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This is to certify that the
dissertation entitled

Waliking a Straight Line: Introductory Discourse on Linearity in
Classrooms and Curriculum

presented by

Christopher Danielson

has been accepted towards fulfillment
of the requirements for the

PhD degree in Mathematics Education

 

 

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To AVOID FINES return on or before date due.
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DATE DUE

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WALKING A STRAIGHT LINE:
INTRODUCTORY DISCOURSE ON LINEARITY
IN CLASSROOMS AND CURRICULUM
By

Christopher Danielson

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Mathematics

2005

ABSTRACT
WALKING A STRAIGHT LINE:
INTRODUCTORY DISCOURSE ON LINEARITY
IN CLASSROOMS AND CURRICULUM
BY
Christopher Danielson

The current curricular reform in US mathematics education has changed many
aspects of classroom teaching. Commonly, discussions about this curricular reform
presume an unproblematic relationship between textbooks and classroom instruction.
This study contributes to the understanding of the relationship between one published
reform curriculum, Connected Mathematics (CMP) (Lappan, Fey, Fitzgerald, F riel &
Phillips, 2001) and classroom instruction. The study characterizes teaching and learning
in terms of communication patterns—discourse—and analyzes the discourse of CMP, of
a traditional US curriculum, Mathematics, Structure and Method (Dolciani, Sorgenfrey &
Graham, 1992), and of two teachers in urban classrooms—focusing on the introductory
lessons on linear relationships in each case. Results include full descriptions of the
introductory discourse on linearity in the textbooks and changes that the CMP textbook

discourse undergoes as the curriculum is implemented in these two classrooms.

For the students of Humboldt Junior High School—past, present and future.

My questions come from you.

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ACKNOWLEDGMENTS

Thanks go first and foremost to Rachel and to Griffin for your love and your
patience. Of course, thanks are also due to the rest of my family: my parents and my in-
laws have all been a great source of support over the long haul.

My committee members—Glenda Lappan, Jack Smith, Jacob Plotkin, Joan
F errini-Mundy and Anna Sfard—have all been essential in this work. Thank you for your
questions, your critiques and your suggestions at every phase of my work.

I need to thank Anne Bartel, without whom none of this work would have begun,
nor would it have been finished.

Judith Martus Miller deserves more gratitude than this page can contain. She
provided support, short—term housing, friendship and countless favors.

Finally, thanks go to fellow graduate students who have been so supportive along
the way: Sarah Sword, Harrison Hwang, Amy Kuiper, Kelly Rivette, Marie Turini and
Deb Johanning. It has been a long road that you each helped to smooth for me in some
way. Thank you.

Ms. M and Ms. H... Je vous remercie mille fois.

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TABLE OF CONTENTS

List of Tables ..................................................................................................................... vi
List of Figures ................................................................................................................... vii
Chapter I: Introduction ........................................................................................................ 1
Chapter 11: Literature Review ............................................................................................. 5
Language ......................................................................................................................... 5
Discourse ......................................................................................................................... 7
Linear Relationships ..................................................................................................... 10
Curriculum and Implementation ................................................................................... 1 1
Chapter III: Theoretical Framework ................................................................................. 16
Background ................................................................................................................... l6
Introductory Notions ..................................................................................................... l7
Discourse ....................................................................................................................... 20
Word Use .................................................................................................................. 24
Visual Mediators ....................................................................................................... 27
Narratives .................................................................................................................. 28
Routines .................................................................................................................... 30
Chapter IV: Research Methods ......................................................................................... 33
Texts .............................................................................................................................. 34
Teachers ........................................................................................................................ 37
School Contexts ............................................................................................................ 39
Research Questions ....................................................................................................... 41
Question 1: What characterizes the introductory textbook discourse on linearity in
Connected Mathematics? .......................................................................................... 42
Question 2: What characterizes each teacher’s discourse on linearity? ................... 61
Question 3: What changes does the discourse of the curriculum undergo in its
implementation by each teacher? .............................................................................. 65
The Researcher .............................................................................................................. 69
Summary ....................................................................................................................... 70
Chapter V: Textbook Discourse ........................................................................................ 71
CMP .............................................................................................................................. 73
Word Use .................................................................................................................. 78
Visual Mediators ....................................................................................................... 85
Narratives .................................................................................................................. 92
Routines .................................................................................................................. 102

Dolciani ....................................................................................................................... l 1 1

Word Use ................................................................................................................ l 15
Visual Mediators ..................................................................................................... 122
Narratives ................................................................................................................ 127
Routines .................................................................................................................. 133
Comparison ................................................................................................................. 138
Chapter VI: Classroom Discourse .................................................................................. 143
Ms. M’s Discourse ...................................................................................................... 146
Word Use ................................................................................................................ 148
Visual Mediators ..................................................................................................... 155
Narratives ................................................................................................................ 177
Routines .................................................................................................................. 185
Ms. H’s Discourse ....................................................................................................... 197
Word Use: Equation ................................................................................................ 198
Word Use: Rate ....................................................................................................... 200
Word Use: Linear .................................................................................................... 201
Visual Mediators ..................................................................................................... 203
Narratives ................................................................................................................ 217
Routines .................................................................................................................. 223
Discussion ................................................................................................................... 229
Ms. M’s Discourse .................................................................................................. 229
Ms. H’s Discourse ................................................................................................... 232
Changes in the CMP Discourse in Implementation ................................................ 234
Chapter VII: Conclusion ................................................................................................. 240
Findings ....................................................................................................................... 240
Textbook Discourse ................................................................................................ 241
Classroom Discourse .............................................................................................. 243
Additional Findings ................................................................................................ 244
Contributions of the Theoretical Framework .............................................................. 247
Systematic Look at Curriculum .............................................................................. 247
Textbooks and Classrooms ..................................................................................... 248
Multiple Perspectives on Teaching ......................................................................... 248
Limitations .................................................................................................................. 249
Student Learning ..................................................................................................... 249
Dolciani classrooms ................................................................................................ 250

A Longitudinal Look ............................................................................................... 251
Conclusion .................................................................................................................. 252
References ....................................................................................................................... 253

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LIST OF TABLES

Table 1: Research Questions and Data Sources ............................................................... 42
Table 2: The Problems in Moving Straight Ahead Investigation 2 .................................. 75
Table 3: Narratives in Investigation 2 of the CMP unit Moving Straight Ahead ............. 94
Table 4: Themes in each discourse on tables ................................................................. 125
Table 5: Themes in each discourse on graphs ................................................................ 125
Table 6: Themes in each discourse on equations ........................................................... 125

Table 7: Narratives in the study of linear relationships in Lessons 8-2 through 8-6 of

Dolciani ........................................................................................................................... 128
Table 8: Themes in each discourse on tables ................................................................. 175
Table 9: Themes in each discourse on graphs ................................................................ 175
Table 10: Themes in each discourse on equations ......................................................... 175
Table 11: Themes in each discourse on tables ............................................................... 204
Table 12: Themes in each discourse on graphs .............................................................. 204
Table 13: Themes in each discourse on equations ......................................................... 204
Table 14: Themes in each discourse on equations ......................................................... 236

vii

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LIST OF FIGURES

Figure 1: An Input-Output table from Ms. M’s instruction with her pilot curriculum,

College Preparatory Mathematics (CPM, 2002) .............................................................. 40
Figure 2: Algebra units in CMP ....................................................................................... 43
Figure 3: Algebra chapters in Dolciani ............................................................................ 45
Figure 4: The model table in CMP Problem 2.2 .............................................................. 57
Figure 5: The model table in CMP Problem 2.2 .............................................................. 97
Figure 6: A visual mediator in Dolciani ........................................................................ 126
Figure 7: Major activities in Ms. M’s Investigation 2 lessons ....................................... 147
Figure 8: Three student produced tables used in Ms. M’s summary of Problem 2.4 159
Figure 9: Graph of the three pledge plans in Problem 2.3 ............................................. 163
Figure 10: Ms. M transforms the graph showing Alana’s pledge plan .......................... 165
Figure 11: Ms. M’s tables from the warm-up on 11/2/05 .............................................. 179
Figure 12: Any representation can be generated from any of the others ....................... 185
Figure 13: Ms. M’s tables from the warm-up on 10/27/05 ............................................ 190
Figure 14: Major activities in Ms. H’s Investigation 2 lessons ..................................... 198

Figure 15: The basic form of the tables in Ms. H’s warm-up preceding Problem 2.3 .. 219

Figure 16: A non-linear table from Ms. H’s instruction ................................................ 220

viii

Chapter I: Introduction

Consider the following two algebra problems, each from an American junior-high
school mathematics curriculum:
Problem I

Use a graph to solve the system of equations;

y—x=4
3y+x=8

(Dolciani, Sorgenfrey and Graham, 1992, p. 284)

Problem 2
In Mr. Goldberg’s gym class, Emile finds out that his walking rate is 2.5 meters
per second. When he gets home from school, he times his little brother Henri, as
Henri walks 100 meters. He figures out that Henri’s walking rate is 1 meter per

second.

Henri challenges Emile to a walking race. Because Emile’s walking rate is faster,

Emile gives Henri a 45-meter head start.

Emile knows his brother would enjoy winning the race, but he does not want to

make the race so short that it is obvious his brother will win.

What would be a good distance to make the race so that Henri will win in a close
race? Describe your strategy, and give evidence to support your answer. (Lappan,
Fey, Fitzgerald, Friel & Phillips, 2001c, p. 21)

Each of these problems asks students to solve a linear system. In this sense, they

have the same mathematical goals. Yet the two problems have profound differences that

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exemplify aspects of the ongoing mathematics curriculum reform in US. classrooms
spurred by the publication of the National Council of Teachers of Mathematics’
Curriculum and Evaluation Standards (NCTM, 1989). The surface difference is an
important one. Problem 1 uses mathematical vocabulary: graph, solve, and equation.
Problem 2 uses everyday vocabulary: walking rate, head start and finish line.

Deeper differences can be found, too. Problem 1 explicitly states the system to be
solved, in the form ax + by = c (called standard form). Problem 2 requires students to
write the system, and the story is more naturally represented in the form y = mx + b
(called slope-intercept form).

The model solution to Problem 1 is demonstrated in the Student Edition, and it
involves finding three solutions to each equation, listing the solutions in a table and using
these solutions to graph two lines. The point of intersection of the lines is estimated, then
checked in the equation.

In contrast, there are three model solutions to Problem 2, and they are in the
Teacher Edition: 1. guessing and checking, 2. generating a table that continues until the
solution is found, 3. making a graph to find the point of intersection.

These two problems suggest some of the changes the curriculum reform has
brought to classrooms. In reform materials, mathematical ideas are commonly introduced
by having students reason about stories (called contexts in this study). Algebra tends to be
taught as a way of understanding changing variables, rather than a set of techniques for
solving equations. Students are expected to develop and choose algorithms, rather than

practice given algorithms.

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In institutionalizing the introduction of contexts before technique, and intuition
prior to formalization, United States middle school curriculum has undergone significant
change. There is empirical evidence that this change has had little effect on the traditional
focus of instruction: algorithmic skill, but that it has had substantial effect on students’
ability to solve problems (e.g. Ben Chaim, Fey, Fitzgerald, Benedetto & Miller, 1998;
Riordan & Noyce, 2001). The quantitative study of student achievement is important, but
in order to truly understand instruction in this new curricular context, with the hope of
improving both published curriculum and classroom instruction, we need to study it
qualitatively as well. This type of understanding of instruction is the focus of the present
study.

In particular, this study is designed to further our understanding of two issues:

1. How, exactly, reform curriculum differs from traditional curriculum, and

2. How classroom instruction of a reform curriculum differs from what is printed

on the page.

The opening examples point to the kinds of differences this study uncovers in
curriculum; the roles of mathematical and everyday vocabulary, the use of mathematical
representations (in algebra—the context for this study—these are representations of
functions—tables, graphs and equations), and the mathematical behaviors encouraged in
texts written for students and for teachers. Additionally, this study finds differences in the
content of introductory algebra instruction.

Together, the analysis of vocabulary, of the use of representations, of content and
of the mathematical behaviors encouraged in curriculum and in classrooms make up a

form of discourse analysis. The details of this analysis are spelled out in Chapters III and

 

 

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meant by discourse, and that analysis of this activity is the focus of this study.

Finally, some unexpected findings are reported about the differences between
published and implemented curriculum. While the major themes of algebra in CMP are
represented in each classroom, the treatment of equations in each classroom differs from
the curricular treatment.

Chapter II of this paper reviews research related to the present study. Chapter III
presents a theoretical framework for the study. Teaching and learning are framed as
problems of communication and the tools of discourse analysis are outlined as they
pertain to the problems in this study. Chapter IV presents the research questions and
outlines the methods that will be used to answer them. Chapter V analyzes one traditional
and one reform curriculum with the goal of reaching a precise characterization of the
reform cum'culum’s approach to introductory algebra instruction. Chapter VI compares
this characterization to the instruction in two classrooms, seeking how the discourse of
the printed page changes as teachers bring it to life in classrooms. Finally, Chapter VII
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Chapter [1: Literature Review

Chapter I introduced this study as concerning the nature of a particular reform
mathematics curriculum—Connected Mathematics (CMP) (Lappan, Fey, Fitzgerald, F riel
& Phillips, 2001) and the relationship between this curriculum and classroom instruction.
The chapter also hinted that the study of communication patterns—discourse—would
form the heart of the analysis. As such, this study sits at the intersection of several large
fields of inquiry in mathematics education, including language/linguistics,
discourse/discourse analysis, the teaching and learning of linear relationships (an algebra
topic), and curriculum and implementation. In this chapter, relationships between the
present study and each of the fields listed above are briefly addressed in order to situate
subsequent chapters.

Language

The language used in teaching and learning mathematics has been much
studied—both empirically and theoretically. One of the fundamental concepts in the
analysis of language in mathematics classrooms is the notion of a register. According to
Halliday (1978),

A register is a set of meanings that is appropriate to a particular function of

language, together with the words and structures which express these meanings.

(p.195)

Halliday defines register in order to discuss how mathematical language differs from
everyday language. While he includes brief mention that a register is not just about

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and creation of words. For instance, set is a word with a technical mathematical meaning
that is related to, but different from, its meaning in the everyday English register.

Halliday’s discussion is theoretical and analytical. Pimm (1987) picks up
Halliday’s ideas and gives them practical application. Pimm too is concerned with words
and their meaning in his discussion of register. He offers carefiil analysis of the use of
such words as any, if-then, and diagonal and suggests that register confusion is the source
of what are often considered mathematical misconceptions. Register confusion involves
participants in conversation having different expectations about the meanings of words
due to operating in different registers. For example, when a student asserts that a
rectangle oriented with one vertex at the bottom of the page has “four diagonals,”
meaning that none of the four sides is horizontal or vertical, the mathematical meaning of
diagonal conflicts with the everyday meaning of the term—this is register confusion.
Pimm is correct that register confusion is one source of misconceptions, but surely there
are others not accounted for by this idea.

The notion of a mathematics register is important, and it applies to parts of the
present study—those parts that seek to understand how mathematical language (a
mathematical register) develops from the everyday language (a non—mathematical
register) that students bring to the classroom, and that CMP exploits through the
contextual problems it presents.

The relationship between everyday language and mathematical language has been
studied by several others, including Herbel-Eisenmann (2000 & 2002) and Mamokgethi,
Adler, Reed and Bapoo (2002). HerbeI-Eisenmann describes the practice of a CMP

teacher who encourages and adopts the use of students’ everyday language for

 

 

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mathematical ideas before introducing more formal mathematical language and suggests
that this practice makes mathematics accessible to a wide variety of students. For
example, this teacher uses the phrase swoopy down curve to describe graphs of
exponential decay relationships. This phrase was introduced by her students.

Mamokgethi, et al., describe a more complex linguistic environment in which the
everyday language of students is different from the language of instruction.
Correspondingly, these researchers’ findings are complicated. They find teachers making
efforts to navigate the “journey” from informal talk in main (home) languages to formal
talk and writing in English (the language of instruction). They find teachers working to
incorporate into their instruction opportunities for students to learn from talk, but they
find that journey to be treacherous, “stranding” many students and teachers. The
researchers introduce the concept of code switching in which the teacher moves between
languages (e. g. formal mathematical English to informal Venda) in order to make
concepts more understandable to students. The linguistic environments in my study are
not as complex as described by Mamokgethi, et al., in that all speakers described are
fluent in English and English pervades the surrounding culture in ways that it does not in
the South African schools they describe. Nonetheless, the metaphor of a journey from the
familiar to the formal is a useful one for the process I wish to investigate in my study—
the process of deriving mathematical meaning from contextual problem solving
experiences.

Discourse
The language studies just cited focus primarily on words and the way that words

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broadly. Certainly many of the concerns of these language studies are also concerns in
the analysis of discourse, but discourse analysis tends to take a larger view and to ask
questions about communication systems as a whole. In educational research, discourse
has been used to characterize teaching and learning (e.g. Greeno, 2003 and Sfard, 2001),
in contrast to more behaviorist notions of transfer (e.g. Thorndike & Woodworth, 1901)
or to cognitive characterizations that focus on acquiring cognitive structures (described
nicely by Sfard, 1998).

Resnick (1988) is among those who argue for such a characterization in
mathematics (as is Lampert, 1990). Resnick argues that mathematics traditionally has
been taught as a “well-structured discipline” in which truths are pre-determined and
organized for the learner. She argues instead that mathematics should be taught as an “ill-
structured discipline,” in which participation in discourse is the learning goal, rather than
the mastery of some set of rote skills. This participation would involve seeking and
organizing mathematical truth. My study does not make assertions about how
mathematics should be taught, but Resnick’s argument about the nature of mathematical
activity resonates with my characterization in Chapter V of CMP’s mathematical
discourse as heuristic—meaning that the discourse is focused on strategies for making
and testing conjectures.

Sfard (2001) characterizes teaching and learning in terms of discourse by
rephrasing the question What is to be learned? as In what ways should students’
communication patterns change? This shift clarifies the question and makes it operative.

In addition, Sfard provides a framework (the communicationalfi'amework) with which to

 

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distinguish mathematical discourse from other discourse genres. Together, these tools
provide the framework for my study, which is detailed in Chapter I [1.

One consequence of the communicational framework is a characterization of an
important (although perhaps not necessary) condition for learning: communicational
conflict. When knowledge is conceived of cognitively, we may refer to cognitive conflict
in which a learner’s conceptions do not match an objective reality or other people’s ideas.
This disagreement, when noticed, can lead to a reformulation of these conceptions. But
there is a difficulty with the concept of cognitive conflict from a research perspective; we
cannot directly observe a learner’s conceptions. In the communicational framework, we
may similarly talk about communicational conflict. This is when a learner’s
communicational patterns disagree with the patterns of others. Without using Sfard’s
terminology, Moschkovich (1996) describes communicational conflict and its resolution
in her study of students trying to decide whether the line for y = x + 5 is steeper than the
line for y = x. One student focuses on the y-intercept, while the other focuses on the
angle the line makes with the x-axis. This narrative conflict arises only when the pair is
asked to show what a steeper line would look like. The first student, using a pen to
represent the new line, moves the pen higher. The second rotates the pen counter-
clockwise.

Sfard and Avgil (2001) detail communicational conflict in the learning of
negative numbers. For instance, they cite a conversation in which students and teacher
disagree about where to find the rules for multiplying integers with opposite signs.
Students argue for extending the rules from addition of integers in which the sign of the

number with the greater absolute value is the sign of the sum, while the teacher tries to

 

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help students to apply more formal rules for deciding the sign of the product. In this case,
both the procedure (how do I multiply integers?) and the modes of justification (how do I
know my procedure is correct?) are in conflict. My study uses a special case of
communicational conflict—narrative conflict. Narrative conflict occurs when two
mathematical assertions are in opposition, such as when the two students in
Moschkovich’s study disagree about whether steeper lines have larger y-intercepts or
greater angles with the x-axis. Teachers, textbook authors and (often) students seek to
resolve narrative conflicts once they are identified and these instances become rich
opportunities for learning.
Linear Relationships

There have been several studies involving the teaching and learning of linear
relationships in reform mathematics classrooms. Because these studies examine students’
ways of speaking and thinking about linear relationships in realistic problem situations,
the analytic focus is on how students make sense of these situations and the tools they are
given to analyze them (tables, graphs and equations). Herbel-Eisenmann’s classroom
study (2000), cited above, documents the use of contextual and transitional language for
slope, for instance. The phrase goes up by, describing the increment in a linear table, is an
example of transitional language. A different classroom study in a high school reform
curricular context (Lobato, Ellis & Munoz, 2003) confirms that this language is
widespread, rather than idiosyncratic and argues that, depending on how teachers focus
student attention, the phrase goes up by can lead to a variety of correct and incorrect
generalizations. A study of CMP student understanding of linear relationships (Smith,

Herbel-Eisenmann, Star & Jansen, 2000) argues that this language is pervasive in student

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talk and thinking. Both the Smith, et al., and the Lobato, et al., studies find that students
develop an understanding of the y = mx + b form of linear equations that involves a
starting point (b) and an amount that x goes up by (m). However, both studies suggest
that when students talk about what a table goes up by, they may see slope as an isolated
increment (a difference) rather than as a ratio of two related values. In the graphical
representation of linear relationships, the importance of steepness as a qualitative (i.e.
descriptive, but non-numeric) measure of slope has been observed in several studies (e.g.
Smith, et al., 2000 and Moschkovich, 1996). Each of these issues—understanding the
ratio, not just the difference nature of a table’s increment; starting points; interpretations
of the equation y = mx + b; and the isolation and identification of steepness—plays a role
in the present study.

Generally, these studies show that students use the contextual problems in reform
curriculum to generalize mathematical ideas, but that this process can be problematic.
The studies further demonstrate that there are various conceptions of the important
aspects of linear relationships (especially slope and y-intercept) that students develop, but
that vary by representation.

Curriculum and Implementation

This is a study of curriculum in classrooms. As is acknowledged by the National
Research Council (2004), the meaning of the term curriculum can vary widely in
research. In some cases, curriculum is a list of district topics; in others it is a set of
realized learning outcomes. This study follows the NRC’s definition.

A curriculum consists of a set of materials for use at each grade level, a set of

teacher guides, and accompanying classroom assessments. It may include a listing

11

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of prescribed or preferred classroom manipulatives or technologies, materials for

parents, homework booklets, and so forth. (p. 38)
The NRC study is a survey of existing studies of the effectiveness of K—12 mathematics
curricula—studies that have in large part been spurred by the curriculum reform that
followed the 1989 NCTM Standards. The NRC study categorizes curriculum
effectiveness studies in four categories: Content Analysis, Comparative Studies, Case
Studies and Syntheses. The study concludes that there is insufficient evidence to support
the effectiveness of particular programs. Instead of arguing for this effectiveness, the
study’s report gives guidelines for judging the quality of current and future studies of
curricular effectiveness. My study is not a study of the effectiveness of curriculum.
Methodologically, my study combines elements of the NRC categories of Content
Analysis and Case Studies. Chapter V of my study compares two curricula in order to do
a careful analysis of the content and instructional support in CMP, but the scope of my
analysis is too limited to meet the NRC guidelines for a Content Analysis study. These
guidelines call for an analysis of the topics for at least a full grade level. My study instead
focuses on depth—looking at the introduction to a single topic. Chapter VI of my study
analyzes classroom instruction in comparison to published curriculum, but here too, the
study is too limited in scope to be considered a Case Study for effectiveness under the
NRC guidelines. In any case, the present study is not concerned with effectiveness per se,
although this is always an important concern. Instead, this study focuses on careful
description of curriculum (as defined by NRC) and of teaching in order to better

understand classroom instruction in a reform mathematics curriculum environment.

12

A study of instruction in US mathematics and science classrooms titled Looking
inside the Classroom (Weiss, Pasley, Smith, Banilower & Heck, 2003) evaluates lesson
quality in these classrooms. Most of the components the Weiss, et al., study evaluates in
each lesson are relevant to my study, including lesson design, lesson implementation and
the mathematics content of the lesson. Weiss, et al., however, are concerned with
evaluating lesson effectiveness on a national scale rather than with rich description of a
small number of classrooms, as in the case of my study. Additionally, Weiss et al. report
that the influence of curriculum on the topics in mathematics and science lessons is
reported by only about half of the teachers in their study, whereas the observations in my
study are of classrooms where the curriculum very much dictates the topics of instruction.

The Senk and Thompson text, Standards-Based School Mathematics Curricula
(2003) includes studies of student achievement in reform curricula at each level:
elementary, middle school and high school. As with the two studies just cited, the
emphasis in the Senk and Thompson text is on evaluation rather than description of
curriculum and classroom practice. The evaluation of CMP in this text (Ridgway,
Zawojewski, Hoover & Lambdin, 2003) is a synthesis of studies and finds that CMP
students outperform students in curriculum that adheres less closely to NCTM’s
Standards and that CMP students make progress in technical skills, but noted a lack of
information about the implementation of the curriculum in the schools in the studies.

Two studies conducted at Michigan State University (also, that is, in addition to
my study) come closer to the focus of my study. Theule-Lubienski (1996) studied a
year’s worth of CMP instruction (her own) in order to understand how students of

different socio-economic backgrounds participated in and learned from the discussions of

13

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mathematical ideas that are part of the C MP Launch—Explore—Summarize instructional
model. She carefully analyzed classroom discussions and raised questions about a variety
of issues. In particular, her questions about how students generalize their solutions in
particular contexts to mathematical algorithms applicable on a wide variety of problems
informed the questions in my study. 7Yet there are significant differences between our
two studies. Theule—Lubienski’s study is concerned with class differences in participation
and learning. These differences are not factors that I analyze. Her data set spans a year’s
worth of instruction while mine spans a few weeks. Theule-Lubienski is concerned with
the social aspect of discourse, and especially how certain groups are privileged in or
excluded from the discourse, while I am concerned with characterizing the mathematical
nature of classroom discourse. Herbel-Eisenmann’s study (2000), cited above, examines
discourse patterns in two CMP classrooms, and in this respect is similar to my own study.
The main results of her analysis relate to the source of mathematical authority in the two
classrooms. This parallels one aspect of discourse analyzed in my study—the routines for
establishing mathematical truth. When the teacher is the source of mathematical
authority, we might expect the routine of telling described in my study to be more
prominent than when the source of authority is individual or class investigation.

In this brief review of related literature, I have shown how the present study
stands in relation to existing theory and empirical research. This study is descriptive, not
evaluative. In looking at communication patterns in general, rather than staying at the
sentence level, this study is in line with other research on discourse in mathematics

classrooms. The content of the lessons studied—linear relationships—means that this

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Chapter III: Theoretical Framework

In Chapter I, I introduced the problem underlying this study: understanding
teaching and learning in US. middle school classrooms in the wake of a particular and
significant curriculum reform. In particular, this study is designed to examine the
teaching part of this problem. In this chapter, I argue in favor of the analysis of discourse
as a construct for making progress on this problem.

Background

Teaching and learning are intertwined in obvious ways and attempts to separate
them are questionable. The questions in this study arise from published curricula, which
by nature make choices about teaching while making assumptions about learning.
Published curriculum is written with the intention of helping students to learn something
new—with the intention of teaching. Authors of curriculum make choices about which
topics to include, the order of these topics, the models to be used in explaining and
understanding, etc. These are all choices about teaching that rest on assumptions about
the students who are learning, e.g. assumptions such as the usefirlness of concrete
models, or the importance of real-life contexts to learners. Because the authors write at
some distance in time from the learners (a distance of years, typically), the authors can
never truly know their students. Instead, authors of curriculum write to idealized
imagined students. In many cases these idealizations are informed by the experiences of
actual students, but they are inarguably idealizations.

In this way, teaching is emphasized in curriculum over learning. The teaching
strategies in curriculum can be planned in substantial detail, but the learning can never be

known in advance. This study examines teaching in the carefully controlled curricular

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sense and in the messier world of real-time classroom interactions. The study takes a
curricular perspective on classroom interactions—looking at instances of teaching
without overt focus on the learning that results, for this is a major question of curriculum
design: what is the relationship between the written and the delivered curriculum?
(Lemke, 1989) Questions of learning are also important (perhaps more so) but
understanding the relationship between curriculum and classroom teaching is an
important goal of its own.

So this study seeks to understand what is being taught in mathematics curriculum
and classrooms. Sfard (2001) has observed that a difficulty in educational research is
unacknowledged disagreement about the meaning of terms such as teaching and learning.
In particular, she notes that these terms are frequently used in non-operative ways—that
definitions are not made in ways that other researchers would be able to pick up and use.
If Sfard’s call for operative terminology in educational research is taken seriously, the
statement, this stuay seeks to understand what is being taught in mathematics curriculum
and classrooms needs to be made operative. This chapter summarizes a vision of teaching
and learning in which participation in a discourse is the goal. The vision makes teaching
in mathematics curriculum and classrooms operative.

This chapter introduces important background material before defining the term
discourse. The analysis in this study rests on a characterization of mathematical
discourse, which is described in detail.

Introductory Notions
Cobb (1994) shares Sfard’s view that difficulties in communication between

constructivists and those adhering to a sociocultural view of mathematics learning are

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exacerbated by different uses of the same words, including the word meaning. He prefers
to see the two perspectives pragmatically—each is useful for understanding certain kinds
of problems and achieving different ends, but neither tells the whole story.

The constructivist view of learning asserts that learners construct understanding
and that they construct meaning in their minds. This view rests on a metaphor: that ideas
are like buildings and learners like construction workers. While this metaphor is useful in
talking about education and is an improvement on that of the tabula rasa’, it does not
easily lead to operative research questions. We cannot directly access people’s ideas: we
can only make inferences about them. These inferences are based on behaviors—
especially communicational behaviors, such as answers people write on test forms, or
responses they give to interview questions. If, instead of attempting to draw inferences
about people’s ideas through their communicational behaviors, we instead examine the
behaviors themselves closely, we may characterize learning in an operative way. This is
the spirit in which the present study uses Sfard’s communicational framework (2001). In
an attempt to describe teaching and learning in an operative way, mathematics knowledge
is characterized as participation in mathematical discourse. Then learning is defined as
change in this participation. A teacher is someone who models a discourse that learners
are attempting to participate in. Teaching is defined as engaging learners in the model
discourse.

In this communicational framework, the question What is being taught? is
tantamount to asking, What is the model discourse? In classrooms, there are typically two

discourses being modeled for students: that of a textbook, which is static and

 

' I.e. that learners are blank slates, upon which the teacher writes new knowledge.

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unresponsive to students, and that of the teacher, which is dynamic and responsive to
students’ participation.

This study uses tools analyzes discourse (in a process referred to as discourse
analysis) in order to characterize the relationship between curriculum and teaching in
reform mathematics classrooms. To summarize, the use of discourse analysis for this
study is rooted in an attempt to operationalize the question what is being taught?

In particular, the goal of this study is to better understand the workings of
classrooms in which a curriculum, Connected Mathematics (CMP, for Connected
Mathematics Project), is implemented. CMP introduces mathematical ideas through the
study of contextual problems, putting mathematical language into a non-standard role in
the curriculum. Mathematical language in C MP tends to be introduced after students have
used the ideas that the language names to solve contextual problems. Slope, for instance,
is introduced only after students have solved many problems with rates and have talked
about the steepness of lines on graphs. As we shall see in Chapter V, many aspects of
CMP that distinguish it from traditional US curricula can be framed as linguistic
differences.

This study seeks to describe part of CMP’s textbook discourse—the introductory
discourse on linearity. A variety of theoretical and pragmatic considerations dictated this
choice; these are detailed in Chapter IV. Linear relationships are the simplest predictable
algebraic relationships: for each unit change in one variable, the other variable changes
by a constant amount. Linear equations can be solved with the four arithmetic operations:
addition, subtraction, multiplication and division. These are among the reasons that

linearity is an introductory algebra topic in CMP.

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In order to keep the study manageable, the focus is not on the entire discourse on
linearity, but only on the introductory discourse on linearity. In a study of this kind, there
is a choice between focusing on one or two features of the discourse over a long stretch
of time and focusing on a larger number of features of the discourse for a short time. In
allocating resources to this study, depth had to be weighed against breadth. The choice
was made to focus on a larger set of features of the discourse for a short time, rather than
fewer features over a longer time span.

The introductory lessons on linearity in two curricula were chosen for study.
There are several reasons that focusing on the introduction to a topic is valuable. The
introductory lessons set the tone for later study. Focusing on the introductory lessons
should give insights on the discourse as a whole. The introduction to a topic tells students
what is valued in the discourse.

Before proceeding to research methods in the next chapter, it is important to make
certain terms operational. In particular, the term discourse needs definition.

Discourse

Discourse is defined as communicative activity and may take many forms: oral
speech, written text and non-linguistic communication among them. This is largely
consistent with other definitions of discourse, including Grimshaw’s (2003):

This...perspective sees discourse as spoken or written text in a language, intended

for use in the accomplishment of social ends of users (speakers, hearers, writers,

readers). (p. 27)

Surely Grimshaw would acknowledge pauses, shrugs, nods and other non-linguistic

communication as part of discourse. Such communicative activity has some standardized

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forms, called genres (ibid), that may or may not be explicitly understood as such by the
users. One problem in the study of discourse is to identify the characteristics of discourse
that distinguish one genre from another in a reliable way. Of particular interest to this
study is the genre of mathematical discourse.

A related perspective to this is that there exist specialized patterns of
communicational activity; each such pattern is a discourse. Thus, within the genre of
mathematical discourse, there may be many patterns of communicational activity—many
discourses. In this sense, mathematical discourse does not refer to a single discourse, but
to a collection of discourses with some set of commonalities. One problem in discourse
analysis is to identify these commonalities.

Earlier, I defined teaching as engagement with learners in the model discourse. By
model discourse, I mean the disciplinary (here mathematical) communication patterns of
the teacher. Thus the everyday discourse in which the teacher engages students in the
hallway is not the model discourse in this study.

This study concerns itself with two types of mathematical discourse that are
closely related: textbook and classroom mathematical discourses. In Chapter IV, I argue
that comparison of these two discourses is warranted on the grounds that textbook writers
have an idealized classroom discourse in mind as they write; their goal is to influence the
classroom discourse. While they are comparable along important dimensions, the two are
indeed different types of discourse. Textbooks are static and unresponsive to students. A
teacher’s classroom discourse is dynamic and co-constructed with students, at least in

part. In the analysis that forms the heart of this study, these differences are clear, as l

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demonstrate that several aspects of the teacher’s discourse are influenced by students’
questions and ideas.

Sfard (2005a) argues that mathematical discourse has four distinguishing features:
word use, visual mediators, (endorsed) narratives and routines. This study uses these
features of mathematical discourse to characterize the discourse on linearity in two texts
and in two classrooms. This presumes a meaningful relationship between disciplinary
discourse and classroom discourse. Lampert (1990) argues that “content” per se is not the
only goal of schooling, but that enculturation into the discourse community of school is
another. If we take seriously the idea of disciplines as communities of discourse, then it
makes sense to investigate the ways in which school discourse resembles mathematical
discourse.

Discourse analysis can attend to many dimensions of discourse. Cazden (1988) is
concerned with the structures of classroom discourse—identifying its patterns in order to
understand schooling as a social and educational phenomenon. In this case, power
relations, turn taking and participation patterns (among other possible dimensions) are
central to the analysis. Herbel-Eisenmann (2000) uses discourse analysis in her study of
two Connected Mathematics classrooms to understand the center of mathematical
authority—whether it is the teacher, the class as a whole or the textbook that decides
mathematical truth. The present study is interested in describing the relationships
between the mathematical discourses of textbooks and classrooms.

Sfard’s description is of formal mathematical discourse, while this study is
concerned with classroom mathematical discourse. The relationship between these is

complex, as is the relationship between any discipline and the school subject bearing the

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same name. Schwab (1978) argues that the structures of a discipline, which may be
roughly described as the relationships between the objects of study and the procedures for
dealing with and verifying these relationships, ought to be preserved when the disciplines
are taught as school subjects, but that too often it is only the fruits of the discipline—the
facts the discipline produces—that are taught. One way of characterizing Schwab’s
structures is as the narratives and routines of discourse in Sfard’s framework. But even in
a classroom that is sympathetic to Schwab’s viewpoint, it is inappropriate to assume the
classroom discourse to be identical to that of the discipline.

Lampert (1990) describes her attempts to teach mathematics with the explicit goal
of developing students as participants in a mathematical discourse with particular
practices, which she contrasts with traditional classroom practices. Without citing
Schwab, her teaching was in this spirit: teaching students to think like mathematicians.
The title of her article, “When the problem is not the question and the answer is not the
solution,” speaks to this emphasis on mathematical discourse practices (posing problems
and making conjectures about solutions) over finding answers to routine questions.

An important difference between formal mathematical discourse and classroom
mathematical discourse is power. The idealized formal mathematical discourse disregards
power relations among participants. This difference is explored in more depth below, in
the section titled Narratives. For now, the obvious difference that a teacher directs the
discourse of a classroom and may share or retain the power to do so in varying degrees
suffices to distinguish classroom discourse from the larger mathematical discourse.
Another difference is the idiosyncratic nature of classrooms. Students and teachers may

use language to refer to prior shared experiences that will be incomprehensible to

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outsiders (e.g. references to “Sean numbers” in Ball’s work (Ball, 1992)). Thus, using the
characteristics of mathematical discourse to study classroom discourse will surely turn up
some mismatches: middle school students and teachers use words less formally from the
way mathematicians would use them, for instance. At the same time, the use of this
analytical tool does indeed reveal that the textbook and classroom discourses under study
are justifiably categorized as mathematical. In particular, the ways of finding or
determining mathematical truth are recognizable from formal mathematical discourse, in
the case of one textbook, and from heuristic (exploratory) mathematical discourse in the
case of the other textbook in this study, and in the cases of the two classrooms in this
study.

The intention of this technique is not to demonstrate that classroom mathematical
discourse differs from formal mathematical discourse, for this is a truism. Instead, the
intention is to treat classroom mathematical discourse as a kind of mathematical
discourse. That is, this study presumes that the discourse in textbooks and classrooms is
mathematical and considers it accordingly. The result is perhaps a different understanding
of classroom discourse than if it is considered as a social phenomenon as in Cazden or
Herbel-Eisenmann.

Word Use

In mathematics discourse more generally, words are used differently from their
use in everyday discourse. This is true in two senses:

l. A particular word may have a different meaning in mathematics,

2. The rules governing word use in mathematics are different, and

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There are words—such as similar—that are used in everyday discourse as well as

in mathematics, but which are used differently in the two domains—that is to say; they
have different meanings. In everyday discourse, we might say that rectangles are similar
to squares. By this we mean that the two shapes have some unspecified properties in
common. In mathematical discourse, the statement is false. The only rectangle
mathematically similar to a square is another square. Similar is an example of word that

is used differently in mathematics than it is used elsewhere, and there are many such

ex amples.
The second sense in which the claim above (words are used differently in

mathematical discourse) is true is that the rules governing word use are different in
math ematics. It is characteristic of mathematical discourse that words are carefully

defi

ned, and then used in ways consistent with the original definition. It is common for
c l aSSroom mathematical discourse to be less rigorous about word use than is formal
mathematical discourse (as the present study will demonstrate). This second sense of
wo rd use is less of a focus in this study. Instead, the focus will be on the use (meaning) of

t . . .
hree words, equation, rate and linear. The ch01ces of these terms are detailed in Chapter

IV-

The way in which each word is used is the major aspect of word use in this study.

A tI'Tiird aspect to word use, relevant to this study, but not unique to mathematical
di SeOlmrse, is identifying which words are used to describe a mathematical idea. This issue
is 0f lesser importance in this study, but there are categories of words that will be useful
for Characterizing aspects of discourse. Herbel-Eisenmann (2000, 2002) describes four

Q . . . . .
ategones of language in Connected Mathematics classrooms: Officral Mathematical

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Language (she uses the acronym, OML), Transitional Mathematical Language (TML),

Classroom Generated Language (CGL) and Contextual Language (CL).

OML consists of mathematical terms that would be recognized by participants in
an En glish-speaking mathematical discourse outside K-12 classrooms. Included here
would be slope, equation and graph. TML is informal language that might be expected to

be understood by a wider audience. The use of steepness for slope is an example of TML.
CG L is idiosyncratic. Herbel-Eisenmann cites a classroom where “swoopy down curve”
iS used to describe graphs of exponential decay relationships; this is an example of CGL.
CL i S vocabulary that arises from the contexts in which mathematics is studied. Walking
rate and head start in Problem 2 in Chapter I are examples of CL. Each may refer to a

more general concept, or it may not, but the vocabulary originates in a contextual

prob lem about a walking race.
For the purposes of this study, the distinction between TML and CGL, namely

" h ether it is idiosyncratic or likely to be in widespread use, is not particularly relevant.
\ S SuCh, the two are combined into a single category: transitional language. This leaves
th r . . . .
ee categories: mathematical language, transztional language and contextual language.

Th ese terms—especially the first and the last—are used in this study to describe word use

i
t) the final sense discussed above; the equivalent use of multiple words. Mamokgethi,
‘d let, Reed & Bapoo (2002) make a similar distinction between OML and CL. They

:-
efel‘ to these two types of language as formal mathematical language and informal

I
anguage.

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Visual Mediators
In everyday discourse, the objects of the discourse are frequently concrete and can
be pointed to in the real world. If we are discussing a movie, we can see the movie. If we
are having a mathematical conversation about functions, though, we are discussing
something that can only be said to exist in a more abstract sense. We cannot point to and
look at fimctions as we talk about themz. Instead, we use representations of these
filnctions: visual mediators such as tables, graphs and equations. In mathematical
discourse, then, the visual mediators are often representations that capture some aspects
of the object of the discourse. In many other discourses, the objects of the discourse are
Concre te; they need not be represented because they can be looked at directly.
In classroom instruction, questions are raised and ideas explored. Visual
medi ators are useful for answering mathematical questions and for testing mathematical
i deas - Closely related groups of questions and ideas are referred to in this study as
the"? es. Themes are different from narratives (discussed below) in that themes raise the
Clue Stions that narratives answer. Narratives are more general than themes and are
dec 1 arative, while themes tend to be interrogative, hypothetical or tentative. A sample
th eme is the idea that rates are represented in tables. There is a corresponding narrative
to describe how the rates are represented.
A simple listing of themes in the discourse gives little information about the
itl“IDOITance of each theme, and thus makes it difficult to characterize the discourse as a
whole- In an attempt to make such principled distinctions, this study uses the language of
recurrent, passing and absent themes. Absent themes are those we might expect to find in
2\

an: ' [athematics is certainly not the only discourse of which this is a feature; theological
d philosophical discourses have this in common with mathematical discourse.

 

27

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a discourse—in comparison with another discourse, for instance—but that are not
present. Passing themes are only briefly or rarely mentioned. Recurrent themes are found
repeatedly in the discourse and are emphasized. Specific measures of these terms vary
slightly between the textual analysis and the classroom analysis; each is discussed in the
appropriate place below.
Narratives
Narratives are the properties of and the relationships between the objects of the
di Scourse, such as The slope ofa line is the change in y-coordinates divided by the
Change in x-coordinates or In a right triangle, the square of the length of the hypotenuse
is equal to the sum of the squares of the lengths of the two legs. In formal mathematical
di Scourse, narratives take particular forms, primarily axioms, definitions and theorems.
Tl'leSe forms, and the degree of absolute truth the discourse attributes to them, distinguish

fomlal mathematical discourse. In particular, formal mathematical discourse does not

a(ll'liit conflicting narratives (as in the Law of the Excluded Middle).

Sfard’s communicational framework (20053), around which the present study is

uses the term, endorsed narratives to identify those relationships and procedures

t . . . .
hat are accepted as true in the mathematical community. In Sfard’s work, endorsed IS
in) . . . . . .
portant in that she IS interested in comparing classroom discourses to formal
lE‘tl'lematical discourses. By using the term endorsed, she points to those narratives in the

lziSSroom discourse that are accepted in the larger mathematical community, “Endorsed

c""‘attves are sets of proposrtions that are accepted and labeled as true by the
:‘atllematical community.” (Sfard, 2005a, p. 6) Thus, mathematicians endorse narratives

i . . . .
n Sfard’s description. In the present study, the correspondence wrth formal mathematical

28

 

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discourse is less important. Indeed, there will be times where narratives under

consideration in classrooms are not endorsed by the mathematical community. For these

reasons, this study refers to narratives rather than to endorsed narratives.
As alluded to earlier, an important distinction between formal mathematical
discourse and classroom mathematical discourse is power. The idealized mathematical
di scourse disregards power relations in endorsing mathematical narratives. There are
routi nes for determining the truth of a narrative and only the true narratives are endorsed.
Th 6 status of the person who produces the narrative is supposed to be irrelevant. In
C l assroom discourses, this is not so. The teacher retains the authority to endorse
narratives. The refrain regarding division of fractions, “Yours is not to question why;
y 0111‘s is to invert and multiply,” is a play on this power dynamic. Indeed, it is not just the
t€331<313er who may endorse classroom narratives by dint of a position of power, as Sfard
( 2 00 S a) demonstrates in a classroom scenario in which a charismatic and intelligent
S t“L‘Cient establishes an incorrect rule for the multiplication of integers, which the class
ac e e1:)ts and which the teacher is unable to dislodge with mathematical questioning.
Because classroom narratives are sometimes not endorsed mathematical
11211j‘atives, this study refers to establishing narratives and to reinforcing them. A teacher
e3 tE‘tlelishes a narrative by indicating for the first time that she believes it to be correct. A
teE‘QI'Ier need not be conscious of a narrative in order to establish it. After establishing a

atlve, a teacher may reinforce rt3. A routine, in the sense described in the next section,

 

3\
Establishing and reinforcing can be seen as two types of endorsing of narratives—

ensummg a more local definition of an endorsed narrative. In thIS alternate definition,
QadOrsement is not just done in the larger community; a classroom, a teacher or a student
I) endorse a narrative. In this case, an endorsed narrative is distinguished from a

n . . . . . .
a“l‘atrve in that the former IS incorporated mto the set of true statements for use in the

29

 

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might be employed to establish or to reinforce the narrative, but it is important to note
that this might be explicit— as in stating a definition— and it might be implicit—as in a
change of topics after a student correctly answers the teacher’s question. A more detailed
version of these possibilities is in Greeno (2003), but the fact that established narratives
mi gh t not be explicitly stated is what is important here.
Routines
Most often, the routines for establishing narratives in a discourse are implicit. In
Order for the participants to be conscious of these routines, they would need to consider
the di scourse, not just to participate in it. It is in this sense that Sfard refers to the rules
that govern the use of routines as meta-discursive (2005a). As an example, CMP asks
q Ll€=Stions to lead students to narratives, but never tells students that answering these
q uestions will help them to understand something. The use of questions to establish
r1211‘l‘zzltives is implicit. One place where these routines are closest to being explicit is when
th er e are conflicting narratives needing resolution, as in the following example, where M

i . . . . . .
S the teacher, R and TA are students, and ellipses indicate that material rs omitted from

t .
h e tl‘anscnpt.

M: Who’s the faster walker. Yeah, R?

R: Yves.
M: How come? Why do you think so?

R: Cause he has whole numbers, not halves.

 

\

i‘Sc0urse. Indeed, Sfard (2005b) indicates that this is the more productive definition. The
Se of narrative together with establishing and reinforcing will serve to make the

heCessary distinctions in the present study.

30

 

 

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TA: The numbers are higher (Ms. M Activity 1, 10/27/04)
In this example, two conflicting narratives are introduced by students: Fractional
numbers in a table indicate smaller rates and Larger numbers in a table indicate greater
rates- Because there exist fractions greater than one, the two clearly conflict with each
other, in addition to conflicting with the curricular narrative, Greater increments in the
dependent variable (given a constant increment in the independent variable) in a table
represent greater rates. In each case, Ms. M asks questions of the student stating the

narrative and of the class. She uses the routine of Asking Questions.

While Lampert ( 1990) does not use the term routine to describe the practices by
wh i Ch she and her students verified mathematical claims in her classroom, the

d ‘ Stinctions she draws are paralleled in the present study.

Knowing mathematics in school... comes to mean having a set of unexamined

beliefs, whereas Lakatos and Polya suggest that the knower of mathematics needs
to be able to stand back from his or her own knowledge... (p. 32)

The generalization in this passage is strong; that school mathematics is about accepting

i deas without question, which stands in stark contrast to Lakatos’s (1976) and Polya’s

( l 9 7 3) characterizations of knowing mathematics, in which reflection and the questioning

()f i deas are paramount. Nonetheless, these two contrasting sets of routines have

relevance in the findings of the present study. One textbook discourse asks students to

accept narratives as given, the other asks them to derive narratives through questions and

re . . . .
aSOning. Two classroom discourses are seen to emphaSize the latter, but also to give

“a - . . .
11.atl VCS In certain Circumstances.

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This chapter has posed the central problem of the study in operative terms with
the use of a communicational framework in which teaching and learning are seen as
statements about participation in discourse. The next chapter states the study’s research
questions in the language of this framework. It also lays out the rules for the analysis of

the textbook and classroom data.

32

 

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Chapter IV: Research Methods

This study arose from questions about classroom instruction in light of a
substantial reform in K-12 mathematics instruction. In particular, these questions were
about the classroom interactions in classrooms using Connected Mathematics (C MP)
(Lappan Fey, Fitzgerald, Friel & Phillips, 2001a). This curriculum is the focus of this
study. The preceding chapter laid out the theoretical framework in which this study takes
place; teaching is conceived of as engagement with learners in a model discourse. In this
framework, the question What is being taught in CMP classrooms? is equivalent to the
qu estion, What is the model discourse in C MP classrooms? The study breaks into three
analytical pieces: analysis of textbook discourse and analysis of teachers’ classroom

di SCourse and the comparison of the two discourse types. These pieces correspond to

three research questions:

I. What characterizes the introductory discourse on linearity in Connected

Mathematics?

2. What characterizes each teacher’s introductory classroom discourse on

linearity?
3. What changes does the textbook discourse undergo in becoming classroom

discourse?

In the first piece (curriculum analysis), a second curriculum Mathematics,

Structure and Method (Dolciani, Sorgenfrey & Graham, 1992), referred to here as
DO [Ciani is used as a contrast in order to better understand CMP. This curriculum was

c1Fi‘OSen under an assumption that the analytical tools being applied in this study would

reVeal sharp differences with CMP. It was also chosen for its reputation as a well-

33

 

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constructed curriculum that has a consistent vision of mathematics and of mathematics
learning. These two curricula are described next.
Texts
CMP is one of several National Science Foundation funded mathematics
curriculum projects developed in response to the National Council of Teachers of
Mathematics Curriculum and Evaluation Standards (N CTM, 1989). CMP was first
published in 1996. It is a grades six through eight curriculum consisting of 8 units at each
grade level and covering 4 strands of mathematics: number, geometry and measurement,
algebra, and probability and statistics. A typical unit is about four to six weeks of
classroom instruction. Each unit focuses on one strand. For instance, the sixth-grade unit
Bits and Pieces 1 is a number unit, while another sixth-grade unit, Shapes and Designs is
a geometry unit. A unit consists of four to seven investigations, which in turn consist of
prOblerns, homework questions and reflection questions. In general, a problem

corresponds to one classroom lesson.

CMP is based on an explicit teaching model, called Launch-Explore—Summarize.
This three-phase lesson structure takes place around “Problems,” which consist of a
Series of tasks and questions, often in context. By context, I mean everyday situations
With Which most students are familiar outside of their school experience, such as bumper
Cars, Walking races and fund-raisers. Janvier (1981) uses the term situation to describe
Conteth, while Boaler ( 1993) refers to contexts. Both researchers are concerned with

Cal ' . . . . . . .
llng attention to the limitations of contexts for mathematics instruction, and these

00 . . .
ncel‘ns are relevant to instruction in CMP.

34

 

 

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In the Launch phase of a lesson, teachers engage students with the problem—both
its context and its mathematical challenge. In the Explore phase, students work on the
problem, typically in pairs or small groups, but sometimes individually or as a whole
class. In the Summarize phase, the teacher directs a discussion of students’ solutions and
helps the class to reach the mathematical goals of the problem.

This study focuses on one seventh-grade unit, Moving Straight Ahead, whose
subtitle is Linear Relationships. Moving Straight Ahead includes six investigations.
Investigation 1 consists of two experiments that students are to perform. The data
collected from these experiments is analyzed with tables, graphs and equations and
students are asked questions that require interpolation and extrapolation based on an
assumption that the relationship in each experiment is linear. Investigation 2 is the focus
of this study and introduces the term linear relationship. Students analyze a series of
contextual problems to explore the properties of linear relationships, especially slope and
y-intercept. Investigation 3 uses graphing calculators to analyze linear relationships that
are not easily modeled by contextual problems, such as with negative coefficients and y-
intercepts. The investigation also introduces solutions to linear equations. Investigation 4
focuses on solving equations. Investigation 5 formally defines slope. Finally,
Investigation 6 consists of a series of contextual problems in which students practice the
ideas of the preceding five investigations.

Dolciani is an older text that has been revised several times over the years. This
study focuses on the 1992 edition. A note at the beginning of the Teacher Edition of the
text explains how the revised text meets the 1989 NCTM Standards. The topics, language

and Presentation are similar to a preceding 1979 edition. As the title, Mathematics,

35

 

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Structure and Method implies, Dolciani’s presentation focuses on mathematical
structure—definitions, theorems and rules—over contexts and real-world applications.
There are two courses to the series, intended for 7'h and 8th grades. This study focuses on
Course 2, which is for 8‘h grade.

The teaching model in Dolciani is less explicit than in CMP, since they do not
directly tell teachers how to structure lessons. But the notes to the teacher suggest that the
American model of teacher exposition followed by student practice (documented in
Stigler (1999)) is assumed. These notes are in a section at the beginning of each chapter
in the Teacher Edition called Teaching Suggestions. There is a set of Teaching
Suggestions for each lesson. The suggestions for each lesson consist of about a page of
text. Objectives are listed for each lesson, while some lessons have additional
commentary or ideas for teachers under such headings as, Exploration, Computer Activity
and Thinking Skills.

This study focuses on Chapter 8 of Dolciani, titled, “The Coordinate Plane”. This
is the third of four algebra chapters in the text. Other earlier chapters have focused on
number (four chapters) and geometry (one chapter). A chapter in Dolciani contains
between 5 and 10 lessons and is roughly two weeks of instruction.

Dolciani is included in this study as a contrast. The study is not interested in the
Dolciani text on its own terms. Instead, the characterization of CMP requires a
contrasting curriculum in order to make the claims about the CMP discourse stand out as
important. In this sense, this study uses Dolciani to better understand the discourse in

CMP. Once Dolciani has been used to characterize the CMP discourse, the study leaves

36

 

 

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Dolciani behind. Attention turns back to CMP, and in particular to the implementation of
CMP by two teachers: Ms. M and Ms. H, who are described in the next section.
Teachers

Ms. M and Ms. H are junior high school teachers in two urban Midwestern public
school districts. My teaching experience is in urban junior high schools, so this was the
context for research in which I was interested. Teachers using CMP were chosen for this
study based on these additional criteria:

1. Experience

2. Time on task

3. Attention to mathematical generality

4. Scheduling

Having identified the geographic area in which the study would take place,
curriculum leaders familiar with one or both districts were consulted and approximately
10 teachers identified as likely candidates for participation in this study. Each of these
classrooms was visited by the researcher at least once in the spring prior to the study.
During these visits, the criteria for selection were considered. The two selected teachers
best met the given criteria and agreed to participate.

The intention of this study is to describe a curriculum, CMP, in practice in a
standard urban mathematics classroom. While there are many important questions about
teachers using this curriculum to learn to teach, or to change their practice, this study has
a different focus.

Participating teachers needed at least three years of experience in junior high

school classrooms and with CMP. The year of the study is Ms. M’s 15th year of teaching

37

 

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mathematics, her 8th year teaching junior high and also of teaching C MP. It is Ms. H’s 7lh
year of teaching junior high math, all of them using CMP.

High time on task was a requirement for this study. A classroom in which a
substantial amount of time were lost to discipline and off-task behaviors would be
unlikely to yield useful information about the workings of the curriculum. Disruptions
were few in each of Ms. M’s and Ms. H’s classrooms during the pre-observations and
each teacher efficiently used class time for mathematics instruction.

Participating teachers needed to show some evidence of attending to the
mathematical goals of the lessons. Ms. M and Ms. H were observed to use mathematical
language and to attempt to elicit prior mathematical vocabulary from their students in the
pre-observations. Finally, Ms. M and Ms. H each had plans to teach the unit Moving
Straight Ahead in the window of time available for data collection for this study: late
September through late November.

In a study of two classrooms, a representative sample of CMP classrooms is
impossible. This minimum set of conditions was important to ensure that the curriculum
Was a major influencing force in these classrooms, but was not intended to create a set of
idealized conditions for maximally effective implementation. Such conditions might have
included an examination of the assignment of students to courses, the professional
development experiences of the teachers, the support for the curriculum in their buildings
and their districts, and other factors (NRC, 2004). Typical classrooms with good
Conditions were sought rather than exceptional classrooms and the result is a description
Of What might very well be happening in many classrooms around the country rather than

an “existence proof” (Lampert, 1990) of what is possible.

38

 

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School Contexts

Ms. M’s school has approximately 450 students in seventh and eighth grade and
follows a seven-period day in which classes last 48 minutes. Ms. M is one of four math
teachers in the school. Students change schedules each semester and they study Moving
Straight Ahead in eighth grade. As a result, some of the students in Ms. M’s class in this
study were her students the previous year, as seventh graders, while some were not. In
addition, the student turnover rate at her school is high: 49% of the student population
transferred in or out of the school in the first semester of the 2003-04 school year (this is
the mid—year mobility rate)4. Students who were at her school as seventh graders studied
CMP, whether with Ms. M or another teacher. Students who were not at her school had a
variety of curricular backgrounds. The school’s enrollment is 86% minority.

Ms. M taught Moving Straight Ahead beginning in the eighth week of school. The
first seven weeks, she taught from a curriculum that her school was piloting, College
Preparatory Mathematics (CPM, 2002) During this time a variety of topics was studied,
including integer concepts, operations on integers, and functions. During the instruction
on functions, the word linear was used in association with graphs. Students wrote
equations and referred to them as rules. These rules were generated from tables of Input-
Output values. A typical input-output table from this period of instruction is in Figure 1.
Ms. M ’3 students therefore came to the study of Moving Straight Ahead with different
instructional experiences than a typical CMP student brings to the unit. Because this
CPM instruction was not observed, these differences cannot be characterized beyond the

citation of topics provided above.

 

 

4 . . .
This is the most recent available data.

39

 

 

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College Preparatory Mathematics (CPM, 2002).

Input

 

 

 

5I3 I9I1 I0 I7 I100 I4 I10
Output I

I I I I I50 I10,0001 I17 101 I

 

 

 

Ms. H’s school is K-8, with approximately 480 students in all grades combined.
The junior high students follow a six-period day with 53-minute classes. Ms. H is the
only math teacher in her building. Student tumover is low (6% midyear mobility rate), so

nearly all of the 8th graders in Ms. H’s classes studied CMP with her in 7th grade. The
school’s enrollment is 44% minority.

Ms. H taught Moving Straight Ahead beginning in the tenth week of school. Prior
to that, she did some introductory and review work with students and taught the 7‘h grade
CMP unit Filling and Wrapping which focuses on volume and surface area of 3-
dimensional figures.

These two classroom contexts—Ms. M’s room and Ms. H’s room—are urban
classrooms with experienced CMP teachers, but I have described several differences
between them. The student populations are diverse, but differently so. Both classrooms
are studying Moving Straight Ahead in the fall of eighth grade, but their immediately
Prior experiences are different—Ms. M’s students studied another curriculum at the
beginning of the year while Ms. H’s student studied CMP. A typical student of Ms. M’s
may have had as many as three junior high mathematics teachers, while a typical student

0f MS- H’s has had only Ms. H for junior high math. These two cases represent common

40

urban school structures. Together they provide a rich context for examining Connected
Mathematics instruction in urban junior high schools.
Research Questions

This study was designed to examine the introduction of mathematical ideas
through work on contextual problems, using discourse to frame the research questions.
The theoretical framework outlined in Chapter III makes operative the research questions
about this process. This section details the research questions and the methods for
answering them.

This study was designed to be a detailed look at one instance of the introduction
of mathematical concepts through work on contextual problems in the CMP curriculum.
The introduction to linearity in Moving Straight Ahead is one place where this process is
cleanest: Investigation 2 contains five contextual problems focused on a small set of
mathematical ideas, but there is little mathematical language introduced until later in the
unit. While there are several other such places in the curriculum, the existing body of
research of research on this investigation and associated content in CMP (e. g. Smith, et
al., 2000; Herbel-Eisenmann, 2000) made this instance particularly appealing for
research.

Table 1 below states the research questions that guide this study, as well as the
sources of data that are consulted in answering each one. Each question will be then be

taken in turn and elaborated.

41

 

 

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Table 1: Research Questions and Data Sources

 

Overarching question: How does the introductory discourse on linearity
change from curriculum to classroom implementation?

 

 

Research question Data source
1. What characterizes the introductory textbook Teacher and Student Edition of
discourse on linearity in Connected Connected Mathematics
Mathematics?

Teacher and Student Edition of
Mathematics: Structure and Method

 

2. What characterizes each teacher’s Videotaped observation

introductory classroom discourse on linearity? Field notes

Teacher Interviews

 

 

3. What changes does the textbook discourse Analyses of research questions 1 and
undergo in becoming classroom discourse? 2

 

 

Question I .' What characterizes the introductory textbook discourse on linearity in
Connected Mathematics?

This study seeks to understand the relationship between the textbook discourse on
linearity in Connected Mathematics and the classroom discourse that results in its
implementation. The first step in understanding this relationship is to understand the
textbook discourse on its own terms. This is the subject of the first research question.
Brief Overview of A lgebra in Each Curriculum

In particular, this study examines the unit Moving Straight Ahead, a unit in grade
7 of the curriculum. This unit is the second in the algebra strand5 and the only unit that
focuses on linear relationships6. More specifically yet, this study focuses most carefully
on the introduction to linearity. The first Investigation of Moving Straight Ahead consists

of two experiments that are intended to produce real-world data that is roughly (although

 

5 The first is also at grade 7: Variables and Patterns.
6 At grade 8, there are also algebra units on non-linear relationships in general, on
quadratic relationships, on exponential relationships and on symbolic reasoning.

42

 

 

 

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imperfectly) linear. The focus is on interpolation and extrapolation of the collected data,
rather than on defining or specifically discussing linearity. For this reason, this study
focuses instead on Investigation 2, in which linear is introduced. Investigation 2 consists
of a series of five contextual problems that introduce students to the major themes of the
unit. The focus on Investigation 2 of Moving Straight Ahead, then, is a choice based on
the topic of the unit and on the introductory nature of the investigation. The location of

this investigation in CMP’s algebra strand is shown in Figure 2.

 

 

Figure 2: Algebra units in CMP
The algebra units in CMP are listed on the left, together with their subtitles. Next are the
Investigations and their titles in Moving Straight Ahead, then the problems and their titles
in Investigation 2.

Variables and Patterns

N Introducing Algebra Predicting from Patterns
8 . Investigation 1 Walking to the Yogurt Shop
3 Movnng Straight Ahead _ Problem 21
Linear Relationships Walking Rates . .
_ Investigation 2 Changing the Walking Rate
Thinking with Mathematical Models\ Ex lorin Lines Promem 2'2
Representing Relationships , p 9 Walking for Charity
g'th EI: C l I 1 Problem 2.3
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3 Problem 2.4
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Quadratic Relationships nvestigation 4 Problem 2.5

, Exploring Slope
Say ll W'th SymbOIS lnvestigation 5

Algebraic Reasoning
Writing an Equation for a Line
investigation 6

 

In early attempts at characterizing the discourse in CMP, it was difficult to do
much more than recite boilerplate phrases from the NCTM Standards. The
characterization felt obvious rather than enlightening. In order to improve the

characterization—to make it new rather than trite—a contrast is needed. In this spirit, I

43

 

 

 

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apply the same analysis to another beginning algebra text, Mathematics: Structure and
Method (Dolciani, Sorgenfrey & Graham, 1992), referred to henceforth as Dolciani. This
curriculum was chosen for being widely used in US. classrooms and for its contrasting
patterns of discourse on linear relationships. Dolciani is a representative of what is often
called traditional curriculum. Also important is the fact that Dolciani shares with CMP an
uncluttered layout, few colors or illustrations and an unwavering focus on the curricular
principles to which it is committed. These two curricula stand in contrast to many of the
widely available published curricula that are lumped, together with Dolciani, in the
traditional category7. The traditional curriculum has been characterized as having a focus
on procedural skills and as taking an incremental approach to the development of these
skills”.

Comparisons between CMP classrooms and Dolciani classrooms also would be
informative, but this study focuses on classroom implementation of CMP and uses the
cun'iculum analysis of Dolciani only as a tool for understanding the CMP textbook
discourse. This contrast is in the spirit of Lemke’s (I998) argument that,

The basis of discourse analysis is comparison. If you are interested in covariation

between text features and context features, you should not collect data only for the

cases of interest, but also for cases you believe will stand in contrast with them.

Instead of investigations, Dolciani is divided into chapters. Correspondingly,

instead of Problems, Dolciani has Lessons. These are roughly equivalent units of

 

7 Indeed, detractors of reform curriculum may deride current versions of traditional
curriculum. Referring to one such textbook, Klein writes, “It is a book that I would wish
on no one...The book is full of pointless color photographs, much like a web page. The
effect is to distract the reader and discourage focused attention.” (n.d.)

8 A side benefit of this study is an improved understanding of the traditional curriculum
resulting from the focus on its language.

44

 

 

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instruction: in CMP one problem generally takes one class period and in Dolciani one
lesson generally takes one class period. There is no chapter with a singular focus on
linear relationships. However, all mentions of linearity are in three lessons in a single
chapter: Chapter 8. I focus on five lessons from this chapter: the three that use the term
linear (Lessons 8-3, 8-4 and 8-6), together with the lesson immediately preceding these
(Lesson 8-2) and the one lesson in Dolciani that includes the term slope (Lesson 8-5).

The location of these lessons in the Dolciani 7-8 curriculum is shown in Figure 3.

 

 

Figure 3: Algebra chapters in Dolciani
The algebra chapters are shown, together with their titles on the left, the lessons in this
study, together with their titles on the right.

Using Variables

is Chapter 2 The Coordinate Plane
% Lesson 8-1
g Solving Equations
Chapter 8 Equations in Two Variables
__ Lesson 8-2
Introduction to Algebra
Chapter 1 Graphing Equations in the Coordinate Plane
Lesson 8-3
Equations and Inequalities , _
00 Chapter 5 Graphing a System of Equations
§ Lesson 8-4
m $23123: rdinate Plane Problem Solving Using Graphs
Lesson 8—5
Polynomials

Graphing Inequalities

Chapter 13 Lesson 8-6

 

The focus of the analysis of each curriculum is restricted to the body of the lesson,
by which I mean the part of the lesson intended for the classroom, around which the
teacher and students interact. In CMP, this means the Problem, together with its
introductory text and follow—ups. Not included are the ACE questions or the

Mathematical Reflections. In Dolciani, the body of the lesson means the explanatory text,

45

 

 

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the examples and the Class Exercises. Not included are the Written Exercises (the
homework questions). Assessments and any supplementary materials are omitted from
the analysis. The focus on the body of the lesson rests on an assumption that the
classroom lesson is the relevant unit of instruction. Research question number 3 will
compare the discourse on linearity in the curriculum with that in classroom instruction.
Accordingly, I analyze the parts of the curriculum that are intended for use in the
classroom lesson.

Data for the analysis of the discourse in the two curricula comes from the Student
and Teacher Editions. The two together provide evidence for the complete picture of the
discourse in each curriculum. There are important surface differences between the two
Student Editions and between the two Teacher Editions. These are outlined next in order
to better understand the context of the cum'cular study. The following consists of a
glimpse at the surface features of each curriculum, not a set of claims about the discourse
of either one. Discourse findings are reserved for the next chapter.

Structure of Each Curriculum

As discussed at the beginning of this chapter the CMP Student Edition consists of
a series of problems, each intended to be worked over the course of a single lesson.
Consider a typical example, Problem 2.3. This problem was chosen because it includes
all of the important features of a typical CMP problem. The problem number and title
come first (Problem 2.3 Walking for Charity), followed by some introductory text about
walkathons. Then there is a set of questions that makes up the problem. These questions

are set off in a shaded box. Under the shaded box is a set of follow-up questions.

46

 

 

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The introductory text is sometimes about the context (as in the example),
sometimes about mathematics. Occasionally, there is an introduction of mathematical
terminology. The problem generally consists of a set of questions, categorized as parts
(A, B, C, etc.) and subparts (Al , etc.). Commonly there are at least three and as many as
a dozen questions. The Follow-Up questions are similar in nature to those in the problem,
but are typically fewer in number. The important point is that there is no standard place in
the CMP Student Edition for explanations or examples. Instead, the mathematical point
of the lesson is communicated through introductory text that focuses student attention,
but does not explain, and through a series of questions in the shaded box.

Dolciani is a contrast to this structure. A typical Dolciani Lesson is Lesson 8-2.
The lesson consists of expository text with examples and is followed by a set of practice
problems. In contrast to the CMP model, Dolciani gives explicit instructions for
procedures in the expository text (e. g. “To find a solution of a given equation in x and y,
we can choose any value for x, substitute it in the equation, and solve for the
corresponding value for y.”) and provides examples for students to follow. The number of
questions included in a Lesson is quite small compared to a CMP Problem. More to the
point, in Lesson 8-2, even including the Written Exercises, there are no questions.
Instead, Dolciani gives instructions and tells students to perform actions (e. g. “Tell”,
“Solve”, etc.)

The two Teacher Editions are also quite different. The wraparound, in which a
reduced-size copy of each student page is surrounded by answers and commentary for the
teacher, is standard in the US. textbook industry, and both curricula include it. Dolciani

overlays answers onto the reproduced student pages, offsetting them in color. CMP does

47

 

 

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not overlay answers, but includes some in the wraparound margins and some on later
pages. CMP and Dolciani each include some notes to the teacher in the wraparound
section. CMP includes at least a page—sometimes as many as three or four pages—of
commentary specifically for the teacher on each Problem. I will refer to this as the LES
section for Launch, Explore, Summarize—the three phases of the CMP teaching model,
and the three main parts of each of these commentaries. The equivalent in Dolciani is
roughly one page of Teaching Suggestions. As with the Student Editions, a notable
difference between these two Teacher Editions is the number of questions. The LES
section for the problems in Investigation 2 averages more than 8 questions the teacher
might ask the students per problem’. The Dolciani Teaching Suggestions for the lessons
in this study include two questions: one in Lesson 8-2 and one in Lesson 8-5. In both
cases, the answers in the Teacher Edition are left out of the analysis.

Next, I turn to methods for the analysis itself. Chapter III laid out four
distinguishing characteristics of mathematical discourse: word use, visual mediators,
narratives and routines. Each of these is taken in turn, focusing on the methods for
analyzing each characteristic.

Word Use. A word use analysis needs to focus on a small set of words with
theoretical importance. The set needs to be small in order to keep the analysis to a
reasonable size, but it needs to be large enough to describe several aspects of the
discourse. This study focuses on the use of three words, equation, linear and rate.
Equation was chosen because any study of linear relationships is an algebraic study and

algebra suggests equations (certainly school algebra does, anyway). The use and

 

9 Forty-four questions over the five problems.

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treatment of equations is also a topic of the visual mediators section below. In this
section, only the word equation") is considered. Under visual mediators, the use of
equations is considered, whether equation is spoken or not. Rate was chosen because it is
a central idea in the CMP approach to linear relationships. Relationships between
variables constitute the focus of CMP algebra instruction. The word rate suggests the
dynamic nature of the functional relationships in CMP. The hypothesis was that the use
of this word would be substantially different in Dolciani, which focuses on solving
equations, rather than on expressing relationships between changing variables. Studying
the use of the word rate brings characteristics of the CMP discourse into sharper focus.
Initially, the third unit for the word use analysis was to be the term linear relationship—
an obvious choice because it is the subtitle of the CMP unit in this study—but this term
does not appear in Dolciani. Instead, the word linear on its own is analyzed, for this term
appears in both curricula. It was hypothesized that studying the use of linear
relationships, or simply of linear in the two curricula would yield useful comparisons
between them, and this study is of the discourse on linearity. The analysis includes
derivatives of linear, such as nonlinear and linearity but not the word’s root: line, which
applies mainly to graphs rather than to relationships more broadly.

There is a perspective that the meaning of a word is equivalent to its use in
communication (Wittgenstein, 1953; Sfard, 2005a). Acceptance of such a position
necessitates the study of the use of the word linear in this study; for the meaning of

linearity is what is sought in two textbooks and in two classrooms. If meaning is

 

'0 In order to distinguish equation, the word, from equation, the thing (e. g. y=2x),
equation will be italicized when it refers to the word and in plain text when it refers to the
thing.

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tantamount to use, use must be studied. This does not constitute a claim that studying the
use of the word linear gives a full picture of the discourse on linearity. On the contrary, it
is one small, but necessary, piece of studying this discourse.

The data for the analysis of word use consists of the sentences in which the words
are used. For each curriculum, these sentences were compiled in three tables; one for
each word. In considering the sentences in isolation, there were two questions: 1. What
patterns arise with respect to other words in these sentences? and 2. What is being said
in this sentence about the word? For example, in the CMP sentence, “How does the
walking rate affect the equations?” (Student Edition, p. 18), rate and equation are in the
same sentence, which did not occur at all in Dolciani, suggesting a pattern in question 1.
With respect to question 2, the sentence suggests that rates do affect equations, and this
implies that rates are represented in equations.

In some cases, a simple count of the number of times a word is used in a
particular way provides a quick, rough contrast between discourses. When this is the
case, word counts are used as an introduction to the finer-grained analysis of word use.

Attention is paid to words that are equivalent to each other in the sense of being
used in the same ways, and to those that are equivalent only in certain situations (partially
equivalent). For instance, function and relationship are equivalent in the introductory text
in Moving Straight Ahead because the relationships studied there form the basis of the
informal definition of function.

From word use, we may also gain clues about the philosophy of the nature of
mathematics that underlies curriculum and instruction. For instance, if a teacher

consistently talks with her students about writing equations, and if she talks about the

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equation as the generalization of a set of computational procedures, we may infer that she
is modeling an operational conception (Sfard, 1991) of equations in algebra. If, on the
other hand, she speaks of finding equations, and if she talks about equations as objects
with parts (e.g. y-intercepts and slopes) that may be separated and analyzed, we may infer
that she is modeling a structural conception of equations (i.e. equations are objects). Each
of these is a well-accepted (while greatly debated) conception of the nature of
mathematical object, so neither will define one teacher’s discourse as more or less
mathematical than another, but such observations will help to characterize each

discourse.

Visual mediators. In mathematical discourse, visual mediators are used to discuss
the abstract objects (functions, in this case) that are not physically present. In this study,
three of these mediators are tables, graphs and equations, the standard algebraic
representations. These mediators are not chosen for the analysis, so much as they are a
product of it. When the non-textual elements of the curricula are considered, these three
representations predominate. Any instances of these representations are analyzed, as are
any diagrams in the text. Also included are any representations or diagrams that the text
explicitly instructs students to make.

The analysis of the use of visual mediators in each curriculum began with a
simple list of everything that is not a word or a number, but that is intended to be part of
the mathematical discourse. In each curriculum, this included tables, graphs and
equations, but ordered pairs also showed up, as well as one diagram in Dolciani. Next,
places were noted where students are instructed to make one of these visual mediators.

Then a list was made of what each visual mediator appeared to be used for each time it

51

 

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appeared. For example, a graph might be included to illustrate how to find slope, to
define linearity, or for one of several other purposes. Finally, patterns were identified
wi thin each curriculum, with special focus on repetitions of the purposes.

Summary and discussion of these patterns are in the visual mediators section of

the analysis of each curriculum. Each of the purposes of each of the visual mediators is
referred to as a theme and the themes are identified as being recurrent, passing and
absent in each textbook’s discourse. Themes describe patterns in the use of visual
mediators in the textbook, and therefore the meaning the authors intend to convey to
students.

Recurrent themes are mentioned through expository text, questioning or examples
in more than one sentence in more than one lesson. The Teacher and Student Editions are
Considered collectively for this purpose, so one mention might be in the Teacher Edition,
anOtl‘l er in the Student Edition. However, the mentions need to be distinct from each
other- If the Teacher Edition mentions that the Student Edition asks about y-intercepts,
for eXample, this is not counted as a second mention, as in this example from Problem
2'5 a “Question 5 informally introduces the concept of the y-intercept by asking questions
about the point where the line crosses the y-axis.” (Teacher Edition, p. 341)

Passing themes might be mentioned more than once in a single lesson, or once in
each of multiple lessons, or a combination of these. Absent themes are not present in the
discourse of the textbook. These terms have corresponding meanings in the teacher
disco‘1I‘Ses, and these are discussed in that section.

In most instances, mentions are straightforward, even when the precise

ma . . . . . .
thematical term 18 not used. In the followmg example, y-intercept is mentioned when

52

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the teacher focuses student attention on a specific part of a graph, “At what points do
Emile’s and Henri’s graphs cross the y-axis?” (Student Edition, p. 22). Here the teacher
mentions y-intercept without using the term by asking a question about the point that is
the y—intercept. However, there are more ambiguous situations. For instance, a follow-up
question to Problem 2.5 asks, “Which of the following equations gives the relationship

between Yvette’s distance from the starting line, d, and the time, t?” (Student Edition, p.
23 ) There are four choices, below (ibid).

i.d=20+2t
ii.d=2+20
iii.d=20t+2
iv.d=20+t

Because this question does not specifically direct students’ attention to the y-intercept’s
representation in the equation, it is not counted as a mention. What makes this ambiguous
is the fact that choices i and iii reverse the roles of slope and y-intercept, making each
idea likely to be raised in student thinking or classroom discussion. In the end, though,
the curriculum does not direct student attention to the concept and so this situation is not
counted as a mention.

To summarize, visual mediators, primarily the algebraic representations of table,
graph and equation, are central objects in the discourse on linearity. The ways in which
these Objects are used by the text, and the procedures the text offers to students for using
these mediators are analyzed as themes in the visual mediators section for each
curricluum. The relationships between these visual mediators are narratives and are
discuSSed in the next section.

Narratives. Narratives are statements about the objects under study, and the

rel ' . . . . .
atlonShips of these objects to each other. In mathematical discourse, narratives are

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statements about the status and relationships of mathematical objects to one another. I use
the term endorsed narrative, to point to those relationships generally accepted as true by
the larger (here, mathematical) community, but the term endorsed may also be applied to
a narrative accepted as true by a subset of a community, including a single person. In
most cases, a narrative in a classroom is endorsed by someone—the teacher, the class or a
single student. Rarely do participants in mathematical discourse knowingly assert
falsi ties. For this reason, this study limits the use of the term endorsed to describe
narratives accepted as true in the larger mathematical community. Definitions and
theorems are examples of endorsed narratives in mathematical discourse.

In the case of linearity, the primary objects of discourse are functions, as
represented in tables, graphs and equations—the visual mediators, but the objects also
include rates, slopes, y—intercepts, etc—the features of linear functions. In the visual
mediators sections, each of these representations is considered on its own. The
relationships between these objects of discourse, their properties and the methods for
Working with them are the narratives in this study. A theme in the preceding section
addresses one aspect of the use of one representation. A narrative in this section
addresses commonalities and differences among the representations and their uses.

There are countless possible narratives in any text. Some, such those that appear
in the form of a definition, are explicit. Some are implicit. Some are important and some
are tri‘Vial. This study set out to identify the major narratives directly related to the study
of linear relationships in each of the two curricula. The themes in the use of visual
mediators are measured for their importance to the discourse. The narratives are not, for

th ‘ . . . . .
6" COmplexny precludes this. Instead, narratives are identified as present or absent,

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without measure of their frequency. If such a measure were devised, we would expect to
find similar patterns in the narratives to the ones found in the use of visual mediators.
CMP is the primary curriculum under study, so these narratives were identified
first. The Dolciani narratives, then, were identified with an eye to making reasonable
comparisons with CMP. For instance, all three algebraic representationsll are studied in
detail in CMP, but they are not in Dolciani. If Dolciani were being studied on its own
merits, there would likely not be a narrative about tables—the topic is not that important
to the Dolciani treatment. Yet, for the purposes of this study, it is enlightening to ferret
out what Dolciani does have to say about this relatively minor topic—because it is
important in CMP.
In general, declarative statements about mathematical objects refer to narratives
(e-g- “[T]here are infinitely many ordered pairs that satisfy this equation.” (Dolciani, p.
276)). Questions, though, can also point to narratives, as in, “How does the change in
Walking rate affect the graph?” This question contributes to the narrative, Rates are
represented in tables, graphs and equations.
Of particular importance in the case of CMP is that narratives should not be stated
in the language of the contexts. That is, faster walkers have steeper lines is not a
mathematical narrative, but rate is represented by the slope of a graph is. Practically, this
is in“Dortant in order to make meaningful comparisons with other curricula. Even another
Carri Culum heavily reliant on contexts might not use the context of walking rates, but
would be expected to address rates and slopes. As a theoretical matter, it is important to
State narratives in mathematical language because this study originated in questions about

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55

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the relationship between contextual problem solving and the formalization of
mathematical ideas in classrooms. It is important, then, to identify in this study the formal
mathematical ideas on which the curriculum focuses. As a result, identifying the
narratives in CMP is a different task from identifying them in Dolciani. To identify them
in CMP, the problems were considered together with their follow—ups. For each part of
each problem, the mathematical goal of the task was identified. The example below is
Part A from Problem 2.2,

In PrOblem 2.1, each student walked at a different rate. Use the walking rates

given in that problem to make a table showing the distance walked by each

student after different numbers of seconds. How does the walking rate affect the

data in the table? (Student Edition, p. 18)

There are two tasks in Problem 2.2: (1) Make a table and (2) Answer a question
about the table. Students are shown the beginnings of a table with a constant increment of
1 in the time column (reproduced in Figure 4 below.) The point of the first task is to
r epl‘ esent the relationship between two variables. The point of the second is that rates are
represented in tables (as the increment in the dependent variable when the independent

Vafi able has a constant increment of 1), which is a narrative in the CMP textbook

d‘SCOUrse.

56

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The introductory text to the Investigation and to each Problem was considered,
Checking for additional narratives. Finally, this list of narratives was compared to the

Teacher Edition Goals. The final list of narratives in Investigation 2 of CMP is given in

Table 2,

To identify the narratives in Dolciani, the process was quite different. Dolciani
asks few questions of students, and the tasks to be performed during the Lesson are also
few- Instead, the expository text communicates the narratives, in a more direct manner
than CMP does through its questions. Each paragraph of the expository text in each of the
five lDolciani lessons was considered. The main points of each paragraph were identified.
when a point was a mathematics concept, it was deemed a narrative. When a point was

r . . . . .
p ocedural, it was noted as a procedural narrative (discussed at the end of this section).

57

 

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When a narrative was not directly related to linear relationships, it was ignored”. Careful
attention was paid to definitions.
As an example of using the main point(s) of a paragraph, this paragraph from
Dolciani establishes the narrative, A table lists solutions to an equation.
The equation in two variables x + 2 y = 6 has infinitely many solutions. The table
of values. ..lists some of the solutions. When we graph the solutions on a
coordinate plane, we find that they all lie on a straight line. (Student Edition, p.
280)
The paragraph reinforced the narratives, A solution to a linear equation is an ordered
pair and A graph is the set of all points that are graphs of solutions to an equation.
The following paragraph was deemed procedural and did not establish any
narratives about the relationships between tables, graphs and equations.
In order to graph a linear equation, we need to graph only two points whose
coordinates satisfy the equation and then join them by means of a line. It is wise,
however, to graph a third point as a check. (Student Edition, p. 281)
HOVvever, the paragraph does reinforce the narrative, A linear equation has a straight-line
graph.
Next. this list of narratives was compared to the purposes of each example, to the
purposes of the Class Exercises and to the goals for each Lesson in the Teacher Edition.

In the end, this produced 9 narratives related to linearity. These are listed in Table 3.

\
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i An example of a narrative in Dolciani, but not included in this study is The point of

reSeniecnon of two graphs IS the solution to a system of equations. This is not directly
evant to the introductory discourse on linear relationships.

58

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The narratives described so far are about relationships between the objects of the
discourse on linearity (e.g. between rates and tables). It has been argued (e. g. Sfard,
2 0053) that another category of narrative comprises the procedural statements about
mathematics. The preceding quotation from Dolciani contains a narrative about graphs:
plotting two points is sufficient for determining the graph of a linear equation. This is a
procedural narrative because it describes how to do something, rather than a relationship
between objects. In particular, the sentence gives the procedure for graphing a linear
equation: plot two points (and join them by means of a line). Procedural narratives are not
analyzed in this study, although the procedures for identifying them would be identical to
those discussed so far.

Routines. In the Student Edition of each curriculum, the pieces of the text that
establish each of the narratives, whether implicitly or explicitly, were considered. In each
case, the basis on which students are to be convinced of the existence and the truth of the
nar‘l‘ative was noted. For instance the narrative, Rates are represented in tables, graphs
and equations, is raised multiple times in CMP through questions of the form, What effect
does the walking rate have on the table/graph/equation? This question form implicitly
suggests that the rate is represented somehow, but that it is for students to figure out how.
Additionally, the process of figuring out how rates appear in each representation involves
inductive reasoning. Students consider three walking rates in Problem 2.2 and three
pledge rates in Problem 2.3. Through six examples”, students are to discern that rates

affect the steepness of a graph.

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Together with a non-example in Problem 2.2,

59

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When narratives conflict, resolution is needed. In classrooms, students often
introduce narratives that conflict with those of the teacher’s model discourse (e.g. Sfard’s
negative numbers example (2005a)). Examining how a teacher resolves this conflict can
reveal routines for establishing and reinforcing narratives. In a textbook, the student’s
voi ce is absent. Teacher Editions often give examples of ideas that students find difficult,
or of student misconceptions. Such mentions of narrative conflict were sought in each
curriculum. An example is this from the Dolciani Teacher Edition, “. . .students should be
wamed to substitute correctly when checking a given number pair in an equation.” (p.
2 7 l b) In each case, I considered the basis for the suggestion (e. g. Why do students need
to be warned about substitution?) and the method suggested for resolving the difficulty
01‘ C onflict (e. g. What is the justification for substituting correctly? and How do students
know whether they have substituted correctly?)

Mathematical discourse is characterized by its word use, its use of visual
mediators, its narratives and its routines for establishing and reinforcing these narratives.
Tex tbook discourse is treated as mathematical discourse and these features are used to
characterize a particular textbook mathematical discourse: that of CMP. A contrasting
cum CulumflDolciani—is used to refine this characterization. While each feature
Contributes to the understanding of the CMP discourse, it is only through a full
examination of the features together that the full picture emerges. Care is taken in the
analyst to avoid getting lost in the minutiae of word counts and prioritization of themes.

Instead, these details are used to contribute to the understanding of the discourse as a

Whole.

60

 

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Question 2: What characterizes each teacher ’s discourse on linearity?

Just as the first research question deals with understanding the textbook discourse
on its own terms, this second question deals with understanding discourse in CMP
classrooms, as modeled by teachers. The same structure is used for analysis, but it must
be acknowledged that textbook discourse and classroom discourse are different from each
other—for instance in the fact that textbook discourse cannot be responsive to students,
while classroom discourse changes in response to student ideas and questions. The
elaboration of the third research question deals with reconciling these differences and
argues for the comparability of the two discourse types.

Prior to the classroom observations, each teacher was interviewed about her goals
for students in the investigation and her experiences in past years with the curriculum.
Each classroom was observed for the duration of the instruction in Investigation 2 of
Moving Straight Ahead. In Ms. M’s classroom, this consisted of 7 lessons. In Ms. H’s
classroom, this was 5 lessons. Each lesson was videotaped with a stationary tripod-
mounted camera that captured approximately 2/3 of the students as well as the teacher
and her visual aids (e.g. overhead projector screen, chalkboard, etc.) Microphones were
placed at two locations on student desks to capture student discussion as well as the
teacher’s language.

Detailed field notes were taken by the researcher during each lesson. These notes
divided each lesson into activities. Every time the nature of student activity changed (e. g.
from listening to writing), a new activity was recorded. Lessons consisted of between 3
and 7 activities. These activities were put into five categories: Launch, Explore,

Summary, warm-up and other. The Launch, Explore and Summary categories correspond

61

 

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to the CMP instructional model in which the Launch engages students with the contextual
and mathematical tasks in a problem, the Explore activities have students answering the
questions in the problem and the teacher leads the whole class in discussion of solutions
and procedures in the Summary. Each teacher used warm-up activities at the beginning of
some periods. These warm—up activities tended to focus on topics from the investigation,
but were of each teacher’s own design. Activities categorized as “other” tended to be
administrative (e. g. collecting homework or listening to school announcements). This
study focused on the warm-up, Launch and Summary activities because these are the
activities richest in teacher language.

Word Use. As with the analysis of word use in the two curricula, the analysis of
word use in each teacher’s discourse began with compiling the sentences using the words
in question. Once these sentences were compiled, patterns of word use from the CMP
discourse were sought. For instance, the association of relationship with the use of the
word equation was a pattern in CMP’s discourse, so the frequency of this association was
analyzed in each teacher’s discourse. After identifying where there were similarities and
differences with CMP, the sentences were examined again to identify additional patterns,
not present in the CMP discourse.

Visual Mediators. The use of visual mediators is more complicated in the
classroom than it is in a text. In a text, one can look at a page and find the non-linguistic
elements at a glance. In a classroom, a visual representation might be present, but not
discussed and it might be discussed but not present. Further, the transcripts from which a
researcher typically works do not include the representations themselves. As a result, the

analysis of visual mediators in classroom discourse had two components. The first

62

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involved working solely with the transcripts; all passages in which each representation
was under discussion were compiled. As with word use, these compiled passages were
examined for the CMP themes and then again for additional themes. The second
component involved watching the videos and capturing what visual mediators the teacher
was focusing students’ attention on. In this component of the analysis, the words a
teacher spoke were less important than what she displayed and pointed to. Again, the
analysis considered first the patterns already identified and then looked for new patterns.

The themes associated with each of the visual mediators were sorted into three
categories as in the curriculum analysis: recurrent, passing and absent. Recurrent themes
are present in more than one teacher turn in each of more than one whole-class activity.
Absent themes cannot be identified in the teacher’s whole-class instruction. Passing
themes may be present in single teacher turns in multiple whole-class activities, in
multiple turns in a single activity, or in a combination of these.

Because the discourse on linear relationships in CMP relies on contexts, the
mathematical topic need not be named to be present in the discourse. For instance, the
theme of identifying a linear relationship from a graph is present when Ms. H discusses
the connection between straight lines and walking at a steady pace. A steady walking
pace is a constant rate, which is the hallmark of linear relationships. As with the
narratives in CMP, mathematical ideas are often stated in contextual language. The
analysis in this paper reflects that.

Narratives. The narratives in each teacher’s classroom discourse were identified
in comparison with the narratives in CMP. That is, the analysis began with the list of

CMP narratives and the lessons were examined for evidence of the presence of each. In

63

 

CMP’s textbook discourse, the identification of narratives involved identifying the
mathematical goal of each part of each problem, and identifying the point of each
paragraph of introductory text. In the teachers’ classroom discourses, the identification of
narratives involved identifying the mathematical point of each teacher turn, whether
interrogative or expository. In general, these two teachers more frequently state narratives
outright than does the CMP cum'culum, making the task of identifying narratives more
direct. A combination of the techniques applied to the two textbooks was employed. As
with CMP, the tasks and questions were analyzed for their mathematical goals. As with
Dolciani, the exposition in each lesson (i.e. teacher assertions) was analyzed for their
main points. As with the textbook discourse, procedural narratives were considered
separately from those that deal with relationships between objects.

Routines. In the transcripts of each lesson, teacher turns were again considered as
the unit of analysis. As in the case of the texts, the basis on which students are expected
to be convinced of the existence and truth of any narratives embedded in the teacher turn
was inferred. In addition, routines can be identified when students introduce conflicting
narratives. When two narratives are in direct conflict, the teacher can be expected to take
steps to resolve the conflict. These steps highlight the routines by which narratives are
established and reinforced. In some instances, the teacher might ask questions to make
the student see the conflict. In other instances, the teacher might ask another to reiterate a
previously established narrative (e. g. “What did we say yesterday?” (Ms. 1H Activity 2,

1 1/09/04)).
Just as textbook discourse is treated as mathematical, teachers’ classroom

discourse is treated as a mathematical discourse as well. In theory this is justified by

64

assumptions about the narratives to be found in mathematics classrooms: whether they
are about procedures or complex relationships between mathematical concepts, we
should expect the discourse of mathematics classrooms to be populated with
mathematical narratives. Also, we might expect mathematical words to be used and for
visual mediators to be used to discuss mathematical ideas. While we also expect
differences from formal mathematical discourse, it is reasonable to expect prior to data
collection that classroom discourse will conform to norms of mathematical discourse in
these ways. This study bears out these expectations, as is argued in Chapter VI.
Question 3: What changes does the discourse of the curriculum undergo in its
implementation by each teacher?

A teacher’s classroom discourse contains narratives just as a curriculum’s
discourse does. In each case, the model discourse establishes relationships among the
objects of the discourse. The narratives may be different in the classroom discourse from
those in the text, and their importance may vary, but the basic nature of a narrative is
unchanged. In this sense, the discourses are comparable.

Similarly, the routines in each type of discourse are of the same nature, although
the presence or absence of specific routines varies across the discourses. The routines
manifest themselves differently in the two types of discourse as well. For instance a
textbook does not respond to student turns. Therefore extended exploration of competing
narratives, in which a student’s narrative is questioned and compared to the model
narrative, is rare in a textbook while it may be common in classrooms.

In this sense, a teacher’s classroom discourse is incommensurate with textbook

discourse. Classroom discourse is co-constructed by teachers and students (e. g. the

65

 

0.

TL

Classroom-Generated Language that Herbel-Eisenmann (2000) found was often coined
by students) while textbook discourse is not. Yet, if each is taken as a kind of
mathematical discourse, it is reasonable to make comparisons. While each discourse is
generated in a different way, and by participants with different roles, each can be
expected to share some features of mathematical discourse.

The problem for the research question at hand, then, is not so much justifying the
comparison of the two kinds of discourse, but ensuring that the descriptions of them are
justifiably commensurate so that meaningful comparisons can be made. This is the task of
this section.

Word Use. In the analysis of the textbook discourses, the terms linear, equation
and rate were analyzed in three ways: an examination of patterns of use with respect to
other words, counts of the appearances of these patterns, and consideration of what is
being said to students about each word when it is used. It is reasonable to expect that
patterns in association in word use would appear in both teacher speech and written
textbooks. In particular, it would be significant if patterns prevalent in a textbook did not
Show up in the teacher discourse and patterns peculiar to a particular teacher’s discourse
are likewise informative.

It is not reasonable, however, to assume that the word counts will correspond
between textbooks and classrooms. The number of words spoken in a classroom lesson is
in general far greater than the number of words on the page. Further, because the
teacher’s discourse is constructed in response to students, the allocation of word use

would be expected to vary greatly. Word counts are a crude tool when the discourse types

66

are alike. When the discourse types vary, they can be meaningful only when the
differences they reveal are great.

Finally, because the question, What is being said about this word in this sentence?
is a way of asking about a word’s meaning, we should expect similarities between
textbook and classroom discourse.

Visual Mediators. As with the use of words, it is reasonable to expect that visual
mediators are put to the same uses in textbook discourse and in classroom discourse. In
each analysis, this study attempts to measure the importance of each use of visual
mediators in each discourse by identifying themes and labeling each theme as recurrent,
passing or absent in each discourse. That these three labels are commensurate in each
type of discourse needs justification.

The purpose of the measure is the same in each discourse. I wish to evaluate the
importance of a theme to a discourse. There is an assumption that the importance of a
theme is related to its frequency. A theme that is mentioned several times in a lesson is
more important than one that is mentioned only once. Similarly, a theme that appears in
multiple lessons is assumed to be more important than one that appears in only a single
lesson. Thus, recurrent themes are assumed to be more important in a discourse than
passing themes, which are obviously more important than absent themes. This is not a
fine-grained measure and it is not used to make subtle distinctions across types of
discourse. Instead, the measure reveals similarities and differences in broad patterns
across discourse types.

The measure in each discourse type is intuitively equivalent. In textbook

discourse, a recurrent theme is mentioned in more than one sentence in more than one

67

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lesson. In a teacher’s classroom discourse, a recurrent theme is mentioned in more than
one teacher turn in more than one lesson. In each case, the goal is to isolate repetition
within a lesson and then across lessons. Because teacher turns are not edited, they often
use multiple sentences and sentence fragments to express an idea that would be stated in
a single sentence in a textbook. Sentences in the two discourses are incommensurate, so
instead a textbook sentence is equated with a teacher turn.

Narratives. Because narratives are the relationships among the objects of the
discourse, these relationships should be stable across discourse types. Indeed it is a
fundamental assumption of mathematics curriculum writing that the relationships among
the objects outlined in textbooks are communicated in classrooms. This is the goal. We
should expect comparison of narratives across the discourse types to be unproblematic
and any differences revealed to be informative.

Routines. Routines are likewise commensurate. Each of the textbooks in this
study gives explicit instructions to teachers for establishing narratives, and pays attention
to the issue of resolving conflicting narratives. The main problem for research, which has
been addressed above, is that these explicit instructions in curriculum do not cover all of
the routines in a textbook and teachers’ discourse is rarely explicit about the routines used
for establishing and reinforcing narratives. Instead, inferences need to be made in each
discourse type. No specific comparative claims will be made about frequency or
importance of these routines in either discourse type, so measurement is not an issue.
Instead, the task is to identify and compare the routines that are present.

This third research question involves a comparison across two discourse types:

textbook and classroom. While there are differences inherent in their nature (especially in

68

the degree of responsiveness to student learning), I have argued that the analytical tools
applied to the two types of discourse can yield comparable data. Where the data are
incommensurate (e.g. fine distinctions in word counts), comparison is not meaningful.
Instead, this study has been designed to use comparable data to make meaningful
comparisons across the discourse types, and to answer the question What is being taught?
in each one.
The Researcher

My background is relevant to this study, so I briefly summarize it here. I am an
experienced CMP teacher. I taught CMP in the Saint Paul Public Schools for six years
prior to starting graduate school. I taught seventh and eighth grade students, but used
nearly all of the sixth, seventh and eighth grade units with these students at various times.
While in the classroom, I conducted professional development workshops for teachers
new to CMP and have continued this work while a graduate student. My graduate
assistantship has been funded by the National Science Foundation grant for the revision
of Connected Mathematics that began in the year 2000. My work for that revision has
included editing, production of pilot units, a small amount of writing for students and
some substantial writing for the revised Teacher Editions. I have conducted and
published one curriculum analysis in a spirit similar to the comparison of CMP and
Dolciani in this study (Danielson, 2005). This analysis focused on differing curricular
approaches to perimeter.

I originally came to Dolciani with fresh eyes. As I worked and analyzed this
curriculum more closely, I became more and more sure that an earlier edition of this

series was the text from which I studied algebra in ninth grade in the mid-1980’s.

69

Summary

In Chapter III, I modified the question, What is being taught in CMP classrooms?
to make it operative by phrasing it in the language of discourse analysis, What is the
model discourse in CA0” classrooms? In particular, this study concerns itself with
understanding the relationship between two model discourses in CMP classrooms—that
of the published curriculum and that of the classroom teacher. This chapter has posed
three related research questions and detailed the methods used to answer them. While the
third research question, in which the two discourse types are compared, is the original
driving force behind the dissertation, each of the other two questions yields useful and
interesting results on its own merits. This is especially true of the analysis in Chapter V,
which is a careful comparison of two curricular discourses in order to characterize the

textbook discourse on linearity in Connected Mathematics. This analysis is next.

70

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Chapter V: Textbook Discourse

The 1990’s saw the introduction of a host of new mathematics curricula written in
response the National Council of Teachers of Mathematics (NCTM) 1989 Curriculum
and Evaluation Standards, most of them with National Science Foundation development
money. These curricula, together often called reform curricula, tend to have more than
surface differences with the standard curricula in place in the 1980’s, called traditional
curricula. These reform curricula, in responding to the Standards changed the emphases
on topics, as well as the approaches to topics across the grade levels. In order to improve
instruction in this new curricular environment—either through revisions to existing
curricula, or throughprofessional development for new and experienced teachers—it is
important to understand the classroom implementation of these curricula. One piece of
this is developing precise language and frameworks for the analysis of curriculum.

Standards documents (e.g. NCTM, 1989 and 2000) make policy recommendations
for curriculum in the form of specifying answers to the questions, “What should be
taught?” and “What should be learned?” Indeed, these questions are the starting point for
any coherent curriculum writing. This chapter and the following one turn the first
question around and ask, “What is being taught?” by looking carefully first at published
curriculum, and then at classroom teaching. A communicational framework (Sfard, 2001)
is used to formulate meanings for terms such as teaching and learning in the language of
discourse. Discourse is defined as communicative activity. Knowledge is characterized in
terms of participation in discourse and learning as changes in that participation. Teaching
is defined as engagement with learners in the model discourse and so the question, What

is being taught? is asked a different way: What is the model discourse?

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In particular, the present chapter seeks to describe the communication patterns in
one curriculum, Connected Mathematics (CMP). These communication patterns are
referred to as the textbook discourse. In earlier chapters, I have characterized CMP as
relying on contexts for the construction of mathematical meaning. In order to study this
process, several places in the curriculum were identified where the process can be most
clearly seen. The introduction to linear relationships in the seventh-grade unit Moving
Straight Ahead was chosen from among these places due to an existing research base on
student knowledge and understanding related to this unit (see Smith, et al., 2000 and
Herbel-Eisenmann, 2000). Thus, the research question underlying this chapter is What
characterizes the introductory textbook discourse on linearity in CMP?

In order to highlight the characterization of the discourse on linearity in CMP, the
analysis continues with a contrasting curriculum, Mathematics, Structure and Methods
(Dolciani, 1992). This curriculum was chosen with the hypothesis that sharp differences
would be seen between it and CMP. In early attempts at characterizing the CMP textbook
discourse, the claims resembled boilerplate reform rhetoric rather than research findings.
In order to bring meaning to the findings, a contrast was necessary; we needed to see how
things could be otherwise. In this sense, Dolciani brings richer meaning to the claims
about the CMP discourse by making salient the fact that the claims about CMP are not
idle generalizations, but real patterns that can be delineated in a systematic and
meaningfiil way. This chapter does not directly speculate on these consequences, nor
bring empirical evidence to bear on them. Instead, the focus here is on the textbooks
themselves. The next chapter looks at how the CMP textbook discourse on linearity

Changes as it becomes teachers’ classroom discourse in implementation with students.

72

A discourse on linearity is a subgenre of discourse on functions. There are two
broad categories of conceptions (and hence, discourse) on functions (see e. g. Kieran,
1993): 1. a relational-dependency conception and 2. a set-theoretic conception. In the
relational-dependency conception, functions are viewed as relationships between
variables. Rates and change are central to this conception. Graphs are considered for their
slopes and tables for their patterns of change. The set-theoretic conception emphasizes
correspondence between x-values and y—values (we are speaking here of functions of one
variable, but there are analogous ideas for multi-variable functions). In each case,
functions may or may not be described by equations. When a function is described by an
equation, the relational conception emphasizes how patterns of change in the two
variables can be identified, while the set-theoretic conception emphasizes finding
solutions (i.e. completing the correspondence).

A central claim of this chapter is that CMP promotes a relational-dependency
conception of functions. This can be seen in every aspect of the CMP discourse on
linearity: in the way words are used (and which words are used) in the discourse, in the
use of visual mediators, in the narratives and in the routines used to establish these
narratives. Dolciani, for its part, emphasizes a set-theoretic conception of functions. The
various aspects of mathematical discourse help to highlight these differences.

CMP

Investigation 2 of Moving Straight Ahead includes five problems. Titles and
descriptions of each of these problems are in Table 4 below. As can be seen in Table 4,
each of the five problems in Investigation 2 is set in a context. Two claims will be

substantiated in this section: I. that CMP has a relational-dependency approach to

73

 

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algebra, and 2. that this approach is founded on the use of contexts to develop
mathematical meaning. In order to facilitate this second claim, an extended description of
the contexts is important.

Problems 2.1 and 2.2 share a common context. Three students—Terry, Jade and
Jerome—are walking to a nearby yogurt shop. Each student has a different walking rate:
Terry walks 1 meter per second, Jade walk 2 meters per second and Jerome walks 2.5
meters per second. The three students begin from the same place and at the same time. In
Problem 2.1 (titled Walking to the Yogurt Shop), students are asked a number of
questions about various times and distances for each walker. These questions can all be
answered with arithmetic computation; there is no explicit call for students to use
algebraic representations or techniques. There are two follow-up questions. The first asks
students to explain their strategies for answering the questions in the problem. The

second follow-up asks whether Problem 2.1 involves linear relationships.

74

 

Table 2: The Problems in Moving Straight Ahead Investigation 2

 

 

 

 

 

 

 

Problem Problem
Number Title Description
Three characters, each with a different walking rate, leave
Walking to
school for the local yogurt shop 750 meters away. Students
2.1 the Yogurt
answer questions about distances covered in given amounts
Shop
of time and about the time taken to cover given distances.
Changing the Students are directed to analyze the situation from Problem
2.2 Walking 2.1 using tables, graphs and equations.
Rate
Three new characters are participating in a walkathon. Each
Walking for has a different pledge plan that relates miles walked to
2.3
Charity money raised. Students are directed to make tables, graphs
and equations to analyze the situation.
Two new characters have a walking race. One character is
Walking to given a head start because he walks more slowly. Students
2.4
Win are asked to decide how long the race should be in order for
the finish to be close.
Crossing the Students make tables, graphs and equations to answer a
2.5

Line

series of questions about the walking race from Problem 2.4.

 

In Problem 2.2 (Changing the Walking Rate), the context is the same, but students

are told to make a table, a graph and an equation for each walker. In the case of each

representation, students are asked a question of the form, “How does changing the

75

 

walking rate affect the [insert representation here]?” Additionally, there are two follow-
up questions based on a different, but related context. In the follow-up, a character named
Abby is examining some time and distance data and wonders whether the relationship is
linear (it is not). The first question asks students to state a rule for knowing whether data
in a table will produce a straight-line graph. The second question asks students to
interpret the data in terms of the story of the race that produced it.

Problem 2.3 (Walking for Charity) introduces a new context—a walkathon.
Again, there are three main characters: Leanne, Gilberto and Alana. Each character
develops a pledge plan for their participation in the walkathon. Leanne decides to ask
each sponsor for 1 dollar per mile that she walks. Gilberto asks for 2 dollars per mile that
he walks and Leanne asks for a five-dollar donation and an additional 50 cents for each
mile that she walks. The questions from Problem 2.2 are repeated; students are asked to
make a table, a graph and to write an equation for each walkathon participant showing the
relationship between the miles he/she walks and the money raised from each sponsor.
Students are asked to identify how increasing the pledge rate affects the table, the graph
and the equation. A series of questions asks students to go back and forth between the
independent and dependent variables, first finding the money raised by walking 8 miles
and then finding the distance required to raise 10 dollars. The effect of the donation on
the table, graph and equation is examined. The follow up questions ask about the
meaning of points on the graph and about the relationship between each equation and the
steepness of its corresponding graph.

Problem 2.4 (Walking to Win) introduces the third new context. In this problem

two brothers, Emile (the older brother) and Henri, are having a walking race. Emile’s

76

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walking rate is 2.5 meters per second, while Henri’s is 1 meter per second. Emile gives
his younger brother a head start of 45 meters. Students are challenged to decide how long
the race should be so that the younger brother, Henri, wins but only by a little bit, so that
it is not obvious that Emile has let him win. As with Problem 2.1, students are not
required to use any particular method for solving the problem and it need not be done
using algebraic tools.

Problem 2.5 (Crossing the Line) revisits the same context, but requires students to
make tables, graphs and equation to solve the problem. Students are asked questions
about the intersection point of the graphs of the two boys’ equations. The follow-up
questions ask students to find other time and distance values and to explain how the
representations can be used to do so; they point student attention to the y-intercept and its
representation in each of the three representations; and they introduce a third walker with
an intermediate walking rate and head start. In this last follow-up question, students make
a table and choose an equation from among several possibilities to represent the
relationship between distance and time for this third walker.

This section analyzes the introductory discourse on linearity in Connected
Mathematics using four distinguishing features of mathematical discourse: word use,
visual mediators, narratives and routines. These features are isolated and addressed in
turn. At the end of this section, the features are considered together in order to
characterize the discourse as a whole. The following section uses the same structure to

characterize Dolciani’s discourse.

77

Word Use

The analysis of CMP’s discourse begins, as with each discourse in this study, with
word use. Three words are analyzed, as described in the Methods chapter, equation, rate
and linear. Equation is analyzed because of the expectation that equations would be
central objects of any algebraic discourse. Rate is chosen for being important to the CMP
approach to linearity, while less important in traditional American approaches, including
Dolciani. Linear is chosen for the quick impression its use should give about the
discourse on linearity. In each case, patterns in the use of the word are identified and
supported with evidence. In some cases, counts of the word’s frequency, or its
appearance in particular situations or in combination with other words will be useful for
demonstrating that the identified pattern is significant. In other cases, quotations from the
discourse are used to substantiate this claim.

Equation

There are two important aspects that describe the use of equation in Investigation
2. The first is the association of equation with the term relationship. The second is the
status of equation relative to that of graph and table. Each of these supports the assertion
that CMP’s approach to functions is a relational-dependency one.

Relationship. The predominant use of equation is as a representation of a
relationship. In the introductory text in the student materials, nearly every use of equation
(5 out of 6) is in the same sentence as the term relationship (or relating), as in the
example below.

How does increasing the average speed at which the van travels affect the table,

the graph and the equation of the relationship between distance and time?

78

(L3

'n
H

(Lappan, et al., 2001b, p. 16; referred to as Student Edition henceforth.)

From this and other uses of the term relationship, it can be inferred that its
meaning is close to the mathematical definition of function”. In the phrase, the
relationship between distance and time, the implication is that for each time value, there
i s exactly one distance value. Yet, relationship is used only to describe continuous,

predictable functions, and in this sense, the terms are not synonymous. So an equation
represents what might be described as a “well-behaved” fiinction, and indeed the term
filnction appears in the introduction to the investigation.

For example, the total cost to rent bikes depends on the number of people on the

tour. We say that the rental cost is a function of the number of people on the tour.

(Student Edition, p. 15, emphasis in original)

H ere, function informally describes a dependency relationship, rather than being
mathematically defined in a formal way. Nonetheless, the introduction of the term here is
important. It sets the tone for the relational-dependency approach to linearity.

This association of equation with relationship continues with Problem 2.2, the

first time in the unit that students write equations. “For each student, write an equation
that gives the relationship between the time and the distance walked.” (p. 18)

In the introductory text and in Problem 2.2, the relationship is the central object of

the discourse. In Problem 2.3, a different aspect of equations is introduced. Here,
equiltion and relationship do not appear together at all. Students are instructed to, “Check
Your equation by graphing it...” (p. 20) The suggestion to students is that the equation

“Self is worthy of study and discussion. There is a subtle shift here as relationships yield

\

l4 . .
Le. a set of ordered pairs such that no two pairs have the same first element, but
distinct second elements.

79

 

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center stage to equations. Leading up to this problem, an equation has been a

representation of a relationship. In this problem, an equation can itself be represented (i.e.

graphed) and studied.

In Problem 2.5, relationship returns in the Student Edition with sentences such as,
‘ ‘Write an equation for each brother showing the relationship between the time and the
distance from the starting line.” (p. 22) At the same time, the problem maintains the idea
that graphs and equations are objects of study without reference to a relationship, “How
can you predict where a graph will cross the y-axis. . .from an equation?” (p. 22) Overall,
43 0/0 of the sentences in which equation appears in Investigation 2 of the Student Edition
81 so include the word relationship. Through the use of the words equation and

relationship, the Student Edition has established a relational-dependency approach to

filnctions.
The Teacher Edition does not use relationship in any sentence with equation.
N Onetheless, the shift from emphasis on relationships as the objects of study early in the
i 1"1"estigation to equations as the objects of study in Problem 2.3 can still be observed
th ere, albeit more subtly. The relationship discussed repeatedly in the Student Edition is
between variables in contextual situations. For this reason, we may consider sentences
using equation that also include the word situation or the names of the contextual
Variables (time, distance, walking rate, etc). Sixty-three percent of the sentences using
equation also use one of these terms in Problem 2.2 (7 out of 11 uses), while 30% use

thetn in Problem 2.3 (3 out of 10 uses). The frequency of allusion to relationships when

diSeussing equations is roughly half what it was in Problem 2.2.

80

Typical of the sentences in Problem 2.2 is, “What effect will increasing your
walking rate have on the table, the graph and the equation?” (p. 34b), while typical of
those in Problem 2.3 is, “How do the tables, graphs, and equations from Problem 2.2
compare to the tables, graphs, and equations for this problem?” (p.34g). The shift from
comparing relationships to comparing equations (as well as tables and graphs) is notable.
In the example from Problem 2.2, the language of the context is included (walking rate),
implying that the equation represents the relationship between the variables there. In the
example from Problem 2.3, the context is stripped away and the equation is considered in

comparison to another equation. That is, the equations are objects of study in their own
right in this example, not just as representations of some contextual relationship.
The greater theme, though, is of equations as representations of relationships,
which is consistent with a relational-dependency approach to functions. A careful reader
m i ght notice that the examples in the preceding paragraph were also statements about

tables and graphs, as well as equations. This is the next topic.

Table and Graph. Related to the preceding discussion is the fact that equation is
but one of three basic algebraic representations of fiinctions, together with table and
gr aph, In a relational-dependency approach to functions, tables, graphs and equations are
Of equal importance—none is said to be closer to the real function. There are two ways
that CMP suggests that the three representations are of equal importance. The first is
tlFlr‘()‘-1gh sentences such as, “The students used tables, graphs and equations to look for
patterns relating variables such as cost, income and profit.” (Student Edition, p. 15) In
the S e sentences, table, graph and equation are listed together. The second way that CMP

Com . . . . .
municates the equrvalent value of the three representations 18 through series of

81

repeated questions, such as, “How does the walking rate affect the data in the table?”,
“How does the walking rate affect the graphs?” and “How does the walking rate affect
the equations?” (p. 18). This repetition suggests that tables, graphs and equations are
equivalent; they receive the same treatment.

These two patterns appear throughout the investigation. Two-thirds of the
mentions of equation in Investigation 2 are of one of these two forms—in a sentence
listing table, graph and equation”; or in a sentence that is one of a series mentioning

each of table, graph and equation.

To summarize, the use of the word equation in Investigation 2 of Moving Straight

A head has two important components:

1. Equation is associated with relationship, and

2. Equation has a status equivalent to that of table and graph.

Each of these components supports the relational-dependency view of functions in

CMP’S introduction to linearity.

Rate

There are three major patterns in the use of the word rate in Investigation 2 of
Mo ving Straight Ahead: association with walking, the assertion that rates affect tables,
g1“"5113hs and equations, and the phrase rate of change. The term rate suggests a
r e l ationship between variables; miles per hour, dollars per mile and dollars per gallon are

a1 1 r ates. In each case, we tend to think of one variable (e.g. miles) changing in response

Ts\

eqf: i Sfing is different from simply being in the same sentence. I do not include “Write an
gra atlon for a pledge plan whose graph is less steep...” (p. 20) in the analysis because
ph and equation are not equivalent items in a list.

 

82

 

p1

to the other (hours). That is, the frequent use of rate supports the relationship pattern in
the use of the word equation.

Each use of rate in the Student Edition of Investigation 2 is associated with
walking. Most of these instances use the term walking rate; others are of a slightly
different form as in, “each student walked at a different rate.” (p. 18) Rates are also
strongly associated with walking in the Teacher Edition; 63% of the mentions of rate are
associated with walking. The rate in Problem 2.3 is not a walking rate, but a fund-raising
rate: dollars per mile. CMP does not use the word rate in Problem 2.3. Instead, the text

refers to the pledge amount and the amount pledged per mile.
The Student and the Teacher Editions suggest that rates affect the three algebraic
representations. Forty-two percent of the mentions of rate in the Student Edition are of
th i 8 form, while 34% of those in the Teacher Edition are. The idea is that greater rates
1' m ply greater increments, steeper lines and greater coefficients. All three of these

as sertions are true for non-negative rates and indeed these are the only rates students see

in I nvestigation 2'6

The Teacher Edition is alone in using the construction rate of change, the

gen eralized form of walking rate. The phrase appears four times, in three of the 32

sen tences using rate. The use of the word rate is consistent with the use of equation in
that each supports CMP’s relational-dependency approach to functions.

Linear

Despite being the central idea of the unit”, the word linear only appears in two

pl ac es in the Student Edition’s Investigation 2: the introductory text, and Problem 2.2. It

M

‘ '7 SteeDness of lines in fact depends on the absolute value of the rate.
ecall that the unit’s subtitle is Linear Relationships.

 

83

l8:

appears three times in each of these places. In nearly all cases, linear accompanies
relationship, although in one instance it accompanies situation instead. In general, use of
the word linear is in the context of testing whether a relationship is linear, as in, “How
can you determine whether a situation is linear by examining a table of data or an
equation?” (p. 16) and, “She wonders whether the data represent a linear relationship.”
(p. 19).
The use of linear is much more frequent in the Teacher Edition, where it is
mentioned 21 times. The contrasting idea of non-linearity is directly introduced here as
well. Teachers are instructed that, “To fully understand linear relationships, students must
also see examples of linear and nonlinear situations.” (p. 34d) In the Teacher Edition
1 1' near modifies situation more frequently than in the Student Edition. Additionally, it
In odifies set of data in the Teacher Edition. All three of these terms are used

syn onymously: relationship, situation and set of data (that is, the terms are equivalents) as

i n the following examples.
How can we use the table to prove that this is not a linear relationship? (p. 34c)
The follow-up raises the question of whether every situation is linear. (p. 34c)

How can we use a graph to determine whether a set of data is linear? (p. 34e)

This section has looked at the word use in CMP. Even at this micro level, the
commumcation patterns are becoming clear: the prevalence of the terms relationship and
ra te point to the relational-dependency conception of functions in the curriculum. The
use of these two terms highlights the fact that CMP is concerned with how one variable

Cha . .
nges in response to another (relationship) and With the measurement and

84

characterization of that change (rate). Further, the second contention about CMP—that it
uses contexts to develop mathematical meaning—can be seen in the discussion of rates.
Rate is a central idea in the CMP discourse on linearity, and the word rate is nearly
always associated with contextual language (especially walking). Thus, the mathematical
conception of function as a dependency relationship between two variables is founded on
students’ classroom experiences with contexts. Similar discourse patterns will be featured
in the next section, which looks at the discourse on a slightly larger scale. The next
section analyzes the use of visual mediators—non-linguistic means of communication.
Visual Mediators
In the student materials, the three algebraic representations (tables, graphs and
equations) are the primary visual mediators. Over the course of the investigation, students
are shown three tables, two graphs and eight equations, while they are asked to make four
tables, six graphs'8 and to write ten equations. One important point here is that students
are more frequently asked to create the visual mediators than they are to consider pre-
m ade visual mediators.
There are two aspects common to the use of all three of the algebraic
representations: they represent contextual information and they reveal linearity (or non-
! i I'I'Barit‘y). The first aspect is broken into three: individual values, y-intercepts and rates.
Together with revealing linearity, these are referred to as themes for CMP’s use of visual
mediators in this study. A theme describes how a visual mediator is used in discourse. It
Cles<31‘ibes the purpose of the visual mediator and the kinds of information it can convey.
W
Three of these tables and three of these graphs represent multiple relationships, so

8
rgdénts make four tables to represent nine relationships and six graphs to represent 1 l
atlonships.

 

85

A theme does not state how the mediator conveys this information, nor do themes address
relationships among the visual mediators. These more complex messages are
communicated by narratives, which are analyzed in the next section.

In this section, the importance of each theme to the discourse is measured using
one of three descriptors: a recurrent theme is mentioned in more than one sentence in
more than one lesson; a passing theme is mentioned only once in each of multiple
lessons, or multiple times in a single lesson (or a combination of these); an absent theme
is not in the discourse. Absent themes are identified by theory (e.g. if a theme appears in
the use of tables and of graphs, the theory would suggest analyzing the corresponding

theme for equations).

Each of the three main contexts in Investigation 2 is based on a linear

relationship. The contextual information represented in tables, graphs and equations is of
3 kinds: individual values (such as the distance for a given time), relationships between
values (such as the walking rate) and initial value (such as a head start). While the initial
Value is of course simply a particular individual value, it has a special status owing to the
S 1 Ope-intercept form of the equation, and so is analyzed on its own. The emphasis on the
1‘ e l ationship between values is evidence of the relational-dependency approach to
filnCtions, as is the fact (to be substantiated in this section) that the three visual mediators
re C eive nearly equivalent treatment in CMP.
The most important of these kinds of contextual information in Investigation 2 is
th e relationship between values. This is the most frequently recurrent theme and it is
re<:‘ll‘l'ent for all three representations. Students are repeatedly asked how the rate affects

aeh representation, as described in the Word Use: Rate section earlier. Problem 2.2, for

86

instance, asks “How does the walking rate affect the data in the table?”, “How does the
walking rate affect the graphs?” and “How does the walking rate affect the equations?”
(p. 18) This is reinforced by the teacher materials, which suggest discussion of the table
for Problem 2.2 along these lines, “Ask the class to indicate where this [the increase of
one second of time causing an increase of 2 meters in a walker’s distance] is shown in the
table.” (p. 34c) The Teacher Edition also refers to graphs,
It should be fairly easy for the class to recognize the effect that different walking
rates have on a graph: the steepness of the lines are different. (p. 34d)
and to equations, “The rate, r, is the coefficient of t; it is the number by which the
variable is multiplied.” (ibid)
This pattern is repeated in Problem 2.3. The Student Edition asks the question,
“What effect does increasing the amount pledged per mile have on the table? On the
graph? On the equation?” (p. 20) The Teacher Edition then reinforces the importance of
the idea by directing teachers to address it in the summary.
The effects of the cost per mile on the representations are similar to the effect of
the walking rate in Problem 2.2. Help students to see this similarity. (p. 34g)
Y-intercepts are a passing theme with respect to all three representations. In
Problem 2.3, students are asked how the y-interceptl9 affects each representation.
Alana suggested that each sponsor make a $5 donation and then pledge 50¢ per
mile. How is this fixed $5 donation represented in the table? In the graph? In the

equation? (Student Edition, p. 20).

_‘

l9 . . . .
Note that the language is contextual, so the questions involve donation and head starts.
The term y-intercept does not appear until the end of Investigation 3.

87

The problem asks one question of each representation, without reinforcement in the
Teacher Edition. Similarly, the y-intercept arises again in Problem 2.5. In this problem,
each representation is briefly mentioned. Graphs are mentioned in two sentences, “At
what points do Emile’s and Henri’s graphs cross the y-axis? What do these points mean
in terms of the race?” (Student Edition p. 22) Tables and equations are each mentioned
once, “How can you predict where a graph will cross the y-axis from a table? From an
equation?” (ibid). Because these themes appear, but are not mentioned more than once, in
more than one lesson, they are passing themes.

For tables and graphs, the topic of finding individual values is a recurrent theme.
In each of Problems 2.3 and 2.5, questions are asked about each representation. The
following examples establish multiple mentions in Problem 2.3.

On the graph of a pledge plan, the point (2,6) means that a student who walks 2

miles earns $6 from each sponsor. On which of the graphs is the point (2,6)?

(Student Edition, p. 20)

If a student walks 8 miles in the walkathon, how much would a sponsor owe

under each pledge plan? Explain how you got your answer. (Student Edition, p.

20)

F 01' part C, discuss, or have students demonstrate, how to use each representation-—
the table, the graph, and the appropriate equation— to answer the question.

(Teacher Edition, p. 34g)

The last example above refers to the preceding one. That is, the combination of the

tlldent Edition question and the Teacher Edition instruction to teachers counts as a

88

 

mention of finding individual values in each of the three representations. Problem 2.5
asks twice about this topic with respect to tables and to graphs.
How far from the starting line will Emile overtake Henri? Explain how you can

use the table and the graph to answer this question. (Student Edition p. 22)

After how many seconds will Emile overtake Henri? Explain how you can use the

table and the graph to answer this question. (ibid)

As in these examples, these questions generally name a representation and ask
students explicitly how to find the information in that representation. Through this kind
of questioning, C MP indicates that these representations are useful for answering
quantitative questions arising from the context. Finding individual values is a passing
theme for equations.

Testing for linearity is a recurrent theme for all three representations. The
introductory text cites having a straight-line graph as a criterion for linearity.

The graphs of C 2 300+ 2n and d = 60t are straight lines. From the graphs, it is

easy to see that the relationships between the number of people and the rental cost

and between the miles traveled and the time are linear relationships. (Student

Edition, p. 16, emphasis in original.)

The same introductory text mentions tables and equations in the next sentence.

In this investigation, you will consider this question: How can you determine

whether a situation is linear by examining a table of data or an equation? (ibid)
Finally, the idea that each representation can be used to determine linearity is reiterated,
establishing multiple mentions for these three themes in the introductory text.

Once you have determined—from the table, the graph, or the equation— that a

89

relationship is linear, you can explore this question, How does changing one of

the quantities in a situation affect the table, the graph, or the equation? (ibid)

In Problem 2.2, students are told, “Abby knows that if the relationship is linear,
the data will lie on a straight line when graphed.” They are then asked whether a table
will generate a linear relationship.

Use the table to determine how the distance changes as the time increases. How

can you use this information to predict whether or not the data will lie on a

straight line when graphed? (Student Edition, p. 19)

The Teacher Edition suggests that this question is important to ask about equations as
well as about tables and graphs, but that it will be more difficult for students to answer in
this investigation than the same question about the other two representations. After
suggesting that teachers ask students, “How can we use an equation [to determine
whether a relationship is linear]?” (p. 34e), the Teacher Edition states,

Most students will not be able to answer this last question for now. . .ask that they

continue to think about it as they work through the next few problems. (ibid)
Nonetheless, the message is that each of the representations can be used to determine
Whether a relationship is linear. Together the Teacher Edition and Student Edition
mention these themes more than once in Problem 2.2, establishing that they are recurrent.

There are also relationships between pairs of the representations in CMP. An
equation can be graphed, for instance. This particular example is discussed above, in the
Section titled Word Use, where it is noted that in some instances in CMP, equation is the
primary ObjeCt of study, while in most instances relationship is. A graph has a direct

relationship With a table as well, “”How can you predict where a graph will cross the y-

90

axis from a table?” In this example, graph is the central object and a table can inform us
about the graph. There is no direct relationship suggested between tables and equations in
the student materials.

There are few other visual mediators in the CMP Student Edition. Ordered pairs
are shown twice (but this vocabulary is not used) and there are several illustrations. The
ordered pairs are referred to as points and students are asked about the meaning of these
points in the walkathon context. The illustrations do not carry mathematical meaning“),
and so are outside the discourse on linearity, except for reinforcing the role of context.

The examination of the use of visual mediators in CMP has taken place at a
slightly higher level than the examination of word use (i.e. it is a somewhat more macro-
view of the discourse). That is to say, the unit of analysis is larger. In considering word
use, we look at the sentence level, while when considering the of visual mediators, we
look at the paragraph level. In addition, the word use analysis only attends to the word
(for example) equation, while the analysis of the use of visual mediators attends to
e(luations, even when the word is not in use. The same patterns emerge at this level.
There is an emphasis on rate and on identifying the relationship between the variables in
each representation. The Word Use section claimed that the treatment of the three
representations in the introductory discourse is equivalent. This is compatible with the
relational-dependency conception of fiinctions, and it is borne out by the evidence in this
SeCtion as well. Using the measures of recurrent, passing and absent we see that three of
the four majOr themes (rates, y—intercepts and linearity) receive equivalent treatment in
ere of a bio I h ' ' ' ' '

. yc e w en the mathematical relationships of a bike trip are

is ' '
Shocjilssted’ Cones of soft serve frozen yogurt in the problem about walking to the yogurt
, e c.

 

91

each of the three representations. The only difference is a decreased emphasis on finding
individual values in equations, which is a passing theme in contrast to being a recurrent
theme for tables and for graphs. The analysis in this section also supports the claim that
CMP has students use contexts to develop mathematical meaning, as all of the visual
mediators in the text and all of the mediators that students are asked to create represent
contexts while the themes have mathematical meaning (i.e. the representation of rates in
equations is a mathematical idea, not a contextual one). Yet, this meaning is created
through students’ interactions with the contextual problems.
Narratives

The word use in the opening section and the themes in the use of visual mediators
in the previous section support the contentions that 1. CMP has a relational-dependency
approach to fiinctions, and 2. CMP builds mathematical meaning on students’
experiences with problem solving in contexts. The narratives identified in this section
bolster these arguments. Recall that a theme in the use of a visual mediator states the uses
of the visual mediator, but does not outline how to go about finding or representing
information. Similarly, themes do not identify relationships among the visual mediators.
These are important roles of narratives. A narrative states a relationship between objects
Ofa discourse (e.g. between rates and equations), a property of an object or a procedure
for Operating on an object. In the CMP student text, narratives are rarely stated explicitly.
Instead the curriculum focuses on asking questions that lead students to establish
narratives themselves. This and other techniques for establishing the narratives in this

S ' . . .
eCtion Will form the focus of the next section, Routines.

92

Investigation 2 includes at least six important narratives related to linearity (listed
in Table 3 below). This section elaborates on the mathematical significance of each
narrative and supports the contention that each narrative is part of CMP’s discourse on
linearity using examples from the Student and the Teacher Editions. The analysis of
narratives takes place at a higher level (i.e. gives a more macro View) than either the word
use or the visual mediator analyses. At this level, the same patterns can be seen: CMP has
a relational-dependency approach to functions in the introductory discourse on linearity

and CMP uses contexts to build mathematical meaning.

93

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fiable 3: Narratives in Investigation 2 of the CMP unit Moving Straight Ahead.
1 Mathematical variables represent contextual variables.
—i2 Tables, graphs and equations represent relationships between variables.
3 Tables with constant increments in the dependent variable (given a constant
increment in the independent variable) represent linear relationships.
4 Straight-line graphs represent linear relationships.
5 Equations of form y = mx + b or y = b + mx represent linear relationships.
6 Rates are represented in tables by the constant increment of the dependent variable
(given a constant unit increment in the independent variable).
7 Rates are represented by the steepness (slope) of a line on a graph.
8 Rates are represented by the coefficient of x in an equation representing a linear
relationship.
8 Y-intercepts are represented in tables by the value of the dependent variable when
the independent variable is zero.
T74ntercepts are represented in graphs by the point where the graph crosses the y-
axis.
l\OYK‘inT-e—Igpts are represented in equations by the constant term.
3%“ tables, graphs and equations can answer contextual questions.

 

 

 

 

\
Mathematical Variables Represent Contextual Variables

This first narrative is a theme throughout the investigation, beginning with the

i
mmducmry text excerpted below.

If we let C be the total cost to rent the bikes and n the number of people who go

on the tour, we can write this equation to show the relationship between the

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number of people on the tour and the total rental cost: C = 300 + 2n. (Student

Edition, p. 15)

By mathematical variable (which is my own term), I mean the independent and
dependent variables that are encoded in tables, graphs and equations. Sometimes, these
variables retain their contextual names in the representation (as in a table with headings
Time and Distance), sometimes they are abbreviations of the contextual names (as in the
example cited just above) and sometimes they are abstractions such as x and y. The
distinguishing feature of a mathematical variable is that it is used in an algebraic
representation. A contextual variable (again, my own terminology) is a changing quantity
around which a story is told. For instance, Emile’s distance in the walking race of
Problem 2.4 is a contextual variable that may be represented variously by the
mathematical variables Emile ’3 distance, E, dEmne, y, etc.

It should already be clear that contextual variables abound in Investigation 2 of
CMP. Mathematical variables also appear frequently. They are in the tables, graphs and
equations that are shown to students, but they are also referred to in the text as in, “Let d
r e3Pl'esent the distance in meters and t the time in seconds.” (Student Edition, p. 18) There
are 24 direct references to mathematical variables in the student materials”. Twenty-one
(88%) Of these refer directly to contextual variables. CMP establishes through repetition
that the variables used in tables, graph and equations represent changing contextual

quantities.

\

21
When two variables are used, as in d = 60t, only one reference is counted.

 

95

Tables, Graphs and Equations Represent Relationships between Variables

This second narrative is supported by combining two aspects of the use of the
word equation discussed above. If equations have the same status (are used for the same
purposes) as tables and graphs, and equations represent relationships, then tables, graphs
and equations all represent relationships.

The next three narratives form the major focus of Investigation 2 of Moving
Straight Ahead; they specify how linearity is represented in tables, in graphs and in
equations. The introductory text to the investigation sets up these narratives by telling
students how to identify linear relationships in graphs, then asking the question, “How
can you determine whether a situation is linear by examining a table of data or an
equation?” (Student Edition, p. 16). The three representations are taken in turn.

Tables with Constant Increments in the Dependent Variable (Given a Constant
Increment in the Independent Variable) Represent Linear Relationships.

This narrative is explicitly stated in one of the Teacher Edition goals, “To
recognize linear relationships from tables: for each unit change in one variable, there is a
constant rate of change in the other variable.” (Teacher Edition, p. 14f) The narrative is
imPlied in the Student Edition. The introductory text to the investigation asks, “How can
you determine whether a situation is linear by examining a table. . .7” (p. 16) The answer
is implied through the introductory text to Problem 2.2, “Your findings will give you
some impOl’tant clues about how to identify linear relationships from tables, graph, and
equations-”(1). 18). Here, CMP suggests to students that the table they make will reveal
characteristics of linear relationships. The curriculum then sets up the table for students,

"h the Independent variable (time in seconds) increasmg by a umt increment. This table

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is reproduced in Figure 5. This assures that the dependent variable will have a constant
increment. By examining three sets of data, students use inductive reasoning to see the

narrative (see Routines, next section).

 

Figure 5: The model table in CMP Problem 2.2

 

Distance (meters)

 

Time (seconds) Terry Jade Jerome

 

O 0 0 O
l l 2 2.5
2
3

 

 

 

 

 

 

 

Straight-Line Graphs Represent Linear Relationships
As with the preceding narrative, this is a Teacher Edition goal, “To determine
Whether a set of data is linear by examining its graph.” (p. 141‘) Note that this is not as
eXplicit as the corresponding goal for tables, which mentions the criterion: constant
increment. Nonetheless, the intention is clear. The criterion is a straight line. In the
i ntroductory text, the Student Edition also does not quite state the narrative.
The graphs of C = 300 + 20n and d = 60t are straight lines. [Graphs are shown]
From the graphs, it is easy to see that the relationship between the number of
people and the rental cost and between the miles traveled and the time are linear

relationships. (p. 16, emphasis in original)

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At this point students are left to piece together that the fact of the graphs being
straight lines implies the relationship being represented is linear. ln Problem 2.2,
however, the principle is stated directly, “Abby knows that if the relationship is linear,
the data will lie on a straight line when graphed.” (Student Edition, p. 19)

Equations of F orm y = mx + b or y = b + mx Represent Linear Relationships

This is explicitly not a goal of this investigation. After suggesting that teachers

ask students, “How can we use an equation [to determine whether a relationship is

linear]?”@. 34c) teachers are told, “Most students will not be able answer this last

question now.” Instead, this narrative is established implicitly through example. All eight

of the equations shown to students in the investigation are of these forms (in two cases,

b = 0). Additionally, this is the general form used in the Teacher Edition to refer to linear
equations, as in the Teacher Edition goal, “To interpret the meaning of the coefficient of
x and the y-intercept of a graph of y = mx + b.” (p. l4f)

Note that y = mx + b is intended here to represent a form where y represents the
dependent variable, m the slope, x the independent variable and b the y-intercept. Thus,
d = 2t is of this form since dis the dependent variable, t the independent variable, 2 is the
slope and O is the y-intercept. The equation d — 2t 2 0, although equivalent, is not of this
form, The use of y = mx + b to refer to the slope-intercept form is not intended to suggest
that only the variables y and x are used.

The next three narratives, like the preceding three, have a parallel structure. The
narratives are alluded to in the Teacher Edition goal, “To recognize how the rate of
Change between two variables is associated with its representations,” (p. 140 and more

direcfl)’ by another Teacher Edition goal, “To recognize that a change in rate will change

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the steepness of a line and the coefficient of x.” (ibid) While tables are not mentioned in
this Teacher Edition goal, the Student Edition indicates that they also represent rates.
Thus, there are three parallel narratives here as well—one for each algebraic
representation:

Rates Are Represented in Tables by the Constant Increment of the Dependent Variable

(Given a Constant Unit Increment in the Independent Variable).
Rates Are Represented by the Steepness (Slope) of a Line on 0 Graph.

Rates Are Represented by the Coefficient of x in an Equation Representing a Linear

Relationship.

These \narratives are established implicitly in the Student Edition through
questioning. Students are repeatedly asked questions of this form, “How does the walking
rate affect the table/graph/equation?” (e.g. in each of Problems 2.2 and 2.3). These

questions also appear repeatedly in the Teacher Edition.

The point of these questions is to help students to notice that the contextual rate is
represented in each of the representations. It is conceivable that a student making a table,
for instance, simply sees it as a list of values. CMP intends for students to notice that the
relationships between these values (e.g. between two consecutive distance entries in the
table, or the vertical distance between points on the graph) are dependent on the rate in

the relationship.

Next in the analysis is another set of parallel narratives. This set deals with the

representation of y-intercepts in tables, graphs and equations.

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Y-lntercepts Are Represented in Tables by the Value of the Dependent Variable When the

Independent Variable Is Zero.

Y-Intercepts Are Represented in Graphs by the Point Where the Graph Crosses the Y—

Axis.

Y-lntercepts Are Represented in Equations by the Constant Term.

These narratives are not stated directly in the Teacher Edition goals. The closest
goal is, “To interpret the meaning of the coefficient of x and the y-intercept of a graph of
y = mx + b.” (p. 140 The goal does imply that y-intercepts are represented in graphs and
equations, but it neglects tables and the focus is on the meaning of the y-intercept, not on
how it is represented. A great deal less time is spent on y-intercepts, in the investigation

as a whole, in comparison to rates. Nonetheless, questions such as, “How is this fixed $5
donation represented in the table? In the graph? In the equation?” (Student Edition, p. 20)
parallel the earlier cited questions about rates, and serve to point student attention to the
concept of y-intercept and to its representation in tables, graphs and equations.
Analysis of Tables, Graphs and Equations Can Answer Contextual Questions

There is no reference to this in the Teacher Edition goals. However, there is a

form of question in the Student Edition that points to this narrative. For example this
Question is asked after students have made tables and graphs and written equations in
Problem 2.3, “For a sponsor to owe a student $10, how many miles would the student
have to walk under each pledge plan?” (Student Edition, p. 20) In case students do not

use one of the three representations to answer the question, there is this question,

On the graph of a pledge plan, the point (2,6) means that a student who walks two

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miles earns $6 from each sponsor. On which of the graphs is the point (2,6)?

(ibid)

Notice that “a student who walks 2 miles earns $6” is the form of answer to the
earlier question about how many miles a student would have to walk to earn $10. With
the question about the graph, CMP is endorsing the graph as a way to find answers to
contextual questions. A question in Problem 2.5 similarly endorses the idea that any of
the three representations can be used to answer a contextual question.

Explain how you can use the table, the graph, and the equations to determine how

far from the starting line each brother will be after 5 minutes. (Student Edition, p.

22)

It should be noted at this point that in the CMP discourse, there is a very close
correspondence between the themes in the use of visual mediators and the narratives.
This is due to the nature of the content. Functions are complex and abstract objects for
which we require visual mediators in order to communicate effectively. Further, the
introductory nature of the discourse necessitates narratives about the representations,
rather than about the abstract objects. At some later point in the curriculum, functions

might be discussed as abstract objects, but at the introductory level, the representations—

being visible and directly accessible—are the focus.

Most of the narratives in CMP have one or more visual mediators as their subject.
The classroom researCh, however, illuminates how the themes in the Visual Mediators
section differ from the narratives. A theme in the use of tables, for instance, is that tables
can be used to identify linear relationships. A narrative says how to do this. In a textbook,

there iS generally only a single narrative. In a classroom, there may be competing

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narratives. In this study, a competing narrative is introduced in one classroom when a
student asks, “Wouldn’t she not be linear?” The student is referring to the table
representing Alana’s pledge plan, which has a non-zero y-intercept. This implicitly
introduces the narrative Linear relationships have zero y-intercept. This competes with
CMP’S (and the teacher’s) narrative, Tables with constant increment represent linear
relationships in which the y-intercept is irrelevant. Competing narratives will be
examined in more detail in the classroom research in Chapter VI. Textbooks do not
generally introduce competing narratives, but the Teacher Editions of the two curricula in
this study give teachers information about likely competing narratives that their students
will establish.

The analysis of narratives in CMP is consistent with the findings at the more
microscopic levels of word use and the use of visual mediators. There is a near
equivalence in the narratives among tables, graphs and equations, as all six of the
narratives deal with the three representations. The only exception is the warning to
teachers that formalizing the form of a linear equation (in order to distinguish it from a

non-linear equation) is not a goal of the investigation. This (near) equivalence of the three
representations is also consistent with a relational-dependency approach to functions, as
iS the first narrative, Mathematical variables represent contextual variables. The
Variables take center stage in this narrative, which allows the curriculum to focus student
attention on the relationship between the variables. This narrative also emphasizes the
role of contexts in developing mathematical meaning. This process is examined in more

detail in the next section.

Routines

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Routines are the ways in which narratives are established and reinforced. Routines
are rarely explicit; they need to be inferred from patterns in texts and from patterns in
behavior when examining classrooms. In formal mathematical discourse, definition and
proof are the major routines for establishing narratives. In less formal mathematical
discourse, there are various forms of heuristics. In middle school classrooms and
curriculum, definition sometimes plays a role, proof rarely. There are three main routines
for establishing and reinforcing mathematical narratives in the CMP Student Edition:
asking questions, inductive reasoning and finding mathematical meaning in a context.
Student Edition: Asking Questions

Many of the narratives in CMP are established through questions. Consider Rates
are represented in tables, graphs and equations as an example. Nowhere is this directly
stated in the Student Edition. Yet, this is clearly a theme throughout the investigation, and
it arises from questions. Indeed, the only questions in the shaded box in Problem 2.2 are
of this form, “How does the walking rate affect the table/graph/equation?” Through

repeated questioning about relationships between the context and the representations,
CMP indicates to students that rates are represented in tables, graphs and equations and
that the only issue is figuring out how.

Other narratives established through questioning include Y-intercepts are
represented in tables, graphs and equations (“How is this fixed $5 donation represented
in the table?” (p. 20)) and Analysis of tables, graphs and equations can answer contextual

Questions (“At what point do Emile’s and Henri’s graphs cross the y-axis? What do these

Points mean in terms of the race? (p. 22)).

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Student Edition: Inductive Reasoning

Some narratives are established through repeated examples; inductive reasoning.

In order to establish that Tables with constant increments represent linear relationships,
students are given the linear relationship between distance and time when walking at a
constant pace in Problem 2.2. Students are asked to complete a table, begun in the text
with a constant unit increment in the time column. In the problem’s introductory text,
CMP alerts students that, “Your findings will give you some important clues about how

to identify linear relationships from tables, graphs, and equations.” (p. 18) Students know

before they make the table that doing so will result in noticing how linear relationships

appear in tables. The essential element is that in a single table, students keep track of the
distances for three walkers (see Figure 5, p. 97). While the distances are different for each
walker, and thus the increments are as well, each walker has a constant increment from
second to second and this increment is the same as the walker’s walking rate. One
example would not be enough to discern a pattern, so three examples are provided.
Students are to use inductive reasoning to conclude that constant increment is a feature of
the table of a linear relationship.

The role of non-example is important to the inductive reasoning. In the follow-up
questions to Problem 2.2, students are shown a table with irregular increments and asked
hOW to determine whether it represents a linear relationship.

The process is repeated in Problem 2.3, when students keep track of the pledge
plans for three walkers in a walkathon. In this case, an anomaly is introduced: a walker

Who aSks for an initial donation. All of the preceding relationships have initial values of

0; the walkers in Problem 2.2 start together at 0 meters and the other two walkers in the

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walkathon in Problem 2.3 each raise a set amount per mile. The initial donation
(corresponding to $5 raised for 0 miles walked) sets up the study of y-intercepts. Without
this anomaly students might draw the conclusion that linear relationships must have
graphs that go through the origin (i.e. have y-intercepts of 0)”.

Student Edition: Deriving Mathematical Meaning from a Context

This last example points to the third routine in the Student Edition: finding
mathematical meaning in a context. I have written the narratives in CMP in mathematical
language. One need not know the curriculum to understand the terms of these narratives.
Further, I have argued that these narratives as written” are goals of Investigation 2 of
Moving Straight Ahead. Yet, much of the writing in Investigation 2 uses contextual
language such as miles, walking rates, pledge plans and the names of the various walkers.
It must be that students are expected to use contextual language to develop understanding
of the mathematical narratives. Head start and initial donation are intended to give

meaning to y-intercepts, for instance and walking rates stand in for the more general
concept of rate. This study makes no claims about the effectiveness of this routine. That
is, it is beyond the scope of this study to claim that the expectation described above is
met. Instead, this is an analysis of the model (i.e. intended) discourse. In this model

discourse, students and teachers use contextual language to develop mathematical ideas.

Teacher Edition: Asking Questions

\
22 Indeed the classroom example cited earlier illustrates this. A student asked, “Wouldn’t
She not be linear, Alana?” and the teacher responded, “It doesn’t matter what it starts at.
glusf matters that it goes up by the same amount every time. So hers is linear.”
With the one exception of the vocabulary item y-intercept. Students are not expected to
learn this term until Investigation 3.

105

The Teacher Edition supports these three routines. Teachers are often told to ask
students questions in order to help them understand something, as in the following
passage.

Ask questions that help students notice that for each unit change in time, there is a

constant change in the distance walked by Terry, Jade, and Jerome. (Teacher

Edition, p. 34c)

In Investigation 2, teachers are told seven times to ask questions of students, either to
help point out an important relationship or to help clarify something with which students
may struggle. Sample questions are provided for these purposes. The LES section for
Investigation 2 includes 47 questions for teachers to ask students.

It is worth parsing the phrase, “Ask questions to help students notice,” for this
phrase differentiates the routine of asking questions. The CMP Student Edition rarely
states narratives explicitly. Instead, the text is full of questions for students to consider.
The Teacher Edition likewise contains far more directions for teachers to ask questions
than to tell students things. The lnitiation-Response-Evaluation pattern of teacher
questioning in classrooms is well documented (e.g. Cazden, 1988) and can be
characterized as a form of teacher telling. In this case, teacher questions depend on the
ensuing evaluation of student responses to communicate meaning, as in:

Teacher: What is 2+2?

Student: 4

Teacher: Very good.

In this simplistic scenario, the question does not establish the narrative, 2+2=4, the

teacher response does. When the Teacher Edition implores teachers to “Ask questions

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that help students notice,” the indication is that the questions themselves (together,
presumably with the ensuing student thinking) will produce the desired narrative (here
that a constant increment in a table is the hallmark of a linear relationship).
Teacher Edition: Inductive Reasoning
In contrast to the Student Edition, the Teacher Edition explicitly advocates the
consideration of examples and non-examples, in the ways described in the above
discussion of induction.
To fully understand linear relationships, students must also see examples of linear
and nonlinear situations. The follow-up questions pose a situation that is not
linear. It is important to discuss this example in class and compare it to linear
situations. (p. 34d)
Teacher Edition: Deriving Mathematical Meaning from a Context
Finally, the Teacher Edition includes an introductory paragraph for the Launch-
Explore-Summary text of each problem. The following, from Problem 2.2 is typical.
In this reprise of Problem 2.1, students examine the effect of different rates on
graphs, tables, and equations, and consider the question of linearity more directly.
(p. 34b)
As in the example, these paragraphs state the mathematical subject of the problem
without using contextual language. Similarly, there is a set of Mathematical and Problem-
Solving Goals for the investigation that states mathematical goals in non-contextual
language. In these ways, the Teacher Edition communicates to teachers that students are

to construct mathematical meaning from the contexts in the investigation.

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The text is rarely explicit about precisely how students are to derive mathematical
meaning from the contexts. Three (of perhaps many) possibilities are:

1. Mathematical vocabulary is defined in contextual terms.

2. A mathematical question is asked (by the teacher or by the student) and the
teacher models or encourages seeking the answer by thinking about the
context.

3. The mathematical language is developed as a generalization of the specific
contextual ideas.24

An example of the first might be, “The y-intercept in a linear relationship is the walker’s
head start.” In an example of the second possibility, a student might ask, “How do I find
the y-intercept in an equation?” and the teacher might respond, “Think of this as an
equation for a walker; where would you find his head start?” In the third possibility, a
teacher might say, “We have seen two examples of y-intercepts; head starts and
donations. In each case, we had to add something to the x term in the equation. In
general, this added term is called the y-intercept.”

The student text focuses almost exclusively on the contexts, and so does not give
indications about deriving mathematical generalities. The Teacher Edition has no
examples of the first two possibilities and only hints of the third in Investigation 2. In
Problem 2.2, there is a follow—up question in the Student Edition that presents a table
with nonlinear data. The Teacher Edition suggests the following,

If none of the students uses the data directly from the table to argue that they do

not represent a straight line, ask: How can we use the table to prove that this is not

 

24 .
Of course, these are conjectural.

108

a linear relationship?(As the time increases 1 unit, there is not constant increase

in the distance. From 0 seconds tol second, the distance increases 2 meters; from

2 to 3 seconds, the distance increases 4 meters.)

Help the class to generalize what they have discovered.

How can we use a graph to determine whether a set of data is linear?... How can

we use a table?.(There will be a constant rate of change between the variables.)

How can we use an equation? (p. 34e)

Teachers are instructed that students are to generalize tests for linearity. The first
time testing for linearity is raised, teachers are told to ask about “this relationship
[emphasis added]”. The second time, they are told to ask about “a set of data [emphasis
added]”. The answer the first time is in contextual language: time and distance. The
answer the second time is in mathematical language: rate of change between variables.

Another example of moving from the specific (contextual language) to the general
(mathematical language) occurs in the summary for Problem 2.5.

It is fiindamentally important that students understand that this [the ordered pair

(20,50)] is a pair of values from the table for Emile that shows how many meters

Emile walks in 20 seconds. From this idea, students can build the understanding

that every pair of values in the table is a point on the graph and that every pair of

values in the table and on the graph is a pair of values that fits the equation.

(Teacher Edition, p. 34k, emphasis in original)

It is reasonable to interpret, “the table”, “the graph” and “the equation” to mean
tables, graphs and equations in general, not just those describing Emile’s distance. Yet, .

the text is indirect about this, in contrast with the preceding example. Later investigations

109

are more frequently explicit about deriving mathematical meaning from contexts, but at
the introductory level that is the focus of this study, the process is implicit in the Student
Edition and infrequently discussed in the Teacher Edition.

Three routines for establishing and reinforcing narratives have been identified in
CMP: asking questions, inductive reasoning and deriving mathematical meaning from a
context. These routines align poorly with formal mathematical discourse in which
exposition, proof and definition are central. The CMP routines, however, do align with a
heuristic mathematical discourse—the kind of mathematical discourse that Lampert
(1990) and Lakatos (1976) argue is essential to the work of mathematicians.

To summarize, in CMP, the study of linearity is a study of relationships. The
more abstract idea of algebraic relationships is based in situations with contextual
meaning. There are multiple ways to represent these relationships whether they are being
considered in context or abstractly. This focus on relationship is at the heart of the
relational-dependency approach to functions.

The discourse on linearity is grounded in contexts. The use of the words equation
and rate is heavily associated with the storylines in the problems. Equation is associated
with relationships between contextual variables. Rate is almost exclusively associated
with walking. All three visual mediators are used to represent and to find contextual
information of various kinds. Most of the mathematical narratives have contextual
equivalents within the investigation. The narrative, Straight-line graphs represent linear
relationships could be rephrased in contextual language as Straight-line graphs represent
walkers with a steady pace, which is the focus of several problems and a follow—up

question. This grounding of classroom mathematical discourse in contexts is not the only

110

form for current textbooks, and the relational-dependency approach to functions is not the
only approach. To highlight these aspects of the CMP discourse, the analysis turns to a
curriculum with a different approach to classroom mathematical discourse, the Dolciani
text.

Dolciani

The introductory discourse on linearity in Connected Mathematics is only one of
several possibilities for such a discourse. To illustrate this point, and to highlight aspects
of the CMP discourse, I now turn to Mathematics, Structure and Method, (Dolciani,
1992). All four components of Dolciani’s mathematical discourse prove to be
substantially different from those of CMP. Each of the words in the word use analysis is
used differently. The patterns in visual mediator use vary greatly among the three
representations of functions. The narratives put tables and graphs in service of equations,
rather than treating each of these three in parallel. The routines are more closely aligned
with those of formal mathematical discourse than the heuristic discourse described in the
section on CMP.

In Dolciani, the introductory lessons on linearity are in Chapter 8, titled The
Coordinate Plane. Chapter 8 of Dolciani consists of six lessons, numbered 8-1 through 8-
6. The chapter’s focus is on graphing the equations and inequalities that are the subject of
study earlier in the text. Lesson 8-1 is omitted from this study because it deals with
content that is not particular to the study of linear relationships: plotting points on the
coordinate plane. This is assumed knowledge in the CMP lessons under study and the

analysis of Dolciani picks up in Lesson 8-2, where point plotting is also an assumed skill.

111

Lesson 8-2 (titled, Equations in Two Variables) introduces equations in two
variables. Students are not told this until the next lesson, but all of the equations in this

lesson are linear equations in standard form (ax + by = c). Students are told that solutions

to these equations are ordered pairs and that there are infinitely many solutions to an
equation in two variables.25 There are three examples. The first is about determining
whether a given ordered pair is a solution to a given equation. This is done by
substitution. The second example is about finding a solution (an ordered pair) to a given
equation. Again, this is done by substitution. The third example is about finding y-values
for given x-values for a given equation. The class exercises are of three forms:
determining whether a given ordered pair is a solution to a given equation, solving a
given equation for y and determining y-values for a given equation for given x-values.

Lesson 8-3 (titled, Graphing Equations in the Coordinate Plane) introduces linear
equations in two variables. All of the equations are given in standard form. Tables of
values are used to determine a set of solutions for an equation. These solutions are
graphed and the text notes that the solutions to linear equations form straight lines in the
plane. As a result, two solutions are needed to draw the graph of an equation, but students
are told to include a third point to check their work. There is one example in this lesson,
concerning graphing a given linear equation in standard form. The class exercises have
students find three solutions for each given equation, including the x—intercept and the y-
intercept for each.

Lesson 8-4 (Graphing a System of Equations) introduces the term system of

equations and has students find the point of intersection of the two graphs of the

 

25 In fact, the statement is made about a particular equation, but it is true for any of the
class of equations for which it is an example: linear equations.

112

equations of the system. There are three examples. In the first example, the solution to a
system is first approximated from the graph, then checked in the equations. In the second
example, the lines are parallel. This justifies the observation that there is no solution to
this system. In the third example, the two equations in the system are equivalent and so
the system has infinitely many solutions. These three examples are summarized at the end
of the lesson:

If the graphs intersect, the system has one solution. The coordinates of the point

of intersection form the solution.

If the graphs are parallel, the system has no solution.

If the graphs coincide, the system has infinitely many solutions. (Student Edition,

p.286)
The class exercises have students determine whether a given ordered pair solves a given
equation and restate several of the salient points of the lesson.

Lesson 8-5 (Problem Solving: Using Graphs) introduces the topic of slope. Slope
is defined as,

[T]he ratio of the change in the y-coordinate to the change in the x-coordinate

when moving from one point on the line to another. (Student Edition, p. 288)
There are two examples, both contextual. In the first example, temperatures are changing
at a constant rate from a given temperature at a given time. The example is about
extrapolating and interpolating temperatures at other times. In the second example
Maryanne is burning calories while she stretches and then at a given rate while she runs.
The example is about using this information to determine how long she must run in order

to burn a given number of calories. An equation is written, a line plotted and the solution

113

is read from the graph. One set of class exercises has students find the slopes of lines that
are already plotted, The other set asks students to plot points, connect the points with a
line, find other points on the line and to determine the slope of the line.

Lesson 8-6 (Graphing Inequalities) considers the graph of a linear equation as the
boundary line between two regions, each defined by an inequality. Inequalities are
defined as relations, and linear equations as belonging to a special class of relations:
functions. There are two examples; each is about graphing an inequality by determining
the boundary line and then one point that satisfies the inequality in order to know which
side of the line to shade (the shading indicates the solutions to the inequality). There are
two sets of class exercises. The first has students rewrite a given inequality in y > mx + b
or y < mx + b form and to state the equation of the boundary line in slope—intercept form.
The second set has students state whether given points satisfy given inequalities.

In the first half of this chapter, the CMP approach to functions is characterized as
a relational-dependency approach in which variables, relationships between variables and
equivalence of algebraic representations are in the foreground. CMP indicates that a
function is a dependency relationship and does not offer a formal definition. The Dolciani
approach is different. Dolciani’s approach to fiinctions is set-theoretic. In this approach,
functions are treated according to a more formal definition, such as, “a set of ordered
pairs such that no two ordered pairs with the same first coordinate have different second
coordinates.” Here, the relationship between the variables is de-emphasized in favor of
careful attention to the correspondence between coordinates (i.e. members of the ordered
pairs). In both approaches, linear functions are described by equations. The focus on

equations in the relational-dependency approach is on how rates (i.e. change) are shown

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in the equation. The focus on equations in a set-theoretic approach is on finding one
coordinate of the pair when the other is given, that is, on solving equations.

This section examines the Dolciani textbook discourse on linearity. One focus of
the analysis will be supporting the contention that Dolciani has a set-theoretic approach
to functions. The same structure as the analysis of CMP’s discourse is followed here;
word use, visual mediators, narratives and routines.

Word Use
Equation

In CMP, the use of equation has two important aspects: equation is used regularly
with relationship and many of its uses are identical to the uses of the words table and
graph. These aspects are both absent from Dolciani. Instead, the mentions of equation in
Dolciani almost always coincide with the term solution.

Relationship. The Dolciani student materials do not use equation in the same
sentence with relationship. The closest the materials come to doing so is quoted below.

Write an equation that relates the total number of Calories (y) that she uses when

she stretches and goes for a run to the number of minutes (x) that she spends

running. (p.288)

There is a subtle distinction here. While CMP frequently uses relationship, a noun,
Dolciani uses relate, a verb. An equation, in CMP, represents a relationship between two
variables. In Dolciani, an equation relates two variables. In CMP, there is a sense that the
relationship is primary and that the equation serves the relationship. In Dolciani, this is
reversed. The equation is primary and the relationship secondary. The CMP sense of

relationship appears once in the Dolciani teacher materials, “When graphs and equations

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of lines are used to represent the relationship between real quantities, the numerical
constants have actual meanings.” (p. 27 1 e) The implication in the word when is that this
representation is not the primary use of equations. This implication is substantiated by the
fact that this is the only use of relationship that parallels CMP’s use.

Once, Dolciani uses the mathematical term relation together with equation, “The
equation y = x +1 is a special kind of relation because it defines a function.” (p. 295) But
here, relation has a technical meaning unrelated to the meaning of relationship intended
in CMP”.

Table and Graph. A common pattern in CMP is to list equation as one of three
representations, with each of the representations having the same usage, or alternatively
to ask the same question three times; once about each of the representations. This never
happens in the lessons on linearity in Dolciani.

There are three sentences in these lessons (all in the Student Edition) that use all
three representations. Each is of this form, “Make a table of values for each equation and
graph both equations on one set of axes.” (p. 285). In each case, the equation is given and
the table is created to generate ordered pairs for graphing. In these lessons, students do
not begin with one of the other representations. Tables are not analyzed. Thus, in contrast
with CMP, equations have a different status from tables and graphs throughout the
introductory lessons on linearity in Dolciani. Equations, unlike the other two
representations, are the starting point for mathematical analysis. This is consistent with

the set-theoretic approach to functions of Dolciani.

 

26 Recall the inference that the CMP meaning of relationship is close to function, but
implies a predictability that is not inherent in the mathematical definition of function. A
relation in Dolciani follows the formal mathematical definition: any set of ordered pairs

whatsoever.

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Solutions

Where CMP uses the word equation as one of three representations of
relationships between variables, Dolciani uses equation in association with solution. At
first glance, 51% of the sentences in the student materials that use equation also use
either solve or solution”. Dolciani says that solutions satisfy equations. If we include this
term (which is equivalent to is a solution of), two-thirds (34 out of 51) of the sentences
using equation are about solutions.

A solution to an equation is an ordered pair that, when substituted into the
equation, makes the equation true. Solution is defined in the first chapter—well before
the lessons in this study—in this way,

When a value of the variable makes an open sentence a true statement, we say that

the value is a solution of, or satisfies, the sentence. (p. 9, emphasis in the

original).
In an earlier lesson, an equation is defined to be a kind of sentence; one which uses an
equal sign to relate two expressions.

Together, these three features of the use of equation in the introductory linearity
discourses of CMP and of Dolciani highlight major differences. In Dolciani, equations
are generally given and their fundamental purpose is to be solved. This is consistent with
a set-theoretic approach to functions. In CMP’s relational-dependency approach,
equations are to be derived from contextual relationships and their fundamental use is to

represent these relationships. In Dolciani, the emphasis is not on representation, nor on

 

27 The proportion is smaller in the teacher materials: 43%, but this is still the major
theme.

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rates, but on correspondence. When we solve an equation, we find the x-value that
corresponds to the given y—value (or vice versa).
Rate

Rate appears twice in the Dolciani Student Edition; three times in the Teacher
Edition. This stands in contrast to the twelve mentions in the CMP Student Edition and
the 32 mentions in the CMP Teacher Edition. In CMP, there were three patterns in the
use of the word rate: walking rates, the effect rates have on the three algebraic
representations, and the use of the phrase rate of change. As with the use of the word
equation, the use of rate differs markedly in Dolciani; none of the CMP patterns is
present.

The three Student Edition mentions of rate are in Lesson 8-5, titled, “Using
Graphs to Solve Problems.” This is the one lesson among those in this study to focus on
contextual problems. The word rate appears in the explanatory text (cited below) and
then twice in an example.

A straight-line graph sometimes expresses the relationship between two physical

quantities. For example, such a graph can represent the conditions of temperature

falling at a constant rate or of a hiker walking at a steady pace. (p. 288)

All three Student Edition citations are about the falling or rising of temperatures.
In a sense, the use of rate here is quite similar to that in CMP. While the context is
different, every use of rate in the Dolciani student materials is based in a context, just as
it was in CMP. In Dolciani, the context is changing temperature; in CMP the context is a
walking race. Yet, the first sentence of this excerpted text states an important difference

between the discourses of the two curricula. The discourse on linearity in CMP is

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founded on the contexts. Dolciani, however, builds a formal28 discourse on linearity and
then tells students that one aspect of this discourse (a visual mediator—a straight line
graph) “sometimes expresses the relationship between two physical quantities.” (p. 288,
emphasis added) As with the Teacher Edition’s use of when to suggest that equations are
not fundamentally about describing contexts, the use of sometimes here reinforces the
notion that contextual relationships are not the primary motivation for the study of linear
equations.

The CMP Teacher Edition uses the phrase rate of change several times. I have
argued that this is a generalization of the contextual rates in the Student Edition, and that
this generalization is not present in the student materials. Dolciani operates in a similar
fashion, i.e. contextual uses of rate in the Student Edition and generalized uses in the
Teacher Edition. The Dolciani Teacher Edition uses the word rate three times in the
Lesson Commentaries for the introductory lessons on linearity, all three in the same
paragraph, part of which is reproduced below.

On the other hand, the 10 is a rate, specifically the rate of energy consumption per

unit of time. The quantity 10x, the energy used in running only, is a variable

amount, depending on the number of minutes. Note that the rate is also the slope
of the line. In general, slopes are rates. (p. 27Ie)

The first mention of rate here is contextual: rate of energy consumption. But the
other two make a more abstract connection: slopes and rates are equivalent. In the

introductory lessons on linear relationships, CMP does not explicitly state this. Indeed the

 

28 In the sense of formalist—the view that mathematics is a game of symbols and rules,
driven by logic and that any correspondence with the world of observation is a side

benefit.

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word slope does not appear in the student materials until Investigation 5, and it does not
appear with rate in the teacher materials in Investigation 2. In the end, rate is rarely
mentioned in the Dolciani discourse on linearity, consistent with the set-theoretic
approach to functions.

Linear

The dominant feature of the use of the word linear in CMP is its use to modify
relationship. The major tasks for students are: to determine whether a given relationship
is linear and to isolate the features of a relationship that help to determine this. As with
the other two words in this analysis, Dolciani’s use of linear is quite different.

Every use of linear in Chapter 8 of Dolciani modifies equation” as in the
following passage.

In order to graph a linear equation, we need to graph only two points whose

coordinates satisfy the equation and then join them by means of a line. (Student

Edition, p. 281)

In the sections on the use of the word equation, I argued that one distinction
between the introductory discourses on linearity in these two curricula is the central
object of study; in CMP this is relationship, while in Dolciani it is equation. The use of
the word linear supports this distinction. In CMP, relationships are linear. In Dolciani,

equations are linear.

 

29 With one exception: in the teacher materials, there is the suggestion that linear
programming is a challenging topic for “abler students” (p. 2711). Curiously, an example
of a linear programming problem is worked out, but a definition of linear programming is

not stated.

120

Where CMP has students spend a good deal of time establishing rules for
determining whether a relationship is linear, Dolciani gives a definition based on the form
of the equation, as in this passage,

In general, any equation that can be written in the form ax + by = c where x and y

are variables and a, b, and c are numbers (with a and b not both zero), is called a

linear equation in two variables because its graph is always a straight line in the

plane. (pp. 280-1)

Note that the definition of linear equation rests on the form of the equation, while the
shape of the graph justifies its name. CMP does not state a formal definition in the
Student Edition. Instead, having a straight-line graph is informally stated as a criterion for
being a linear relationship. Again, the contrast between the two approaches to algebra is
clear.

Following this definition (above), the other uses of linear in the Dolciani student
materials involve graphing linear equations. The teacher materials add the idea of non-
linearity to the discourse, “To strengthen students’ understanding of linear equations, you
might point out that not all equations are linear.” (p. 271d).

As with CMP, the microscopic view of the Dolciani discourse afforded by the
close examination of word use reveals a distinctive approach to functions. Equation is
associated with solving—that is, with completing the correspondence between x and y at
a particular point, rather than on the dynamic relationship between the two variables.
Rate is nearly absent in the teacher and student materials. This too points to the de-
emphasis on the relationship between the variables. Finally, linear modifies equation in

Dolciani, rather than modifying relationship, pointing to the increased emphasis on

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equations over the other representations, which is to be expected in a set-theoretic
approach to functions. This differential emphasis on the representations will be seen in
more detail in the next section.
Visual Mediators

As in CMP, tables, graphs and equations are important visual mediators in
Dolciani. Additionally, Dolciani includes ordered pairs as a frequently used visual
mediator. In contrast with CMP, though, students are rarely asked to generate the
representations. Students are shown 10 tables, 19 graphs and 39 equations30 in the
classroom portion of the Dolciani Lessons in this study“. Students are asked to make five
graphs, but are not asked to make tables, nor to write equations”. That is, students are
shown all three of the algebraic representations, but in class are asked only to generate
one: graphs. This observation supports the contention in the word use section that
equations are generally given, rather than derived from contexts, in Dolciani. Of course,
the much greater frequency of equations in the student materials supports the contention
that equations are the central objects of study in Dolciani’s discourse on linearity.

This section first examines the themes in the use of visual mediators previously
identified in CMP and then highlights themes unique to Dolciani. The analysis begins
with the representation of rate in tables, graphs and equations.

R

30 The figure for equations does not include equivalent equations generated in the process
glf solving equations.
32 Reeall that this compares to three tables, two graphs and eight equations in CMP.

This rs partly, but not entirely, an artifact of this study’s analysis method. A Dolciani
lesson has a different structure from a CMP lesson, so it is plausible that these numbers
Would look quite different if we include Dolciani’s written exercises, which are intended
as homework (as are the ACE questions in CMP). Indeed, a student who did all of the
Written exercises would make 117 graphs and write 14 equations. Such a student would

not be asked to make a table.

122

In contrast to their use in CMP, the tables in Dolciani are used to keep orderly
lists of solutions to equations. These solutions are often used to create a graph, as in
Lesson 8-3.

The table of values lists some of the solutions. When we graph the solutions

on a coordinate plane, we find that they all lie on a straight line. (Student Edition,

p.280)

Tables themselves are not analyzed for patterns of change as they are in CMP, and they
are not discussed as a representation. Instead, they are intermediaries between equations
and graphs. Because tables are used to keep track of solutions, rather than to represent
change, there is no reference to increments in tables, and in fact the solutions are not in
any consistent order within the tables. Not surprisingly, then, the theme of the
representation of rates in tables is absent in Dolciani. The corresponding themes for
graphs and equations are passing, being mentioned only in Lesson 8-5, the one lesson that
focuses on the real-world applications of linear relationships.

Consistent with the set-theoretic approach of Dolciani, the identification of
individual values is a recurrent theme for each of tables, graphs and equations. Tables are
organized lists of individual values and appear multiple times in each of Lessons 8-3 and
8-4. Each of these tables corresponds to a graph, on which are labeled the values from the
table. In addition, graphs are used to solve systems of linear equations. In this case, the
individual value is the point of intersection of the two lines on the graph. An estimate is
made from the graph and then the value is checked in the corresponding equations.
Lessons 8-2 through 8-4 each have examples where multiple equations are solved

(generating individual values).

123

 

As in CMP, the themes of the representation of y—intercepts in tables, graphs and
equations are less emphasized than most of the others. The theme is absent for tables and
passing for graphs and equations. The y-intercept is defined and discussed in terms of the
graph in Lesson 8-3, “The y-coordinate of a point where a graph crosses the y-axis is
called the y-intercept of the graph.” (Student Edition, p. 280, emphasis in original)
Equations are used in this lesson to find y-intercepts in order to make a graph. The one
other place that y—intercepts appear, although not by name, is in Lesson 8-5, “We can see
that the line crosses the vertical axis at —I .” (p. 289)

The final theme identified in CMP’s use of visual mediators is the identification
of linearity in each of the three representations. Consistent with the use of tables
discussed in relation to rates, linearity is an absent theme in Dolciani’s use of tables.

Lin earity, in Dolciani, pertains mainly to graphs. There are multiple mentions of the
straight-line graphs and connecting points by means of a line in Lessons 8-3 and 8-5, so
the theme is recurrent. With respect to equations, the only mention of linearity is in the
definition of a linear equation in two variables in Lesson 8-3. This theme is passing.

The treatment of these themes in the two curricula is summarized in Tables 6—8.
From these tables, it can be seen that the two curricula have similar emphases on the
themes with respect to graphs, but vary greatly on their treatment of tables and of

equations.

124

 

 

 

 

Table 4: Themes in each discourse on tables.

Rate Individual Value Y—intercept Linearity
CMP passing
Dolciani absent absent absent

 

 

 

 

 

Table 5: Themes in each discourse on graphs.

 

Individual Value

       

Y-intercept Linearity

    
 

 

   

   

Dolciani passing passing

 

 

 

Table 6: Themes in each discourse on equations

 

1 Rate

    

Individual Value Y-intercept Linearity

     
 
   

  

 

 

 
 
     

passing passing

 

     

   

    
 

Dolciani passing passing passing

 

 

 

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I have made the argument several times that equations are the central objects of
study in Dolciani’s introductory discourse on linear relationships. A further piece of
evidence is that four of the five lessons in this study begin with an equation, as in Lesson
8-2, “The equation x + y = 5 has two variables, x and y.” (p. 276)

A fourth major visual mediator in Dolciani is ordered pairs. Over the course of the
five lessons, students are shown 89 ordered pairs33 of the form (x,y) where x and y are
usually integers, but may be rational numbers or variables. In general, these ordered pairs
are solutions to given equations, reinforcing again that equations are the central objects of
discourse in Dolciani. The use of ordered pairs to display solutions to equations is a
recurrent theme in Dolciani.

There is one last visual mediator in these lessons. The diagram reproduced in
Figure 6 shows a hill with varying steepness. The diagram is included to give students an

example for thinking about steepness, and then slope.

 

 

 

Figure 6: A visual mediator in Dolciani

 

To summarize, the use of visual mediators in Dolciani reinforces the set-theoretic

approach to functions in several ways. First, the three algebraic representations of table,

 

33 Recall that this compares to two ordered pairs in CMP. The figure for Dolciani does
not include the ordered pairs that label points on graphs.

126

 

graph and equation receive differential treatment, with much more emphasis on equations
than the other two. Second, the theme of identifying and representing rates in the
representations is at most passing, being mentioned in one lesson. Third, the abundance
of ordered pairs is reminiscent of the formal definition of function in which a function is
defined as being a set of ordered pairs with particular properties. Similar patterns emerge
at a higher level in the next section, when the narratives in Dolciani are considered.

Narratives

In contrast with CMP, Dolciani’s narratives are more explicitly stated in the
student materials (see Table 9). CMP tends to point to true mathematical statements
through repeated questioning, while Dolciani tends to do so through definition. As an
example of this second characteristic, consider the case of the first of these narratives, A

linear equation is of the form ax + by = c.

127

 

Table 7: Narratives in the study of linear relationships in Lessons 8-2 through 8-6 of

Dolciani. Italics indicate narratives that are verbatim from the text.

 

1 A linear equation is an equation of the form ax + by = c.

 

2 A solution to a linear equation is an ordered pair.

 

3. Solutions are found by substituting values into an equation.

 

4 A table lists solutions to an equation.

 

5 A graph is the set of all points that are graphs of solutions to an equation.

 

6 A straight-line graph can represent a relationship between contextual variables.

 

7 A straight-line graph represents a constant rate.

 

8 A linear equation has a straight-line graph.

 

 

9 Slope is the ratio of the change in x-coordinates to the change in y-coordinates on a

graph.

 

 

A Linear Equation is of the Form ax + by = c
Recall that in CMP, we have to infer the form of a linear equation in the
introductory lessons on linearity. While this form is made explicit later in the unit, the
introductory lessons suggest that linear equations are of the form y = mx + b or
y = b + mx through examples and the emphasis on how rate and y-intercept are
represented in the equation. Dolciani is more direct.
In general, any equation that can be written in the form ax + by = c where x and y
are variables and a, b, and c are numbers (with a and b not both zero), is called a
linear equation in two variables because its graph is always a straight line in the
plane. (pp. 280—1, emphasis in original)

This is the first mention of the term linear, and the definition is explicit.

128

 

A Solution to a Linear Equation Is an Ordered Pair
This narrative is stated explicitly in the Dolciani Student Edition, in the paragraph
cited below.
The equation x + y = 5 has two variables. A solution to this equation consists of
two numbers, one for each variable. The solution can be expressed as an ordered

pair of numbers (x, y). (p. 276)

Once this fact is established, Dolciani relies on it throughout the remaining lessons on
linearity, asking students to determine whether given ordered pairs are solutions to given
equations (or systems of equations) in Lessons 8-2, 8-3 and 8-4. Ordered pairs are
discussed as solutions in Lessons 8-5 and 8-6 as well.
Solutions Are Found by Substituting Values into an Equation

Lesson 8-2 is titled Equations in Two Variables. The Objective stated in the
Teacher Edition is, “To find solutions for an equation in two variables.” (p. 271b) The
introductory paragraph in the Student Edition establishes that a solution of a linear
equation is an ordered pair. These facts establish that this lesson is about solving
equations. There are three examples worked out in the text. The central aspect of each of
these examples is a substitution. There is one paragraph between Example 2 and Example
3, cited below.

To find the value of y corresponding to any given value of x, we could substitute

the value of x into a given equation and solve for y, as in Example 2. An easier

method is to solve for y in terms of x first, and then substitute, as in Example 3.

(Student Edition, p. 277)

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The essential message here is that substitution is the technique necessary to find
solutions. The implication in this paragraph is that the only issue worth debating is at
what stage in the problem-solving process to make a substitution.

A Table Lists Solutions to an Equation

As discussed above, under the heading Visual Mediators, tables are used
differently in Dolciani than they are in CMP. There are four narratives in CMP involving
tables; that they represent relationships, that rates and y-intercepts can be discerned from
them and that they can be analyzed to answer contextual problems. None of these is
suggested in Dolciani. Instead the Student Edition states,

The table of values lists some of the solutions. When we graph the solutions

on a coordinate plane, we find that they all lie on a straight line. (Student Edition,
p.280)
In addition, each of the 10 tables in the Student Edition is an unorganized list of
solutions, usually generated in order to graph an equation. The only consistency in the
choices of the x-values in these tables is that they are single digit integers. Consider these
three lists of x-values—each is in the order that the values appear in a table—{-9, -3, 0, 3,
6}, {0, 6, 3} and {-1, 5, 2}. In each case, the corresponding y-values are listed as well,
but these are unorganized lists of solutions, not representations of a relationship. Students
are not expected to extract meaning from these tables; they are expected to use the tables
to make graphs.
A Graph is the Set of All Points that Are Graphs of Solutions to an Equation

This narrative is pretty much the definition of graph of an equation in Lesson 8-3.

The contrast with CMP is strong. In CMP, a graph is a representation with its own status.

130

In Dolciani, a graph depends on an equation. More fundamentally, a graph is composed
of solutions in Dolciani, while solutions are not introduced in CMP until students have a
great deal of experience with graphs.
A Straight—Line Graph Can Represent a Relationship between Contextual Variables
Lesson 8-5 is titled Problem Solving: Using Graphs. This lesson is the only one
among the five lessons in this study whose examples include contextual variables. The
relationship between mathematical variables and contextual variables is essential to the
CMP approach, but is much less essential in Dolciani. Yet, the relationship is not
neglected. Students are told, “A straight-line graph sometimes expresses the relationship
between two physical quantities.” (p. 288) The word sometimes is an acknowledgment of
the difference between this lesson, which emphasizes contextual variables, and the rest of
the chapter, which does not.
A Straight-Line Graph Represents a Constant Rate
This is most directly suggested in Chapter 8 of the Student Edition in the
following sentence.
For example, such a graph can represent the conditions of temperature falling at a
constant rate or of a hiker walking at a steady pace. (p. 288)
This is less direct than most Dolciani narratives, which tend to be definitional. But this
one suggests that temperature falling at a constant rate and hiker walking at a steady
pace are among the things that can be represented by straight-line graphs, rather than
defining straight—line graphs as necessarily being about constant rates and steady paces.
The narrative appears implicitly again in this passage from the same lesson.

Since we know that the temperature climbed at a constant rate, we can first graph

I31

the two points and then draw a straight line through them. (p. 289)
A Linear Equation Has a Straight-Line Graph

This narrative is directly stated in Dolciani’s Lesson 83 when the term linear is
defined,

In general, any equation that can be written in the form ax + by = c where x and y

are variables and a, b, and c are numbers (with a and b not both zero), is called a

linear equation in two variables because its graph is always a straight line in the

plane. (pp. 280-1)
Slope is the Ratio of the Change in Y-coordinates to the Change in X-coordinates on a
Graph

In these lessons, slope is a matter with relevance only to graphs. This is in
contrast to CMP, where rates (contextual rates, although these are later tied to slope) are
studied with respect to tables and to equations, as well as to graphs. There is one lesson in
Dolciani that concerns itself with slope, Lesson 8-5.

The slope of a line is the ratio of the change in the y-coordinate to the change in

the x-coordinate when moving from one point on the line to another. (p. 288)

In CMP, all narratives but one have visual mediators as subjects. In Dolciani, all
of the narratives have visual mediators as their subject. CMP’s narratives have a
parallelism that was closely related to a pattern noted in the use of the word equation:
tables, graphs and equations have roughly equivalent status. This shows up in the CMP
narratives in the repeated construction, “tables, graphs and equations,” and in the
narrative Linearity is represented in tables, graphs and equations 4, 5 and 6, which state

how to identify linearity in each representation. This is consistent with the relational-

132

dependency approach to functions. No such parallelism is present in the Dolciani
narratives. Equations and graphs are the major subjects of narratives, with tables
appearing only once. There is no narrative that describes all three representations. In the
CMP Narratives section, the argument is made that the narratives have the visual
mediators as subjects because of the abstract nature of functions. This claim is also trued
for the Dolciani narratives. Dolciani gives a much different treatment of linear
relationships and yet the narratives are still about the representations. This is due to the
nature of the content, rather than indicating other similarities in the two discourses.
Routines

As with the other aspects of the Dolciani discourse on linearity, the routines for
establishing narratives differ markedly from those of CMP. The three main discourse
routines of CMP— asking questions, inductive reasoning and finding mathematical
meaning in a context— are mostly absent from Dolciani. Instead, Dolciani has four main
routines: defining, telling, calculating and examples. Three of these— defining, telling and
examples— are prominent in the Student Edition. Two are prominent in the Teacher
Edition, defining and calculating.
Student Edition: Defining

Defining is the most common routine for establishing the narratives in this study.
Dolciani definitions follow the mathematical form quite closely. In this form, the defined
term is highlighted (here in bold type), followed by the word is, and then the meaning of
the term, as in:

The slope of a line is the ratio of the change in the y-coordinate to the change in

the x—coordinate when moving from one point on the line to another. (Student

133

Edition, p. 288)
In this example, the Dolciani form is distinguished from the formal mathematical form in
that “when moving from one point to another” is an operation that is supposed to have an
intuitive meaning rather than a strictly mathematically meaning. A definition that follows
the mathematical form more closely is this one,
In general, any equation that can be written in the form ax + by = c where x and y
are variables and a, b, and c are numbers (with a and b not both zero), is called a
linear equation in two variables because its graph is always a straight line in the
plane. (pp. 280-1)
This is equivalent to “A linear equation is any equation...” This example establishes two
narratives: A linear equation is of the form ax + by = c and A linear equation has a
straight-line graph. The first of these narratives is established by definition. The second
of these narratives is slightly different. The narrative A linear equation has a straight line
graph tells students about the relationship between two defined objects. This is the
second routine in Dolciani.
Student Edition: Telling
Definitions in Dolciani establish the meaning of a term the first time it is used.
Telling involves terms that have already been defined34 and are already in use. Three
narratives are established by telling students a relationship between mathematical objects:
A solution to a linear equation is an ordered pair, A straight-line graph can represent a
relationship between variables and A linear equation has a straight-line graph. In each

case, two previously defined objects (solution and ordered pair, graph and relationship,

 

34 Although possibly very recently.

134

and linear equation and graph) have a relationship that is told to students. This is akin to a
theorem in formal mathematical discourse. Dolciani does not prove the told narratives, as
must be done with theorems in a more formal discourse, but the emphasis on the
relationship between two defined objects is analogous to the structure of a theorem. One
example of telling is in the preceding section, immediately after linear equation is
defined. Dolciani justifies the term linear by stating that linear equations always have
straight-line graphs. Note the difference between telling and defining in that passage.
Linear equation is defined in terms of its form as an equation; students are told that such
equations have straight-line graphs.
Student Edition: Examples

In contrast to CMP, every Dolciani Lesson in this study has at least one example.
The five Lessons together have 11 numbered examples”. A Dolciani example has a
particular form, and the use of the term example is different from its use in the analysis of
CMP and inductive reasoning. In Dolciani, an example is a well-defined task with the
answer given together with a set of steps leading to the answer. In general, many of the
Written Exercises (which are not analyzed in this study) are tasks similar to the examples.
The examples can be used by students as a reference when working on the Written
Exercises. In the analysis of CMP’s use of inductive reasoning, examples are instances of
a relationship that has not been explicitly stated. The CMP examples are not labeled as
such. They are often the product of a task, and their form changes depending on the

relationship to be demonstrated.

 

35 As with questions in the CMP problems, these Examples often have subparts, so there
are really more than 11 problems worked out in the 11 Examples.

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In Dolciani, examples are used to establish two narratives. The first, Solutions are
found by substituting values into an equation, is established in Lesson 8-2. Here, the task
in the example is to, “Tell whether each ordered pair is a solution of the equation

2x + y = 7.” (p. 276) Under the heading “Solution,” the procedure is to, “Substitute the

given values of x and y in the equation.” (ibid) There are 6 Examples that explicitly
involve solutions. Of these, three use the word substitute, while the remaining three
demonstrate substitution without using the term.

The second of the narratives established by example is A straight-line graph
represents a constant rate. In the Narratives section, I argue that this narrative is
established indirectly. This happens in the midst of an Example with this task,

The temperature at 8 am. was 3°C. At 10 am, it was 7°C. If the temperature

climbed at a constant rate from 6 am. to 12 noon, what was it at 6 am? What

was it at 12 noon? (p. 288)

This task involves extrapolation. Two data points are given, along with the fact that the
rate of change is constant. It has not previously been established that constant rates and
straight-line graphs have a close relationship, but the text indicates that this is so.

Since we know that the temperature climbed at a constant rate, we can first graph

the two points and then draw a straight line through them. (p. 289)

This is an indirect way of saying “Straight-line graphs represent constant rates.”
Teacher Edition: Defining
When establishing or reinforcing the narratives in this study, the Dolciani Teacher

Edition sometimes refers to previous definitions, as in this passage, which gives teachers

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a strategy for reinforcing an important aspect of the narrative, A solution to a linear
equation is an ordered pair.
By this time students find it quite natural to substitute for variables. They need to
be reminded, however, that we have agreed to the convention that ordered pairs

are (x, y) pairs. (p. 271b)
The text continues to demonstrate that (4,—1) is a solution to 2y + x = 7, but that (—l,4)

is not. In order to fully establish the narrative about ordered pairs, the definition
(convention) of an ordered pair is reinforced.
Teacher Edition: Calculating

The Word Use section demonstrates that in Dolciani, equations are number
sentences. Thus, the various statements about equations in Dolciani are statements about
equality of expressions that represent numbers. Therefore, finding the numerical value of
these expressions is one way to verify statements about equations. Calculation is a
common strategy for establishing narratives in the Teacher Edition. As an example,
consider the follow-up to the example above, under Definition. That example revisited
the definition of an ordered pair in order to reinforce the narrative, A solution to a linear
equation is an ordered pair. The routine of calculation immediately follows,

Conversely, students should be warned to substitute correctly when checking a

given number pair in an equation. In Example I, (4,—1) is a solution, but (— 1,4) is

not because 2(—l)+ 4 i 7. (p. 271b)

In this example the assertion is that the order of the pair matters, and this is established

through calculation.

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These routines point to Dolciani’s set-theoretic approach to functions, but also to
the mathematically formal nature of the discourse in Dolciani. A formal definition of
function, which has been stated here, involves sets of ordered pairs. It is sensible, then,
that Dolciani’s set-theoretic approach to functions would involve more formal
mathematical routines for establishing narratives; especially the routine of defining. It is a
notable contrast that Dolciani offers seven definitions in the five lessons in this study,
while CMP offers none.

Comparison

To summarize, the contrast between CMP’s discourse and Dolciani’s discourse is
strong. The use of all three words in the Word Use analysis of Dolciani differs markedly
with the use of these words in CMP. Equation is associated in Dolciani with solution,
rather than with relationship. Rate is rarely mentioned in Dolciani, while it is a central
object in CMP. Linear is restricted to a single lesson in Dolciani. Each visual mediator in
Dolciani has a unique purpose and use, in contrast to CMP, where the representations are
nearly equivalent. Dolciani’s narratives focus on equations, while CMP’s narratives
encompass all three representations equally. The routines for establishing narratives are
more formal and direct in Dolciani than in CMP.

Another major difference that has consequences in each of these categories is the
relationship between contexts and mathematics in the two curricula. In Dolciani, contexts
are introduced in a single lesson: Lesson 8-5 and students are told that linear equations
sometimes represent relationships between real-world variables. In CMP, the analysis of
change in contexts is the focus of the discourse. The study of linear relationships in CMP

begins with the study of contexts. In Dolciani, the study of linear relationships mentions

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contexts. This can be seen in the routine Deriving mathematical meaning from a context,
which is absent in Dolciani and it can be seen in the differing uses of equations;
describing relationships versus solving.

At the beginning of this chapter is a claim that equations are important objects in
the study of algebra, and this has certainly been confirmed in the analysis of these two
curricula. I conclude the chapter using an equation, together with contrasting material
from Dolciani to characterize the nature of the mathematical discourse in CMP.

The equation y = mx + b is the general form of linear equations in CMP. In

Dolciani, the form is ax + by = c. In the CMP form (called slope-intercept), it is easy to

find the slope (or rate of change) and the y-intercept. The slope is m, the y-intercept is b.

There is a direct relationship with the contexts in which the study of linearity
takes place. There are basically two contexts in CMP around which the introductory
discourse on linearity in CMP takes place: walking races and raising money in a
walkathon. One important feature of each context is highlighted by b and one by m. The
y-intercept in a walking race is the starting point. If everyone has the same starting point,
then all of the equations will have the same b. In a walkathon, it is the initial donation—
the amount of money someone raises without walking a step. In a walking race, m is the
walking rate; in a walkathon m is the pledge rate. Important contextual information can
be read directly from the slope-intercept form of the equation.

Recall that CMP has a relational-dependency approach to functions. In practice,
this means that instruction should be focused on the relationship between variables, and
that one should think of one variable depending on the other. The chosen contexts

support this way of thinking, and together the contexts and this way of thinking support

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the slope-intercept form of linear equations. I am not arguing that a curriculum with a
relational-dependency approach to equations could not use the standard form for linear
equations. Instead I claim that there is a natural connection between the approach CMP
has taken to functions, the contexts in which students learn about linear functions and the
form of linear equations that CMP endorses. The curriculum is consistent in this
approach.

It is also notable that CMP endorses the slope-intercept form of equations, in this
investigation, through example and through the contexts, rather than through definition.
Formal mathematical discourse—as captured in textbooks, journals and lectures—has a
character different from the everyday work of mathematicians. Lampert (1990) makes
this argument in the study of her classroom, and the mathematical activity described by
Lakatos (1976) is quite different from the linear definition-theorem-proof pattern of
formal mathematical discourse. Behind the clean formality is difficult messy conjectural
activity. CMP encourages students to engage in this messy activity by having them write
equations before explicitly stating a general form. Students are guided to a particular
form by the nature of the problems in the curriculum, but as in a mathematician’s work,
the definition comes later. I have argued that Dolciani holds a contrasting view of the
nature of mathematical activity. Dolciani lessons tend to begin with a definition and with
theorem-like statements in the manner of mathematics journal articles or lectures.

A final way to characterize these differences is to examine assumptions about
learning in the twb discourses. Learning a discourse involves imitation to some degree

(enculturation). The'teacher is modeling a discourse and the student’s job is to participate

in this discourse, which she can only do by imitating the teacher’s discourse patterns. In

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particular, students imitate (and to a certain degree negotiate) the routines for establishing
narratives. In the Dolciani form, students imitate procedures for using and verifying pre-
established truths——calculating and the working of problems similar to examples. In
CMP, students imitate routines for finding mathematical truths that may be unknown—
asking questions, reasoning inductively and finding mathematical meaning in real-world
problems (a form of metaphor) are all heuristic techniques that advanced students of
mathematics use when attempting to solve new problems. This is the justification for
considering these two textbook discourses as mathematical discourses, as discussed in
Chapter IV. These two textbook discourses encourage different types of mathematical
discourse: Dolciani the formal discourse of journals and lectures and CMP the informal
discourse of mathematical discovery.

This chapter has described the introductory discourse on linearity in two curricula,
using the discourse in Dolciani to highlight important features of the CMP discourse. The
analysis in this chapter is instructive about the differences in reform curriculum. The
theoretical framework involving communication has turned up substantial differences in
the content and teaching procedures of CMP as compared with Dolciani. Also, the
analysis has been instructive for future study of reform curriculum. It has revealed that
assumptions about the content of two curricula—even when the topic is the same—may
be invalid. Finally, this chapter has set the stage for answering the larger question about
the relationship between published curriculum and classroom instruction by
characterizing the published curriculum CMP. The next chapter shifts to an analysis of

the discourse in two classrooms using Investigation 2 of Moving Straight Ahead. The

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ways in which the textbook discourse of CMP changes in classroom implementation will

be the focus of the next chapter.

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Chapter VI: Classroom Discourse

The curriculum Connected Mathematics (Lappan, et al., 2001a) is part of a larger
curriculum reform effort in US K-12 mathematics instruction. Lively debate has taken
place surrounding this reform and its goals. Much of this debate has been based on
presumed correspondence between the printed curriculum and classroom instruction. This
chapter examines this correspondence in detail, using one set of lessons (those analyzed
in Chapter V) as realized in two classrooms. There are two research questions addressed
in this chapter: What characterizes each teacher ’s discourse on linearity and What
changes does the discourse of the curriculum undergo in its implementation by each
teacher? Together, these questions are designed to develop our understanding of
instruction in the wake of a significant curriculum reform, and to provide information that
will be useful in improving mathematics instruction in this context.

The preceding chapters have characterized teaching and learning in a
communicational framework using the language of discourse analysis. Knowledge has
been characterized as participation in a discourse; learning as change in that participation
and teaching as engagement with learners in the model discourse. These definitions are
offered in order to make the question, What is being taught in reform mathematics
classrooms? a researchable question. Rephrased as What is the model discourse? the
question is operative in the sense that the terminology can be turned into straightforward
research tools. These tools are detailed in Chapter IV.

Chapter V examines the Connected Mathematics curriculum (CMP; Lappan, et
al., 20013), specifically the curriculum’s introduction to linearity. Another curriculum,

Mathematics, Structure and Method (Dolciani, 1992) is used as a contrast in order to

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highlight important aspects of the CMP discourse. The introductory textbook discourse
on linearity of CMP is characterized by its relational-dependency approach to functions
and adherence to a heuristic—rather than formal—mathematical discourse.

CMP’s approach to functions differs from Dolciani’s approach, which is set-
theoretic. The relational-dependency approach emphasizes the variation underlying the
term variable, the relationships between variables, and multiple representations of these
relationships. This is contrasted with a set-theoretic approach that emphasizes
correspondence between variables, variables as unknowns, solving equations and places a
priority on equations and ordered pairs over graphs and tables.

A heuristic mathematical discourse involves the techniques people use when they
are trying to solve novel problems in mathematics. For instance, reasoning inductively
from examples, asking questions and employing various forms of metaphor are ways that
mathematicians and advanced students of mathematics attempt to understand new
domains. This is contrasted with formal mathematical discourse, where the heuristics,
once used, are hidden and in their place, terms are given careful definitions and theorems
are stated and proven. Formal mathematical discourse is the discourse of lectures and
journals. Heuristic mathematical discourse is what takes place around the chalkboards in
the offices of mathematics professors and graduate students.

The question that drives the present study is, What is being taught in reform
mathematics classrooms? Having phrased this question in terms of discourse, it is
important to note that there are two model discourses in most mathematics classrooms:
that of the textbook and that of the teacher. The present chapter has two goals that

correspond to the research questions;

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(1) Characterize the introductory classroom discourses on linearity in two

classrooms, and

(2) Examine the changes that the textbook discourse undergoes as it is translated

into classroom discourse by each teacher.
Chapter IV addresses some of the difficulties inherent in comparing these two discourse
types (e. g. that students have input into the classroom discourse, but not the textbook
discourse), and these difficulties will be kept in mind and addressed again in the
comparison section of this chapter.

The two classrooms in this study were chosen based on the following
characteristics: they are located in city schools with diverse student populations, the
teachers have experience with the student population and with the curriculum, there is
high time on task in the classrooms, the teachers pay attention to the mathematical
generality behind the contextual problems in the curriculum, and the time of year the unit
Moving Straight Ahead (CMP, 2001b) would be taught. These conditions were intended
to frame a study of typical classrooms with good conditions, rather than ideal classrooms.
This study does not seek to provide an “existence proof” of what is possible (Lampert,
1990), but to describe situations that are likely to be replicated in many classrooms in this
country.

Sfard (2001) argues that four aspects distinguish mathematical discourse from
other discourse genres: word use, the use of visual mediators, endorsed narratives and the
routines for endorsing these narratives. These aspects are analyzed in the textbook
mathematical discourses of CMP and of Dolciani in Chapter V. This chapter uses the

same set of analytical tools (modified for the different discourse form of classrooms

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instead of textbooks) to characterize the classroom mathematical discourse of the
teachers in this study. This chapter has three sections. Each teacher is considered
separately (two sections), and then the changes in the textbook discourse as it is
implemented in the two classrooms are detailed. In each of the first two sections, the
characteristics of mathematical discourse (word use, visual mediators, narratives and
routines) are considered in turn, giving a close, medium and long range view of Ms. M’s
discourse, and then Ms. H’s discourse. The chapter ends with a comparison of the
textbook discourse on linearity (in CMP) with the classroom discourses in this study.
Ms. M’s Discourse

Ms. M teaches in a school with approximately 450 students in 7’h and 8th grades.
The school’s enrollment is 86% minority. The school follows a seven-period day in
which classes last 48 minutes. Students change schedules each semester, staying with
some of their previous teachers and getting some new ones in an unsystematic way.
Moving Straight Ahead, the CMP unit that introduces linearity, is studied in 8th grade.
Some of Ms. M’s students were in her classes the previous year, as 7th graders, while
some were not. The student mobility rate is high: 49% of the student population
transferred in or out of the school in the first semester of the 2003-04 school year, the
most recent year for which figures are available. Ms. M’s students who were at her
school as 7th graders studied CMP, either with Ms. M or another teacher. Students who
were not at her school had a variety of curricular backgrounds.

Ms. M. spends seven class periods working through the problems of Investigation
2. Her instruction includes the three phases of a CMP lesson (Launch, Explore,

Summarize), but generally not all three in the same day. The sequencing of these phases

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is shown in Figure 7. The typical sequence is to Launch and Explore a problem one day
and to have the Summary near the beginning of class the next day, followed by the

Launch for the next problem.

 

Figure 7: Major activities“ in Ms. M’s Investigation 2 lessons.

 

 

 

Key
warm-up El

launch
explore

summary .
other

 

 

 

 

 

In addition, she spends a significant amount of time on warm-up activities
(referred to as warm-ups, these are indicated by the white areas in Figure 7). These
warm-ups are related mathematically and contextually to the problems in the
investigation, but the tasks are of her own design. In general, Ms. M uses these warm-ups
to discuss individual narratives from the list of narratives in CMP. The major contribution

that warm-ups make to the classroom discourse is to establish narratives that are absent

 

’6 Activity is defined in Chapter III. A new activity is recorded whenever the answer to
the question, What are students supposed to be doing now? changes.

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from the Launch-Explore-Summary (LES) part of the lessons. In the following analysis,

claims are first made about each aspect of Ms. M’s discourse using evidence from the

LES sequence for the problems from the text. When appropriate, separate claims follow

about how the warm-ups contribute to each aspect of the discourse. That is, the warm-ups

are addressed here only when they contribute to the understanding of Ms. M’s discourse.
Word Use

Equation

In this study, word use refers to the patterns in the appearance of words in
discourse. Occasionally, the frequency of these patterns is telling, so frequency counts
appear in parts of this section. Frequency counts, though, are a crude tool—particularly
when comparing across discourse types. The main foci of the analysis of word use are
identifying the patterns, considering what these patterns say about the meaning of the
words, and considering how the patterns contribute to the characterization of each
classroom mathematical discourse. The first patterns to be analyzed are the patterns that
appear in the CMP textbook discourse, then any new patterns that appear in the teacher’s
classroom discourse, but not the textbook discourse, are analyzed.

Ms. M’s use of the word equation shares with CMP’s use the association with
tables and graphs, but she talks about these differently. She puts less emphasis on the
term relationship than does CMP. In addition, there is a pattern unique to Ms. M’s use of
equation: that of possession.

Relationship. In CMP’s discourse, the predominant use of equation is as a
representation of a relationship. In contrast, Ms. M is less explicit about this. Where 43%

of the Student Edition sentences using the word equation also used the term relationship,

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only 16% of Ms. M’s sentences using equation use either relationship or the term
situation.

The association between equation and relationship is less prominent in Ms. M’s
discourse at least in part because she discusses the writing of equations at length. Even if
the number of sentences associating equation and relationship were identical in the two
discourses, the extended discussions of writing equations in Ms. M’s discourse drives
down the percentage (more on this in the next section, Visual Mediators).

Table and Graph. Ms. M. opens two of her summaries by discussing tables,
graphs and equations, as in this passage,

Yesterday what you were asked to do was to make a table, graph and come up

with an equation to represent this walking race situation where Henri walked a

meter per second, Emile walked two and a half meters per second. (Ms. M.

Activity 3, 11/02/04)

Additionally, she follows the lead of the text in discussing tables, graphs and equations in
turn, but her questions tend to be different in nature for each of the three representations.
Ms. M. asks questions that suggest students will be able to notice and articulate how rates
affect tables, as in”:

You also looked at the table between the three of them. Like we saw in the warm-

up today, how can you tell in a table which person has the faster rate? (Ms. M.

Activity 2 10/27/04)

 

’7 When transcript excerpts involve multiple turns, M indicate Ms. M’s turns, while
students’ initials (or Student when the student is not identified) indicate students’ turns.
Ellipses within turns indicate pauses. Ellipses between turns indicate that content has
been omitted from the transcript.

I49

This is a rephrasing of the text’s question, How does the walking rate affect the graph? in
which Ms. M asks for a general principle (“how can you tell in a table...?”) that might be
applied to any table of walking rates. The question she asks about graphs is different,
“Looking at those three graphs, where is Jerome’s graph? What color is it?” (Ms. M.
Activity 2, 10/27/04) In this case, the students had remembered aloud that Jerome was the
fastest walker and Ms. M asks about locating this particular walker, rather than about a
general principle. She asks students to say which is Jerome’s graph, rather than to say
how to tell who has the fastest rate when looking at a graph.

Finally, Ms. M tends to ask questions that lead to the writing of equations, rather
than to the comparison of equations. This is in contrast to her questions about tables, in
which students are asked to state principles for rate comparisons, and to her questions
about graphs, which implicitly ask students to identify graphs with particular properties
(i.e. with the fastest rate).

Now the final part, part C, you were asked to find the equations that displayed the

relationship between time and distance for these three walkers, and that’s always

kind of the challenge part, but at this point, you guys are really good at finding

equations.

What can you tell me about time and distance? (Ms. M Activity 2 10/27/04)
The line of questioning that begins with, “What can you tell me about time and
distance,” continues as students state that time and distance are the same for one walker

(hence the equation d = t), that distance is twice the time for another walker (d = 2t), etc.

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Once the equations are written in this summary, she finishes the conversation by
saying, “So we have tables, graphs and equations, all that can represent this linear
situation.” (ibid). This final statement reiterates the text’s message that tables, graphs and
equations are equivalent representations. Yet it comes at the end of lengthy discussion in
which the representations have been used in non-equivalent ways.

In CMP, the same questions are asked of students about tables, graphs and
equations, requiring parallel answers: larger rates yield larger increments in tables,
steeper lines in graphs and larger coefficients in equations. In Ms. M’s discourse, the
comparison of the coefficients is left out. Equations get written, but they do not get
compared to each other in the way that tables and graphs do.

Possession. The most common pattern in Ms. M’s use of equation is the use of
possessive adjectives. In Problem 2.2, she refers to “Jade’s equation.” In Problem 2.3, she
asks students, “...what’s a good equation for Leanne?” and then, “So what’s her
equation?” This use pattern shows up in one problem in the text, Problem 2.5, “Write an
equation for each brother showing the relationship between the time and the distance
from the starting line.” (CMP Student Edition, p. 22) By contrast, Ms. M uses a
possessive adjective or talks about an equation for someone in each of Problems 2.2
through 2.5, and the CMP construction showing the relationship is absent, as can be seen
in the preceding examples.

The tension between relational-dependency and set-theoretic approaches to
functions has been summarized by Kieren (1993) as a process-object tension. The
process-object distinction captures some, but not all of the differences between the two

approaches. When a function is seen as a process, the computational procedures that it

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encompasses are highlighted; these are the rules for moving from values of the
independent variable to values of the dependent variable. This is Sfard’s (1991)
operational description of functions. When a function is seen as an object, the
correspondence and ordered pairs are highlighted. This is Sfard’s structural description
of functions. The way the word function is used, too, can give clues about whether
functions are seen as processes or objects. It is in this light that Ms. M’s possession
pattern is important. When she speaks of “Alana’s equation,” equation is being used
linguistically in the manner that an object would, such as Alana’s shoe or Alana’s home.
At the microscopic level of the analysis of word use, this is evidence of an object view of
functions, which is consistent with a set-theoretic approach. Thus, we can see the
process-object tension at play in Ms. M’s classroom at the level of word use.

The term tension applies in Ms. M’s discourse because the two approaches
coexist, but are not reconciled. That is, equations have somewhat different treatment from
tables and graphs, but Ms. M does not discuss this differential treatment. The two
approaches are unresolved; they are in tension. In CMP, the relational-dependency
approach is the norm. Problem 2.3 briefly places equations at the center as objects of the
discourse, suggesting set-theoretic notions. Like Ms. M’s discourse, this transition is not
directly discussed. However, unlike Ms. M’s discourse, this occurs once and briefly; this
second approach does not linger. Instead, the discourse highlights set-theoretic ideas in
the follow-up questions and then returns to the relational-dependency approach in the
next problem. There is no ongoing tension between the two approaches.

One question about the classroom implementation of CMP is how a traditionally

trained teacher teaches a reform curriculum. One might conjecture that a teacher with

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experience in a traditional curriculum (and thus with a set—theoretic approach to
functions) would struggle to use tables and graphs with students—that equations might
continue to take the prominent place in the teacher’s discourse. This is not the case with
Ms. M’s discourse. Another conjecture might be that the use of equation would correlate
with solving and correspondence (set-theoretic notions) over variation and relationship
(relational—dependency notions). This second conjecture is a bit closer to the mark in Ms.
M’s discourse. While she does not heavily emphasize the set-theoretic ideas, she does
speak of functions as objects in at least one way and she notably de-emphasizes the
relational-dependency ideas. The conjecture that this is related to her teaching experience
prior to teaching CMP is one for future research.

Ms. M’s use of equation reveals two co-existing patterns: she asks questions
about tables and graphs together with her questions about equations, and so emphasizes
some form of equivalence among the three representations, but she does not emphasize
the relationships that CMP does, and she speaks of functions as objects in at least one
way. This points to a tension in her discourse between the set-theoretic and the relational-
dependency approaches to functions.

Rate

There are two patterns in Ms. M’s use of the term rate: association with walking,
and the identification of rates in the three representations. Recall that in CMP, the use of
rate is associated with walking, with affecting the three algebraic representations, and
with the term rate ofchange.

Sixty-nine percent of the sentences (25 out of 36) that include rate include the

word walking or a variation. This usage is comparable to the discourse in the CMP text,

153

 

where all of the mentions of rate in the Student Edition are associated with walking and
63% of those in the Teacher Edition are.

The use of rate in Ms. M’s instruction parallels the use of equation. When
discussing equations, Ms. M alters the CMP discourse. Where CMP focuses on
comparing equations, Ms. M emphasizes writing them. Similarly, the CMP discussion of
how rates affect tables, graphs and equations is downplayed in favor of identification of
rates in tables, graphs and equations. The comparison task is changed to an identification
task in each case. In the Student Edition, 42% of the mentions of rate asked how the rate
affects one or more representations, while the figure is 34% in the Teacher Edition. By
contrast, Ms. M’s figure is 17% (6 out of 36). Of these six mentions, she is quoting
directly from the Student Edition five times—only once does Ms. M suggest on her own
that rates affect tables, graphs and equations. Instead, she more frequently identifies rates,
or has students identify rates in the representations.

A lesser pattern in the CMP discourse is the use of the term rate of change as a
generalized form of walking rates. Ms. M does not use this term. Ms. M’s use of rate is
consistent with CMP’s use in its association with walking—rate is a contextual term in
Ms. M’s discourse, just as it is in the CMP Student Edition. Yet, Ms. M deviates from
CMP’s use of rate in a manner similar to her deviation in the use of equation; comparison
tasks are changed to identification tasks.

Linear

The term linear arises in four passages (9 sentences altogether) in Ms. M’s

lessons, in contrast to the Teacher Edition of CMP, which uses the term 21 times. Ms. M

links linearity strongly with graphs, as in, “They make linear graphs. Are all three of

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them linear? Are all three of these graphs linear?” (Ms. M Activity 5, 10/28/05) While
Ms. M takes time to identify rates in tables, graphs and equations, the only representation
associated with her use of linear is graphs. She mentions linear relationships and linear
situations one time each. Ms. M does not use the term nonlinear.

In the microscopic view afforded by this word-use analysis, differences between
CMP’s textbook discourse on linearity and Ms. M’s classroom discourse begin to
emerge. It is relevant and important that the differences noted are not inherent in the two
types of discourse; instead, they point to different approaches to functions and to
linearity. These differences speak to differences in discourse patterns. In particular, Ms.
M’s use of the word equation, with its de-emphasis on relationships and emphasis on
possession, points subtly in the direction of a set-theoretic approach to functions
characteristic of traditional American algebra curriculum, where CMP consistently takes
a relational-dependency approach. Yet, Ms. M is also consistent with the curriculum in a
number of ways, most notably in her use of rate and in the association of equation in
sentences with tables and graphs. The comparison with CMP, of course, highlights where
there are deviations in Ms. M’s word use from those we would expect in a strict
relational-dependency approach to fiinctions. Overall, it is more accurate to characterize
Ms. M’s discourse in terms of deviating from a relational-dependency approach in the
particular ways outlined here, than to characterize it as an example of a set-theoretic
approach.

Visual Mediators
In mathematical discourse, we use visual mediators to communicate about

abstract objects that cannot be physically present. In the case of functions, three algebraic

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representations—tables, graphs and equations—are the primary visual mediators. In
CMP, these three representations are used in nearly equivalent ways. In Chapter V, this
claim is substantiated by considering themes in the use of the representations and
characterizing them as recurrent, passing or absent. In textbook discourse, a recurrent
theme is mentioned in more than one sentence in more than one lesson. Textbooks are
carefully edited while teacher talk is not, so this measure is too fine for the analysis of
classroom discourse. Instead, when analyzing teachers’ classroom discourse, a recurrent
theme appears in more than one teacher turn in more than one lesson. A passing theme
appears in a single teacher turn in multiple lessons, or in multiple turns in a single lesson,
or a combination of these. Absent themes are not present”.

Ms. M uses the same visual mediators as CMP: tables, graphs and equations. She
shares the curriculum’s expectation that students will create these mediators more often
than they will be shown them. In the Launch-Explore-Summary sequence, the only time
she shows students tables, graphs or equations is during the summary—after students
have created their own visual mediators. Further, she frequently uses the student-created
mediators for her summaries. Ms. M also shares the two uses of the representations with
the curriculum: representing contextual information and revealing linearity.

Tables

There are three important kinds of contextual information that can be found in
each representation: rates, individual values and the starting value (or y-intercept). The
use of tables to identify rates and find individual values are recurrent themes in Ms. M’s

discourse, but the role of y-intercepts in tables and the use of tables to identify linear

 

38 This is a non-trivial matter, as the themes identified in the curriculum are considered in
the classroom discourse. Thus, the set of absent themes is finite.

156

relationships are both passing themes. The first time she discusses the representation of
rates in tables is in her summary of Problem 2.2.

M: ...how can you tell in a table which person has the faster rate?

Student: bigger numbers

M: Bigger numbers. OK. Bigger numbers, and where else can you look besides

just the individual number itself? Because here, yeah, we end with the bigger
numbers here. Think about what E was doing when she found the rate in the
table.

Remember you can also look in between. What’s the change here?

Student: 2.5

M: 2.5. What’s the change here?

Students: 2

M: 2. What’s the change here?

Students: 1

M: 1. OK. So a faster rate in a table is going to show up in a larger change

between values. Let’s say it like that. (Ms. M Activity 2, 10/27/04)

In this case, Ms. M’s first question suggests that tables can be used to compare
rates. When the student answers are not quite mathematically correct, she focuses the
students’ attention on the important feature of the table (“What’s the change here?”) and
then states the relationship, “a faster rate is going to show up in a larger change between
values.”

Ms. M revisits the topic of rates in tables in her summary for Problem 2.4, where

she asks questions of a group whose table is displayed on poster paper at the front of the

157

classroom. In this case, she adopts students’ language for increment (“adding each time”)
and uses this to elicit from students an explicit statement about how rates appear in tables.
This establishes the theme as recurrent.

In Problems 2.3 and 2.4, Ms. M uses the tables to identify individual contextual
values. In Problem 2.3, Ms. M has students identify points that would be on each
walker’s graph. The discussion begins on the graph, but moves to the table when the
points are difficult to read on the graph on the overhead projector.

In Problem 2.4, Ms. M has groups of students tell how long the race between
Emile and Henri should be. Of particular importance for the present analysis is that each
of these groups has used a table to solve the problem and Ms. M focuses the conversation
on the part of the table that shows the finish line determined by each group. These two
examples show that finding individual values in tables is a recurrent theme.

The role of y-intercepts in tables, however, is a passing theme for Ms. M
discusses it in a single problem. She has students use tables to find the initial value in
Problem 2.3. She asks, referring to Alana’s pledge plan, “Where’s that five dollar
donation represented in the table?” (Ms. M, Activity 5, 10/28/04). There are several turns
making up this part of the discussion, but the topic is not raised again in the investigation.

The representation of rates in tables is a recurrent theme. Ms. M has students
compare the rates across multiple tables in the investigation. However, in the Launch-
Explore-Summary sequence of the lesson, she does not directly discuss the idea of a
constant rate, nor of linearity with respect to tables. This theme would be absent if only
this part of the lesson were analyzed. Instead, it appears in the warm-up on 11/2/04, so

the theme is passing. This episode is examined in more detail in the Narratives section.

158

In Problem 2.4, there is a subtle difference between Ms M’s use of tables and
CMP’s use. In the problem, students are looking for the point in the race where Emile
catches Henri. Ms. M has student groups put their solution strategies on poster paper and
Ms. M conducts the entire summary using these posters. In all five cases, the students
used a table to solve the problem. It should be noted that these students had prior
experience from a pilot unit in another curriculum39 in making tables and writing
equations from the patterns in tables. This other curriculum emphasized the relationship
between x and y in each pair, rather than the increment in the table. Many students made
tables that did not involve a constant increment for the independent variable. Three

student tables are reproduced in Figure 8.

 

 

 

Figure 8: Three student produced tables used in Ms. M’s summary of Problem 2.4

 

 

 

 

 

 

 

Guess Henri Emile
Number +45 x2.5
Time Emile Henri Seconds meters meters Sec. Henri Emile
0 ] 0 l 45 10 55 25 1 45 o
10 25 55 20 65 50 2 46 2.5
20 50 65 30 75 75 3 47 5.0
25 63 70 29 i 74 72.5 ...4"
29 72 74 29 [blank] [blank]
30 [blank] [blank]

 

 

 

 

39 College Preparatory Mathematics, (CPM, 2002). See Chapter II for a more detailed
description of these students’ curricular path.

40 These students continued the unit increment all the way to 30 seconds. They continued
keeping track of Henri’s distance through 20 seconds and Emile’s distance through 9
seconds.

159

 

This type of table (with non-constant increment) is only implicitly sanctioned by
the Student Edition, which for this problem tells students to, “[m]ake a table showing the
distance each brother is from the starting line at several different times during the first 40
seconds.” (p. 22). This indicates that students are free to choose any values they wish for
time, and that there need not be a pattern to their choices. The answer in the Teacher
Edition, however, shows a table with constant increment of 10 (i.e. the time values are 0,
10, 20, 30 and 40). Further, a follow-up question introduces a third walker, with a
different rate and different head start and asks students to complete a table started in the
text. As with the other tables shown in the Student Edition, this one has a constant unit
increment for the independent variable. In this instance, Ms. M’s discourse seems to vary
from that of CMP; Ms. M explicitly endorses tables with non-constant increments“,
while the CMP Student and Teacher Editions model only tables with constant increments.
Graphs

The role of rates in graphs, the finding of individual values in graphs and the
representation of the y-intercept in a graph are all recurrent themes in Ms. M’s discourse.
The use of graphs to identify linearity is also a recurrent theme. Ms. M adds one aspect to
CMP’s treatment of graphs: a manipulation that helps to highlight the meaning of
steepness.

Ms. M addresses the topic of rates in graphs in three Problems: 2.2, 2.3 and 2.5. In
Problem 2.2, Ms. M not only indicates that rates are represented in graphs, she tells
students how.

It’s steeper, OK? And that’s a word that I want you to keep thinking about. That

 

4’ By using such tables as the focus of her summary, for instance.

160

the person who had the fastest walking rate... what kind of line did he have? He

had a much steeper line than the other two. (Ms. M, Activity 2, 10/27/04)

She returns to this theme in Problem 2.3. Faster does not apply in that problem, but the
idea of an increased rate (i.e. more money per mile) showing up as a steeper line is the
focus in this problem.

Finally, in Problem 2.5, Ms. M discusses the follow-up question, which addresses
a third person who walks faster then Henri, but slower than Emile and who gets a smaller
head start than Henri does. She directly addresses how the rate will affect the steepness of
the graph.

Ms. M raises the subject of representing individual values in the graph in two
Problems: 2.3 and 2.5. In Problem 2.3, a student indicates that he found dollar amounts
for given distances using his graph. Ms. M pursues this conversation for several turns to
elicit an explicit statement of the student’s procedure.

In her summary for Problem 2.5, Ms. M uses the graph to locate the point where
Emile overtakes Henri. Students have not raised the topic, so Ms. M uses several turns to
focus student attention on this point. This establishes the theme of using tables to identify
individual values as recurrent.

The representation of y—intercepts in graphs is raised by Ms. M in two Problems:
2.3 and 2.5. In Problem 2.3, Ms. M asks, “Where’s that five dollars represented on the
graph?” (Ms. M Activity 5, 10/28/04) From prior instruction, her students know the term

y-intercept, so she follows this question with another, “What’s that point called?” (ibid)

161

Ms. M discusses a follow-up question to Problem 2.5 involving a third walker
with intermediate rate and head start. The brief discussion of the graph begins by
determining where the line should start (i.e. the location of the y-intercept).

M: How would that equation fit in up here? Where would it be starting on the x-
axis [sic]?

Student: At 20

M: At 20 meters, good. So we start it right here. (Ms. M Activity 3, 11/02/04)

Ms. M introduces the topic of linearity in graphs in Problem 2.1. In the Launch
for that problem, she points her students’ attention to a set of graphs on chart paper at the
back of the room. These are graphs that students made in their instruction with the CPM
curriculum (see footnote, page 156). In that experience, students made graphs and learned
names for the patterns in the graphs, linear among them. Ms. M asks which of these
graphs is linear. Students answer correctly. She asks, “How do you know a linear
relationship?” and students answer, “Straight line.” (Ms. M Activity 2, 10/25/04) In
Problem 2.3, Ms. M revisits the t0pic of linearity in graphs. She asks a question about the
relationship between the rate and the graph.

M: Ma, can you keep going with how does it affect the graph?

Ma: It just makes a linear... (Ms. M Activity 5, 10/28/04)

Ms. M follows up with several turns, asking questions to clarify whether all three graphs
are linear (they are).

Ms. M adds to the CMP discussion of graphs a manipulation of this visual
mediator. In the following passage, she is struggling to help her students verbalize a
distinction between higher points on a graph and a steeper line. Figure 9 reproduces the

graph under discussion. In the figure the bold line represents Leanne’s pledge plan, the

162

narrow line represents Gilberto’s pledge plan and the dotted line represents Alana’s

pledge plan.

 

 

Figure 9: Graph of the three pledge plans in Problem 2.3

Money raised (dollars)

20
18
16
14
12
10

8

 

/

ON-bCD

012345678910
Distance walked (miles)

 

\l
.3???

But what we’re trying to figure out is how come these lines
look different?

Because one’s higher and the other one’s lower.

Why is this green one higher than the red one? Why is it
steeper than the red one?

Because I think that’s more money.

More money! You’re there, A. What was the pledge for this
green graph? How much did they get per mile?

Two dollars.

Two dollars. How much did Leanne get per mile?

One dollar.

One dollar. And imagine everybody, if I take this black line,

163

 

10

ll

12

13

14

15

16

I7

18

I9

20

21

A:

M:

Student:

M:

and let’s forget that they got that five-dollar jump. If I would
just move it down to here,

It’d be less.

It’d be even less steep than this one. The only reason it seems
higher for a little bit is it got that five dollar jump in there.
But if it wouldn’t have had the five dollar jump, it would be
even less.

Say that again.

But if it wouldn’t have had that five-dollar jump, it would be
less.

Right. Because this was only 50 cents per mile, right there. So
M, back to your question. How does that dollar per mile
change the graph? What happens to it as that dollar per mile
increases?

The black line decreases...

The black line is flatter. What happens to one dollar per mile?
(whispers to Ma) It gets higher.

What’s that word we used yesterday everybody?

Steeper

Steeper. One dollar per mile it gets steeper. (Ms. M Activity 5,

10/28/04)

When Ms. M says, “If I would just move it down to here,” in turn 9, she indicates

the transformation in Figure 10. Her students’ language (e.g. in turn 2) suggests they are

164

seeing only the height of the line, rather than its slope. This is an improvisation in
response to her students’ repeated use of higher. After she performs the transformation,
she seems to be convinced that the problem is no longer students not seeing the
phenomenon she wants them to see, but the words they are using to describe it. She asks,
“What’s that word we used yesterday...” to clarify that she is seeking a particular word.

Transforming graphs is a passing theme, appearing only this one time.

 

Figure 10: Ms. M transforms the graph showing Alana’s pledge plan.

20 ., ~
18 "
16
14
12
10

(I,
,/ /
/

1' ,4
/
/

Money raised (dollars)

I

//
/

 

 

omamoo

012345678910:

‘ /
Distance walked (miles)

 

 

 

There are two main differences between CMP’s treatment of graphs and Ms. M’s
treatment, but neither points to fundamental differences in the two discourses. Ms. M puts
more emphasis on the representation of y-intercepts in graphs than does CMP and she
introduces the theme of transformations, which is absent in CMP. This is not the case
with equations. Ms. M uses equations differently from CMP. She indicates to students
that equations represent contextual information, but she downplays the discussion of how
equations can be used to decide whether a relationship is linear.

Equations

165

In Ms. M’s discourse involving equations, rates and y-intercepts are recurrent
themes. The finding of individual values is a passing theme. Discussion of identifying
linear relationships from equations is absent from Ms. M’s discourse in the problems of
the investigation, but it does appear in the warm-ups.

Ms. M addresses the topic of rates in equations in three Problems: 2.2, 2.3 and
2.5. In Problem 2.2, she uses the table students have worked on to help them to notice the
relationship between the two variables; distance and time.

Ms. M revisits the representation of rates in the equation in Problem 2.3. In this
case, the equations have already been written. She asks students to articulate how the rate
appears in each. This consumes several turns.

The follow-up question to Problem 2.5 has been analyzed in the two preceding
subsections, tables and graphs. In this follow-up, Ms. M also raises the topic of rates in
equations for a final time in the investigation.

M: Ok, if Paulette is at 20 meters head start and walks 2 meters per second, what

would be her equation? Yeah, T?

T: [inaudible]

M: Talk a little louder. d equals 20 plus 2 t [inaudible; presumably the student is

saying each part of this equation, but it cannot be heard] Excellent.

M: 20 is the amount added on. Two meters is our rate.

The topic of using equations to find or represent individual values is a passing
theme. It surfaces in one problem in Ms. M’s instruction: 2.3. After she quotes the text’s

question, For a sponsor to owe a student IO dollars, how many miles would a student

I66

have to walk under each pledge plan? (CMP, Student Edition, p. 20), a student tells how
she computed her answers and Ms. M summarizes,

...she was using these equations here. Because she was saying, well my distance is

10, one times 10 is gonna give me my money. For this one, my distance is 10, two

times 10 is gonna give me my money. Then for Alana, the trickier one, she went

50 cents times 1 0 is five dollars and added on the 5. (Ms. M Activity 5, 10/28/04)

The y-intercept’s relationship with equations is a recurrent theme in Ms. M’s
discourse. She raises the question of how the y-intercept is represented in the equation in
two Problems: 2.3 and 2.5. In Problem 2.3, as the class is discussing the writing of
Alana’s equation, Ms. M presses a student to state the constant term in the equation in a
voice loud enough for the whole class to hear, then draws attention to this term by
discussing its form.

M: Keep talking, R. What’d you say?

R: m equals point fifty d. ..

M: point fifty d...

K: plus five

M: Plus five. Yeah, K, yeah, Ma. Now when I wrote it here, I didn’t write point

fifty, I just wrote point five. Does everybody agree that five tenths is the same
as fifty hundredths?
Students: yeah
M: Point five is the same as point five zero? OK. d plus that five dollars. All

right. (Ms. M Activity 4, 10/28/04)

167

Collapsing Ms. M’s last turn in the excerpt to read, “point five d plus that five
dollars,” shows that one purpose of the sentence is to summarize the place of the y-
intercept (in context, the initial donation in the pledge plan) in the equation. The y-
intercept in an equation is raised again in Problem 2.5.

In the Launch-Explore-Summary sequence of the five problems of Investigation
2, Ms. M does not discuss identifying linear relationships from equations. Equations are
discussed in four of the five summaries, but the word linear does not appear in these
summaries.

As indicated in the section on the use of the word equation, the main focus of Ms.
M’s instruction involving equations is on writing them, with a lesser emphasis on
comparing them. This emphasis on writing equations is revealed in an analysis of her
discussion of equations as visual mediators (as opposed to the analysis of her sentences
using the word equation). In these discussions, the focus is on how the rate appears in the
equation, with questions and statements such as the following:

What’s the relationship between those two columns? (Ms. M Activity 2,

10/26/04)

So the rate this time shows up in the equation as well. (ibid)

Where is the amount per mile in this equation? (Ms. M Activity 4, 10/28/04)

Keep that in mind, that rate is always multiplied to the variable, OK? (ibid)

d equals 20 plus 2t. Excellent. Twenty is the amount added on. Two meters is our

rate. (Ms. M Activity 3, ll/02/04)

168

These examples come from three different problems: Problem 2.2, 2.3 and 2.5. These are
the three problems in which equations were a focus of the summary. This means that
every time Ms. M discusses equations, she raises the issue of how the rate is represented
in equations. In the first example, “What’s the relationship between those two columns?”
Ms. M asks the question to help students notice that the number of seconds Jerome (in
Problem 2.2) walks is multiplied by 2.5 to find the number of meters he walks in that
time. This leads to writing the equation d = 2.5t.

An extended example helps to clarify how Ms. M discussed equations differently
from tables and graphs. Consider the following transcript, from Ms. M’s summary of
Problem 2.3 (Ms. M Activity 5 10/28/04).

M: We had this group over here, M, E and A, your question is, what effect does

increasing the amount pledged per mile; what does the effect have on the

table, on the graph and on the equation ?42

E: The table, it, uh... it increases, so if... the numbers it increases...

E: OK, so... the amount pledged on the graph, per mile, it goes up and it
increases.

M: OK

E: And that’s what the effect has, it increases for the pledge.

M: So you’re saying if the amount pledged per mile increases, the amount on the

table is...

 

42 The italics here indicate that she is quoting from the text.

169

E: increases

M: Increases as well. OK. Does Alana’s throw anything off at all?

E: It increases.

M: It increases, but she’s only getting fifty cents a mile, where these guys are

getting two dollars a mile.

A: It’s decreasing?

E: Well, I think it’s the same cause she has the five dollars.

M: Cause she had that five dollar jump. OK. It’s still going up, but not quite as

fast, maybe.

To this point, Ms. M is pushing the student (E) to distinguish between the number
in the table and the increment. When she asks, “Does Alana throw anything off?” she
indicates that the student’s seeming assertion that greater rates yield larger numbers in the
table is incorrect. When this doesn’t work, she supplies the answer, “It’s still going up,
but not quite as fast.” Throughout this segment on tables, Ms. M keeps the focus on
comparison of the tables for walkers with different rates. The conversation then turns to
graphs.

M: Ma, can you keep going with how does it affect the graph?

Ma: It just makes a linear...

M: They make linear graphs. Are all three of them linear? Are all three of these

graphs linear?

A: No.

[the class revisits the definition of linear]

M: There are three straight lines, so M they’re all linear. How does the

170

changing...

[interruption]

M: But what we’re trying to figure out is how come these lines look different?

A: Because one’s higher and the other one’s lower.

M: Why is this green one higher than the red one? Why is it steeper than the red

one?

Here, Ms. M introduces the term that she wants students to use, steeper, and
pushes students to explain the term in context (i.e. to state what the steepness of a line
says about the rate). As with the discussion of tables, the conversation stays focused on
comparing the three graphs. The conversation continues a few moments later,

M: 80 Ma, back to your question. How does that dollar per mile change the

graph? What happens to it as that dollar per mile increases?

Ma: The black line decreases...

M: The black line is flatter. What happens to one dollar per mile?

A: (whispers to Ma) It gets higher.

M: What’s that word we used yesterday everybody?

Student: Steeper

M: Steeper. One dollar per mile it gets steeper. Two dollars per mile,

A: Gets higher

M: It gets even more steep. It gets even steeper than before. As you increase that

amount per mile, the graph seems to be getting steeper.

171

Even though the nature of the questions is different here than it was for tables
(e.g. she is asking for word recall), the focus remains on comparison. The comparison is
lost when the conversation moves to equations.

M: All right, and then... the equations. Where do you see that amount per mile in
the equation everybody? Or, sorry, you guys over here, that’s still your
question.

Where is the amount per mile in this equation?
In the graph?

: The l.

: Where’s the amount per mile in this equation?
The 2?

: Where’s the amount per mile in this equation?

The two-fifty.

Z??Z?.>ZZ?.>

: The fifty. Right there, the point 5. That number that is always multiplied to
that (I. That distance. Keep that in mind, that rate is always multiplied to the
variable, OK?

This extended example shows a profound change in discourse patterns from the
patterns in CMP. Recall that CMP asks the same questions about each of the three
representations, either in a list as in, “How can you tell whether a set of data is linear by
examining a graph? A table? An equation?” (Teacher Edition, p. 34g), or by asking the
same question three times; once for each representation. This example has demonstrated

that Ms. M’s discourse has a different pattern: tables and graphs are compared, equations

are written, but not compared.

172

In the four warm-ups Ms. M conducts over the course of this Investigation, these
patterns of use of visual mediators are largely consistent. One consistent difference is that
Ms. M shows students visual mediators more frequently in warm-ups than she does in the
course of the lessons. In her warm-up on 10/27/04, she asks students to use two given
tables and two given graphs to determine which of two walkers is faster. On 10/28, she
gives students a table and asks them to make a graph. On 10/29, she gives students a
graph and has them make a table and write an equation. Finally, on 11/2, she shows
students 2 tables, 3 graphs and 3 equations, and asks students to identify which
representations represent linear relationships. In the course of these four warm-ups,
students are shown 5 tables, 6 graphs and 3 equations while they make one of each. This
contrast to the numbers in the lessons speaks to the different use of warm-ups in Ms. M’s
room. She uses the warm-ups to discuss one feature of linear relationships at a time. On
10/27, she focuses on comparing rates in tables, on 10/28, she focuses on using a table to
identify a linear relationship, on 10/29, the warm-up is focused on making a table and
writing an equation for a relationship when the graph is given and on lI/2, it focuses on
identifying linearity in all three representations. She keeps these warm-ups brief by
showing the representations rather than having students make them.

The warm—up on 11/2 is significant in that it adds to the classroom discourse a
theme that would otherwise be absent. There is an extended discussion in this warm-up of
how to identify linear relationships from their equations. As a result, this is a passing,
rather than an absent theme.

Ms. M’s use of visual mediators, in comparison with the themes in CMP’s use, is

summarized in Tables 10-12. The significant difference is in the use of equations. There

173

are two themes that receive different treatment from Ms. M than from CMP: y-intercepts
and linearity. Ms. M places more emphasis on y-intercepts than does CMP, but less
emphasis on identifying linearity in equations. It is not clear that these differences point
to fundamental differences in the two discourses. However, in the Word Use section, I
observed that Ms. M’s differential use of the word equation from the words table and
graph could signify a tension between the set-theoretic approach to functions of the

traditional American cuniculum and the relational-dependency approach in CMP.

I74

 

Table 8: Themes in each discourse on tables.

 

Rate Individual Value Y—intercept Linearity
CMP
passing
Ms.
passing passing
M .

 

 

 

Table 9: Themes in each discourse on graphs.

 

Rate Individual Value Y-intercept Linearity

C MP

 

 

Table 10: Themes in each discourse on equations

 

     

Individual Value Y-intercept Linearity

 

 

 

 

CMP

   

passing passing

 

passing

I75

 

 

    
 
 
   
 

This observation is bolstered by the use patterns of the three representations.
Graphs, which receive scant attention in the set-theoretic approach to functions (e. g. in
Dolciani, 1992, see Chapter V), receive equal attention through each of the four themes:
rates, individual values, y-intercepts and linearity. The themes vary in their emphasis with
respect to equations. In a pure set-theoretic approach, we would expect the theme of
finding and representing individual values in equations (i.e. solving them) to be recurrent.
This is not the case in Ms. M’s discourse. Instead, slopes and y-intercepts are recurrent
while individual values and linearity are passing. This does not support a claim that Ms.
M’s approach to functions is set-theoretic, but this assertion is not being made.

Instead, the claim is that one dimension along which Ms. M’s discourse varies
from that of CMP is the relative importance of set-theoretic and relational-dependency
approaches to functions. On the one hand, Ms. M’s approach seems to lack some
qualities that would be important to the relational-dependency approach, such as
equivalent treatment of tables, graphs and equations. On the other hand, her approach has
some elements—lacking in CMP—that contribute to the relational-dependency approach,
such as asking questions about the contextual meaning of the transformation of a graph,
and consistency in treatment of the y-intercept across the three algebraic representations.

One more aspect of Ms. M’s use of visual mediators deserves comment. When
Ms. M endorses the use of tables with non-constant increments in her summary of
Problem 2.4, the emphasis shifts from using tables to identify rates (a relational-
dependency notion) to using tables to identify the correspondence between x and y (a set-

theoretic notion).

176

Narratives

Narratives express the properties of and the relationships between the objects in a
discourse. Narratives are different from the themes of the previous section in that themes
in the use of visual mediators state that visual mediators have particular uses, but do not
outline how to use them, nor do themes express relationships. In formal mathematical
discourse, narratives are generally definitions and theorems. In classroom mathematical
discourse, the forms for expressing relationship between objects of the discourse are less
standardized.

The narratives in teachers’ classroom discourse are first compared to the CMP
narratives. That is, evidence is sought for the presence of the CMP narratives in each
teacher’s discourse. Second, transcripts are analyzed for the presence of additional
narratives related to linearity. In this analysis, tasks, questions and exposition are studied
and the main mathematical points compiled. The overall result is that each of the CMP
narratives is also a narrative in Ms. M’s discourse, but additionally there is one narrative
in Ms. M’s discourse that does not appear in CMP. Each of these narratives is listed next,
together with at least one example from Ms. M’s instruction.

Mathematical Variables Represent Contextual Variables

Recall the distinction made in the CMP Discourse section between mathematical
variables and contextual variables. The former are used in mathematical representations
such as tables, although they may retain contextual names (as in the heading “distance”),
while the latter is a changing quantity in the context (as in a walker’s distance). Recall
too that Dolciani’s mathematical variables tend to be unrelated to contexts. That is, x and

y are largely studied on their own terms in Dolciani, while they tend to represent

I77

contextual variables in CMP. In this, Ms. M’s discourse closely mirrors that of CMP. as
in, “ m = 2d. The money equals two times the distance.” (Ms. M Activity 4, 10/28/04).
She writes the mathematical variables on the overhead projector, m and d, and names the
contextual variables that they represent. There are no examples in Ms. M’s summaries of
mathematical variables that are not tied to contextual variables.
Tables, Graphs and Equations Represent Relationships between Variables

As in the case of CMP, combining two patterns in the use of the word equation in
Ms. M’s discourse yields this narrative. The fact that equations represent relationships
together with the fact that equation is used similarly to graphs and tables implies that all
three representations represent relationships between variables. This syllogism is true in
CMP and it is true in Ms. M’s discourse. The difference in Ms. M’s use of equation—the
emphasis on writing over comparing—does not pertain to this narrative, for she still
speaks of equations representing relationships.
Tables with Constant Increments Represent Linear Relationships

As discussed in the Visual Mediators section, this narrative is absent from Ms.
M’s discourse in the Launch-Explore-Summary sequence. Instead, Ms. M establishes it in
a warm-up. On 1 1/2, the final day of the investigation, Ms. M opens class with a warm-
up that shows students 2 tables, 3 graphs and 3 equations. For each representation,
students are asked to decide whether the relationship represented is linear. The case of the
tables (reproduced in Figure l 1) gives an opportunity to see how Ms. M’s warm-ups
contribute to the classroom discourse on linearity. Without this warm-up, this narrative

would not be a part of the discourse. Ms. M has a discussion with students in which she

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asks students to determine which of the tables or equations represents a linear

relationship.

 

Figure 11: Ms. M’s tables from the warm-up on 11/2/05.

2468 dM

 

 

yl-4 -8 -12 -16 0 10

112
215
318
417

 

 

 

 

A student, E, initially suggests the narrative by saying, “it [the (x,y) table in
Figure 11] has an even pattern.” Ms. M later refines this by equating the term increment
with continuous pattern, which is equivalent to even pattern. In turning to the nonlinear
table, Ms. M establishes first that there is a constant increment for the independent
variable by asking, “What’s happening in the x-column?” She walks students through the
increments in the dependent variable, observing that they are non-constant, “Do we
always add two?...We add three, then we add..?” With her use of the student’s language
of pattern and the nonlinear example she has chosen here, she leaves the constant
increment condition implicit in this discussion. The narrative, Tables with predictable
patterns represent linear relationships is an incomplete narrative. There are non-linear
relationships in which we can identify predictable patterns in the dependent variables;
quadratic relationships, for instance. This idea is revisited a few minutes later, when the

C lass uses tables to check for linearity of the equation, y = x2.

I79

M: Do we have a linear pattern here?

Ma: No

M: Why do you say no, Ma?

Ma: Because they don’t go, like in a constant rate. Because... I don’t know. Yeah.
M: You’re saying they don’t go in a constant rate? What’s the increase here?

Ma: Because the change is like 3 right there, and then it goes like 5, and then...

M: OK, it’s not linear. So this one is not.
In this sequence, the language of constant rate is introduced by a student, Ma. Ms. M
pushes her to be explicit about what this means and so establishes that linear relationships
must have a pattern in the table, and that this pattern is a constant increment in the
dependent variable (assuming a constant increment in the independent variable).
Straight-Line Graphs Represent Linear Relationships

In her launch for Problem 2.1, Ms. M introduces linear relationships by referring
to a set of x-y graphs displayed at the back of the room“. She asks students which of the
graphs is linear and then asks, “How do you know a linear relationship?” (Ms. M Activity
2, 10/25/04), accepting the answer straight line. In Problem 2.3, Ms. M revisits the
question of linearity in graphs when a student struggles to decide whether all three graphs
in the problem are linear. When she asks, “Who can explain what a linear graph looks
like” (Ms. M Activity 5, 10/28/04), Ms. M re-introduces the criterion for linearity: the
graph is a straight line.

Equations of Form y = mx + b or y = b + mx Represent Linear Relationships

 

43 Due to technical difficulties, this conversation was not audio-recorded. It has been
reconstructed from field notes.

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As with CMP, Ms. M does not use the form of an equation to distinguish linear
relationships from nonlinear ones in this investigation. However, given that a relationship
is linear she does explicitly work on this preferred form. In addition to writing equations

in slope-intercept form, y = mx + b“, Ms. M pushes students to do so also, as in Problem

2.2,
So Jade’s equation, distance equals t times 2. What’s the other way we could

write that equation?

[U]sually in our rules, we like to put the variables after the number, so you could

rewrite them as d = It, d = 21, d = 2.51. (Ms. M Activity 2, 10/27/04)

In this passage, Ms. M encourages the writing of equations in the form y = mx instead of
y = xm. Nowhere does she write or discuss equations in standard form.

Consistent with this discussion of the slope-intercept form, the y-intercept is
added on, as b, in Ms. M’s discourse. In Problem 2.3, she discusses what equation would
have a graph that is less steep than the ones in the problem. A student suggests m = 0.5d
where m is the money raised by walking d miles. Ms. M responds, “Oh, that’s kind of
like Alana’s plan. But without the added five?” (Ms. M Activity 5, 10/28/04) In Problem
2.5, she discusses a follow-up question with her students, who have suggested writing the
equation d = 20 + 2t. She summarizes, “20 is the amount added on. Two meters is our
rate.” This discussion is based in the slope-intercept form of a linear equation. While Ms.
M uses neither the name nor the general form, she models it and reinforces its structure.

Rates Are Represented in Tables, Graphs and Equations

 

44 Recall from the discussion of this narrative in the case of CMP that y = mx + b does
not imply that the variables need to be x and y.

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Ms. M’s discourse parallels CMP’s discourse with respect to this narrative, except
that she is more explicit than CMP in endorsing it. Where CMP repeatedly asks, “How
does the rate affect the table/graph/equation?” Ms. M asks this question and follows it up
with specific statements and questions about each representation. For example, in
Problem 2.2, she says, “Think about what E was doing when she found the rate in the
table. Remember you can also look in between. What’s the change here?” (Ms. M
Activity 2, 10/26/04) In Problem 2.3, she asks:

How does that dollar per mile change the graph [in comparison to a rate of 0.5

dollars per mile]?
and:

Where is the amount per mile in this equation? (Ms. M Activity 5, 10/28/04)

In each of these examples, Ms. M is not discussing generalized comparisons, but the
representation of a specific rate. The implication is that the rate can be found in the table,
the graph and the equation because it is represented there. There is a slight difference
between graphs on one hand and tables and equations on the other in Ms. M’s discourse.
With graphs Ms. M asks how an increase in a specific rate affects the graph. She is
looking for a qualitative description (e.g. steeper), not a quantitative measure. She does

not ask for students to identify the rate in the graph, only to make a comparison. That is,

she does not expect students to be able to find slope, as the ratio $1, from a graph at this

point in the unit, while she does expect them to find the rate in a table (as the change
between successive values) and in an equation (as the multiplier of the variable).

Y—intercepts Are Represented in Tables, Graphs and Equations

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As in the CMP discourse, the representation of y-intercepts in tables, graphs and
equations is less central than the representation of rates, but Ms. M does address the
question. In Problem 2.3, Ms. M asks these questions,

Where’s that five-dollar donation represented in the table?

Where’s that five dollars represented on the graph?

...y-intercept. OK. Where is it represented in her equation? (Ms. M Activity 5,

10/28/04)

Although CMP does not use the vocabulary of y-intercept, but rather refers to the
y-intercept in contextual language, it is a term that Ms. M had used with students in the
cuniculum she was piloting. She revisits it here. The use of y—intercept was initiated by
Ms. M, not by her students.

Analysis of Tables, Graphs and Equations Can Answer Contextual Questions

Ms. M paraphrases the text’s questions that establish this narrative. In Problem
2.3, she emphasizes when a student has used the equation to find a contextual value, Ms.
M quotes this question from the text, and “...for a sponsor to owe a student 10 dollars,
how many miles would a student have to walk under each pledge plan?” (CMP Student
Edition, p. 20) In the ensuing conversation, Ms. M addresses all three representations.

There is little difference between the narratives in Ms. M’s introductory discourse
on linearity and those of CMP. All of the CMP narratives appear in Ms. M’s discourse,
although one of them does not appear in the Launch-Explore-Summary sequence of her
lessons.

The observation was made in Chapter III that any mathematical text, whether

written or spoken, contains many narratives. One difficult job for research is determining

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which of these is significant and worthy of inclusion in analysis. Correspondingly there
are many narratives in Ms. M’s discourse that have not yet been listed, but most will not
be analyzed here, just as there were many narratives that were left out of the analysis of
the CMP and Dolciani discourses. All of Ms. M’s tasks, questions and assertions related
to linearity are represented by one of the CMP narratives, together with one more. There
is one narrative in Ms. M’s discourse that is closely related to the other narratives, but
which is absent from CMP. Because this narrative, Any of the three representations of a
linear relationship can be generated from any of the others is related topically to the
others and is of similar intellectual power, it is included here and shows one dimension
along which Ms. M’s discourse differs from that of CMP.
Any of the Three Representations of a Linear Relationship Can Be Generated from Any of
the Others

Three of Ms. M’s warm-ups contribute to this narrative. On 10/28, Ms. M shows
students a table and has them make a graph. On 10/29, she shows students a graph and
has them generate the table and the equation. On 1 1/2, she leads students through the
generation of a table from given equations in order to test for linearity. Figure 12 shows a
diagram of these interrelationships. As a result, we can begin with an equation, make the
table and then use the table to make the graph (as in Dolciani). We can begin with a
graph, make a table and use this table to write an equation. Finally, we can move directly
from a table to either an equation or a graph. Note from the diagram that the direct moves

between equation and graph are not made in Ms. M’s discourse.

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Figure 12: Any representation can be generated from any of the others.

10/28

 

10/29

11/2

E

1 0/29

 

 

 

The narratives in Ms. M’s discourse closely parallel those of CMP, with the
exception of an added narrative that affords transitions between representations. This is
not inconsistent with the findings in the Visual Mediators section. In that section, all of
the themes in the use of visual mediators from CMP are identified in Ms. M’s discourse,
as are the narratives in this section. The themes are measured for their importance to the
discourse, which the narratives are not. Narratives are identified as present or absent,
without measure of their frequency. If such a measure were devised, we would expect to
find similar patterns in the narratives to the ones found in the use of visual mediators.

The final section analyzing Ms. M’s discourse pays attention to the ways of
establishing and reinforcing these narratives—routines.

Routines

The routines for establishing and reinforcing narratives in discourse are often
implicit. The problem for research is identifying routines through inference. The major
question in the analysis of teachers’ classroom discourse is, On what basis are students to

notice a narrative or to accept it as true?

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The routines for establishing narratives in CMP’s discourse are also present in
Ms. M’s discourse. Ms. M asks questions, she provides multiple examples to lead
students to conclusions through inductive reasoning and she expects them to find
mathematical meaning in the contexts in the investigation. These are all consistent with
the heuristic mathematical discourse described in the analysis of CMP. In addition to
these heuristic routines, Ms. M explicitly states (tells) some narratives.
Asking Questions

One important difference between textbook discourse and classroom discourse is
the presence of students in the latter. When CMP asks questions, students do not respond
in the text; student responses occur in the classroom. The Initiation-Response-Evaluation
(IRE, see Cazden, 1988) sequence of classroom discourse is absent from textbook
discourse. As a result, the routines asking questions and telling (telling is described
below) may be more similar to each other than the corresponding textbook routines. In
considering the routine of asking questions in classroom discourse, I include instances of
the IRE sequence, even when it veers close to telling. The distinguishing feature is
whether the teacher is attempting to have students establish a narrative, or whether the
teacher is establishing the narrative on her own. When Ms. M asks, “Where is that five
dollar donation represented in the table?” (Ms. M Activity 5, 10/28/04), she is attempting
to get students to establish the narrative, Y-intercepts in tables are represented by the
value of x when y is zero, and she is using the routine of asking questions. When students
respond correctly, the evaluation can be seen as a form of telling. When students respond
incorrectly, Ms. M sometimes employs explicit telling in which she states the narrative

herself, rather than evaluating a student’s narrative. In any case, the use of questions to

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push students to state narratives is a routine that is different from the routine of telling,
even though the routines may be closely associated in many classroom situations.

Ms. M asks all of the same questions as in the shaded boxes for the problems in
the Student Edition, either quoting directly or paraphrasing. As a result, the implicit
endorsement of narratives through questioning that is outlined in the section on CMP’s
routines applies to Ms. M’s discourse as well. She also asks many questions that are not
in the curriculum. Of particular note, though, is that Ms. M tells narratives when her
questions do not lead students to them.

Telling

The following transcript excerpt was used as an example for the narrative
Analysis of tables, graphs and equations can answer contextual questions in Ms. M’s
discourse.

M: And we could check all these points on our table, too. Here’s the point (3,3).

Here’s the point (2,4) for Gilberto. What was your other one? (4,6)? Four
miles...

Student: Six dollars.

M: Ooh...four miles would be how much? (Ms. M Activity 5, 10/28/04)

In this excerpt, Ms. M establishes the narrative through telling. While she asks the
question, “four miles would be how much?” she has already told students that the answer
can be found by analyzing the table. Note that she asks for the contextual interpretation of
the answer (six dollars), but she tells students that the table can be used to answer

contextual questions, “[W]e could check all these points on our table, too.”

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A common pattern in Ms. M’s instruction is to ask a series of questions and then
to tell when students do not provide a correct and complete answer. Two examples
follow. The first is from Problem 2.2.

M: Looking at those three graphs, where is Jerome’s graph? What color is it?

Students: black

M: Black. Where is it in relation to the other two?

Student: Above?

M: OK, it’s above the other two graphs. Imagine everybody starting to ride your
bike. You’re down at this point. You have three hills to go up. This hill, this
hill and this hill. How does Jerome’s hill compare to Jade’s and Terry’s?

G: Steep

M: It’s steeper, OK? And that’s a word that I want you to keep thinking about.
That the person who had the fastest walking rate... what kind of line did he
have? He had a much steeper line than the other two. (Ms. M Activity 2,
10/27/04)

Students are paying attention to the height of the points on the graph, rather than to the
steepness. Ms. M introduces the word steep, since the students did not provide it, and
then tells them, “...the person who had the fastest walking rate...lrad a much steeper line
than the other two.”

Inductive Reasoning

Ms. M follows the same structure as CMP in this regard. She has students make
the same tables showing multiple walkers in order to highlight the importance of constant

increment. She uses the same non-examples to ensure that students do not over generalize

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the patterns they detect. She reinforces with telling the inductive reasoning that she sets
students up to do (see preceding discussion of the routine, telling).
Deriving Mathematical Meaning from a Context

Just as CMP does, Ms. M has mathematical goals for students that are not based
in contexts, yet much of her instruction takes place using contextual language, such as
walking rate and head start. Recall three possibilities from Chapter V for how teachers
might encourage students to derive mathematical meaning from a context: defining terms
in contextual language, answering math questions by referring to the context and
generalizing. Because the bulk of the language in Ms. M’s lessons, like that in the student
text, is contextual, only the third possibility is plausible. As with CMP, there are a few
places in Ms. M’s discourse where generalizing is explicitly encouraged. One example is
the following.

T: Where’s the five-dollar donation?

DE: Under her table.

T: Under the table, where under the table? There’s lots of points here under hers.

DE: Across from the zero.

T: Across from the zero. Because remember she gets that donation of 5 bucks

even if she doesn’t walk. Where’s that 5 dollars represented on the graph?

DE: Um... on the y-axis.

T: On the y-axis. Which point?

DE: 5

T: At 5. What’s that point called? ER told us a little bit ago.

Student: y-intercept.

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In this case, Ms. M asks students to give the general name, y-intercept, to the
contextual value, donation. In this sense, she is asking them to generalize. Note, though,
that y-intercept is previously established mathematical vocabulary so the process is
fiinctioning differently in Ms. M’s classroom than in the CMP text. Ms. M asks students
to see donation as an instance of a known pattern: y-intercepts. CMP asks students to use
instances (e.g. donations and head starts) to construct the general pattern.

In the preceding examples of routines, Ms. M has been establishing new
narratives or reinforcing established ones. It is also illuminating to observe the routines
used when two or more conflicting narratives are under consideration in a classroom. In
the first example of conflicting narratives, Ms. M has students working on a warm-up that
asks which of two walkers whose time and distance data are displayed in a table is faster.

The two tables are reproduced in Figure 13.

 

 

 

Figure 13: Ms. M’s tables from the warm-up on 10/27/05.
Distance (meters)
Time (sec) Xavier Yves
0 0 0
l 1.5 2
2 3 4
3 4.5 6
4 6 8
5 7.5 10

 

 

 

 

 

In the following excerpt, four conflicting narratives arise:

190

1. Tables with whole numbers represent greater rates than do those with
fractions.
2. Tables with greater numbers represent greater rates.
3. The value of the dependent variable when the independent variable is l is the
rate.
4. Tables with greater increments for the dependent variable (given a common
increment for the independent variable) have a greater rate.
The first three are raised by students, while the fourth is by Ms. M. Note that this fourth
narrative is part of the narrative, Rates are represented in tables, graphs and equations.

In the transcript that follows, the turns are numbered to facilitate the discussion.

I M: Who’s the faster walker. Yeah, R?

2 R: Yves.

3 M How come? Why do you think so?

4 R: Cause he has whole numbers, not halves.

5 M He has whole numbers not halves. 2, 4, 6, 8, 10, yup there’s
some halves in here. Anybody agree with R? You shook your
head, why not, TA?

6 TA: 1 went like this!

7 M: OK, so you agree. Why do you agree? Same reason? TA, same
reason?

8 TA: (unintelligible)

9 M: Xavier? Yves?

10 TA: The numbers are higher

191

ll

12

l3

I4

15

l6

17

18

19

20

21

22

23

24

25

26

27

Student:

M:

E:

1"."

.3. .3.

F1?

These person’s numbers are higher. OK. These numbers are
higher. Are they always higher than what Xavier’s walking at?
Yeah.

Yeah. OK. Anything to add. E?

It’s because Yves walks 2 meters per second and Xavier only
walks... walks 1.5 per second.

How do you know that Yves walks two meters per second?
Because if you look at one second, it equals two.

And after one more second...

4

And after one more second.

6

What’s happening each second?

It goes up by two.

It goes up by two.

What’s happening over here?

It goes up by 1.5

It goes up by 1.5 each time. OK, so we saw bigger numbers,
we saw the rates, E actually picked out, of 1.5 meters per
second vs. 2 meters per second. We saw that change here of
going by 2 each time, here we’re only going by 1.5 each time.
So, R, I just want to get back to you. Halves vs. whole

numbers. I don’t, I don’t quite see what you were at, what you

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were saying.

28 R: (unintelligible)... is going up by halves...

29 M: So this first guy, the X guy, goes up by 1.5? OK, so not really
a half but by one and a half. And this guy’s going by..

30 R: two

31 M: Two, so a whole number which happens to be a bigger number
than this one. Not so much that they’re going up by half, half
numbers, but that there’s a bigger increase.

32 All right, so Yves seems to be the faster walker. (Ms. M
Activity 1, 10/27/04)

In turn 4, R says, “Cause he has whole numbers, not halves.” Rewritten in
language equivalent to Ms. M’s narratives, this narrative says, Tables with whole
numbers represent greater rates than do those with fractions. Clearly, this competes with
the narrative Ms. M seeks to establish, Tables with greater increments for the dependent
variable (given a common increment for the independent variable) have greater rates, as
whole numbers are not necessarily larger than mixed numbers and the conflicting
narrative refers to the values, rather than to the increment. After examination of other
conflicting narratives, Ms. M returns to R, who has now adopted the language of the
intervening conversation, “...is going up by halves...(emphasis added)”

Ms. M tells the student (turns 29 and 31) that Xavier’s increment is 1%, not i, and

that the fractional nature of the increment is irrelevant in any case. It is not clear whether
this resolves the narrative conflict for the student, as he does not speak again, but it is a

clear attempt by Ms. M to resolve the conflict by telling.

193

The second conflicting narrative, Tables with greater numbers represent greater
rates, is introduced by a different student, TA, in turn 10. Again the narrative conflicts
with the one Ms. M is trying to establish (in the general case, although the two are
consistent when the y—intercept is zero). This can be seen in the table for Problem 2.3, for
instance, where many of Alana’s pledge totals are greater than Leanne’s totals for the
same distances, although Alana’s pledge rate is less. Only when two linear relationships
have a common y-intercept does a greater rate correspond directly with greater values for
the dependent variable. Ms. M’s attempt to resolve this conflict is subtler than in the
previous example. She does not say that the student is wrong. Instead she pursues another
narrative introduced by another student, E, in turn 14.

Ms. M signals that this narrative, The value of the dependent variable when the
independent variable is I is the rate is more valued (presumably because it is closer to
the desired narrative) by the time she spends on follow-up questions. In fact, it rests on
the same assumption as TA’s narrative: a zero y-intercept, but E uses the language of
walking rates. Ms. M uses turn 17, “And after one more second...” to shift the focus from
the individual value corresponding to 1 second in the table to the increment between
consecutive values in the table. This is an example of the routine asking questions,
although in this case the question, “What would the value be after one more second?” is
implied. Ms. M uses her questions to guide the student to the desired narrative. When she
asks, “What’s happening over here?” in turn 24, E’s reply, “It goes up by 1.5” indicates
that Ms. M has successfully shifted this student’s narrative.

In sum, like CMP’s discourse on linearity, Ms. M’s discourse is grounded in

contexts and visual mediators. Ms. M’s discussions of equations are focused on writing

194

them for the contextual problems in the investigation. She tends to attribute possession of
the equations to the characters in the contexts, as in a discussion of “Jerome’s equation.”
Rates are primarily associated with walking rates.

Ms. M’s use of visual mediators differs more markedly from CMP’s use in that
she treats tables, graphs and equations differently. She uses tables to identify rates and
individual values, with a lesser emphasis on y—intercepts. She steps outside the problems
of CMP in order to discuss linearity in tables. She consistently uses graphs to identify
rates, individual values, y-intercepts and linearity. She emphasizes rates and y—intercepts,
consistent with her focus on writing equations, and deemphasizes using equations to find
individual values and to identify linearity. While the emphasis on each representation is
different, she draws connections across the representations in a way that CMP does not,
as evidenced by the narrative unique to her discourse, Any of the three representations of
linear relationship can be generated from any of the others.

CMP’s textbook discourse on linearity is characterized in Chapter V in two ways;
by its approach to functions, and by the nature of its mathematical discourse. CMP takes
a relational-dependency approach to functions, and evidence of this could be seen at the
microscopic level of word use as well as with a more macroscopic view of the narratives.
Classroom discourse, of course, is more complicated.

In the microscopic view afforded by word use, Ms. M’s introductory classroom
discourse on linearity diverges from CMP’s textbook discourse. A major pattern in CMP,
fundamental to the relational-dependency approach to functions, is the association of
equation with relationship. This pattern is not absent from Ms. M’s discourse, but it is

less prominent than in CMP. The prominence of relationship is lessened in part because

195

of an increased emphasis on the writing of equations in Ms. M’s discourse. Ms. M has
lengthy discussions with students about strategies for writing equations. Some of these
conversations focus on computing one value (e.g. distance) when the other (e. g. time) is
known, and others focus on using a given rate (e.g. 2 m/s) to write an equation. These
two examples point to the dual nature of functions; in Ms. M’s discourse, functions can
be seen both as processes and as objects. At a more macroscopic View, we may consider
equations as visual mediators. Of the three representations, equations in Ms. M’s
discourse differ the most from their role in CMP’s discourse. Here, too, the discourse
cannot be neatly characterized as relational-dependency or as set-theoretic. At the top-
level analysis of narratives, there is more consistency with the CMP discourse. Ms. M
and CMP have similar goals for students, and these reflect a relational-dependency
approach to functions; all three representations come into play in Ms. M’s discourse.

Chapter V gives us a second curriculum to compare against, and doing so makes
clear that Ms. M is closer to a pure relational-dependency approach to functions than to a
set-theoretic approach. While correspondence appears in Ms. M’s classroom discourse on
linearity, it is not a major idea. While Ms. M does not treat the three representations
exactly equivalently, all three are used for a variety of purposes and there is no special
priority afforded to functions, in contrast with the Dolciani discourse.

Finally, returning to the idea of learning a discourse as changing one’s
participation patterns in the discourse, we should consider the characteristics of
participation in the model discourse. Chapter V asserted that the CMP model discourse is
akin to a heuristic mathematical discourse, in which questions are asked, examples tested,

generalizations made, and forms of metaphor employed. These techniques are present in

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Ms. M’s model discourse. She asks questions to get students to state narratives before she
states them herself. Ms. M employs the examples from CMP, and creates her own in the
warm-ups in order to lead students to use inductive reasoning. She encourages
generalization from the contexts in the unit. Ms. M does employ the routine of telling
when students do not state narratives in response to her questioning. Similarities and
differences with the CMP discourse are examined in more detail in the third section of
this chapter. First, the analysis turns to Ms. H’s discourse.

Ms. H’s Discourse

Ms. H teaches at a K-8 school that has approximately 480 students. The junior
high (i.e. grades 7 and 8) students follow a six-period day in which each class last 53
minutes. Ms. H is the only math teacher in her building. This, together with a low
turnover rate— the midyear mobility rate was 6% in 2003-2004— means nearly all of the
8th graders in Ms. H’s classes studied CMP with her in 7th grade. The school’s enrollment
is 44% minority.

Ms. H spends 5 class periods working through Investigation 2 (see Figure 14).
Prior to the lessons in the study, Ms. H opened the unit by having her students measure
their own walking rates in yards per second, and then convert these to meters per second
to match the walking rates in the investigation.

Unlike Ms. M’s instruction, Ms. H typically fits the Launch-Explore-Summary
sequence into a single class period. The one exception is with Problem 2.3, where she
begins the summary at the end of the period and finishes it at the beginning of the next
day’s period. In addition, Ms. M has a warm-up three out of the five days. These warm-

ups are related mathematically, and sometimes contextually to the problems in the

197

 

investigation. As with the analysis of Ms. M’s discourse, the main analysis for Ms. H
relies primarily on the Launch—Explore-Summary sequence and incorporates evidence

from the warm-ups as necessary to fully describe the discourse.

 

 

Figure 14: Major activities in Ms. H’s Investigation 2 lessons.

 

day1

 

 

day2 I .

Key ,
warm-up [:l day 3
launch;jj
explore i
summary I
other day 5

 

 

 

 

 

 

day 4

 

 

 

 

 

Word Use: Equation

Recall that word use refers to the patterns in the appearance of words in discourse.
Frequency counts appear in parts of this section, when the crude comparisons they afford
are useful and informative. Because the main comparison in this chapter is across
discourse types (i.e. the comparison is between carefully edited textbooks and real-time
teacher talk), frequency counts are used sparingly. The main thrust of the analysis is
identifying the patterns. Once identified, the patterns are considered for what they say
about the meaning of the words, and for how they contribute to teacher’s classroom
mathematical discourse. The first patterns to be analyzed are the patterns that appear in
the CMP textbook discourse, then any new patterns that appear in Ms. H’s classroom

discourse, but not the textbook discourse, are analyzed.

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Ms. H’s use of the word equation shares with CMP frequent listing with table and
graph. Ms. H does not use the words relationship or situation in any sentences together
with equation. She shares with Ms. M the tendency to attribute possession of equations to
the characters in the curriculum’s contextual narratives.

Relationship

In the lessons of Investigation 2, Ms. H does not use the word relationship, nor
the word situation when talking about equations. Together with the next pattern,
possession, the absence of relationship when discussing equations points to the same
tension in Ms. H’s discourse that was present in Ms. M’s discourse; the tension between a
relational-dependency approach to functions and a set-theoretic approach.

Table and Graph

Ms. H frequently lists table, graph and equation in a manner similar to CMP.
Indeed, the frequency is high enough for her students to be able to recite the trio, as in
this example from her Launch of Problem 2.5.

H: Today we’re all going to make a...

Student: Graph

H: And a...

Student: Table

H: And a...

Student: Equation.

H: All three of those. (Ms. H Activity 1, 11/10/04)

In the repeated use of the words table, graph and equation, Ms. H’s word use is

consistent with CMP’s word use.

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Possession

Ms. H shares with Ms. M the tendency to attribute possession of equations to the
contextual characters. In passages cited above, Ms. H refers to “Jerome’s equation” and
“Emile’s equation,” among others.

In the analysis of Ms. M’s discourse, I pointed out that speaking about equations
as being possessions is a small indication that equations are objects, rather than
processes. Equations as objects are consistent with a structural description of functions,
and the set-theoretic approach. This pattern is present in Ms. H’s discourse as well. By
the same token, the absence of the term relationship in association with equation in Ms.
H’s discourse downplays the relational-dependency approach of CMP. At the
microscopic view afforded by the word use analysis, Ms. H does associate table and
graph with equation and so does not privilege equation over the other two terms—a
pattern that is consistent with CMP’s approach.

Word Use: Rate

Ms. H frequently associates walking with rate, as do CMP and Ms. M. She
addresses how walking rates relate to equations, but does not use the CMP language of
affecting. Finally, Ms. H does not use the term rate of change.

Eighty-five percent of Ms. H’s sentences including the word rate include a
reference to walking (recall that for CMP, the figure is 100% in the Student Edition and
for Ms. M the figure is 69%), as in the following two examples.

If Jerome is wondering how long it’s gonna take people to get to the yogurt shop

if they walk at different rates, then that means that they really have to walk at

those different rates. (Ms. H Activity 2, 1 1/03/04)

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We started off class talking about walking rate and speed. (Ms. H Activity 5,

1 1/04/04)

In her Summary of Problem 2.2, Ms. H asks questions about how the walking
rates affect equations as in CMP, but she uses the word relate instead.

What...how...anything connecting the walking rate to the equation?

Can we relate the walking rate to the equation?

So just for Terry. Just for Terry, how does his walking rate relate to his equation?

(Ms. H Activity 5, 11/04/04)

Note that relate leads to considering one equation at a time, while affect leads to
consideration of a class of equations. That is, Ms. H asks how one (“his”) rate affects one
(“his”) equation. The text asks how changing a walking rate affects the (general)
equation, without reference to a particular rate or equation.

Finally, in contrast to both CMP and Ms. M, Ms. H does not use the word rate
after Problem 2.245. For example, when discussing the two boys’ graphs in Problem 2.5,
she says that one boy’s line is, “Steeper. He’s going... he walks... faster.” (Ms. H
Activity 3, 11/10/04) In this case, rates are referred to, but the word rate is not used.

Word Use: Linear

Ms. H uses the word linear in two problems in Investigation 2, Problems 2.3 and

2.5. In the summaries for these problems, she uses the term seven times. The term arises

from student use in Problem 2.3, when a student asks during discussion of Alana’s table,

 

 

45 Although the word is vocalized when a student reads Problem 2.4 aloud.

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“Wouldn’t she not be linear?” Ms. H corrects her, referring to the previously
established46 fact that, “It doesn’t matter what it starts at, it just matters that it goes up by
the same amount each time.” (Ms. H Activity 2, 1 1/09/04)

In each of Ms. H’s uses of the word linear in Problem 2.5, the term modifies
relationship, as in the example below.

So you could tell from the graph that Henri had, that his distance was a linear

relationship with his time. (Ms. H Activity 3, 11/10/04)

As exemplified in this passage, her use of linear is associated with identifying linearity in
tables and graphs.

To summarize, Ms. H’s use of the word equation differs from CMP’s use in that it
is not associated with relationship and she regularly discusses functions as possessions
(i.e. objects). However consistent with CMP, she uses equation together with table and
graph. The next section, Visual Mediators, affords a more macroscopic view which
allows us to see greater differences in the treatment of equations than is revealed by
examining only the word equation. Ms. H’s use of rate is closer to CMP’s use, especially
the use in the Student Edition, in that it is largely contextual and related to walking. Ms.
H’s use of linear mirrors that of CMP, being associated with relationship and with a
lesser emphasis on the term nonlinear.

The word use analysis, then, reveals some attention to both approaches to
functions that were seen in curriculum: a relational-dependency approach and a set-

theoretic approach. As is the case with Ms. M however, Ms. H’s word use is best

 

46 This was established not in a problem, but in a warm-up. More on this below.

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characterized as deviating somewhat from a pure relational-dependency approach to
functions than as a set-theoretic approach.
Visual Mediators

Taken as a group, Ms. H uses visual mediators similarly to CMP; the only
deviation from CMP’s use of visual mediators is in the use of equations. The themes in
Ms. H’s use of visual mediators are summarized in Tables 13—15. She uses the visual
mediators to identify and represent three kinds of contextual information: rates,
individual values and y-intercepts. She also uses visual mediators to identify linear
relationships. Taken individually, the mediators have substantially different roles.

In the problems for the investigation, Ms. H is consistent with Ms. M and with
CMP in showing only representations that she expects students already to have made. She
does not show students tables, graphs or equations in the Launch, but reserves them for
the Summary. Less frequently than Ms. M does, Ms. H sometimes displays a
representation made by a student (e. g. as a poster or overhead transparency). Ms. H has
one warm-up task that is not from the investigation, and as Ms. M’s warm-ups do, it
includes the display and analysis of the representations, but made by the teacher, not by

the students.

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Table 11: Themes in each discourse on tables.

 

Rate Individual Value Y-intercept Linearity

 

 

 

Table 12: Themes in each discourse on graphs.

 

Rate Individual Value Y—intercept Linearity

 

CMP

 

 

Table 13: Themes in each discourse on equations

 

 

  
  
   

  

Individual Value Y-intercept Linearity

   

 

 

   
   
 

passing passing

 

absent

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Tables

Identifying rates in tables is a recurrent theme in Ms. H’s discourse. The
identification of individual values and y-intercepts in tables are passing themes. The use
of tables to identify linear relationships is a recurrent theme.

In Problems 2.3 and 2.5, Ms. M has extended conversations with students about
how rates appear in tables. In Problem 2.3, the discussion is about comparing the rates in
tables, as in this passage.

H: All right. I want to see if from the table I can tell who’s getting the most per
mile. What’s this going up by? Well, you tell me. How can I, How can I look
at the table and know who’s getting the most money per mile?

Student: Whoever goes up the most.

H: What do you mean goes up the most? This one, I mean they’re all going up.

How can I tell which one goes up the most? Just explain a little more.

H: They all go up by some number, but Gilberto’s number... the number that he
goes up by is the highest, which means he has the highest pledge amount, and
[Alana’s] is the smallest. (Ms. H Activity 5, 11/04/04)

This passage introduces the phrase goes up by which becomes the standard way

for Ms. H and her students to discuss the increment in a table. The relationship between

What a variable goes up by and the rate is clarified in Problem 2.5, where Ms. H uses a

Constant increment of 4 for the independent variable. Ms. H says, “I’m gonna go by 43”

as she makes a table on the overhead projector, referring to the increment in the time

Column (independent variable). This means that the rate will not be the increment in the

205

distance (dependent variable) column. Accordingly, she challenges students to convert
the rate to the correct increment in turn 2, “How should I do this here? I went by different
numbers than you did.”

This substantiates the claim that identifying rates in tables is a recurrent theme in
Ms. H’s discourse. Ms. H spends less time discussing the appearance of individual values
of the variables in tables than she does discussing the rates; it is a passing theme. She
discusses individual values in a single turn in Problem 2.2, telling students, “[W]e can try
to read exact values on the graph; sometimes it’s difficult, so then we can always rely on
our table.” (Ms. H Activity 5, 1 1/04/04) The topic is the center of the summary of
Problem 2.4.

Nearly the entire summary of Problem 2.4 focuses on finding individual values in
tables. Yet, the fact that it does not reappear in multiple teacher turns in other places
keeps it from being recurrent. Indeed, Ms. H is explicit about her preference for finding
individual values in graphs in Problem 2.3.

OK. On to letter C. And I like using the graph for these. I noticed a lot of people

were using their table when they were doing their warm up. The question was,

letter C, how far does each person walk in... how much money do they get if they

walk 8 miles? (Ms. H Activity 2, 11/09/04)

The role of the y-intercept in a table is also a passing theme in the Launch-
IEXplore-Summary sequence of Ms. H’s instruction in Investigation 2. At the end of an
eXtended discussion of equations in Problem 2.3, including how rates and y-intercepts

appear in equations, Ms. H asks about tables.

H: Ooh look back at your table that someone got a five-dollar donation. Si.

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Si: Um, all the zeroes. Leanne, Gilberto, they got zero, but for Alana, they got
five... she got five dollars when the distance was zero. (Ms. H Activity 2,
11/09/04)

All of the tables that Ms. H discusses or shows students have constant increments
for the independent variables, with constant unit increments being the general rule. An
exception to the latter pattern is the discussion from Problem 2.4 cited above, in which
Ms. H uses a constant increment of 4 for the independent variable. In the overall pattern,
this use of tables with constant increments is consistent with CMP, where the Student
Edition shows only constant unit increments and the Teacher Edition shows one table
with a constant non—unit increment. Recall that this contrasts with Ms. M, who sanctions
tables made with non-constant increments. Tables with constant increments focus student
attention on the rates—the relationships between successive values of the dependent
Variable. This is representative of a relational-dependency approach to functions.

The issue of identifying linearity in tables is recurrent, arising in the summaries
for two problems in the investigation. Both times the theme is addressed, however, the
e)(change is brief. In the summary of Problem 2.3, one student has in mind that linear
relationships have y-intercept equal to zero.

B: Wouldn’t she not be linear, Alana?

H: Wouldn’t she not be linear? Well if it goes... what did we say yesterday?

Student: If it goes up by the same amount, that’s linear.

H: It doesn’t matter what it starts at. It just matters that it goes up by the same

amount every time. So hers is linear. And it looked linear, right? (Ms. H

Activity 2, 11/09/04)

207

Recall that Alana’s pledge plan includes a donation, which results in a non-zero y-
intercept. Ms. H refers to the criterion for linearity in a table: a constant increment in the
dependent variable (assuming a constant increment in the independent variable). She also
briefly refers to the criterion for linearity in a graph when she says, “it looked linear,
right?” Here, she refers to the fact that the graph of Alana’s pledge plan is a straight line.
The topic arises again in the summary for Problem 2.5, at the end of a discussion of rates.

H: So, everybody, what was Emile’s distance going up by here?

Student: 10
H: Every time?
Students: Yup.

H: Yup. So does Emile have a linear relationship?

Students: yes.

H: Ok. Is it still linear? Here, they were both going up by four. Here this is going
up by four and this is going up by 10. That’s OK?. .. Yes. As long as they both
go up by something and stick with that, then the relationship is linear. (Ms. M
Activity 3, 11/02/04)

Three aspects to the use of tables are important to the relational—dependency
approach to functions: that they are used to represent functions (as in CMP}—and not
Only to keep an organized list of solutions (as in Dolciani)—that rates can be identified in
them, and that they can be used to determine the type of function being represented. All
three are emphasized in Ms. H’s discourse. The lesser aspects of Ms. H’s discourse—
iderrtifying and representing y—intercepts and other individual values—are also less

essential to CMP’s relational-dependency approach to functions.

208

Graphs

The representation of rates and individual values in graphs are recurrent themes in
Ms. H’s discourse, while y-intercepts are a passing theme. Identifying linear relationships
in graphs is a recurrent theme in Ms. H’s discourse. In addition, she performs a
manipulation with graphs that is quite similar to the one Ms. M performs.
Ms. H discusses the representation of rates in graphs in three problems; Problems
2.2, 2.3 and 2.5. In Problem 2.2 the discussion is brief. One of Ms. H’s students expresses
the same idea about rates in graphs as Ms. M’s students do: higher points are related to
faster rates.
H: I would like some observations about these graphs. S.
S: Uh. . .The faster they walked, the higher they went up, like that Terry, he had
really, he was far away from everybody else and Jerome had the tallest graph.
(Ms. H Activity 5, 11/04/04)
Following this student’s observation, several turns are spent verifying the implied claim
that Jerome is the fastest walker. The fact that faster walkers have steeper, rather than
higher graphs remains implicit in this discussion, which then turns to whether the graphs
I‘epresent linear relationships.
A more substantial discussion of the relationship between rates and graphs is in
1PI‘Oblem 2.3. The term steeper is introduced first by a student in response to Ms. H’s
r equest for observations about the graphs. Ms. H reinforces this language and later ties
the steepness of the lines to the rates.
H: [W]ho had the highest pledge amount?

J: Gilberto.

209

H: And what was it?

Student: Twenty, at ten

H: No his amount. His pledge amount... per mile

Students: Two dollars

H; Two dollars per mile. And how can I tell that from the graph that Gilberto has

the highest pledge amount? J?

J: Cause it keeps going up.

H: It’s the steepest one. Yup. And whose is the least steep?

Students: Leanne

Student: Alana (ibid)

Although Ms. H, like the CMP text, does not use the term rate in the discussion of this
problem, this is the point of her question about “pledge amount...per mile.” Notice that
the term pledge amount leads a student to respond with an individual value. Amount per
mile is a term for rate and Ms. H ties it to steepness in this passage.

Ms. H returns to the topic a third time later in the summary. In this case, she poses
an alternate scenario in which one of the walkers, Leanne, decides to ask for an initial
donation in addition to her $1 pledge per mile. Ms. H plots the points on the overhead
projector as she discusses them with the class.

I H: How’s her graph gonna change?
2 Si: It’s gonna start at three

3 Student: Won’t it be steeper?

4 H: Will it be steeper?

5 Student: No

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10

11

12

l3

14

J:

Student:

H:

Students:

H:

It’ll be bigger

It’ll start at three.

So, say she doesn’t walk at all, C, how much money is she
gonna get?

Three dollars.

Three dollars

Three dollars for the three dollar donation, so I’m gonna go
right up here, and then she walks a mile, she gets...

Four dollars.

Another buck, that’s four dollars. She walks two miles, she
still gets her three-dollar donation, plus two, she’s up at five.

Let’s look at it. Is it steeper?

They’re parallel lines, so they have the same steepness, but it

starts up higher because of that donation. (ibid)

In turn 3, a student suggests that the donation will make Leanne’s graph steeper.

Ms. H makes a graph in order to demonstrate that, in fact, the donation has no effect on

the steepness of the graph.

M s. H discusses finding individual values in two Problems: 2.2 and 2.3. In each case,

there is an extended use of the graphs to identify individual values of the contextual

Vari ables.

211

The relationship between the y-intercept and the graph is a passing theme in Ms.
H’s discourse. In Problem 2.2, she asks, “What do all the graphs have in common?”
(ibid) One response from a student prompts this brief exchange,

K: They all start at zero.

H: They all start at zero. What else, N? (ibid)

This is a single teacher turn. The topic is discussed at more length in Problem 2.3, where
Ms. M asks,
How do we see the five dollar donation on the graph? How can I look at the graph
and know that somebody got a five-dollar donation and somebody didn’t? (Ms. H

Activity 2, 1 1/09/04)

Several teacher and student turns follow, where the discussion is about where each graph
“starts” and the amount each walker has “at zero [miles]”.
Ms. M uses the form of the graph as a criterion for the linearity of a relationship.
This topic is recurrent in Ms. H’s use of visual mediators; she discusses it in two
Problems: 2.2 and 2.5. While the term linear does not come up in Problem 2.2, linearity
is the focus of a brief conversation touched off by a student observation about the graphs
in the problem—that they all form straight lines. The theme is more explicit in Problem

2.5.

H: Now you already graphed Henri’s line. Was it a line?

Students: yeah

H: I just called it a line, so yeah it was a line. So you could tell from the graph

that Henri had, that his distance was a linear relationship with his time. (Ms. H

Activity 3, 1 1/10/04)

212

To summarize the role of graphs in Ms. H’s discourse, their use in finding two
kinds of contextual information: rates and individual values are recurrent themes, as is
their use for identifying linearity. The use of graphs in finding y-intercepts is a passing
theme. This nearly matches the use of tables in Ms. H’s discourse. Therefore, a consistent
pattern in the use of visual mediators is emerging: with tables and graphs, the aspects
essential to a relational-dependency approach to functions are emphasized. Equations
have a different role.

Equations

In her discussion of equations, rates and y-intercepts are recurrent themes. The
identification of individual values is absent. Likewise, the identification of a linear

relationship from an equation is absent from Ms. H’s discourse.
The topic of identifying rates in equations comes up in two problems in Ms. H’s

classroom. The first time this happens, she asks students to express how rates are related

to equations, which her students struggle to do.

H; Can we relate the walking rate to the equation?

Does everybody see a I here?

Students: yes.

H: And a I here? and a 2 here? 2 here? 2.5? 2.5?

What’s going on?

H: So just for Terry. Just for Terry, how does his walking rate relate to his

equafion?

213

S: Uh because. the distance he’s going is I second... meter
C: Going the same distance.

D: One second per meter

S: yeah.

H: They both have, it’s... we’ll come back to this on Monday. (Ms. H Activity 5,
11/04/04)

While the student who says, “One second per meter” is technically correct, this is
not generalizable (i.e. the rate as expressed in meters per second, not seconds per meter in
the equations) and thus the answer is not accepted as correct. Yet it is the closest to a
correct answer that the students produce. Ms. H reinforces the importance of the theme in
Problem 2.3. In the summary for that problem, a student suggests that the equation
m = 0.5 + 5d could express Alana’s pledge plan. Ms. H replies, “So here, I’m gonna take
the distance I walk... For every mile I walk, do I get five dollars?” (Ms. HI Activity 7,
11/08/04) With this question, she suggests to the student that the coefficient in the
equation (5 in the student’s example) is the rate. She makes this explicit a moment later,
“Point five and the d are next to each other because that’s what I multiply. That’s what
the plan is. It’s fifty cents a mile.” (ibid)

Ms. H discusses the role of the y-intercept in two problems in the investigation,
Problems 2.3 and 2.5. In Problem 2.3, Ms. H asks, “What if Leanne does not get out of
the house that day, does not get off the couch, she walks zero. How much money is she
going to get?” (Ms. M Activity 7, 11/08/04) The answer, five dollars, brings to the
conversation the first non-zero y-intercept that the class has discussed. This takes place in

the context of discussing the equations for the three pledge plans. Just prior, the class has

214

sorted out the coefficients and Ms. H has asked, referring to the equations for Gilberto
and Leanne’s pledge plans, “How come these guys don’t have anything added on?”
(ibid). She then goes on to ask, “So why did these guys not have a plus in the equation?”
This discussion is targeted at the fact that the y-intercept shows up as the constant term in
the slope-intercept form of a linear equation.

Finally, in Problem 2.5, Ms. H relates the y-intercept in that problem with the one
in Problem 2.3, through discussion of the equation.

H: So, you’ve seen having a plus five it was the other day it was a donation. Here

a plus 45 is a... What is this plus 45?

Students: Head start. (Ms. H Activity 3, 1 1/10/04)

In the cited discussions of rates in equations, Ms. H, like Ms. M, does not have
students compare equations. Students compare tables (“the number that [Gilberto] goes
up by is the highest, which means he has the highest pledge amount”) and they compare
graphs (with the language of steeper), but the discussion of equations is about writing
them, not about comparing them.

In the analysis of Ms. M’s discourse is a conjecture about how a traditionally
trained teacher approaches the topics of tables, graphs and equations when teaching a
reform curriculum; that such a teacher would struggle to use tables and graphs as
extensively as CMP does and that equations might continue to take the prominent place
in the teacher’s discourse. This is not the case with Ms. M’s discourse and it does not fit
Ms. H’s discourse either. A second conjecture, that the use of equation would correlate
with solving and correspondence (set-theoretic notions) over variation and relationship

(relational-dependency notions) more closely matches the evidence in each of these

215

 

cases—especially the lessened emphasis on relationships and the increased emphasis on
equations as objects, and on the writing of equations to express a correspondence. Neither
teacher strongly emphasizes the set-theoretic notions of fiinction, but each shows a set-
theoretic influence when addressing equations that she does not show when discussing
the other two representations.

In summary, 1 reinforce this point. Ms. H consistently uses the three
representations equivalently to identify rates. Beyond that however, she asks different
kinds of questions about equations than she does about tables and graphs. When
discussing tables in her Summaries, Ms. H asks questions that involve finding contextual
information. When a graph proves difficult to read, she suggests that they turn to the table
for a more precise answer. When discussing graphs in the summaries, Ms. H focuses on
finding contextual information, as has just been suggested, but she also spends time
having students use this information to identify the relationship that a graph represents. In
Problem 2.2, she asks students to determine which of the unlabeled graphs on the
overhead projector represents which walker. Neither of these questioning pattems—
finding contextual information, or identifying characters— is present in Ms. H’s
discussion of equations. Instead, the focus is on writing equations, as in the following
example.

H: Uh... equations... dequals...

Student: 1t

H: t for Terry. Some people wrote It. . .and, um... Jade’s equation?

Student: d equals ...2

H: 2t. Jerome’s equation?

216

Student: 2 point 5 t(Ms. H Activity 5, 1 1/04/04)

In this passage, the contrast in the use of the representations can be seen. Tables and
graphs represent aspects of the context, including rates, and time is spent discussing this
fact. Equations, however, are written and then the conversation moves on.

Narratives

In the Word Use and Visual Mediators sections, 1 have argued that Ms. H’s
introductory discourse on linearity alters the CMP discourse—especially on the topic of
equations. In the use of tables and graphs as visual mediators, Ms. H’s discourse is
similar to CMP’s textbook discourse, but equations are treated differently from these
other representations in terms of their emphasis. This differential emphasis also appears
in the wider view of the discourse afforded by the focus on narratives.

Most of the CMP narratives are also narratives in Ms. H’s discourse, with one
exception, Linearity is represented by tables, graphs and equations. Each narrative is
listed next, together with at least one example as evidence of this claim.

Mathematical Variables Represent Contextual Variables

This is a theme throughout Ms. H’s instruction in this investigation. There are no
examples in the summaries of mathematical variables that do not represent contextual
variables. Instead, mention of variables is rooted in the contexts, as in this passage.

The money’s based on the distance. So it’s always gonna be.. money equals, the

money equals, the money equals. For Leanne, her money equaled her distance: a

dollar per mile. Does that remind you of somebody else’s equation from last

week?(Ms. H Activity 7, 11/08/04)

217

Tables, Graphs and Equations Represent Relationships between Variables

In the cases of CMP’s and Ms. M’s discourses, two patterns in the use of the word
equation establish this narrative: equations represent relationships and equations have a
status roughly equivalent to tables and graphs. In Ms. H’s discourse, the second of these
patterns holds: equations are equivalent (in some ways) to tables and to graphs as
representations, but the first does not hold: Ms. H does not use the term relationship.
Instead, the narrative is implicit in how Ms. H discusses the representations. In the
previous example, when Ms. H says, “The money’s based on the distance,” she is
pointing to the relationship between the two variables, money and distance. This leads
into a discussion of how to represent the relationship with an equation. Indeed, this is a
common theme in Ms. H’s discussions of equation writing; the goal is to express how one
variable is based on, or found from the other. In a similar vein, Ms. H establishes through
the discussion of rates in each of the representations that relationships between variables
are under study. Nonetheless, the narrative is implicit in Ms. H’s discourse.
Tables with Constant Increments Represent Linear Relationships

Ms. H explicitly states this narrative in response to the student who asks in
Problem 2.3, “Wouldn’t she not be linear?” (Ms. H Activity 2, 11/09/04). The student is
referring to the walker with a non-zero y-intercept. Ms. H elicits the working definition of
linear from a student, then restates it herself.

Student: If it goes up by the same amount, that’s linear.

H: It doesn’t matter what it starts at. It just matters that it goes up by the same

amount every time. So hers is linear. And it looked linear, right? (ibid)

Straight-Line Graphs Represent Linear Relationships

218

As discussed in the visual mediators section, Ms. H uses graphs to identify linear
relationships, and the criterion is that the graph must be a straight line, as in this example.
I just called it a line, so yeah it was a line. So you could tell from the graph that
Henri had, that his distance was a linear relationship with his time. (Ms. H
Activity 3, Il/10/O4)
In this teacher turn, the condition of being a line is equated with a linear relationship.
The major work on linearity in Ms. H’s instruction takes place in a warm-up
before Problem 2.3, where she also introduces the idea that a relationship can be “not
linear”. For this warm-up, Ms. H displays six tables. Each table is of the form in Figure
15, with independent variable values I, 2, 3 and 4, with dependent variable values for the
first three of these and the instructions, “Find the 4th term and the equation.” In the
discussion of the warm-up, Ms. H addresses several aspects of linearity: how to identify

linear relationships from tables and from graphs, and the idea of non-linearity.

 

 

Figure 15: The basic form of the tables in Ms. H’s warm-up preceding
Problem 2.3

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The discussion of identifying linearity in a table uses the language of goes up by.
This refers to the increment in either the independent variable or the dependent variable,
as in this passage,

It went by one point five every time. There’s my equation... again, this is a linear

relationship. Every time this went up by one, this went up by one point five. Did

they go up by the same number? Did t and d go up by the same number? (ibid)
Similarly, Ms. H points out that non-linearity can be identified in a table (Figure 16) by
what the variables go up by, “...it’s not linear. It [the dependent variable] can’t go up by
different amounts here when this [the independent variable] is going up by the same

amount every time. “(Ms H Activity 2, 11/08/04)

 

 

Figure 16: A non-linear table from Ms. H’s instruction
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Ms. H also focuses on identifying linearity in a graph. Here the requirement is that
the points lie on a line. In the following example, Ms H plots the three points given in one
of the tables from the warm-up as she speaks.

This one is not linear. I graphed it. I tried doing this last hour. At 1, it’s at 4. At 2

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it’s at 5, 6, 7, 8, 9, 10. And then at 3, all ofa sudden it’s only at...12. Look at this

graph. Is that a line? (ibid)

Because the graph is not a line, the relationship is deemed not linear. Later in the warm-
up, Ms H reminds students of the importance of studying nonlinear relationships.

we talked about this. We said, if our whole book is about linear relationships,

sometimes we’re gonna need to look at some that. . .are not linear to help us

recognize to appreciate the ones that are. (ibid)
Even in this warm-up, however, the identification and representation of linearity in
equations is absent.
Equations of F orm y = mx + b or y = b + mx Represent Linear Relationships

As noted in the Visual Mediators section, the representation of linearity in
equations is an absent theme in Ms. H’s discourse. It follows, then, that how linearity is
represented must also be missing, and indeed it is.

However, all of the equations that Ms. M writes and discusses are of these two
forms. She writes y = mx + b, but she explicitly endorses a student’s suggestion that one
could also write y = b + mx in this passage,

Si: Igot m=5+.5d

H: You could, you could um you could put the donation first and then the rate,

that would be fine too.
Therefore, Ms. H introduces the appropriate forms for linear equations, but she never
directly addresses the idea that this form is relevant to determining linearity. In doing so,
she leaves Open the possibility for competing narratives, such as All equations must be of

theform y=mx+b or y=b+mx.

221

Rates Are Represented in Tables, Graphs and Equations

As discussed in the visual mediators section, Ms. H’s discourse includes rates as a
recurrent theme with all three representations. Additionally, the means of determining
rates in all three representations are addressed.
The y-intercept in a table is represented by the x-value when y is zero
The y-intercept in a graph is where the graph starts
The y-intercept in an equation is the amount added on

While these three narratives receive less emphasis than those about rates, these
too are discussed in the visual mediators section, where y-intercepts were shown to be at
least passing themes in each representation.
Analysis of Tables, Graphs and Equations Can Answer Contextual Questions

While the three representations are not equivalent in this narrative, it is the case
that each representation can be used to answer some contextual questions according to
Ms. H’s discourse. Analysis of tables can answer questions about individual values, for
instance, “[W]e can try to read exact values on the graph; sometimes it’s difficult, so then
we can always rely on our table.” (Ms. H Activity 5, 11/04/04) This example also
establishes that graphs can answer questions about individual values. This idea, however
is absent from the discussion of equations. Instead, equations are useful for identifying
starting values, “How can I tell in my equation that someone got a five dollar donation?
And that someone else didn’t?” (Ms. H Activity 2, 11/09/04)

Ms. H does not establish the final narrative that appears in Ms. M’s discourse,
Any of the Three Representations of a Linear Relationship Can Be Generated from Any of

the Others. Instead, like CMP, she treats the three representations in parallel. Each has

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several purposes, many of them overlapping, but there is not an indication that one can be
used to generate the others. The only exception is the warm-up on 11/08, in which the
final line of each table calls for the writing of an equation (see Figures 15 and 16).

The narratives in Ms. H’s parallel those of CMP, with the exception of the
narrative describing how to identify linear relationships from equations. This is consistent
with the findings in the Visual Mediators section, where equations are observed to be
given different treatment from the other two representations. It is impossible to know the
reasons for this difference. The speculation above about an interaction between Ms. H’s
mathematical education and experience prior to teaching CMP (which is also unknown,
but likely involved a set—theoretic approach to functions) and the relational-dependency
approach in CMP is just that: speculation. Regardless of the origins of the noted
differences, they are clear. Equations have a different role from tables and graphs in Ms.
H’s discourse. Equations are less strongly associated with relational-dependency notions
of function, while these notions are emphasized in her treatment of tables and graphs.

Routines

The routines for establishing and reinforcing narratives in CMP’s discourse are
also present in Ms. H’s discourse. Ms. H asks questions, provides multiple examples to
support inductive reasoning and she expects students to find mathematical meaning in the
contexts of the problems. Additionally, Ms. H mirrors Ms. M in explicitly stating, or
telling, narratives in a way that the CMP Student Edition does not.

Asking Questions
Ms. H asks the questions in the shaded boxes in the problems in the investigation,

sometimes quoting directly, as in, “letter B. What effect does increasing the amount

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pledged per mile have on the table, on the graph and on the equation?” (Ms. H Activity 2,
1 1/09/04) and sometimes paraphrasing, as in, “Letter D, what if they each just wanna get
10 dollars? I’m just gonna walk till I get 10 dollars, then I’m quitting. OK?” (ibid) As in
the case of Ms. M, it is notable that Ms. H often tells at the end of a series of questions,
but it is also important to distinguish between asking questions and the telling that may
follow.
Telling

Recall that the routine of telling differs from the routine of asking questions in
that the latter routine gives students the opportunity to state narratives in their own terms;
to contribute their ideas to the discourse. Telling states narratives in the teachers’
language. When a teacher tells, the usual result is a lack of further narratives. When a
teacher asks questions, multiple competing narratives often enter the discourse. Ms. H
commonly asks a series of questions, and then tells at the end. Frequently, as in the
following example, Ms. H’s telling is a response to her students not stating it, or to their
giving incorrect answers to her questions. In this example, Ms. H is trying to get students
to compare equations. When her initial question goes unanswered, she asks a series of
questions intended to highlight a part of the narrative, Rates are represented in tables,
graphs and equations, namely: The larger coefficient represents the greater rate. She
ends the exchange by stating this in contextual language.

11: How in the equation can I see who has the steepest... who has the highest

pledge amount?

Ii: Does this two mean... where does this 2 come from?

224

 

 

Students: The pledge!
H: Two dollars per mile. And this 1 comes from?
J: From the one dollar
H: One dollar per mile, and this point five comes from?
J: F ifiy cents per mile
H: Fifty cents per mile. So who has the highest pledge amount?
J: Gilberto!
H: Gilberto and you can see it in his equation because you see this 2 next to the (I.
And this only has a

Students: One

H: One next to the d, which is a smaller number. And this has a point five next to
the d, which is the smallest number of all. So even if you had never seen the
graph, to know that Gilberto’s was the steepest line, you could look at the
equation and see the biggest number that you’re multiplying times the
distance. So that means that he is getting the most per mile. (Ms. H Activity 2,
11/09/04)

In this example, Ms. H establishes a narrative by telling, Larger rates are
represented by larger coeflicients in the equation. She also tells when reinforcing a
narrative. This is best seen when there are conflicting narratives, as in an example cited
earlier (in the Visual Mediators section, under the subheading Tables).

B: Wouldn’t she not be linear, Alana?

H: Wouldn’t she not be linear? Well if it goes... what did we say yesterday?

Student: If it goes up by the ‘same amount, that’s linear.

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amount every time. So hers is linear. And it looked linear, right?

H: Good point. They’re all linear. They all go up by some number. (Ms. H
Activity 2, 11/09/04)

The student introduces the narrative, Linear relationships have zero y-intercept. This
conflicts with Ms. H’s intended narrative, Tables with constant increments represent
linear relationships, which does not depend on the y-intercept. In order to resolve the
conflict, Ms. H asks a student to restate the rule, “it goes up by the same amount,” which
is a rephrasing of the intended narrative. Ms. H then restates it herself; she tells.
Inductive Reasoning

Ms. H follows the same structure here as CMP. Students make the same tables,
graphs and equations and she uses the same non-examples of linearity that are in the
student texts. The preceding example, under the routine Telling demonstrates this. In her
attempt to lead students to the observation that the coefficient is the same as the rate, she
raises the three examples in turn before she tells students the narrative.
Deriving Mathematical Meaning from a Context

Ms. H’s instruction follows the student text quite closely. One result of this is that
asking questions is one of the routines for establishing narratives in her discourse.
Another result is that much of the discourse in her classroom is grounded in contexts, just
as is the case with CMP. Yet, also like CMP, she has mathematical goals for her students.
For instance, in the interview, Ms. H states that in addition to knowing how to write an

equation,

226

 

 

The other goals would be, I guess what I was just saying about them recognizing,
you know, not having formal language for y-intercept, but recognizing that when
the graph has a y—intercept other than zero, that means that there’s something
that’s being added on independent of the variable. And again, they wouldn’t have
that formal language, but they might think of it as the head start in the race, or the
donation part of the pledge sheets. (Ms. H Interview, 10/21/04)

Logically, then, it follows that she expects students to derive mathematical
meaning from the contexts. In the problems in this investigation, this process remains
implicit. Ms. H follows CMP’s lead in not introducing the mathematical vocabulary of
slope and y-intercept, so she cannot define these in terms of the contexts. She also
follows the text’s lead in asking questions about the contexts in the investigation, so there
was no opportunity to refer back to a context in order to answer a non-contextual
question. There is no place in the investigation that the student text refers to purely
mathematical goals and there is no place where Ms. H does either.

Because Ms. H follows the CMP text closely, her mathematical discourse is
founded on the contexts in the curriculum, and the three algebraic representations of
functions play a prominent role as well. The detail-oriented analysis of Ms. H’s word use
reveals some differences that appear at the other levels of analysis—in particular the
differential role played by equations in her discourse. Ms. H focuses on writing equations
over comparing them, and spends no time on using them to identify linearity. The use of
the word equation is not associated, in Ms. H’s discourse, with relationship.

While this difference is an important one, it should not be overstated. The goal of

the analysis is to characterize each teacher’s introductory classroom discourse on

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linearity. This necessitates characterizing the approach to functions in each discourse.
Using a standard characterization of this topic, in which there are two approaches to
functions—a relational-dependency approach and a set-theoretic approach—the
commonalities of Ms. H’s discourse with the relational-dependency approach attributed
to CMP in the previous chapter are the stronger result than the differences with the CMP
approach. A set-theoretic approach would privilege equations over the other two
representations, which is clearly not the case in Ms. H’s discourse. A set-theoretic
approach would use the other two representations in service of equations, which again is
not the case in Ms. H’s discourse. Instead, Ms. H could be said to have a relational-
dependency approach to functions when represented by tables and graphs, but a more set-
theoretic approach to functions represented by equations.

Finally, learning is characterized in this study as changing one’s participation
patterns in a model discourse. We may consider the characteristics of participation in Ms.
H’s model discourse. In chapter IV the CMP model discourse is related to a heuristic
mathematical discourse, in which questions are asked, examples tested, generalizations
made, and forms of metaphor employed. This is in contrast to a formal mathematical
discourse in which the heuristics are hidden and the focus is on straightforward, logical
development. The heuristic techniques are present in Ms. H’s model discourse. She asks
questions to get students to state narratives before she states them herself. Like Ms. M,
Ms. H uses the examples from CMP, and creates her own in the warm-ups in order to
lead students to use inductive reasoning. The exception to the heuristics characterization
is that Ms. H tells when students do not state narratives in response to her questioning.

Telling is a classroom equivalent to stating a theorem in formal mathematical discourse.

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Discussion
This final section has three parts. The first two parts address the research
question, What characterizes each teacher ’s introductory classroom discourse on
linearity? by recapping and summarizing the classroom discourses of each teacher. The
focus is on characterizing the study of linearity, the use of contexts and the mathematical
character of each discourse. The last part addresses the research question, What changes
does the discourse of the curriculum undergo in its implementation by each teacher?
Ms. M ’s Discourse
The introductory lessons on linearity in Ms. M’s classroom clearly indicate that
the study of linearity is a study of relationships. At the level of word use, linear is not
frequently used, but when it is, it is associated with relationship and with situation”.
Rate is a central idea, mentioned 36 times by Ms. M in these lessons. As I argue in
Chapter V, use of the term rate implies a focus on relationships, since rates involve two
variables (e. g. time and distance). At the level of visual mediators, relationships appear
through the identification of rates in discussions of each of the three representations. The
narratives in Ms. M’s discourse similarly reinforce the idea that linearity is about
relationships through an emphasis on rates. Rates, in Ms. M’s discourse, affect graphs by
changing their steepness. They show up as increments in tables and multipliers of the
dependent variable in equations. With respect to each of these three narratives, Ms. M
uses language that includes both variables (e. g. money and distance), as in the example,
“Where’s the amount per mile in this equation? That number that is always multiplied

to that d.” (Ms. M Activity 5, 10/28/04) This contrasts, for instance, with a student turn in

 

47 Relationship and situation are equivalent-used in the same way—as is the case in
CMP.

 

229

the same lesson that mentions only one variable in trying to account for the differences in
two graphs, “Because I think that’s more money.” (ibid)

Sfard’s (2005a) four distinguishing features of mathematical discourse—word
use, visual mediators, (endorsed) narratives and routines—are used in this study primarily
to complement each other in characterizing each teacher’s discourse and to compare
discourses. The cases of equation as a word and equations as visual mediators provide an
opportunity to make comparisons within a discourse. That is, this study considers the use
of the word equation and it considers the use of equations as representations of functions.
These give different views of the role of equations in Ms. M’s discourse. Her use of the
word equation is associated with the use of table and of graph—at the level of word use,
there is an equivalence among the three representations, as evidenced in, “So we have
tables, graphs and equations, all that can represent this linear situation.” (Ms. M Activity
2, 10/27/04) Yet, when the use of equations as visual mediators is considered, there are
differences among the three representations. Most notably, when Ms. M discusses
equations, she focuses on writing them in contrast to CMP, where the focus is on
describing how equations are affected by changes in walking rates. Ms. M asks long

series of questions about the writing of specific equations, but does not ask these kinds of

 

questions about making tables or graphs. Similarly, the themes in her use of equations as
visual mediators differ from her themes in the uses of tables and of graphs. Without
considering the uses of equations as mediators, in addition to the use of the word
equation, these differences might go undetected.

Ms. M’s introductory discourse on linearity also relies heavily on contexts—the

same contexts as in CMP. In closely following the sequence of problems in CMP’s

 

230

Investigation 2, Ms. M introduces slope through contextual language (e. g. walking rate)
in each of the three algebraic representations. She also uses this contextual language to
reinforce the mathematical term y—intercept, which had been introduced in a pilot
cuniculum prior to the teaching of Moving Straight Ahead. The contexts are not mere
artifacts of the curriculum, however. When Ms. M steps out of the curricular framework
of Launch-Explore-Summarize and gives students tasks of her own design (in the phase
of the lesson she calls warm-ups), she uses the walking rate context to address I
mathematical topics. That is, this context is a central aspect of Ms. M’s discourse on
linearity that is retained outside the specific structure of the curriculum’s tasks. There is
one warm-up in which Ms. M uses the mathematical variables x and y instead of d and t.
But in this case, there is no reference to the walking rate context. In each of the other
warm-ups, walking rates are either an explicit part of the original task or Ms. M asks
students to compare the tasks to the walking rate tasks from the curriculum.

Lampert (1990) describes her attempts to bring the practices, rather than only the
products, of the discipline of mathematics to her students through conjecturing, testing
and asking questions. Ms. M’s discourse has elements of this kind of classroom

mathematical discourse; she asks many questions and sets students up to use inductive

 

reasoning (i.e. to generalize from examples). In Ms. M’s discourse, there is also an
indication that mathematics consists of some previously established facts; she asserts
mathematical relationships (narratives) through telling, as in, “[T]he person who had the
fastest walking rate...had a much steeper line than the other two.” (Ms. M Activity 2,
10/27/04). The routine of telling is distinguished from the evaluation of a student’s

answer to a question by the person who states the narrative; when Ms. M asks a question

 

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and then confirms a student’s answer, the student has stated the narrative. When Ms. M
states a narrative, she is telling. This routine is a pervasive pattern in Ms. M’s discourse
and it is generally associated with the routine of asking questions—Ms. M tends to tell
only when her questions do not result in students stating narratives she wishes to endorse.
As a result of the mixing of the CMP routines with the routine of telling, Ms. M’s
discourse has characteristics of a heuristic mathematical discourse, with some elements of
a formal mathematical discourse. The more consistent emphasis is on the former.
Ms. H ’s Discourse

Ms. H’s introductory discourse on linearity is also about relationships between
variables. In her word use, this shows up in her frequent use of the term rate (20
mentions in the five lessons), and in the association, in Problem 2.5, between linear and
relationship. Ms. H’s use of visual mediators demonstrates the importance of
relationships as well, with the identification and/or representation of rates being a
recurrent theme for each of the three algebraic representations.

As with Ms. M’s discourse above, we may take the opportunity to compare the
use of the term equation with the use of equations as visual mediators in Ms. H’s
discourse. At the level of word use, Ms. H treats table, graph and equation equivalently.
She lists these three terms together often enough for her students to chorally recite them
in this order. When we pay attention to the use of equations, the objects, not just to the
use of the word equation, we see differences that might not be detected at the word use
level. Ms. H has students compare tables to each other (“The number that Gilberto goes
up by is the highest”), and compare graphs to each other (“Let’s look at it [the graph], is

it steeper?”). But Ms. H does not have students compare equations to each other. Instead,

232

she focuses on writing equations. There are two contrasts here. The comparison task is
missing from Ms. H’s discourse, but also the writing of equations is emphasized in a way
that the making of tables and graphs is not. Further differences between the use of
equation and the use of equations emerge by considering the themes in the use of the
visual mediators. The identification of individual values is a recurrent theme for tables
and for graphs in Ms. H’s discourse, but the corresponding theme is absent from her uses
of equations. The same pattern holds for her the theme of linearity—the theme is
recurrent for tables and for graphs, but absent for equations.

Ms. H’s introductory discourse on linearity relies heavily on contexts because she
closely follows the sequence of problems in the curriculum, resulting in the use of
extensive contextual language: walking rate, donation, head start, etc. The ideas of slope,
y-intercept and solutions are introduced by Ms. H in this series of lessons, but they are
discussed by students and teacher alike in contextual language. Ms. H uses three warm-
ups over the five lessons; two of them rely on the walking rate context, while one of them
does not, except for the use of the variables d and t. These suggest distance and time, but
the discussion of the task does not rely on this interpretation.

Ms. H’s routines for establishing and reinforcing narratives are similar to Ms.
M’s. The heuristic nature of mathematical discourse in CMP’s textbook discourse, and
advocated by Lampert (1990) is present in the questions Ms. H asks, and in the examples
used to encourage inductive reasoning. These examples (and corresponding non—
examples) appear in the Launch-Explore-Summary sequence from the CMP text, but they
also appear in the warm-ups that Ms. H creates to establish and reinforce narratives

outside of this sequence. Ms. H indicates to students that some mathematical

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relationships are previously established by asserting (telling) narratives, as in this
example, “It doesn’t matter what it [the dependent variable] starts at [in a table]. It just
matters that it goes up by the same amount every time.” (Ms. H Activity 2, 11/09/04) Ms.
H tends to tell when asking questions does not result in students stating narratives she
endorses. As with Ms. M, the emphasis in Ms. H’s discourse is on the heuristic nature of
the CMP mathematical discourse, but the use of the routine of telling is an element of a
more formal mathematical influence.

Changes in the CW Discourse in Implementation

The introductory discourse on linearity in Connected Mathematics is founded on
contexts. Students are expected to derive mathematical meaning from these contexts. The
linear relationships embedded in these contexts are expressed through tables, graphs and
equations. Each of these three visual mediators can be used interchangeably to convey
contextual information of various kinds, including rates and individual values. The two
teachers in this study, Ms. M and Ms. H similarly base their discourse on contexts—the
ones in CMP. Each teacher follows the student text with few departures, but there are
several notable changes that the curricular discourse undergoes as it is implemented in
these classrooms.

There are differences in the format of the lessons—differences from the CMP
format that are strikingly similar in the two classrooms. Not every narrative in these two
classrooms is established or reinforced through the problems in Investigation 2. Each
teacher uses warm-ups to establish narratives that are not established in the Launch-
Explore-Summary (LES) sequence, or to reinforce narratives. It is not clear whether the

decision to establish narratives outside of the LES sequence is intentional for either

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teacher, but it is again remarkable that the teachers independently use the warm-ups,
which are a deviation from the form of the curriculum, to keep their discourses on
linearity consistent with the discourse of the text. Without these warm-ups, for instance,
Ms. M’s discourse would lack the narrative, Linearity is represented in tables by a
constant increment in the dependent variable. By contrast, this narrative is prominent in
Investigation 2 of Moving Straight Ahead.

Ms. M and Ms. H both use what they call warm-up activities. This name is a
generic term for mathematical tasks given to students prior to the main body of the
lesson. But for both, the warm-ups are used to introduce important content, not just to
prime students for the ensuing instruction. Both teachers use the curriculum’s contexts
most of the time for their warm-ups (Ms. M: 4 out of 5 warm-ups, Ms. H: 2 out of 3), but
each teacher has one warm-up that does not rely on a context.

In the analysis of curriculum and of classrooms, this study attempts to measure
the importance of each use of visual mediators in each discourse by identifying themes
and labeling each theme as recurrent, passing or absent in each discourse. Recurrent
themes are more important than passing themes, which are obviously more important
than absent ones. The most striking change in the textbook discourse occurs in each
classroom: Themes in the use of equations have an importance different from these same
themes in the use of tables and of graphs.

In CMP, the uses of the three representations have nearly identical importance—
with the one exception being that finding individual values in equations is de-emphasized
relative to finding individual values in the other two representations. In both Ms. M’s and

Ms. H’s discourses, tables and graphs are used for the same purposes, and these uses have

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similar importance as in CMP. Table 14 shows that the treatment of equations in each
teacher’s classroom differs markedly from the treatment in CMP. When we consider the
two approaches to functions outlined in this study, we see that tables and graphs play a
much more prominent role in the relational—dependency approach than in the set-theoretic
approach, while equations are prominent in both. Due to the lack of emphasis on the
other two representations, equations take center stage in a set-theoretic approach to
functions. The set-theoretic approach, which is equation-centric, is standard in the
traditional curriculum, such as Dolciani, while the relational-dependency approach
(which is variable-centric) is more common in newer curriculum like CMP. Equations,
while not center-stage, have different uses in a relational-dependency approach from their

uses in a set-theoretic approach.

 

Table 14: Themes in each discourse on equations

 

    
  

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In these two teachers’ discourses, we see the expansion of the roles of tables and

graphs from the traditional approach, but not the hill expansion of the role of equations.
Recall the conjecture about approaches to functions by teachers who did not learn algebra

in a reform curriculum. The initial conjecture was that such teachers would de-emphasize

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tables and graphs, with which they were conjectured to be less familiar, while
emphasizing the more familiar representation: equations. This conjecture is not supported
by the data from these two classrooms. The uses of equations are, in fact, more limited in
one of the classrooms. Ms. H does not address the finding of individual values in
equations (which corresponds to solving—the main purpose of a traditional approach like
Dolciani’s), nor does she address the identification of linear relationships from their
equations. Consider that this (identifying linear relationships in equations) depends on the
form of the equation, and that in the traditional approach exemplified in this study by
Dolciani, this form is the defining feature of linearity. In this sense, Ms. H’s treatment of
equations is in opposition to the set-theoretic approach, but also leaves out an important
aspect of CMP’s relational-dependency approach—identifying linear relationships from
the form of equations.

In addition to the themes in the uses of equations as visual mediators, the results
in this chapter show that each teacher emphasizes the writing of equations, while de-
emphasizing the comparison tasks in the cuniculum. Some form of comparison is
highlighted for tables and for graphs in each teacher’s classroom; for instance the graphs
for three walkers are displayed on the same set of axes in both classrooms, and each
teacher uses the word steeper. The discussions of equations in both classrooms focus on
Writing equations rather than on comparing them. It is unclear why this is so. Perhaps the
teachers have made a choice to develop the comparison of equations in later
investigations, perhaps they each perceive that their students struggle with equation

Writing and so focus their efforts on this topic.

237

 

Ms. M and Ms. H each make a change to the graph in Problem 2.3, either putting
in or removing a donation to a walker’s pledge plan and examining the effect of doing so
on the graphs. In each case, the teacher does this in order to isolate the steepness of a line
from its height. Each teacher demonstrates the change and then asks a form of the
question, “How does this affect the situation?” In so doing, each teacher models two
routines from the CMP discourse: asking questions and the non-example part of inductive
reasoning. The student-generated narrative, Larger rates yield higher points on the graph
can be seen as an overgeneralization from examples with zero y-intercept. By providing
an example that has the same steepness, but different height, each teacher introduces a
counterexample to the implicit inductive reasoning. It is remarkable that these two
teachers make the same improvisational instructional move—CMP does not include
instructions to do so.

Two aspects of the teachers’ instruction—the use of warm-ups and the
improvisation with graphs in Problem 2.3 —demonstrate that the teachers see the CMP
discourse as the model for their own discourse. They modify the form of instruction to
keep this model discourse intact.

Recall the research question, What changes does the discourse of the curriculum
undergo in its implementation by each teacher? I have argued that answering this
question is important because the writing and public discussion of curriculum often
presumes an unproblematic relationship between published curriculum and classroom
teaching. This study turns up ways in which this relationship is indeed unproblematic.
The content of CMP, as expressed through the narratives in discourse, is largely

unchanged in these two classrooms, for instance. Ms. M adds a narrative about transitions

238

among the three algebraic representations and Ms. H leaves out the form of linear
equations, but otherwise the major narratives in CMP’s textbook discourse on linearity
are the same as the narratives in the classroom discourses.

Each teacher models the routines by which mathematical narratives are
established and reinforced in CMP’s textbook discourse. There is one important change
to these routines that appears in each classroom, however. Each teacher uses the routine
of telling—primarily at the end of a series of questions in which students do not state a
narrative that the teacher endorses. When the routine of asking questions does not
produce a student-stated narrative, or when it produces only narratives that conflict with
the teacher’s narratives, Ms. M and Ms. H tell. Under these conditions, each teacher
states the desired narrative herself.

A significant goal of the introductory instruction on linearity in CMP, however, is
changed in each teacher’s classroom; the treatment of equations. Given the traditionally
central role of equations in US cuniculum (as exemplified in Dolciani), this may not be a
surprising finding. But it is a significant finding. Some major changes in algebra
instruction in the current curricular reform have taken hold in the two classrooms studied:
an increased use of tables and graphs and the study of rates and relationships between
variables among them. One major change has not: the equivalent status and use of tables,
graphs and equations as representations of linear relationships. Full implementation of the
curriculum reform would require this. The study of the consequences of this difference,
and the problem of addressing it in the classrooms of practicing teachers, is a topic

addressed in the concluding chapter.

239

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Chapter VII: Conclusion

This study began with a question about classroom instruction in the wake of
significant reforms in US. mathematics cuniculum. Chapter 111 frames this question in
the language of discourse analysis, using a communicational framework in which
knowledge is characterized as participation in a discourse. The question, What is being
taught? becomes What is the model discourse? Chapter IV turns this question into two
research questions, What characterizes the textbook discourse? and What characterizes
the classroom discourse?, and it adds a third research question, What changes does the
discourse of the curriculum undergo in its implementation by each teacher? That same
chapter makes an argument that these two discourse types—textbook and classroom—
have some different properties, but that they can be reconciled and compared in a
principled way. Chapter V uses a traditional American curriculum, Mathematics,
Structure and Method (Dolciani, 1992) as a way of understanding the claims made about
the reform curriculum under study, Connected Mathematics (CMP; Lappan, et al.,
2001a). Finally, Chapter VI characterizes the teacher’s model discourse in each of two
classrooms and compares these to the textbook discourse of CMP.

Findings

This section recaps the important findings of this study. It points out the audiences
who should care about these findings and suggests ways in which some of these findings
might be put to use in the service of improving teaching in reform mathematics

classrooms.

240

Textbook Discourse

The introductory textbook discourse in CMP is characterized as having a
relational-dependency approach to functions in which variables and the relationships
between them are the primary foci of instruction. This is contrasted with the set-theoretic
approach to functions found in Dolciani, in which equations and their solutions are the
primary foci. Most important among the differences between these two curricula (and
therefore most important in the characterization of CMP) are those in the narratives and
in the routines for establishing and reinforcing narratives.
Narratives

The narratives in CMP’s introductory discourse on linearity have a parallel
structure: tables, graphs and equations are treated as equivalent representations. Each can
be used to represent rates and y—intercepts and to determine linearity. The equivalence of
these representations pervades Investigation 2 of Moving Straight Ahead. Absent from
the CMP narratives are relationships among these three representations. Dolciani, for its
part, emphasizes particular relationships between the three representations, but these
relationships establish priorities—equations are the most important representations,
graphs have a lesser importance and tables are used only to create graphs.
Routines

Narratives depend on the content of the curriculum in that a unit on quadratic
relationships would necessarily have different narratives (although their structure might
be similar to those described here). Routines, on the other hand, can be expected to apply

more generally. We should expect to see the routines identified in CMP in this study to

241

 

appear in other CMP units when students study other content, and likewise for the
Dolciani routines. The differences between these two cunicula likely exist throughout.

The routines in CMP model the actions that advanced knowers of mathematics
take when working on a new and non-routine problem, and for this reason I have
characterized CMP as having a heuristic mathematical discourse. CMP students are asked
questions, encouraged to make conjectures from sets of examples and given contexts in
which to find mathematical meaning—a form of metaphor. The routines in Dolciani, by
contrast, model the formal presentation of mathematics in lectures, texts and journals.
Students are given definitions and worked-out examples.

These differences in narratives and routines should be of interest to anyone
selecting cuniculum for classroom implementation. Principled selection pays attention to
the mathematical content of a curriculum. The characterization of this content through
narratives and routines provides a richer look at curriculum than a list of correspondence
to state standards. An analysis such as this one could make a useful contribution to a
curriculum selection process.

In the current curriculum reform, teachers—like the two in this study—are
generally being asked to teach mathematics in a way different from how they learned it.
Teachers are aware of this, but often discuss these differences in pedagogical language.
They identify the use of calculators and the increased use of small groups in reform
mathematics instruction as important differences, for instance. The analysis of cuniculum
in this study makes precise some of the differences between the mathematics teachers
learned as students and the mathematics they are being asked to teach when they adopt

new curriculum. Such insight should allow teachers to develop their own mathematical

242

knowledge. It should help them in their instruction by allowing them to make decisions
about when to bring their own prior knowledge to bear and when to work within the
cunicular expectations. Without a precise understanding of the differences between what
they learned as students and what the curriculum is trying to teach, teachers are unable to
make such decisions consciously.
Classroom Discourse

Classroom discourse is not so neatly characterized as textbook discourse is, but
each teacher in this study is seen to encourage a relational-dependency approach to
functions, with modifications. Each teacher treats tables and graphs in a manner
consistent with a purely relational-dependency approach to functions. Each teacher also
treats equations differently from tables and graphs. This would not be predicted in a
classroom with a relational-dependency approach to functions. In both classrooms, the
goal in discussions of equations is to get to the equation—to write it. This contrasts with
the teachers’ goals in discussions of tables and graphs, where getting to the tables and
graphs is assumed to be unproblematic and instead, there is comparison. When there are
three walkers in a race, all three tables are shown simultaneously and the class discusses
what each walker’s distance goes up by. All three graphs are shown simultaneously and
the class discusses which line is steeper. By contrast, equations are considered one at a
time and there is no corresponding language for comparing them. Ms. H uses equations
for fewer purposes than she uses tables and graphs; she omits discussion of identifying
linearity and of finding individual values in equations. She uses tables and graphs for

both of these purposes.

243

These results suggest some important considerations for several audiences. While
the full analysis undertaken in this study is too detailed to be practical for practicing
teachers to use in considering their own teaching, the results from these two classrooms
might serve as case studies that point teachers to areas for reflection and possibly small
action-research projects. For example, knowing some dimensions along which Ms. M and
Ms. H treat equations differently from tables and graphs should lead teachers to consider
how they might help students make comparisons among equations.

Curriculum developers can learn from the changes that curriculum undergoes as it
is implemented in classrooms. In the introduction, 1 suggested that curriculum writing
addresses idealized Ieamers. In a similar way, it addresses idealized teachers as well. This
study’s findings add to the body of knowledge about teachers available to curriculum
developers and so refine their image of the model teacher. In particular, the findings
suggest that teachers of reform curriculum need some form of support in addressing the
role of equations in the introduction to linear relationships. This support might come in
the form of restructured student tasks, or in additional or restructured writing in the
teacher materials.

Finally, many teachers new to CMP receive some form of professional
development. The findings of this study are informative for those who provide this
professional development. In identifying where teachers deviate from the curricular
vision, this study should help to focus instructional time with teachers. For example, a
leader of professional development might give an explicit model for comparing
equations, since this is something that was lacking in each teacher’s discourse.

Additional Findings

244

This section considers findings from this study that do not neatly fit as answers to
the original research questions. Instead, these are observations, supported by evidence, on
issues tangentially related to the research questions, and which could be developed
fiirther with the data from the study, or with some additional data collection.

Role of Warm- Ups

It is observed in Chapter VI that Ms. M and Ms. H each use warm-ups in their
classrooms to begin a typical day of instruction. Some form of warm-up— in which
students work on a small set of practice problems before the main lesson— is common in
US. math classrooms. Warm-ups can be used in many ways, including practice for
standardized tests, review of previously learned material and preparation for the present
day’s lesson.

CMP does not suggest the use of warm-ups in either the Student Edition or the
Teacher Edition. Thus, it is remarkable that these two teachers use warm-ups in very
similar ways. Ms. M uses four warm-ups over seven lessons, while Ms. H uses three
warm-ups over five lessons. Each of these teachers uses her warm—ups to reinforce ideas
in the curriculum and to introduce ideas that did not surface in the main body of previous
lessons. For instance, Ms. M uses a warm-up to address how to identify and represent
linearity in a table. While the other themes involving the use of tables as visual mediators
are discussed in the lessons, she does not discuss this one until the last day of the
investigation. On that last day, Ms. M shows students tables, graphs and equations on the
overhead projector and has students determine which represent linear relationships and
which do not. This leads to establishing the criterion Linear relationships are represented

by constant change in a table.

245

This use of warm-ups to establish themes and narratives that would otherwise be
absent from a teacher’s discourse suggests insight and reflection on the part of these
teachers. These seem not to be teachers blindly following a curricular path, but teachers
who understand the path well enough to adjust their lessons in ways that keep them on
the path. Ms. M does not use the warm-up described above only for practice and
reinforcement; she uses this warm—up— at this time— to establish an important narrative.
That both teachers use warm-ups in this way is remarkable. This similarity raises a
question, as well. One may wonder how widespread this use of warm-ups is, and at a
higher level, one may wonder how common the reflection attributed to Ms. M and to Ms.
H are among American mathematics teachers.

The Relational-Dependency Approach to Functions in the Classroom

CMP’s textbook discourse on linearity is representative of a relational-
dependency approach to functions in which variables and rates are center stage and the
three algebraic representations are of equal importance. In the two classroom discourses
in this study, the relational-dependency approach to functions is evident, but there are
modifications. In particular, the role of equations changes as the textbook discourse
becomes classroom discourse.

Equations do not take the prominent role that they would in a set-theoretic
approach to functions, but they are also not treated equivalently to the other
representations. Equations, in both Ms. M’s and Ms. H’s classrooms, are less strongly
associated with the representations of relationships than are tables and graphs, less
strongly associated with linearity than the other two representations and less likely to be

analyzed in comparison tasks than the other two. In Chapter VI, I speculate about the

246

causes for this—that it might stem from an interaction between these teachers’ traditional
training and the expectations of reform curriculum, for instance—but it remains
speculation. At its root, this speculation asks about the relationship between teacher
knowledge and classroom instruction—a deep and lively current research domain in
mathematics education (see, e.g. Ball, 2002).

The modifications of the relational-dependency approach to functions in the
transition from published cuniculum to classroom are remarkably similar in the two cases
examined in this study. As with the preceding topic, further research could investigate
questions about whether this is coincidental or indicative of some more widespread
phenomenon by looking at a larger set of classrooms.

Contributions of the Theoretical Framework

I have characterized the decision to adapt Sfard’s communicational framework for
this study as a pragmatic one. Characterizing teaching and leaming in terms of discourse
makes these terms operative and means that the data for the study is directly observable.
Communicational behavior in the classroom can be observed and captured (in large part)
on audio and videotape in a way that structures in the mind cannot. Yet, the framework
brings other advantages.

Systematic Look at Curriculum

From my classroom experience, I know the CMP unit Moving Straight Ahead
very well. The analysis of this unit was guided by substantial prior knowledge about its
structure, style and content. Using the four characteristics of mathematical discourse——

word use, visual mediators, narratives and routines—provided a way to write about this

247

unit that could be repeated by someone without my prior knowledge. The framework
provided a sense of validity to the CMP analysis.

Just as important, the framework gave me a principled way to examine a
cuniculum I did not know well: Dolciani. The four characteristics of mathematical
discourse are independent of my deep knowledge of CMP so this knowledge did not taint
my examination of Dolciani. Instead, the communicational framework gave me the same
tool to apply in the two cases, producing comparable results.

Textbooks and Classrooms

On a related note, the framework provided a way to compare textbook discourse
to classroom discourse by considering them as two instances of the same phenomenon:
discourse. Because textbook writing was considered as communicational behavior in this
study, it could be examined in the same way as the communicational behavior of the two
teachers, using the same characteristics to describe it. While the precise methods differed
to account for some inherent differences between the two discourse types (e. g. that
students help to shape the classroom discourse, but not textbook discourse), the four
characteristics of mathematical discourse could be applied equally well to describing
each discourse type.

Multiple Perspectives on Teaching

1 have argued that three of the four characteristics of mathematical discourse in
this study give views at different levels—word use is a close-up view of discourse, the
use of visual mediators is at a medium range, and narratives gives a long-range view of
discourse. These three views provided some useful contrasts. For instance, I have

analyzed the use of the word equation in each teacher’s discourse and the use of

248

equations as representations in each teacher’s discourse. In this case, the word use is not
consistent with the visual mediator use. That is, the word equation is used in ways similar
to graph and to table, but equations themselves are put to different uses than are tables
and graphs, as discussed above. The communicational framework gave me a way to find
and describe these differences.
Limitations

This study investigates the relationship between published curriculum and
classroom instruction. There are three major limitations of this study: student learning is
not examined, teaching is not examined in a Dolciani classroom, and the research
questions are not examined longitudinally.

Student Learning

In the analysis of teachers’ classroom discourse, I have noted several times that
this discourse differs from textbook discourse in that textbook discourse is unaffected by
students and their contributions to classroom discourse. While the teacher models the
mathematical discourse, there is some degree of negotiation between the students and the
teacher that can be seen in this study when teachers adopt student language such as goes
up by or constant pattern for the increment in a table. More fundamentally, though, this
study has left out a close examination of the changes in student discourse participation
over the course of the investigation—an examination of student learning.

In Chapter III, I argue for the importance of studying teacher discourse in
isolation from student discourse on the grounds that a major question of curriculum
design is how the written curriculum impacts teachers’ instruction. The results of this

study are informative for curriculum design, but it would also be useful to know what

249

 

these students have learned. Data was collected in the course of this study that could be
brought to bear on this question. The classroom audiotapes contain student talk during the
Explore phase of each of the lessons in this study, and there is a small amount of pencil
and paper assessment data that could be analyzed and used to understand students’
discourse patterns in these two classrooms.

This study is important because it helps us to understand how to improve teaching
in a reform mathematics curriculum environment and we can expect this to lead to
increased student learning. A limitation of this study is the lack of analysis of student
learning in these classrooms and with these textbooks.

Dolciani classrooms

Chapter V provides a meaningful set of contrasts in curricular approaches. In
particular, CMP’s relational-dependency approach to functions contrasts with Dolciani’s
set-theoretic approach and differences between these appear in each of the four aspects of
mathematical discourse. This study does not examine classroom practice using the
Dolciani cuniculum. There are two further questions that could be answered with data
from Dolciani classrooms. How does the discourse on linearity in Dolciani change as it
is implemented? is the analog of research question 3 about the changes from the CMP
text to the classroom. How does classroom discourse differ in a Dolciani classroom from
that in a CMP classroom? is the classroom analog of the textbook comparison. Each of
these questions is pertinent to the important issues of mathematics curriculum change that

have been raised elsewhere in this dissertation.

250

A Longitudinal Look

The routine of deriving mathematical meaning from a context is what sparked the
original ideas for this study. When I propose, in Chapters V and VI, that there are at least
three possibilities for how students are to derive mathematical meaning from a context,
these possibilities are based on theory, observation and reflection on my own teaching
practice. Recall that the three suggested possibilities are,

1. Mathematical vocabulary is defined in contextual terms.

2. A mathematical question is asked (by the teacher or by the student) and the
teacher models or encourages seeking the answer by thinking about the
context.

3. The mathematical language is developed as a generalization of the specific
contextual ideas.

In Investigation 2 of Moving Straight Ahead, little new mathematical language is
generally developed. In particular, the vocabulary items slope and y-intercept are not
developed until later in the unit. This means that inferences needed to be made for this
study about the direction the text and the teacher might be headed as the contextual
language is developed. A study of a classroom that lasted through the entire unit would
be informative about this process. If the same research techniques were applied to 4—6
weeks worth of instruction, the researcher would expect to find moments where
mathematical vocabulary is introduced, or where contextual language seems to be
pointing to a concept that goes beyond the context in question. In a week’s worth of
instruction in which all of the problems are tied to contexts, these processes were not

directly observed. Thus, it is a limitation of this study that it does not look at a larger

251

section of the curriculum. In Chapter II I describe the National Research Council (NRC)
report on studies of curriculum effectiveness (2004). This report calls for studies that
examine at least an entire grade. This study is much more limited.

Another potential benefit of studying a larger section of the curriculum would be
investigation of the hypothesis that the discourse patterns in Moving Straight Ahead are
representative of those in the cuniculum as a whole. My informal knowledge of the
curriculum suggests that this is the case, but I have not applied the techniques of this
study to other portions of the cuniculum.

Conclusion

This chapter has summarized the important findings of this study as a whole,
pertaining to two types of classroom discourse—textbook and teacher. A small number of
additional findings have been addressed. The chapter has suggested how the
communicational framework contributed to progress on the study’s central problem of
describing the use of contexts to develop mathematical meaning in a reform mathematics
cuniculum. Finally, limitations of the present study have been considered, including the
singular focus on the teacher in the classroom data and the small segment of the
curriculum that is studied closely. These limitations are described in a way that points to
important analyses of existing data, as well as to future data collection and analysis. Such
future research should expand our understanding of the mathematical goals of curriculum

materials and how these materials and goals interact with classroom instruction.

252

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