 

 

Referring to Appendix D and noting the exercises Karen assigned in

Investigation 1 from Table 5.1 provides a particular example of Karen’s
coverage of an investigation in Covering and Surrounding. To provide a
sense of how Karen's coverage of the investigations she taught compares
to the number of problems in the unit, Table 5.2 below shows the number

of Problems, Follow-Up Questions, and ACE items that Karen assigned

together with the total number (in parentheses):

 

 

 

 

 

Table 5.2: Karen: Assigned number of exercises compared to total.
Number of # of Problems # of Follow- # of ACE
Invest. Assigned Up Assigned assigned and
and (total) and (total) (total)

1 3 (3) 14(1) 0 (10)

6 2 (2) 2 (2) 2 (6)

7 3413) 1 (l) 1 (4)

5 3 (4) 1 (4) 0 (8)

 

 

 

 

 

 

Table 5.3 below gives the percentage of Problems, Follow-Up Questions,
and ACE items Karen assigned from each investigation she taught in

Covering and Surrounding. Comparing the proportions shows that Karen

emphasized the Problems and Follow—Up Questions, but did not assign many

114

 

 

 

 

 

 

 

 

 

ACE items:
Table 5.3: Karen: Percentage of exercises assigned in each covered
investigation.

Number of % Problems % Follow-Up % ACE

‘Investigation Assigned Assigned Assigned
1 100% 100% 0%
6 100% 100% 33%
7 100% 100% 25%
5 75% 25% 0%

 

 

i . I . . i E . E E l] i 1 . .
Wists

Table 5.1 shows that Karen taught the investigations in Covering
and Surrounding in a different sequence than they appear in the unit.
Karen taught the four investigations she covered in the sequence 1, 6,
7, and 5. She taught the investigations in that order because this is
the sequence in which her students chose to study the investigations.
After teaching the first investigation, ”Measuring and Designing Rooms”,
Karen allowed her students to vote on which investigation to study next.
Her students voted to study Investigation 6 as the second investigation.
After Investigation 6, Karen's students voted for Investigation 7 and
then for Investigation 5 (in that order).

Karen decided to allow her students to choose the sequence of the
Covering and Surrounding investigations by majority vote because she
wanted to maintain their interest. Observing Karen's class at the
beginning of the second week of Covering and Surrounding, I was puzzled
to find that, having just finished Investigation 1, the students were
now working in Investigation 6. I asked Karen about her decision to

allow her class to study Investigation 6 having just finished

Investigation 1:

Karen:

Karen:

Karen:

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Now, the students are doing Investigation 6?

Yes, as the second investigation. And they chose number 6
because of all the shapes in there —- they want to get into
it. And especially the parallelograms, they really like
those.

When you decided to let the class choose what investigation
to do next, how were you thinking about that?

Well, we looked at all of them. And again, it has to do
with their interest. If they're not interested in it, I
didn't want to dwell on it -- you know, OK, we go through
Investigation 1, then Investigation 2, and so on -- that's
just like a textbook. So I threw it out to them and said
”What should we do next?” And their interest was to do
number 6. It was practically unanimous to do 6, although
someone wanted to do the beehive [i.e., Investigation 7].

So by letting them choose the investigations --
It seems to keep their interest up on the project, rather

than me saying to them that we're going to this one and then
the next one, and the next.

During her instruction of Investigation 6 in Covering and

Surrounding, Karen did not address the class as a whole. As in the

other investigations she taught in covering and Surrounding, Karen spent

most of the class time circulating around the room and working with

students individually or in small groups. On the third day of working on

Investigation 6, a group of three students, Matt, Trevor, and Aaron,

were eager to share something with Karen. They were very excited about

a discovery they had made while working on Problem 2 in Investigation 6

(CMP, 1992a, p. 44):

 

 

Problem.2: Orlando wonders whether there is a relationship between triangles
and parallelograms. Look back at the parallelogram in Problem 1. Can you find a
way to cut a parallelogram into 2 triangles that match each other? Be sure to
show how you know the 2 triangles are the same.

Now draw a triangle. Cut out 2 copies of your triangle. What figures can you
make by putting your triangles together so that an edge matches?

How does the area of the original triangle compare to the figures you made?

Describe as many ways to find the area of a triangle as you can. Draw any
diagrams that help explain your methods.

 

 

116

Matt, Trevor, and Aaron showed Karen two congruent right triangles which

were pushed together to form a rectangle:

Figure 5.3: Matt, Trevor, and Aaron's rectangle for proving triangle
area.

 

 

 

 

The three students told Karen that they had discovered why the area of a
triangle was ”one-half the base times the height.” Trevor said that his
mother had showed him the area formula for a triangle the night before,

and he and his two classmates were puzzled about ”why it worked.” Karen

listened closely as the students explained their reasoning to her:

Matt: Basically, you can make a rectangle from two equal
triangles. The base times the height gives you, like, the
area of the rectangle.

Trevor: But we want the area of the triangle, not the area of the
whole thing -- I mean the rectangle. So you just take half
of the base times the height because the triangle is half of
the rectangle.

Aaron: Yea. You can see that it's [indicating the triangle] half
of it [indicating the rectangle] so you only need half of
it, like half of the area of the rectangle.

Karen: This sounds very interesting -- good work! What I would
like each of you to do is sit down and write your own
explanation -- in writing and showing me this diagram [i.e.,

the rectangle constructed from two congruent triangles] --
of this discovery. Maybe tomorrow I'll have you share this
with the class.

Clearly excited at having their discovery praiseds, Matt, Trevor, and

 

SMatt, Trevor, and Aaron’s discovery is a special case of a more general justification
for the area formula for triangles. For any triangle (not just right triangles), two
congruent copies can be put together to form a parallelogram (e.g., a rectangle in the
case of right triangles as in Matt, Trevor, and Aaron's discovery). The area of the
parallelogram is the product of the base and height of the triangle since the
parallelogram can always be transformed into a rectangle with dimensions equal to the
base and height of the triangle. Since two copies of the triangle form the
parallelogram, the area of the triangle is one-half the product of its base and
height, i.e., half of the area of the parallelogram.

117

Aaron went back to their desks and began writing descriptions of the
reasoning they had just shared with Karen.

A few minutes after Matt, Trevor, and Aaron had shared their
discovery with Karen, I drew her aside and asked her about what she

thought the three boys were learning about problem solving:

AR: What did you think of Matt, Trevor, and Aaron's discovery?

Karen: Oh, very good! They're really into it and I liked the way
they figured that out about the area of a triangle. I'll

have them share it with the class tomorrow.

AR: Karen, what would you say they learned about problem solving
in making their discovery?

Karen: Well, really nothing significant.
AR: [pause] Hmm... What would you say they did learn about?
Karen: Well, they're doing hands—on. I mean, they're doing stuff

with the triangles and rectangles, cutting and moving them
around, and now they understand the area formula. I'd say
that they're doing hands-on work with the shapes rather than
problem solving.

Karen's decision to allow her students to determine the sequence
of the Covering and Surrounding investigations and her perception that
students were not learning problem solving are not consistent with the
teacher materials. The Covering and Surrounding teacher materials (CMP,
1992b) describe how the sequenced investigations presented in the unit
are intended to ”...build a deep understanding of what it means to
measure area and what it means to measure perimeter” (p. 1). While the
materials do not explicitly say that the investigations must be taught
in the presented sequence, the teacher edition does outline how the
developers expect students' understanding of perimeter and area to
develop through study of the sequenced investigations (see Chapter 4;
CMP, 1992b). The teacher materials also state that it is expected that

students will gain ”...insight and skill in ... problem solving” (p. 2).

118

In fact, the teacher edition explicitly addresses the kind of discovery
made by Matt, Trevor, and Aaron as a specific problem—solving strategy

that students should develop in Covering and Surrounding:

Cutting and rearranging parts of figures to see
relationships between kinds of figures, in particular,
parallelograms, triangles, rectangles, then devising
strategies for finding areas by using the relationships
observed (CMP, 1992b, p. 2).

All five of the problem-solving strategies in the teacher materials (see
Chapter 4) were highlighted by Karen in her copy of the teacher edition.
Although Karen did not perceive Matt, Trevor, and Aaron's discovery as
problem solving, from the perspective of the curriculum developers their
discovery is an excellent example of the kind of problem-solving skill
that the developers intend students to develop in Covering and
Surrounding. The triangle area discovery illustrates differences
between Karen's beliefs about problem-solving activity and the
conception of problem-solving activity underlying the Covering and
Surrounding'materials.

How Karen chose to sequence the unit investigations illustrates
the power of her beliefs about student learning. While Karen's decision
about sequencing investigations by letting students vote on which
investigations they find the most interesting or appealing is not
consistent with the Covering and Surrounding unit, it is consistent with
her beliefs. Recall that Karen places heavy emphasis on student
learning being directly tied to whether or not students are interested
in the lessons or activities. This belief completely overshadowed
teaching the investigations in the existing sequence intended by the

curriculum developers. Karen's concern with maintaining students'

119

interest was apparently not influenced by other factors, such as how the
enacted investigation sequence would affect student learning.

Karen's beliefs about problem solving also clearly overrode the
teacher materials in shaping her perception of what her students were
learning about problem solving in Covering and Surrounding. Karen’s
beliefs about problem solving are oriented around standard word problems
like in the textbook, not the non—standard or puzzle problems that
appear in Covering and Surrounding. Karen's perception of Matt, Trevor,
and Aaron's discovery as hands-on is consistent with her prior beliefs
about Covering and Surrounding being hands-on. Karen's decision about
sequencing investigations and her perceptions about students' (not)
learning about problem solving sharply illustrate the power of her
beliefs in shaping her use of Covering and Surrounding. The teacher
materials had little or no impact on these decisions and perceptions.

Karen used Covering and Surrounding to provide her students with
hands-on activities, but she did not feel that the unit was sufficient
to maintain their computational skills. However, Karen also taught
problems from Covering and Surrounding in ways that gave her students
hands-on opportunities to wrestle with perimeter and area.

Karen's teaching of Problem 1 in Investigation 1 is a
representative example of how she used Covering and Surrounding to give
her students open-ended learning experiences with perimeter and area.
Karen began Investigation 1 by having her class read the first part of
of the investigation that precedes Problem 1 (see Appendix D for

Investigation 1 in its entirety):

120

Figure 5.4: Mr. Dull’s room.

—-!==l——l==l——

 

l=l

indicates a window

 

 

 

 

—l:=i——_idoor '—

As an architect, Mr. Dull uses grid paper to design his
plans, like in his design of Mrs. Hide's room. Architects
also use models in their work. Physical or pictorial models
are used to represent and learn about other things that are
too big or small to study easily. In designing rooms it is
much easier to build models of rooms to make different
designs than to build all of the actual rooms.

Take 12 tiles, which will be used to model carpet sections,
and arrange them in the same design as Mr. Dull's (CMP,
1992a. Pp. 1-2).

As a student read the above excerpt from the unit aloud to the class,

Karen

circulated around the room giving each student about 15 l-inch

square tiles.

After distributing tiles to each student, Karen discussed the

perimeter and area of the room Mr. Really Dull designed for Mrs. Hide:

Karen:

Wendy:

Karen:

How many of your tiles does it take to cover Mr. Dull's
design? [As Karen asks this question all the students count
the tiles they have used to arrange in the shape of Mr.
Dull's room.]

Twelve!
And how many tile edges does it take to surround the room?

[As Karen asks this question every student begins quietly
counting around the edge of the room.]

Marcus:

Karen:

121

Fourteen!

OK. Now, according to the unit [Karen begins reading from
the text (CMP, 1992a, p. 2)], ’the number of square tiles
needed to cover the floor is a measure of the area and the
number of wall sections around the edge of the room is a
measure of the perimeter’ [emphasis in the text and in
Karen's recitation.] So, the area of Mr. Dull's room design
is 12 and the perimeter is 14.

At this point in the lesson, Karen paused as she noticed several

students still counting the number of tile edges surrounding the room.

After a momentary pause she reiterated the analogy of covering with

tiles to measure area and counting the tile edges surrounding the room

to measure perimeter:

Karen:

Wendy:

Karen:

Class:

Karen:

Trevor:
Karen:
Trevor:

Karen:

Okay. I want all of you to place your tiles exactly on top
of Mr. Dull's room design in your unit -- put the tiles
right on top of it. [pause while students place their tiles
over the above illustration of Mr. Dull's room design in
their unit] Now, what did you just do with your tiles?

We covered up the room with the tiles.

Right! You covered the room with the tiles. And the number
of tiles it takes to cover the room gives you what
measurement?

Area!

Good! And how many wall sections are there around the edge
of the room and what's that a measure of?

There are 14 wall sections.

Okay, so that’s a measure of what?

The perimeter!

Right! Now does everyone understand this? You cover with

tiles to find area and surround with wall sections to find
perimeter.

Karen’s intent in the above excerpt was to make sure that her

students understood the carpet tile and wall section analogy for area

and perimeter. To this point in the lesson, Karen's teaching somewhat

reflects her ”abstract” teaching mode -- she is trying to illustrate a

procedure.

Although Karen is not telling her students how to use a

122

formula, she is telling them how to use the tiles to measure perimeter
and area. But Karen’s teaching more sharply reflects the intentions of
the curriculum developers than her mode of teaching abstractly. For
example, Karen was clearly concerned that her students understand the
representation of carpet tiles and wall sections and how it connects to
measuring perimeter and area -— the developers also stress that
understanding the representation is very important (see Chapter 4).
Karen's emphasis on hands-on experiences with the tiles for her students
is also reflected in her teaching, especially in directing them to
physically cover Mr. Dull’s room with the tiles.

After the exchange with her class to clarify the analogy for area
and perimeter, Karen had a student distribute l-inch grid paper (the
squares on the grid paper and the manipulative tiles are both 1-inch
square) and told the class that Problems 1, 2, and 3, and Follow-Up
Question 1 on page 9 would be due at the end of the week (i.e., four
days later). Karen stressed to the class that each problem should be
read carefully and that they should trace their room designs made with
tiles on the grid paper for Problem 1. Karen's students went quickly to
work for the rest of the class time.

Problem 1 in Investigation 1 asks students, in the context of
designing a room, to make a rectangular figure with 1—inch tiles that

has an area of 12 and a perimeter of at least 14:

123

 

room.

a)

b)

 

Problem 1: Mrs. Hide likes the amount of floor space in Mr. Dull’s
design, but she wants more windows and a more interesting shape for her

Remember that a window or a door can go into each section of wall space.
Mrs. Hide tells you that she wants at least 14 sections of wall space,
including windows and doors, in the room you are designing.

drawing to show the location of the door and where each window is.
Write a paragraph to tell Mrs. Hide why your design is better than Mr.
Dull's

She asks you to help her out.

Use 12 square tiles to create a floor plan design for Mrs. Hide.

After you have designed the floor plan for the room make a

 

(CMP, 1992a,4p. 2).

 

At this point in the lesson, students began using the tiles to explore

perimeter and area. For example, as Shakaya was working on Problem 1

she had a question for Karen. She had designed this room using 12

tiles:

Figure 5.5:

Shakaya’s room.

 

 

 

 

 

 

 

 

 

 

Shakaya was confused about how to find the perimeter of her room.

Specifically, she wanted to know if she should count the perimeter of

the inner ”courtyard” as well as the distance around the figure. Karen

and Shakaya had this conversation:

Shakaya:
Karen:
Shakaya:
Karen:

Shakaya:

Should I count the (pause) perimeter around the inside?
Would this room satisfy the requirement of 12 tiles?

Yea, cause I did use (pause) yea, I used all 12 tiles.

Do you have 14 walls?

[Long pause as the student counts the exterior tile edges,

then pausing on, but not counting, the courtyard perimeter]
Yea. The tiles are the shape of the room.

124

Karen: What's this [indicates the courtyard] -- a hallway?
Shakaya: Yea. Sort of a hallway, would that be OK for the perimeter?
Karen: [Hesitates] I don't know. [long pause] This might be OK,

but would you want a room like that [indicating the
courtyard]?

Shakaya: Hmm. (pause) I don't know.

Karen: OK. [looks at clock and sees that the class period is almost
over] Well, let's look at that again tomorrow.

The next class period, by which time students had finished
designing their rooms, Karen had each member of the class present his or
her room to the rest of the class. Each student showed his or her room
design traced on grid paper and shared their answer for part b) of the
problem, i.e., their explanation of why their room was better than Mr.
Dull's. For example, Joe made his room with 12 tiles and a perimeter of
17, including three doors. Joe said that he included three doors
because ”The more doors I put in would be more easy to get from place to
place, and is safer in case of fire.” Shakaya had changed her design to
a rectangular shape without a courtyard. When I asked her why she had
eliminated the courtyard, she replied that the room looked ”more
complete” without the ”space (i.e., the courtyard) in it”. Shakaya’s
new room was a 2 x 6 rectangle. Presenting her room to the class, she

explained:

My room requires 12 squares of carpet to cover the floor and
16 sections for the wall. Since my room has more wall space
Mrs. Hide can hang more pictures than in Mr. Dull's room.

Karen complemented Shakaya on her explanation but did not ask her to
talk about why she had changed her room design by eliminating the
courtyard.

The responses given by Joe and Shakaya to Problem 1 illustrate how

students developed varying room designs and explanations which Karen

125

felt were legitimate. Karen's class collectively produced a wide range
of rooms and explanations for them. The students clearly enjoyed the
problem and Karen was very pleased at their ”high level of interest.”
Karen used Problem 1 in Investigation 1 to give her students a
hands-on activity with perimeter and area. Moreover, Karen’s use of the
problem is open-ended because students were given the freedom to create
a solution rather than trying to find the answer. This view is
reinforced by Karen's encouragement to her students to think about their
questions to resolve them for themselves. Karen's perception, however,
was not that her students were doing open-ended investigations with
perimeter and area concepts, but that they were doing hands-on activity
with the tiles to find perimeter and area of their rooms. Karen
considered the most important facet of the room design problem not to be
the open endedness of the activity, but the hands-on work with tiles to

correctly find perimeter and area:
AR: How would you say the students are doing with this problem?

Karen: Oh, very well! They have a high level of interest in this
problem and are really into it.

AR: What do you think is most important for them to get out of
this lesson?

Karen: To do the hands-on work with the tiles to find the perimeter
and area of their rooms. And they’re doing a great job --
all their rooms that I've seen have the correct perimeter

and area.
AR: How about the room design part of the problem and their
explanations -- how do you see that fitting with what

they're learning?

Karen: Well (pause), I want them to tell me why they designed their
rooms the way they did, but the most important thing for me
is for them to use those tiles. I think that the hands-on
of doing that is important. Yes I want them to design a
room they like, but I really want them to know how to find
the right perimeter and area of whatever room they come up
with.

126

Problem 1vin Investigation 1 in Covering and Surrounding unit did
influence Karen’s teaching to the extent that, in her terms, she did
hands-on instead of ”teaching abstractly”. Her use of the room design
problem also may have provided students with more opportunity to develop
conceptual understanding about perimeter and area than working with area
and perimeter formulas, which is consistent with the intentions of the
curriculum developers. But Karen's perception of what was most
important in this lesson, that students find the correct perimeter and
area of their room, is consistent with her prior emphasis on doing
hands-on activities to develop students' proficiency at getting right
answers.

Over the next two class periods Karen's students completed their
assignment in Investigation 1 and then studied Investigation 6, which
they worked on for a week. At the beginning of the third week of
Covering and Surrounding, students were working in Investigation 7. On
the first day of Investigation 7, students were spending part of their
class time working from the textbook. As well as the assignment from
Covering and Surrounding, students were also doing whole number
multiplication and division exercises. The exercises were ”naked
number” drill exercises (see Chapter 4) which do not appear in Covering
and Surrounding. I asked Karen why she had decided to have her students

working out of the textbook. She replied:

Skill maintenance. They're doing exercises in computing
with whole numbers and using the calculator to keep their
skills up. Some of the parents were happy to know [i.e., at
recent parent-teacher conferences] that I'm still holding
them to a textbook. The parents like the unit [i.e.,
Covering and Surrounding] but we don't want to be so far out
here with this project that they've [i.e., the students]
forgotten basic concepts.

127

Karen noted that while her students were learning about perimeter and
area in Covering and Surrounding, she felt that ”these kinds of
materials don't give students practice with computation skills.” Karen
said that her students need to maintain skills because they will be
tested on them when they take standardized tests later in the year.
Moreover, Karen said that she felt skill maintenance in mathematics was
as important as hands—on experiences. Karen continued to supplement
Covering and Surrounding, both during classwork and in homework
assignments, with computational practice from the textbook for the
duration of the unit.

Karen's remarks highlight an important facet of her use of
Covering and Surrounding. While she does use the unit to provide hands-
on activities for her students, as shown in her teaching of Problem 1
and her students' reasoning about triangle area, she does not believe
that the unit adequately helps her students maintain computational
skills. Karen's dual focus on Covering and Surrounding hands-on
activities and skill maintenance did not emerge until halfway through
the unit, but it remained constant after beginning Investigation 7.

Recall that Karen’s views and beliefs about mathematics and
problem solving, and her views about problem—solving activity, are
oriented around the textbook. Karen makes extensive use of the
textbook, including using it to address problem solving. Noting how
Karen's practice of teaching mathematics is linked to the text helps to
understand why she began bringing it back into her teaching after two
weeks of covering and Surrounding. In particular, Karen's use of the
textbook along with Covering and Surrounding was more than just a

relapse into old habits. Even before she began teaching the unit, in a

128

conversation during a summer interview, Karen revealed that using the

textbook along with Covering and Surrounding was on her mind:

Karen: Ah, will the Covering and Surrounding unit be related to the
textbook they're [i.e., the students] using at all?

AR: If you feel that need to bring the textbook in at some
point, or if there is something that the unit doesn’t do
very well, feel free.

Karen: Well, that's what I meant, because when I'm teaching Factors
and.Mu1tiples [Fitzgerald et al., 1986], sometimes I have to
go back to the textbook and lay the groundwork for what I'm
about to introduce to them.

AR: This is something that I'm very interested in.

Karen: Good, and I sometimes use the book as homework and they get
back to me to make sure that they understand what‘s there.

Karen's above comments from the summer prior to teaching Covering
and Surrounding emphasize her orientation around the textbook in her
mathematics teaching. Karen views the material in the textbook,
particularly on skill maintenance, as important for her students. Since
she does not believe Covering and Surrounding addresses skill
maintenance like the textbook does, she brought the textbook into her
use of the unit. For Karen, the textbook helps students maintain their
computational skills while the Covering and Surrounding unit is hands-
on. Since Karen believes that students need hands-on experience to
understand mathematical concepts like perimeter and area, and she also
believes that it is important for her students to maintain computational
skills, it is perfectly reasonable for her to use the textbook and
Covering and Surrounding simultaneously in her mathematics classroom.

Recall that Karen's use of her textbook teacher edition and the
Factors and Moltiples unit teacher materials was mainly to obtain

answers to check students' work. Karen's predominant use of the

129

Covering and Surrounding teacher materials was similar to her use of the
textbook teacher edition and Factors and MUltiples. Karen used the
Covering and Surrounding teacher materials mainly to obtain answers to
problems, although her teaching sometimes reflected the spirit of the
materials in asking her students to explain their reasoning.

The intention of the developers of Covering and Surrounding is
that the teacher materials are not just a source of answers to
problems, but are a resource for teachers to develop and refine modes of
teaching practice that are consistent with the problem-solving
orientation of the unit (see Chapter 4). Suggestions for teaching are
presented frequently in the Covering and Surrounding teacher materials
and are intended to provide the teacher with opportunities to plan for
or rethink how a lesson might be taught or organized.

Karen, in general, did not make use of the suggestions for
teaching provided in the teacher's edition of Covering and Surrounding
in her classroom practice. Although the unit provides suggestions for
how to represent concepts and processes and how to question students to
further develop their understandings of concepts, Karen did not use
these ideas in her teaching. For example, the Covering and Surrounding
teacher edition suggests to the teacher that she might want to make a
set of illustrations to make the wall section and carpet tiles analogy
for perimeter and area, respectively, more explicit to students (CMP,

1992b, pp. 23-24):

The story of designing rooms in the student text can be used
to launch the investigation. It is important that students
get a clear picture of the relationship between the physical
tiles representing squares of carpet and the prefabricated
wall sections that are the same width as the edge of a
carpet tile. The prefabricated sections can each contain
either a window or a door, but may also be blank wallspace.

130

You may want to prepare a few cut outs from cardboard with a
door or a window drawn on to show students a floor plan:

Figure 5.6: Diagrams of door and window wall sections.

 

 

 

 

 

 

 

 

 

 

door window

l-inch square (tile size) wall sections

In her teaching of this investigation, as discussed earlier, Karen was
clearly concerned that her students understand the analogy but she did
not make use of any such illustrations in her teaching.

Another example is Karen's interaction with Matt, Trevor, and
Aaron about their discovery justifying the area formula for triangles in
Investigation 6. Recall that Karen listened carefully to their
explanation, complimented them, and then had them return to their seats
to express their reasoning in writing. Karen did not ask Matt, Trevor,
or Aaron probing questions about their discovery. However, in
Investigation 6, the Covering and Surrounding teacher edition (CMP,

1992b) suggests that:

As students work through the problems of finding the
relationships between rectangles and parallelograms and
parallelograms and triangles, you might ask them questions
about how they are dividing up their parallelograms or
rectangles. For example, when a student discovers that when
she cuts a parallelogram along one of the diagonals two
congruent triangles are formed, you might ask her how the
area of the triangles relates to the area of the original
parallelogram. You might ask her further, then, to think
about how the area of a triangle could be expressed in terms
of the area of the original parallelogram (p. 64).

While the teacher edition cites instructional events like Matt,

Trevor, and Aaron's discovery as an opportunity to help students develop

131

their understandings of perimeter, area, and relationships between
shapes, Karen did not pursue this with the three boys in-depth. She did
not probe their understanding, for example, to help them extend their
conjecture to parallelograms that are not rectangles and thus triangles
that are not right triangles. This does not mean that Karen's use of
the triangle area problem with Matt, Trevor, and Aaron was ineffective.
She did ask them, for example, for a detailed explanation of their
conjecture both in words and in writing. To this extent, Karen used the
problem in the spirit suggested by the Covering and Surrounding teacher
edition. But counter to the suggestions of the teacher materials, Karen
did not take this opportunity to push her students’ understanding toward
a more general statement about general relationships between areas of
triangles and parallelograms, not just right triangles and rectangles.

One explanation for Karen's omission of the wall section
illustrations, or why she did not question her students' discovery in
Investigation 6 further, is that she was not aware of the suggestions in
the teacher materials. This, however, was not the case. Throughout her
teaching of Covering and Surrounding, Karen read the teacher materials
carefully. She remarked to me during Investigation 6 that ”This
material is new for me, so I've been putting in some extra time with the
teacher edition.” That Karen was reviewing the teacher notes carefully
was evident from looking at her copy of the teacher materials in which
she highlighted, as she said, ”important points and answers.” For

example, Karen had highlighted in her copy of the teacher edition

132

portions of the above passage from Investigation 6 in the teacher

materialsé:

As students work through the problems of finding the
relationships between rectangles and parallelograms and

parallelograms and triangles, you might eek_them_oneetione
WWW
rectangles- For example. Wanner:
WWW

. .
WW: 1 ! . 1 J !] E l . . 1
petellelootom. You might ask her further, then, to think

about how W
W- These kinds of

questions help students develop their own problem-solving
skills and understanding of perimeter and area ... (p. 64).

So Karen was aware of these suggestions for teaching Investigation 6 and
she emphasized them in her own copy of the teacher materials. Another
explanation for why Karen did not make use of suggestions in the teacher
materials emerges from a conversation with Karen after her interaction
with Matt, Trevor, and Aaron.

After class, I asked Karen about Matt, Trevor, and Aaron's
explanation for triangle area and if she thought it might be helpful to
ask them more questions about their discovery. She replied, ”Not really
-- they know that the area of a triangle is half base times height.”
When I pressed Karen further about asking the three boys about what
problem solving skills they used to make their discovery and relating
parallelograms and other kinds of triangles besides right triangles she
said

I don't know about doing any problem solving yet What I
want them to know is one-half base times height -- that’s
(i.e., the formula) what they'll use to do problem solving
because now they have the formula to use to solve problems.

 

6The underlined portions of the passage indicate what Karen highlighted in her copy of
the teacher edition (CMP, 1992b) in yellow highliner.

133

As Karen's response indicates, she didn’t ask her students about problem
solving because she didn’t perceive that problem solving was happening.
From her perspective, the point of the problem was for students to learn
one-half base times height as a formula to use for solving other
problems, not developing problem solving skills or a more general
relationship between triangles and parallelograms. Karen did not pursue
suggestions from the Covering and Surrounding teacher materials because
the materials suggested pursuing goals that she did not have for her
students. Knowing the area formula for triangles was what Karen wanted
her students to know. When Matt, Trevor, and Aaron demonstrated this,
Karen asked them to explain their reasoning but did not dig deeper
because the formula had been discovered.

During the same conversation with Karen after class, I asked her
about the wall section analogy for perimeter and area and the
illustrations provided in the teacher edition. Karen said that she
didn't feel the wall section illustrations were ”really necessary”
because her students ”got the idea of using the tiles on the rooms” with
little difficulty. So Karen was also aware of the Covering and
Surrounding teacher edition suggestion about using the illustrations of
the wall sections (see Appendix E) but chose not to use it. Karen's
decision not to use the wall section illustrations is congruent with the
suggestion of the teacher materials which suggest that the illustrations
be used if students are having difficulty understanding the analogy (see
Appendix E).

Karen did use the Covering and Surrounding teacher materials to
see if her students had found the correct areas and perimeters of

figures. For example, Problem 1 in Investigation 6 presents students

134

with different parallelograms on a grid and students are to develop and
describe a strategy to find the area of each parallelogram. An example
is given below:

Figure 5.7: Parallelogram on a grid (CMP, 1992b, p. 43).

L

 

 

 

 

 

 

 

 

 

 

In the teacher materials (CMP, 1992b), Karen had marked the answers to
the parallelogram area problems and used them in checking her students’
work. As I observed from looking over students' corrected papers on the
above problem, their answers were graded by Karen as right or wrong,
depending on whether or not their answer matched the answers in the
Covering and Surrounding teacher materials. Karen did not comment on
any of the students’ papers about their strategies for finding area.
Overall, Karen's use of the Covering and Surrounding teacher
materials was similar to her prior use of other teacher materials (e.g.,
her textbook teacher edition, Factors and MUltiples) -- she used the
Covering and Surrounding teacher edition primarily as a source of
answers to problems. However, she also read the teacher edition more
carefully than just to obtain answers, even highlighting some passages.
But while Karen was aware of suggestions for teaching provided in the
teacher edition, these suggestions usually did not appear in her
practice. Sometimes this was the result of Karen filtering the

suggestions through her own beliefs (e.g., the triangle area problem)

135

and at other times was based on decisions she made that were consistent
with the teacher materials (e.g., the wall section analogy).

Karen felt that her teaching using covering and Surrounding was
different from her usual mode of practice -— she believed that she had
taught perimeter and area doing hands-on activities instead of teaching
abstractly. Contrasted with her usual mode of teaching perimeter and
area ”abstractly”, Karen did teach differently while using Covering and
Surrounding. However, while she felt that the unit had pushed her to
teach differently, Karen expressed and exhibited a tendency to teach
abstractly after teaching Covering and Surrounding.

During the fourth and final week of teaching Covering and
Surrounding, Karen's students studied Investigation 5 which they had
chosen as the last investigation. Investigation 5 explores the
relationship between perimeter and area of rectangular figures when
perimeter is held constant and area is varied. Karen said that she was
glad the class picked Investigation 5 because it would give them more
hands-on work with the tiles. For example, Karen assigned her class

these two Problems from Investigation 5 (CMP, 1992a, p. 37):

 

Problem 2: What's the Smallest? Construct the original pentomino
again.

 

 

 

 

 

What is the smallest number of tiles you can add and have a perimeter of
18 units? Explain and illustrate your answer.

 

 

 

136

 

Problem 3: What's the Largest? Construct the original pentomino
again.

 

 

 

 

 

What is the largest number of tiles you can add and have a perimeter of
18 units? Explain and illustrate your answer.

 

 

 

Students worked through Investigation 5, including the two problems
above, at their own pace.

During this final week of Covering and Surrounding, I asked Karen
to share her thoughts and impressions about teaching the unit. In
summarizing her experience teaching Covering and Surrounding, Karen
said:

It forced me to use tiles, and I'm not a tile person -- I
think abstractly and I memorize formulas and things ... it
made me use the tiles and grids to give kids the hands-on
they need to understand. I would actually prefer to use
formulas and teach abstractly, but I'm doing more hands-on.
I'm trying to change but it's tough!

In her above reflection on teaching Covering and Surrounding, Karen
provided confirming evidence about what she believed to be perhaps the
most important feature of the unit for her students -- doing hands-on
activities. Moreover, while Karen felt that teaching Covering and
Surrounding had pushed her to teach differently, she expressed a
preference for teaching abstractly.

Karen's remarks are especially interesting given her use of
Covering and Surrounding. Karen believes that she should do more hands-
on activities rather than teach abstractly. The fact that she taught

Covering and Surrounding for a month is evidence of the strength of this

137

belief. Yet, Karen also claims she would rather teach abstractly, and
this claim was reinforced by a lesson on circumference of circles which
she taught after most of the class had finished Investigation 5.
Beginning with this lesson, Karen resumed working from the textbook
while students who were not quite finished with Investigation 5
completed it. Karen's lesson was on circumference of circles and was

taught on 12/2/92. The following excerpt is from the beginning of the

lesson:

Karen: [addressing the class] Do you know the reason why we're
using u? [some confused looks from the class] When you're
looking for circumference, according to your author [i.e.,
Abbott & Wells, 1985], your supposed to use n, to the
measurement of --

Class: Three point one-four.

Karen: Or --

Class: Twenty-two sevenths!

Karen: OK. Now when you start this off, they usually will give you
a formula. And you will see both of the formulas at the
bottom of page 344. Can anybody remember why there are two
different formulas?

Sheila: There's one for the diameter and one for the radius.

Karen: [to the class] Hmm. He said that there's one for the
diameter and one for the radius. Everyone look at it and
see if you agree. Do you think that's what that is? [nods
from the class] Well, what's the C for then?

Class: Circumference!

Karen: Right. So we're saying that if the diameter is given, which
formula do you use? (pause) Give me the formula if the
diameter is given in the problem. Which formula would you
use?

Carlos: Two r?

Karen: If the diameter is given? Look again now on 344, don't go

to 345 yet, that's a whole different story there -- stay on
344. Suppose a problem is given and they said ”The diameter
of this circle is 14”. Which of these formulas do you think
you would use? (pause) There are two formulas -- they're
right down there at the bottom of 344 where it says ”step 1
and step 2”. Which formula would you use?

Carlos:

Karen:

Darren:
Karen:
Darren:

Karen:

Becky:

Karen:

Melissa:

Karen:

Melissa:

Karen:

Melissa:

Karen:

138

Step one?

Give me the formula. Give me the whole formula -- read what
you see.

C = Ed.
OK. Now what does nd mean to you?
Three point one-four diameters.

Three point one-four diameters? I disagree with you. What
is nd? There's something implied there but there's
something left out.

Three point one-four times the diameter!

Alright! So when the diameter is given you must understand
that that nis represented by three point one-four. Now, in
some cases, it gives you a reason at the top of page 344 as
to when you're to use the fractional equivalent. Now you
would use this one for nlwrites 3.14 on the chalk board] if
you're given a whole number or a decimal. But suppose you
were given that the radius is, say, 2 1/2. What would you
have to do with this is?

You would have to multiply that by twice and then multiply
it again by three point one-four.

Alright, I wouldn't use 3.14 with this one. I could if I
changed 2 1/2 over to a decimal, but if it's like this I'd
use the other one [writes 22/7 on the chalkboard]. First
I'd multiply 2 1/2 by 2 because you need two what?

Two radiuses make a diameter!

Alright. So you have to have two radiuses to equal one
diameter. Now if the diameter is given such that it's say,
26, I would use this [points to 3.14] C = Rd, this being
three point one-four times whatever number is given. You
have to know which one [i.e., which formula] to use. Now
suppose you're given a radius of, say, 6. Do you know what
to do? What would you're formula be if you're just given
the radius?

C = an.
C = 2nr [writes on the board]. Which means that you take

this two, multiply it by 6, and then you have to still
multiply by 3.14.

The above transcript provides a good example of what Karen means

when she describes herself as an ”abstract” teacher -- she focused on

teaching her students formulas and how to use them. The formulas Karen

139

taught were not conceptually motivated nor concretely illustrated, they
were simply presented to the students and then demonstrated. But the
lesson also illustrates how different teaching Covering and Surrounding
was from Karen’s ”abstract” teaching. Compare the circle circumference
lesson above with, for example, Karen's use of the room design problem
with Shakaya or her interaction with Matt, Trevor, and Aaron about
triangle area. Comparing these three snapshots of Karen’s teaching
yields striking differences between her ”abstract” mode of teaching and
her ”hands-on” mode using Covering and Surrounding.

What is striking about Karen's use of covering and Surrounding
when compared to the circumference of circle lesson above is the
coexistence of her ”hands—on” and ”abstract” modes of teaching perimeter
and area. For example, the Covering and Surrounding unit does contain
an investigation about the area and perimeter of circles (see Chapter
4). Karen's ”abstract” lesson on circumference of circles, rather than
teaching the ”hands-on” investigation from Covering and Surrounding,
coupled with her remarks about preferring to teach abstractly
illustrates the strength of her beliefs about teaching abstractly. Her
use of Covering and Surrounding simultaneously exhibits her belief that
teaching ”hands-on” is important. Karen's comment that ”I'm trying to
change but it's tough!” may reflect a struggle to move away from
”teaching abstractly” to ”hands-on” teaching, or perhaps that she is
trying to somehow balance these two modes of practice.

Summary

Karen's use of covering and Surrounding reflected her views and

beliefs. Her views and beliefs were predominant in shaping her teaching

and overshadowed the intended curriculum when her views and beliefs did

140

not correspond with the intended curriculum. Karen's sequencing of
investigations to maintain her students' interest, her concerns about
skill maintenance and computation, and her perception that her students
were not learning about problem solving illustrate this. These beliefs
were constant throughout Karen's use of Covering and Surrounding and did
not seem to be challenged or change while teaching the unit.

Karen's mode of teaching perimeter and area while using Covering ‘
and Surrounding did change. Karen used Covering and Surrounding to
teach perimeter and area using hands-on activities rather than teaching
abstractly. Congruent with the intentions of the curriculum developers,
Karen had her students using tiles, grids, and cutting up and
rearranging shapes to help them solve perimeter and area problems.
Karen's use of the Covering and Surrounding teacher materials was
similar to her use of prior materials, but her teaching did reflect the
spirit of the covering and Surrounding materials in terms of having
students explain their reasoning and allowing students to explore
problems. Karen's perceptions about what her students were learning in
the unit, however (e.g., about problem solving), contrast sharply with
the intentions the curriculum developers have for students' learning.
Karen perceived her own experience teaching Covering and Surrounding as
pressing her to teach differently, although she still expressed views
and taught perimeter and area in ways that matched her abstract mode of

teaching.

CHAPTER 6

THE CASE OF BETTY WALKER

A Profile of Betty Walker

As Betty turned on the overhead projector to begin class, her

students quickly found their seats and opened their math journals.

Everyone in the class was writing the date, 11/5/92, on a clean sheet of

paper in their spiral-bound notebooks as Betty drew their attention to

the problem on the overhead projector:

Betty:

Stuartl:

Betty:

Andrea:

Betty:

Dwayne:

Betty:

Alright everyone, now listen closely. The problem we’re
going to start with today is to [reading the problem on the
overhead] ’Find the prime factorization of the number
represented by the year of your birth.’ Now, not everyone
is going to have been born in the same year. Who was born
in 1981? [most of the class raises hands] Who was not born
in 1981? [the remainder of the class raises their hands].
OK, so not everyone was born in the same year so you may be
working with different numbers in this problem.

What’s the prime factorization? Should we -—

Remember the clue yesterday? The end of the word sounds
like what? Let’s see the hands of who can remember what
factorization sounds like. Does anyone remember the trick I
told you yesterday to remember? [About half of the class
raises their hands]

Factorization sounds like multiplioation!

Right! Factorization sounds like multiniioation, so that
means we’re going to have a multiplieation problem. Now,
how do we find a prime factorization -- what were we doing
yesterday?

Making factor trees of the prime numbers that divide into
the number.

Right! Very good! So what everyone needs to do is find the
prime factorization, by making a prime factor tree, of their
birth year.

 

1All students' names are pseudonyms. In choosing pseudonyms, I have attempted to
represent each student’s ethnicity to provide a sense of the diversity of Betty’s

class.

141

142

Directing her students to work with a partner if they would like, Betty
began to circulate around the room as her students grouped themselves
and started working on the problem. Betty’s students were obviously
used to working together. As Betty moved around the room she answered

students’ questions about getting started on the problem:

Stephan: How do we know if our birthday, I mean our birth year, is
prime or not?

Betty: Stephan, use your calculator and what you know about prime
factors to figure that out. Remember? Start out by finding
2

the prime whose square is bigger than your birth year .
Does that help?

Stephan: Oh yea. Okay, I think I got it now [presses some keys on
his calculator]. So, like, 47 times 47 is (pause) 2209, so
I check the numbers --

Betty: Not the numbers, the ptime numbers.

Stephan: Yea, I mean the prime numbers, less than 47?

Betty: That’s it —- got it? [Stephan nods] Compare your strategy
with someone else and see if they’re finding the prime
factorization like you are. [Betty continues circulating].

After the class had worked on the problem for about 15 minutes,
Betty called for everyone’s attention back up at the overhead. Betty
asked if anyone wanted to share their solution. Raising her hand,
Angela said, ”I think that the prime factorization of 1981 is 7 times
283.” There were nods of agreement from about one-third of the class.
Writing 1981 = 7 x 283 on the overhead, Betty asked the class, ”What do
other people think? How do we know that this is the prime
factorization?” Stephan raised his hand and says, ”Well, cause 7 x 283

is 1981 and both of those numbers are primes, so they work.” Betty

 

2Prior to this lesson, Betty’s students had completed an activity in finding prime
factorizations of numbers from Factors and.Multiples. One of the intentions of the
activity was for students to understand that to find the prime factorization of a
given number, only the primes whose squares are smaller than the given number need to
be tested (see Fitzgerald et. al., 1986).

143

replied, ”Does everyone agree with that? Is 283 really prime?” Some
students punch buttons on their calculators and the rest of the class
nods. Betty then said, ”Since both of these numbers are prime, yes,
this has to be the prime factorization, because, remember, a number only
has one prime factorization.”

The preceding vignette is from a lesson on factors, multiples, and
primes that Betty taught four days before beginning to teach Covering
and Surrounding. Betty often poses problems to her students at the
beginning of the class period, which they then work on for the remainder
of class time. As her students work on a problem, Betty moves briskly
about the classroom, answering students’ questions, challenging them to
come-up with another solution or strategy, and encouraging them to
compare answers and ideas with one another.

An experienced and dedicated middle school mathematics teacher,
Betty brings tremendous enthusiasm and energy to her sixth-grade
mathematics instruction. Betty has taught middle school for 20 years.
She taught seventh and eighth grade as a substitute teacher for 4 years,
and has been teaching sixth-grade full-time for the last 16 years.
Although she has also taught English, social studies, and reading during
her career, Betty especially loves to teach mathematics. Betty feels
that while students seem to struggle more with mathematics than any
other subject, helping students understand and enjoy mathematics is
challenging and rewarding.

Betty traces her own interest in mathematics to her father who is
a mechanical engineer. Originally majoring in mechanical engineering in
college, she switched her major to elementary education by the end of

her first year. Difficulty with chemistry, as well as with adjustment

144

being a woman in the engineering school, and an increasing interest in
teaching, were the factors that eventually led Betty to switch her
career goal to teaching. Betty completed her master’s degree in
education in 1976.

Betty teaches in a middle school that enrolls about 550 students
in grades six, seven, and eight. The class periods at Betty’s school
are 41 minutes long. The short class periods are an ongoing concern for
Betty, who emphasized on a number of occasions that 41 minutes ”just
isn’t enough time to do mathematics!” Betty teaches mathematics two
class periods per day and she taught Covering and Surrounding in both of
her mathematics classes.

Betty’s school district does not have a tracking program for
mathematics in grade six. In grades seven through twelve, however, the
district does track students in mathematics. Betty describes her
students as a ”heterogeneous group” and does not believe that tracking
is beneficial to students’ learning.

Betty and the Domains of the Framework

During the summer prior to teaching Covering and Surrounding, I
interviewed Betty to learn about her views, knowledge, and beliefs
within the four domains of the analytic framework developed in Chapter 2
(see Appendices A, B, and C for interview protocol and items). Some
observational data of Betty prior to teaching Covering and Surrounding
is also used in this section. Important landmarks which help define the
terrain of Betty’s views, knowledge, and beliefs are explored in detail
in this section with respect to each of the domains of the framework.
Key conceptions within Betty’s views, knowledge, and beliefs about

mathematics and problem solving, problem-solving activity in classrooms,

145

subject-matter knowledge, and perceptions and beliefs about student
learning are identified.
1!. i E 1' E E] | H ll . 1 E l] E J .

Betty views problem solving as central to teaching and learning

mathematics and computation as only a minor part of school mathematics:

We try to solve problems every day! That’s what mathematics
is all about! Why bother to learn all this stuff if you're
not going to use it to solve problems? It’s a waste of time
just to do a bunch of drill and computation, other than the
fact that, I guess, you learn, you know, your times tables
that way and things like that, but what good do the times
tables do you if you never use them? We really do try to do
lots of problem solving, and I think, gosh, how often do we
do it? I'm trying to think -- I mean, you’re asking them
questions all the time, asking them to solve problems!

Betty believes that she teaches problem solving as an integral part of
her practice and that problem solving is routine in her mathematics
teaching.

Betty describes herself as trying to teach mathematics so that her
students don’t just memorize formulas but understand how to solve
problems. Betty says that wants her students to learn mathematics so

that

mathematics is interesting. It’s intriguing, and there's
more there than just a right answer. And it’s OK to not
always use paper and pencil in mathematics. A lot of
exploration and experimenting is done without a paper and
pencil. To me, you don’t need to memorize formulas and
practice them. Mathematics is exploratory and creative --
it’s not just a bag of tricks and rules! If you just
memorize formulas then you really don’t even know when to
use them or what they mean - they’re just there! This way
[i.e., exploring and experimenting]. if they don’t remember
what the formula is, they can figure it out.

Betty believes that learning mathematics in school should include
opportunities for students to explore and experiment with mathematics

and to formulate and test their own ideas.

146

The snapshot of Betty’s lesson on factors, multiples, and primes
at the beginning of this chapter, when compared with her expressed
beliefs about mathematics above, is somewhat paradoxical. While Betty
believes that mathematics is not ”just a bag of tricks and rules”, she
did give students a mnemonic device in this lesson. The mnemonic was
not connected to the concepts of factors or multiples but was intended
to help her students remember that factorization means a multioiioation
problem. Still, this does not mean that she teaches rulebound
mathematics despite her stated beliefs. Rather, Betty's own views and
beliefs about mathematics may be enacted to varying degrees in her
mathematics teaching. While Betty’s expressed views and beliefs about
mathematics and problem solving are oriented around learning mathematics
as ”exploratory and creative”, her practice includes problem solving as
solving both standard word problems and non-standard, more open-ended
problems.

Betty’s use of the mnemonic in the birth year problem illustrates
problem solving as solving standard word problems -- Betty gave her
students the problem and then helped them translate it into an operation
(i.e., check for divisibility by all primes whose square is less than
the birth year) to solve it (see Chapter 4). In contrast, two days
after the birth year problem, Betty assigned this ”mystery number”

problem to the class:

I’m thinking of a number that’s between 2000 and 3000. This
number has the prime factors 2, 3, 7, and 17. What are all
the possibilities that this number could be?

Betty supplied her students with no hints or explicit guidance as she
had in the birth year problem. She assigned the problem as part of a

homework assignment and instructed her students to work on the problem

147

independently and record their strategy for solving it in their math
journals along with their solution. The mystery problem was, for
Betty’s students, what Hembree (1992) would classify as a ”non-standard
(process or open-search) problem” which ”encourage(s) the use of
flexible methods; the solver possesses no routine procedure for finding
an answer” (p. 249). This kind of ”open-search” problem is more
congruent with Betty’s beliefs about mathematics being exploratory and
creative.

Betty’s expressed views and beliefs about mathematics and problem
solving place emphasis on representing mathematics as a domain of
inquiry that is more than ”just a bag of tricks.” However, in Betty’s
teaching, she represents problem solving as both implementing rules and
procedures to solve standard word problems (e.g., the birth year
problem) and as solving more open-ended, non—standard problems (e.g.,
the mystery number problem). Therefore, while Betty’s expressed views
and beliefs about mathematics center on teaching mathematics and problem
solving as exploratory, creative, and open-ended, her practice includes
a range of representations. These representations vary from mnemonics
and rules to more open-ended problems.

Betty conceptualizes problem-solving activity in her classroom as
”exploring and experimenting.” By exploring and experimenting, Betty
means investigating problems unencumbered by rigid guidelines or rules
to learn about mathematical concepts and processes. Exploring and
experimenting involves formulating strategies and approaches to solve

problems and also making sense of and interpreting data:

148

AR: What kinds of activities would you say you and your students
do when you’re working on problem solving?

Betty: They may be experimenting, measuring -- they may be
manipulating things such as tiles and stuff like that to try
to find out, say, how do you find the area of this table,
what’s one way they could do it? One group might decide to
use tiles, another group might decide to use a ruler, you
know, depending on what they’re doing. (pause) They do a
lot of work with graphs, reporting the data -- so you found
this data, what form can you put it in, and then once you do
that what does it tell you? They do some writing, trying to
explain things, like corresponding with other groups of
people to help them learn how to do something. I’ve sent
them home to do problem solving -- for example, finding out
the height of the people in their family. First of all they
had to figure out what kinds of units they were going to use
and then what kinds of tools they would need.

Betty’s conception of problem-solving activity as exploring and
experimenting is further illustrated by her comments about her students’

work on the mystery number problem discussed earlier:

I’m thinking of a number that's between 2000 and 3000. This
number has the prime factors 2, 3, 7, and 17. What are all
the possibilities that this number could be?

Betty said that in solving the mystery number problem

The students experimented with different approaches. Like
one kid, he tried to list all the numbers between 2000 and
3000 divisible by 17 and others tried to work backwards.
Like, another student, she said that 2 x 3 x 7 x 17 = 714 so
you can’t use the 7 or the 17 again because the number would
be too big. They were trying different things and
explaining their approaches -- that’s what I mean by
exploring -- they come up with a strategy and then see if
they can solve the problem with it. Doing stuff like that
is how kids should be learning math!

Betty’s comments emphasize that she believes exploring and experimenting
is open-ended, allowing students to develop problem-solving processes as
well as understandings about mathematical concepts.

While Betty characterizes problem-solving activity as exploring
and experimenting, she also describes classroom activity that was not as
open-ended as problem solving. For example, Betty described the mystery

number problem as an example of problem solving and she also referred to

149

the lesson on the birth year problem as problem solving. However, as
the vignette at the beginning of this chapter illustrates, the birth
year problem was less open-ended than the mystery number problem. In
the birth year problem, students were instructed to solve the problem
using factor trees and were reminded that factoring involves
multiplying. The mystery number problem, by comparison, is more open-
ended. The mystery number problem leaves students to explore and
experiment more broadly than in the birth year problem where firm
guidelines were established at the outset (e.g., make a factor tree).

Comparison between the birth year and mystery number problems does
not imply that the latter was problem solving and the former was not.
Rather, the contrast shows that while Betty describes problem-solving
activity in terms of exploring and experimenting, her teaching practice
includes more structured experiences with problems that she
characterizes as problem solving. As with her views and beliefs about
mathematics and problem solving, Betty’s teaching demonstrates a wider
range of conceptions of problem-solving activity than her expressed
views and beliefs. Along with problem-solving activity in Betty's
classroom including non-standard (process or open-search) problems like
the mystery number problem, standard word problems like the birth year
problem play a role also. As the birth year and mystery number problems
illustrate, some of the problems Betty provides for her students involve
more ”exploring and experimenting” than others. However, this variation
in practice is not completely represented in her expressed views and

beliefs about problem-solving activity.

150

SMDiSCL;MsLLSI_KnQEl§dQ§

Betty’s subject-matter knowledge of perimeter and area is rich and
connected. She is able to use both symbolic representations and
conceptual relationships to understand and work with students’ non-
standard algorithms and conjectures. For example, Betty's subject-
matter knowledge helped her to unpack a non-standard technique for
finding perimeter of rectangles. This was demonstrated in her response
to a question about a student’s non—standard way of calculating
perimeters of rectangles prior to teaching Covering and Surrounding (see
Appendix B). In making sense of the student’s work from two examples
provided, Betty derived an expression that was a generalization of the
student’s method for finding perimeter. Below is one of the two
examples of student work and Betty’s comments as she constructed a
mathematical explanation of what the student was doing:

Figure 6.1: Student’s work on finding perimeter of a rectangle:

 

 

 

 

9
P = 2 x(18/9 +18/2): 2 X (36/18 +162/18)

= 2 x198/18 =198/9 = 22

Well, 2 times the quantity -- and this would be A divided by
b, the base, A being area -- plus area again [begins writing
a formula out on paper] divided by the height and we can get
rid of the times sign [continues to write and produces the
formula P = 2 x (A/b + A/h) where A = b x h. Betty then
writes the formula as P = 2[(Ah + Ab)/bh]. Well, that would
be the rule. That’s good enough!

151

While Betty did not prove that the student's rule was valid3, she was
able to obtain a general expression for how the student was computing
perimeter. In this case, Betty's subject-matter knowledge enabled her
to work flexibly with symbols to uncover how the student was most likely
thinking about finding perimeter of rectangles.

Directly related to her subject-matter knowledge, Betty
demonstrated a conceptual understanding of relationships between units
for measuring area. Moreover, she drew on pedagogical content knowledge
to craft a concrete learning experience to address a student’s
understanding of relationships between square units. Below is an
interview excerpt (see Appendix B) where Betty addressed a student’s
conjecture about converting from square yards to square feet in
measuring the area of a triangle:

Figure 6.2: Student’s conjecture about square units for area:

6 yards

 

8 yards

A = 1/2 x B x H = 1/2 x 6 x 8 = 1/2 x 48 = 24
square yards

3 feet = 1 yard so Area = 24 x 3 = 72 square
feet

AR: Now, suppose that during your instruction on perimeter and
area one of your students raises his hand and is very
excited. He says that he has figured out how to convert
between different units used to measure area. And he shows
you his solution to this area problem [referring to Item 2
above] on this triangle and that he came up with 24 square
yards. Now, he says that since there are 3 feet in a yard
he can convert the area from square yards to square feet
How would you respond to this student?

 

3Replacing area in Betty’s expression with bh yields: P = 2 x [(Ah + Ab)/bh] = 2 x
[(bhh + bhb)/bh] = 2 x [bh(h + b)/bh] = 2 x (h +b) = 2h + 2b = h + b + h + b, which is
the perimeter of a rectangle of height h and base b. 80 Betty’s expression is a valid
formula for finding the perimeter of a rectangle of dimensions b and h.

152

Betty: Well, first of all, I’d congratulate him for putting those
two things together. Then I think I would ask him if he
could describe what a square foot was -- could he draw a
picture for me of a square foot. If he could not I’d help
him draw, you know, a square and we could measure it one
foot by one foot. And then we would talk about what would
be the area of that -— that’s one square foot. And then I’d
say, well if that works for that, does it stand then that if
I take a square yard and do some the same thing, have it one
yard on each of the sides, what would it’s area be -- well,
one square yard. And then I would ask him to look -- I
might even have him out it out. Some times if you have them
cut out a square foot and then (pause) cut out a square
yard, it’s much easier to see that, very obviously, it’s
going to take more than three of these [i.e., square feet]
to fill-up this [i.e., the square yard]. Well, how is that
possible? Sometimes the touching, the feeling of it makes
it - no matter what you would say -- that makes it so much
simpler. My guess is I usually have somebody who says ’But
wait a minute, you have to measure along the side and find
it’s length first in the same units.’ That means if you’re
going to talk about feet, well that one over there, you
should say instead of being one yard long that it’s three
feet long on a side. Oh, well then you could say, that’s
like three times three -- so 1 square yard is really 9
square feet!

Betty drew on her pedagogical content knowledge in describing how she
would represent the student's conjecture in a way that would allow him
to explore the relationship between square feet and square yards in
greater depth. Rather than telling the student he is wrong and showing
him the correct conversion factor (i.e., multiply square yards by 9 to
convert to square feet), Betty stressed the need to assess the student’s
current understanding and then connect him concretely with the two
different area units. The instructional experience Betty describes
draws on her knowledge of area concepts, relationships, and beliefs
about how students learn to connect the student with the relationship
between square yards and square feet This blend of subject-matter
knowledge and pedagogy is what some researchers refer to as pedagogical

content knowledge (see Chapter 2).

153

Taken together, the two excerpts suggest that Betty’s knowledge of
perimeter and area concepts facilitates her ability to unpack students’
understandings about perimeter and area as well as inform her
pedagogical reasoning. Betty’s ability to move between symbolic (e.g.,
deriving the perimeter formula) and pictorial representations (e.g., her
description of comparing actual square yards and square feet) of
perimeter and area concepts further emphasizes the depth of her
pedagogical content knowledge about perimeter and area.

Betty believes that students, particularly sixth-graders, best
learn about mathematical concepts through concrete learning experiences
that utilize physical objects (e.g., manipulatives, like tiles). For
example, Betty described how important she thought it was for middle
school students to learn about perimeter and area by physically placing
square tiles on figures to measure perimeter and area. Betty argued
that most curricular materials merely describe how area and perimeter of

figures might be found by using square tiles and that

a lot of kids at sixth—grade are not ready for this just
being described. They have to have the kinesthetic, they
have to pick it up, touch it, set it down, touch each tile
as they count. And then after they have some time with
that, to see it, then it’s a piece of cake, because now they
understand what perimeter and area are.

Elaborating on how she believes sixth-graders learn mathematics for
understanding, Betty maintained that students must be given ”a chance to
discover and experiment around, and try to pull out some concepts, some
patterns, looking for those kinds of things.” Betty believes that this
is what students have opportunities to do by using manipulatives, like

tiles, to learn about perimeter and area.

154

Betty’s belief that students learn about perimeter and area
concepts by working with manipulatives is supported further by her prior
teaching. For example, although Betty had not taught Covering and
Surrounding before, she had previously taught a unit called Mouse and
Elephant: Measuring Growth (Shroyer & Fitzgerald, 1986) several times.
The Mouse and Elephant unit is an activity-oriented unit that makes use
of a variety of manipulatives (e.g., 1-inch square tiles, cubes, grids)
and representations (e.g., tables, graphs) to teach students about the
concepts of area, perimeter, surface area, and volume and the
relationships among them4 (Lappan, 1983; Shroyer & Fitzgerald, 1986).
Betty liked the fact that Covering and Surrounding also makes use of
tiles to teach perimeter and area concepts and was interested in how the
unit explores perimeter and area of irregular figures and circles, which
the Mouse and Elephant unit does not address. Betty’s past use of
curricula that use manipulatives reinforce her stated belief that
students learn mathematics by having concrete learning experiences where
they can work with physical objects.

Another facet of Betty Walker’s perceptions and beliefs about
student learning is that students learn mathematics by working together.
Betty says that cooperative groups are an ideal setting for students to
”experiment with mathematical ideas” because ”they share those ideas,

they talk over those ideas.” Betty emphasized that ”I believe kids

 

4TheMouse and Elephant unit was developed by the Middle Grades Mathematics Project
(MGMP). The MGMP was a middle school curriculum development project at Michigan State
University that developed a series of problem-solving oriented units in the early
19808 (see Lappan, 1983). Among the MGMP developers were William Fitzgerald, Glenda
Lappan, and Elizabeth Phillips. These individuals are also co-directors of the
Connected Mathematics Project and contributed to the development of Covering and
Surrounding.

155

learn so much from one another -- it doesn’t bother me when there is a
whole lot of good talking going on in my room.”

It was obvious from watching Betty’s students prior to Covering
and Surrounding that they routinely work together in groups. For
example, during her preparation period immediately preceding mathematics
class on 10/12/92, Betty moved all of the desks in her classroom
together into clusters of four. On one of the desks in each cluster she
taped a different color index card. As her students entered the room,
they reached into a paper bag Betty was holding and (randomly) drew out
an index card. Each student then sat down at the desk cluster with the
same color as the card they had drawn. Betty explained that this was ”a
non-biased way of grouping the kids so they can really learn from one
another in class and not just their friends.” One of Betty’s students
remarked that they chose groups like this ”about once a week” and that
”we work together in groups all the time.”

Students needing concrete learning experiences with manipulatives
to learn and understand mathematics, and students learning from one
another, are two predominant themes in Betty's perceptions and beliefs
about student learning. These perceptions and beliefs are evident in
Betty's talk about her own teaching and are clearly enacted in how she
teaches mathematics in the classroom. Betty’s past use of curricular
materials that use a wide range of manipulatives and her routine use of
cooperative groups are congruent with her expressed beliefs about how
students learn. Betty’s comments suggest that she is predisposed to
teaching a unit like Covering and Surrounding because it does make

extensive use of manipulatives to teach perimeter and area concepts.

156

Summer!

Betty believes that mathematics is an interesting and intriguing
subject and that problem solving is ”what mathematics is all about!”
Betty’s expressed views about problem solving and problem-solving
activity center on students developing strategies and approaches to
learn problem-solving skills as well as understanding mathematical
concepts. However, in her teaching Betty demonstrated use of standard
problems (i.e., implementing a specific procedure) as well as open-ended
problems. Problem-solving activity in Betty’s classroom is more
variable than her characterization of ”exploring and experimenting”
would indicate. This is because some of the problems are more (or less)
open-ended than others (e.g., the birth year problem as compared to the
mystery number problem), entailing different degrees of exploring and
experimenting.

Betty’s subject-matter knowledge of perimeter and area is rich and
flexible. Betty’s pedagogical content knowledge allows her to address
students’ understandings about perimeter and area concepts consistent
with her emphasis on exploring and experimenting. Betty’s perceptions
and beliefs about student learning are oriented around students using
manipulatives to understand concepts like perimeter and area, and the
belief that students learn mathematics from one another. Both of these
perceptions/beliefs about student learning appear in Betty’s classroom
instruction, shaping the way she structures her class (e.g., into
cooperative groups) and in her choice of curricula (e.g., prior teaching

of the Mouse and Elephant unit).

157

Betty's Use of Covering and Surrounding

This section begins by providing a summary of Betty's coverage of
Covering and Surrounding. A synopsis and brief analysis of the
exercises Betty assigned is given to convey which problems she
emphasized most in her teaching of the unit. The subsequent sections
each describe and analyze a specific characteristic of Betty’s use of
Covering and Surrounding. I focus on three aspects: (1) Issues
surrounding Betty’s characterization of problem-solving activity as
exploring and experimenting; (2) use of teacher materials, and; (3)
changes in practice while teaching Covering and Surrounding. Throughout
these sections, Betty’s views, knowledge, and beliefs in the four
domains of the framework are employed to unpack and understand her use
of Covering and Surrounding.
W

Betty began Covering and Surrounding on 11/9/92 and finished
teaching the unit on 12/14/92. Betty began with Investigation 1 and
sequentially covered the first five investigations of the unit.
Throughout her instruction, she emphasized the Problems and Follow-Up
Questions in each investigation, with ACE items being assigned if she
felt that time permitted. Betty believed that her students were
”exploring and experimenting” (i.e., doing problem solving) throughout
Covering and Surrounding, but the degree to which students had
opportunities for open-ended exploration and experimentation varied.
Moreover, Betty did not use the Covering and Surrounding teacher
materials much and this influenced the degree to which the teacher

materials had the potential to shape her teaching. After teaching the

158

unit, Betty’s perception was that teaching Covering and Surrounding did

not influence her to teach perimeter and area differently, nor did she

perceive a need to change her teaching. For Betty, teaching Covering
and Surrounding involved nothing unfamiliar and was perceived by her to
be congruent with her prior teaching of perimeter and area.

Table 6.1 below summarizes the sequence in which Betty taught the

 

 

 

 

 

 

 

 

 

 

investigations in Covering and Surrounding and the Problems, Follow-Up
Questions and ACE items she assigned from each:

Table 6.1: Betty: Summary of Problems, Follow-Up Questions, and ACE
items assigned.

Investigation Assigned Assigned Follow- Assigned ACE
Title Problems Up Questions Items

1 - Measuring

and Designing 1, 2, and 3 l 1 thru 6

Rooms

2 - Areas and

Perimeters of 1 and 2 1-3 (Problem 1) 1 thru 3
Figures with 1-4 (Problem 2)

Irregular Edges

3 - Going Around 1, 2, and 3 1, 2, and 6 none

in Circles through 11

4 - Constant

Area, varying 1 1 thru 5 and 7 1 thru 5, 9 and
Perimeters 10

5 - Constant

Perimeter, 1, 2, 3, and 4 1 thru 4 none
1va£ying Area

 

 

Referring to Appendix D and noting the exercises Betty assigned in
Investigation 1 from Table 5.1 gives a particular example of her

coverage of an investigation in Covering and Surrounding. To provide a
sense of how Betty’s coverage compares to the number of exercises in the
unit, Table 6.2 below shows the number of Problems, Follow-Up Questions,

and ACE items she assigned in each investigation she covered and the

total number (given in parenthesis):

159

 

 

 

 

 

 

Table 6.2: Betty: Assigned number of exercises compared to total.
Number of # Problems # Follow-Up # ACE items
Investigation Assigned and Assigned and Assigned and
(total) (total) (total)

1 3 (3) 1 (1) 6 (10)

2 2 (2) 7 (7) 3 (4)

3 3 (3) 8 (ll) 0 (6)

4 1 (l) 6 (7) 7 (14)

5 4 (4) 4 (4) 0 (8)

 

 

 

 

 

 

Table 6.2 shows that Betty assigned all of the

Problems in each of the

investigations she covered and most of the Follow—Up Questions as well.

There is, however, variance in the proportion of ACE items Betty

assigned her students.

Problems,

Table 6.3 below illustrates the proportion of

Follow-Up Questions, and ACE items Betty assigned her students

in each investigation in percentages to more clearly illustrate her

emphasis on Problems and Follow-Up Questions:

 

 

 

 

 

 

 

 

 

 

 

 

Table 6.3: Betty: Percentage of exercises assigned in each covered
investigation.
Number of % Problems % Follow-Up % ACE
Investigation Aseigned Assigned Assigned
1 100% 100% 60%
2 100% 100% 75%
3 100% 72% 0%
4 100% 85% 50%
5 100% 100% 0%
' o- ’u o H l. ’ ‘r l“. u 0 10 0 :0 ..I9 .19‘ 11‘! 9-

Betty believed she emphasized ”exploring and experimenting” in

Covering and Surrounding, but the degree to which students had

opportunities to explore and experiment varied.
her teaching of some of the investigations,

what she wanted them to do.

In other investigations, however,

Betty was directive in
explicitly telling students

Betty

allowed her students to wrestle with problems, guided them, and helped

them reach their own conclusions.

Regardless of the the opportunities

160

for exploring and experimenting in lessons, Betty believed her students
were learning about problem solving. Two lessons from Betty’s teaching
of covering and Surrounding are illustrative of these points.

Betty began Covering and Surrounding with Problem 1 in
Investigation 1. To begin the unit, Betty asked the class to open up
their copies of Covering and Surrounding to the first page and look
carefully at the room Mr. Dull designed for Mrs. I Wanna Hide (see also
Appendix D):

Figure 6.3: Mr. Dull’s design (CMP, 1992a, p. 1).

—l=l——l==l—

 

l==l

indicates a window

 

 

 

 

—i=i—__ldoor i"—

As her students were looking at Mr. Dull's design, Betty clarified the
representation by explaining how doors and windows are indicated on the

room design:

When it [i.e., Mr. Dull’s design above] shows a window, if
you were actually seeing the window itself, you’d be looking
at the wall with the window in it. If you were looking at
the door wall, the section of the wall that had a door in
it, you would actually have a door. What you are seeing on
the plan is like the edge -- the edge. You’re actually
seeing like this line right here on the floor [points to a
section of the base board of the classroom where it meets
the floor]. That's what you’re seeing on the plan.

161

Sometimes walls can be blank. In this particular case, Mr.
Dull has designed this room so it can get lots of sunshine.
So there are lots of windows.

Betty then distributed about 15 one-inch tiles and a sheet of grid paper
to each student and instructed the class to cover Mr. Dull’s design with
tiles. Betty emphasized to her students as they began placing the
tiles, ”Remember that the tiles must touch all the way.” Betty quickly
held up two tiles and showed how they should touch fully edge-to-edge,
like the grid squares in Mr. Dull's design. After the students placed

their tiles, Betty and her class had this conversation:

Betty: All right, now how many tiles did it take to cover Mr.
Dull’s room?

Jaime: Twelve!

Betty: Good. And how many tile edges does it take to surround Mr.
Dull’s room?

Class: [pause while students count] Fourteen.

Betty: So it takes 12 tiles to cover Mr. Dull’s room and 14 tile

edges to surround his room. Now, another name for covering
is area, and another name for surrounding is perimeter. So
Mr. Dull’s room has an area of 12 and a perimeter of 14.
Does everyone see that? [about half of the class nods.]

Now in this first problem, you’ll be designing your own
room. Read the problem carefully before you get started and
remember to make your room with tiles first and then trace
it on grid paper like Mr. Dull traced his -- okay? [more
nods from the class].

Betty’s students begin reading Problem 1 (CMP, 1992a) from the Covering

and Surrounding unit:

 

Problel.1: Mrs. Hide likes the amount of floor space in Mr. Dull’s
design, but she wants more windows and a more interesting shape for her
room. She asks you to help her out.

a) Use 12 square tiles to create a floor plan design for Mrs. Hide.
Remember that a window or a door can go into each section of wall space.
Mrs. Hide tells you that she wants at least 14 sections of wall space,
including windows and doors, in the room you are designing.

b) After you have designed the floor plan for the room make a
drawing to show the location of the door and where each window is.
Write a paragraph to tell Mrs. Hide why your design is better than Mr.
Dull’s (p. 2).

 

 

 

162

Within two minutes, most of the students in Betty's class were
moving tiles around on their desks forming designs for rooms. As her
students began Problem 1, Betty circulated around the room, watching
what students were doing and answering their questions. Max was one of
the first students to ask Betty a question about a room he had designed.
Max showed Betty his room which had a ”courtyard” in it. He was
confused about whether he should count the perimeter of the courtyard as
part of the perimeter of his room. Betty looked carefully at Max’s room
before responding:

Figure 6.4: Max's room:

 

 

 

 

 

 

 

 

Betty: If there can be window or a door, than that's included in
the perimeter [i.e., each exterior tile edge of the 'room’
can have a ’door’ or 'window’ built into it]. But now, be

careful. You’re talking about two different kinds of
perimeter. This looks to me like if you do this you have an
inside courtyard, am I correct? You have plants, and trees,
and things like that?

Max: Yea, that’s what I meant.

Betty: Let's look at the outside perimeter and let’s compare the
outside perimeter of the room with the courtyard. When we
do these [flipping back to rooms A - J in Problem 2,
Investigation 1 -- see Appendix D] we’ll compare the outside
perimeters. Would that be fair? Remember, perimeter is
surrounding and are you surrounding the room if you count
around the courtyard?

Max: Oh! OK, so we shouldn’t include courtyards.

Betty: Right. Because, if we did, we might not always be
consistent in how we count perimeter.

163

Max seemed satisfied that his question had been answered, deciding not
to count the perimeter of the courtyard. He did not change his design,
but crossed out ”perimeter is 24” on his paper and wrote ”perimeter is
16” instead. After talking with Max, Betty continued to circulate
around to individual students for the remaining five minutes of class,
during which she also collected tiles. Betty then announced to her
students that they would be finishing their room designs the next day in
class. After class, Betty commented on what she thought her students

had learned about problem solving:

Well, they got to work with the tiles and began to design
their rooms. They’re exploring and experimenting with
different shapes to be creative and make a room with 12
tiles that has a perimeter of at least 14. I’d say they’re

doing a lot of problem solving as they're experimenting with
the tile rooms.

In her use of the designing room problem, Betty began the lesson
by trying to make sure her students understood the wall sections and
carpet tiles analogy of perimeter and area, respectively. Betty was
then very explicit with her students about what perimeter and area are
and connected the concepts of perimeter and area back to the wall
sections and carpet tiles analogy by measuring Mr. Dull’s room with the
tiles. In her teaching to this point in the lesson, Betty had provided
little opportunity for her students to do any ”exploring and
experimenting.” For example, Betty did not have her students cover Mr.
Dull’s room with tiles, ask them to interpret what perimeter and area
are, and then discuss students' conjectures. Instead, Betty was clearly
concerned that her students understand how perimeter and area are
represented with the tiles and how they are used to measure the

perimeter and area of rooms. While this is a defensible instructional

164

approach (e.g., students need to understand how to use the tiles before
measuring perimeter and area with them), Betty telling her students what
to do with the tiles leaves little room for exploring and experimenting.

As students began working on Problem 1, however, they did begin to
explore and experiment. The problem of designing your own room with 12
tiles such that the perimeter is at least 14 is a problem with multiple
solutions and students need to decide, at some point, if their room
design met the problem conditions. This is precisely what Max was
struggling with when he asked Betty about whether he should count the
perimeter of the courtyard in his room as part of the perimeter. Max’s
question had the potential to provide an interesting context for
exploring and experimenting. Although Betty simply told Max that he
should not count the perimeter of the courtyard, this position could be
challenged. For example, if Max had to purchase baseboard for his room
design he would need to count the wall sections enclosing the courtyard
as part of the perimeter. Investigating Max's question actually seems
well-suited to the kind of exploring and experimenting activity that
Betty emphasized in her interviews prior to Covering and Surrounding.
However, in this case, she did not pursue the opportunity.

On 11/30/92 Betty began Investigation 4 in Covering and
Surrounding with her students. Betty’s use of the Problem and its
Follow-Up Questions in Investigation 4 (there is only one Problem in
Investigation 4), relative to her use of Problem 1 in Investigation 1,
provided greater opportunity for students to explore and experiment.
Betty said before she started Investigation 4 that she was going to have
her students do the investigation in cooperative groups. In the Problem

and Follow-Up Questions, the students are to assist Jennifer in making a

165

recommendation to the City Council of Clean City, Michigan about what

kind of 24 square meter room should be added on to the city

administration building (CMP, 1992a, pp. 31-32):

Betty

 

 

Problem: Jennifer has decided that the room must be a rectangle to
keep costs down. Using 24 square tiles, build every possible
rectangular room that Jennifer could suggest. Cut a rectangle out of
grid paper to show each room that you found. Label the bottom edge and
side edge of each.

Make a table that shows the data from all of the rooms.

 

 

Follow-Up Questions:

1. Order your rectangles from the largest perimeter of the smallest
perimeter on your desk. How are they changing? What patterns do you
see? How does the rectangle with the largest perimeter and the
rectangle with the smallest perimeter compare in shape?

2. Working with your group, figure out how to make a graph that shows
how the side edge of the rectangle changes as the bottom edge gets
larger.

3. Make a graph that show how the perimeter changes as the bottom edge
gets larger.

4. Look carefully at your two graphs and describe all the patterns that
you see. Look back at you data table. What patterns do you see in the

data?

5. What do you think Jennifer should recommend to the City Council and
why?

7. Describe a way to find the area of a rectangle. Now try to describe
a different way to find the area of a rectangle.

explained how each group would represent their findings in

answering the Problem and its Follow-Up Questions above:

After

I’m going to have each group create a poster. And on that
poster they're going to have to include all the rectangles
that they cut out. They’re going to have to include their
tables to keep their data. And it also has to include their
written recommendation to the City Council. ”What should
Jennifer recommend to the Council” -- so they're actually
going to make a poster that represents their group's
findings.

going over the problem with the class, Betty emphasized that

The purpose of doing this problem in groups is so that you
can explore and learn about ideas together with your
classmates -— there is not one right answer, what is
important is that you need to work together to formulate a
recommendation that makes sense.

166

All of Betty’s other comments to the students to start the Problem
focused on organizing their groups, not on the Problem itself.

During their second day on Investigation 4, one of the groups in
Betty’s class encountered a problem in deciding if they had found all of
the possible rectangular rooms that could be made with 24 tiles. One of

the members of the group, Hillary, asked Betty about this:
Betty: You aren’t sure about how many rooms there are?
Hillary: Yea. Because we need to find all the rooms with a (pause)

with 24 tiles and put them on the table. But we aren’t sure
if we have them all.

Betty: Hmm. Well, let me ask you this -- think about the rooms you
have found -- what rooms have you found so far?

Hillary: Well, we’ve got one with a bottom edge of 12 and a side edge
of 2, and another with a bottom edge of 8 and a side edge of
3 -- I don’t know the others.

Betty: Okay. Well, why don’t you go back to your group and look

carefully at the rectangles you do have and see if you can
find any relationships. You just talked about the bottom
edge and side edge -— work in your group to see if the
bottom edge and side edge might help you decide how many
rooms there are with 24 tiles.

The next day, Hillary’s group had resolved the question of finding all
of the rectangular rooms that can be made with 24 tiles. The group had
determined that the connection that the rectangles with area 24 have
with the bottom edge and side edge is that the rectangles were really
”the areas of factor pairs of 24” (Hillary’s words). Hillary said that,
after ”thinking hard” with her group, they decided that, ”When we got

all the factor pairs of 24 we got all the rectangles.5”

 

5For example, 1 and 24, 2 and 12, and 3 and 8 are all factor pairs of 24 because their
products are 24. One interpretation of the product of two numbers is the area of a
rectangle that has length and width (i.e., bottom edge and side edge) corresponding to
the two numbers in the factor pair. Hillary and her group recognized that since
factor pairs correspond to rectangles, one way of finding the bottom edge and side
edge of all the rectangles with area 24 is to find all the factor pairs of 24.

167

After the lesson, Betty noted that the poster project was an
example of ”group problem solving” and that along with problem solving
through using tiles and graphs, the students learned to problem solve

together:

One of the things is that they're learning interdependence.
They don't have to rely on only themselves for every step of
the problem. That they do have a responsibility to one
another in their group, in order to have the group solve the
problem. It isn't just one person doing it, it's the
group's responsibility to explore and experiment around and
that everyone in the group knows what's going on and that
they understand too.

Betty’s reflections show that she believed her students were
”exploring and experimenting” as they investigated both the problem of
designing their own room and the problem of helping Jennifer decide
which room to recommend to the City Council. Betty’s characterization
of students’ activities in the two lessons as ”exploring and
experimenting” is congruent with her belief that students learn about
problem solving as they explore and experiment with problems, learning
about mathematical concepts and processes. But although Betty’s
perceptions of the two lessons share these commonalties, the degree to
which students had opportunities to explore and experiment varied
between the two lessons. This variation is consistent with her teaching
prior to Covering and Surrounding, as the different degrees of exploring
and experimenting parallel differences between the birth year problem
and the mystery number problem.

Betty’s use of the Problem in Investigation 4 contrasts sharply
with the room design problem. Betty explicitly framed the problem of
helping Jennifer with her recommendation to the City Council as open-

ended and not having just ”one right answer.” Perhaps more significant,

168

however, was how Betty interacted with Hillary when she asked her
question about determining all the different rectangles that could be
made with 24 tiles. Rather than telling Hillary to find all the
rectangles corresponding to the factor pairs of 24, Betty suggested to
her that she and her group try to find a relationship between the bottom
edge and side edge of the rectangles they had already constructed. In
contrast to Max's question where exploring and experimenting was not
pursued, Betty used Hillary’s question as a means to motivate exploring
and experimenting. Both the room design problem and the group poster
project are, for most sixth-graders, non-standard problems that are
potentially open-ended. The variance in how Betty used the problems in
her teaching led to different degrees of openness.

The varying opportunities in Betty’s teaching for exploring and
experimenting were similar in that each experience provided students
with opportunities for problem solving, albeit different kinds of
problem solving. For example, Betty’s teaching motivated Hillary and
others in her group to construct relationships between factor pairs and
the rectangles connected to the factor pairs. P. Thompson (1985), for
example, notes that ”creating relationships is the hallmark of
mathematical problem solving” (p. 190). In her interactions with Max,
Betty provided him with an explanation to not count the perimeter of the
courtyard and he changed his answer accordingly. Betty’s teaching can
be viewed as giving Max an explanation for finding perimeter so he could
use a ”guess and check” strategy (see Charles & Lester, 1982) to see if
his room met the conditions of the problem. After designing his room
(i.e., a ”guess” at a suitable room design), Max used Betty’s way of

finding perimeter (i.e., don’t count the perimeter of the courtyard) to

169

”check” to see if the room met the problem conditions (i.e., area of 12
and perimeter of at least 14). Betty’s teaching illustrates that
varying degrees of exploring and experimenting can create different
kinds of problem-solving opportunities for students -- in these examples
constructing relationships for Hillary and using a guess and check
strategy for Max (c.f., Charles & Lester, 1982; P. Thompson, 1985).

s o ..

Betty's use of the Covering and Surrounding teacher materials
corresponded to her prior use of teacher materials -- she hardly used
them at all. Betty did not believe she had the time to use the Covering
and Surrounding teacher edition. Moreover, she did not perceive a need
to use the teacher materials. Not using the teacher edition shaped
Betty’s encounters with covering and Surrounding, and the potential of
the material to influence how she taught problem solving with perimeter
and area.

After she had taught the first investigation in Covering and
Surrounding and was midway through the second investigation, I asked
Betty about how often she used the teacher edition. Betty quickly
replied, ”I haven’t ever looked at that!” I then asked her about why
she hadn’t looked at the teacher edition so far in her teaching. Betty
went on to talk about how the constraints she was under in teaching
mathematics don't allow her the time needed to consult the Covering and

Surrounding teacher materials:

To be totally honest, I haven’t used the teacher materials
at all so far, except to get the masters to xerox grid paper
and the table for the kids to record their room data [see
Labsheet 1.2 in Investigation 1 teacher materials in
Appendix E]. I’m always frazzled for time! I've pretty
much gotten a feel for the unit as I've been teaching it.
With almost 30 kids in each of my two math classes, and only

170

a 41—minute period -- not to mention spelling, social
studies, assemblies and activities, correcting papers, and
everything else —— I really don’t have the time to sit down

and look at the teacher's guide!
Betty also cited that there had been two assemblies in the last two
weeks which had taken away from her time for teaching and preparation.

About four days after Betty made her above remarks, I asked her
how the Covering and Surrounding teacher materials compared to other
teacher materials she had used. One of my intentions in this
conversation was to discover whether Betty didn't have time to use the
Covering and Surrounding teacher materials in particular, or whether she

found little time to use teacher materials of any kind:

AR: How do the Covering and Surrounding teacher materials
compare, in terms of how helpful they are, with other
teacher materials you've used?

Betty: Well, actually they really aren’t that helpful. Now I know
that the current stuff is just a working draft, and I hope
that you’ll eventually go back to the three—column MGMP
format6 like in the Mouse and Elephant. But I don’t even
use the MGMP teacher stuff anymore really.

AR: Why not? I mean, are the MGMP teacher materials not useful
anymore or --

Betty: Oh, they’re fine, and the MGMP materials were useful when I
first started with them. But now I’m pretty familiar with
it, having taught MGMP a number of times, and I don’t need
the teacher stuff anymore.

AR: What about the adopted textbook you use -- is the teacher
edition for the text something you use or --

Betty: [laughs] I didn’t use the textbook 20 times last year! I
use different kinds of materials during the year and meet
our curriculum goals, but I generally do it with a variety
of outside materials.

 

6The Middle Grades Mathematics Project materials provide the teacher with a three-
column ”script” which describes in detail how each lesson might be taught. The
columns, ”teacher action”, ”teacher talk”, and ”expected response”, each
simultaneously represent a different aspect of the lesson. The teacher action column
specifies materials needed, how to display data, and suggestions for illustrating
concepts. The teacher talk column provides important questions for the teacher to ask
students to help develop understandings of concepts and problem-solving strategies.
The expected response column gives the teacher correct answers to the questions in the
teacher talk column, as well as anticipated common errors students might suggest (see
Shroyer & Fitzgerald [1986] and Lappan [1983]).

171

AR: And how are the teacher materials for these outside
materials helpful to you -- I mean MGMP stuff and the other
things you use too?

Betty: Well, I generally don’t need to use them. (pause) I mean,
I might glance at them once in a while for an answer or to
get a blackline master to xerox, but that's about all I need
from the teacher guides. It isn’t the teacher guides that
are most important anyway. Now to me -- everything that I
do -- it isn’t the materials so much as how I do it that
makes it effective or not effective.

The above conversation reveals that as well as not having the time to
use the Covering and Surrounding teacher materials, Betty doesn’t
perceive the need to use teacher materials in general. Betty’s use of
the Covering and Surrounding teacher materials is congruent with her use
of other teacher materials -- she doesn’t use them much.

Not using the Covering and Surrounding teacher materials shaped
the opportunities Betty had to learn from the material. There were
occasions where suggestions for teaching provided in the teacher
materials might have been useful to Betty in teaching Covering and
Surrounding. However, because she did not use the teacher materials,
Betty was not always aware of these suggestions and was therefore not
able to take advantage of them in the classroom. An example of this
kind of situation occurred when Betty taught her students about areas of
circles and min Investigation 3.

On 11/23/92 Betty and her class worked on Problem 2 in
Investigation 3. This problem presents students with a circle and a
square that has edge length equal to the radius of the circle (CMP,

1992a, p. 24):

 

Problem.2: How many of the large shaded squares would it take to
cover the circle? You may want to make copies of the square on grid
paper and cut them out to see how much you can cover or trace the square
out on our trans arent centimeter rid.

   

172

Figure 6.5: Shaded square and circle diagram:

 

Betty began this problem by asking the class to predict how many of the
shaded squares it would take to cover the circ1e7. Betty had reproduced
the circle and the shaded square on separate transparencies so that the
shaded square could actually be moved around on top of the circle:

Betty: So, how many of these squares do you think it would take to
cover the circle?

 

7A = nrz is the well-known formula for finding the area of a circle where r is the
radius of the circle. One way to interpret the formula is that n squares of area r2
(i.e., squares with edge length equal to the radius of the circle) are needed to cover
the circle (i.e., to measure the area of the circle). The intent of the problem is to
help students begin to develop an understanding of n by seeing that there are always
more than 3, but less than 4, squares needed to cover the corresponding circle.

173

John: Four -- see, you can cover the circle all the way with four
squares.
Betty: But aren't we, then, also covering this extra part of the

grid outside the circle [Betty indicates a corner of the
grid that is outside the circle]?

Dana: I think you only need 3 1/2 squares to cover the circle
[murmurs of agreement and nods can be seen around the
class].

Betty: OK. Well, would three squares be enough, do you think, to

cover the circle? [class seems uncertain] Well, if 4
squares is too much, than lets see if 3 is enough. How
should we do that? (pause) Well, it so happens that I have
4 squares [Betty produces a total of four squares on
transparency material]. We just said that 4 squares would
be too much [Betty places the four 6 x 6 squares so that
they cover up the entire 12 x 12 grid] and we can see that
[Betty points to where the squares cover the parts of the
grid outside the circle and then removes one of the four
squares]. Now, how could we figure out if these 3 are
enough? (pause) Well, how about this -- let's just trim
those pieces off the three squares!

At this point, Betty produced a pair of scissors and trimmed off the
parts of the three squares that lied outside the circle; she then
replaced them so that exactly 3/4 of the circle was covered with the 3
pie-shaped wedges. Betty then put the pieces she had cut off back on

the overhead next to the circle:

Betty: Now let's see. (pause) If I use these pieces to cover up
the remaining part [i.e., the uncovered 1/4] of the circle
and there is some area I can't cover, than that means I'll
need more than three squares -- right? [the class is quiet
and some students look confused] Because everything I've
got here came from just three squares, so whatever I can't
cover I'll need to get from the fourth square [Betty holds
up the whole fourth square on transparency material]. Now
let's see if I can do this.

Betty began cutting and arranging, with some difficulty, the cut-off
pieces from the three squares to cover the remaining 1/4 of the circle.
During this time, which lasted several minutes, the class was restless
and the students didn't seem to grasp what Betty was doing -- one

student remarked to another, 'Why's she doing that?‘ Betty eventually

174

managed to place all the pieces on the uncovered portion of the circle,

with an area of about 5 square centimeters left uncovered.

Betty: Okay! I used all of the area of three of the squares to
cover the circle and look! There's some area of the circle
that I couldn't cover -- that means that we do need more

than three squares to cover the circle. But not too much,
just (pause) about 5 more square centimeters. [the class
looks confused] Does everyone see that? [some students
nod]. Okay -- well [Betty glances at the clock] we're
almost out of time so we'll do more with this tomorrow.
What we're doing here has to do with finding the number n!
Has anyone heard of that? [a few hands go up] Well, a few
of you have -- nturns out to be a very important number in
mathematics and we'll explore it more tomorrow.

Within two minutes the bell rang and the students left. I asked Betty

how she thought the lesson had gone. She replied:

They need more stuff with the tiles. I'm not sure that the
unit gives them enough practice with the tiles -- they need
to play with the tiles and put them on the rooms more, I
think. ... The circle stuff is in the wrong place. We
should work through squares and rectangles, and maybe even
triangles before we get to the circle stuff. ... While it
might seem nice that they right away realize that not
everything is perfect and has nice neat corners and stuff, I

think that for some kids it was very confusinge.
Betty perceived that her students had difficulty understanding her
lesson. She believed that the source of their difficulty was in not
being able to connect their understandings of perimeter and area
developed from using the tiles to the circle because they hadn't had
enough experience using the tiles yet While this is congruent with

Betty’s belief that students learn about perimeter and area through

 

8Covering and Surrounding is sequenced to introduce students to measuring perimeter
and area (Investigation 1), then measuring perimeter and area of irregular figures
(Investigation 2) and circles (Investigation 3) to immediately reinforce that the
concepts of perimeter and area apply to figures with irregular or curved edges as well
as straight edges. The unit then returns to rectangles to examine relationships
between perimeter and area (Investigations 4 and 5), connects perimeter and area of
rectangles to parallelograms and triangles (Investigation 6), and then links these
ideas to a real-world context (Investigation 7). See Table 4.1 in Chapter 4 for a
detailed summary of vaering and Surrounding.

175

using tiles, it doesn't address the possibility that her use of the
transparency materials was what her students found confusing.

The Covering and Surrounding teacher edition provides two
suggestions to teachers for teaching the circle problem. One suggestion
is similar to Betty's use of the transparencies, but doesn't specify
cutting up and rearranging the squares like Betty did. The alternate
strategy suggested in the Covering and Surrounding teacher edition is
for students to estimate the area of the circle in the manner they
learned to estimate area in the second investigation9 and then compare

it to the area of the shaded square which is 36 square centimeters:

note that the area of the square is 36 sq. cm and the
[estimated] area of the circle is ... about 112 sq. cm. 112
divided by 36 = 3.11 ... a good estimation of n. Another
way to get at this is to note that since the square has area
36, three squares have area 36 + 36 + 36 = 108, which is 4
less than the estimated area of the circle, so three squares
is not quite enough, but four squares, with total area 144,
would be too much (CMP, 1992b, pp. 40-41).

So the Covering and Surrounding teacher's edition suggests that another
way to get at the problem is to estimate the area of the circle and
numerically compare it to the area of the square.

The alternate strategy for teaching the circle problem outlined in
the teacher edition may or may not have helped Betty's students. What
is clear, however, is that Betty didn't have the opportunity to use the
strategy in her teaching because she was not aware of it. She was not

aware of it because she didn't use the Covering and Surrounding teacher

 

9Investigation 2 in vaering and Surrounding examines perimeter and area of irregular
figures. Problems include developing strategies for estimating perimeter and area of
irregular figures like marshes and shoeprints (CMP, 1992a). In Investigation 2
Betty’s students developed a strategy of 'piecing together' area —- first count the
'whole square units' of area that cover the figure and then 'piece together other
whole units' from the partial square units covering the figure; the sum of the whole
units and the 'pieced together' units is the area of the figure (phrases in quotes
were used by Betty's students to describe their strategy).

176

materials to any significant extent. Had Betty used the teacher edition
in thinking about teaching the circle problem, she might have had
another option open to her in teaching the lesson. Had she still chosen
to use the transparencies, she would still have had another
representation readily available. The central issue Betty's teaching of
the circle problem raises is that her use of the Covering and
Surrounding teacher materials shaped the opportunities she had to change
her teaching through awareness of other approaches to teaching the unit.
MW

Betty perceived that her prior education and experience with other
curricula (e.g., Mouse and Elephant) had already prepared her to use
curricular materials like Covering and Surrounding. This belief
supported her non-use of the Covering and Surrounding teacher materials.
Additionally, Betty did not believe that teaching Covering and
Surrounding would lead to significant changes in her future practice.

After concluding her teaching of Covering and Surrounding, I asked

Betty to reflect on her experience of teaching the unit:

AR: Has teaching the unit led you to reexamine or reaffirm any
of your thinking or assumptions about teaching mathematics
or area and perimeter concepts?

Betty: I think that maybe I'm a bit biased since I've done
activity-based kinds of things like this before. Maybe for
me it didn't really cause change like it would for some
other people who might not have done a lot of this kind of
thing before, but I've done this kind of stuff before. And,
in fact, I've been trying to do work like this all along
when I look back at my own experience and what I had when I
was in undergraduate school.

AR: So you feel that your experience is congruent all along with
this kind of curricular material?

Betty: When I look back at what my math methods courses, we did a
lot of stuff with Cuisenaire rods, we played around with
pattern blocks and geoboards. 'What you're doing now, we did
then. I can see where stuff like this [i.e., the Covering

177

and Surrounding unit] developed from. So I think that, and
I'll put this in quotes, that I was already 'trained'. So I
think that I'm kind of biased because I've already done a
lot of this stuff. And I've been looking for more of this
kind of activity-based, hands-on kinds of things, compared
to somebody who came from a different kind of background.

Teaching Covering and Surrounding did not motivate Betty to reexamine
her beliefs or assumptions about teaching because she perceives that her
past education and experiences have already prepared her to teach
mathematics with materials like Covering and Surrounding. For example,
recall that Betty had taught the Mouse and Elephant unit several times.
Betty noted that Covering and Surrounding is 'a lot like Mouse and
Elephant -- using tiles and grids and stuff like that.‘ Because she
perceived the two units to be similarlo, Betty's prior use of Mouse and
Elephant supported her belief that Covering and Surrounding incorporated
little that was new for her teaching. For Betty, teaching Covering and
Surrounding was essentially business as usual.

Betty’s beliefs about what Covering and Surrounding has to offer
her teaching are consistent with her use of the unit's teacher
materials. Since she perceives Covering and Surrounding as bringing
little, if anything, new into her teaching or classroom, there seems to
be no reason for Betty to spend already scarce time reviewing the
teacher's edition. What this perpetuated, however, is that there were
few instances for Covering and Surrounding to provide Betty with
opportunities to reflect on her approach to teaching this content. It
is unlikely she would have a chance to seriously consider any

suggestions for teaching or reflect on instructional problems or

 

10Of the seven investigations in Covering and Surrounding, three (i.e., l, 4, and 5)
are similar to activities in the Mbuse and Elephant. But while Mbuse and Elephant
also focuses on surface area and volume of rectangular solids, four of the
investigations in Covering and Surrounding examine perimeter and area of irregular
figures, circles, parallelograms, and triangles.

178

situations to supply motivations for change from the teacher materials
if she doesn’t use them. The Covering and Surrounding unit, screened
through Betty's beliefs that she had already ”done a lot of this stuff"
led directly to her lack of using the teacher materials.
Summary

Betty's use of Covering and Surrounding paralleled her prior
teaching and use of curricular materials. Different degrees of
exploring and experimenting were present in her teaching both prior to
and during Covering and Surrounding. These different degrees of
exploring and experimenting corresponded to different opportunities for
students to learn about problem solving. Moreover, Betty does not, in
general, make extensive use of teacher materials and this was also the
case with Covering and Surrounding. Betty’s beliefs about teaching were
not challenged by Covering and Surrounding as she perceived the unit as
congruent with her prior teaching and experience. Betty’s beliefs not
being challenged by vaering and Surrounding are linked to her use of
the teacher materials -- since she did not use them, there were really
no instances for the unit to shape her practice. Betty would have, of
course, been doing different problems and activities with her students
had she not used Covering and Surrounding. But, from her perspective,
she still would have taught perimeter and area by having her students
explore and experiment with situations similar to those found in

Covering and Surrounding.

CHAPTER 7

KAREN .AND BETTY:
TEE INTERPLAY BETWEEN TEACHER AND CURRICULUM

The cases of Karen's and Betty's uses of Covering and Surrounding
depict the interplay between each teacher and a piece of problem-solving
oriented curriculum. Both cases describe how Karen and Betty shaped
Covering and Surrounding and the extent to which the unit materials
influenced their teaching. In this chapter, I first unpack the notion
of interplay and how this concept describes Karen’s and Betty's uses of
Covering and Surrounding. I then analyze Karen’s and Betty’s uses of
Covering and Surrounding from the perspective of the curriculum
developers by carefully examining how the enacted curriculum was similar
to, and also different from, the intended curriculum. Finally, I
revisit Covering and Surrounding and use the four-dimensional framework
developed in Chapter 2 to analyze the unit from Karen's and Betty’s

perspectives. The chapter concludes with a synopsis of study findings.

Interplay: Reciprocal Influence Between
Teacher and Curriculum

Some research has shown that curricular materials can have little
influence on teachers' practice (e.g., Porter & Freeman, 1989) while
other studies reveal that curriculum can play a powerful role in shaping
how teachers teach (e.g., Remillard, 1991a). In the cases of Karen and
Betty, their uses of Covering and Surrounding were shaped by a mixture
of influences from both the materials and their own views, beliefs, and
knowledge. For example, Karen’s teaching was shaped by Covering and
Surrounding -- the unit 'forced' her to use tiles and teach perimeter
and area using 'hands-on' activities instead of 'teaching abstractly.”

179

180

But Karen's use of Covering and Surrounding was also shaped by her own
beliefs. Because she believes it is important to maintain her students’
interest, she allowed them to determine the sequence of investigations
as a way of keeping them interested in the unit.

Betty's nonuse of the teacher edition represents a facet of her
use of Covering and Surrounding that is a component of interplay between
her and the unit. Since she did not use the teacher edition, Betty’s
use of the unit was based primarily on her own views and beliefs, not
the recommendations of the developers. For example, Betty had her
students complete Investigation 4 as a group poster project because she
believes students learn from one another, not because this was suggested
in the teacher edition. But not using the teacher materials also shaped
Betty's teaching of Covering and Surrounding. The circle problem
investigating n, for example, put Betty in a teaching situation where
her own approach left students confused. Had she used the teacher
materials, Betty would at least have had the alternate numerical
strategy available to her and would have had the opportunity to teach
the lesson differently. Karen's and Betty's teaching was thus shaped by
both the unit and their own beliefs -- their enactments of the unit was
the product of the interplay between teacher and curriculum.

To further unpack the notion of interplay in Karen's and Betty's
uses of vaering and Surrounding, consider the room design problem.
Recall that Shakaya's question to Karen and Max's question to Betty were
the same —- whether or not to count the courtyard perimeter as part of
the perimeter of their rooms. For both teachers, the unit shaped their
teaching with these students. Covering and Surrounding, by putting

students in the open-ended situation of designing a room without

181

direction whether or not to include courtyards, provided the context for
Shakaya's and Max’s question to emerge. The unit shaped Karen's and
Betty's teaching by placing them in the situation where they needed to
address the question of whether or not to include the perimeter of
courtyards as part of the perimeter of the rooms. But Karen's subject-
matter knowledge about perimeter and area and Betty's prior experience
with other curricular materials also shaped their teaching with Shakaya
and Max.

Karen’s uncertainty about her own subject-matter knowledge about
perimeter and area shaped her decision to respond to Shakaya by asking
her to think about the perimeter of her room further. Recall that Karen
was hesitant and uncertain in her responses to perimeter and area items
on the interview prior to teaching Covering and Surrounding. Karen also
remarked in her conversation with Shakaya 'I don’t know' when Shakaya
asked her if she should count the perimeter of the courtyard as part of
the perimeter of her room. Since Karen herself was not sure whether or
not to count the perimeter of Shakaya's room, she was not prescriptive
in her answer, but instead left the question open. Karen's use of the
room design problem, shaped by uncertainty in her own subject-matter
knowledgel, resulted in a learning experience that gave Shakaya the

opportunity to explore the concept of perimeter further.

 

1Karen saying 'I don't know' could indicate other kinds of uncertainty besides
uncertainty in her subject-matter knowledge about perimeter and area. For example,
Karen’s comment could also reflect unfamiliarity with the materials, uncertainty as to
where different interpretations of measuring perimeter (i.e., whether or not to count
the courtyard perimeter) might lead, or perhaps she was hesitant because she had not
anticipated the situation. In any case, the instruction with Shakaya was a product of
interplay -- the materials put Karen in an open-ended situation that was unfamiliar
and Karen’s use of the room design problem was shaped by her uncertainty.

182

Betty's decision to tell Max not to include the perimeter of
courtyards when finding the perimeter of his room was shaped by her
prior use of other curricular materials. As Betty noted on multiple
occasions, she had previously used the Mouse and Elephant unit (Shroyer
& Fitzgerald, 1986) to teach perimeter and area. Mouse and Elephant
uses a different representation to help students learn about perimeter
and areaz. In using the representation, the materials explicitly direct
teachers to have their students construct figures without 'holes' in
them3, i.e., the rectangular figures designed by students in Mouse and
Elephant are specified to not have 'courtyards' (see Shroyer &
Fitzgerald, 1986). Betty's use of the room design problem was
consistent with Mouse and Elephant to the extent that she directed Max
not to count the perimeter of his courtyard. In her comments about that
lesson, Betty noted that in the Mouse and Elephant, 'holes aren’t part
of the perimeter.’ While this decision is consistent with the Mouse and
Elephant representation, it is not necessarily valid in the Covering and
Surrounding representation. For example, if perimeter is interpreted as
the number of wall sections in the room, then it is arguable that Max
should have counted the perimeter of the courtyard in his room.

The room design problem shows how Karen’s and Betty's teaching is
shaped both by Covering and Surrounding and by their own knowledge and

prior experiences. But as well as illustrating how Karen's and Betty's

 

2In Mouse and Elephant, banquet tables are used as a representation for perimeter and
area. Each 1-inch tile is interpreted as a square table that can seat one person per
edge. Banquet tables are formed by pushing together square tables. The area of a
rectangular figure, or 'banquet table', is the number of square tables. The perimeter
is the number of people who can be seated around it.

3In Mouse and Elephant relationships between perimeter and area of rectangular figures
are explored. The unit focuses students on rectangles (without holes/courtyards) to
prepare them for study of perimeter and area relationships in rectangles (see Shroyer
& Fitzgerald, 1986).

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instruction is the product of the interplay between teacher and
curriculum, this study reveals that interplay can result in potentially
unexpected teaching. For example, in the room design problem Karen was
open-ended in her response to Shakaya and Betty was more directive in
her interaction with Max. Given Karen's prior teaching of perimeter and
area focusing on formulas and ”teaching abstractly', and Betty's
continued emphasis on open-ended ”exploring and experimenting', one
might expect the situation to be reversed. While Karen’s and Betty's
instruction with the room design problem may be surprising at first,
studying the interplay between them and Covering and Surrounding sheds
light on how their teaching with Shakaya and Max unfolded.

In the next two sections I take a closer look at the interplay
between Karen and Betty and Covering and Surrounding. I first examine
Karen's and Betty's uses of the unit from the perspective of curriculum
development. I then turn the tables and survey Karen's and Betty's uses
of Covering and Surrounding from their perspectives. These contrasting
viewpoints on Karen's and Betty's teaching of Covering and Surrounding
provide findings from the dual perspectives of curriculum developers and

teachers.

Unpacking Karen's and Betty's Uses of Covering and
Surrounding: A Perspective from Curriculum Development

Through their uses of Covering and Surrounding, Karen and Betty
both involved their students in learning about perimeter and area in
ways that reflected multiple goals of the unit developers. At the same
time, however, each teacher exhibited practices or perceptions and
beliefs that, to some extent, go against the grain of the developers'

goals or intentions.

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Karen and Betty both used Covering and Surrounding to engage their
students in problem solving as they investigated and learned about
perimeter and area. Karen's use of the triangle area problem with Matt,
Aaron, and Trevor, and Betty’s use of the problem of recommending a room
to the city council with Hillary are examples of each teacher involving
her students in problem-solving activity rich in mathematical
connections and content. These instructional snapshots of Karen and
Betty are consistent with learning goals the curriculum developers
outline for students in Covering and Surrounding (see Chapter 4). The
two lessons are also congruent with the underlying philosophy of the
developers that students learn mathematics and problem solving skills by
investigating problems and wrestling with the mathematical ideas and
patterns embedded in problems (see Fitzgerald et al., 1991).

At least in the case of Karen, Covering and Surrounding was
essential in changing her mode of instruction from teaching perimeter
and area 'abstractly' to teaching 'hands-on'. For example, she noted on
multiple occasions that the unit made her use tiles to do hands-on
activities. Not only her comments, but also her actual practice provide
evidence of this change in Karen's teaching. Prior to Covering and
Surrounding Karen taught lessons that focused on’rules and procedures
(e.g., the lesson on divisibility laws). After Covering and Surrounding
she taught a lesson on the perimeter of circles that was formula-
oriented and exemplified her mode of "teaching abstractly." When
contrasted with these two lessons, Karen's Covering and Surrounding
instruction that occurred between them is clearly different. In fact,
while Karen wanted her students to learn formulas in Covering and

Surrounding, she resisted telling them to her students, waiting for the

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students to develop them (see Chapter 5). Karen's lesson on triangle
area with Matt, Trevor, and Aaron is an example of this. These changes
in Karen’s practice while teaching Covering and Surrounding sharply and
clearly reflect the kinds of shifts in practice the unit developers
would like to see teachers make through using the materials (see
Fitzgerald et al., 1991).

Another feature of both Karen's and Betty's uses of Covering and
Surrounding is their heavy use of concrete materials. Both teachers,
for example, used tiles and transparent grids extensively as they taught
the unit. While using manipulatives does not guarantee that students
will learn about problem solving or make connections among mathematical
ideas, manipulatives can be used effectively as tools to help accomplish
these kinds of learning goals with students (c.f., Ball, 1990b; NCTM,
1989, 1991; Putnam et al., 1992). The extensive and routine use of
mathematical 'tools' such as tiles and transparent grids is one of the
skills the developers of Covering and Surrounding intend students to
develop (Fitzgerald et al., 1991). While this study did not assess
students' learning, Karen's and Betty's use of concrete materials while
teaching Covering and Surrounding seems to be on the trajectory of the
developers' for students to use tools (e.g., transparent grids) and
models (e.g., tile models of rooms) to '... represent, simulate, and
manipulate patterns and relations in problem settings' (Fitzgerald et
al., 1991, p. 6).

While Karen's and Betty's uses of concrete materials, their
engagement of students in investigating mathematically rich problems,
and Karen's change from abstract to hands-on teaching while using

Covering and Surrounding all reflect intentions of the curriculum

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developers, other aspects of their teaching do not. Karen's integration
of arithmetic drill exercises from the textbook while teaching Covering
and Surrounding, for example, clearly does not match the developers'
vision of how middle school students should learn mathematics. In their
proposal for the Connected Mathematics Project, Connected Mathematics,

the developers state:

The predominant method of mathematics instruction asks
students to learn facts and skills by listening, imitating,
and practicing routine exercises. This skill-oriented
curriculum and the drill-and—practice instructional style
with which it is commonly 'delivered' are particularly
inappropriate for middle school students (Fitzgerald et al.,
1991, p. 2).

Another aspect of Karen’s use of Covering and Surrounding that varies
from the philosophy of the developers is her perception of problem
solving. While Karen did not believe that her students were learning
about problem solving in Covering and Surrounding, the developers
definitely view the unit as problem-solving oriented (see CMP, 1992b).
While it is not clear in this study to what extent Karen's perception of
her students' learning about problem solving influenced her teaching,
problem solving is a component of her views and beliefs that contrasts
sharply with the perspective and intentions of the developers.

How both Karen and Betty used the Covering and Surrounding teacher
materials is another aspect of their practice that varied from the
intentions of the developers. While the developers intend the teacher
materials (i.e., CMP, 1992b) to be a rich resource for teaching COvering
and Surrounding (see Chapter 4), Betty hardly used the teacher materials
at all. Although Karen highlighted passages, she seemed to use the
teacher materials in practice only for answers to problems. The

developers view the Covering and Surrounding materials as a sharp

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departure from the usual curricula predominantly used in middle school
mathematics (Fitzgerald et al., 1991). Covering and Surrounding was
also a new unit for both Karen and Betty. Despite this, Karen and Betty
appear to use the Covering and Surrounding teacher materials as they
have used other curricular materials.

The aspects of Karen's and Betty's uses of Covering and
Surrounding that reflect intentions of the curriculum developers may be
encouraging news to curriculum developers. The features of their
practice that vary from the developers' perspectives, however, warrant
closer examination. I now look at Karen's and Betty's uses of Covering
and Surrounding from their viewpoints to shed more light on the aspects
of their practice that varied from the intentions of the curriculum

developers.

Unpacking Covering and Surrounding:
Karen's and Betty's Perepectivee

The cases suggest that two mechanisms were particularly powerful
in shaping Karen's and Betty's uses of Covering and Surrounding: (l)
Filtering the unit through powerful views, beliefs, and knowledge; and
(2) use of the teacher materials in a manner consistent with prior use
of teacher materials. But as well as examining mechanisms present in
their practices, it is also critical to unpack what the Covering and
Surrounding unit did or did not provide for Karen and Betty to use in
their teaching from their perspectives. This is because the interplay
between Covering and Surrounding and Karen and Betty is influenced by
how the unit addresses their beliefs and the issues they feel are
important in teaching. Unpacking Covering and Surrounding through the

four-dimensional framework (see Chapter 2) sheds more light on Karen's

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and Betty's uses of the unit and their role in the interplay between
teacher and curriculum.

Even though Karen had read the five problem—solving goals in the
Covering and Surrounding teacher edition, one of which was congruent
with an event in her classroom, she perceived her students as doing
'hands-on' activities rather than problem solving. Why did Covering and
Surrounding not affect Karen's perception that her students were not
learning about problem solving? A strong possibility is that neither
the student materials nor the teacher edition addressed Karen's beliefs
about problem solving.

Problems in Covering and Surrounding are almost all non-standard
(process or open-search) problems, real-world problems, or puzzle
problems (see Chapter 4). Standard word problems are a rarity in
Covering and Surrounding, yet solving these kinds of problems is how
Karen conceptualizes problem solving. Nowhere in the Covering and
Surrounding materials is the range of problems, or the absence of
standard word problems, made explicit to the teacher. In fact, the five
problem-solving goals supplied in the teacher edition (CMP, 1992b) are
the only explicit statements about problem solving in the materials.
The five goals specify problem-solving strategies and skills that are
not descriptive of solving standard word problems (see Chapter 4). As
discussed in Chapter 4, the range of types of problems in Covering and
Surrounding imply different kinds of problem-solving for students.
While some problems (e.g., standard word problems) require translating
information from words into symbols or computations, other kinds of

problems (e.g., non-standard, open-search problems) may involve cutting

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up or manipulating objects as Karen’s students did in exploring the area
relationships between rectangles and triangles.

Since the perspective on problem solving of the curriculum
developers was not clear and the kinds of problems in Covering and
Surrounding are not standard word problems, from Karen's perspective
problem solving was not happening. No scheme or framework was supplied
in the Covering and Surrounding teacher edition for classifying problems
and associated problem-solving processes in the unit (as in Chapter 4).
Such a framework could conceivably illustrate that problem solving can
occur for students in Covering and Surrounding even though students
encounter few standard word problems. Providing an explicit, more
detailed portrait of different kinds of problems and problem solving is
no guarantee that the teacher materials would have changed Karen's
perception about her students' learning about problem solving. However,
at least the opportunity for change would have existed for Karen to have
a clearer picture of what kinds of problem-solving opportunities the
curriculum developers were trying to provide.

In Betty's use of Covering and Surrounding, she structured
problem-solving activities with different degrees of 'exploring and
experimenting' for students. However, she perceived that all the
problem-solving activity in her teaching of vaering and Surrounding was
very open-ended. While problems in Covering and Surrounding may have
different degrees of openness (see Chapter 4), this is another issue
that the teacher edition does not explicitly address. The teacher
materials do state that the problems in Covering and Surrounding are

intended to provide students with rich and engaging mathematics (see

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CMP, 1992b). However, comparisons or suggestions addressing the
relative openness of problems are not given.

The situation for Betty with respect to the degree of openness of
problems is analogous to beliefs about problem solving for Karen -- the
teacher edition does not address Betty's beliefs about 'exploring and
experimenting” in problem-solving activity. Betty may have relied on
her own beliefs about problem solving being universally open-ended
because this aspect of problem solving is not discussed in the teacher
edition4.

WW

As detailed earlier, both Karen and Betty bumped-up against the
situation of whether or not to count the perimeter of a courtyard as
part of the perimeter of a room. While the situation of distinguishing
between the 'inner' (i.e., courtyard) perimeter and the 'outer' (i.e.,
exterior) perimeter emerged for both teachers, the Covering and
Surrounding teacher edition does not address it. The teacher edition
provides suggestions to help students understand the wall section and
carpet tile analogy for perimeter and area (see Appendix E), but does
not help teachers in addressing the courtyard dilemma.

Karen's and Betty's use of the room design problem and responses
to their students illustrates their reliance on their own subject-matter
knowledge of perimeter and area to address students’ questions whether

or not they were open to using suggestions from the teacher edition. If

 

4Since Betty did not use the Covering and Surrounding teacher edition it is
questionable whether the teacher materials would have had an impact on her conceptions
of openness in problem solving activity even if they were addressed. However, for
Betty or other teachers with similar beliefs about problem-solving always being open-
ended, the opportunity to bring this issue to the surface in the teacher materials
does not exist.

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either teacher looked to the Covering and Surrounding teacher edition
for suggestions on how to deal with courtyard perimeter, none would be
found. It is perhaps this kind of situation that led Karen to remark,
when I asked her how the Covering and Surrounding teacher materials
could be made more useful, ”By doing more about giving suggestions and
examples for what counts as good student answers and responses.” Karen
went on to connect this comment to her own subject—matter knowledge by
explaining that ”The kids came up with a lot of interesting ideas in the
unit [i.e., Covering and Surrounding], but I don't always know whether
or not they're [i.e., the students] right!”

The issues of computation and skill maintenance contrast sharply
in Karen’s and Betty's uses of Covering and Surrounding. Karen believed
that computation is an important part of what her students should be
learning in mathematics and felt that the unit was not sufficient for
her students to build and maintain computational skills. Betty did not
assign computation exercises in her use of Covering and Surrounding
whereas Karen had her students do computation practice from the
textbook.

The Covering and Surrounding teacher materials do not address
computation or make the perspective of the curriculum developers on
computation explicit (see CMP, 1992b). While the CMP funding proposal
makes clear that rote computation practice (e.g., ”naked number” drill
exercises -- see Chapter 4) will not be included in the CMP materials
and provides a rationale, this stance is not communicated to teachers in
Covering and Surrounding (c.f., CMP, 1992b; Fitzgerald et al., 1991).

From Karen's perspective, the unit did not address an important

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component of her students’ learning. She therefore brought in the
textbook which did address her belief that students need computational
practice and skill maintenance. From Betty's perspective, that the
Covering and Surrounding unit did not explicitly address computation was
not a factor in her teaching because she either did not view computation
as important for her students or felt that computation was sufficiently
addressed. For example, Betty noted that she felt her students were
doing computation practice in Investigation 4 finding factor pairs of 24
and calculating the perimeter of all the rectangles of area 24 for their
graphs. However, she also stressed prior to teaching COvering and
Surrounding that doing ”naked number” computation exercises (like Karen
assigned) were really ”not useful” for students.
Commentary

One possible implication for the Covering and Surrounding teacher
materials that could emerge from the above discussion is to simply make
them more explicit in suggesting to teachers how they should teach. The
situation, however, is more complex than that. Researchers have argued
persuasively that teachers necessarily make instructional decisions
based on their own views, beliefs, knowledge, and context (Ball, 1990a,
1990b; Ball & McDiarmid, 1990; NCTM, 1991; Putnam et al., 1992; A.
Thompson, 1989, 1992; Wilcox et al., 1991, 1992). This growing body of
research implies that providing more structured, explicit, or directive
materials for teachers won't necessarily result in teaching that is more
closely aligned with the intentions of the curriculum. No matter what
the curriculum, in a classroom situation a teacher will still teach it.

A major finding of this study that emerges from Karen's and

Betty's uses of Covering and Surrounding is that curriculum developers

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may not be able to put teachers on the trajectory of the intended
curriculum by ignoring their perspectives or developing more explicit or
directive materials. As this study shows, Karen and Betty are
necessarily partners with the curriculum in the interplay from which the

enacted curriculum emerges.

Synopeie of Findings

Karen's and Betty's uses of Covering and Surrounding were not
simply a matter of each teacher implementing the unit in her classroom.
Both teachers' uses of the unit involved an interplay between teacher
and curriculum which shaped the instruction I observed in their
classrooms. As the cases of Karen and Betty show, this interplay
resulted in some teaching that was perhaps unexpected (e.g., the room
design problem) but is understandable when the interplay between teacher
and curriculum is examined.

Karen and Betty provide a contrast in the degree to which teaching
Covering and Surrounding corresponded to changes in their practices.
Karen changed her mode of teaching perimeter and area from abstract to
hands-on. Betty, however, maintained what she felt was her teaching
mode of ”exploring and experimenting”, perceiving no reason to change
her practice during Covering and Surrounding. This comparison between
Karen and Betty readily demonstrates the importance of both sides of the
interplay between teacher and curriculum: Karen's use of the unit
provides evidence for how the curriculum shapes classroom teaching;
Betty's use of covering and Surrounding shows how teachers' views and
beliefs can influence how instruction plays out in the classroom.

Moreover, as the cases suggest (see Chapters 5 and 6), teaching

covering and Surrounding did not appear to change Karen's or Betty's

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views or beliefs even when their views and beliefs ran counter to the
intentions of the curriculum developers. There are at least two
plausible explanations for the resilience of Karen's and Betty's views
and beliefs. One is that teaching Covering and Surrounding was simply
not a long enough or sufficiently extensive intervention to challenge
and/or change Karen's and Betty's views or beliefs. Some researchers,
for example, have argued that it takes time for teachers to change their
views and beliefs, especially those related to problem solving (e.g., A.
Thompson, 1989, 1992). Another explanation is that the beliefs Karen
and Betty maintained that are, at least to some extent, counter to
intentions of the curriculum developers were not addressed by the
teacher materials. Since teaching Covering and Surrounding is an
interplay between Karen and Betty and the unit, if the unit materials do
not address aspects of their knowledge and beliefs, it is not surprising
that they may remain intact.

Having discussed the conclusions that can be drawn about Karen's
and Betty's uses of Covering and Surrounding, I next turn to the broader
implications of this study. In the next chapter I will provide
arguments for what this study holds for curriculum development and, to a
lesser extent, teacher education. I will also indicate some possible

directions for future research this study points to.

CHAPTER 8

WHAT LIES AHEAD?! CHALLENGES AND DILENIAS
FOR CURRICULUN DEVELOPHENT

The Covering and Surrounding unit influenced the teaching which
occurred in Karen's and Betty's classrooms. This study shows that
Karen’s and Betty's uses of Covering and Surrounding were shaped by
interplay between their respective views, beliefs, and knowledge, and
the unit. As a result, each teacher's instruction reflected her views,
beliefs, and knowledge, and some of the intentions of the curriculum
developers. This study shows how teachers’ views, beliefs, and
knowledge can interact with a piece of problem-solving oriented
curriculum to shape teaching.

Studying how teachers like Karen and Betty use curricular
materials like Covering and Surrounding may hold implications for
teacher education as well as curriculum development. For example, a
relevant issue for teacher education is to conceptualize how experiences
such as teaching a unit like Covering and Surrounding can be used to
assist mathematics teachers in changing their beliefs (see Dewey
1938/1963; Paley, 1989) and teaching to align with reforms. A. Thompson
(1992) suggests that such research could examine relationships and
interactions between teachers and teacher educators that can be crucial
in supporting teachers and helping them change their practices and
beliefs to implement mathematics education reforms in their classrooms.

But interaction with a teacher educator and Karen or Betty did not
occur in this study. As discussed in Chapter 3, my role was that of an

interviewer and an observer. My intent was to study teachers’ use of a

195

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problem-solving oriented piece of curriculum, not to intervene or show
Karen or Betty how they should be using the Covering and Surrounding
unit. Since no interventions were present besides the Covering and
Surrounding student and teacher materials, the implications of this
study are limited for teacher education and are more suited to
illuminating issues in curriculum development.

However, while this study does not imply specific practices or
recommendations for teacher education, the implications I will discuss
for curriculum development in this chapter do contribute to teacher
learning and change. As I argued earlier, problem-solving oriented
curricula are viewed by reformers as a lever for helping change how
teachers predominantly think about and teach mathematics. Since this
study does inform the role such curricula can play in shaping how
teachers teach, implications for curriculum development of this study
are also relevant to teacher education. Karen's and Betty's uses of
Covering and Surrounding being the product of interplay between teacher
and curriculum further suggests that the arenas of curriculum
development and teacher education can both be informed by examining
issues connected to teachers' use of problem-solving oriented curricula.

In the first section of this chapter, I outline a fundamental
dilemma facing curriculum developers and how Karen's and Betty's uses of
Covering and Surrounding reflect the dilemma. The next section digs
deeper into specific challenges and dilemmas facing curriculum
developers by using snapshots of Karen's and Betty's teaching with
Covering and Surrounding to illustrate the complexity of four major

issues. For each of the issues I discuss, anomalies embedded in them

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point to areas for future research that are connected to teachers’ use
of problem-solving oriented mathematics curriculum.
A Dilemma for Curriculum Developers

Designing mathematics curriculum that helps teachers bring new
roles for themselves and their students into classrooms (see Chapter 4)
involves balancing a dilemma between reform and practice. On the one
hand, curriculum developers view mathematics curricula as not only
providing teachers and their students with opportunities to engage in
rich mathematics (NCTM, 1989), but also to push teachers to develop
their teaching along the trajectory of reforms (Cohen & Ball, 1990).
However, curriculum developers and reformers also argue that teachers
are professionals who necessarily shape their teaching and make
decisions based on their own context and beliefs; therefore, it is not
possible to prescribe to teachers how to teach (The Holmes Group, 1990;
NCTM, 1991). Mathematics curriculum developers are confronted with the
dilemma of how to design mathematics curriculum that will move reforms
into classrooms while simultaneously respecting the teacher's role which
includes shaping curriculum to meet the needs of his or her students.

Traces of the above dilemma can be found in Karen's and Betty's
uses of Covering and Surrounding. As discussed in the last chapter,
engaging students in problem solving and using concrete materials were
two characteristics of both teachers' uses of Covering and Surrounding
that are consistent with the intentions of the curriculum developers and
reformers. Karen's use of the unit to change her mode of teaching
perimeter and area from ”abstract” to ”hands—on” is another example of
using Covering and Surrounding in a manner that is in line with the

vision of the developers and reforms. But Karen's use of drill

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exercises from the textbook along with teaching Covering and Surrounding
and her perception that her students were not learning problem solving
in the unit clearly vary from the intentions of the developers.
Similarly, Betty's belief that the materials had nothing new to offer
her teaching are in tension with the intentions of the developers and
reforms. In the case of these facets of Karen's and Betty's uses of
Covering and Surrounding, how might the developers balance the dilemma
of pressing Karen and Betty to reexamine their beliefs and use of the
unit and yet respect their role in shaping curriculum in their
classrooms?

Managing the dilemma of pushing teachers to reexamine their
beliefs and practice while still respecting their role in shaping
curriculum is complicated further because it is not clear if the vision
of teaching and learning mathematics of curriculum developers is
necessarily ”right” in all cases or circumstances. After all, it seems
possible that teachers could be justified in their use of curricula like
Covering and Surrounding even when their use of the materials varies
from the intentions of the curriculum developers. For example, could
Karen be justified in her insistence on supplementary work and practice
on computational skills while teaching Covering and Surrounding? Are
both the curriculum developers and Karen somehow justified in their
different stances on computation and skill maintenance?

From the perspective of the developers, students will gain more by
engaging with rich problems in Covering and Surrounding than the rote
computation drill Karen assigned. But, at the same time, students do
not all have the same learning needs; it is conceivable that at least

some students in Karen’s class did need explicit computation practice

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which Covering and Surrounding does not provide. Moreover, the concerns
of teachers, their students' parents or guardians, and their schools and
communities may vary from the intentions of curriculum developers --
e.g., some teachers, parents, schools, and communities may view rote
computation drill as important to students’ learning. Given this
variance and the necessary role teachers play in shaping curriculum, it
seems inevitable that some teachers, like Karen, will assign computation
drill. From Karen's perspective, since computation is important for her
students to learn, it seems reasonable to spend some instructional time
on computation practice and skill maintenance.

This example of Karen's use of computation drill along with
Covering and Surrounding shows that determining the rightness or
wrongness of a particular use of curriculum is highly problematic. In
the next section, I will unpack some specific issues that demonstrate
the difficulty of unraveling teachers’ use of problem-solving oriented
curricula and illustrate them with snapshots of Karen's and Betty's uses
of Covering and Surrounding. My analysis is based on the premise that
Karen’s and Betty's uses of Covering and Surrounding are shaped by

interplay between the teacher and the unit.

Karen and Betty: Teachers Who Provide
Snapshots of Dilemmas for* Curriculum. Developers

To what issues must curriculum developers attend as they design
problem-solving oriented curricula intended to help teachers teach
mathematics as problem solving? What do curriculum developers need to
know, or know more about, to design curricular materials that are more

effective in helping teachers reexamine their beliefs and practice as

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they use them? Schwab (1978), for example, notes that in designing

curriculum the developers must have

...knowledge of what ... teachers are likely to know and how
flexible and ready they are likely to be to learn new
materials and new ways of teaching (p. 367).

But researchers have provided detailed portraits of what teachers know
about mathematics (e.g., Ball, 1988; Ball & McDiarmid, 1990), what
teachers believe about teaching mathematics and how they are prepared to
teach problem solving and mathematics for understanding (e.g., A.
Thompson, 1992; Wilcox et al., 1991, 1992), and the match between
teachers' expressed and enacted conceptions of problem solving and how
these conceptions can change (e.g., Charles, 1989; A. Thompson, 1985,
1989). Curriculum developers must now turn to the challenge of using
this extensive knowledge of teachers to design mathematics curriculum
that will help teachers reach new goals for teaching school mathematics
and address the dilemmas of attaining these goals (e.g., NCTM, 1989,
1991). This study suggests that conceptualizing teachers' use of
curriculum as interplay between teacher and curriculum sheds light on a
number of dilemmas that illustrate challenges facing curriculum
developers in the context of mathematics education reform.

Although Karen used Covering and Surrounding to provide her
students with open-ended, ”hands-on” problems to learn conceptually
about perimeter and area, she also divided her students’ time between
the CMP unit and computational practice with calculators. Recall that
Karen assigned whole number and decimal multiplication drill exercises
to be done with paper and pencil and then checked with a calculator

during the second half of Covering and Surrounding (see Chapter 5).

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Even though rote drill of the kind Karen assigned clearly varies from
the intentions of the curriculum developers, skill maintenance is an
important part of what Karen believes her students should be doing in
her mathematics class. Betty more clearly represented the vision of the
curriculum developers with respect to computation. She did not assign
drill exercises in computation, but maintained her focus on Covering and
Surrounding. Betty believed that her students were getting sufficient
computation practice through working on problems in the unit. Who is
”right” in her approach to computation along with teaching Covering and
Surrounding -- Karen or Betty?

It may be tempting to view Betty as more faithfully representing
the ”right” stance on computation of the curriculum developers and
dismiss Karen as subscribing to the focus of the traditional, procedure-
oriented mathematics curriculum. However, the contrast in stances on
computation and skill maintenance of teachers like Karen and curriculum
developers designing materials like Covering and Surrounding is more
complex than determining who is ”right.”

From the perspective of a classroom teacher like Karen using
materials like Covering and Surrounding, how to handle computational
practice may not be clear. While curriculum developers and reformers
clearly deemphasize computation, they have not given it up entirely.
Reformers and curriculum developers often refer to, for example, ”the
recent emergence of calculators” (Fitzgerald et al., 1991, p. 8) as an
argument against incorporating computation practice into problem-solving
oriented materials. By emphasizing the wide availability of
calculators, curriculum developers and reformers argue that less time

needs to be spent on developing students' computational skills. This

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argument also aims to ease fears that students studying the materials
will be unable to perform computations when they need to because
students will be able to use calculators (see also NCTM, 1989). In
fact, reformers advocate building students' computational skills with
calculators (e.g., Fitzgerald et al., 1991; NCTM, 1989).

Since Covering and Surrounding does not include computation drill
exercises nor explicit use of calculators or calculator exercises (see
CMP, 1992a, 1992b), Karen's use of calculators as part of computation
practice can be interpreted as a reasonable instructional decision that
is arguably aligned with reforms. Karen could conceivably view having
her students check their answers with a calculator as building their
skills in calculator use. As noted above, reforms do imply that
computation, especially with calculators, is worth spending some time
on. From this analysis, it is more apparent that deciding who —- Karen
or Betty and the curriculum developers -- has the ”right” stance on
computation and skill maintenance is problematic. Either Karen or Betty
can interpret her approach to computation as aligned with reforms.

Karen's use of computation along with Covering and Surrounding is
also significant because her perspective is common among teachers. For
example, while standardized tests are slowly changing to decrease
emphasis on computational proficiency, such proficiency is still a major
evaluative criterion. Some teachers, therefore, feel a need to focus on
computation in their teaching (Kulm, 1991). Karen, for instance, did
not believe that computation embedded in problems in Covering and
Surrounding was sufficient for her students; she believed that explicit

work with computation (i.e., drill) was needed (see Chapter 5).

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If the Covering and Surrounding developers wish to see teachers
like Karen not spend instructional time on computation practice, then
perhaps computation should be directly addressed in the materials. For
example, should the developers include computation practice that is
somehow related to the content to address Karen’s beliefs about
computation? Or, on the other hand, should the developers include
arguments in the teacher materials to convince teachers like Karen that
it is not worth spending time doing supplementary computational drill?
These are difficult questions, especially considering teachers like
Betty who may not believe that they need to use the teacher materials
and teachers like Karen who may use the materials mainly for answers.

The dilemma of computation for curriculum developers thus leads to
another challenge -- how can curriculum developers address computation
in the curricula they design when teachers might not use the materials
or not use them as intended? To explore this in the context of this
study, I next unpack the difficulty Karen’s use and Betty's nonuse of
the Covering and Surrounding teacher edition presents to curriculum

developers.

Karen's and Betty's uses of Covering and Surrounding reveal
contrasts in how two teachers used the same piece of problem-solving
oriented curriculum (c.f., computation, sequencing of investigations).
Given the variance in views, beliefs, and use of a single unit between
just two teachers, trying to develop materials that are usable and
address the needs of teachers is clearly difficult. In this sense,
interplay between teacher and curriculum complicates the charge of

curriculum developers to move reforms into classrooms. Teachers have a

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wide range of views and beliefs and trying to address them all in a set
of curricular materials is a formidable undertaking. However,
confronting this challenge seems to be part of the agenda of at least
some curriculum developers. The developers of Covering and Surrounding,
for example, note that while inservice and professional development with
the CMP curriculum is desirable and strongly recommended, they intend to
develop the materials so that ”a well motivated teacher can teach the
materials with little or no inservice” (Fitzgerald et al., 1991, p. 15).
It is understandable why the CMP or other curriculum developers
would undertake designing a curriculum that could, if need be, stand on
its own in supporting teachers. Other forms of teacher support, like
inservices or workshops, are often too infrequent to have significant or
lasting meaning for teachers, providing only scant glimpses of complex
issues (A. Thompson, 1992). Some teacher support efforts are ongoing
and involve teacher collaboration and sustained interaction with teacher
educators and other teachers (e.g., The Holmes Group, 1990; A. Thompson,
1989). But, in most cases, teachers work in isolation and continued
professional development is rare due to, for example, institutional and
fiscal constraints (Lipsky, 1980; Lortie, 1975; Sarason, 1982). Given
the limitations and constraints of teacher support mechanisms like peer
collaboration, contact with support staff, workshops and inservices, it
seems reasonable to expect curriculum materials to provide support to
teachers. Yet Karen's, and particularly Betty's, uses of Covering and
Surrounding raise questions about whether curriculum materials alone are
sufficient to help teachers align their practices with the intentions of

curriculum developers and the reforms embedded in the materials.

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Betty reflected many of the intentions of the curriculum
developers in her use of Covering and Surrounding. However, she did not
use the teacher materials much and did not believe that she needed to.
Yet, as her lesson on circumference of circles illustrates, the Covering
and Surrounding teacher materials had potential to offer Betty support
in her practice. How can the curriculum support Betty if she does not
use the teacher materials and does not believe she needs support?
Consider Karen's use of the Covering and Surrounding teacher materials
as she taught the unit. While Karen did use the Covering and
Surrounding teacher edition, she used it mainly for obtaining answers to
problems and not as a resource for her practice as the developers
intend.

The Covering and Surrounding teacher edition was a form of support
available to Karen and Betty, whereas other potential means of support
were not at hand. For example, I did not support them as they taught
the unit, I only observed. Neither Karen nor Betty attended inservices
or workshops on teaching Covering and Surrounding. None of the other
teachers in either Karen’s or Betty’s building taught Covering and
Surrounding, so neither of them had other teachers to talk to about
teaching the unit. Yet, Betty did not use the teacher edition at all
and Karen used it almost exclusively for answers. Although Karen's and
Betty's uses of Covering and Surrounding reflected many of the
intentions of the developers and teaching with the materials helped
Karen change her teaching of perimeter and area, the point here is that
the teacher materials did not seem influential or supportive for either
teacher. Therefore, while this study provides evidence that using

curricula like Covering and Surrounding can help teachers change their

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teaching, it also raises questions about the extent to which the
materials actually supported Karen and Betty in their teaching.
Limitations of this study should be considered before inferring
that teacher materials would be ineffective in supporting teachers in
their use of the CMP curriculum or similar curricula. For example, this
study examined Karen's and Betty’s uses of only one unit over a period
of about one month. Covering and Surrounding, in actuality, is not an
isolated unit as it was for Karen and Betty in this study, but is part
of a complete sixth-grade curriculum. Might Karen's or Betty's uses of
CMP teacher materials change over the course of teaching the entire CMP
sixth-grade curriculum? During a full year of CMP instruction would
Betty begin to use the teacher materials? Would Karen continue to allow
her students to vote to determine the sequence of investigations in each
CMP unit and use the teacher materials mainly for answers? These
limitations are emphasized further by the context of Karen's and Betty’s
uses of Covering and Surrounding. For both teachers, the unit was an
isolated piece of their curriculum that did not represent a long-term
investment to change their practice. Since Karen's and Betty's teaching
of Covering and Surrounding entailed only a four to five week
commitment, it is not surprising that both maintained beliefs and
patterns of practice that have been part of their prior teaching.
Another factor to consider is that the Covering and Surrounding
teacher’s edition, along with the student and teacher materials for all
other CMP units, are still in the preliminary stages of development.
Later versions of the Covering and Surrounding teacher materials might
prove more useful to Karen and Betty than the version they used in this

study. For example, the Covering and Surrounding student and teacher

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editions used by Karen and Betty are formatted separately. While
neither teacher mentioned the two editions being separate as
problematic, it is possible that the separate format shaped how they
each used the teacher materialsl. It is not clear in this study,
however, to what extent reformatting the teacher materials (e.g., to
include the student text) would help support Karen and Betty in teaching
with the CMP curriculum.

The limitations of this study preclude dismissing teacher
materials as a form of support for Karen and Betty in using the CMP
materials. At the same time, Karen’s and Betty's uses of the Covering
and Surrounding teacher edition put into question the adequacy of
curricula alone as a teacher's sole source of support.

Ball and McDiarmid (1990) emphasize that little is known about how
teachers' subject-matter knowledge is shaped from using curricula, or
how teachers learn subject matter from curriculum. Since teachers’
subject-matter knowledge of mathematics can be fragmented and rulebound,
using problem—solving oriented curricula will push at the boundaries of
some teachers' subject-matter knowledge. This was also the case with
the ”New Math”, a national K-12 mathematics curriculum reform effort
during the late 1950s and early 1960s post-Sputnik era. The new math
curriculum was designed to increase the mathematics preparation and

achievement of K-12 students and was based largely on the axioms of set

 

1Trials of the CMP materials in other sites have indicated that at least some teachers
have found juggling both versions to be cumbersome. Some teachers report reading the
teacher materials once, making notes in a student copy, and then teaching from the
student book. These teachers would prefer a teacher edition that includes the student
text. Currently, the CMP is reformatting the teacher editions for all units so that
the teacher edition displays for the teacher the corresponding text in the student
edition.

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theory and the field properties of real numbers. Although the new math
was intended to provide students with a solid foundation to study higher
mathematics in college, the movement floundered. The failure of the new
math is usually attributed to unrealistic expectations for teachers to
teach new kinds of mathematics with little or no subject-matter
preparation (Sarason, 1982).

The new math reform era showed that it may be quixotic to expect
teachers to teach mathematics that is unfamiliar or not well-understood.
In the present context of problem-solving oriented reforms, the failure
of the new math suggests that curriculum developers may need to be
careful to provide teachers with subject-matter support. Curricula like
Covering and Surrounding have the potential to put teachers in
mathematical situations that are unfamiliar. In retrospect, the new
math may suggest that curriculum developers need to supply teachers with
extensive subject-matter support. For example, curriculum developers
might provide detailed solutions to problems that have the potential to
be problematic for teachers. In light of the previous discussion of the
possible limitations of teacher support through curricula, perhaps video
tape segments or even teaching consultants should be supplied to give
teachers examples of how to handle students' ambiguous questions,
misconceptions, or conjectures that may be unanticipated.

However, this study also suggests that the issue of subject-matter
knowledge need not be a matter of providing lots of comprehensive
support to teachers. For example, it may be possible that subject-
matter knowledge is not a major issue for some teachers in teaching with
problem-solving oriented curricula. Karen’s use of Covering and

Surrounding suggests that some teachers may be able to use problem

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solving oriented materials effectively with students even if the subject
matter is unfamiliar and subject-matter support is scarce. Betty’s use
of the unit shows that providing detailed subject-matter support in
teacher materials, such as described above, may still not address
teachers' instructional problems, even for teachers with deep and
connected subject-matter knowledge.

Karen exhibited limited subject-matter knowledge about perimeter
and area in preliminary interviews. Moreover, in using Covering and
Surrounding, she was hesitant in responding to Shakaya's question about
whether or not to count the perimeter of the courtyard in a room she had
designed as part of the room's perimeter. Karen’s interaction with
Shakaya, coupled with the interview data, raises questions about whether
her subject—matter knowledge was sufficient for her to unravel the
perimeter situation. However, the outcome of this lesson (i.e., Shakaya
deciding for herself to redesign the room without a courtyard) is
actually congruent with at least some aspects of problem-solving
oriented reformsz. Moreover, there is no discussion in the Covering and
Surrounding teacher edition about Shakaya's question. Although the
subject-matter appeared to be unfamiliar to Karen and the teacher
materials offered no assistance, Shakaya's question was resolved and
Shakaya appeared to have a conceptual understanding of perimeter and
area upon completion of designing her room.

Curricula like Covering and Surrounding, by providing students

with open-ended problems like the room design problem, do have the

 

2For example, students addressing their own questions and determining the validity of
their own mathematical ideas, as Shakaya did with her question about perimeter, is
part of ”mathematical power” which reformers advocate students should develop in K-12
mathematics (see NCTM, 1989, 1991).

210

potential to provide students with opportunities to explore their own
ideas about mathematical concepts. Karen's use of the room design
problem shows how a teacher's use of an open-ended problem can
contribute to providing an opportunity for students to pursue their own
questions. However, when coupled with her thin subject-matter knowledge
about perimeter and area, Karen's use of the room design problem raises
the issue of how curriculum can support teachers in encouraging student
exploration when the teacher herself is not necessarily familiar with
the mathematical terrain students may wander into.

In contrast to Karen, Betty demonstrated rich and connected
subject-matter knowledge of perimeter and area in the interviews prior
to teaching Covering and Surrounding. However, a growing body of
research argues that subject-matter knowledge is a necessary but not
sufficient component of the knowledge teachers need to teach mathematics
for understanding; teachers should also possess knowledge of how to
represent and make content accessible to learners (Shulman, 1987; Wilson
et al., 1987). This point is illustrated in Betty's use of the problem
on circumference of circles. She attempted to get at the concept of n
by using transparencies to show that slightly more than three squares of
edge length equal to the radius of the circle are needed to exactly
cover the circle. While Betty appeared to understand the content of the
problem, her way of representing it was confusing to at least some of
her students. What is also significant is that the Covering and
Surrounding teacher edition did provide another approach for teaching
the circle circumference problem of which Betty was not aware. This
example shows that subject-matter knowledge alone may not be sufficient

for teaching with problem-solving oriented materials and, perhaps more

211

alarming from a curriculum developers’ perspective, even if needed
pedagogical support is provided, teachers may not use it.

Betty's use of the circle problem also points to the importance of
teachers' pedagogical content knowledge, i.e., the knowledge teachers
draw on to represent and connect content to learners (Shulman, 1987;
Wilson et al., 1987; Wilson, 1989). Curricula like Covering and
Surrounding employ representations extensively. Teaching with the
representations provided in problem-solving oriented curricula like
Covering and Surrounding will require teachers to know about how to
connect representations with learners as well as understanding subject
matter (Fitzgerald et al., 1991; Shulman, 1987; Wilson et al., 1987).
For example, Covering and Surrounding uses the circle problem as a
representation to connect students with the concept of n . As Betty's
use of the circle problem shows, however, it may be difficult for some
teachers to use the representation effectively. While Betty understood
the subject matter embedded in the circle problem, she struggled in her
teaching because she was not able at the time to use the representation
to connect her students with the content she herself understood.

Betty’s use of the circle problem also speaks to how curricula
like Covering and Surrounding may influence teachers' pedagogical
content knowledge. Although Betty did not take advantage of it, the
teacher edition does provide suggestions for how to teach the circle
problem and Betty was unfamiliar with at least one of the suggested
approaches. Even though Betty did not learn about the alternate
approach to the circle problem from the Covering and Surrounding
materials, the fact that the approach was new to her and that it is

detailed in the teacher edition is significant. It suggests that

212

curricula like Covering and Surrounding have the potential to help
teachers who do use the teacher materials develop their pedagogical
content knowledge. For example, if Betty had used the teacher materials
in planning her lesson on the circle problem, she would at least have
had at her disposal another approach to using the representation.

Karen's and Betty's uses of Covering and Surrounding illustrate
two facets of subject-matter knowledge for curriculum developers. One
is the perhaps encouraging finding that even when on potentially
unfamiliar mathematical ground, teachers might use problem-solving
oriented materials in ways that lead to conceptual understanding in
students. Raising a thorny dilemma, however, is the possibility that
some teachers who are in familiar mathematical territory may still have
difficulties teaching with problem-solving materials and may not use the
materials for assistance even when it is provided.

The dilemma of Betty's nonuse of teacher materials may also be
connected to subject-matter knowledge. Covering and Surrounding
addresses perimeter and area, concepts that are both part of the
standard sixth-grade mathematics curriculum and which Betty had taught
before with other nonstandard curricula. Betty's familiarity with
teaching perimeter and area with other materials may have contributed to
her belief that she did not need to consult the Covering and Surrounding
teacher materials. From Betty's perspective, she may have perceived
Covering and Surrounding as mapping out familiar mathematical territory
she had traversed with other students before, and therefore did not feel
the need to consult the teacher materials. However, this might not be
the case for Betty with other curricular materials that address less

familiar content. For example, if the content of Covering and

213

Surrounding had been unfamiliar to Betty -- e.g., if she had taught a
CMP unit on statistics, probability, or spatial visualization -- she
might have been more inclined to use the teacher materials to get ideas
for teaching problems or for help in answering students' questions.

Covering and Surrounding is part of a complete curriculum. The
varying (and perhaps unfamiliar) subject matter of the other sixth-grade
CMP units might contribute to Betty changing her use of CMP teacher
materials. Of course, Betty could conceivably maintain her nonuse of
teacher materials even when teaching unfamiliar content. However,
Betty's familiarity with the content of Covering and Surrounding
nonetheless raises the issue to curriculum developers of how teachers
may change their use of teacher materials when confronted with teaching
unfamiliar subject matter. Another open research question connected to
teaching unfamiliar content is to examine how teachers learn from
mathematics curricula3. For example, what might Karen have learned
about perimeter through her use of Covering and Surrounding and her
interaction with Shakaya? What might Betty have learned about teaching
circumference of circles? Materials like Covering and Surrounding seem
to hold potential to help teachers learn about mathematics and teaching
mathematics. But how this learning might occur, and to what extent it
shapes subsequent teaching, are areas needing further investigation.

Researchers have argued persuasively that subject-matter knowledge
and pedagogical content knowledge shape practitioners' teaching (e.g.,
Ball & McDiarmid, 1990; Shulman, 1987; Wilson et al., 1987), and

curriculum developers seek to support teachers in these areas (e.g.,

 

3Ball and McDiarmid (1990) point to this same line of inquiry as an area for future
research in teacher education.

214

Fitzgerald et al., 1991). However, snapshots of Karen's and Betty’s
uses of Covering and Surrounding show that anomalies can exist between
teachers' knowledge and their use of curricula. Karen shows that
teachers can use problem-solving oriented curricula in ways powerful for
students even if subject—matter knowledge is thin. Betty shows that
teachers can experience instructional difficulties even if the content
is familiar and knowledge of the content is rich and connected. These
snapshots emphasize the complex role subject-matter knowledge and
pedagogical content knowledge can play, or not play, in using problem-
solving oriented curriculum.

Finally, this study suggests that teachers' perceptions and
beliefs about student learning may not only shape how teachers use
problem-solving oriented curriculum, but also what they teach to
learners. In agreeing to participate in this study, Karen specified
that she would teach Covering and Surrounding to her Enriched
Mathematics class, but not her General Mathematics classes (see Chapter
5). Karen’s reason for this decision was that she perceived her
Enriched Mathematics students as being more likely to be interested in
the unit and succeed with it. She also felt that she had to know how
her ”best students could handle the unit” before teaching it to the
General Mathematics classes. After teaching Covering and Surrounding,
Karen noted that she would probably teach parts of the unit (e.g., the
first investigation) to the General Mathematics classes.

Although the developers of Covering and Surrounding clearly intend
the materials to be a curriculum for all students (see Fitzgerald et

al., 1991), Karen did not initially perceive the unit as suitable for

215

all of her students. While Karen's Enriched Mathematics class had over
four weeks of studying Covering and Surrounding, her General Mathematics
students most likely received far less. Karen’s perception of student
learning between her Enriched and General students leads to an
instructional decision (i.e., what parts of the unit should be taught to
whom) that is inconsistent with the intent of the curriculum developers
and illustrates contemporary arguments about inequity in learning
opportunities in school (e.g., Anyon, 1981; The Holmes Group, 1990;
NCTM, 1989, 1991; Oakes, 1985).

Another facet of Karen's use of Covering and Surrounding related
to perceptions and beliefs about learners is that throughout teaching
the unit she maintained that her students were not learning about
problem solving. Although there are different ways that problem solving
can be conceptualized for teaching and curriculum, it seems surprising
that Karen did not, for example, view Matt, Trevor, and Aaron's argument
about triangle area as involving problem solving. Should curriculum
developers address Karen -- e.g., try to convince her that students can
learn about problem solving in Covering and Surrounding? If so, how?
The problem solving objectives for students in the unit (which Karen
highlighted) did not convince her that problem solving occurred in her
classroom. This suggests that Karen's beliefs about problem solving may
not be easily challenged.

One approach could be to try and change Karen's perceptions about
her students’ learning about problem solving -- e.g., engage her in
thinking more about problem solving by, for example, including a
framework in the teacher materials discussing a range of problems.

Karen would then have a resource to fit her conception of problem

216

solving into a spectrum of problems and processes (e.g., see Hembree,
1992; Chapter 4). These could range from standard word problems which
fit her conceptions of problem solving and problem-solving activity, to
non-standard, real—world, and puzzle problems which constitute most of
the kinds of problems in Covering and Surrounding (see Chapter 4). Such
a revision of the teacher materials would perhaps help Karen see more
happening in Covering and Surrounding than just ”hands-on” activities.
But another approach that curriculum developers could take would be to
do nothing, i.e., in a sense accommodate Karen’s beliefs by not
challenging them. After all, whether Karen believes her students are
learning about problem solving or not is perhaps not so important to the
curriculum developers if her students are doing what they envision as
problem solving.

It is unlikely that there is a single ”right” way to address
Karen’s perception that her students were not learning about problem
solving in Covering and Surrounding (see Chapter 2). Before either of
the above approaches -- change or tacit accommodation -- should be
considered, however, an issue embedded in both warrants further
scrutiny. To what extent does how the teacher conceptualizes problem
solving shape what her students learn about problem solving when using
problem solving oriented curricula? This is a critical research issue
to consider, especially since reforms put problem solving at the core of
what students should be learning about in school mathematics. If
teachers’ conceptions of problem solving do shape students' learning
about problem solving, then perhaps curriculum developers should be more
concerned about how teachers conceptualize problem solving. For

example, a framework such as suggested above might be included in

217

problem-solving oriented curricula to help teachers understand how
problem solving is envisioned and embedded in the materials.

Another perspective on the extent to which teachers' perceptions
of what their students are learning about problem solving shapes
students' learning would be to consider what curriculum developers have
to learn from teachers like Karen. For example, perhaps the developers
should incorporate more standard word problems into Covering and
Surrounding. Since it is not clear what role standard word problems
play in students' learning about problem solving, the developers might
be overlooking a type of problem in Covering and Surrounding that is key
to students' learning about problem solving. While these multiple
perspectives do not provide answers to how teachers' perceptions about
student learning should be addressed, research in this area could help
guide curriculum developers' future decisions about the substance and
design of problem-solving oriented curricula.

We

This chapter has raised a host of issues and dilemmas that are
related to teachers’ use of problem-solving oriented curricula.
Furthermore, the four major areas discussed in this section are by no
means exhaustive. As a researcher, I feel that this study has raised
more questions than it answers. In retrospect, however, I think that
this is desirable. Amid the current flurry of mathematics education
reform and curriculum development, it would be a mistake to lose sight
of the complexity of these endeavors. Most importantly, I hope that
those who are involved in reforming K-12 mathematics education,

especially through developing problem-solving oriented curriculum,

218

continually keep teachers like Karen and Betty in mind -- for teachers,

some like Karen or Betty, will be the final arbiters of reform.

APPENDICES

APPENDIX A

Below is the interview protocol used for the vignette about a lesson on
problem solving. Following the protocol is the vignette that was

provided for the teacher participants:

Provide teacher participants with the ”Mr. Fern” vignette. Allow the
teacher adequate time to read the vignette and then ask the following

questions:

What stands out to you about this lesson?

What are the strengths and weaknesses that you see in this

lesson?

Do you think that Mr. Fern chose a good problem for his

students to solve? Why or why not?

How would you structure a lesson in your math class, using
the same or a different problem, to teach the problem

solving strategy of ”guess and check”?

Altogether, this lesson took about two-thirds of the class

,period. Do you think that the lesson was a good use of

classroom time? Why or why not?

219

220

Vignette:

The following passage is a vignette of a sixth-grade mathematics
lesson. Mr. Fern, the teacher, was observed by a researcher who wrote

this description of Mr. Fern’s class:

An example of how Mr. Fern teaches specific problem solving
strategies can be drawn from a lesson he taught at the beginning of the
year. He told the researcher before the lesson that his students were
going to learn about the problem solving strategy ”guess and check'.
This particular strategy, where a potential solution to the problem is
guessed and then checked for validity against the conditions of the
problem, is a problem solving tool that is often incorporated into
classroom problem solving materials. Mr. Fern presented students with

the following problem:

If you have 9 coins that are together worth 58
cents, what are the kinds of coins you can have?

Mr. Fern illustrated the ”guess and check” strategy to the
students by asking them how they might solve the problem if one of the
coins was a half-dollar. Many students quickly volunteered that 1 half-
dollar and 8 pennies would be the solution. After noting the students'
”guess”, he ”checked” the solution by seeing if it met the conditions of
the problem - does 1 half-dollar and 8 pennies make 9 coins and 58
cents? After noting how the solution was validated by the check, Mr.
Fern imposed an additional condition on the problem - find a solution to

the problem withggt using a half-dollar. He instructed the class,

221

working in their cluster groups (the students' desks are arranged in
clusters of 3 or 4) to find two solutions to the problem.

The students quickly began working on the coin problem. Mr. Fern
said later to the researcher that although none of the groups used a
half-dollar in their solutions, at first most attended to only one of
the other two conditions. Students generally either used 9 coins and
neglected their value (e.g., one group's initial ”guess” solution used 2
quarters, l nickel, and 6 pennies) or had a number of coins different
than 9 with a value of 58 cents (e.g., another group's first ”guess” at
a solution used 1 quarter, 3 dimes, and 3 pennies). Mr. Fern circulated
among all the groups and in each case asked if the group's ”guess” at a
solution ”checked” with all the conditions of the problem. Groups whose
solution was revised and validated by meeting all the problems
conditions were challenged to find another solution. Groups whose
solutions did not meet all conditions were encouraged by Mr. Fern to
revise their solutions and check again. After 15 minutes of working on
the coin problem, all the groups had at least one valid solution and
most groups had two.

Mr. Fern brought the class back together through a class
discussion where two groups were each asked to give a solution and
describe why it was valid. Mr. Fern emphasized to the class that this
was an example of a mathematics problem which has more than one solution
and could be solved using the problem solving strategy of ”guess and
check”. He also mentioned that the ”guess and check” strategy would be
useful for many other kinds of problems.

After class, Mr. Fern said that he had several goals for students

in the lesson. First, he wanted his students to understand and

222

successfully implement the ”guess and check” problem solving strategy.
This includes formulating a solution and then checking the solution
against the conditions of the problem; a solution is valid if it meets
all of the problem conditions. Mr. Fern also wanted students to work
with a math problem that has more than one answer, can be investigated
cooperatively with peers, and draws on a variety of prior knowledge
(e.g., knowledge of money as well as knowledge of multiplication and

addition).

APPENDIX 3

Below are the interview protocols used for Items 1, 2, and 3 in the
teacher interview to assess subject-matter knowledge. Following the

protocols are the individual items provided for teachers to react to:

ma:
Suppose that while you are teaching your class about
perimeter and area, one of your students comes up to you.
She says that she has discovered a new theory that you never
told the class. She says that she has found that as the
perimeter of a figure increases its area also increases.
She shows you these pictures she has made, and says that
they prove her theory [Show item 1 sheet (i.e., Figure A.Bl)
to the teacher participant]. How would you respond to this
student?

mu:
Suppose that during your instruction on perimeter and area a
student raises his hand and is very excited. He says that
he has figured-out how to convert between units used to
measure area. He shows you his solution to this area
problem [Show item 2 sheet (i.e., Figure A.BZ) to the
teacher participant]. He says that the area of the triangle
is 24 square yards. He also says that since there are 3
feet in a yard, he can convert the area of the triangle to
square feet [indicate students' calculations on the
interview item]. How would you respond to this student?

Item:
Suppose that one day after school a fellow sixth-grade teacher
stops by your room. She has a student’s assignment on perimeter
with her and shows it to you [Show item 3 (i.e., Figure A.B3) to
the teacher participant]. She says that she isn't sure if the
student’s method is valid, or what the student’s method is, even
though he got right answers. She would like you to help her make
sense of the student’s work. How would you try and explain the
student’s work to her?

Probe: Can you generalize the student’s rule? and/or Can

.you show whether the student’s method is valid or invalid
for finding the perimeter of rectangles?

223

Item I

 

 

 

 

Perimeter : 14
Area = 12

 

 

Perimeter = 12

Area = 6

224

 

 

 

 

Perimeter = 18

Area = 20

 

12

Perimeter = 36

Area :: 54

 

225

Item 2

6 yards

 

 

8 yards

A=I/2xBxH=1/2x6x8=l/2x48=24squareyards

3 feet: i yard

Area = 24 x 3 = 72 square feet

226

Item 3

 

 

 

 

6

Perimeter: 2 x (24/4+ 24/6) = 2 x (72/12 + 48/12) = 2 x 120/12

:120/6 = 20

 

 

 

 

Perimeter: 2 x(18/9+18/2)=2 x(36/IB+162/18)= 2 x198/18

:198/9 = 22

APPENDIX C

Below is the interview protocol used for the vignette about structuring
problem-solving activity in classroom settings. Following the protocol

is the vignette that was provided for the teacher participants:

Provide the teacher participant with the ”Mrs. Beacham” vignette. Allow
the teacher participant adequate time to read the vignette. Read the

following passage as the vignette is provided:

Karen Beacham is a sixth-grade mathematics teacher. This is
a description of a lesson from one of her math classes and
some of her views on teaching. Please take a few minutes to
read this vignette, and then I’d like to ask you some
questions about what you think about Mrs. Beacham’s teaching

[Provide the teacher participant with the vignette].

After the teacher participant has read the vignette, ask these

questions, probing if necessary:

Do you have any first reactions to Mrs. Beacham and her classroom

teaching? [Probe for reasons behind the teacher participant's views]

What do you think about how Mrs. Beacham is structuring the lesson
(i.e., the three students illustrating how the class is divided up into
three different groups) for Dana — Jessie - Pat? [Probe for reasons

underlying the teacher participant's views]

227

228

What would you say Dana - Jessie - Pat is learning from this lesson?

What beliefs about mathematics do you think Dana’s — Jessie’s - Pat’s

assignments promote? Why?

would you structure your class lessons like Mrs. Beacham? Why? If not,

Why not and How would/do you structure your lessons differently?

Vignette:

Karen Beacham is a sixth-grade middle school teacher. Her
schedule includes teaching two math classes. Mrs. Beacham emphasizes
the diversity of her students' mathematical abilities in her two
mathematics classes. She notes that some students catch on to math very
easily while others have math skills that are below grade level. Mrs.
Beacham says that to accommodate the diverse needs of her students, she
organizes her classes around themes like geometry, number concepts, or
statistics. Structuring the class around themes, Mrs. Beacham notes,
allows her to provide students with different learning tasks on the same
topic that address their specific mathematical strengths and weaknesses.
Mrs. Beacham says she generally divides the class into three groups.
What follows are excerpts from one of Mrs. Beacham's lessons where
number concepts is the common topic for the entire class.

After answering questions at the beginning of the period, Mrs.
Beacham gives out three assignments and the students quickly go to work.
A few minutes later, Dana, who is busily working on a drill sheet of

decimal multiplication exercises, is confused about where to put the

229

decimal point in the product of 2.63 x 4.8. ”Remember Dana,” Mrs.
Beacham says, ”count up all the places behind the two decimals and make
sure that that is how many places you have behind the decimal point in
your answer.” Dana says ”OK” and goes back to work. Looking over
Dana's shoulder, Mrs. Beacham says ”Be sure to work all of the problems
Dana, because practice makes perfect.”

Mrs. Beacham next walks over to Jessie, who has constructed 1x1,
2x2, 3x3, and 4x4 squares with l-inch tiles. Jessie’s assignment was to
find numeric and geometric representations of the number sequence 1, 4,
9, l6, ... . Mrs. Beacham asks Jessie what the squares mean. Jessie
says that the 1x1 square, which has area 1, corresponds to the number 1,
the 2x2 square, which has area 4, corresponds to the number 4, and so
on. Mrs. Beacham asks Jessie ”What would you make for the eighth term
in the sequence?” ”An 8x8 square” Jessie replies. ”Why?” asks Mrs.
Beacham. ”Because an 8x8 square has area 64 and that’s the eighth
number in the sequence 1x1, 2x2, 3x3, 4x4, and on and on.” Mrs. Beacham
asks Jessie if there are other connections between the squares and the
numbers in the sequence besides area. Jessie says ”The sides of the
square are factors of the numbers. Like in the 8x8 square, 8 times 8 is
64.” Mrs. Beacham smiles and says ”You've provided some good
justifications, Jessie.” She then asks Jessie to next try using cubes to
represent the number sequence 1, 8, 27,

Near the end of the class period, Mrs. Beacham is circulating
around the room and notices an error on Pat’s paper. Pat is working on
an assignment which consists of story problems that involve fractions,
decimals, and percents. Pat is using a calculator to check answers.

One problem asks for a solution in percent form, and Pat found the

230

answer to be 13/20. Mrs. Beacham notices that in converting 13/20 from
fraction form to decimal form, Pat put .75 down as an answer. ”How many
203 are there in 130?” Mrs. Beacham asks Pat, pointing to the paper
where Pat is dividing 20 by 13. ”Six with 10 left over” answers Pat.
”Then how can you have .15?” Mrs. Beacham says, emphasizing the number
7. Mrs. Beacham asks Pat to check the answer on the calculator. Pat
punches in 20 divided by 13 and says ”Oh, I messed up! I should have
.55, not .15.” Mrs. Beacham tells Pat to be careful so as not to make
arithmetic mistakes, and Pat continues using the calculator to check
other problems.

After working with Pat, there are about two minutes left before
the end of the period. Mrs. Beacham asks the class to clean-up and

finish their assignments for homework.

 

APPENDIX D
This appendix is a complete investigation from the student edition of
Covering and Surrounding (CMP, 1992a, pp. 1-12). This investigation is
representative of the format and kinds of questions/tasks present in all

of the investigations in Covering and Surrounding:

Investigation 1: Measuring and Designing Rooms

Mathematics is used in the world around us in many ways. Some of these
important uses of mathematics involve measuring things. This unit is
about measurement. The questions ”How big is it?” ”How small is it?”
”What is the biggest or most?” and ”What is the smallest or least?” will
be asked again and again in this unit. You are going to be asked to
think about what it means to measure something and to plan how you can

actually measure properties of objects or events.

Architects and builders deal with measurement in all aspects of their
work. In this first investigation we are going to help an architectural

firm measure and design interesting rooms.

Mrs. I. Wana Hide is a friend of yours. She has asked an architect, Mr.
Really Dull, to design a special room to add onto a covered walkway
attached to her house. Mr. Dull has divided the plan of the room up
into squares because the carpeting needed to cover the floor of the room
comes in large squares. The sections of wall that are used to build the
room may have either a door or a window made into them. In Mr. Dull's
design, each wall section has a door or window. This is the design Mr.

Dull produces for Mrs. Hide:

231

232

 

I=l

_ _ indicates a window

 

 

 

 

 

—I=I————idoo, I———-!=i—

As an architect, Mr. Dull uses grid paper to design his plans, like in
his design of Mrs. Hide’s room. Architects also use models in their
work. Physical or pictorial models are used to represent and learn
about other things that are too big or small to study easily. In
designing rooms it is much easier to build models of rooms to make

different designs than to build all of the actual rooms.

Take 12 tiles, which will be used to model carpet sections, and arrange
them in the same design as Mr. Dull’s. Note that it takes 12 carpet
tiles to cover the room and there are 14 wall sections, or tile edges,
in the room. In Mr. Dull's design each wall section has a door or
window, but this may not be the case for every room. The names that
mathematicians use for the kinds of measures you have computed for each
room are the area and perimeter. You have probably already realized that
the number of square carpet tiles needed to cover the floor is a measure
of area and the number of wall sections around the edge of the room is a
measure of the perimeter. Because units which measure perimeter are
straight, like the tile edges, units used to measure perimeter are also
called linear units. Because units which are used to measure area are

square, like the tiles, units used to measure area are also called

233

square units. Therefore, the room Mr. Dull designed has an area of 12

tiles or square units, and a perimeter of 14 tile edges or linear units.

In the following problems you will use the tiles to make models to help

you to measure and design rooms.

 

Problem.l: Mrs. Hide likes the amount of floor space in Mr. Dull's
design, but she wants more windows and a more interesting shape for her

room. She asks you to help her out.

a) Use 12 square tiles to create a floor plan design for Mrs. Hide.
Remember that a window or a door can go into each section of wall space.
Mrs. Hide tells you that she wants at least 14 sections of wall space,

including windows and doors, in the room you are designing.

b) After you have designed the floor plan for the room make a drawing

to show the location of the door and where each window is. Write a

 

paragraph to tell Mrs. Hide why your design is better than Mr. Dull's.

 

234

 

Problem.2: The Add-a-Room Architectural Firm works with its
customers to meet their special needs when designing a room. Some
customers want rooms that have a great deal of window space to let
in lots of light. Others need a room with less sunlight. The
floors of all the rooms the architectural firm produces are
covered with large square carpet tiles. The wall sections are as
long as the edge of a carpet tile; a window or a door may be put

into each wall section.

Julio and Julianna work for the firm. They drew the following

floor plans for rooms labeled A-J.

a) A customer needs at least 10 square units of floor space, which

rooms are possible?

b) A customer wants at least 15 windows and one door (16 sections

of wall) in the room. Which plans are possible?

c) Examine each floor plan and find the area and perimeter.
Organize your answers so that for each room you record the
letter of the room design, the number of carpet tiles needed to
cover the floor (the area), and the number of wall sections in
the room (the perimeter). One way of organizing your data

would be to make a table.

 

 

 

These are the floor plans for Rooms A—J. Here is the picture of one of
the large square carpet tiles. As you can see, your small square tiles
can be used to represent the carpet tiles to cover the floor plans of

the rooms.

 

 

 

 

235

Room A

 

 

 

 

Room B

 

 

 

 

 

 

 

 

236

Room C

 

 

 

 

 

 

 

 

 

Room D

 

 

 

 

 

 

 

237

Room E

 

 

 

 

 

 

 

 

Room F

 

 

 

 

 

 

 

 

 

 

 

 

238

Room G

 

 

 

 

 

 

 

 

 

 

 

Room H

 

 

 

 

 

 

 

 

 

 

 

 

 

239

Room I

 

 

 

 

 

 

 

 

 

 

240

Room J

 

 

 

 

 

 

 

 

 

 

 

 

Follow-Up Question:

1. Use the data in your table to decide which three designs you think
are the ”best”. You may want to consider such things as what you think
each room is to be used for as you make your decisions. Be sure you can

give a convincing argument to support your choices.

241

 

 

Problem.3: The Add-a-Room Architectural Firm is always looking

for clever designers. The following questions give you a chance to

be a designer. Build your answers using the tiles, but be sure to

make a record of your response in your journal. One easy way to

record your work is to draw a picture on a piece of grid paper to

represent what you built with the tiles. Each question refers to

one of Julio and Julianna’s rooms.

a) Can you make a room from tiles with the same area as Room A,

but with a smaller perimeter? Why or why not?

b) Can you make a room from the tiles with the same area as

Room C, but with a smaller perimeter? Why or why not?

Room E can be made from Room D by removing 3 tiles. This makes
the area of E three less than the area of D. How does the

perimeter of D compare to E? Explain why.

Look at Room G and Room H. Without using your tiles, find the
area and perimeter of the new figure that can be made by

pushing Room G and H together to form a rectangle.

Rooms F and I have the same perimeter. Cover F with your tiles.
Can you rearrange the tiles to match the shape of Room I? Why

or why not?

Make a tile pattern to match Room B. Moving only one of the

tiles make a new figure that has a perimeter of 14.

 

 

242

Applications - Connections - Extensions

Applications

1. Stephanie claims that she can add tiles to the plan for Room J to
create a new plan with an area of 6 square units more than Room J,
but with a perimeter of 22 units. Using your tiles, make Stephanie's
figure. Why are the perimeters of Room J and Stephanie's room so

different?

2. Look back at Room B. Arrange tiles that match B. Move exactly 2 tiles

and make another figure with perimeter 14. Record your figure.

3. Arrange your tiles to match Room B. Can you rearrange tiles to make a

figure with a perimeter of 30? Why or why not?

4. In tile units what is the area and the perimeter of the rooms below:

 

 

a) b)

 

 

 

 

 

 

5. Copy rooms a) and b) above on a sheet of paper. Use a centimeter
ruler to measure each edge of the rectangles a) and b). What is the

perimeter of each room in centimeters?

6. Use a centimeter ruler as a guide to draw in the square centimeters

to show the area of the rooms in problem 4 in square centimeters.

243

Connections

7. Suppose you have 18 square tiles. The edge of a tile is one unit
long. What are the perimeters of the rectangles you can make with an
area of 18 square tiles? Draw a picture on grid paper of each
rectangle you found. How do you know you have found all possible

rectangles?

8. If you walk around the perimeter of a rectangle, through how many

degrees have you turned when you get back to where you started?

Extensions

9. Using 12 tiles build the floor plan for a rectangular room that will
provide space for the most wall sections. Now rearrange the tiles to
make a floor plan that will provide space for the fewest wall

sections. Record your solutions.
10. Look back at Room I.

a) Find the area and the perimeter of the room. Use the square tile
below as the square unit to measure the area and perimeter of

Room I:

 

 

 

 

b) How would your answers to part a) change if this

 

is the square unit for measuring area?

APPENDIX E
This is the supporting material provided for teaching Investigation 1 in
Covering and Surrounding. The material is from the Covering and

Surrounding Teacher’s Edition (CMP, 1992b, 22-29):

Investigation 1 - Measuring and Designing Rooms
The context for this investigation is measuring and designing rooms.

The mathematical and problem solving goals for students are:

* Developing awareness of the differences between area and perimeter.

* Developing concept images (i.e., mental images) that help distinguish

between area and perimeter.

* Finding area and perimeter through covering with tiles and counting

edges and numbers of tiles.
* Understanding that two figures with the same area may have different
perimeters and two figures with the same perimeter may have different

areas .

* Visualizing what changes occur when tiles forming a figure are

rearranged, added, or subtracted.
* Organizing information in a table.
Iaterials Needed
* Square l-inch tiles, 24 per student (if you have different size tiles, the

figures will need to be redrawn to match the tile)

* 1-inch grid paper (Labsheet 1.1).

244

245

The story of designing rooms in the student text can be used to launch
the investigation. It is important that students get a clear picture of the
relationship between the physical tiles representing squares of carpet and the
prefabricated wall sections that are the same width as the edge of a carpet
tile. The prefabricated sections can each contain either a window or a door,
but may also be blank wallspace. You may want to prepare a few cut cuts from

cardboard with a door or a window drawn on to show students a floor plan:

 

 

 

 

 

 

 

 

 

 

 

 

door window

l-inch square (tile size) we]! sections

Once the students understand how the floor plans are modeled by the tiles,
then they are ready for the first problem, designing a floor plan with square
units of floor space but with more windows and doors. This part of the

investigation can be done individually or in pairs.

The second and third problems involve the set of floor plans A-J, given
in the student text. These problems provide an excellent opportunity to help
students begin to build flexible understandings of perimeter and area and to
help students see the power of organizing information in a table. The
questions in problem three also involve visual reasoning as the students

explore the changes that occur when tiles are moved, added or removed.

In the ACE, problem 4, 5, and 6 are designed to help students begin to
connect measuring dimensions and finding perimeter and area. Some students
may quickly see short cuts or create formulas for rectangles. Other students
will need to continue to cover and count (i.e., cover figures with tiles and
count them to find area) or surround and count (i.e., count the edges of the

tiles that surround figures to find perimeter).

246

Problem 1: Room designs with 12 square tiles will vary; the perimeter

should be at least 14 tile edges long.

Problem. 2

a)

b)

Rooms D(a=12), G(a=11), I(a=10), and J(a=16) are possible rooms
with at least 10 square units of floor space.
Rooms C(p=20), F(p=l6), G(p=24), H(p=20), I(p=16), and J(p=34) are

possible rooms with at least 15 windows and one door.

8mm Area Estimate:
A 9 12
B 7 12
c 9 20
D 12 14
E 9 14
F 9 16
c 11 24
H 9 20
I 10 16
J 16 34

Follow-Up: Students’ answers will vary — what is important is they

have defensible arguments for selecting the three designs that they

think are best. For example, a student might choose buildings G, I,

and J because they are constructed with at least 10 tiles and have 16

windows so they are roomy, bright working environments. Other

students might go with the three buildings using the least number of

tiles because they are the least expensive to construct.

Problem 3

1.

a) No-—Room A is a square, with the largest area possible for its

given perimeter.

b) Yes--Examples are given in Room A, E, and F.

The perimeter will still be the same, because removing the tiles

does not affect the perimeter.

. The new rectangle Room will have an area of 20 and a perimeter of

18.

. No--The area of I = 10, the area of F = 9. An area of 9 can't be

arranged into an area of 10.

247

5. Here is one solution; others are possible.

 

i —

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Applications

1. The perimeters are so different because the 6 tiles are used to

”fill-up” the middle of the spiral. Although the area has
increased by 6 square units, the perimeter decreases because the

interior of the figure has no ”inside” perimeter to count.

Here is one solution; others are possible.

 

 

 

 

——>

 

 

 

 

 

 

 

 

 

. No--Since B has an area of 7 square units, the greatest perimeter
would have to be less than 7 x 4 = 28. In other words, the
perimeter of each individual tile is 4, so a figure made up of 7

tiles could never have a perimeter larger than 28.

4. a) area = 3.75 square units, perimeter = 8 units

b) area = 3.75 square units, perimeter = 8.5 units

5. a) perimeter = 21 cm

248

b) perimeter = 21.5 cm

 

 

a) b)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Students do not need to label a) and b) precisely - what is most

important is that they are able to draw a centimeter grid over each

rectangle.

Connections

7.

There are six different rectangles possible: 1x18, 2x9, 3x6, 6x3,
9x2, 18x1. These rectangles have perimeters of 38, 22, 18, 18, 22,
and 38, respectively. Some students may only find three
rectangles, because each of the six rectangles has a reversal,
e.g., 1x18 and 18x1. However, the rectangles can still be
considered different because they have different bottom and side
edges, i.e., they are interchanged. In future investigations in
this unit students will need to record reversal rectangles
separately. Students should know that they have found all of the
possible rectangles because they have used all the different factor
pairs of 18 as edges of the rectangles.

You turn through 360 degrees. This is because, if a student traces
his or her path, he or she will turn through a complete circle.

Alternately, students might reason that four 90-degree turns are

249

made walking around the rectangle for a total of 4x90 = 360

degrees.
Extensions
9. A 1x12 or a 12x1 rectangle room--with a perimeter of 26--would
provide space for the most windows(25) and one door. A 3x4 or a
4x3 rectangle room-—with a perimeter of 14--would provide space for
the fewest windows(13) and one door.
10. a) The area of the building is 10, the perimeter is 16.

b) With the new unit, the area is 40 and the perimeter is 32. This

is because there are four of the smaller units for every one of the
larger units, so the area with the smaller unit is 10 x 4 = 40.
Using the new unit, there are 2 units of perimeter for every unit
of perimeter with the old unit, so the perimeter with the new unit

equals 16 x 2 = 32.

250
Labsheet 1.1 - Tile Grid Paper

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

251

Labsheet 1.2 - Room Table

 

Room

Perimeter

Area

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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