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DISCOURSE ANALYSIS AS A TOOL TO INVESTIGATE

THE RELATIONSHIP BETWEEN WRITTEN AND ENACTED CURRICULA:

THE CASE OF FRACTION MULTIPLICATION IN A MIDDLE SCHOOL
STANDARDS-BASED CURRICULUM

By

Jill Newton

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Curriculum, Teaching, and Educational Policy

2008

 

 

 

ABSTRACT
DISCOURSE ANALYSIS AS A TOOL TO INVESTIGATE
THE RELATIONSHIP BETWEEN WRITTEN AND ENACTED CURRICULA:
THE CASE OF FRACTION MULTIPLICATION IN A MIDDLE SCHOOL
S TANDARDS-BASED CURRICULUM
By
Jill Newton
In the 19905, the National Science Foundation (NSF) funded the development of
curricula based on the approach to mathematics proposed in Curriculum and Evaluation
Standards for School Mathematics (National Council of Teachers of Mathematics, 1989).
Controversy over the effectiveness of these curricula and the soundness of the standards
on which they were based, often labeled the “math wars,” prompted a plethora of
evaluative and comparative curricular studies. Critics of these studies called for
mathematics education researchers to document the implementation of these curricula
(e.g., National Research Council, 2004; Senk & Thompson, 2003) because “one cannot
say that a curriculum is or is not associated with a learning outcome unless one can be
reasonably certain that it was implemented as intended by the curriculum developers”
(Stein, Remillard, & Smith, 2007, p. 337). Curriculum researchers have used a variety of
methods for documenting curricular implementation, including table-of—content
implementation records, teacher and student textbook use diaries, teacher and student
interviews, and classroom observations. These methods record teacher and student
beliefs, extent of content coverage, in—class and out-of-class textbook use, and classroom
participation structures, but do little to compare the mathematics presented in the written

curriculum (the student and teacher textbooks) and the way in which this mathematics

plays out in the enacted curriculum (that which happens in classrooms).

In order to compare the mathematical features in the written and enacted
curricula, I utilized Sfard’s Commognition framework (most recently and fully described
in Thinking as Communicating: Human Development, the Growth of discourses. and
Mathematizing published in 2008). That is, I compared the mathematical words, visual
mediators, endorsed narratives, and mathematical routines in the written and enacted
curricula. Each of these mathematical features provided a different perspective on the
mathematics present in the curricula. The written curriculum in this study was
represented by Investigation 3 (Multiplying with Fractions) included in Bits and Pieces
1]: Using Fraction Operations in Connected Mathematics 2 (Lappan, Fey, Fitzgerald,

F riel, & Phillips, 2006). Videotapes of this same Investigation recorded in a sixth grade
classroom in a small, rural town in the Midwest were used as the enacted curricula for
this case.

The study revealed many similarities and differences between the written and
enacted curricula; however, most prominent were the findings regarding objectification in
the curricula. Sfard defines objectification as “a process in which a noun begins to be
used as if it signifies an extradiscursive, self-sustained entity (object), independent of
human agency” (Sfard, 2008, p. 412). She proposes that objectifying is an important
process for students’ discursive development and that it serves them particularly well in
the study of advanced mathematics. Both objectification itself and the opportunities
present for objectification were more prevalent in the written curriculum than in the

enacted curriculum.

 

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ACKNOWLEDGMENTS

I owe a great debt to many individuals without whom I can hardly imagine
completing this dissertation: For your friendship as we traveled this road together and for
your reviews of my work during this project and others that preceded it, thanks to my
CSMC colleagues. For helping me to see myself as a writer and for constant guidance
and intellectual support, thanks to my writing mentor and dear friend, Ann Lawrence.
For friendship, encouragement, laughter, and helping me keep it all in perspective, thanks
to my partners in crime (graduate student colleagues) in Teacher Education and the
Division of Science and Mathematics Education. For guidance and support, thanks to the
faculty of Teacher Education and the Division of Science and Mathematics Education.
For countless favors and generous hugs and encouragement, thanks to Jean Beland,
Margaret Iding, Lisa Keller, Judi Miller, and Becky Murthum. For encouraging me to
pursue a Ph.D. in mathematics education at MSU and providing support along the way,
thanks to Cass Book. For teaching me calculus in 1984 (during my first year as a .
Spartan), providing funding for my Ph.D. 20 years later, and for always taking time to
discuss “big” ideas with me, thanks to Betty Phillips. Finally, for endless patience,
kindness and guidance, I thank my committee members: Joan Ferrini—Mundy, Beth
Herbel-Eisenmann, Glenda Lappan, Anna Sfard, and Jack Smith. Each of you offered
something unique to my project and I have gained so much from my work with you. You
pushed me to think, write, and make claims carefully, and you never let me easily off the

hook. I greatly appreciate your confidence in me — for this I am forever in your debt.

 

"Ty—N

 

TABLE OF CONTENTS

LIST OF TABLES ........................................................................... viii
LIST OF FIGURES .............................................................. '. .......... x
NOTES ON TEXT ........................................................................... xiii
CHAPTER 1
INTRODUCTION ........................................................................... 1
CHAPTER 2
THEORETICAL BACKGROUND ....................................................... 5
Review of Relevant Literature .................................................... 5
The Development of Standards-based Curricula ...................... 5
The Phases of Curricula ................................................... 7
The Complex Path between the Written and the Learned
Curricula ..................................................................... 9
The Importance of Implementation in establishing Curricular
Effectiveness ............................................................... 1 2
Documenting the Implementation of Curricula ........................ 15
The Discursive Nature of the Written and Enacted Curricula ....... 16
Conceptual Framework ............................................................ 20
Commognition ............................................................. 20
Objectification ............................................................. 23
A Step Forward ............................................................ 24
CHAPTER 3
METHODOLOGY .......................................................................... 26
Written Curriculum ................................................................. 26
Enacted Curriculum ................................................................ 27
Fraction Multiplication ............................................................ 29
Design of the Study ................................................................ 32
CHAPTER 4
GOALS OF THE WRITTEN AND ENACTED CURRICULA ...................... 38
Goal 1: “Estimate products of fractions” ....................................... 41
Written Curriculum ........................................................ 41
Enacted Curriculum ........................................................ 44
Goal 2: “Use models to represent the product of two fractions” ............ 46
Written Curriculum ........................................................ 46
Enacted Curriculum ....................................................... 48
Goal 3: “Understand that finding a fraction of a number means
multiplication” ...................................................................... 50

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Written Curriculum ........................................................ 50

Enacted Curriculum ....................................................... 51
Goal 4: “Develop and use strategies and models for multiplying
combinations of fractions, whole number, and mixed number to

solve problems” .................................................................................... 53
Written Curriculum ........................................................ 53
Enacted Curriculum ............................................ ' ........... 56
Goal 5: “Determine when multiplication is an appropriate operation” 60
Written Curriculum ........................................................ 61
Enacted Curriculum ....................................................... 61
Goal 6: “Explore the relationships between two numbers and their
product” .............................................................................. 62
Written Curriculum ........................................................ 62
Enacted Curriculum ....................................................... 64
Goal 7: “Develop and use algorithms for multiplying combinations of
fractions, whole numbers, and mixed numbers” ................................. 66
Written Curriculum ........................................................ 66
Enacted Curriculum ........................................................ 67
Goal 8: “Develop and use an efficient algorithm to solve any fraction
multiplication problem” ............................................................ 68
Written Curriculum ........................................................ 68
Enacted Curriculum ........................................................ 69
Summary .............................................................................. 70
CHAPTER 5
MATHEMATICAL WORDS IN THE WRITTEN AND ENACTED
CURRICULA ............................................................................ 74
Words Signifying Algorithm ...................................................... 80
Words Signifying Estimation ...................................................... 85
Words Signifying Fractions ....................................................... 9]
Uses of “fraction” in the Curricula ...................................... 93
Uses of “piece” and “part” in the Curricula ............................ 96
Uses of “number” in the Curricula ....................................... 98
Fractions in the Curricula ................................................. 99
Words Signifying Multiplication ................................................. 109
Uses of “product” in the Curricula ...................................... 110
Uses of “group” in the Curricula ........................................ 111
Uses of “times” in the Curricula ......................................... 113
Uses of “multiplication” in the Curricula .............................. 115
Uses of “of” in the Curricula ............................................. 118
Summary ............................................................................. 122
CHAPTER 6
VISUAL MEDIATORS IN THE WRITTEN AND ENACTED
CURRICULA ................................................................................. 129
Words Signifying Visual Mediators .............................................. 131

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Words Signifying Concrete Mediators .................................. 134
Words Signifying Iconic Mediators ..................................... 134
Words Signifying Symbolic Mediators ................................. 142
Verbs Associated with Visual Mediators ............................... 142
Use of Visual Mediators ............................................................ 149
Concrete Mediators ........................................................ 152
Iconic Mediators ............................................... ‘ ............. 155
Types of Iconic Mediators ....................................... 155
Use of Iconic Mediators .......................................... 160
Symbolic Mediators ....................................................... 177
Summary ............................................................................. 180
CHAPTER 7
ENDORSED NARRATIVES IN THE WRITTEN AND ENACTED
CURRICULA ................................................................................ 186
Types of Narratives ................................................................. 188
Object-level Narratives ................................................... 1 91
Meta-level Narratives ..................................................... 200
Recalling Narratives ............................................................... 204
Constructing Narratives ........................................................... 205
Construction of Object-level Narratives .............................. 206
Construction of Meta-level Narratives ................................. 216
Substantiating Narratives ......................................................... 218
Summary ............................................................................ 230
CHAPTER 8
MATHEMATICAL ROUTINES IN THE WRITTEN AND ENACTED .
CURRICULA ............................................................................... 238
Leading Words of Questions ..................................................... 242
Elicited Answers to Questions ................................................... 244
Mathematical Processes addressed by Questions .............................. 251
Miscellaneous Questions .......................................................... 255
Summary ............................................................................. 258
CHAPTER 9
DISCUSSION ................................................................................ 262
CHAPTER 10
CONCLUSION .............................................................................. 271
REFERENCES .............................................................................. 275

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LIST OF TABLES

Table 1: Goals of Problems 31-35 in Investigation 3: Multiplying Fractions ...38
Table 2: Investigation 3 Problems in the Written and Enacted Curricula ......... 40
Table 3: Goals of Problems 3.1-3.5 in Investigation 3: Multiplying Fractions

(as stated in the Written Curriculum) with “Multiplication” and

“Fractions” and their Derivatives underlined ............................... 77
Table 4: Goals of Problems 31-35 in Investigation 3: Multiplying Fractions

(as stated in the Written Curriculum) with Key Mathematical Words

and their Derivatives underlined .............................................. 78

Table 5: Examples of the Use of “Algorithm” in the Written and Enacted
Curricula ......................................................................... 81

Table 6: Examples of Uses of “Estimate” and its Derivatives in the Written
and Enacted Curricula ......................................................... 86

Table 7: Examples of Categories of “Fraction” Use in the Written and
Enacted Curricula ............................................................... 93

Table 8: Fractions representing at least 5% of the Total Number of Fractions

in the Written and Enacted Curricula ........................................ . 102
Table 9: Examples of Uses of “group” in the Enacted Curriculum ................. 112
Table 10: Summary of “Mathematical Words” Analysis .............................. 123

Table 11: Examples of General and Specific References to Visual Mediators
in the Written and Enacted Curricula ........................................ 131

Table 12: Examples of General Words referencing Iconic Mediators in the
Written and Enacted Curricula ............................................................. 136
Table 13: Examples of Functions of References to Iconic Mediators in the

Written and Enacted Curricula ................................................ 138

Table 14: Verbs associated with Visual Mediators in the Written and Enacted
Curricula ......................................................................... 143

 

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Table 16:

Table 17:

Table 18:

Table 19:

Table 20:

Table 21:

Table 22:

Table 23:

Table 24:

Table 25:

Table 26:

Examples of Uses of Verbs associated with Visual Mediators in the
Written and Enacted Curricula ................................................ 144

Summary of “Visual Mediators” Analysis .................................. 181

Relative Frequencies of the Types of Numbers involved in
Object-level Narratives in the Written and Enacted Curricula ........... 191

Relative Frequencies of three Forms of Simple Object-level
Narratives ........................................................................ 195

Relative Frequencies of Types of Meta-level Narratives in the
Written and Enacted Curricula ................................................ 201

Summary of “Endorsed Narratives” Analysis .............................. 231

Rankings of Common Question Words in the Written and Enacted
Curricula ........................................................................ 239

Sample Questions eliciting Particular Response Types in the Written
and Enacted Curricula .......................................................... 246

Sample Questions eliciting Particular Actions in the Written and
Enacted Curricula .............................................................. 248

Sample Questions addressing Particular Mathematical Processes
in the Written and Enacted Curricula ........................................ 251

Sample Questions addressing Comparison of Mathematical Methods
in the Written and Enacted Curricula ........................................ 254

Summary of “Mathematical Routines” (i.e., Questioning) Analysis 258

 

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Figure 4:

Figure 5:

Figure 6:

Figure 7:

Figure 8:

Figure 9:

Figure 10:

Figure 11:

Figure 12:

Figure 13:

Figure 14:

Figure 15:

LIST OF FIGURES

Model of four curricular phases ............................................... 8
Model of three curricular phases .................................. ' ........... 9

Model for the analysis of the relationship between the written and
enacted curricula ................................................................. 35

F our-stage model of the development of word use ........................ 74

Relative frequencies of uses of “algorithm(s)” in the written and
enacted curricula ................................................................. 80

Relative frequencies of the parts of speech of “estimate” and its
derivatives in the written and enacted curricula ............................. 88

Relative frequencies of categories of “fraction” use in the written
and enacted curricula ............................................................ 95

Relative frequencies of uses of “number(s)” in the written and
enacted curricula ................................................................. 99

Frequencies of distinct proper and improper fractions and mixed
numbers in the written and enacted curricula ................................ 100

Relative frequencies of the total number of appearances of proper
and improper fractions and mixed numbers in the written and

enacted curricula ................................................................. 101

Relative frequencies of words signifying multiplication in the written
and enacted curricula ............................................................ 110

Relative frequencies of “multiplication” and its derivatives in the
written and enacted curricula .................................................... 1 15

Relative frequencies of the parts of speech of “multiplication” and
its derivatives in the written and enacted curricula ......................... 1 17

Relative frequencies of mathematical uses of “of” in the written and
enacted curricula ................................................................. 119

Relative frequencies of general and specific references to visual
mediators in the written and enacted curricula .............................. 132

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Figure 18:

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Figure 20:

Figure 21:

Figure 22:

Figure 23:

Figure 24:

Figure 25:

Figure 26:

Figure 27:

Figure 28:

Figure 29:

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Relative frequencies of references to concrete, iconic, and symbolic
visual mediators in the written and enacted curricula ...................... 133

Words signifying iconic mediators in the written and enacted
curricula .......................................................................... 135

Relative frequencies of categories of the uses of iconic mediators in
the written and enacted curricula ............................................. 139

Relative frequencies of references to iconic visual mediators in the
written and enacted curricula ................................................... 140

Relative frequencies of verbs associated with visual mediators in the
written and enacted curricula .................................................. 145

Relative frequencies of the types of visual mediators in the written
and enacted curricula ............................................................ 150

Relative frequencies of iconic mediators in the written and enacted
curricula ........................................................................... 156

Types of narratives in the written and enacted curricula ................... 190
Distribution of “Fraction x Fraction” type object-level narratives
across the five days of Investigation 3 in the written and enacted

curricula ........................................................................... 192

Distribution of “Other” type (i.e., all types except Fraction x Fraction)
object-level narratives across the five days of Investigation 3 in the

written and enacted curricula ................................................... 193
Types of object-level narratives in the written and enacted curricula. . .. 194
Distribution of “A x B = C” across the five days of Investigation 3

in the written and enacted curricula ........................................... 196
Distribution of “A of B = C” across the five days of Investigation 3

in the written and enacted curricula ........................................... 196
Distribution of “A x B ~C” across the five days of Investigation 3

in the written and enacted curricula ........................................... 197

Relative frequencies of narratives associated with contextual and
non-contextual Questions in the written and enacted curricula ........... 207

xii

 

 

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Figure 32:

Figure 33:

Figure 34:

Figure 35:

Figure 36:

Figure 37:

Distribution of narratives associated with contextual Questions across
the five days of Investigation 3 in the written and enacted curricula. . 208

Distribution of narratives associated with non-contextual Questions
across the five days of Investigation 3 in the written and enacted

curricula .......................................................................... 209

Relative frequencies of leading words of questions in the written
and enacted curricula. . . . . . . . . .. ................................................ 243

Relative frequencies of questions asking student to “say” something
versus those asking students to “do” something ............................ 245

Relative frequencies of categories of elicited answers to questions
in the written and enacted curricula .......................................... 247

Relative frequencies of “doing” question responses in the written
and enacted curricula ............................................................ 250

Relative frequency of mathematical processes addressed in questions
in the written and enacted curricula .......................................... 253

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NOTES ON THE TEXT

“Unit,” “Investigation,” “Problem,” and “Question” with upper—case first letters reference
the particular components of Connected Mathematics. When these words appear with
lower-case first letters, they are being used colloquially.

WC is used to designate “Written Curriculum” in examples
EC is used to designate “Enacted Curriculum” in examples

Examples from the written curriculum are cited using the following format (TG, p. 23) or
(SO, p. 32) where TG indicates “Teacher’s Guide” and SG indicates “Student’s Guide”

Examples from the enacted curriculum are cited by day (e. g., “Day 4”). In addition, the
following abbreviations are used to indicate the speaker:

T Teacher

S Student (used only in cases in which one identified student speaks)

S1, SZ, etc. Student (used in cases in which at least two identified students
speak)

S? Student (used in cases in which the speaking student can not be

identified because he/she does not appear on the videotape and the

voice is not recognizable)
83 Students (used in cases in which multiple students are speaking, at
least one of which does not appear on the videotape and the voices

are not recognizable)

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CHAPTER 1: INTRODUCTION
The mathematics education research community continues to grapple with the
elusive question: Which mathematics curricula are the most effective for promoting
student learning? While this question has long been of interest nationally, debates around
and intensive study of this question were most recently prompted by a series of three
events beginning in the 19805: (1) the release of national reports (e.g., A Nation at Risk
[National Commission for Excellence in Education, 1983]) which claimed that students
in the United States were not adequately prepared to compete with their counterparts
around the world in mathematics; (2) the publication of Curriculum and Evaluation
Standards for School Mathematics (1989), Professional Standards for Teaching
Mathematics (1991), and Assessment Standardsfor School Mathematics (1995) by the
National Council of Teachers of Mathematics (N CTM); and (3) the National Science
FOllndation’s (NSF) funding of the development of 15 elementary, middle, and high
School mathematics curricula which were stipulated to use NCTM’s standards documents
as the foundation for their approach to mathematics teaching and learning in the early
1 9903 (hereafter referred to as “standards-based curricula”).
The standards documents and the standards-based curricula promoted a new
Vision of what it meant to teach and learn mathematics. That is, NCTM advocated for a
l"rlore participatory form of mathematics education that valued a wider range of
mathematical skills, including not only traditional computational procedures but also the
e()l’l’lmunication, problem solving, and reasoning required to engage collaboratively with
Qther students on rich mathematical tasks. These curricula and the debates surrounding

them have stimulated research studies from professional organizations (e. g., American

 

 

 

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Association for the Advancement of Science, 2000) as well as individual researchers
(e.g., Webb & Dowling, 1997; Wu, 2000). One of the most significant challenges in the
attempt to study these new curricula has been measuring the “quality” and “quantity” of
their implementation in classrooms. The question is whether and to what extent the
materials are actually being used in a particular classroom that is part of a study. Are the
materials to some degree faithful to the intent of the authors of the particular curriculum
in question? Can a curriculum be assigned credit or blame for student achievement
without a careful investigation of the ways in which the curriculum is enacted in the
classroom?

This phenomenon prompted the call for the study of the implementation of
SIandards—based curricula as a precursor to making claims regarding the effectiveness of
these curricula (e.g., Senk & Thompson, 2003; National Research Council, 2004). In
Other words, learning outcomes of students in classrooms in which standards-based
cul‘ricula are “used” cannot be attributed to the particular curriculum in use unless some
degree of fidelity to the curriculum has been established. Studies of standards-based
cl-lrricula have documented their implementation in many ways, including stating that

“the teacher used the curriculum,” interviewing teachers regarding their use of curricular
l'l'laterials, asking teachers to keep curriculum logs (i.e., recording when they use the
ellI‘riculum), and conducting classroom observations detailing features of the environment
c"filled for in NCTM’s standards documents (e.g., cooperative learning, questioning
te(ll’miques) (e.g., Post et al., 2008; Tarr et al., 2008). These methods provide important

i Ilformation; however, they do not examine the details of the mathematics as it is

presented in standards-based curricula or classrooms using these curricula. This study,

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which compares the mathematical features in the written and enacted curricula. is a step
in this direction.

Given the discursive nature of both the written and enacted curricula (i.e., both
involve “talk” in different modes), discourse analysis in the form of the‘Commognition
framework described in Thinking as Communicating: Human Development, the Growth
of Discourses, and Mathematizing (Sfard, 2008) is utilized to investigate the relationship
between the written and the enacted curricula.l C ommognition requires a careful

examination of key mathematical features, including (1) use of mathematical words, (2)

use of uniquely mathematical visual mediators in the form of symbolic artifacts that have

been created specifically for the purpose of communicating about quantities, (3) special
mathematical routines with which the participants implement well-defined types of tasks,
and (4) endorsed narratives, such as definitions, postulates, and theorems produced
throughout the discursive activity. The Commognition framework provides a new lens
through which to compare the relationship between the mathematical features of the
Written and enacted curricula.

In Chapter 2, I position this project among studies addressing curriculum in
mathematics education in general and studies that examine the relationship between the
Written and enacted curricula in particular. In addition, I describe the Commognition
fr"=11‘nework in detail including a description of each of its four key mathematical features

and important phenomena highlighted in this framework, including objectification.

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{3 Ithough this framework has been described and used in other publications (e.g., Ben-Yehuda, Lavy.
th‘f‘chevski, & Sfard, 2005; Sfard, 2007). Sfard (2008) will be used as the primary reference throughout
Q '3 Study because it represents Sfard’s most elaborated rationale for and detailed description of

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Chapter 3 describes the methodology of the study, including depictions of the written and
enacted curricula, fraction multiplication, and an outline of the design of the study.
Chapters 4-8 contain the results of the analyses conducted in this study along with
interpretations of the findings. In Chapter 4, the stated goals of the written curriculum
are examined in both the written and enacted curricula in an effort to compare the
opportunities associated with each goal in the curricula. The key mathematical features
associated with Commognition (i.e., mathematical words, visual mediators, endorsed
narratives, and mathematical routines) are compared in the written and enacted curricula
in Chapters 5-8 respectively. These analyses (Chapter 4-8) provide a detailed picture of
the relationship between the written and enacted curricula using a Commognitive lens.
Finally, Chapter 9 provides a discussion of the findings across the five analyses
highlighting the threads that run throughout, and Chapter 10 summarizes the study’s

cOntributions to the field, its limitations, and suggestions for future action.

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CHAPTER 2: THEORETICAL BACKGROUND
Review of Relevant Literature
The Development of Standards-based Curricula
During the 19805, several national reports (e.g., A Nation at Risk [National
Commission for Excellence in Education, 1983], Educating Americans for the Twenty—
first Century [National Science Board Commission on Pre-College Education in
Mathematics, Science, and Technology, 1983]) and results from the Second International
Mathematics Study (SIMS) suggested that students from the United States were lagging
behind students from other developed countries in mathematics. These reports indicated
that this deficit was a threat, not only to the everyday functioning of US. schools and
businesses, but to our national security as well. This was certainly not the first
mathematics education crisis in the United States. In fact, Fey and Graeber (2003)
Characterize the direction of curricula and teaching in elementary and secondary school
mathematics as having a “predictable rhythm of crisis-reform-reaction episodes” (p. 521).
However, it was the “crisis-reform—reaction episode” catalyzed by these reports in the

1 9805 that is of particular interest here. These reports can be said to have communicated
the “crisis” of the episode.

The episode’s “reform” was largely taken up by the National Council of Teachers
of Mathematics (N CTM) beginning with the publication of An Agenda for Action:

R ecommendations for School Mathematics of the 19805 in 1980 that outlined eight
reCommendations for improvement in school mathematics, including a call to emphasize

problem solving, expand the meaning of “basic skills,” and encourage the use of

CalCulators and computers at all grade levels. As both a response to the reports

 

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mentioned earlier and in an attempt to expand the recommendations proposed in An
Agenda for Action, NCTM initiated the development of a set of standards for
mathematics education in 1986. A series of landmark publications resulted: Curriculum
and Evaluation Standards for School Mathematics (1989), Professional Standardsfor
Teaching Mathematics (1991), and Assessment Standardsfor School Mathematics
(1995).2
A new approach to teaching school mathematics was put forth in these standards
publications. This approach challenged the transmission model of the learning-teaching
process which was thought to be prevalent at the time in mathematics education and
Suggested a more participatory form of learning. The standards emphasized that “the
discourse of a classroom - the ways of representing, thinking, talking, agreeing and
di sagreeing - is central to what students learn about mathematics as a domain of human
inquiry with characteristic ways of knowing” (NCTM, 1991, p. 34). This new
conceptualization of mathematics education offered multiple discursive entry points for
Students which addressed a key goal of the standards: to make mathematics meaningful
for all students. That is, mathematics education was expanded to value more than the
SIDeed and accuracy with which a student could state basic arithmetic facts or manipulate

algebraic symbols. This new approach also valued mathematical skills such as

Communication, reasoning, and problem solving.

The national reports of the 19805 and NCTM’s publication of Curriculum and

Evaluation Standards for School Mathematics (1989) prompted the National Science

FOUndation (NSF) to fund the development of 15 mathematics curricula with the goal of

2\
A r"ore recent standards document, Principles and Standards for School Mathematics, was published by

M in 2000. This newer version advocates for many of the same tenets found in the original standards

 

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alignment with NCTM’s standards documents.3 These “standards-based” mathematics
curricula included development at the elementary, middle, and high school levels.
Following the development of these curricula, the problem of proving curricular
effectiveness was brought to the fore in mathematics education. This demand for proof
of the curricular effectiveness of standards-based curricula can be conceptualized as the
“reaction” in this crisis-reform-reaction episode.

It is not surprising that the burden of proof for curricular effectiveness seemed to
rest on those involved in the development of standards-based curricula because it is
common for the developers of a new product to have the responsibility of convincing the
public that their new product is better than the older, more familiar one.4 Resources were
granted to the authors to carry out small formative and summative evaluations. These
evaluations were promising, but not convincing to critics. It became clear that providing
the necessary evidence to justify a change to new and significantly different mathematics

curricula was not going to be without its challenges (Kilpatrick, 2003; Romberg, 1992).
In fact, the very nature of standards-based curricula complexifies the process of proving
their effectiveness (Stein, Remillard, & Smith, 2007).
The Phases of Curricula
The factors contributing to the complex problem of proving curricular
effectiveness of standards-based curricula are most easily conveyed through the use of a

rnoCiel of curricular phases. Figure 1 provides one such model.5 Clearly, this

3\\

4 N SF has a history of funding innovative mathematics and science curricula.

o f 0 clarify, “standards-based” mathematics curricula are those funded by NSF following the publication

to NCT M’s 1989 standards and “traditional” curricula are those with editions that were on the market prior
‘ e development of NCT M’s 1989 standards. These categorizations are used in Stein, Remillard, and

s m‘th (2007) except that they use “conventional” instead of “traditional.”

th t Should be noted that there are many versions of this model; this was used because it seemed to best suit
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Figure I . Model of four curricular phases.
representation is a simplified version of the phases of curricula. However, it provides
ways to talk about curriculum as it is translated from the minds of the authors (the
intended curriculum) to that which is learned by students and teachers (the learned
curriculum).6 Here, the written curriculum is represented by the textbooks (both teacher
and student editions) and any ancillary materials provided by the publisher. The enacted
curriculum is the curriculum as it plays out in classrooms with teachers and students.7
This model can be used to frame the problem of proving curricular effectiveness
of standards-based curricula. The goal of the author(s) of any curriculum, regardless of
whether it is traditional, standards-based, or a hybrid of these approaches, is that the
intended curriculum (i.e., their vision) becomes the learned curriculum.8 However, the
intended curriculum, in most cases, is only accessible to the users of the curriculum by
proxy of the written curriculum. Therefore, the intended curriculum will not be discussed
I:UI'ther here. This decision, although it makes the model even less representative of the
cOlirlplex curricular process, leaves a version of the model focused on the key elements
f 01' this discussion. Figure 2 provides a representation of this revised model.
6\
Ofien the learned curriculum is reserved for student learning; however, given the breadth of research

teevoted to teacher learning through curriculum use it seems appropriate to include both students and

7 achers in this definition.

th fien the enacted curriculum is used to describe the work of teachers; however, this study conceptualizes

8 e enacted curriculum as being co-constructed by teachers and students in classrooms.

St have included the notion of a hybrid of the two approaches because since the publication of the
an\dards in 1989 there have been efforts on the part of some curriculum developers to balance these
lblbtoaches in their curricula.

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What is it that is represented in the model that makes proving curricular
effectiveness of standards-based curricula so complex? The authors’ goal in this new
revised framework would be that the written curriculum becomes the learned curriculum.
This leaves the enacted curriculum, along with the transitions between these forms of
curricula (represented by the arrows in the model which are arguably at least bi-
directional) as the paths or the obstacles to the authors’ goal. The complex nature of
these three components (i.e., the transition between the written and the enacted curricula,
the enacted curriculum itself, and the transition between the enacted and the learned
curricula) will be described briefly here. Two arguments will be made: (1) These
components are complex in nature regardless of which curricula are studied and (2) The
nature of standards-based curricula makes these components even more complex.

The Complex Path between the Written and the Learned Curricula

First, the transition between the written and enacted curricula is most often
addressed in the literature as the ways in which teachers make sense of the written
curriculum. That is, the teacher is seen as the agent who facilitates this transition.
Remillard (2005), in a review of the literature addressing teachers’ use of curriculum,
categorized those studies according to the way in which the researchers framed teachers’
curricular use: (1) Curriculum use as following or subverting the text, (2) Curriculum use

as drawing on the text, (3) Curriculum use as interpretation of the text, and (4)

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Curriculum use as participation with the text. The multiple perspectives on the ways in
which teachers’ use of curriculum is framed in research indicate the complex nature of
this transition in the process. In addition, if the notion of this meaning making is
extended to students because they also produce their own meaning from the written
instructional materials, the transition becomes even more complex. This transition is
complex regardless of the type of curriculum involved; however, Remillard (2005)
stresses the increased complexity with standards-based curricula given that the new
written materials are “foreign in form” to many teachers and students (p. 212).

Second, the enacted curriculum, defined here as the curriculum co-constructed by
teachers and students in classrooms, is equally complex in nature. Lampert (2001)
developed an “elaborated model of teaching practice” in which she began with the
interaction between the teacher, an individual student, and the mathematical content, and
expanded that three-dimensional model over a class of 20-30 students. She added the
teacher’s interaction with small groups and the whole class, student-student interactions,
and located all of these interactions within mathematical content. In addition, she pointed
out how both the temporal and social characteristics inherent in the classroom context
further increase the complexity because the relationships between students, teachers, and
content “have a history and project into future encounters” (p. 425). This “elaborated
model” provides a compelling representation of the complex nature of the enacted
curriculum. In standards-based enacted curricula, this model is potentially more
elaborate given the different roles of both teachers and students proposed in NCTM’s
standards. Remillard (2005) substantiates this saying, “They [standards-based curricula]

require the teacher to play a substantially different role in the mathematics classroom

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than has been typical among teachers in the United States” (p. 21 1). Similarly, the role of
students in classrooms using standards-based curricula requires a shift in the students‘
role from that of passive recipient to active participant (NCTM, 1989, 2000). Sfard
(2007) reinforces this message regarding the complexity of standards-based enacted
curricula, saying “Our helplessness as researchers is aggravated by the fact that the
current reform, promoting the pedagogy of talking classrooms and of communities of
inquiry, makes learning processes not only more visible, but also much more intricate and
messy” (p. 568).

Finally, the transition between the enacted curriculum and the learned curriculum
is knotty as well, particularly because this project frames both of these phases of curricula
in terms of teachers and students alike. That is, teachers and students co-construct the
enacted curriculum, and both teachers and students are learners in the curricular process.
The ways in which the transition from enacted to learned curriculum is conceptualized
depends largely upon the learning theory adopted by the individual responsible for the
conceptualization. Sfard (1998) problematizes the dichotomous relationship often
assigned to the acquisitionist and participationist theories of learning and discusses the
affordances and limitations of each theory as well as the dangers in developing too great
a devotion to one of them. This leads to the conclusion that the variety of available
learning theories and the relationships among these theories contributes to the complex
nature of this final transition in the model in Figure 2.

In summary, the complex nature of these three components of the curricular
model contributes to the difficulty of evaluating curricular effectiveness of standards-

based curricula. Although all three of these components are of great importance, this

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project focuses primarily on the first two (i.e., the transition from the written to the
enacted curriculum and the enacted curriculum itself). The study of the final transition
(i.e., the transition from the enacted curriculum to the learned curriculum) is particularly
tenuous currently because the development of outcome measures for the foci of
standards-based curricula (e.g., mathematical reasoning, communication) are not well
developed in the field.

The Importance of Implementation in Establishing Curricular Effectiveness

Curricular implementation, as it will be used here, combines the two components
addressed previously.9 It includes both the ways in which teachers and students make
sense of the written curriculum as well as the ways in which they enact these curricula in
mathematics classrooms. An alternate conceptualization is that implementation “turns
on” the relationship between the written and enacted curricula. The importance of the
documentation of this implementation in curricular studies is well documented in recent
literature.

Senk and Thompson (2003) edited a collection of studies that addressed the
effectiveness of standards-based curricula. They were interested in the simple question,
“How well do these standards-based instructional materials work?” (p. x). Early in the
volume, the editors state the requirement that the researchers address the “fidelity of
treatment” which they define as the “extent to which a curriculum is used in the way it
was intended” (p. 20). That is, in order to associate achievement with a particular

curriculum there must be a positive relationship between the intended curriculum (by

 

9 I don’t actually use “implementation” in this study; rather I use “the relationship between the written and
enacted curricula.” However, I use “implementation” in the literature review because it is the term most
often used in the field. Again, my choice not to use “implementation” comes from the fact that it has been
used most often to refer to the teacher’s agency (teacher as implementer), whereas I am conceptualizing
both teachers and students as agents in curricular implementation in the classroom.

 

proxy of the written curriculum) and the enacted curriculum. The book contains sections
committed to evaluations of standards-based elementary, middle, and high school
curriculum projects. The need to document curricular implementation is echoed in each
of these sections as well as in the volume’s final conclusion.

First, in his summary of the elementary curriculum projects, Putnam (2003) states
that “It is important to attend to such implementation issues, because without information
about how curricula are being implemented, it is difficult to know what is being
compared in student outcome studies” (p. 170). Chappell (2003), in her reaction to the
middle school curriculum projects in the same volume reiterated this concern saying,
“One shortcoming pertains to monitoring how the curriculum is actually implemented in
the classrooms” (p. 292). Finally, Swafford (2003), in her reflections on the high school
curriculum projects added that, “Future research on the impact under routine
implementation also has to take into account the degree or faithfulness of the
implementation and the relationship of the degree to which the materials are implemented
to student outcomes” (p. 467). Kilpatrick (2003), in the final conclusion of this volume,
summarized the findings of the evaluations:

Students studying from standards-based curricula do as well as students

studying from traditional curricula on standardized mathematics tests and

other measures of traditional content. They score higher than those who

have studied from traditional curricula on tests of newer content and

processes highlighted in the Standards document. (p. 472)

He called the evidence “promising and substantial,” but went on to point to the

difficulties in “evaluating something as complex as curriculum” (p. 472). He emphasized

l3

 

that “two classrooms in which the same curriculum is supposedly being ‘implemented’
may look very different” (p. 473).

Similar conclusions regarding the importance ofconsidering curricular
implementation were drawn in the National Research Council’s 2004 report, 0n
Evaluating Curricular Effectiveness: Judging the Quality of K-1 2 Mathematics
Evaluations. The committee’s charge was twofold:

First we aim to examine evidence currently available from the evaluation

of effectiveness of mathematics curricula. Second, we will suggest ways

to improve the evaluation process that will enhance the quality and

usefulness of evaluations and help guide curriculum developers and

evaluators in conducting better studies. (p. 15)

The major finding reported was that due to the number of and nature of the studies for
any one given curriculum it was impossible to determine the effectiveness of individual
programs. The report’s major contributions to the mathematics education community
included a detailed outline of the required evaluations needed to scientifically establish
the effectiveness of a curriculum program and three essential components to be addressed
in the evaluations. The second of these components is of primary importance here:
“Evaluations should present evidence that provides reliable and valid indicators of the
extent, quality, and type of the implementation of the materials” (p. 194).

These two documents call for careful consideration of curricular implementation
for “one cannot say that a curriculum is or is not associated with a learning outcome
unless one can be reasonably certain that it was implemented as intended by the

curriculum developers” (Stein, Remillard. & Smith, 2007, p. 337).

 

Documenting the Implementation of Curricula

The studies described in Senk and Thompson (2003) account for curricular
implementation in a variety of ways and to varying degrees. Interestingly, five of the 12
chapters (descriptions of studies of individual curriculum projects) say only that the
curriculum was being “used” or that the students were “taking the course” that utilized
the curriculum. They went no further in accounting for the implementation of the
curriculum in the classrooms.'0 Two of the remaining seven chapters also mentioned that
the teachers in the study received training in the use of the curriculum.

The remaining five chapters include a more thorough description of the extent and
nature of the implementation. Of these five chapters, three include some measure of the
content coverage. That is, they document (through surveys or interviews) the specific
chapters which the teacher covered during the course. Four of the five chapters include
classroom observations in their data collection methods. The purposes of these
observations were “to establish that instruction was aligned with the program being
taught” (p. 123) and “to provide a systematic basis for comparisons across classrooms”
(p. 363). In the former case, the chapter does not provide a description of what it means
to be “aligned” with the program. In the latter case, an observation instrument which
provided a rating system was included as an appendix (pp. 372-3 73). The instrument
addressed five major classroom components: (1) Structure, (2) Cooperative Student
Learning, (3) Types of Thinking, (4) Gender Interaction, and (5) Other Striking Features.

Similar methods of documenting curricular implementation have continued to be

used more recently. For example, Tarr, Chavez, Reys, and Reys (2006), in their study of

 

'0 It is possible that the original studies addressed implementation more thoroughly and that this
information was not included in the book due to space constraints.

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the impact of middle school teachers on students’ opportunities to learn, reported use of
teacher surveys (asking questions about education, teaching experience, beliefs about
teaching and learning mathematics, and practices regarding textbooks), textbook-use
diaries (teachers’ reports of their use of the textbooks and other curricular materials),
classroom observations (documenting curricular use), teacher interviews (focusing on
what and how to teach mathematics and the extent to which their textbook influenced
these decisions), and table-of-content implementation records (recording the amount of
the textbook “covered” during the school year). Using similar methodologies, Tarr et al.,
(2008) investigated textbook fidelity and its relationship to student outcomes by
documenting content coverage and the amount of time spent on specific topics. In
addition, they used classroom observations to describe the extent to which the classroom
activities reflected a “standards-based learning environment” (SBLE).

Although these methods for documenting curricular implementation provide
useful information, one is left wondering about the mathematics taking place in the
classrooms and the alignment between the mathematics highlighted in the written
curriculum (both student and teacher textbooks) and the mathematics enacted in the
classrooms. In this study, I utilize a discursive framework to document the curricular
implementation by carefully exploring the mathematical features of both the written and
enacted curricula and describing the relationship between them.

The Discursive Nature of the Written and Enacted Curricula

Discourse has been defined in many ways, ranging from notions as simple as

“language” or “talk” to much broader conceptualizations. For example, Fairclough

(1992) wrote, “Discourse is, for me, more than just language use: it is language use,

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whether speech or writing, seen as a type of social practice” (p. 28). Similarly, Jaworski
and Coupland (1999) propose an expanded notion of discourse as, “language use relative
to social, political and cultural formations — it is language reflecting social order but also
language shaping social order, and shaping individuals’ interaction With society” (p. 3).
For each of these definitions, the method of discourse analysis takes a different form. For
example, Fairclough, based on his broad conceptualization of discourse developed a
three-dimensional framework for studying discourse which mapped three separate forms
of analysis onto one another. These forms of analysis included: (1) analysis of language
texts, (2) analysis of discourse practice, and (3) analysis of discursive events as instances
of social practice. Using this framework, he considers three levels of “language in use”:
details of the language itself, language patterns, and the implications of language for
broader social contexts.

Regardless of whether the simple definition of discourse as “talk” or a more
expanded notion such as those of F airclough or Jaworski and Coupland is adopted, both
the written and enacted curricula can be conceptualized as discursive constructs. The
written curriculum contains “talk” in the form of written text as well as messages
regarding social practices. For example, if a textbook always uses the pronoun he and
never she, this may carry a message to the reader regarding who can do mathematics.
Similarly, the enacted curriculum (that which occurs in classrooms) certainly contains
“talk” as well as carries messages regarding social order. An example of this may come
from something as simple as which students are chosen to answer the most complex

problems. Additionally, the enacted curriculum can shape an individual’s interaction

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with society either by reinforcing the messages that students are receiving from outside
the classroom or by providing alternative messages for them to carry into the world.

Treating the enactment of mathematics curriculum as discursive is certainly not
without precedent. Halliday (1978) introduced the notion of the “mathematical register”
to highlight the ways in which some new and some everyday words are used in particular
ways in mathematical discourse. It is through the use of these words in conventional
ways that one is said to be participating in mathematical discourse. In related work,
Pimm (1987) proposed that mathematics might be construed as a language. He stated
that “part of learning mathematics is learning to speak like a mathematician, that is,
acquiring control over the mathematics register” (p. 76). He suggested three purposes for
increasing student communication in mathematics classrooms:

Within the educational context of a mathematics classroom there are two

main reasons for pupils talking, namely talking to communicate with

others and talking for themselves. There is also a further justification, »

namely for the teacher to gain access to and insight into the ways of

thinking of the pupils. (p. 23)
In addition, Pierre van Hiele, in his model of geometric thinking proposed in 1957,
posited that each level of geometric thinking has its own language and symbols (i.e., its
own discourse). For example, a “square” may or may not also be a “rectangle”
depending on a students’ level of geometric thinking. Finally, although the research of
Lemke (1990) was conducted in the context of science education, he suggests that it is
applicable to all scientific and technical subjects, and arguably therefore to mathematics.

He analyzed science classroom discourse and identified discursive patterns used to

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communicate scientific content. Halliday, Pimm, and van Hiele all talk about the
difficulties inherent in mathematical communication (and Lemke the difficulties inherent
in communication in science classrooms) as justification for the need for its careful study.
Similarly, one need not look far in order to access examples in which mathematics
curricula in written form has been treated as a discursive construct. A recent example is
found in Herbel-Eisenmann’s 2007 article, From Intended Curriculum to Written
Curriculum: Examining the “ Voice of a Mathematics Textbook. Here, she investigated
the roles of the authors and readers of a mathematics curriculum by examining particular
language forms and concluded that shifting authority toward students’ reasoning is
difficult given the predominant tradition of authoritative “voice” in mathematics
textbooks. Danielson (2005) also analyzed written curriculum materials as discursive
constructs. He compared the communicational characteristics of two middle school
mathematics textbooks, highlighting the distinctive features of each and finds, among
other things, that tables, graphs, and equations serve one another in one curriculum
whereas tables and graphs are in service of equations in the other curriculum. In a final
example, Morgan (2005) investigated the notion of definition in secondary mathematics
textbooks using systemic functional linguistic tools and found that “the ways in which
definitions appear in school mathematics texts vary significantly with the type of
mathematics involved and with the age of the intended student-readers” (p. 111).
Therefore, for the purposes of this project, the written and enacted curricula are
conceptualized and analyzed as discursive constructs. Sfard’s (2008) Commognition

framework, with its focus on mathematical features, is utilized to examine the discursive

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relationship between the mathematical features of the written and enacted standards-
based curricula.ll
Conceptual Framework
C ommognition
Commognition (created by merging communication and cognition) treats
communication (interpersonal exchange) and cognition (intrapersonal exchange) as two
forms of the same phenomenon. It was developed to emphasize the close relationship
between these two processes. In the spirit of Lave and Wenger (1991), commognition
recognizes learning as occurring through legitimate peripheral participation. That is,
learning is participatory in nature and requires social interaction, particularly with
“oldtimers” (i.e., those with full membership) in the discourse community. The
framework recognizes a broad definition of discourse (e. g., includes non-verbal gestures);
however, it identifies linguistic commognition as the primary source of human
uniqueness. When using the commognition frame, careful attention is paid to defining
terms; it is seen as a matter of conceptual accountability as well as an important step to
creating common language to talk about learning. In that spirit, the Commognition

framework proposes the following: (1) mathematics is the discourse about mathematical

objects and (2) learning mathematics is a change in participation in mathematical

discourse. '2

Commognition suggests a detailed analysis (i.e., a search for patterns) of the use

of discursive features of mathematics, including (1) Mathematical Words, (2) Visual

 

11 . . . ,, . . . .

In this pr0ject, “curricula 15 used to refer to the written curriculum and the enacted currlculum.

It 15 noteworthy that mathematical objects (e.g., numbers, shapes) are discursive objects themselves and
therefore part of the discourse.

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interest in an analysis of mathematical discourse are primarily those that signify
quantities and shapes (e.g., number, triangle) and those that highlight relationships
between these quantities and shapes. Of particular interest are the Ways in which the
words are used. The Commognition framework proposes four phases in the development
of word use, including (1) passive use, (2) routine-driven use, (3) phrase-driven use, and
(4) object-driven use in which the word is used as a noun as if it has a life of its own.

Second, visual mediators are artifacts created for the primary purpose of
mathematical communication, including but not limited to algebraic symbols, diagrams,
and graphical representations. These visual mediators may be conventional or
individually designed. They may be drawn on a chalkboard, built from toothpicks, or
operated on in the mind. The power of these objects, however, does not reside in the
objects themselves. Rather, their power is created as they are used in classrooms and
discursively linked to other related mathematical experiences. Sfard (2008) categorizes
visual mediators into concrete (e. g., fraction strips, geoboards), iconic (e. g., pictures,
graphs), and symbolic (e. g., arithmetic number sentences, algebraic expressions).

Third, narratives include any text, spoken or written, which is framed as a
description of objects, of relations between objects or processes with or by objects, and
which is subject to endorsement or rejection (i.e., being labeled true or false).
Definitions, axioms, theorems, and proofs are commonly endorsed narratives in
mathematics. Endorsed narratives hold a special place in mathematics because “the

overall goal of mathematizing is to produce narratives that can be endorsed, labeled as

 

'3 The mathematical features of commognition are described briefly here and more fully in Chapters 5-8
respectively.

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true, and become known as mathematical facts" (Sfard, 2008, p. 289). Sfard argues that
much of mathematical discourse consists of recalling. constructing, and substantiating
narratives. Recalling entails bringing a previously-constructed narrative to mind,
constructing results in a new narrative, and substantiating is the process that leads to the
decision whether or not to endorse a narrative.

Finally, routines are repetitive characteristics of mathematical discourse. Both the
how and when of routines are important in mathematics. For example, knowing when to
find a common denominator is every bit as important as knowing how to find one. Sfard
(2008) argues that the emphasis in school mathematics is too often on the how to the
virtual exclusion of the when of routines. Closely related to the when is the why of
routines. That is, understanding why a routine works is fundamental to assessing a
situation in order to decide whether or not the routine is appropriate in a particular
context. Mathematical routines are divided into three categories: (1) explorations, (2)
deeds, and (3) rituals. Explorations are those routines whose goal is the creation of
endorsed narratives about mathematical objects. Rituals and deeds are developmental
predecessors to explorations. Sfard (2008) stresses that, “As long as school teaching
focuses on the issue of how routines should be performed to almost total neglect of the
question of when this performance would be most appropriate, it is more likely to result
in the discourse of rituals than of explorations” (p. 289).

It should be noted that the four categories of mathematical features (mathematical
words, visual mediators, endorsed narratives, and mathematical routines) are intricately
related to one another. That is, a visual mediator (e. g., area model) has particular words

associated with it (e.g., part, piece, fraction), is used in routine ways (e.g., partitioned),

22

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and can be described by a narrative (e.g..

four mathematical features in the written and enacted curricula provides new ways to talk
about curricular implementation that highlight the mathematics in standards-based
classrooms.
Objecti/ication

One phenomenon that is particularly sensitive to this type of analysis is
objectification, which will be discussed here in some detail because it is a major factor in
the analysis of these curricula. Objectification is defined as “a process in which a noun
begins to be used as if it signifies an extradiscursive, self-sustained entity (object),
independent of human agency” (Sfard, 2008, p. 412). The process consists of two closely
related sub-processes: reification and alienation. Reification and alienation are defined
as follows:

0 Reification: Replacement of talk about processes with talk about objects

0 Alienation: Using discursive forms that present phenomena in an impersonal

way, as if they were occurring of themselves, without the participation of
human beings

These two processes, when taken together indicate that what was previously something to
“do” (processual) becomes a discursive “object” (structural). Mathematical and
scientific discourses are particularly dependent upon objectification for their successful
evolution.

The case of whole numbers illustrates this importance. Children first approach
whole numbers processually; whole numbers are the result of counting sets of objects. If

a young child is asked how many marbles are in a bag, he will probably count them and

his answer will be the last number he states in his counting. Later, he may use numbers

23

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as adjectives; he may state that he has “four” marbles. It becomes more probable that he
has “objectified” whole numbers when he responds “four” to the same question. In this
case, it is possible that he is using “four” as a noun, an object in its own right.'4 This
process is important because it is only once a number is used as a noun (i.e., is
objectified) that operating on it outside of contextual situations makes sense. Such
objectification facilitates an individual’s mathematical communication, and therefore,
their learning of mathematics as defined here.

It is not easy to detect objectification in mathematics, but several discursive clues
provide evidence. First is the use of “is” instead of “are” (i.e., a singular verb instead of a
plural verb). For example, “four is greater than three” would indicate that “four” is being
used as the noun (and therefore objectified). In contrast, “four are more than three” may
indicate that the child is still imagining four objects (“four as an adjective”). In this case
it is likely that the number has not been encapsulated. Encapsulation is a process
associated with objectification in which objects previously seen as separate entities are
encapsulated into one object. Here, what used to be four objects (i.e., marbles) for the
child are now encapsulated into a single object, the number four. Again, this is evidenced
by a change in verb use from plural to singular. Objectification was investigated in
association with each of the four mathematical features in this study.

A Step Forward

This investigation does not take the position that standards-based curricula are

more effective than traditional curricula. Rather, a standards-based curriculum will be

used in the study because as noted earlier, the construct of curricular implementation is

 

'4 It is important to note here that this use of“object” is not Platonic in nature. That is, the implication is
not that the child is discovering a previously existing object. Rather that the student speaks about the
number as if it is an object. Therefore, mathematical objects are theoretical constructs and nothing more.

24

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particularly problematic for researchers who seek to study the effectiveness of these
curricula. The question of which curricula provide the best learning opportunities for
students remains unanswered, although many in the mathematics education community
and beyond have strong feelings in one direction or the other (i.e., standards-based
curricula or traditional curricula). This study provides a step in the process of answering
that question by examining a new approach to investigate curricular implementation
using a framework that aligns with the theoretical assumptions of the standards-based
curricula themselves; that is, mathematics learning is participatory and therefore
discursive in nature. This framework is promising as a means to study the
implementation of standards-based curricula because its foundations are consistent with
that of NCTM’s standards (1989, 2000). It allows us to move beyond the structural
features of classrooms and the content coverage described in teacher surveys used
traditionally to describe curricular implementation to using mathematically discursive
features (i.e., mathematical words, visual mediators, endorsed narratives, and ‘
mathematical routines) of both the written and enacted curricula to provide a detailed
look at the ways in which the mathematics is addressed both in the written text materials
and in the classroom.

This leads to the question guiding this study: What does an investigation of the
key features of mathematical discourse, using the Commognition framework, in the

written and enacted curricula reveal?

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CHAPTER 3: METHODOLOGY

The purpose of this project is to explore discourse analysis as a tool to describe
the relationship between written and enacted standards-based mathematics curricula. To
this end, I made use of two primary data sources, (1) the written version of a standards-
based mathematics curriculum and (2) an enacted version of the same standards-based
mathematics curriculum. I describe these sources in greater detail here followed by the
methods used to analyze these data sources.

Written Curriculum

The Connected Mathematics Project (hereafter referred to as Connected
Mathematics) (Lappan, Fey, Fitzgerald, F riel, & Phillips, 2006) was selected as the
standards-based curriculum in this project because in the late 19905 it was named as one
of five “exemplary” mathematics curricula by the US. Department of Education and was
ranked first in a curricular evaluation of 13 middle school curricula by the American
Association for the Advancement of Science (AAAS). The curriculum has also achieved
substantial penetration of the middle school textbook market nationally. Connected
Mathematics provides 24 Units for grades 6-8 (8 units at each grade level).15 Each Unit
is composed of a series of Investigations, each Investigation contains a sequence of
Problems, and each Problem consists of a set of Questions.

Investigation 3, Multiplying with Fractions, included in Bits and Pieces 1]: Using
Fraction Operations (the second of three Units in grade 6 that address fractions) was

utilized in this study. This Investigation (pp. 32-47 in the Student’s Guide and pp. 59-88

 

'5 Note that “Unit” here has an upper-case “U” 50 as not to be confused with the colloquial use of unit (this
will appear with a lower case “u”). The same treatment will be given to “Investigation,” “Problem,” and
“Question.” That is, when they are used in reference to particular components of the written curriculum,
they will be capitalized. When their use is colloquial, they will not be capitalized.

26

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in the Teacher’s Guide) contains a sequence of five Problems and has a “suggested
pacing” of 5.5 days.16 The Connected Mathematics instructional model consists of three
major components, (1) Launch, (2) Explore, and (3) Summary. The authors propose that
each lesson begins with a problem that introduces the topic to the whole class (i.e.,
“Launch”), followed by time for students to work on a series of problems individually or
in small groups (i.e., “Explore”), and concludes with a whole class discussion in which
the teacher guides the students to reach the mathematical goals of the problem and
connect their new understanding to prior mathematical experiences and knowledge (i.e.,
“Summary”). In addition, “Applications,” “Connections,” and “Extensions” (ACE) are
included at the end of each Investigation. For the purposes of this study, only the five
Problems in the Investigation, Problems 31-35, were included in the analysis (i.e., ACE
was not considered for this analysis).
Enacted Curriculum

I utilized videotapes of this classroom enactment of Investigation 3 in" a sixth
grade classroom recorded on five consecutive days in October 2006. This particular
class, consisting of 14 girls and 9 boys, is heterogeneous in mathematical ability (i.e., the
students are not tracked).l7 It is located in a middle school (grades 6-8) in a small rural
town (population 3,800) in the Midwest approximately 30 miles from the state capitol.
The district serves nearly 2,000 students, of whom 97% are Caucasian and 15% receive
free or reduced lunch. In 2005, the school made Adequate Yearly Progress (AYP) and

79% of the students were proficient in mathematics compared to 69% in the state. The

 

‘6 The last 0.5 day in the written curriculum is designated for a reflection on the work completed in the
Investigation.

'7 I will use present tense verbs in reference to both the written and enacted curricula throughout this
investigation as I am conceptualizing them both as texts with eternal form.

27

 

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teacher of this particular class is a veteran Connected Mathematics teacher. She has
taught in a middle school for 20 years, the last 13 of those using Connected Mathematics
at this particular middle school. She has attended and conducted professional
development for Connected Mathematics and verbally endorses the curriculum. She
represents a good starting point for this work due to her familiarity with the curriculum.
That is, I expected to find a relationship between the written curriculum and the enacted
curriculum in her classroom. This provided an opportunity to focus on the description of
this relationship.

In this classroom, the Connected Mathematics instructional model varies
depending on the day. The sequence of the components, Launch-Explore-Summary, is
consistent; however, some variation exists in the time allotted to each component. For
example, Day 4 consists only of a Launch-Explore followed on Day 5 by an Explore-
Summary. The Explore on Day 5 is a continuation of the Explore on Day 4. In contrast,
Day 3 contains two Launch-Explore—Summary sequences. Day 1 is the only'day that
contains one Launch-Explore-Summary sequence. The written curriculum recommends
that the Launches and Summaries are whole class discussions and the Explorations are
carried out either individually or in pairs or small groups. The teacher leads the Launch
and students answer posed questions, working at the whiteboard upon request. During
exploration, the teacher walks around interacting with individuals and groups. Finally,
students present their work on posters or overhead transparences for teacher-facilitated
whole class discussion during the Summary.

The analysis of the enacted curriculum required transcribing the spoken language

from the classroom into written language for the purposes of comparison. The goal in

28

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producing these transcriptions was to represent the classroom discourse as completely
and accurately as possible. Within the sections of transcripts included in the analysis
chapters, I used “T” to indicate that the teacher is speaking and “S” to indicate that a
student is speaking. “SI”, “82,” “S3,” etc. are used to designate specific students when
more than one identified student has a turn in a particular conversation. Each section of
transcript is treated separately (i.e., “SI” may represent different students in different
sections of transcript). “S?” is used in situations in which the speaking student is off-
camera and therefore unidentifiable. Finally, “85” is used when more than one student is
speaking simultaneously. These are often situations in which the speaking students are
off-camera as well. In situations in which the teacher or a student refers to another
student by name, a pseudonym is used.

In the same way that the union of the discourse of the Teacher’s Guide and the
Student’s Guide are considered the “written curriculum” for the purposes of this study,
the discourse of the classroom (i.e., the union of the words and actions of the teacher and
the students) is considered the “enacted curriculum” for the purposes of this study. Sfard
(2008) supports this, saying “Discursive rules of mathematics classrooms, rather than
being implicitly dictated by the teacher through her discursive actions, are an evolving
product of teacher’s and students’ collaborative efforts” (p. 262).

Fraction Multiplication

I selected fractions as the topic for study because it has traditionally been a
difficult topic to teach and learn. Both the first and second handbooks of research on
mathematics teaching and learning (published in 1992 and 2007 respectively) support this

statement, encouraging research addressing rational numbers. In the first handbook,

29

Behr, Harel, Post, and Lesh (1992) state that. “There is a great deal of agreement that
learning rational number concepts remains a serious obstacle in the mathematical
development of children” (p. 2%). Fifteen years later, in the Second Handbook of
Research on Mathematics Teaching and Learning. Lamon (2007). makes an even stronger
statement:
Of all the topics in the school curriculum, fractions, ratios, and proportions
arguably hold the distinction of being the most protracted in terms of
development, the most difficult to teach, the most mathematically complex, the
most cognitively challenging, the most essential to success in higher mathematics
and science, and one of the most compelling research sites. (p. 629)
She goes on to comment on the state of research addressing this topic:
In the last decade, researchers have made little progress in unraveling the
complexities of teaching and learning these topics [fractions, ratios, and
proportions]. Worse yet, the number of scholars pursuing long-term research
agendas in the field of rational numbers is disproportionate to the mathematical
richness of the domain. (p. 629)
These statements regarding the problematic nature of the domain of fractions combined
with research that emphasizes the difficulties associated with fraction operations (e.g.,
Erlwanger, 1973; Graeber, 1993; Ma, 1999) served to justify the selection of fraction
multiplication as an important topic for study.
The use of “fraction” in this study was a conscious choice; however, I could also
have argued for the use of “rational number.” This distinction is a contentious one with

the possibility of unfavorable consequences for mathematics students. A rational number

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15 a number that can be expressed as a ratio 1n the form of E where a and b are 1ntegers

and b does not equal zero, whereas a given fraction is one of infinitely many possible
representations of a rational number. “Fraction,” perhaps because of its colloquial use as
a part, is often used in school mathematics when speaking about the number represented
by the fraction. This discursive confusion between mathematical discourse communities
may contribute to the difficulty of the transition to rational numbers later in the school
mathematics curriculum. I chose to use “fraction” because it is the word used both in the
written and enacted curricula. That is, “rational number” was not present in either
curriculum on the five days under investigation in this study.

Regardless of which word is used, objectification of rational numbers (called
“fractions” throughout this study) is important. If we extend the earlier argument
regarding the objectification of whole numbers to include fractions, several of the same
principles apply. First, objectification of fractions involves “encapsulation.” For

example, to objectify fractions, two whole numbers (and later integers) (e.g., “2” and “3”)

need to be "encapsulated” into a fraction (“3 ”). Second, 51m1lar types of ev1dence can

be utilized to detect objectification of fractions. If a student says, “two thirds are. . .,” the
chances are that she has not objectified the fraction. That is, for her, “two” is an adjective
describing the number of “thirds.” However, if she says, “two thirds is. . .,” then “two
thirds” (actually “two-thirds” in this case) has been objectified because it is being treated
as a singular noun (evidenced by a corresponding singular verb). Often, it is only after
such objectification has occurred that operating on fractions as numbers makes sense to

students. The objectification of fractions is important for the study of advanced

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mathematics because as students are expected to operate within other number systems

(e.g., rational numbers, integers, real numbers), the objects representing numbers become

more complex (e.g., “J: ”). In later years, students will be expected to use “J;— ,”
not as something to “do,” but rather as something that “is" (i.e., a number in its own
right). In sum, the objectification of fractions is but one such objectification expected in
students’ ever expanding domain of numbers.

The objectification of fractions is related to the various interpretations (or
constructs or personalities) of fractions described in the literature (e.g., Kieran, 1976,
1980, 1988; Behr, Lesh, Post, and Silver, 1983; Nesher, 1985; Ohlsson, 1988). Lamon
(2007) summarizes these descriptions, saying “When these analyses were reconciled,
measure, quotient, ratio, and operator were recognized as distinct subconstructs and
there was some disagreement about whether part-whole was distinct from the measure
sub-construct” (p. 630). She goes on to say that, “Whatever the number of subconstructs,
the overriding issue was that current instruction, which addresses only one of them, is
inadequate” (p. 630). Here, she challenges the overuse of the part-whole or measure
subconstruct in school mathematics. The overuse of the part-whole subconstruct is
relevant to this study because it may not promote the objectification of fractions.

Design of the Study

In the tradition of case study, I observed one curriculum in one classroom with
one teacher. This choice was particularly appropriate for several reasons. First, utilizing
commognition to describe the relationship between written and enacted mathematics
curricula represents a new application for its use. Therefore, an in-depth analysis of one

classroom’s mathematical features provides a useful starting point. Second. it is

32

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important to first establish the usefulness of the Commognition framework for this
particular purpose before using it to compare cases or to generalize to a broader set of
curricula or classrooms. Stake (2005) would classify this study as an “instrumental case
study” because “the case is of secondary interest, it plays a supportive role, and it
facilitates our understanding of something else” (p. 445). In this case, that “something
else” is the implementation of standards-based curricula and the usefulness of the
Commognition framework to investigate this implementation.

I can imagine many further applications of the Commognition framework: (1) to
compare the mathematical features of several curricula (either standards-based curricula
with traditional curricula or several standards-based curricula), (2) to describe the
mathematical features in classrooms of teachers with varying amounts of teaching
experience, (3) to compare the ways in which the mathematical features of a particular
curricula play out in various contexts (e.g., urban, rural) and with particular student
populations. However, given the complex nature of both the written and enacted
curricula and the novel use of this framework to examine these forms, a case study such
as the one described here provides the first step toward these further investigations. Of
course the use of a case study methodology will limit my ability to make claims about
either the written or enacted curricula as a whole.

My data analysis entails, broadly speaking, applying Sfard’s Commognition
framework to both the written and enacted curricula, and using the results of these
analyses to investigate both the relationship between the two curricula as well as the
usefulness of the framework to describe the results. Here, I outline the steps of the

analyses in greater detail. First, I compared the treatment of the Investigation 3 goals

33

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(stated in the written curriculum) in the written and enacted curricula to determine the
amount of emphasis placed on each goal in each curriculum (see the description of both
the particular methods utilized and the results of this analysis in Chapter 4). Second, I
identified the mathematical features in the written curriculum in Investigation 3 (both the
Student’s Guide and the Teacher’s Guide) and the enacted curriculum during the five
days of videotaping using the four categories in the framework: ( 1) Mathematical words,
(2) Visual mediators, (3) Endorsed narratives, and (4) Mathematical routines (see the
descriptions of both the particular methods utilized and the results of these analyses in
Chapters 5-8 respectively).

The Venn diagram in Figure 3 provides one way to represent the expected results.
Mathematical features found in the written curriculum but not found in the enacted
curriculum are located in Area A. Conversely, Mathematical features located in Area C
are found in the enacted curriculum but not in the written curriculum. Finally, features
located in Area B are found in both the written and enacted curricula. In a classroom
using a particular curriculum, many features are expected to be located in Area B
although it would be surprising to not also find features in Areas A and C. In addition,
detailed descriptions of the use of the mathematical features in the written and enacted

curricula are included in each analysis.

34

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Figure 3. Model for the analysis of the relationship between the written and enacted
curricula.

The descriptions of the goals and mathematical features serve two purposes: (1) to
describe the relationship between the written and enacted curricula, and therefore
curricular implementation in this particular case, and (2) to describe the use of the
Commognition framework as a way to investigate this relationship.

The comparison of written and enacted forms of curricula brings with it inherent
challenges since these two forms of curriculum differ considerably in nature. For
example, the written curriculum is in written mode while the enacted curriculum contains
both written (e.g., posters) and spoken modes (e.g., classroom dialogue). At times, I
made specific analytic decisions in an attempt to address these differences; these are
outlined in the chapters describing particular analyses. In addition, I conjecture several
times in the analyses when it seems that the difference in curricular form impacts the
results. This said, it is impossible to know precisely to what extent the difference in

curricular form has influenced the findings of the analyses presented here.

35

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Because this case study investigates the relationship between two forms of
curricula, written and enacted, the descriptions include comparative discourse such as
“similar” and “different.” Due to the qualitative nature of this study, I have not
determined the statistical significance of these differences. Rather, I provide here some
guidelines for my use of comparative discourse. First, in all cases of comparisons of
counts of various mathematical features, either the raw numbers and/or the relative
frequencies have been provided in the text. Second, I considered less than 10%
difference in relative frequency as “similar” and less than 5% difference in relative
frequency “very” or “quite” similar. Finally, I used differed “greatly” or “substantially”
in cases in which there was at least a 20% difference in relative frequencies.

Throughout this process, I have acted as a Commognitive researcher. This label
carries with it important assumptions and responsibilities. First, I acknowledge that
research is a form of communication. That is, I communicate my perception of the
analysis to the reader. Therefore, my research is as much about howl see the world as it
is about the world itself because I bring my own perspective and biases with me to this
study.

Second, my principal object of attention in this project is discourse. I am guided
by the principle of “utmost verbal fidelity” in my transcriptions while recognizing that
“perfect” transcription is an impossible feat. I have used the voices of my participants
whenever possible to allow readers the opportunity to draw their own conclusions. Third,
I recognize that interpersonal communication is observable, while intrapersonal
communication is not, therefore, I do not have access to the whole story since I do not

have direct access to individual’s intrapersonal communication. Finally, I consciously

36

fluctuated between an insider’s and an outsider’s point of view, recognizing the benefits
and limitations of each and noting the particular difficulties of taking on the eyes of an
outsider to mathematical discourse because I have been involved, as a student and/or a

teacher, in mathematics education for nearly forty years.

37

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CHAPTER 4: GOALS OF THE WRITTEN AND ENACTED CURRICULA
The primary goal of this chapter is to describe the relationship between the goals

stated in the written curriculum and their enactment in the classroom on the five days
analyzed in this study. This study was limited to five days as this is the “suggested
pacing” for Investigation 3 .' Multiplying Fractions in the written curriculum. That is,
what happens before and after these five days is unknown and not considered in this
analysis. Table 1 includes the title of each Problem in Investigation 3 in the written
curriculum along with the goals for each Problem as stated in the written curriculum.l8
Table 1.
Goals of Problems 3. I -3.5 in “Investigation 3: Multiplying Fractions " (as stated in the

Written Curriculum)

 

Problem Title Goal

 

3.1 How Much of the Pan Have We ' Estimate products of fractions
Sold? (A Model for Multiplication) 0 Use models to represent the product
of two fractions
° Understand that finding a fraction of

a number means multiplication

 

 

 

'8 “Investigation,” “Problem,” and “Question” will begin with upper case letters when they refer to
components of the written curriculum. They will begin with lower case letters when they are used
colloquially.

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Problem Title

Goal

 

3.2 Finding a Part of a .Part (Another

Model for Multiplication)

3.3 Modeling More Multiplication

Situations

3 .4 Changing Forms (Multiplication

with Mixed Numbers)

' Estimate products of fractions

0 Use models to represent the product

of two fractions

° Understand that finding a fraction of

a number means multiplication

Estimate products of fractions

' Develop and use strategies and

models for multiplying
combinations of fractions, whole
numbers, and mixed numbers to
solve problems

Determine when multiplication is an
appropriate operation

Explore the relationships between
two numbers and their product
Develop and use algorithms for
multiplying combinations of
fractions, whole numbers, and

mixed numbers

 

39

Table 1 (cont’d)

 

Problem Title Goal

 

3.5 Writing a Multiplication Algorithm ° Develop and use an efficient
algorithm to solve any fraction

multiplication problem

 

Note. The information included in this table was collected from pp. 60, 65, 71, 75, and 81 in the Teacher’s

Guide for Investigation 3.

Each problem is intended to be completed during a single day; therefore, Problem 3.1
will be associated with Day 1, Problem 3.2 with Day 2, and so on in the written
curriculum.l9 Three of the eight goals are included on more than one day (e. g., “Estimate
Products of Fractions” is included on Days 1-3). In addition, multiple goals are included
in the written curriculum for all but Day 5. Table 2 summarizes the Questions included
in the written curriculum and those enacted in the classroom for each day.20

Table 2.

Investigation 3 Problems and Questions in the Written and Enacted Curricula

 

Written Curriculum Enacted Curriculum

 

Day Problem Question(s) Problem Question(s)

 

l 3.1 A-D 3.1 A&B
2 3.2 A-D 3.1 C&D
3 3.3 A - D 3.2 A2, A3, B, C

 

 

‘9 This is not exactly true as the written curriculum explicitly states that Problem 3.2 may be summarized
on the following day; however, the curriculum’s pacing allows for one day per problem in this
Investigation.

20 Recall that only the problems completed during class are included in these analyses (i.e., not ACE
problems or other problems given for homework).

4O

Table 2 (cont’d)

 

Written Curriculum Enacted Curriculum

 

Day Problem Question(s) Problem Question(s)

 

4 3.4 A-D 3.3 A—D

5 3.5 A—C 3.3 A—D

 

Table 2 indicates that three of the five Problems (Problems 3.1-3.3) are enacted in the
classroom during the five days included in this analysis. That is, Days 1 and 2 in the
classroom are used to complete Problem 3.1 and Days 4 and 5 in the classroom are used
to complete Problem 3.3. The only Problem that is enacted in a single day is Problem 3.2
on Day 3 in the classroom. The Questions in the written curriculum for Problems 3.4 and
3.5 are not enacted in the classroom during these five days. In the remainder of this
chapter, I compare the experiences associated with each goal in the written and enacted
curricula followed by a categorization of the experiences associated with each goal as
“similar,” displaying a “low level of discrepancy,” or “displaying a higher level of
discrepancy” in the written and enacted curricula. In each case, I explain how the goal
was categorized. Finally, I discuss briefly the possible importance of these differences
through the lens of objectification.
Goal 1: “Estimate products of fractions”
Written Curriculum
The goal, “Estimate products of fractions,” is included in Problems 3.], 3.2, and

3.3 (Days 1, 2, and 3) of the written curriculum in Investigation 3. In particular, Question

41

D1

 

 

 

D in Problem 3.] (see Example I) and Question A-l in Problem 3.2 (see Example 2)

explicitly address estimation:

Example 1 (WC).

 

D. Use estimation to decide if each product is greater than or less than 1.
To help, use thes‘ of” interpretation for multiplication. For example, in
part (1), think“ - “of

S

 

 

 

. I 25 5 3
LEX-z- 2.'6-X XI 3'6X2 4.7X X2

3 3 I 9 I 10 9 10
S'ZXZ 5.5X ‘5 7.5X-7- 8,-1-6X-7—
ExampleZflVC).

 

A. 1. For parts (3)-(d), use estimation to decide if the product is
greater than or less than 5.

I l 2 I
3.§X§ b.’X§ C. d. X

OOIH

 

X
Unla-
O‘I'JI
Alt»

 

 

These Questions instruct students to use estimation to decide if a product of two fractions

is greater or less than a given value (e.g., “% ”). In Problem 3.3 (see Example 3), the

written curriculum contains two experiences with estimation. First, four non-contextual
mixed number multiplication problems are included in the “Getting Ready” (SG, p. 36)”:

Example 3 (WC).

 

 

Getting Ready for Problem w

Estimate each product to the nearest whole number (1. 2.3. . . .).
1 9 l
r X 21—0 1'2'
Will the actual product be greater than or less than your whole number
estimate?

9 l 4 I ll
XZE 2§X7 3:)(21—2'

 

 

 

 

 

2' Recall that Problems 3.2, 3.3, and 3.4 begin with a “Getting Ready” problem. Excerpts from the
Teacher’s Guide (TG) and Student Guide (SG) of Investigation 3 will be cited in the form presented here.

42

The expectation in Example 3 is slightly different than in Examples 1 and 2. Instead of
being asked to determine if the product will be greater than or less than a particular
number, here students are asked to both estimate the product “to the nearest whole
number” and decide whether this will be an overestimate or an underestimate. Example 4
presents Questions A-D from Problem 3.3 in which the class is asked to estimate answers

to four contextual fraction and mixed number multiplication problems:

Example 4 (WC).

 

For each question:

0 Estimate the answer.
0 Create a model or a diagram to find the exact answer.

0 Write a number sentence.

A. The sixth-graders have a fundraiser. They raise enough money to reach
§of their goal. Nikki raises 9, of this money. What fraction of the goal
does Nikki raise?

B. A recipe calls for g of a l6ounce bag of chocolate chips How many
ounces are needed?

C. Mr. Flansburgh buys a 2%-pound wheel of cheese. His family eats % of

the wheel. How much cheese have thev eaten?
D. Peter and Erin run the corn harvester for Mr. MeGreggor. They

harvest about 2:;- acres each day. They have only 10% days to harvest

the corn. How many acres of corn can they harvest for
Mr. McGreggor?

 

 

 

The experience with estimation provided in Example 4 is different yet from the previous
three examples. In the Questions in Example 4, students are expected to estimate
answers to contextual problems including different combinations of fractions, whole

numbers, and mixed numbers. (i.e., Question A includes two fractions, Question B

43

ill

ii

OI

includes a fraction and a whole number, Question C includes a fraction and a mixed
number, and Question D includes two mixed numbers).

Enacted Curriculum
Question D from Problem 3.1 (Example 1 on p. 42) is enacted on Day 2. The teacher
asks the students to follow the instructions given with the problem, “Use estimation to
decide if each product is greater than or less than 1.” The second problem of this type
(i.e., where students are expected to estimate the magnitude of fraction products),
Question A-l from Problem 3.2 (Example 2 on p. 42), is not completed by the class.
Instead, they complete Question A—2 from Problem 3.2 in which they “solve” the
problem rather than estimate the answer. In the enactment of Problem 3.3 on Day 4, the
class completes the “Getting Ready” (Example 3 on p. 43) which contains, as mentioned
previously, four non-contextual mixed number multiplication problems. Much emphasis
on estimation is present during the discussion around these four problems. Example 5
provides an excerpt from that discussion.22 The bolded words indicate language
associated with estimation (i.e., “estimate,” “round”) used by both the teacher and
students. More will be said about the use of these words in Chapter 5 .' Mathematical
Words.

Example 5 (EC):
. . . I I 1
( I ) T: Could we just get an estimate [pomts to 37x 2;? on the overhead]?

(2) S .' Yeah.

 

22 Recall that “T” is used to indicate that the teacher is speaking. “S” is used to indicate that a student is
speaking — numbers will be used to distinguish between students (e.g., SI, 82). The same number is
associated with a particular student within an example. “S?” is used when the identity of the student is
unknown (i.e., they are out of the camera’s view). “85” is used when several students in the class answer
simultaneously. For emphasis, bolding will be added to particular words in examples throughout the
chapter.

44

(3 ) T: Could we figure out an estimate? Is there anything I could round up a
bit to make it easier to work with? Steven. what should we do?

(4) S: Round the eleven twelfths up to twelve twelfths so the two would be
three.

(5) T: Would be a three. Would that help us?

(6) S: Yeah. yes. six.
. . 1
(7) T: So, what, what, what if I just got rid of this too [points to “j "j?
Could you guys/ind three groups of three?

(8) S: It's nine.

(9) T: It 's nine? So, would nine be a halfivay fair estimate?

(10) S: Yes.

Example 5 illustrates a discussion that took place on Day 4 in the enacted curriculum in
which estimation was the focus in the context of multiplying mixed numbers.

The second opportunity for estimation in Problem 3.3 involves four contextual
Questions (see Example 4) that ask students to first “estimate the answer,” then “create a
model” and finally to “write a number sentence.” Example 6 indicates the way in which
the problem is presented to the class:

Example 6 (EC):

T: “Okay. You need to do all three of those things. So you need to estimate
what you think the answer should be, about how much, you need to create
some sort of model or diagram, so that you can show how you solved it, and
then when you're done try to write a number sentence. Okay? So you need to
estimate, model, write a number sentence for A, do all three for B. C and D. "

During the Summary, however, no mention is made of the estimates; only the models and
number sentences are discussed.

Given the information presented here, I argue that there are more experiences

with estimation present in the written curriculum than in the enacted curriculum in

45

 

Investigation 3. I also argue that opportunities for estimating fraction products contribute

to the objectification of fractions because estimation has the potential to encourage

. 5 , I ,, . .
students to make statements such as “g (S greater than — .’ (As mentioned prev10usly,
the use of “is” instead of “are” may indicate that students have encapsulated the “5” and

the “6” or the “5” and the “Sixth” mto a tractron “g as an entity in and of itself. Recall

that encapsulation involves taking mathematical objects that were previously seen as

separate (e.g., “5,” “6”) and seeing them as together forming a new mathematical object

5 . . . . . .
(e.g., “g ”). In addition to encapsulation, wh1ch represents a form of rerficatron, the use

of “is” also works to fulfill the other requirement for objectification, alienation. Recall

that alienation is depersonalization of the mathematics. Here, the use of “is” alienates the

99

. . . . . , 5 . I .
mathematics from the act1on of dorng it. That 15,“; is less than 3 15 stated as a

mathematical fact rather than as an action carried out by an individual. Much more will
be said about the role of estimation in objectification throughout this analysis.
Goal 2: “Use models to represent the product of two fractions”
Written Curriculum
The goal, “Use models to represent the product of two fractions,” appears in
Problems 3.1 and 3.2 (Days 1 and 2) in the written curriculum. Students are expected to
construct iconic mediators to represent the products of two fractions in Problems 3.1

(“Getting Ready.” A-l, A2 13.1. B-2, C-1, C-2. 03. and C-4) and 32 (“Getting Ready,”

46

A-2, B-l, B-2, and B-3).23 Example 7 presents Question A-l in Problem 3.1 which
contains explicit directions regarding the construction of an iconic mediator to represent
fraction multiplication:

Example 7 (WC).

 

A. Mr.Williams asks to buy% of a pan that is § full.

1. Use a copy of the brownie pan model Model Of a Brownie Pan

 

shown at the right. Draw a picture to show
how the brownie pan might look before
Mr. Williams buys his brownies.

2. Use a different colored pencil to show the
part of the brownies that Mr. Williams
buys. Note that Mr. Williams buys a part of
a part of the brownie pan.

 

 

 

 

 

 

Note the level of detail in the directions in Example 7 regarding the construction of an
iconic mediator representing fraction multiplication. This same level of detail is not
present in later problems. In Example 8, Question A-2 in Problem 3.2 simply instructs
students to “use the brownie-pan model or the number-line model”:

Example 8 (WC).

 

2. Solve parts (a)-(d) above. Use the brownie-pan model or the
number-line model.

 

The brevity of the directions for the use of iconic mediators in Example 8 seems to

indicate an expectation that by this point in the Investigation students have developed

 

23 In an effort to use language consistently throughout this project, I will use “iconic mediator” (see
description on p. 130) in the text instead of “model” because in all cases except one in the enacted
curriculum, “model” refers to this type of visual mediator. However, examples lifted directly from the
written and enacted curricula will maintain their original wording.

47

skills for constructing iconic mediators for fraction multiplication without detailed
instructions.
Enacted Curriculum

All of the aforementioned Questions that address the construction of iconic
mediators in the written curriculum are enacted in the classroom. That is, the students are
expected to construct iconic mediators for all of these Questions and therefore are
exposed to the intended opportunities for this goal. One difference is that these
Questions are completed over the course of three days in the enacted curriculum instead
of the two days suggested in the written curriculum. In addition, the amount of time
taken to describe the method of iconic mediator construction lessens over the course of
the days in ways similar to the written curriculum. For example, if we consider the
enactment of the same two Questions mentioned above from the written curriculum
(Examples 7 and 8), the difference in time devoted to discussing the construction of the
iconic mediators is striking. In the enacted curriculum, Question A-l in Problem 3.1
(Example 7) results in an eight-minute discussion of iconic mediator construction on Day
1. The discussion presented in Example 9 follows students’ previous interactions with
iconic mediators that represent fraction multiplication. The brief excerpt is associated
with Question A-2 in Problem 32.“:

Example 9 (EC).

(1) T: Okay. Alright now, you're going to really have to listen 'cause each
group of, each couple of tables only had to do one of these, so they're
going to be a couple up here that you haven 't looked at yet. So be sure
that you really listen, okay? Could, the first one was one third of one half

So, whoever did that one, could you come over and talk about that one,
please?

 

24 The student in this example is referring to an iconic mediator that was constructed on the whiteboard and
not saved. Therefore, it is not presented here.

48

(2) SI : [filled in half because I knew it was going to be half of a bar, and
then I made three parts of that half and I filled in one half er, one part of
that half and I got one sixth. because if you still had this half. there 'd be
these three pieces.

(3 ) T: What do you guys think?

(4) S5: Yeah.

In both the written and enacted curricula, descriptions of the process of constructing
iconic mediators decreased in length over the course of the week except in cases where a
new type of iconic mediator was being introduced.

Given the fact that all Questions in Problems 3.1 and 3.2 addressing this goal are
enacted in the classroom, I argue that similar emphasis is given to representing fraction
multiplication in iconic mediators in the written and enacted curricula. In terms of
objectification, most of the talk surrounding the construction of iconic mediators in both
curricula is personalized. This is evidenced in Example 9 in which the student says “I
filled in half, “I made three parts,” and “I got one sixth.” In addition, the use of models,

although potentially serving students well for making sense of fraction multiplication,

does not necessarily encourage the reification of fractions. For example, in the statement,

“Mr. Williams asks to buy % of a pan that is —:- full.” (see example 7), g and —:- are

used as descriptors of “a pan” and “full” rather than as reified objects in their own right.
A much more in-depth comparison of models and their use in the written and enacted

curricula can be found in Chapter 6: Visual Mediators.

49

Goal 3: “Understand that finding a fraction of a number means multiplication”
Written Curriculum

The goal, “Understand that finding a fraction ‘of’ a number means
multiplication,” is stated for Problems 3.1 and 3.2 (Days 1 and 2) in the written
curriculum. Again, I was interested in whether or not the enacted curriculum emphasizes
this goal in ways similar to the expectations in the written curriculum. In Problems 3.1
and 3.2 in the written curriculum, the notion that “of” means multiplication is addressed
implicitly and explicitly in several places. Explicit statements like those included in
Examples 10-12 are found in the Investigation (specifically in Problems 3.1 and 3.2).

Example 10 (WC).

“When you multiply a fraction by a fraction, you are finding ‘a part of a part’.”

(SG, p. 33)

Example 11 (WC).

“Understanding that ‘part of a part’ means ‘X’ is raised in Question C.” (TG, p.

60)

Example 12 (WC).
“To help, use the ‘of’ interpretation for multiplication.” (SG, p. 33)

Examples 10-12 include explicit reference to the fact that “of ’ often indicates the need
for fraction multiplication or that fraction multiplication can be thought of as a “part of a
part.” Implicitly, Questions in the written curriculum in which students are writing
number sentences using “X” when confronted with contextual problems containing “of”
provide opportunities to make sense of the connection between “01” and fraction
multiplication. Example 13 contains a Question (SG, p. 35) that exemplifies a case in
which “of” in the contextual problem can be interpreted as multiplication to aid in writing

the number sentence.

50

Example 13 (WC).

 

B. Solve the following problems. Write a number sentence for each.

1. Seth runs % of a %—mile relay race. How far does he run?

 

X

col—-

The number sentence stated for this Question in the written curriculum is “

Al'—
Nl—t

(TG, p. 70). The opportunities for considering the relationship between “of" and fraction
multiplication in the written curriculum occur in Problems 3.1 (“Getting Ready,” A, B, C,
D) and 3.2 (“Getting Ready” and B).
Enacted Curriculum

Both explicit and implicit opportunities to examine the relationship between “01”
and fraction multiplication are also found in the enacted curriculum. For example, the
interaction in Example 14 containing an explicit reference to this goal occurs on Day 1 in
the classroom:

Example 14 (EC).

(1) S: Does the "of" mean like times?

(2) T: It can mean "times, ” yes. I worry about saying that "of" is every
single possible time always multiplication, but yeah.

Although, the teacher seems hesitant to suggest that “of” always implies multiplication,
she answers the student in the affirmative. Also explicit in nature, the statements

included in Examples 15 and 16 are made by the teacher on Day 2:

Example 15 (EC).

T: “Okay. We 're going to come back and look at all these [a series of
mathematical statements of the form A of B = C, where A. B, and C are
proper fractions ], 'cause you know what I 'm going to do? I 'm going to change
that word "of”[points to “of" in “-}20—of g = 6% "] to a mathematic symbol.
What you guys have done here. I love the way that you're all looking at me. I
can tell you're listening. Thank you very much. When I take a part [points to

51

2 2 I 2 _ 1
“— " in “—o — = — " 0 another art )oints to “— in
7 7 f 2] ] .t p [I 3
u 2 . I 2 ,, . . . . - u n
70] 3- = 27 ] . I can write that as a mlllllpllCCllIOl’I problem [writes X
ll ’ ' ‘t 2 I 2 '! I,
over 0 m —o — = — .
f 7 f 3 7 I

Example I 6 (EC).
T: “ Instead of saying nine tenths of a sixth, I can also say nine tenths times a

sixth. Go ahead, Trevor. "
In Examples 15 and 16, the teacher refers to six problems that included “of” in the

l = _2_ ”y She writes “X” on top of “of” in each example,

2
number sentence e. ., “—0
( g 7 f3 2]

2..

 

producing a new number sentence (e.g., “éxé = 2—1 ). Example 17 illustrates a sample
result of this process:
Example I 7 (EC — Student Work).
w—SI

 

..L - .2;
“'7 K a: .11.

 

 

 

In addition to explicit statements regarding the relationship between “of” and fraction
multiplication, students complete the same contextual problems included in the written
curriculum in which “of” is embedded in the context and students are expected to write
number sentences using “X.”

Given the explicit and implicit references to this relationship provided in both
curricula and the fact that students enact all of the Questions that provide opportunities to
develop this relationship, I conclude that the emphasis on this particular goal,

“Understand that finding afraction of a number means multiplication ” is similar in the

52

 

written and enacted curricula. In terms of objectification, this goal provides a
combination of personalization and alienation in both the written and enacted curricula.
This goal from the written curriculum states that “a fraction of a number means

multiplication.” This statement is void of human agency which indicates alienation to

. . . I 1
some extent; however, other statements. such as “When mathematiCians write 50f Z,

they mean the operation of multiplication, or %x :11— .” (SG, p. 33) include human agency,

albeit the agency of professional mathematicians. This goal seems to be providing

99

. . . . . . l 1 l 1
opportunities for students to reify fractions because the move from “50f Z”to “5x 2

66199

. . . . . . .. . ., . 1 ,,
can serve to faCilitate reification given the encapsulation of “l and “2’ into “:2— and

and “4” into “E " as entities themselves (i.e., numbers) that can be multiplied.

Goal 4: “Develop and use strategies and models for multiplying combinations
of fractions, whole numbers, and mixed numbers to solve problems”

Written Curriculum

The goal stated in the written curriculum, “Develop and use strategies and models
for multiplying combinations of fractions, whole numbers, and mixed numbers to solve
problems,” extends fraction multiplication, which has thus far been largely limited to
fractions (with the exception of Question D in Problem 3.] in the written curriculum
which includes whole numbers), to include combinations of fractions, whole numbers,
and mixed numbers. This goal is stated only for Problem 3.3 (Day 3) in the written
curriculum. I also argue that several activities in the written curriculum in Problem 3.4

(Day 4) are closely related to this goal, including opportunities for discussion of the

distributive property and changing mixed numbers to improper fractions in order to
facilitate multiplication. On Day 3, Questions A, B, C, and D in Problem 3.3 ask students
to “estimate the answer,” “create a model or a diagram to find the exact answer,” and
“write a number sentence” for four contextual problems involving mixed numbers and
whole numbers (see Example 4 on p. 44). These problems require students to develop
strategies for multiplying combinations of fractions, whole numbers, and mixed numbers
and to construct iconic mediators. In addition, sample “Suggested Questions” from Day
3 in the written curriculum that solicit responses regarding strategies are provided in
Examples 18 and 19:

Example 18 (WC).
“How do you find g of something? " (T G, p. 72)

Example I 9 (WC).
“Does anyone have a different way to think about the problem? ” (TG, p. 72)

The questions included in Examples 18 and 19 are “suggested” in the written curriculum
to encourage students to share their strategies for multiplying combinations of fractions,
whole numbers, and mixed numbers.

As mentioned previously, Problem 3.4 (Day 4) also provides opportunities
directly related to this goal. The “Getting Ready” (SG, p. 37) and Question C (SG, p. 38)
are presented in Examples 20 and 21 respectively. They introduce two important
strategies for multiplication involving mixed numbers: (1) changing a mixed number to

an improper fraction and (2) using the distributive property.

54

Example 20 (WC).

 

Yuri and Paula are tiying to find the following product.

2; X l
3 4
Yuri says that if he rewrites 2%. he can use what he knows about multiplying
fractions. He writes:
§ 1
3 X 4

Paula asks. “C an you do that? Are those two problems the same?"

What do you think about Yuri's idea? Are the two multiplication problems
equivalent?

 

 

 

Example 21 (WC).

 

C. Takoda answers Question A part (I) by doing the following:
I l I
(2 X 13)+(§ X13)
1. Do you think Takoda‘s strategy works? Explain.

2. Try Takoda‘s strategy on paits (2) and (5) in Question A. Does his
strategy work? Why or why not?

 

 

 

Examples 20 and 21 illustrate opportunities for students to discuss strategies for

multiplying combinations of fractions, whole numbers, and mixed numbers using two

particular strategies proposed by fictitious students in the written curriculum.
Questions A and B in Problem 3.4 also address this goal. These Questions are

provided in Example 22.

55

Example 22 (WC).

 

A. Use what you know about equivalence and multiplying fractions to
first estimate. and then determine. the following products.

I l 4 i 3 .
1.25XIK 2.3‘5-X1 -3.ZXI6
5 I 6 l 9

B. Choose two problems from Question A. Draw a picture to prove that
your calculations make sense.

 

 

 

Question A instructs students to multiply six combinations of fractions, mixed numbers,
and whole numbers using “what you know about equivalence and multiplying fractions.”
This requires the use of “strategies” as mentioned in Goal 4. Question B expects students
to draw pictures for two of the problems from Question A to “prove that your
calculations make sense.” This Question emphasizes the use of “models” as mentioned
in Goal 4. Therefore, I argue that Day 3 and Day 4 in the written curriculum emphasize
this goal even though it is only explicitly stated for Day 3.
Enacted Curriculum

In the enacted curriculum, this goal is addressed on Days 4 and 5. After some
introduction, the students are asked to complete Questions A-D in Problem 3.3. As
discussed previously, these four problems directly address both the “strategies” and
“models” components of this goal (see Example 4 on p. 44). Student work included in

Example 23 illustrates a possible strategy for solving Question D regarding the harvesting

of 2%acres of corn on each of 10% days.

56

Example 23 (EC - Student Work).

 

- ‘ “N
D—Q‘fli 2‘ 3.9.1:? 11" y ll.‘ \_ a 5 (Jr
will] gill) {did 3?) fill-WWI P1 1” .mw
\/ ‘\__/
9’13 (IT/.1 . If?!

.... .._. '19,! 5 0.; M /2 00 it
nan-n . gm “hi-iii]: (as. m: . . .i.’
\f \‘ "r/ M]

R

9‘7, 735.

9"” 0‘1 ”‘1

6'92? r ‘19/3; ‘1 7; .. ‘1 ”/a
sit; + 1757}; 9 '/3 .i 9%
735‘? “0):; 55l&

9’76"? al.,-=13"), + 5 5x. @210

 

 

 

This student work illustrates both the use of a “strategy” and a “model” for multiplying
mixed numbers as called for in Goal 4.

As noted previously, the Questions in Problem 3.4 (Day 4 in the written
curriculum) are not enacted in the classroom during these five days. Therefore, the
Questions in Problem 3.4 related to this goal (i.e., “Getting Ready” and Questions A, B,
and C) are not included in the enacted curriculum. It is interesting to note that although
these specific exercises are not included, important points made in these exercises emerge
in the enacted curriculum. For example, in the “Getting Ready” from Problem 3.4 (see
Example 3 on p. 43), Yuri, a fictitious student, suggests the strategy of converting a
mixed number to an improper fraction in order to facilitate fraction multiplication. A
second fictitious student, Paula, questions this strategy asking, “Can you do that?”
Although there is no discussion of “Yuri” and “Paula” in the enacted curriculum, a

similar strategy emerges from a student in the classroom (see Example 24):

57

 

I'D

£13..

Example 24 (EC — Student Work).

 

\_ p.
A, 3-5-fl

 

 

Ch 9%,

1" ..
A 3” ff _--.)..
IO“: 2-'-""/9 {b
v0 '47,,
I" ‘3
gifg
’ h
'J r-
¢ ”"
x 7
h‘I—N

 

 

 

This student uses the same strategy as Yuri and Paula (i.e., converting a mixed number to

an improper fraction for the purposes of fraction multiplication); therefore, this strategy is

discussed in the enacted curriculum.25 Likewise, the distributive property suggested by

“Takoda” in Question C of Problem 3.4 (see Example 21 on p. 56) emerges in the

. . . . l
enacted cumculum. In the written curriculum, Takoda suggests decomposmg the “2— ”

in “21 x11”
2 6

to facilitate multiplication of mixed numbers. Example 25 presents

student work in which a similar strategy emerges, albeit with different numbers:

 

25 This conversation will be further discussed later in the chapter.

58

Example 25 (EC - Student Work).

 

 

\
: 9°” 313 it“ Danni‘s amt II. from
‘ +0 ‘4/5

I
7: a; a “link '1‘ figs cfiansld JIS

3“? added ”(a to '1', gal got 5/6.

 

 

 

In this case, the student decomposed the “2;— ” in “% of 2%” to facilitate multiplication.

Therefore, I conclude that the primary strategies and “models” for multiplication of
combinations of fractions, whole numbers, and mixed numbers addressed in the written
curriculum also emerge in the enacted curriculum. That being said, the additional
discursive opportunities provided by Question A in Problem 3.4 that are not enacted in
the classroom could potentially further develop the use of these and other strategies and
“models” for the multiplication of combinations of fractions, whole numbers, and mixed
numbers. Whether or not objectification (i.e., alienation and reification) is promoted
depends on both which “strategies” and “models” are used as well as how they are used.
For example, the use of fictitious students to present the strategy of converting mixed

numbers to improper fractions for the purpose of fraction multiplication (see Examples

59

20 and 21 on p. 56) promotes the personalization of mathematics; however, the use of

this strategy and the equ1valence employed in it promotes the reification of “23” and

8 9 .. . ~- ' ’ ' I
“3 " because “equivalence" is used in numerical discourse.

Goal 5: “Determine when multiplication is an appropriate operation”

The goal, “Determine when multiplication is an appropriate operation,” is
included in Problem 3.3 (Day 3) of the written curriculum. There are at least three
possible interpretations of this goal: (1) the goal is closely related to Goal 3,
“Understand that finding a fiaction of a number means multiplication. ” (2) the goal
expects students to decide whether to use fraction multiplication in a particular situation
(versus other fraction operations), and/or (3) the goal expects students to make sense of
the transition from multiplication of fractions to multiplication of combinations of
fractions, whole numbers, and mixed numbers.

The first possibility was addressed earlier in the discussion of Goal 3 in which the
argument was made that the written and enacted curricula give similar emphases to the
goal. The second possibility seems unlikely because all contextual problems in
Investigation 3 are appropriate situations for multiplication (versus other operations). It
is difficult for me to imagine how students are to gain understanding of when
multiplication is the appropriate operation without a series of problems in which they
must decide on the appropriate operation for the situation. The third possibility is

addressed briefly in this section.

60

Written Curriculum

Because Problem 3.3 addresses modeling situations involving combinations of

fractions, whole numbers, and mixed numbers, it is possible that this goal is implying that

students will need to move past the notion of “part of part” given these new

combinations. If this is the case then “suggested questions” in the written curriculum,

such as “What does it mean when it says ‘ % of a 16-ounce bag’?” (TG, p. 72) may

provide support for this transition.

Enacted Curriculum

A related discussion in the enacted curriculum (see Example 26) occurs when

. , 1
students are presented With “3 4 x 2 —- .

ll”.
12

Example 26 (EC).

(1)

( 2)
( 3)
( 4)
(5)

( 6)
( 7)
(8)
( 9)

T: You told us how that you think you might solve that [3 :3 x 2%], but

I'm not sure we even understand what this is asking. If it were just three
times two, what would it mean?

SI: Three groups of two.
T: It would mean three groups of two?

S 1 .' Yeah.

T: So, what does it mean now? I need to get some more of you involved.
Chloe, what does it mean now?

52: Three and one fourth groups of two and eleven twelfths.
S ?: So maybe you have to
T: Say that again, Chloe. What does it mean?

S2: Three and one fourth groups of two and eleven twelfths.

6l

(10) T: Let ’s look - hold onjorjust a second. Let's look at thisfor a minute.
Does this just mean I have more than three groups of more than two?

(I l) Ss: Yeah.

This discussion, like the “suggested question” in the written curriculum addresses the
“meaning” of the multiplication of these new combinations of fractions, whole numbers,
and mixed numbers. Therefore, if the goal expects students to make sense of multiplying
combinations of fractions, whole numbers, and mixed numbers, I argue that the emphasis
on this goal is similar in the written and enacted curriculum because all Questions in
Problem 3.3 are enacted in the classroom.

Goal 6: “Explore the relationships between two numbers and their product”

Written Curriculum
The goal, “Explore the relationships between two numbers and their product,” is

included in Problem 3.4 (Day 4) in the written curriculum. There are several Questions
in the Investigation that address the magnitude of the product of combinations of
fractions, whole numbers, and mixed numbers. Fewer Questions, however, explicitly
address the relationship between the magnitude of the product and the magnitude of the
factors as this goal seems to demand. In Problem 3.1, Question D instructs students to
estimate the size of the product of eight fraction multiplication problems; however, the
Teacher’s Guide suggests that teachers “be sure to focus the conversation on how
students decided if each product was greater than or less than I and greater than or less
than each of the factors” (TG, p. 62). Question A in Problem 3.2 asks students to
estimate the magnitude of products of fraction multiplication problems. In this case,
however, the Teacher’s Guide makes no suggestion regarding a comparison to the

magnitude of the factors. Example 27 presents Question D in Problem 3.2. This

62

Question represents the first time in the Student's Guide that a Question explicitly
references the comparison between the product and factors:

Example 27 (WC):

 

D. Ian says. “When you multiply. the product is greater than each of the
two numbeis you are multiplying: 3 X S = 15. and 15 is greater than
3 and 5.“ Libby disagrees. She says. “When you multiply a fraction by
a fraction. the product is less than each of the two fractions you
multiplied.“ Who is correct and why?

 

 

 

Note that in Example 27 from the written curriculum, students are expected to discuss the
relationship between the magnitude of the factors and the magnitude of the product.

Example 28 illustrates the Teacher’s Guide that suggests several questions for discussion:

Example 28 (WC):

 

For Question D. have students present their
reactions to Ian and Libby‘s conversation.

0 Does multiplying two numbers always lead to
a greater product? Why?

0 Can you give an example that shows what
Libby is talking about?

0 Does multiplication with fractions always lead
to a product that is less than each factor?

 

 

 

The Teacher’s Guide goes on to state, “Do not expect students to resolve this last
question. They will explore this idea in the problems that follow.” (TG, p. 67). Indeed
the written curriculum does come back to the issue on Day 4. The introduction to
Problem 3.4 in the Student’s Guide reminds students to consider the magnitude of the
product, saying “Before you begin a problem, you should always ask yourself: ‘About
how large will the product be?’” (SG, p. 37) Finally, the last two parts of Question D in

Problem 3.4 (see Example 29) ask students to:

63

Example 29 (WC):

 

4. Describe when a product will be less than each of the two factors

5. Describe when a product will be greater than each of the two
factors.

 

 

The Teacher’s Guide emphasizes the need to address the facts that the product will be
less than both factors when the factors are both less than one and that the product will be
greater than both factors when the factors are both greater than one. Therefore, even
though the written curriculum states this goal only for Day 4, the relationship between the
magnitude of the factors and product is also addressed relatively substantially on Day 2 in
the written curriculum.
Enacted Curriculum
In the enacted curriculum, the issue of product magnitude is addressed several
times. For example, Question D in Problem 3.1 is enacted in the classroom and
discussions addressing the magnitude of the product occur in conjunctiOn with that
Question. Examples 30-32 contain statements from the enactedcurriculum referencing
the magnitude of the products of fractions are included here:
Example 30 (EC).
T: “ It was five sixths of one half and we 're trying to figure out, is five sixths of
one half going to be more or less than the whole thing. ” (Day 2)
Example 31 (EC).
T: You know what. I’m noticing that every time. I’m noticing, that, ooh, I got a
little two sixths, or a third I got three eighths, I got two twenty oneths. My

answer got so little. Why? Why am I getting a little answer? Yeah?

S: Because the denominator is getting larger and that makes the pieces smaller?
(Day 1)

64

 

Example 32 (EC).
T: “ Can anybody think ofa situation where I would multiply and it wouldn 't get
bigger?” (Day 1)

Example 30 asks students to compare the product to “the whole thing” and therefore
addresses the magnitude of the product, but does not relate its magnitude to the
magnitude of the factors. Examples 31 and 32 address the relationship between the
factors and the product using comparative language (e. g., “larger,” “bigger”). A
generalized discussion regarding when the product is less than or greater than the factors,
as described in the written curriculum, does not occur in the enacted curriculum. In
addition, the majority of the problems in the written curriculum that address this issue
(i.e., Questions A and D in Problem 3.2, and Question D in Problem 3.4) are not enacted
in the classroom. Therefore, I conclude that there is less emphasis on this goal in the
enacted curriculum than in the written curriculum.

This goal, as several others, presents mathematics as a combination of
personalized and alienated. For example, fictitious students, Ian and Libby, are again
used to present mathematical conjectures. In this case, the conjectures address the
relationship between the magnitude of the factors and products. The use of these students
may personalize the mathematics. However, Question D-4 provides an example of a
question that alienates the mathematics, “Describe when a product will be less than each
of the two factors.” (see Example 29 on p. 65). In this example, the use of “a product
will be” implies that there is a mathematical fact that can be used to determine the answer

to this Question. The ways in which this goal is presented in the written curriculum

provide opportunities for reification of fractions. That is, if students are asked about the

. . . 1
relationship between the magnitude of the factors and the product, statements such as “-3-

65

13 less than —2— and 3 may occur. Statements that use language such as “is less than"
possibly indicate reification of fractions because this type of language is used with

numbers. If reification has not taken place. indivrduals are more likely to say “3 is

2
smaller than % and 3 .” Because this goal is less emphasized in the enacted curriculum

than in the written curriculum and it provides opportunities for objectification, it again
appears (as with Goal 1) that opportunities for objectification (and reification more
specifically) are less prominent in the enacted curriculum.

Goal 7: “Develop and use algorithms for multiplying combinations
of fractions, whole numbers, and mixed numbers”

Written Curriculum

“Develop and use algorithms for multiplying combinations of fractions, whole
numbers, and mixed numbers,” is stated as a goal for Problem 3.4 (Day 4) in the written
curriculum. “Algorithm” is defined in the Student Guide as “a reliable mathematical
procedure” (p. 38). The line between “strategy” and “algorithm” seems somewhat
blurred here. As described earlier, Problems 3.3 and 3.4 address strategies associated
with the multiplication of combinations of fractions, whole numbers, and mixed numbers.
These strategies include: (I) converting mixed numbers or whole numbers to improper
fractions and then applying the traditional algorithm for fraction multiplication and (2)
using some method of decomposing the mixed numbers and applying the distributive
property. Are these strategies or algorithms, or both? It is interesting that this goal
addressing algorithms is included in Day 4 and “algorithm(s)” is entirely absent in the

written curriculum for this day. The Teacher’s Guide discusses both of the above-

66

mentioned strategies (p. 75) and goes on to discuss the “case-specific" nature of these
strategies (p. 77). The notion that particular strategies serve some situations and not
others leads into Goal 8, which stresses the development of an “efficient” algorithm.
Enacted Curriculum
When the class begins their discussion of multiplication of combinations of

fractions, whole numbers, and mixed numbers in the enacted curriculum, the teacher
stresses that she is only interested in strategies, not algorithms, for the time being (see
Example 33):

Example 33 (EC)

T: “I 'm not so interested in an algorithm today. I'm interested in how can we
literally solve through a drawing or something a problem like this. How can
we do it? And it looks like some of you have some good ideas of ways to start.
Maybe you're going to have a picture of something and be fractioning off
parts or whatever. We 're not on some mission to find an algorithm. We're
on a mission to make sense of what is this really saying, and how can we do
that. ”(Day 4)

The enacted curriculum contains discussions of the same two primary strategies for
multiplying combinations of fractions, whole numbers, and mixed numbers found in the
written curriculum. In the context of these problems, “algorithm” comes up in reference
to the traditional algorithm for use afier whole numbers and mixed numbers have been
converted to improper fractions. Examples 34 and 35 are two such statements from Day
5:

Example 34 (EC).

T: “Oh my gosh! Now wouldn't that be nice if we could multiply any fraction by

a whole number just by doing that [giving it a denominator of I], and we
could still use our algorithm? ”

67

Example 35 (EC).

T: “She didn't like the idea of tnixed numbers, so she switched them over
[converted them to improperfractions] so now she has two fractions. And
then did she just used our algorithm, multiply the numerators and multiply the
denominators? "

Because algorithms for use with multiplication of combinations of fractions, whole

numbers, and mixed numbers are largely discussed as strategies on Day 4 in both the
written and enacted curriculum, and in fact, the same two strategies are emphasized, I
conclude that the emphasis on this goal is similar in the written and enacted curricula.

Because both the written and enacted curricula expect students to develop their

own strategies/algorithms for multiplying combinations of fractions, whole numbers, and
mixed numbers, I argue that the mathematics associated with this goal is largely
personalized in both curricula. However, the movement toward algorithms for operating
on fractions seems to provide an opportunity for reification because algorithms are

largely performed on numbers.

Goal 8: “Develop and use an efficient algorithm
to solve any fraction multiplication problem”

Written Curriculum
The goal, “Develop and use an efficient algorithm to solve any fraction
multiplication problem” is included in Problem 3.5 (Day 5) of the written curriculum.
The focus on this goal in Problem 3.5 is evidenced by the title of the problem, “Writing a
Multiplication Algorithm” as well as an additional 23 appearances of “algorithm(s)”
throughout the Problem. Examples 36-38 illustrate several uses:
Example 36 (WC).

“Test your algorithm on the problems in the table. If necessary. change your
algorithm until you think it will work all the time. ” (SG, p. 39)

68

Example 3 7 (WC).
“Several students may offer the traditional algorithm of multiplying the
numerators together and multiplying the denominators together. Be sure they talk

about how this is done when one or both of the factors is a mixed number or
whole number. ” (TG, p. 81)

Example 38 (WC).

“Evaluating whether each algorithm is useable and helpful, and how it compares
with other algorithms, will further students’ understanding of multiplication of
fractions, mixed numbers, and whole numbers. ” (T G, p. 82)

In addition, all of the Questions (i.e., A, B, and C) in Problem 3.5 address algorithm

development, albeit in slightly different ways. Question A instructs students to “write,”

“test,” and, if necessary, “change” an algorithm. Questions B and C provides

opportunities in which to “use” the developed algorithm; Question C contains the

additional task of noting patterns in the products obtained from algorithm use.

Enacted Curriculum

In the enacted curriculum, the final day of the week (i.e., Day 5) ends with the

discussion included in Example 39:

Example 39 (EC).

(1)

T: Okay, now. This is what I see, and tell me if you see this. I see a
couple of diflerent ways to think about this. I see this way [points to an
example in which a mixed number was changed to an improper fraction
and then the traditional algorithm was utilized] where we take any mixed
number or any whole number and write it as a fi'action, as an improper
fraction, and then just use our algorithm - hold on a minute - and then I
see another way. I see kind of picking it apart [points to an example in
which numbers are decomposed, multiplied, and then put back together]
and saying ”Ten and a half twos, ten and a half of thirds, " or really
picking it apart. I see this sort of as one way [points to example where
mixed number was changed to improper fraction], and this as another
way [points to example in which mixed number is decomposed]. What are
you thinking about those two strategies at this point? What are you
thinking about them? Is there one that you’re like, "Whoa, I like that one.
I'm going to use that one. " Or are you kind of like, "Hmmm, I still want to
think about it. " Tell me where you're at.

69

(2) SI .' I think thefirst one is more efficient.
( 3) Ss: Yeah. yes.

(4) T: This one [points to example in which the mixed number is changed to
an improperfraction]? More efficient?

(5} Ss: Yeah.
(6) S2: I like the ten and a half-

( 7) T: You like this one [points to example in which mixed number is
decomposed]? Okay?

In this discussion, two “different ways to think about” multiplication of combinations of
fractions, whole numbers, and mixed numbers are compared. Students develop these
strategies/algorithms and a student proposes “efficient” when explaining why he prefers
one strategy/algorithm to the other. Therefore, I argue that the enacted curriculum
addresses this goal in spite of the fact that Problem 3.4 is not enacted in the classroom. I
would also argue that the development of the goal is more extensive in the written
curriculum. In terms of objectification, the same comments from Goal 7 apply also to
Goal 8 because both address algorithm development.
Summary

The goal of this chapter is to provide evidence to address two main questions: (1)
What does an investigation of the goals in the written and enacted curricula allow us to
see? (2) What do we know now about the relationship between the written and enacted
curricula that we did not know before? This investigation allows us to see whether or not
the goals stated in the written curriculum receive similar emphasis in the enacted
curriculum. What we know now that we did not know before is that the mathematics in

some of the goals receive similar treatment in the written and enacted curricula while the

70

mathematics in other goals receive less emphasis in the enacted curriculum than in the
written curriculum.

The statements in this summary and throughout this analysis are made with
several caveats. First, this analysis compares the written text with an enactment of the
written text (one of infinitely many possible enactments); this should be kept in mind
when reading these statements as some results may be attributed to this difference in
curricular form. Second, similarity (and difference) here is through my eyes only. That
is, another person (e. g., a teacher, a textbook author) using their own lens may see things
quite differently. Finally, the evidence for my claims is gleaned from five days in the
written and enacted curricula. That is, none of my statements can be generalized either to
the written curriculum as a whole or the enacted curriculum as a whole. Rather, my
statements highlight insights gained through the use of this framework regarding the
relationship between the written and enacted curriculum on these five days that may be of
interest to teachers, curriculum developers, and mathematics education researchers.26

Of the eight goals included in the written curriculum for this Investigation, four
receive “similar” emphases in the written and enacted curricula. That is, students are
given opportunities to interact with these goals in ways closely aligned with the
suggestions in the written curriculum:

Goal 2: Use models to represent the product of two fractions

Goal 3: Understand that finding a fraction ‘of’ a number means multiplication

Goal 5: Determine when multiplication is an appropriate operation

Goal 7: Develop and use algorithms for multiplying combinations of fractions,
whole numbers, and mixed numbers

 

26 These caveats will be included as a footnote in each of the chapter summaries and the final discussion
(Chapter 9) to emphasize the importance of these statements.

7l

The remaining four goals receive greater attention in the written curriculum than in the
enacted curriculum. The degree of discrepancy varies among these four goals. One of
these goals displays a “low level of discrepancy” between the written and enacted
curricula. That is, qualitatively the goal is addressed in similar ways; however, the
amount of time spent on the goal is slightly less in the enacted curriculum than in the
written curriculum:

Goal 4: Develop and use strategies and models for multiplying combinations of
fractions, whole numbers, and mixed numbers to solve problems

Finally, three goals display a higher level of discrepancy between the written and enacted
curriculum. That is, several experiences that address these goals in the written
curriculum are not enacted in the classroom on the days examined in this study:

Goal 1 : Estimate products of fiactions

Goal 6: Explore the relationships between two numbers and their product

Goal 8: Develop and use an efficient algorithm to solve any fraction

multiplication problem

The differential treatment of these goals may provide different opportunities for learning
mathematics than was intended by the authors of the written curriculum.27 In particular,
the lack of emphasis on Goals 1 and 6 in the enacted curriculum, which address
estimating products of fractions and the relationship between the magnitude of the factors
and the product of fraction multiplication, may limit students’ access to opportunities for
the objectification of fractions. That is, both of these goals encourage the use of fractions
as numbers and promote discourse associated with numbers. As mentioned earlier in this

chapter, objectification of fractions (i.e., their use as numbers) provides an important step

in the process of expanding the domain of numbers in which students are able to function.

 

7 ‘ . . ,, . . . . . . . .
2 Recall that ‘learning mathematics here IS conceptualized as a change In partrcrpation in mathematical
discourse.

72

This investigation of the goals has set the stage for a more detailed examination of
the discursive practices present in the written and enacted curricula. Subsequent chapters
address the relationships between four key features of mathematical discourse in the
written and enacted curricula: (1) Mathematical words, (2) Visual mediators, (3)

Endorsed narratives, and (4) Mathematical routines.

73

CHAPTER 5: MATHEMATICAL WORDS
IN THE WRITTEN AND ENACTED CURRICULA

Mathematical words, as described previously, are largely words that signify
quantities and shapes (e. g., “number,” “triangle”). These words describe the
mathematical objects or the products of mathematical discourse. In addition to these
mathematical “product” words, here I also include words that signify mathematical

59 G6

processes (e.g., “multiply, estimate”) for consideration in this analysis. Because
“learning mathematics” is conceptualized here as changes in participation in

mathematical discourse, a close look at the uses of mathematical words seems essential

for this analysis. Figure 4 provides one way to represent the process of word

 

  
   
 

    
 

  

development:
Passive Routine- Phrase- Obj ect-
use driven driven driven
use use use

 

 

 

 

 

 

 

 

Figure 4. Four-stage model of the development of word use. (Sfard, 2008, p. 236)

An individual engages in “passive word use” when she can respond to questions about
the word, but does not actually utter the word herself. The final three stages are
characterized by “active word use.” That is, the individual uses the word in her own
utterances. She may begin by using the word within very particular discursive sequences
- “routine-driven use.” As her experience with the word increases, she may begin to use
it more flexibly, but still primarily in a limited number of phrases (i.e., “phrase-driven

use”). Finally, she will use the word as if it has a life of its own (object—driven use). This

74

is the primary goal of word development as described here, so more must be said about
this stage.

The “object-driven use” stage of this model is characterized by the objectification
of the word in question (i.e., using it as a noun). Recall that objectification is defined as a
process in which a word begins to be used as if it signifies an extra-discursive, self-
sustained entity (object), independent of human agency. The process of objectification
consists of two closely related, but not inseparable sub-processes: reification and
alienation. Reification involves the replacement of talk about processes with talk about
objects. Alienation is the use of discursive forms that present phenomena in an
impersonal way, as if they are occurring of themselves, without the participation of
human beings. That is, objectification, the final stage of word use, includes both talking
about products rather than processes and describing the mathematical objects void of
human agency. For example, “I multiplied the numerators,” is a processual, personalized
use of “multiply.” It describes something that the student is doing. In contrast,
“Multiplication is the opposite of division,” uses “multiply in a structural (i.e.,
objectified) way because multiplication is a noun arid the sentence is void of human
agency.

Throughout this analysis of mathematical words, it is important to keep in mind
that written and enacted forms of curricula are being compared. This may impact the
results of this analysis. For example, the words in the written curriculum are solely in
written form while the words in the enacted curriculum are primarily spoken. The words
used in written text may differ from those in spoken text due to the nature of the mode; it

is impossible to predict the impact of this difference on the findings presented here. In

addition, the voice of the written text is largely that of the textbook authors whereas the
classroom voice includes that of the teacher and students. It might be expected that these
voices would differ in their word use (e.g., formal word use may be more common in
textbooks than in classrooms).

Because it would be impossible to examine each and every mathematical word in
the written and enacted curricula, I analyzed only the mathematical words that seem
“central” to this Investigation. The title of the Investigation, “Multiplying with
Fractions,” provided two important words: “multiplying” and “fractions.” I investigated
all derivatives of “multiplying” (e. g., “multiplication”) and other words that signify
multiplication (e.g., “times”) that were present in either the written or enacted curricula.

For “fractions,” I also investigated the singular version as well as any specific case of a

99 ‘6

fraction (e.g., “two thirds," “3’ ) and words such as “fifths, parts, and “pieces." I

used the goals of the Investigation as stated in the written curriculum to make further
decisions regarding the key mathematical words.28 Table 3 presents these goals by
Problem. Words signifying multiplication and fractions have been underlined to illustrate

their prevalence.

 

28 This process of choosing key words using the goals ofthe Investigation is but one possible method for
word selection. I argue, however, that this process provides a set of important words in the Investigation. 1
do not argue that my key word list is comprehensive.

76

Table 3.

I Goals ofProblems 3. I -3.5 in “Investigation 3: Multiplying Fractions " (as Stated in the

Written Curriculum) with “Multiplication and “Fractions and Their Derivatives

Underlined

 

Problem

3.1

3.2

3.3

3.4

3.5

Goal
Estimate products of fractions
Use models to represent the product of two fractions

Understand that finding a fraction gt” a number means

 

multiplication

Estimate products of fractions

Use models to represent the M of two fractions

Understand that finding a fraction of a number means
multiplication

Estimate products of fractions

Develop and use strategies and models for multiplying
combinations of fractions, whole numbers, and mixed numbers to
solve problems

Determine when multiplication is an appropriate operation
Explore the relationships between two numbers and their Mg
Develop and use algorithms for multiplying combinations of
fractions, whole numbers, and mixed numbers

Develop and use an efficient algorithm to solve any fraction

multiplication problem

 

77

Words signifying “multiplication” (appearing 14 times in the goals) and “fraction(s)”
(appearing 10 times in the goals) figured prominently throughout the goals. In addition,
“estimate” appears as the leading word in the first goal in Lessons 3.1, 3.2, and 3.3,
“models” appears in the goals of Lessons 3.], 3.2, and 3.3, “number” appears nine times
within the goals, and “algorithm” appears in the goals for Lessons 3.4 and 3.5.
Therefore, these words were also selected for analysis. Table 4 illustrates the goals
again, this time with these additional key mathematical words underlined.

Table 4.

Goals of Problems 3. I -3.5 in Investigation 3: Multiplying Fractions (as Stated in the

Written Curriculum) with Key Mathematical Words and Their Derivatives Underlined

 

Lesson Goal

 

3.1 ° Estimate products of fractions
° Use models to represent the product of two fractions

° Understand that finding a fraction Q[ a number means

 

multiplication

3.2 ' Estimate products of fractions
0 Use models to represent the product of two fractions

° Understand that finding a fraction Q[ a number means

 

multiplication

 

78

Table 4 (cont’d)

 

Lesson Goal

 

3.3 ° Estimate products of fractions
° Develop and use strategies and m for multiplying
combinations of fractions, whole numbers, and mixed numbers to
solve problems
° Determine when multiplication is an appropriate operation
3.4 ° Explore the relationships between two numbers and their M
0 Develop and use algorithms for multiplying combinations of
fractions, whole numbers, and mixed numbers

3.5 ° Develop and use an efficient algorithm to solve any fraction

multiplication problem

 

In this chapter, I will address these words in increasing order of frequency in the goals

9, 66

(i.e., from “algorithm, estimate,” “number,” “fraction,” to “multiplication”). “Number”

is included in Words Signifying Fractions because most of the numbers in this analysis

are fractions. In addition, specrfic fractions (e. g., “3 ”) Will be discussed in the same

section. “Models” will be addressed in the next chapter, Visual Mediators in the Written
and Enacted Curriculum, when both references to models and the use of models will be

analyzed.

79

Words Signifying Algorithm
“Algorithm(s)” appears 30 times in the written curriculum and 24 times in the
enacted curriculum.29 Twenty-six of the occurrences of “algorithm(s)” in the written
curriculum (87%) are in Lesson 3.5 (Day 5). In contrast, uses of “algorithm(s)” are
spread fairly evenly across Days 3-5 (8, 7, and 8 occurrences respectively) in the enacted
curriculum. The three most common actions associated with “algorithm(s)” in both the
written and enacted curriculum were writing/developing, using, and testing/checking.

Figure 5 indicates the relative frequencies of each action in the respective curriculum.

 

I Written Curriculum
[3 Enacted Curriculum

Relative Frequency
N
U1

    

 

 

Write/Deve lop Use Test/Check

 

Figure 5. Relative frequencies of uses of “algorithm(s)” in the written and enacted
curricula.
Figure 5 indicates that “writing/developing” algorithms is the most prevalent action in the

written curriculum whereas “writing/developing” and “using” algorithms occur in nearly

 

29 It is important to remember that the word use described here reflects only the five days under
consideration in these analyses and cannot be generalized to statements regarding the overall word use in
the written or enacted curriculum. That is, it is not possible to know whether this word use is “typical” for
the textbooks or the classroom outside of these five days.

80

equal proportions in the enacted curriculum. Table 5 provides examples to demonstrate

the use of “algorithm(s)” in these three categories in each curriculum.

Table 5.

Examples ofthe Use of ”A lgorithm " in the Written and Enacted Curricula

 

Action Written Curriculum

Enacted Curriculum

 

Write/ “Write an algorithm that will work
Develop
for multiplying any Mofractions.
including mixed numbers. (SG, p.

39)

Use "Some students may use the
traditional algorithm but also
develop special rules for special

situations. ” (TG, p. 83).

Test/ “Test your algorithm on the
Check

problems in the table. ” (SG, p. 39)

T: “We’re going to try to write an
algorithm, an algorithm for
multiplying, but we 're going to kind
of do this one as a big class. So, if
you finish B early, come on over and
help us get started to write an
algorithm for what we think so far. "
(Day 3)

T: “Well, if we, if we use our
algorithm, we said if I have any two
fractions, I can multiply across the
numerator and multiply across the
denominator. so that would be
twenty halves, right? " (Day 5)

T: “Can, can I just make sure that
our algorithm worked here? Did,
did you guys check that? Did our

algorithm work? ” (Day 5)

 

8l

In addition to the explicit use of"algorithm(s),” both the written and enacted
curricula contain references to algorithms without including the word itself. For
example, both curricula include descriptions of the traditional (and other) algorithms for
multiplying fractions. The examples included here illustrate such references:

Example 40 (WC).

“Double the denominator because you are making pieces one-half'as large, or

two times smaller. ” (TG, p. 67)

Example 41 (EC).

T: “Do you think that these are the steps that we should tape, take? Multiply the

numerators and multiply the denominators, then you put them together as a

fraction. ” (Day 3)
Comparing or evaluating algorithms is evidenced in both curricula as well. In fact, the
goal of Problem 3.5 in the written curriculum states “Develop and use an efficient
algorithm to solve any fraction multiplication problem " (TG, p. 81). The use of
“efficient” here seems to indicate something beyond developing an algorithm that works.
Rather it alludes to the need for an algorithm that has specific characteristics. The
following example from the written curriculum indicates that the authors see comparing
algorithms as a useful enterprise:

Example 42 (WC).

“Evaluating whether each algorithm is useable and helpful, and how it compares

with other algorithms, will further students ' understanding of multiplication of

fractions, mixed numbers. and whole numbers. ” (T G, p. 82)
Here, the written curriculum advocates for the evaluation of each algorithm and

comparing algorithms. A discussion on Day 5 in the enacted curricula compares two

possible algorithms:

82

Example 43 (EC).

(1)

(2)
( 3)
(4)

(5)
( 6)
( 7)

T: Okay, now. This is what I see, and tell me if you see this. I see a
couple ofdifferent ways to think about this. I see this way [points to an
example in which a mixed number was changed to an improper fraction
and then the traditional algorithm was utilized] where we take any mixed
number or any whole number and write it as afraction, as an improper

fraction, and then just use our algorithm - hold on a minute - and then I

see another way. I see kind of picking it apart [points to an example in
which numbers are decomposed, multiplied, and then put back together]
and saying "Ten and a half twos, ten and a half of thirds, " or really
picking it apart. I see this sort of as one way [points to example where
mixed number was changed to improper fraction], and this as another
way [points to example in which mixed number is decomposed]. What are
you thinking about those two strategies at this point? What are you
thinking about them? Is there one that you're like, "Whoa, I like that one.
I ’m going to use that one. " Or are you kind of like, "Hmmm, I still want to
think about it. " Tell me where you’re at.

S I .' I think thefirst one is more efficient.
Ss: Yeah, yes.

T: This one [points to example in which mixed number is changed to an
improper fraction ] ? More eflicient?

Ss: Yeah.

S2: I like the ten and a half -

T: You like this one [points to example in which mixed number is
decomposed]? Okay?

As in the written curriculum, “efficient” is used in the enacted curriculum as a way to

describe one of the possible algorithms. Neither the written nor the enacted curricula are

explicit about what makes an algorithm “efficient” in the days I considered in this

analysis.

On Day 2 in the enacted curriculum after the class has completed several fraction

multiplication number sentences using models, Trevor notices the pattern that leads him

to suggest the traditional algorithm:

83

Example 44 (EC).
S: "I think it 's times the numerator and numerator and then times the
denominator and denominator. "
From this moment forward, Trevor’s name is often attached to this algorithm in the
enacted curriculum. In contrast to the written curriculum which uses “algorithm” or
simply describes the process without naming it, the traditional algorithm takes on several
other labels in the enacted curriculum:
Example 45 (EC).
T: “Okay, guys. Okay. Listen up, please, okay. What we 're going to do now is
we 're going to take a peek at these, keeping in mind T revor's idea that he
brought up earlier about "Hmmm, I wonder if we can just multiply and

multiply. " (Day 2)

Example 46 (EC).
S: “I know that Trevor's way works. ” (Day 3)

Example 47 (EC).
S: “Yeah, but you can 't, Trevor's method doesn't apply with wholes. " (Day 2)

In Examples 45-47, the algorithm is referred to as “Trevor’s idea,” “Trevor’s way,” and
“Trevor’s method.” This is one of many examples in the enacted curriculum in which
mathematics is portrayed as personal and happening in the here and now. This personal
nature of mathematics seems to take precedence over the historical nature of mathematics
(i.e., that someone discovered this algorithm prior to Trevor’s observation). As
mentioned previously, this personalization of mathematics is also found in the written
curriculum when fictitious students suggest strategies. One such example is presented

here:

84

Example 48 (WC).

 

C. Takoda answers Question A part (1) by doing the following:
I l l
(2 X 15) + (z X '5)
1. Do you think Takoda's strategy works? Explain.

2. Try Takoda‘s strategy on parts (2) and (5) in Question A. Does his
strategy work? Why or why not?

 

 

 

This reference to the distributive property is presented as a students’ proposed strategy
rather than as a historically established mathematical property. This is in sharp contrast
to the way in which the distributive property might be presented in a more traditional
mathematics textbook. Recall that depersonalization is a requirement for objectification.
Therefore, this personalization of the traditional algorithm and the distributive property,
although no doubt used for particular purposes, might be seen as unsupportive of the
process of objectification. As mentioned in Chapter 4, the movement toward algorithms
for operating on fractions seems to provide an opportunity for reification because
algorithms are largely performed on numbers. Therefore, the use of “algorithm” in both
curricula is personalized and offers opportunities for reification; the former does not
necessarily support objectification but the latter may.
Words Signifying Estimation

“Estimate” and its derivatives (e.g., “estimation”) are present in both the written
and enacted curricula, appearing 50 and 23 times respectively, more than twice as often
in the written curriculum as in the enacted curriculum. Table 6 includes the derivatives
of “estimate”, the parts of speech they represent, and examples of their use in each

curriculum.

85

Table 6.

Examples of Uses of “Estimate " and its Derivatives in the Written and Enacted

 

 

Curricula
Part of
Word Speech Written Curriculum Enacted Curriculum
Estimate Noun “Remind students that they S: “Yeah. That's what I
can use their estimate as a got, too. I got sixteen for
guide for a reasonable my estimate. ” (Day 4)
answer when they
compute. " (TG, p. 66)
Verb “Estimate each product to S: “How did they estimate
the nearest whole number to get the same answer? ”
(I, 2, 3, . . .). ” (SG, p. 36) (Day 4)
Under- Noun “It would be an -----
estimate
underestimate because I
1 ,”
rounded I g down to 1.
(TG, p. 76)
Over- Noun .. ‘7. 1 .....
estimate Would the estimate of 23

be an underestimate or an

overestimate? ”(T G, p. 76)

 

86

Table 6 (cont’d)

 

 

Part of
Word Speech Written Curriculum Enacted Curriculum
Estimated Verb “ Who can explain how . T: “We estimated how
they estimated 1 g x much it would be if we put
a couple of fractions or a
2 % ? ” (TG' P- 72) couple of decimals
together. ” (Day 1)
Estimating Verb ----- T: “Graham, how could we
think about estimating that
answer? (Day 4)
Estimation Noun “F or parts (a)—(d). use -----

estimation to decide if the

product is greater than or
1 ., ..
less than 3. (SG, p. 35)

Adjective Finish up by having -----
students share their
estimation strategies in

Question D. (T G, p. 62)

 

Note. Dashes (i.e., “ ----- “) indicate that the word does not appear as this part of speech in the designated
curriculum.

It is interesting to note that only four of the eight forms of “estimate” present in the
written curriculum appear in the enacted curriculum. That is, there are no explicit

references to “underestimate,” “overestimate,” or “estimation” (as a noun or an adjective)

87

in the enacted curriculum. It is also notable that, although not captured in Table 6, 19 of
the 23 instances of “estimate” and its derivatives (83%) in the enacted curriculum occur
on Day 4, whereas occurrences in the written curriculum occur over the first four days
(Problems 3.1-3.4) in relatively consistent numbers (ranging from 8 to 14). Recall that
“object-driven” word use, the final stage in the word development process, is found only
when words are used as nouns (i.e., “object-like”). Figure 6 summarizes the relative

frequencies of the parts of speech presented in Table 6.

Written Curriculum Enacted Curriculum
Adjective Adjective
6% 0%
x/‘f (I /

,/ Noun
43%

Verb .

Noun i.

o .

62% 5M \

 

Figure 6. Relative frequencies of the parts of speech of “estimate” and its derivatives in
the written and enacted curricula.

Figure 6 indicates that objectification of “estimate” (i.e., nouns) occurs more often in the
written curriculum than in the enacted curriculum, representing 62% and 43% in each
curriculum respectively. This seems to indicate that “estimate” is objectified more often
in the written curriculum than in the enacted curriculum. That is, for the class,
“estimation” is largely something you “do” and not an object to be discussed in its own

right.

The enacted curriculum contains other less formal words and phrases associated
with estimation. This is not surprising since, as mentioned earlier, it is more common to
use formal language in spoken than in written mode. Examples of this other language

99 6‘

(i.e., “about, almost,” “round up") are provided here:

Example 49 (EC).
T; “So you would call this about three? " (Day 4)

Example 50 (EC).
S: “Well, two thirds is pretty much almost half ” (Day I )

Example 51 (EC).
S: “Round the eleven twelfths up to twelve twelfths so the two would be three. ”
(Day 4)

Closely related to “estimation” and its derivatives are situations found in the written
curriculum in which questions are posed regarding whether, for example, the product of a
fraction multiplication problem is greater or less than 1. In fact, the written curriculum
recommends that students “use estimation to solve” such problems (SG, pp. 33, 35). An
example of this type of discussion is included here from Problem 3.1 (Day 1) of the

written curriculum along with a related discussion from Day 2 of the enacted curricula:

Example 52 (WC).

“What is g of 1 whole? (part of a whole, or —3)

Does this help you estimate whether I:— x I is greater than or less than I ? (Yes, it

is less than I)

Now think about I x g . Is this greater than I ? (no)

Is it greater than 5? (no)

89

, 5 I I . , . 5 1 . .
So ts g x 3 greater than 3 ? (No. Iou have/11st g of 3, which ts less than

fl

5) ”3” (TG, pp. 60-61)

Example 53 (EC).

( I ) T: Let me give you an example. I have, mmmm, a half of a submarine
sandwich left. and I eat a third of it. I eat one third of one half [writes

“ éxé "j. Okay. Did I eat more or less than a whole submarine

sandwich [spreads hands apart to indicate the length of a whole
submarine sandwich]?

(2 ) Ss: Less.

(3 ) T: Okay. Now I need you to explain why. How did you know I ate less?
Connor, how did you know it was less than a whole?

( 4) S I : Um -
(5) T: Not sure about that one? How do we know? Maria?

(6) $2: Because. um, when you times the denominators you get kind of like a
big, um, well

( 7) T: Without multiplying it can I know? Or do I have to multiply it? Do I
have to do Trevor 's idea and solve it, or can I know just by looking at it?

(8) Ss: Just by looking at it.

( 9) T: Just by looking at it?

(IO) S3: Just the whole, you only had half left in the first place.
( I I ) T: Tell Maria. What do you think about that?

(12) S3: She said out of a whole, but then there was half gone, there was only
half left out of the whole, and if you're, if you have, if you're taking

( I 3) S4: Even if you don 't eat any of it, it would still be less than a whole
anyway because there was only a half left.

 

3° The parenthetical components in this example are possible question responses provided in the written
curriculum.

90

(l 4) T: Does that make sense? I started with a half of a sub, and I 'm eating a
part of that part. Am I ever going to get the whole sub?

(I 5) S4: No, because you have half already, and, uh. then you have to. uh,
take away more, so you'd have less than half.

( I 6) T: Okay. Ashle i, did you want to add something to Grace 's?

( I 7) S5: You're just taking the half and cutting it into thirds and taking one
out of the three thirds.

(I 8) T: Exactly.

( I 9) S5 : You're taking less than

(20) T: Yeah. I ’m taking apart of what 's already a part.
Example 52 and 53 illustrate the use of estimation strategies for determining the
magnitude of the product of two fractions. Again, differences are present in terms of
objectification. Example 52 (from the written curriculum) uses primarily language
associated with numbers (i.e., reified fractions), such as “greater than 1” and “less than
1.” Example 53 (from the enacted curriculum) uses language not associated with
reification, such as “more than a whole, “less than a whole.” The uses in the written
curriculum treat fractions as numbers by comparing them with numbers. The uses in the
enacted curriculum treat fractions as pieces or parts. This supports our finding earlier in
this section that objectification of “estimation” is more common in the written curriculum
than in the enacted curriculum.

Words Signifying Fractions

As discussed previously (see pp. 30-31), “fraction” has been historically

problematic because it is used in many different ways.3 ' For example, it is used when

referring to a rational number itself (i.e., as a number) or as one of many representations

 

3‘ When fraction appears within quotation marks (i.e., “fraction”), 1 am referring to the word, when it
appears without quotation marks, I am referring to the fractions themselves.

91

of a rational number (i.e., as a numeral). The glossary of the Student’s Guide of Bits and
Pieces 1 (the first unit addressing fractions in Connected Mathematics) provides several

definitions explaining how “fraction” is used in the written curriculum. The glossary

, . .. .. . a
defines a “fraction" as a number (quantity) of the form b where a and b are whole

numbers,” an “improper fraction” as “a fraction in which the numerator is larger than the
denominator” and as “a fraction that is greater than 1,” and a “mixed number” as “a

number that is written with both a whole number and a fraction” (p. 74-75). By defining
“fraction” as “a number (quantity) of the form g” and defining improper fractions and

mixed numbers in terms of this definition, it is not clear whether “fraction” is used as a
number or a particular “form” of a number (i.e., a numeral). Since there is no mention in
either the written or enacted curriculum of alternative “forms” or representations of these
numbers (e.g., decimals) nor is there any mention of “rational number,” it seems that both
curricula are treating fractions as numbers rather than as numerals. For the purposes of

this study, I will extend the definitions provided in the written curriculum to include
. - 3 - ' cc ° 99
fractions equivalent to one (e. g., 3) as improper fractions and use proper fraction only

when referring to fractions less than 1.

The analysis of words signifying fractions begins with an examination of the

99 ‘6 99

words “fraction, piece, “part,” and “number” and their derivatives in the written and
enacted curricula. “Number” is included here for several reasons: (1) The numbers in
question in this Investigation are primarily fractions or parts of fractions, (2) The written
curriculum defines a fraction as a number, and (3) The fraction literature (e. g., Kieran,

1976; Lamon, 1999) characterizes fractions in several ways including as parts of a whole

92

and as numbers. The final analysis in this section will examine the use of proper and
improper fractions and mixed numbers in the written and enacted curricula.
Uses of “fraction ”in the Curricula
Four common categories of the use of “fraction” and its derivatives (e. g.,
“fractioning”) emerged from the written and enacted curricula, including fraction as a

number, a part, an adjective, and a verb. Table 7 provides examples of each category of

use in both curricula.

Table 7.

Examples of Categories of “Fraction " Use in the Written and Enacted Curricula
Category Written Curriculum Enacted Curriculum
Number “The strategy introduced in the S: “One thing I don 7 get is, you

Getting Ready involves changing the know how when we said eight times
form of mixed numbers and whole seven and we showed like the
numbers so students can operate in picture. Like, that wouldn't work
the same way as when both factors with fractions, would it? ” (Day 2)

are fractions. ” (TG, p. 75 )

 

93

Table 7 (cont’d)

 

Category Written Curriculum

Enacted Curriculum

 

Part “What fraction of the goal does

Nikki raise? " (SG, p. 36)

Adjective ”Sometimes, they have to find a

fractional part of another fiaction. "

(so, p. 32)

Verb -----

S: "Um, well, there, we knew that
she, there was, it was half full, so
we split it into half, and then we,
she bought two thirds of it, so we
shaded, we made thirds first, and
then we shaded in two, and then,
um, she, it said what fraction of the
pan she bought, and she would buy
two sixths of the pan because if you
split all these into thirds you'll get
two sixths. (Day I)

T: “ You can buy any fractional
part of a pan of brownies and pay
that fraction of twelve dollars. "
(Day 1)

T: "Maybe you’re going to have a
picture of something and be
fractioning off parts or whatever. "

(Day 4)

 

Note. Dashes (i.e., “ ----- “) indicate that the category of use of“fraction” is not present in the designated

curriculum.

94

I distinguished between the "number” and "part” categories by determining which word
(i.e., “number” or "part") could replace “fraction” in each sentence. Even though
“fraction" is used as a noun in both of these categories. the “number” category is further
along in the word development process (see Figure 4 on p. 75) because when “fraction” is
used as “part,” it is used only in specific types of phrases (e.g., “fraction of the whole
pan,” “fraction of a fundraising goal”). Therefore, it is only when “fraction” is used as a
number that it is objectified and discussed as if it has a life of its own. The fraction as
number example from the written curriculum refers to fractions as factors and certainly
factors are objects in their own right. In the fraction as number example from the enacted
curriculum, the student is comparing fraction multiplication to whole number
multiplication which reifies “fraction.” In contrast, the other examples use “fraction” and
its derivatives as processes or descriptors which does not fulfill the reification
requirement of objectification. Figure 7 summarizes the relative frequency of each

category in the curricula.

Written Curriculum Enacted Curriculum
Verb
_ _ Verb Adjective o
Adjective 0% 4% V0

[3%

  

  
  

Number
53%

Number \\
72%

Figure 7. Relative frequencies of categories of “fraction” use in the written and enacted

curricula.

95

Figure 7 indicates that “fraction“ as number is the most common use of “fraction” in both
the written and enacted curricula; however, its use as number represents nearly three-
fourths of the uses in the written curriculum compared to slightly more than half in the
enacted curriculum. “Fraction” as part is more common in the enacted curriculum than in
the written curriculum (42% compared to 15%) which indicates that “fraction” and its
derivatives are objectified more often in the written curriculum than in the enacted
curriculum.
Uses of “piece” and “part” in the Curricula

Both “part(s)” and “piece(s)” are present in the written and enacted curricula.32
“Part(s)” was much more common than “piece(s)” in the written curriculum (72
appearances compared to 11 appearances). The opposite was true in the enacted
curriculum; “Piece(s)” was nearly twice as prevalent as “part(s)” (238 appearances
compared to 120 appearances). Both “part(s)” and “piece(s)” are included in the
following excerpts from the written and enacted curricula:

Example 54 (WC).

“T he é-sized section of the thermometer is divided into four equal parts. When

the rest of the thermometer is sectioned into pieces of the same size, there are six

pieces. Then g of g on the thermometer is g of the distance fiom 0 to I. ” (T G,

p. 65)

 

32 Only instances of“part(s)” and “piece(s)” in mathematical context were included in this analysis.

96

Example 55 (EC).

S: “ What we did is, it was in one sixths, so we split it up into six pieces, and then
they had just this part, and then all this part was gone, and they had, we split
this up into ten pieces. and then they had nine of them, they bought nine of
them. and then there was this one, and then we figured to figure out how
many, um, we would have for the denominator, we had this in ten pieces, so
we split each of the other six into ten pieces, so we got sixty, and then we had
nine in here and then all the rest wasn't there so we got nine sixtieths. " (Day
2)

The significance of this distinction is not clear; however, it provides another example of
differences in fractional discourse between the written and enacted curricula. It seems
that “piece(s),” more prevalent in the enacted curriculum, is less obviously “fraction” talk
than “part(s).” That is, “part” might more often be used with a question such as “What
part of the cake was eaten?” which elicits a fraction answer; whereas, “piece” is more
likely to be used in a question such as “How many pieces?” which elicits a whole number
answer. If this conjecture is true, then the fractional discourse is more advanced in the
written curriculum than in the enacted curriculum because “pieces” is more common in
the enacted curriculum than “parts.” Closely related to pieces and parts are instances of
words such as “thirds,” “fifteenths,” etc. The written curriculum contains 7 distinct
words of this type with 30 total appearances. The enacted curriculum included 14 words
of this type with 217 total appearances.33 Examples are included here from both
curricula:

Example 56 (WC).

“The alternative approach below leads to twelfths as the unit rather than sixths.

In this approach, you first partition a thermometer or number line into thirds and
label the thirds. ” (T G, p. 65)

 

33 The only instances counted here were those with no number (e.g., “two,” or “2”) in front ofthe word
(e.g., “thirds”).

97

Example 5 7 (EC).
S: “I think they 're all in sixths because you had them into fifths and then you cut
them into thirds so now they're all in sixths so you have, let's, er. fifteenths. "

(Day 1)
The prevalence of these types of words in the enacted curriculum seems consistent with
the prevalence of “piece(s)” in the enacted curriculum because these words indicate a
particular fractional piece.
Uses of “number” in the Curricula
Uses of “number(s)” in the written and enacted curricula occur in the word phrases

99 6‘

“number line,” “number sentence, whole number,” and “mixed number,” as well as
several instances of “number(s)” standing alone. When “number(s)” is used outside of
the word phrases it refers to numbers more generally. Examples from the written and
enacted curricula are included here:
Example 58 (WC).
“Question C provides additional multiplication practice and looks at the result of
multiplying a number by its reciprocal. ” (T G, p. 81)
Example 59 (EC).
T: “Is it? Is multiplication always commutative, or just with some numbers? ”
(Day 4)
In Example 58, the “numbers” include fractions, whole numbers, and mixed numbers. In

Example 59, the teacher is asking about how broadly the commutative property of

multiplication can be applied. The Question under consideration when this question is

. l 4 . . .
asked is “2 2 x 7 Statements/Questions such as these may serve to facrlitate the

objectification of fractions because they are included here under the broad heading of
“numbers.” Figure 8 summarizes the relative frequencies of these uses in the written and

enacted curricula.

98

 

 

i WrierfCEiculum

Blinded giggling.

Relative Frequency

 

Number Mixed Whole Number Number
Number Number Litre Sentence

 

 

 

Figure 8. Relative frequencies of uses of “number(s)” in the written and enacted
curricula.

Figure 8 indicates several similarities and differences in the use of “number(s)” in the
written and enacted curricula. “Mixed number” and “whole number” occur in similar
proportions in the curricula (ranging from 22% to 30%). In contrast, “number line” and
“number sentence” are present in quite different proportions in the curricula (ranging
from 0% to 29%).34 In the case of “number(s)” standing alone, which as described above
may facilitate the objectification of fractions, the relative frequencies in the written and
enacted curricula are similar.

F ractions in the Curricula

. . . . . . 3 4
The final analysrs in this section examines the use of fractions (e.g., “4 ,” “3 ,”

“3 % ”) in the written and enacted curricula. Examples of proper fractions, improper

fractions, and mixed numbers are present in both curricula. Nearly all cases of fractions

 

34 These differences will be expanded upon in Chapter 6: Visual Mediators in the Written and Enacted
Curricula.

99

in the written curriculum are in symbolic form, whereas the enacted curriculum includes
both symbolic form (written on paper, transparencies, posters, etc.) and spoken form. In
spoken form, it is often impossible to determine whether “two thirds” refers to “two
thirds” (i.e., two pieces that are thirds) or “two-thirds” (i.e., a rational number). For the
purposes of this analysis, all instances of the form “two thirds” are included as fractions;
however, a closer investigation of this problem is included later in this section. I
recorded each distinct proper and improper fraction and mixed number and noted the
number of times they appeared in each curriculum. Figure 9 indicates the number of

distinct fractions in each of the three categories included in the curricula.35

  
   

  

I Written Curriculum
E1 Enacted Curriculum

 

Number of distinct forms

Proper Fraction Improper Mixed Number
Fraction

 

 

 

Figure 9. Frequencies of distinct proper and improper fractions and mixed numbers in
the written and enacted curricula.

Figure 9 indicates that in both the written and enacted curricula the number of distinct
proper fiactions is the greatest, followed by the number of distinct mixed numbers, and

finally the number of distinct improper fractions. The largest discrepancy between the

 

. . 2 6 . . . . . .
35 Equrvalent fractions (e.g., 3,3) are consrdered distinct forms in this analysrs.

100

written and enacted curricula is in the number of distinct proper fractions, 72 in the
written curriculum compared to 56 in the enacted curriculum. Figure 10 summarizes the
relative frequencies of the total number of appearances of these forms in the written and

enacted curricula.

      

Written Curriculum Enacted Curriculum
Mixed
Number
. 0
Mixed Improper 13A)
Number Fraction
35% 4%
Proper
Fraction
57%
Improper
Fraction Proper
8% Fraction

83%

Figure 10. Relative frequencies of the total number of appearances of proper and
improper fractions and mixed numbers in the written and enacted curricula.

Figure 10 indicates that the relative frequency of the appearance of each type varies
substantially. For example, proper fractions represent 57% of all fractions present in the
written curriculum compared to 83% in the enacted curriculum, whereas mixed numbers
represent 35% of all fractions present in the written curriculum compared to just 13% in
the enacted curriculum. This may indicate that in this Investigation, students have more
experience with proper fractions and less experience with improper fractions and mixed
numbers than intended by the authors of the written curriculum. In terms of
objectification, the extension from multiplication of fractions to include improper

fractions and mixed numbers may facilitate reification (and therefore objectification) by

101

problematizing the “part of part” discursive pattern. For example. it is possible to speak

about M%x% as "-;—of 4” (i.e., part of a part) indefinitely. Therefore, if only proper

fractions are provided as examples, the reification of fractions may never happen.

However, it is much more problematic to speak about “%x%” as “%of § .” This

problematizing is important because it may facilitate the reification of fractions.

Table 8 summarizes particular fractions in the three categories which represent at

least 5% of the total number of fractions in each curriculum.

Table 8.

Fractions Representing at Least 5% of the Total Number of Fractions in the Written and

Enacted Curricula

Written Curriculum

Enacted Curriculum

 

 

Relative Relative
Category Fraction Frequency Fraction Frequency
Proper Fractions 1 l
— 1 4% —- 27%
2
2
— 1 0% — l 7%
3
1 l
— 7% — 14%
3 3
l l
— 6% — 6%
4 4
3 3
— 5% — 6%
4 4

 

|02

Table 8 (cont’d)

 

 

 

Written Curriculum Enacted Curriculum
Relative Relative
Category Fraction Frequency Fraction Frequency
Improper Fractions No improper fraction represented more than 1% of

the total number of fractions in either the written or

enacted curriculum.

Mixed Numbers

6% 5%

Nl—‘Nl—‘

6%

 

Table 8 indicates that the most common proper fractions in the written and enacted
curricula are the same. In fact, even their order of prevalence is the same. It is notable;
however, that the three most common proper fractions are approximately twice as
prevalent in the enacted curriculum as in the written curriculum. When taken as a whole,
these common fractions make up 42% of all proper fractions that appear in the written
curriculum and 70% of all proper fractions included in the enacted curriculum. This
statistic seems to indicate that students are having less exposure to a variety of proper
fractions than the authors intended. Particular improper fractions make up a very small
percentage (less than 1%) of the total number of fractions in either curriculum. Particular
mixed numbers are also not very prevalent and those that appear most often in the written
and enacted curricula are different.

As mentioned previously, reification of fractions is dependent upon the

encapsulation of the two numbers in the fraction (e. g., “4” and “7”) into one number

103

6

(e.g., ‘ 7 ”). Evrdence of such reification in assocration With proper and improper

fractions and mixed numbers is rare in the enacted curriculum because nearly all fraction
discourse is of the form “X of Y” where X is a fraction (e.g., two thirds) and Y is either
another fraction or a contextual object (e. g., acre of land). Examples such as the ones

presented here are ubiquitous throughout the five days:

Example 60 (EC).
S: “Because he wants to buy one half of the pan that is two thirds full. ” (Day 1)

Example 61 (EC).
T: “One third of a half, right? So what are you starting with, if you have one
third of a half, what are you starting with? ” (Day 3)
Example 62 (EC).
S: “A nd then, then we cut each of them into thirds, and colored in two thirds of
each sixteenths. ” (Day 5)
This prevalence of “X of Y” is also found in the written curriculum in Problems 3.1-3.3;
however, Problems 3.4 and 3.5 seem to contain less of this use. This may be partly
attributed to the fact that the problems on the last two days are much less contextualized.
One feature of both the written and enacted curricula that might lead to the
reification of fractions is the use of number sentences. That is, once the contextual
situation (representing “X of Y”) is translated into a number sentence the fractions can be
treated primarily as numbers. However, this does not seem to be the case in Problems
3.1-3.3 in the written curriculum or in the enacted curriculum because the number
sentences are so closely associated with the contexts and the iconic mediators. When the
discussion in the enacted curriculum turns to the algorithm, the iconic mediators are used

to demonstrate its reasonableness and this leads to additional references to fractions as “X

of Y.” The discussion of how the algorithm is represented in the iconic mediator for the

104

number sentence “%x% = % " taken from Day 2 in the enacted curriculum provides a

typical example of these discussions. The student work (including the iconic mediator

and number sentence) and conversation associated with the Question are included in

Example 63. Recall that “Trevor’s idea" is the traditional algorithm for multiplying

fractions.

Example 63 (EC - Student Work).

 

 

 

 

 

(1)

(2)
(3)

( 4)
(5)

( 6)
( 7)

T: Does that make sense? Thanks, Grace. So if I think about Trevor ’s
idea, in every one of thefourths [points to each of thefourths in the
model], how many pieces did she split it into? How many pieces are in
each?

Ss: Three.

T: Three. So could that explain the, four groups of three in our
denominator [points to the “12”in the number sentence]?

Ss: Yeah.

T: How about in each of those threefourths [points to the three one-
fourth pieces in the model ], how many did she color in?

Ss: One.

T: One here, one here, and one here [points to each of the one-fourth
pieces in the model]. Would that be one times three?

105

( 8) Ss: Yeah.
Note that this discussion includes “pieces,” “coloring,” and “groups of” which are more
indicative of the “X of Y” use of fractions. Further evidence that number sentences are
closely linked to their associated iconic mediators is provided by statements in the written
curriculum that advocate drawing connections between the number sentence, algorithm,
and iconic mediator:

Example 64 (WC).

“If students should happen to notice that they can multiply the numerators and

multiply the denominators, ask them to use their drawings to show why they

think this works. ” (T G, p. 60).

Example 65 (WC).

“Students might point out that the numbers give the right answer. If students do,

direct them to consider both the numbers and the brownie pans. ” (T G, p. 61).
It can certainly be argued that using iconic mediators to establish the traditional algorithm
for fraction multiplication serves important purposes mathematically; however, it does
not seem to facilitate the objectification of fractions in terms of reification or alienation
because the discussions addressing algorithms are so closely linked to the actions of
students.

In both the written and enacted curricula, estimation discussions seem to facilitate

the objectification of fractions. For example, the following “Getting Ready” in Problem

3.3 from the written curriculum provides such opportunities:

106

Example 66 (WC).

 

 

Getting Ready for Problem®

Estimate each product to the nearest whole number (1. 2. 3. . . .).

i 1 1 2. i :1 ' i u
5x210 l2xzio 22x7 34X212
Will the actual product be greater than or less than your whole number

estimate?

 

 

 

 

Here, the question asks “Will the actual product be greater than or less than your whole
number estimate?” Numbers are “greater than or less than,” therefore, the use of these
words promotes the objectification of fractions. This type of language is used in several
places in the written curriculum in association with estimation. Another example is
included here:

Example 67 (WC).
“gm 3 and é—of 3 is I g which is greater than I. ” (T G, p. 62)

. . l . . I . . 1
Again in Example 67, 15 “is greater than” 1 seems to imply the reification of “l — ” as a
mathematical object in its own right rather than as a whole number (i.e., l) and a fraction
(i.e., 2 ). This is not the case for all estimation problems in the written curriculum;

however, it is notable that such instances are common in the written curriculum in
association with estimation.
Some language used in association with estimation in the enacted curriculum also

indicates the possible reification of fractions. The following statement is made by a

student that is rounding up 2% to 3:

107

Example 68 (EC).
S: “Because if I was going to round up the nine tenths like Graham did so " (Day
4)

“Rounding up” is language associated with numbers. That is, 2% (and 120- itself) seems
to be used here as a number. Another example comes from a student who is trying to

. 1
multiply 42 by 1:

Example 69 (EC).
S: “Well I think you’d have to times it 'cause one times any number is always
itself so I think it'd be about four and a half because ” (Day 4)

This student states explicitly that 4;— is a number and therefore multiplying it by 1 would

“be about four and a half.” The next excerpt, taken from the same day indicates the
complexity of determining whether or not reification has occurred. The Question being

discussed is “—1— x 21”:
2 10

Example 70 (EC).

(1) T: Could you help me with an estimate? If I said I was going to get one
half of two and nine tenths, Graham, how could we think about estimating
that answer?

(2) S1: Well. nine tenths, that's close to a whole, so -

(3 ) S2: One.

(4) S3: Yeah, so

(5 ) T: So what, Graham?

( 6) 82: Nine tenths is close to a whole.

( 7) T: Okay. So you could round that up into a whole and then two would be

a three. So you would call this [points to 2% ] about three?

108

. . . . . . 9
This excerpt 1S interesting because the student seems to be reifying “E” because he says

“nine tenths is” in Line 6. This is in contrast to saying “nine tenths are” in which the
plural verb is used because the nine is plural. In addition, the teacher uses “round that
up” in Line 7 which also tends to be used in cases where a fraction is reified because
numbers are rounded up. The complexity comes from these indications of reification
combined with the use of “whole” in Lines 1, 5, and 6 by both the teacher and the

students. Discourse indicative of reification would use “one” instead of “whole” to

. . 9 . .
indicate that “a,” because it IS a number, should be compared to another number.

Words Signifying Multiplication
In the analysis of “multiplication,” all versions of the words “multiplication” (e.g.,

“multiplying”) were considered as were several other words that signified multiplication.

99 66 9’ 66

These included “times, product, group,” and “of.” In all cases, only uses of each
word directly related to multiplication were included in the analysis. Figure 11 presents a
summary of the relative frequencies of the use of mathematical words that signify

multiplication in the written and enacted curricula.

109

Written Curriculum Enacted Curriculum

Multiplication
Multiplication 10%
32%

   
 
 

   

Times
Of 17%
44%
Product
’ Times Of 0%
_ 2% 65% GrouPS
Groups Product 3%

0% 22%

Figure 1 I . Relative frequencies of words signifying multiplication in the written and
enacted curricula.
As Figure 11 indicates, “of” is the most common word signifying multiplication in both
the written and the enacted curricula; however, proportionally it is more prevalent in the
enacted curriculum. Forms of “multiplication” and “product” are common in the written
curriculum (32% and 22% respectively), while forms of “multiplication,” “times,” and
“groups” occur in substantial numbers in the enacted curriculum (10%, 17%, and 8%
respectively).
Uses of “product” in the Curricula

“Product,” while used actively in the written curriculum is used passively in the
enacted curriculum (see Figure 4 on p. 75). That is, the class (both students and teacher)
access statements using “product” from the written curriculum throughout the lessons;
however, the only utterance of “product” in the enacted curriculum occurs when a student
is reading aloud from the written curriculum. Also interesting is that “product,” being a
noun, is objectified. The uses of “product” can be characterized as both reified and

alienated. For example, “Use estimation to decide if the product is greater than or less

110

than %. " occurs in Problem 3.2 in the Student Guide (p. 35). In this sentence, “product”

is presented as an object and is void of human agency. Therefore, when comparing the
written curriculum and the enacted curriculum, it is notable that “product” is used
passively (the first stage in the word development model) in the enacted curriculum,
while objectified (the final stage in the word development model) in the written
curriculum.
Uses of “group ”in the Curricula

“Group” and its derivatives are virtually absent in the written curriculum,
occurring only once in the teachers guide, “Other students group the I 6 oz into sets of 3
and take two out of each set. (T G, p. 74) In the enacted curriculum, “group” (also
appearing as “groups” and “grouped”) occurs 75 times. Of these, 73 instances (97%)
occur in one of three primary contexts: (1) Drawing connections between multiplication
of whole numbers and multiplication of fractions, (2) Confirming the algorithm for
multiplying fractions, and (3) Discussing the multiplication of mixed numbers. In all
three of these contexts, “group” is used in the following form: “X groups of Y” where X
and Y are numbers. It is notable that it is always used in conjunction with “of.” This is
most likely a residual discursive pattern from whole number multiplication where it is
commonly used (e. g., “four groups of six”). Table 9 shows the relative frequency of

these instances along with an example of their use.

111

Table 9.

Examples of Uses of ”group in the Enacted Curriculum

 

 

Relative
Context Frequency - Example
Multiplication of whole numbers 23% S: “Because it 's zero groups of

six so you don ’t have any. ”
(Day 1)
Algorithm for multiplying fractions 15% T: “Where do I have ten
groups ofsix? (Day 2)
Multiplication of mixed numbers 59% T: How could we find three
and a quarter groups of two

and eleven twelfths? (Day 4)

 

These three categories appear at distinct times in the enacted curriculum. The instances
related strictly to the multiplication of whole numbers occur during the Launch on the
first day of the Investigation.36 The second category occurs within a discussion of how
the algorithm for multiplying fractions is represented in buying parts of brownie pans
primarily on the second day of the Investigation. The instances in the final category
occur primarily on Days 4 and 5 while multiplying combinations of fractions, whole
numbers, and mixed numbers. The frequent use of “group” in the enacted curriculum and
the lack of its use in the written curriculum indicate a difference in the ways in which

multiplication is being talked about in the two curricula. Again, in this framework in

 

36 Recall that the Launch is the first component ofthe Launch-Explore-Summary sequence of Connected
Mathematics lessons.

112

which learning mathematics is defined as a change in participation in mathematical
discourse, such differences may play a role in the change.
Uses of “times” in the Curricula

“Times” and its derivatives are much more prevalent in the enacted curriculum
than in the written curriculum (17% compared to 2%). Four of the seven instances of
“times” in the written curriculum are statements in which “times” is in between two
objects being multiplied. Provided here are two examples:

Example 71 (WC).

“Yes. Takota broke the 2% into two parts and multiplied Ig- separately by each

part. First he multiplied I-é— by 2, and then he multiplied 1% timesé. ”(TG, p.

80)

Example 72 (WC).

“Creating models for multiplication problems that involve combinations of
fractions, whole numbers, and mixed numbers is a bit more complex than fraction

times fraction situations. " (T G, p. 71)

In the other three instances, “times” follows a number. Two examples are provided here:

Example 73 (WC):

“It is four times as great as the answer to g of g. (T G, p. 6 7)

Example 74 (WC):
“You can repeatedly add 2% 10 times for the 10 days. ” (TG, p. 74).

Both of these categories of use (Examples 71-74) seem to indicate a lack of
objectification of fraction multiplication. In all four cases, the uses are operational (i.e.,
not structural) and personalized (i.e., not alienated). For instance, Example 74 describes

a procedure (i.e., “repeatedly add”) carried out by a person “You.”

113

Of the 151 instances of “times” (including two instances of “timesed” and four
instances of “timesing”) in the enacted curriculum, 1 15 (76%) are of the first form
mentioned above (i.e., an object “times” an object). These objects are present as number
words (e. g., “two” or “two thirds”) or names of numbers (e. g., “numerator,” “mixed

number”). Two examples are included here:

Example 75 (EC).

T: “Oh, I just set you up. It 's like I paid you to say that. That's exactly what
we 're going to figure out tomorrow. What happens if I don 't just have a
fi‘action of a fraction, if I don 't have a part of a part? What if I have a
fraction of a whole number? What if I have a fraction of a mixed number?
What if I have a mixed number times a mixed number? ” (Day 3)

Example 76 (EC).

T: “Okay. I tell you what. Let's look at one more and let's see if we can just
figure out what it’s saying. Not necessarily how to solve it, but what is it
saying if I have three and a fourth times two and eleven twelfths? What does
that mean? What does that mean? ” (Day 4)

The two most common types of objects being “timesed” in the enacted curriculum are
whole numbers (74 instances) and fractions (1 8 instances)” The second type mentioned
above (i.e., “times” following a number) occurs 26 times (17% of all instances of
“times”) in the enacted curriculum. Two examples are given here:

Example 77 (EC).
S: “It's like maybe we have to add two and eleven twelfths like, three times, and
then another fourth time? ” (Day 4)

Example 78 (EC).

S: “Um, I knew I had to do two and a third ten and a half times, and I found that
one half of two and one third was one and one sixth, so, um, I added all the
wholes, which got me twenty-one, and then I added all the thirds, and three
thirds is one whole, and I had one whole, two wholes, and then I had three
wholes and a sixth, and a, one, a third and one sixth, and um, I can 't add those
so I changed the third to two sixths and I added those and it was three sixths.
so altogether I got twenty-four and three sixths. ” (Day 5)

 

37 This whole number multiplication occurred either during the Launch which led into fraction
multiplication or as a component of fraction multiplication (e.g., multiplying a whole number numerator
times a whole number numerator).

114

In summary, the qualitative nature of the use of “times” in the written and enacted
curricula is quite similar, primarily taking one of two forms, either between two numbers
(e. g., three “times” four) or after a number (e.g., three four “times”). However,
proportionally, “times” is much more common in the enacted curriculum than in the
written curriculum (17% compared to 2%). Again, both of these forms are operational
rather than structural and personal rather than alienated, and therefore are not likely to
facilitate objectification of fraction multiplication.
Uses of “multiplication ”in the Curricula

“Multiplication” and its derivatives are more common in the written curriculum
than in the enacted curriculum (32% compared to 10%). Four forms appear in both
curricula: “multiply,” “multiplied,” “multiplying,” and “multiplication.” Figure 12

summarizes their relative frequencies in each curriculum.

Written Curriculum Enacted Curriculum
Multiplication Mult lication
36%) Multme 1P

27°/ 20%
0

  

Multiply

_ _ 48%

Multiplied Multiplying
8% 25% '

    

Multiplying Multiplied
29% 7%

Figure 12. Relative frequencies of “multiplication” and its derivatives in the written and

enacted curricula.

115

Figure 12 indicates that the operational forms of “multiplication” (i.e., “multiply,”
“multiplied,” and “multiplying”) are more prevalent than the structural form (i.e.,
“multiplication”) in both curricula; however, the relative frequency of the structural
forms varies, 36% and 20% in the written and enacted curriculum respectively. Recall
from Figure 4 (p. 75) that object-driven word use is the final stage in word development
and therefore the desired result. The fact that the written curriculum includes the
objectified form of the word (i.e., “multiplication”) more frequently than the enacted
curriculum seems notable. Two examples of the use of “multiplication” from the written
curriculum are included here:

Example 79 (WC).

“In this investigation, you will relate what you already know about multiplication

to situations involving fractions ” (SG, p. 32).

Example 80 (WC).

“This helps students decide whether multiplication will help solve a problem in

other situations ” (TG, p. 60).
In both instances, multiplication is not presented strictly as a prOcess, rather as an entity
in and of itself to be discussed. In these cases (and others like them in which
multiplication is a noun), the word is objectified. Remember that the other requirement
for objectification is alienation (i.e., no person needed). This requirement is also fulfilled
here because multiplication is not presented as something to be “done” by a person in
these instances. Similar examples of objectified use of “multiplication” occur in the

enacted curricula:

Example 81 (EC).
S: “Um, division is like the opposite of multiplication ” (Day 5)

Example 82 (EC).
T: “Is multiplication always commutative. or just with some numbers? " (Day 4)

116

A closer look at the word use in terms of the part of speech represented revealed
statements in the written and enacted curricula in which “multiplication” served as an
adjective rather than a noun. For example, the following statements in which
multiplication functions as an adjective occur in the written and enacted curricula
respectively :

Example 83 (WC).
“Question C provides additional multiplication practice and looks at the result of
multiplying a number by its reciprocal " (T G, p. 81).
Example 84 (EC).
T: “Just like I wrote three groups of eight, I wrote like this, groups of I put a
multiplication symbol in” (Day I).
Figure 13 quantifies the relative frequencies of the parts of speech (i.e., noun, verb, and

adjective) of each use of “multiplication” and its derivatives in the written and enacted

curricula.

Written Curriculum Enacted Curriculum
NO?" Adjective Noun Adfétwe
174’ 20% 17% °

    

     

Verb Vegb
63% 80 /o

Figure I 3. Relative frequencies of the parts of speech of “multiplication” and its

derivatives in the written and enacted curricula.

Figure 13 indicates that multiplication is objectified (used as a noun) in equal frequency

in the two curricula.

117

Uses of “of” in the Curricula

When all uses of “of” in the written and enacted curricula were counted, it was the
third most common word in both curricula making up nearly 3% of all words in each text.
For this analysis, however, only mathematical uses of “of” were considered. Even when
considering only these instances, Figure 11 (p. 111) indicates that “of” was the most
common word associated with multiplication in both the written and enacted curricula,
44% and 65% of all references to multiplication respectively. In fact, a stated goal in
Problems 3.2 and 3.3 in the written curriculum is “Understand that finding a fraction of a
number means multiplication.38 Three statements explicitly addressing the meaning of
“of” in fraction multiplication are present in the enacted curriculum. For example,

Example 85 (EC).

T: “Okay. W e 're going to come back and look at all these [a series of

mathematical statements of the form A of B = C, where A, B, and C are proper
fractions], 'cause you know what I'm going to do? I'm going to change that word

"of'[points to “of" in “—of1 g— = 6—90- ”] to a mathematic symbol. What you guys
have done here. I love the way that you're all looking at me, I can tell you're

listening. Thank you very much. When I take a part [points to “£— ” in

2 l
— —=—— 0 another art oints to “—" “—o —=— ’ ,Ican
‘70f3 21”]f plp 3 7f} 22]]
wr:ite that as a multiplication problem [writes “X” over “of” in
_ _ = — ,9 H Da 2
:01f3 221 ] ( ay )

In Example 85, “of” is replaced by “X” in a series of fraction multiplication number
sentences to indicate the relationship between “of” and multiplication in these cases.
Approximately half of all instances of “of” in the written and enacted curricula

are related to multiplication, 47% and 53% respectively. When examined for discursive

 

38 For a discussion ofthis goal in the written and enacted curricula, see pp. 50-54.

118

patterns, several categories of use emerged. These categories, along with their relative

frequencies in the written and enacted curricula are provided in Figure 14.

 

I Written Curriculum
[Enacted Curriculum

Relative Frequency

 

Number“of' Number"of‘ "(‘toups of” "Fractionol" “Partof'
Number Non-Number

 

 

 

Figure 14. Relative frequencies of mathematical uses of “of” in the written and enacted
curricula.
Figure 14 indicates Number "of" Number is the most common usage of “of” in both the
written and enacted curricula, representing 40% and 47% of all mathematical uses of “of”
respectively. The majority of these (65% in both curricula) are Fraction “of” Fraction,
followed by Fraction “of” Whole Number (25% and 18% in the written and enacted
curriculum respectively). Examples of these types from both curricula are provided here:

Example 86 (WC).

“How much is éof é ? ” (SG, p. 32)

Example 87 (EC).

S: “Two sixths of half ” (Day 1)

Example 88 (WC).
“Since 2% is almost 3, a reasonable estimate would be éof 3 or 1%. " (T G, p.
71)

119

Example 89 (EC).
T: “So what would one third of sixteen be? " (Day 4)

Figure 14 also indicates that the use of “of” in Number “of" Non-Number
situations is quite common in both the written and enacted curricula, 19% and 24%
respectively. The most common way in which this appears in both curricula is as a
Fraction “ofa(n) " Contextual Word (86% and 70% respectively). Examples from the

written and enacted curriculum are provided here:

Example 90 (WC).
“Mali owns % of an acre of land ” (SG, p. 35)

Example 91 (EC).
S: “That is nine tenths of a mile long. ” (Day 3)

Two other categories of Number “of’ Non-Number situations occur in reasonably
substantial numbers in the enacted curriculum. Approximately 14% and 16% of the
instances are Fraction “of a part ” and Fraction “of a whole ” respectively. Examples of
these two types from the enacted curriculum are provided here:

Example 92 (EC).

T: “You want one third of just this orange part. How can we think about doing

that? ” (Day 3)

Example 93 (EC).
T: “Half of the whole thing, right? ”(Day 3)

The “Part of” category is much more common in the written curriculum than the
enacted curriculum (26% compared to 10%). The most common form of this category is
“part of a part " in both the written and enacted curricula (33% and 44% respectively).

Also common in the written curriculum are “part of a whole ” (30%) and “part of a(n) ”

120

Contextual Word (24%). Examples of these three types are provided here from the
written curriculum:

Example 94 (WC).

“As you work on this problem, think about the size of the answer when you are

finding a part of a part. ”(T G, p. 60)

Example 95 (WC).
“How many parts of the whole are being bought?” ( T G, p. 61)

Example 96 (WC).
“ Use a different colored pencil to show the part of the brownies that Mr.
Williams buys. ” (SG, p. 33)
Besides “part of a part, " two other categories are present in more than 10% of the “Part
of” category in the enacted curriculum: “part of ’ fraction (20%) and “part of a whole”

(14%). Examples of all three types from the enacted curriculum are given here:

Example 97 (EC).

S: “They got part of a part. ”(Day 2)

Example 98 (EC).

S: “Ifilled in half because I knew it was going to be half of a bar, and then I
made three parts of that half and [filled in one half er, one part of that half
and I got one sixth, because if you still had this half there 'd be these three
pieces. ” (Day 3)

Example 99 (EC).
T: “Did they take a part of the whole? ” (Day 2)

“Fraction of ” which is not very common in either curriculum (10% and 6% in
the written and enacted curricula respectively) is nearly twice as prevalent in the written
curriculum as in the enacted curriculum. The most common form of this type in the
written curriculum is “Fraction of a(n) ” Contextual word, representing 69% of the
appearances in this category. In the enacted curriculum, the most common form (also
representing 69% of the instances) is “Fraction of a whole.” Examples of the most

common types in each curriculum are included here:

121

Example 100 (WC).
“What fraction of the goal did Nikki raise? " (SG, p. 36) "

Example 101 (EC).
T: “So then what fraction of this whole thing are they? ” (Day 3)

Interestingly, “Groups of” which represents 9% of the mathematical instances of
“of” in the enacted curriculum is absent in the written curriculum. Examples of its use in
the enacted curriculum are provided here:
Example 102 (EC).
T: “W e 're not, Trevor, we 're not subtracting. We ’re taking groups - I told you
this was going to be tough today, right? Now we 're not even having part of a
part. Now we have groups of a part. W hooo. I told you we were going to

have to think extra hard today. Elliot?” (Day 4)

Example 103 (EC).
T: “Three and one fourth groups of two and eleven twelfths. ” (Day 4)

Overall, the most striking similarity between the use of “of” in the written and
enacted curriculum is the sheer quantity of its use in both curricula. Beyond this, the
categories of use of “of” in the curricula are also quite similar. [It seems that the use of
“of” does little to facilitate objectification of fraction multiplication except in that it has
the potential to move students toward situations in which fractions become encapsulated
as mathematical objects.

Summary

The question addressed in this summary is, “What does an investigation of the

mathematical words in the written and enacted curricula allow us to see?” That is, “What

do we know now about the relationship between the written and enacted curricula that we

I22

did not know bef‘or‘e?"39 Table 10 summarizes the findings from the investigation of
mathematical words for the purposes of examination for overall conclusions.

Table 10.

Summary of “Mathematical Words" Analysis

 

 

Word Written Curriculum Enacted Curriculum
“Algorithm” 30 appearances 24 appearances
87% on Day 5 Spread evenly across Days 3-5
“Writing” algorithms most “Writing” (25%) and “Using”
common (43%) (29%) algorithms most common

Both include references without use of “algorithm”
Both expect comparison or evaluation of algorithms

(including “efficient” as an adjective)

Suggests strategies for Uses strategies suggested in the
facilitating the discovery of the written curriculum and student
traditional algorithm notices the traditional algorithm

Personalizes proposed strategies Personalizes traditional algorithm
through the use by connecting

of fictitious students student’s name to it

 

 

39 The statements in this summary and throughout this analysis are made with several caveats. First, this
analysis compares the written text with an enactment of the written text (one of infinitely many possible
enactments); this should be kept in mind when reading these statements as some results may be attributed
to this difference in curricular form. Second, similarity (and difference) here is through my eyes only.
That is, another person (e.g., a teacher, a textbook author) using their own lens may see things quite
differently. Finally, the evidence for my claims is gleaned from five days in the written and enacted
curricula. That is, none of my statements can be generalized either to the written curriculum as a whole or
the enacted curriculum as a whole. Rather, my statements highlight insights gained through the use of this
framework regarding the relationship between the written and enacted curriculum on these five days that
may be of interest to teachers, curriculum developers, and mathematics education researchers.

123

Table 10 (cont’d)

Word Written Curriculum Enacted Curriculum
“Estimation” 50 appearances

23 appearances
Spread evenly across Days 1-4

83% on Day 4
Eight forms of “estimate”

Four forms of “estimate”
62% objectified

43% objectified
When using estimation strategies,

When using estimation strategies,
uses discourse that promotes

uses discourse supporting
fraction reification

fraction as a descriptor
(e.g., “greater than”)

(e.g., “whole”)

Includes other discourse

associated with estimation

(e.g., “round up”)
“Fraction” Most often used Most often used
as number (72%) as number (53%) and part (42%)
“Piece” 72 appearances of “part” 1 l appearances of “part”
”Part” 120 appearances of “piece” 238 appearances of “piece”
“Halves”
“Thirds” 7 distinct words with 30 14 distinct words with 217
“F ourths” appearances
etc.

appearances

 

124

Table 10 (cont’d)

 

 

Word Written Curriculum Enacted Curriculum
“Number” Most common in “mixed Most common in “whole number
number” (29%) and “number (30%) and “number
line” (26%) sentence” (29%)

Similar frequency (18% compared to 16%) of use of “number” alone
(i.e., not in phrase such as “number line”)
Fraction 72% of distinct fractions 56% of distinct fractions

(e.g., £6 99) are pI'Oper fractions are proper fractions

wIN

(remainder are improper fractions (remainder are improper fractions
or mixed numbers) or mixed numbers)
57% of all fractions are proper 83% of all fractions are proper
fractions (remainder are improper fractions (remainder are improper
fractions or mixed numbers) fractions or mixed numbers)

Both include the same five most common proper fractions

Five most common fractions Five most common fractions
represent 42% represent 70%
of all proper fractions of all proper fractions

Prevalence of “X of Y” (X and Y Prevalence of “X of Y” (X and Y
are fractions) primarily on Days are fractions) throughout all five

1-3 days

 

125

Table 10 (cont’d)

 

 

Word Written Curriculum Enacted Curriculum
“Product" 22% of words signifying 0% of words signifying
multiplication multiplication
Objectified use Passive use
“Group" 0% of words signifying 8% of words signifying
multiplication multiplication

----- All uses in “X groups of Y”
----- Most common use is
with multiplication
of mixed number (59%)
“Times” 17% of words signifying 2% of words signifying
multiplication multiplication
Both include uses of two types
(i.e., “three times four” and “three four times”)
“Multiplication” 32% of words signifying 10% of words signifying
multiplication multiplication
Both include four forms of “multiplication”
Both include more operational use than structural

(i.e., objectified use represents 17% in both curricula)

 

126

Table 10 (cont’d)

 

Word Written Curriculum Enacted Curriculum

 

“Of " Third most common word
in the written and enacted curricula
44% of words signifying 65% of words signifying
multiplication multiplication
Both explicitly and implicitly relate “of” to multiplication
Both use “of” most often in “X of Y” where X and Y are numbers
(40% and 47% respectively). In 65% of the cases of “X of Y”

in both curricula, X and Y are fractions

 

Note. Dashes (i.e., “ ----- “) indicate that the category is not applicable to the designated curriculum.

Table 10 and the more detailed analysis included within the chapter highlight many
similarities and differences in the uses of particular mathematical words between the
written and enacted curricula. The problem with such a table is that some of the richness
and interpretation present within the text of the chapter is lost. However, the summary
provided within the table allows a look at the data as a whole.

With object-driven word use (see Figure 4 on p. 75) as a stated goal for the use of
mathematical words, it seems that, according to the evidence presented in the chapter and
summarized in Table 10, the written curriculum affords objectification in association with
more words than the enacted curriculum. That is, although some words are used and
objectified in similar ways in both curricula (e.g., “algorithm,” “multiplication), other

word use that affords objectification (e.g., “estimation,” “product”) are more common in

127

the written curriculum whereas word use that seems to offer less opportunities for

. . . ,, ,, , . .. . . 4o
objectification (e.g., group, ‘ times ) are more prevalent in the enacted curriculum.

 

40 These differences are highlighted in Table 10.

128

CHAPTER 6: VISUAL MEDIATORS
IN THE WRITTEN AND ENACTED CURRICULA

In this chapter, I describe the use of visual mediators in the written and enacted
curricula. Recall that visual mediators are defined as symbolic artifacts created for the
purposes of mathematical communication. Given the ubiquitous nature of visual
mediators in mathematics at all levels, it is critical to investigate their use as part of this
analysis. Sfard (2008) recognizes three categories of visual mediators: (1) Concrete
(e.g., blocks, fraction strips), (2) Iconic (e.g., pictures, diagrams), and (3) Symbolic (e.g.,
“3 x 4 = 12,” “2x + y = 14”). These three types may be seen as a trajectory through
which students move in school mathematics. That is, concrete mediators are much more
common in early elementary school, whereas symbolic mediators are the norm in most
high school mathematics courses. Iconic mediators are often present beginning in early
elementary through high school, increasing in complexity over the years. The decision to
use “visual mediator” instead of the more commonly used “representation” comes from
the fact that “representation” implies that there is a “real” mathematical object which is
being represented. In this analysis, I am conceptualizing mathematical objects strictly as
discursive constructs. Therefore, the visual mediator is as much the mathematical object
as anything else.

Several additional features of visual mediators are worth noting. First, the power
of visual mediators lies in their use, rather than in the artifact itself. For example,
marshmallows and toothpicks serve little mathematical purpose until they are put
together in such a way that allows an individual to make sense of mathematical
terminology such as “vertex” and “edge.” Second, visual mediators are extraordinarily

useful for facilitating discussions around particular mathematical topics. That is,

129

examining a graph, table, and equation allows an individual to examine distinctive
features of a function. Each of these visual mediators serves certain purposes of their
own and making connections between them can be even more enlightening. For
example, discussions regarding the domain and range of a function may be facilitated by
relating their symbolic and graphical forms. Third, all visual mediators are not created
equal. Some visual mediators are more or less useful for particular students with
particular topics in particular situations. That is, the usefulness of visual mediators is
context dependent. Finally, the use of visual mediators in curricula involves many
questions leading to pedagogical decisions, both by curriculum developers and teachers,
such as: (1) Which visual mediators should be used to introduce and develop a particular
mathematical topic? (2) How should the specific visual mediators be used? (3) When
should students be expected to transition between visual mediators (e.g., concrete
mediators and symbolic mediators)?

Sfard (2008) introduces two important goals for visual mediation in mathematical
discourse. The first goal for students is “mediational diversity.” That is, the ability to
use a variety of visual mediators flexibly. She argues that students who display
“mediational diversity” are better prepared to learn mathematics. Metaphorically, the
mediationally flexible student can be compared to a person building a house with an

extensive toolkit rather just a hammer and nails. The second goal for visual mediation is
to move students toward symbolic form (e.g., 4 x 12 = 9 ). Other Visual mediators

(e. g., concrete objects) are used to make sense of mathematical topics, but the goal in
school mathematics is for individuals to operate mathematically in symbolic notation as

much as possible. This is a great advantage over carrying marshmallows and toothpicks

130

or unifix cubes with them from place to place. That being said, concrete mediators
remain the norm in many situations outside of formal mathematics.

In this chapter, I will examine both the words associated with visual mediators (an
extension of the previous chapter) and the actual use of the visual mediators themselves
in the written and enacted curricula.

Words Signifying Visual Mediators

Both the written and enacted curricula include references to visual mediators in
general as well as references to specific types of visual mediators. Table 11 provides
examples of each category from the curricula.

Table 11.
Examples of General and Specific References to Visual Mediators in the Written and

Enacted Curricula

 

 

Category Written Curriculum Enacted Curriculum
General “Create a model or diagram to find T: “As - whatever model you feel
the exact answer. ” (SG, p. 36) comfortable with, that 'sfine. (Day
4)
Specific “Draw a number line under the S: “So can we show it like anything,
thermometer. ” (TG, p. 65 ) like brownie pans or anything?”
(Day 4)

 

There are many cases in which the context provides information that indicates that the
statement which appears to be a “general” reference when only the utterance is
considered is actually a reference to a specific type of visual mediator. Such an example

from the enacted curriculum is provided here:

131

Example 104 (EC).
T: “Are you showing that anywhere in your drawing " (Day I)

In this particular excerpt the teacher is referring to a drawing of a brownie pan model
presented by a student. In these instances, the reference is categorized as “specific.“
Figure 15 illustrates the relative frequencies of “general” and “specific” references to

visual mediators in the written and enacted curricula.

 
      

Written Curriculum Enacted Curriculum
General
General 14%
" 26% “‘\\
\g .

   
 

\

l

Sim“C Specific
74% 86%

Figure 15. Relative frequencies of general and specific references to visual mediators in
the written and enacted curricula.

Figure 15 indicates that both the written and enacted curricula have more references to
specific visual mediators (e.g., number line) than visual mediators in general. The
difference between the two types, however, is more pronounced in the enacted
curriculum. It seems that more general references to visual mediators would facilitate the
goal of mediational diversity especially when the general references mention the option

of choosing an appropriate model.

132

As mentioned previously, Sfard (2008) describes three types of visual mediators,
concrete, iconic, and symbolic. Figure 16 provides a summary of the relative frequencies

of references to each of these types in the written and enacted curricula.“

Written Curriculum Enacted Curriculum
Symbolic Concrete Concrete
6% 0% Symbolic 1%

18%

    

   

Iconic
81%

Iconic

94%
Figure 16. Relative frequencies of references to concrete, iconic, and symbolic visual
mediators in the written and enacted curricula.
Figure 16 indicates that the vast majority of references to visual mediators in the written
and enacted curricula refer to iconic mediators; however, this difference is more marked
in the written curriculum than in the enacted. In fact, the written curriculum includes no
mention of concrete mediators and only six references to symbolic mediators in this
Investigation.42 In the remainder of this section, I provide details of the analysis of
references to each of the three categories of visual mediators and will end with a

summary of the verb use associated with these visual mediators.

 

4' Remember that these are word references and do not include the actual visual mediators themselves.
42Remember that the analysis of the enacted curriculum relied on videotapes of the lessons. As mentioned
previously, in most cases the video camera followed the teacher. Therefore, it is unknown whether other
groups used concrete mediators in their work. Here, as throughout this analysis, 1 relied completely on
what was captured on the videotapes and in student work.

133

Words Signifying Concrete Mediators
Only one reference to a concrete mediator appears in the enacted curriculum (i.e.,
“blocks”) and no such reference exists in the written curriculum. The reference on Day 4
of the enacted curriculum is included here:
Example 105 (EC).
T: “How could we do that? How could we get two thirds of all of these wholes?
We have sixteen whole blocks here. How much, could you get a half of
them? ”
Words Signifying Iconic Mediators
Ninety-four percent and 81% of the references to visual mediators in the written
and enacted curricula respectively refer to iconic mediators (see Figure 16). The general
words used to signify the iconic mediators include “diagram(s),” “drawing(s),”
“model(s),” and “picture(s).” The written curriculum and enacted curriculum contain 85

and 95 appearances of these words respectively. Figure 17 summarizes the relative

. . . . 4 '
frequencres of these words in the written and enacted curricula. 3

 

43 This figure only presents the uses ofthese words as nouns/objects. Verb use associated with visual
mediators will be described later in the section.

134

Written Curriculum Enacted Curriculum

P icture Diagram Diagram
9% | 5% P icture 3%

‘ ‘\\ _ 24% - ‘

\ Drawrng
12%

   

   
 

Drawing
45%

Model
28%

Mode l
64%

Figure 17. Words signifying iconic mediators in the written and enacted curricula.
Figure 17 indicates that the written curriculum uses “model(s)” most often to signify an
iconic mediator, while the enacted curriculum most often uses “drawing(s).” The written
curriculum uses the other words relatively infrequently; “diagrarn(s)” is the second most
common representing 15% of the words signifying iconic mediators in the written
curriculum. In contrast, two other words, “picture(s)” and “model(s),” each represent
approximately 25% of the words signifying iconic mediators in the enacted curriculum.
The significance of these findings is not obvious; however, because learning mathematics
is conceptualized here as changes in mathematical discourse, the differences in the ways
of referring to iconic mediators may impact such change. It is interesting to note that
“representation(s),” a particularly common word used in mathematics to signify iconic
mediators does not appear in either curriculum. Table 12 provides examples of the use of

each of these words in the written and enacted curricula.

135

Table 12.

Examples ofGeneral Words referencing Iconic ll/Iediators in the Written and Enacted

 

 

Curricula
Word Written Curriculum Enacted Curriculum
Model(s) “Check to see if the context is one T: "Now, what I want us to think
where it would make sense to use a about today is a fundraiser
number line model. " (TG, p. 70) situation for sixth graders, so I
want to share this with you and see
what you think about a different
kind of model. ” (Day 3)
Drawing(s) “One student makes the drawings T: “ Whoever did that drawing,
shown below. ” (SG, p. 34) can you explain your drawing real
quick? ” (Day I )
Diagram(s) “F or Question A, how did you first T: “What we want to do now is we

mark your brownie pan? (into
thirds, as in Part I of the diagram

in the next column). (TG, p. 6])

want to take a look at what we did
and see, is this what you did or
different from what you did, did
you get the same answer, maybe a

different diagram, okay? ” (Day 5)

 

I36

Table I2 (cont’d)

 

Word Written Curriculum Enacted Curriculum
Picture(s) “Draw a picture to show how the S: “One thing 1 don't get is, you
brownie pan might look before know how when we said eight times

9

Mr. Williams buys his brownies. ’ seven and we showed like the
(SG, p. 33) picture. Like, that wouldn't work

with fractions, would it?” (Day 2)

 

Table 12 includes examples in which the word is used alone (e.g., “picture”) as well as
examples in which the word is used in conjunction with a specific type of iconic mediator
(e.g., “number line model”). In addition to these two types, the curricula also include
references to specific iconic mediators in which none of the general words mentioned

above are present. An example from the written curriculum is provided here:

Example 106 (WC).
“These thermometers are like number lines. ” (SG, p. 3 4)

Example 106 references two iconic mediators (i.e., thermometer, number line) without
the use of the general words mentioned earlier. Instances of all three types (i.e., “model,”
“number line,” and “number line model”) were included in the remainder of the analysis
of iconic mediators.

When all instances were examined, four primary categories of the functions of
these references to iconic mediators emerged: (1) Constructing visual mediators
(including both original construction and operating on the visual mediator), (2) Using
visual mediators for various purposes, (3) Sharing visual mediators with the class, and (4)

Mediational Diversity (including selecting, comparing, and translating between visual

137

mediators). Table 13 summarizes the categories and provides examples of each type
from the written and enacted curricula.

Table 13.

Examples ofFunctions ofRe/erences to Iconic Mediators in the Written and Enacted

Curricula

 

Category Written Curriculum Enacted Curriculum

Construct “Draw a picture to prove that T: ” We 're gonna use one
your calculations make brownie pan, two different
sense. ” (SG, p. 38) colors. One color to show

what ’s in the pan, and another
color to show what we
actually did. (Day I)
Use “If students should happen to T: “Kristen used just

notice that they can multiply numbers and she used a

the numerators and multiply picture. ” (Day 5)

the denominators, ask them to

use their drawings to show

why they think this works. ”

(T G, p. 60)

Share “When students share their T: “Okay. Are you guys
models and exact answers, following his drawing? ”
focus on their reasoning. " (Day 1)
(TG, p. 72)

 

138

Table 13 (cont’d)

 

Category Written Curriculum Enacted Curriculum
Mediational Diversity “Both models are useful . T: "Kelsey’s trying it linear.
because each seems more like the longer model, and you
natural in diflerent try it the square and see if one
situations. ” (T G, p. 65) is easier than another. “ (Day
3)

 

The analysis does not reveal any patterns regarding when particular words (e.g., drawing,
diagram) are used. That is, particular words referencing general iconic mediator do not
occur more often in one category than another. The words seem to be used
interchangeably. Figure 18 illustrates the relative frequencies of each category of iconic

mediator use in the written and enacted curricula.

 

8888

lI Written Curriculum
Enacted Curriculum

Relative Frequency
N
O

 

15
10
5
0 ‘ 7,,
Corstruct Mediational Use Slnre
Diversity

 

 

 

Figure 18. Relative frequencies of categories of the uses of iconic mediators in the

written and enacted curricula.

I39

Figure 18 shows that references involving constructing visual mediators and mediational
diversity are the most common in both curricula. The largest discrepancy between the
written and enacted curricula are the references to sharing iconic mediators; these
references are much more common (more than three times as prevalent) in the enacted
curriculum than in the written curriculum. This is not surprising given that the written
curriculum may include one suggestion to encourage students to share their iconic
mediators and this suggestion may be enacted multiple times in the classroom.
References to three primary types of iconic mediators are found in the written and
enacted curriculum: rectangular area models (a partitioned rectangle of dimensions a x b
where a and b are both greater than 1), linear area models (a partitioned rectangle of
dimensions a x b where either a or b is I), and number lines. Figure 19 summarizes the

relative frequencies of references to these types of iconic mediators in the curricula.

 

l—F_ ____._ ._ _ ._____. -_-

80 t":?""—".'.—‘ ' T‘s? .‘.“--'_.~"‘.. .‘"~—".-"."*-‘-7 -, f“ ‘r'rr‘f' ._. .' "" ‘i'fi'll

7o .
60 j
50
4o 7
30 .
20 '
10
o

   
 

 

l; Written Curriculum
._.... lCl Enacted Curriculum

 

 

Relative Frequency

Rectangular Linear Area Number
Area Model Model Line Model

l_____ __

 

 

 

 

Figure 19. Relative frequencies of references to iconic visual mediators in the written

and enacted curricula.

140

Figure 19 indicates that both the written and enacted curricula reference rectangular area
models most often. These usually refer to models of brownie pans. Examples are
provided here from each curriculum:

Example 107 (WC).

“Can someone share a way to mark the brownie pan so it is easy to see what part

of the whole pan is bought? ” (TG, p. 61)

Example 108 (EC).
S: “But we put it all on one brownie pan? ” (Day 1)

Less common in both curricula are references to linear area models. These most often
refer to fundraising “thermometers.” Examples are provided here from each curriculum:
Example 109 (WC).
“To figure out the new length, the student divides the whole thermometer into

pieces of the same size. ” (SG, p. 35)

Example 110 (EC).
T: “I'm going to use a long, skinny model. " (Day 3)

Finally, references to number lines represent 28% of the references to iconic mediators in
the written curriculum, but such references are absent in the enacted curriculum. An
example from the written curriculum is provided here:
Example 1]] (WC).
“If you partition the number line this way, there will be four sections in each
third or 3 x 4 = 12 parts in the whole. " (T G, p. 65)
It is certainly notable that the enacted curriculum makes no reference to number lines. In
terms of objectification, this is very important because number line models are more

effective facilitators of the reification of fractions than area models. More will be said

about this later in the chapter.

l4l

Words Signifying Symbolic Mediators
The only words signifying symbolic visual mediators in the written and enacted
curricula come in the form of the phrase “number sentences(s).” This phrase appears in
both curricula; however, it is more prevalent in the enacted curriculum than in the written
curriculum (18% compared to 6% of all references to visual mediators). Examples are
included here to illustrate its use:

Example 112 (WC).
“It is helpful to write the number sentence by the models. ” (T G, p. 61)

Example 113 (EC).
S: “T hat's what I was thinking. but 1 don't how I do, like do the number
sentence.” (Day 4)
In both the written and the enacted curricula, the vast majority (83% and 86%
respectively) of the references to symbolic mediators address writing a number sentence.
The remaining references involve either a comparison of two or more number sentences
or a comparison of the result of a number sentence to an answer acquired using a
different strategy.
Verbs Associated with Visual Mediators

Many of the verbs associated with constructing iconic mediators in this

Investigation are derivatives of the nouns discussed earlier. Table 14 summarizes these

verbs as they are found in the written and enacted curricula along with “write,” a verb

associated with the construction of symbolic mediators.

I42

Table 14.

Verbs associated with Visual Mediators in the Written and Enacted Curricula

 

Verb Written Curriculum Enacted Curriculum
Model Model(s) ‘ Model
Modeling Modeled
Draw Draw Draw
me me
Drawing Drawing
Drawn
Represent Represent Represent( s)
Representing
Write Write(s) Write
Writing Writing

 

Table 14 indicates that virtually the same set of verbs associated with visual mediators is
present in both curricula. It is notable that although “representation” does not appear in
either curriculum as a noun, derivatives of “represent” as a verb are present in both the
written and enacted curricula. Table 15 provides examples of the use of each family of

verbs.

I43

Table 15.

Examples of Uses of Verbs associated with Visual Mediators in the Written and Enacted

Curricula

 

Verb

Model

Draw

Represent

Write

Written Curriculum
“You have developed some
strategies for modeling
multiplication and finding
products involving fractions. ”
(SG, p- 3 7)
“In the last problem, we drew
models to multiply with mixed

numbers. ” (TG, p. 75)

I
“How would you represent a x

gar: a number line? " (SG, p. 34)

“ What number sentence could I

write for Question A ?

x

)” (TG, p- 61)

WIN
||
cxlm

MIN

Enacted Curriculum

T: “Can you model it? ” (Day 4)

S: “You have to draw ten of these.

They did this and that. " (Day 5)

T .' “Well, you need sixteen
something. i What do you want to

represent your ounces? ” (Day 4)

T: “T hat ’s how much they ended
up with. But help me write my

number sentence. ” (Day 2)

 

As mentioned previously, three of the four verb families presented in Tables 14 and 15

(i.e., “model,” “draw,” and “represent”) are used in reference to iconic mediators whereas

“Write” is used in reference to symbolic mediators. Figure 20 summarizes the relative

frequencies of these verbs in the written and enacted curricula.

I44

I Written Curriculum

[1 Enacted Curriculum

Relative Frequency

 

Model Draw Represent Write

 

 

Figure 20. Relative frequencies of verbs associated with visual mediators in the written
and enacted curricula.
Figure 20 indicates that 75% of the verbs associated with visual mediators in the written
curriculum are used in conjunction with iconic mediators (i.e., “Model,” “Draw,” and
“Represent”) This frequency increases to 83% if the less common verbs are included
because they are all associated with iconic mediators, leaving just 17% of the verbs
associated with symbolic mediators in the written curriculum. In contrast, the
corresponding frequencies in the enacted curriculum are 55% associated with iconic
mediators and 45% associated with symbolic mediators. This makes sense given that
references to “Number sentence(s)” are also more common in the enacted curriculum,
18% compared to 6% in the written curriculum. Figure 20 further indicates that “Draw”
and its derivatives are the most common verbs associated with iconic mediators in both
the written and enacted curricula (33% and 34% respectively).

Recall that one of the goals of visual mediator use is mediational flexibility (i.e.,

the ability to use multiple forms of visual mediators and to move flexibly between them).

145

Several types of references to this issue are found in the written curricultun. First, there
are instances in the Student Guide in which students are asked to construct a “model” or
draw a “picture.” The use of these general words seems to imply acceptance of a variety
of types of iconic mediators. Two examples are provided here to illustrate this
phenomenon:

Example 114 (WC).

“Choose two problemsfrom Question A. Draw a picture to prove that your

calculations make sense. ” (SG, p. 39)

Example I I 5 (WC).

“Your group is responsible for creating a model that you can use in the summary

to talk about your assigned problem. ” (TG, p. 76)
Second, the written curriculum suggests that the teacher encourage students to share a
variety of models. Two examples illustrating such suggestions are included here:

Example 116 (WC).

“Lee used a brownie pan model for part (1 a). Who can share a number line

model? " (TG, p. 6 7)

Example I I 7 (WC).
“Look for various models to use in the summary. ” (TG, p. 72)

The enacted curriculum contains many statements in which the relationships and
differences between visual mediators are highlighted. In addition, the teacher selects a
variety of visual mediators to be presented by the class. Examples of statements
addressing these issues in the enacted curriculum are included here:
Example 118 (EC).
T: “So you need to estimate what you think the answer should be, about how
much, you need to create some sort of model or diagram, so that you can

show how you solved it, and then when you're done try to write a number
sentence. ” (Day 4)

I46

Example I I 9 (EC).

T: “Cool. Okay. Now we 're all going to switch gears over there, and we have
three different drawings. Maybe they're similar, maybe they 're different.
We 're going to take a look at them and see. ” (Day 5)

Example 120 (EC).
T: “Could, could hers be a picture of Katie 's? ” (Day 5)

In Example 118, the teacher is asking the students to create “some sort of model or
diagram.” Her language here indicates openness to several possible types of visual
mediators. Examples 119 and 120 indicate drawing connections between iconic
mediators presented by several students.

In addition to these references to mediational flexibility between iconic mediators,
references to the relationship between iconic mediators and symbolic mediators are also
present in both the written and enacted curricula. Examples of such statements from both
curricula are provided here. Recall that “Trevor’s way” is the traditional algorithm for
fraction multiplication:

Example 121 (WC).
“It is helpful to write the number sentences by the models. By doing this, students

typically notice that you can multiply the numerators and denominators to find the
product. ” (T G, p. 61)

I47

Example 122 (EC - Student Work).

 

'3 .

  

 
 

...- »-M-o—o -.

 

.._..

 

 

 

5:2,-.-“ Cf]

S: “ What we did is, we knew that one fourth meant there was four pieces, so we
divided it up into four pieces [ points to each fourth]. And then it says that we
had one out of the four pieces, so we made that [ points to shaded fourth], and
then we had to divide that up into three pieces [indicates thirds], so we
divided that up into three pieces, and we had one of those three pieces, which
is there [points to shaded third]. And then we divided the rest of them up like
if they were there [indicates dashed lines], we divided those up into three, and
we ended up get, that gave us a total of twelve, and so we ended up getting
one twelfth [points to double-shaded piece], which we also figured out like if

. . . 1 1 ,,
you did it Trevor's way. too [pomts to number sentence “—x— - — .

(Day 2)
Finally, both curricula include references to the relationship between two or more
symbolic mediators. Examples are included here:

Example 123 (WC).

“To raise the issue, list the number sentences solved so far on the board (e. g.

éx-g- = g), then ask: What patterns do you see between the fiaction numerators

and denominators and their products? ” (TG, p. 6 7)

Example 124 (EC).

T: “They're the same or they 're, you're saying they 're opposite and you're saying
they’re the same thing. those two number sentences? (Day 5)

I48

Example 123 from the written curriculum suggests listing a series of number sentences in
hopes that students will recognize the pattern of the traditional algorithm for fraction
multiplication. Example 124 also addresses the relationship between symbolic mediators,
but in this case the teacher asks the students about the? equivalence of two number
sentences.

References to a variety of carefully selected visual mediators and discussions
regarding the relationships between them in both the written and enacted curricula
enhance the possibility of accomplishing both of the visual mediation goals put forth here
(i.e., mediational flexibility, fluency in the use of symbolic mediators).

Use of Visual Mediators

To clarify, the difference between this section and the previous one is that the
previous section highlighted the uses of words associated with visual mediators. For
example, it answered questions such as “How many times does “number sentence”
appear in each curriculum and how is the phrase used?” In contrast, this section
addresses the actual use of visual mediators in the curricula. It answers questions such
as, “How many number sentences are present in each curriculum and how are they
used?”

The primary use of all visual mediators in this Investigation in both curricula is to
construct narratives (i.e., mathematical statements) addressing fraction multiplication.
This use of visual mediators will be detailed in the next chapter (Endorsed Narratives in
the Written and Enacted Curricula); therefore, this section will address their use more
generally. All three types of visual mediators (i.e., concrete, iconic, and symbolic) are

present in the enacted curriculum, whereas the written curriculum does not contain

149

concrete visual mediators. It may seem somewhat odd to expect a written curriculum to
contain concrete mediators (e. g., cubes); however. some written curricula (e. g., Bridges
in Mathematics. Everyday Mathematics) include a variety of such materials. In fact,
Connected Mathematics provides some concrete mediators (e. g., fraction strips) for use in
other Investigations. In any case, the written curriculum does not make reference to
concrete visual mediators in this Investigation.44 It should be noted; however, that many
“general” references to visual mediators are included and it would certainly be possible
for students to choose to “model” a problem using concrete objects. In fact, it is one of
these “general” references that led to the one instance of concrete visual mediator use in
the enacted curriculum. Figure 21 summarizes the relative frequencies of the three types

of visual mediators in each curriculum.

    

Written Curriculum Enacted Curriculum
concrete Concrete
0% Iconic 1%
24% f/
/ Iconic
K 41%
Symbolic Symbolic \\

76% 58%

Figure 21. Relative frequencies of the types of visual mediators in the written and

enacted curricula.

 

44 The truth ofthis statement can be debated because the use of some ofthe iconic mediators prevalent in
this Investigation (e.g., model ofa brownie pan) is enhanced by students’ possible experiences and/or
imagining of the actual concrete object itself. Due to the fact that it is impossible to know whether students
are operating on pictures of brownie pans or an image ofthe actual brownie pans, brownie pans and the like
are not included as concrete mediators for the purposes ofthis investigation.

150

Figure 21 indicates that symbolic mediators (i.e., number sentences) are the most
prevalent type of visual mediator in both the written and enacted curricula. Their
prevalence, however, is more pronounced in the written curriculum than in the enacted
curriculum. That is, the discrepancy between symbolic and iconic visual mediators is
greater in the written curriculum than in the enacted curriculum, a difference of 51% and
17% respectively. As with many statistical discrepancies in this analysis, it is difficult to
determine how much of this difference is attributable to the differences between the two
forms of curricula, written and enacted. In this case, it seems likely that the consideration
of the number of pages in the textbooks would influence the decision to include less
iconic mediators because they take up more space in the text.

It is interesting to note that the relative frequencies of references to iconic and
symbolic mediators in the written and enacted curricula differ substantially from the
presence of the visual mediators themselves in the curricula (see Figures 16 and 19). The
relative frequencies of references to iconic mediators in the written and enacted curricula
are 94% and 81% respectively compared to 24% and 41% here. The relative frequencies
of references to symbolic mediators in the written and enacted curricula are 6% and 18%
respectively compared to 76% and 58% here. That is, it is more common to reference
iconic visual mediators than to include them in the curricula, whereas it is more common
to include symbolic visual mediators in the curricula than to reference them. Again, this
may be related to space limits in written text materials.

The remainder of this section provides details of the categories of these types of
visual mediators (i.e., concrete, iconic, and symbolic) and their use in the written and

enacted curricula.

lSl

Concrete Mediators
Remember that concrete visual mediators never explicitly appear in this
Investigation in the written curriculum and they are used only once on Day 4 in the
enacted curriculum. Here, I present that use. The students were asked to complete the
following problem from p. 36 in the Student Guide:

Example 125 (WC).

 

For each question:

0 Estimate the answer.
0 Create a model or a diagram to find the exact answer.

0 Write a number sentence.

A. The sixth-graders have a fundraiser. They raise enough money to reach
% of their goal. Nikki raisesg of this money. What fraction of the goal
does Nikki raise?

B. A recipe calls for g of a l6-ounce bag of chocolate chips. How many
ounces are needed?

C. Mr. F lansburgh buys a Zé-pound wheel of cheese. His family eats % of
the wheel. How much cheese have thev eaten?

D. Peter and Erin run the corn harvester for Mr. McGreggor. They
harvest about 2% acres each day. They have only 10% days to harvest

the corn. How many acres of corn can they harvest for
Mr. McGreggor?

 

 

 

Four students are using 16 multi-colored wooden cubes to model Problem 3.38 when the
teacher walks up to the group:

Example 126 (EC).

(1) T: They're saying if I took, what you want to do is, you want to get two
thirds of all of these [indicates the I 6 cubes].

(2) 51: Okay.

152

(3)

(4)
(5)
( 6)
(7)
(8)
(9)

(10)
(11)
(12)
(13)
(14)
(15)

(16)

(17)
(18)
(19)
(20)
(21)

(22)

T: How could we do that? How could we get two thirds of all of these
wholes [indicates the 16 cubes]? We have sixteen whole blocks here
[indicates the 16 cubes]. How much, could you get a half of them?

S2: Yeah.

T: Get a half Show me a half

S]: It would be eight of them [moves the 16 cubes into two groups of 8].
T: How did you know it was going to be eight of them?

S2: 'Cause half of sixteen 's eight.

T: Okay [moves cubes back into one group of 1 6 ]. So now what if I
wanted you to get a fourth of them? Could you do that?

S l : Four [moves the 16 cubes into four groups of four ].
T: Why would it be four?

SZ: 'Cause four divided by sixteen is four.

T: Sixteen divided by four is four, you mean?

SZ: Yeah.

T: Okay [moves cubes back into one group of I 6]. So, how could I think
about getting thirds now?

S2: Well, it would be an odd number, because you never, 1 mean it would
have a remainder.

T: Could we do that? Could we have parts of an ounce?

S3: Yeah.
T: Why not?
S1: Yeah.

T: Can we have parts o/‘ounces?

S]: Yeah.

153

( 23 ) T: So think about that [teacher is interrupted by another student — she
turns her attention to him].

In Example 126. the students use the 16 cubes to represent the 16 ounces in the problem.
The teacher, perhaps thinking that finding two-thirds‘of the cubes is a difficult place to
start, begins with “a halt” which is a more familiar fraction. She then asks the same
question about getting “a fourth of them” and then asks about finding a third. This is
when one of the students realizes that unlike half and fourth, finding a third would give
an “odd” number, adding that it would “have a remainder.” The teacher interprets this to
mean “parts of an ounce” and proceeds to ask if it is possible to have “parts of ounces.”
Recall that the problem actually asks for two-thirds, so even if the students figure out
one-third of 16, their work is not finished. The teacher is interrupted by another student,
but the group continues their discussion:

Example 127 (EC).

(24) S2: Yeah. Or six or seven, so try over here. Let's try one. I don 't think
that you can divide sixteen by four, er, three.

(25) S]: It 'd be harder.

(26) S2: Well, here. Let's try and divide it by three and see what the
remainder is [Student 4 divides the cubes into five groups of 3 which
leaves one block alone]. Let's see - yeah, ’cause if we split it - yeah, we
have, well, what if we divided all of them into halves?

(27) SI: Divided them into five. So. it would be three wholes and

(28) 52: If we divided this one intofifths, every, everybody would have like,
um, three and oneftfth.

(29) SI: What?

(30) 52: I think I get it.

154

In this excerpt, the students divide the cubes into five groups and see that one is left
alone. After some discussion and looking at and touching the cubes, but not rearranging
them, one of the students gives the answer of “three and one-fifth.” It is unclear if the
other students make sense of this answer, because the fourth student in the group
recommends using an iconic mediator and the cubes are not used again by this or any
other group.

Because we are examining the relationship between the written and enacted
curricula, it is notable that this use of concrete mediators is a direct result of the request in
the written curriculum to “Create a model or diagram to find the exact answer.” (SG, p.
36). The students in the group illustrated here choose blocks to model this situation.
Blocks (and other concrete objects) are available to students in the room at all times to
use when they decide it is appropriate; however, most students (all but this group on Day
4 for this particular problem) choose not to use concrete mediators to solve the
problems.45 Iconic mediators are much more common.

Iconic Mediators
Types of Iconic Mediators

As mentioned previously, 24% and 41% of the visual mediators in the written and
enacted curricula respectively are iconic mediators. These represent 34 iconic mediators
in the written curriculum and 42 iconic mediators in the enacted curriculum. The four
types of iconic mediators present in the curricula are (1) Contextual pictures, (2)
Rectangular area models, (3) Linear area models, and (4) Number lines. Figure 22
summarizes the relative frequencies of these types of iconic mediators in the written and

enacted curricula.

 

45 Again, this refers to groups which appear on the videotapes.

155

 

I Written Curriculmn
D Enacted Curriculum

Relative Frequency

 

Contextual Rectangular Linear Area Number Line
Picture Area Model Model

 

 

Figure 22. Relative frequencies of iconic mediators in the written and enacted curricula.
Rectangular area models are the most common in both the written and enacted curricula,
representing 53% and 60% respectively. Provided here are examples of rectangular area
models from both the written and enacted curricula.

Example 128 (WC).

 

2. Possible model:

 

 

 

 

(TG, p. 64)

156

 

Example 129 (EC - Student Work).

390””wa j
Jitter,

 

‘ 1" put: "4;, ”I...

 

 

 

(Day 2)

 

The other type of iconic mediator present in the enacted curriculum is the linear area
model. This type is also present, although not common, in the written curriculum.
Provided here is an example of a linear area model from the written curriculum.46

Example 130 (WC).

 

 

- "l i ; l

 

5|-
9,
WlN

 

 

 

(SG, p. 34)
Finally, two additional types of iconic mediators are present in the written

curriculum but absent in the enacted curriculum. The first of these is contextual pictures,

 

”6 All student work from the enacted curriculum containing linear area models was completed on the
whiteboard and not saved, however as will be discussed later in the section, the linear area models in the
enacted curriculum look very similar to the one from the written curriculum included here.

157

representing 9% of the iconic visual mediators in the written curriculum. An example is

provided here:
Example 131 (WC).

 

 

(36, p. 35)

 

 

 

Example 131 accompanies a problem about painting two-thirds of a stripe that is nine-

tenths of a mile long. It could be argued that this contextual picture (and the others) is

accessed in the enacted curriculum because the students complete these problems.

The final type of iconic mediator is the number line model. It is absent in the

enacted curriculum, but is represented by 29% of the iconic mediators in the written

curriculum. An example from the written curriculum is provided here:

Example 132 (WC).
1 1
3
(TG, p. 66)

o
Wb—v—J
il-of this third g of this third

When comparing the relative frequencies of the actual use of iconic mediators to

 

wIN

 

 

 

the relative frequencies of references to iconic mediators (see Figures 19 and 22), the

158

prevalence of the actual and referenced iconic mediators are quite similar in most cases.
For example, the references to the number line in the written curriculum represent 28% of
all references to iconic mediators compared to representing 29% of the actual visual
mediators. The largest discrepancies occur with linear area models. In the written
curriculum, 22% of the references to iconic mediators referred to linear area models
compared to only 9% of the actual iconic mediators. In the enacted curriculum, 26% of
the references to iconic mediators referred to linear area models compared to 40% of the
actual iconic mediators. That is, linear area models are referenced more than twice as
often as they are included in the written curriculum, whereas they are included more than
four times as often as they are referenced in the enacted curriculum.

Probably most notable in this section is the absence of number lines in the enacted
curriculum. As mentioned previously, number lines may facilitate the reification of
fractions as mathematical objects (i.e., numbers) in their own right in contrast to the “part
of a whole” use of fractions because students need to make decisions regarding the
location of a particular fraction (i.e., between which two numbers). The placement of a
fraction or mixed number between whole numbers affords discussions regarding whether
the fraction is greater than or less than particular whole numbers. This language (i.e.,
“greater than,” “less than”) is reserved for numbers and therefore may serve to facilitate
the reification of fractions. NCTM (2000) made the following statement regarding the
affordance of number lines in particular and multiple representations of fractions in
general:

Different representations often illuminate different aspects of a complex

concept or relationship. For example, students usually learn to represent

159

fractions as sectors of a circle or as pieces of a rectangle or some other

figure. Sometimes they use physical displays of pattern blocks or fraction

bars that convey the part-whole interpretation of fractions. Such displays

can help students see fraction equivalence and the meaning of the addition

of fractions, especially when the fractions have the same denominator and

when their sum is less than 1. Yet this form of representation does not

convey other interpretations of fraction, such as ratio, indicated division,

or fraction as number. Other common representations for fractions, such as

points on a number line or ratios of discrete elements in a set, convey

some but not all aspects of the complex fraction concept. Thus, in order to

become deeply knowledgeable about fractions—and many other concepts

in school mathematics—students will need a variety of representations

that support their understanding. (p. 69)
Here, NCTM promotes multiple representations and uses of fraCtions, mentioning
number line as a possible alternative to area models to serve other “aspects of the
complex fraction concept.” In the next section, I describe the ways in which iconic
mediators are used in the written and enacted curricula.
Use of Iconic Mediators

The written curriculum provides suggestions for the use of all types of iconic
mediators except for the contextual pictures. It is likely that these pictures have been
included in the written curriculum to assist students in understanding particular contexts.
For example, the picture of the truck shown previously (see Example 131 on p. 159) may

help students imagine how a truck paints a stripe on a road.

160

Several statements in the written curriculum provide guidance regarding the use

of rectangular area models. For example:

Example 133 (WC).

“Draw a picture to show how the brownie pan might look before Mr. Williams

buys his brownies. Use a different colored pencil to show the part of the

brownies that Mr. Williams buys. ” (SG, p. 33)

Example 134 (WC).

“Can someone share a way to mark the brownie pan so it is easy to see what part

of the whole pan is bought? (Here is an opportunity to suggest that using

horizontal and vertical lines makes it clear what is happening. ” (TG, p. 61)

Examples 133 and 134 refer to ways to use the brownie pan model to illustrate fraction
multiplication, the first suggesting the use of two colors and the second suggesting the
use of horizontal and vertical lines. Rectangular area models are often used in similar
ways in the enacted curriculum. In fact, the enacted curriculum contains 158 references
to coloring either a rectangular area model or a linear area model. Two examples are
provided here:

Example 135 (EC).

T: “Okay. I used red to Show how much is left in the pan. I'm gonna use a
different color to show what we ’re gonna buy of it. Natalie, could you come
up here on this, could you show us what three fourths of that would look
like? ” (Day 1)

Example 136 (EC).

S: “ Well, I knew we had one half of the bar left, so I had to color in one half and
then I had to divide each half into three pieces, and then 1 colored in two
pieces of the one half and 1 got two sixths. ” (Day 3)

There is also discussion in the enacted curriculum regarding drawing lines on the
rectangular and linear area models. In fact, 40 statements addressing drawing lines on
these models are present in the enacted curriculum. In addition, 85 statements address

cutting (i.e., using lines to “cut” the rectangular or linear area model into pieces) and 66

statements address dividing (i.e., using lines to “divide” the rectangular or linear area

161

model into pieces). Examples of statements associated with rectangular area models that
involve lines, cutting, and dividing are provided here with additional explanations and
student work. Example 137 illustrates student work presented in association with
Question C -4 from Problem 3.1:

Example 137 (EC - Student Work).

 

 

 

 

 

 

 

 

 

4140

Ly? < 5

-U .

7135526

 

 

 

S: “Do the rest of them, the vertical lines. " (Day 2)
The statement included in Example 137 is made by a student encouraging the student
constructing the rectangular area model to draw all of the vertical lines clearly.
Originally, the student had only drawn the two leftmost vertical lines with marker. The
rest were drawn lightly with pencil. This statement expresses the students’ desire for all
of the vertical lines to be drawn clearly for the group’s presentation to the class. Example

138 illustrates student work associated with Question A from Problem 3.1:

162

Example 138 (EC - Student Work).

 

3. OQ

 

 

 

 

 

S: “We drew two thirds and then we cut the two thirds in half to see what he
would buy. And then, he would buy two sixths, because if you cut it in half he
would get three, and then three which is six. And that’s equivalent to one
third. So he would either, he would get two sixths or one third. ” (Day 1)

The student’s explanation of the work provided in Example 138 emphasizes “cutting” the
rectangular area model using vertical and horizontal lines. The final student work and

associated student explanation illustrates the use of “dividing”:

Example 139 (EC - Student Work).

 

 

 

 

 

163

S: “Well. um, he had one, there was one third of the pan left over, so we had to
divide it into seven pieces, and he bought two sevenths, two sevenths of the
one third, and if you divide all the rest into seven pieces, it would be twenty
one pieces, so all together he bought two twenty oneths of the whole pan. "
(Day 1)

The explanation included in Example 139 is similarto the previous one. Again a student
is describing the use of lines except that here the student talks about “dividing” it into
pieces rather than “cutting” it.

It is important to note here that in all of these cases and many others, the
discussions of these models contained many hand movements in a vertical and horizontal
direction as the students and teacher talked about the lines drawn on the models. Also
notice that in these examples of student work, some of the lines have been extended in
order to see the total number of pieces in the model. The students often make these line
extensions dotted or dashed as illustrated by the student work in Examples 138 and 139.

Although the way in which these three students draw lines and color is fairly
typical in the enacted classroom (i.e., dividing the rectangle into (A x B) pieces where A
and B are the denominators of the two fractions), this is not always the case. In fact,

several long discussions take place in which students choose to use lines and coloring

differently. An example of one such discussion is provided here (along with the

99

corresponding poster of student work). The problem under consideration is “%x—:— .

Emma and Katie have just explained their model which uses the strategy illustrated above

99

. 2
and stated their answer as “E.

164

Example 140 (EC - Student Work).

 

 

 

 

 

 

L 1
4";

r Ll ‘-‘ 1:1

 

 

 

Note that they have used the method described above (i.e., dividing the rectangle into (A
x B) pieces where A and B are the denominators of the two fractions). Another group is
asked to explain their model (included here in Example 141) of the same problem, and
the discussion ensues.

Example 141 (EC - Student Work).

 

 

 

 

 

 

 

 

165

(I)

(2)

(3)

(4)
( 5)
( 6)
( 7)
(8)

('9)

(10)

(11)

(12)
(13)
(14)

T: Okay. How about this one? You've got afourth of two thirds? Could
you guys, somebodyfrom this one and someonefrom this one, talk about
what you did?

S] : In this one, we had the two thirds [indicates thefour shaded pieces].
which is what we had to work with, and then we split it up into fourths
[indicates the bold horizontal line and bold vertical line], and then we
took one of them away [points at double-shaded piece], and then we made
a line right here [indicates solid portion of horizontal line], and then that
would, continuing right here [points at dashed portion of horizontal line],
and then if you had a line down here [points at non-bold vertical line],
then it would make it into sixths, so we ’ve made it one sixth, because there
was one piece taken out of these six.

T: So, you guys disagreed with that at your table [addressing Emma and
Katie who constructed the first model]?

S2: Well, we didn't really get

S3 : It was a different way.

SZ: Yeah, we didn’t really get how they got it, like how she kind of knew
T: So ask her, ask her then what you didn't understand.

SZ: Because we didn't get, um, how you could just break this [points to
shaded portion] up into fourths because that really doesn't make it into
fourths, it kind of makes it into sixes, like we

T: Hold on, hold onfor just a second. Why doesn't it split that purple
section into fourths [points to shaded portion]? 'Cause it looks like
fourths to me.

S3 : 'Cause you don 7 look at this part [points to unshaded part].

S2: 1 know, but I don 't like it. We didn't get how they got it, like how they
just knew to draw that line into fourths.

T: So ask her, how did she just know?

S2: How did you know?

S1: Because in the problem it says, um, one fourth times two thirds, 1
think, so we have the two thirds right here [points to two shaded thirds],

and then we had the one fourth so we split these two pieces [points to
shaded two thirds] into fourths.

166

(15)
(16)
(17)
(159
(19)
(20)
(21)
(32)
(23)
(249
(25)
(26)

(279

(259

(299

(30)

(31)

S2 : Oh, okay.

S1: And then we had the one.

S4: What did you get/or your answer, Emma?

S2: We got, um, two twelfths.

T: [s that a different answer than one sixth?

SZ: Yeah, but they're -

T: Is it?

S2: They're common, they 're, they're equal to each other.
T: So is your, did you get a different answerfrom them?
S2: No.

S3 : Um, what -

T: Go ahead.

S3 : Um, we have to split each third into four pieces and you only split
each third into two pieces.

T: But why do you have to do that, Katie?

S3 .' Because, um, it only shows six pieces, and you have two thirds, like, if
you had only one third you could do that, I think, but you need, you have
two thirds, so you have to split each third into four pieces.

82: Because we didn't really get how they couldn’t, 'cause we thought that
you had to split them all into four pieces, and, 'cause you have three,
which we got how they got the three, and then, um, how they just kind of
like just split it.

T: Okay. So if 1 look at yours. this right here is a fourth, right, and this is
a fourth, and this is a fourth, and this is a fourth [indicates each of the
fourths in the two-thirds]. If 1 look at theirs, isn't this a fourth [indicates
one of the fourths in the two-thirds]? Didn't they split theirs into four
pieces, too? They just split it up differently. But, but 1think, Katie,
correct me if 1 'm wrong, but what I hear you saying is you thought you had
to cut it like this [gestures horizontally with hands]? Okay. Do I have to?

167

She ended up with a sixth. and you said you got two twelfths. Can 1 write
your, are those two answers differentfrom each other?

(32) SZ: No.

(3 3) T: Are they the same amount?

(34) S3 : Yeah.

(35 ) T: Does the same amount just look a little bit different?

(3 6) S3 : Yeah.

(3 7) T: In every other problem we 've done, we have cut it like this [points to

Katie and Emma 's poster]. I think what these guys saw was, gosh, maybe
I'll take advantage of that cut I already have [indicates bold vertical line]
and get my pieces that way. Is it okay if they both ended up with one
sixth? It would just mean that maybe this notion of going across like this
[gestures horizontally] we didn't really do. In her drawing she didn't take
each third and split it into four pieces. They just did their drawing a little
bit different.

This excerpt exemplifies one of several discussions that took place over the course of the

week in the enacted curriculum in which questions arose about the appropriate types and

uses of iconic mediators.

There are similarities between the use of rectangular area models and the use of
linear area models, but there are distinctions to be made as well. One distinction is that
the partitioning suggested for the linear area model in the written and enacted curricula
only uses vertical lines. That is, the linear area model is partitioned in ways similar to a
number line. The use of two colors, one to denote each fraction so that the overlap of
colors indicates the answer, is continued from the work with the rectangular area model.
Similar language is also used (e.g., “lines,” “cutting,” “dividing”). Example 142 from the

written curriculum includes both a sample linear area model and a statement addressing

its use:

168

Example 142 (WC).

 

Goal

 

 

 

 

 

 

 

 

 

 

 

"The student above divides the fraction of the goal (é) that is met in four days

into fourths to find the length equal to éof %. T 0 figure out the new length, the

student divides the whole thermometer into pieces of the same size. ” (SG, p. 35)
Note the use of vertical lines and two colors.47 The written curriculum also recommends
that students use the linear area model with the number line. Example 143 illustrates this
recommendation:

Example 143 (WC).
”Draw a number line under the thermometer. ” (T G, p. 65 )

Students are encouraged to locate O and l on a number line drawn beneath a thermometer
and to use it to assist in the partitioning of the thermometer (i.e., linear area model).

The primary discussion regarding the use of the linear area model in the enacted
curriculum took place on Day 3. Example 144 includes the problem that was displayed
using the overhead projector and the class discussion that ensued:48

Example 144 (EC).
“One sixth-grade class raises éof their goal in four days. They wonder what

fraction of the goal they raise each day on average. To figure this out, they find
10,2 .,
4 ‘ 3 '

(1) T: So, let 's say that a sixth grade class, maybe it '5 us, let 's say that we
raise two thirds of our goal. Instead of making our goal this big square

 

47 Here the colors (originally red and blue) display as two shades of gray.
48 It should be noted that this problem occurs verbatim in the written curriculum (SG, p. 34).

169

box like a brownie pan, I'm going to use this kind of a model. I'm going to
do this [Draws a long. narrow rectangle. ] I'm going to use a long, skinny
model. Do you guys think we can handle that today?

 

 

(2) Ss: Yes.

(3 ) T: Okay.

(4) SI : No.

(5 ) T: No? Where 's that good attitude? [Laughter] Okay. This is the whole
goal, right? So it says that wow, after four days, we reached two thirds of
that, who can come up and show what that would look like on a long,
skinny model? Steven, you want to do that for us? What does two thirds
look like on this kind of a model?

( 6) S2: Something like that [Divides linear area model into thirds]

 

l l

( 7) T: Can you show us where two thirds would be, though? Fill it right up
to two thirds. Does everybody agree?

[Shades two of the thirds on the linear area model using red marker]

 

l l
( 8) Ss: Yeah, yup.

( 9) T: So, so far, so good. Okay. Thanks, Steven. They wonder what
flaction of the goal they raised each day on average [points to problem
displayed by overhead projector], so to figure this out they said, let's
figure out what one fourth of that two thirds is. So this [points to vertical
line at right end of shading] is all they’ve reached, is this much of their
goal. What if I want to figure out what one fourth of that red part [the
shaded part] is? Steven, will you pick somebody to help us do that?

(10) S2: Michael.

(11) T: Here, Michael [hands Michael a marker]. How can we figure out
what one fourth of that is up there [points to model]?

[Student 2 divides the shaded two thirds into two parts]

170

 

l l l l

(12) T: Remember, you want onefourth ofthe red part [shaded part].
(13) S3: Oh, onefourth of it?

[Student 3 erases lines he has added]

 

l
(1 4) T: Y up.
(15) S3 : The red part?
(1 6) T: Yup.
(17) S3 : 1 don't know. I can 't
(18) T: You can ask for help if you want to askfor help.
(19) S3 : I don't want to mess up on it, though.
(20) T: Do you want to ask somebody to help you out, Bud?
(21) S3: Yeah.

(22) T: Okay. There 's lots of people with their hands up and they 'd be glad to
help you.

( 23 ) S3 : Graham.

(24) T: It 's a little trickier looking at a model like this instead of that big box,
isn't it?

(25 ) S4: Okay. So now one fourth of two thirds

[Student 4 redraws the lines cutting each third in half]

 

l l l l
(26) T [to Student 3]: 1 think that 's just what you were doing, right?

(2 7) S4: Now, here would be onefourth [colors in one-fourth of the two-thirds
with black marker].

171

 

 

(28) T: 1 think that 's exactly what Michael was doing, wasn't it?

(29) Ss: Yeah.

(30) T: So we just want afourth of just that red part. So we ’re kind of
ignoring the part that 's not there. Thanks for helping, Graham. So if this
isn't there [covers up the unshaded part with arm], here '5 four parts
[points to shaded parts] and he got one of them, right? So you were
absolutely doing the right thing there. So now the question is, though,
what fraction of the whole goal did we get each day? So, Chloe, what
fraction of that whole bar is this double piece that we have there [points to
double-shaded piece]?

(31) S5: One fourth?
(32) T: One fourth [writes “é ” under drawing]? Would you guys agree?

(33) Ss: No.
(34) T: Okay. Talk to Chloe if you disagree.

(35 ) S6: 1 think it should be one fifih because, uh. you have to add an extra,
no, wait, I think it should be

(3 6) S? : I think it should be two twelfths because

(3 7) S?: Yeah.

(38) T: Is this picture showing two twelfths?

(3 9) S?: I think. no

(40) T: What is this picture [points to linear area model] showing?
(41) S7: Four times three

(42) T: Well, but 1 want, 1 want to talk about this drawing though.
(43 ) S? : It shows one sixth.

(44) S?: Yeah, one sixth.

172

(45) T: It shows one szxth or onefourth [writes ”3 under drawzng]?
[Many students speaking at once]

(46) T: Okay. Hold on. One at a time. One at a time. What is this drawing
showing? Is this shaded part onefourth of the whole bar, or one sixth of
the whole bar?

(47) C: One sixth.

(48) T: But prove that, though. Okay. Go ahead. Can you prove that this is
one sixth?

(49) S8: [Student walks to board, gets marker from teacher] Because, if you,
just like on the one big bar that we had, if you just end up splitting that
one [divides the unshaded section in half], it ends up giving you six pieces,
and so, like, if we use those big bars, it would just show you like you had
one sixth.

 

r——i 1 l l |

 

(50) S?: I told you it was two twelfths. That's equivalent. [Laughter]
(51) S?: That's what it is.

(52) T: Chloe, what do you think about that? Do you think that it's one fourth
or do you think it 's showing one sixth of the whole bar?

(53) S5 : One sixth.
In this construction of the linear area model, the students use techniques similar to those
suggested in the written curriculum (i.e., vertical lines and shading). In fact, such
techniques are applied in all uses of the linear area model in the enacted curriculum.
Twenty-nine percent of the iconic visual mediators included in the written
curriculum are number lines whereas number lines are absent in the enacted curriculum.
The written curriculum acknowledges that number lines can be partitioned as described

for the linear area model above; however, they also point out an “alternative approach”

173

for partitioning number lines for fraction multiplication. The Teacher’s Guide (p. 66)
poses the problem and demonstrates the “alternative approach:

Example 145 (WC).

 

0 Let's look 0% X %. How would you start if

you wanted to use a number line model?

(Draw a number line. and label 0 and 1. Next.
partition the number line into thirds. and

1 2
mark 3 and 3.)

-r l
‘r i i'
0

l t ‘

--

0 How could you repartition to find % of %?

(You could break each third into five equal
parts)

1 2
0 3 3 1

0 What is % of two thirds? (Each % of one third
is fi. so twice that would be [—25.)

 

nrrlrrrrl-
WIIIIIIIIT
1 2

k Ag

 

 

r} of this third lot this third

 

 

This approach parallels the most common use of the rectangular area model because the
model is partitioned into A x B pieces where A and B are the denominators of the
fractions. Even though the enacted curriculum does not use numbers lines per se, this

approach is used with the linear area models in several situations. In fact, the

174

. . . . . 1 2 .. . .
conversation included earlier in which "Zof 5 is modeled on a linear area model (see
Example 144) continues with a student suggesting this “alternative approach.” Recall
. . 1 ,, ' .
that the final agreed upon solution 18 “—6— . ‘ Example 146 presents the linear area model

as it appeared at the conclusion of the discussion in Example 144 along with the
continuing conversation:

Example 146 (EC).

 

 

 

J l l l

(54) T: Okay, so we '11 rethink this guy then. right? Okay, now, Elliot, why two
twelfths? You said two twelfths.

(55) S1: 1 was imagining the whole bar in thirds, as it was first

[The teacher draws the original model of two-thirds on the board above the other
model]

 

l l J

 

- l l i 1

(56) T: Okay.

 

(5 7) S1: Then you cut it intofourths and then cut each of those thirds into four
pieces.

(58) T: You want to cut each of these into four pieces?

[The teacher divides each of the shaded thirds into four pieces in the top model]

 

l l l l l l l I l

 

 

l l i

(59) S1: Yes.
(60) T: Oh. So, if] split this [points at leftmost third] intofour pieces and this

[points at the other shaded third] intofour pieces, how is that going to
help me get onefourth of the two thirds? How is that going to help me?

175

(61)

(629

(63)
(64)

(65)

(66)

(67)

(68)

SI: Well, 1 was actually, I wasjust thinking of the one third. 1 wasn 7' just
thinking of one fourth of both of them, but

T: Oh, okay. So you were just going to get a fourth of this one [points at
leftmost third]? .

S1: Yeah.
T: And how is that going to help us get a fourth of two thirds?

S 1 : 'Cause it 's just, if you double that then it would be your answer for
that.

T: Oh.
S? : ’Cause that 's a third and get a fourth from another third.

T: Oh. So I've got a fourth from this third [shades in one-fourth of the
leftmost shaded third] and a fourth from this third [shades in one-fourth of
the other shaded third], so together, and I’m going to do what Karen just
did down here [divides the unshaded third into fourths using dashed
vertical lines], I would have two of the twelve pieces?

 

 

 

 

 

(659

(70)
(71)
(72)
(73)

(74)

(75).

(76)

 

Ss: Yeah.

T: Are those different, two twelfths and one sixth?

Ss: No.

S? : They 're not different.

T: They're not different?

S ?: They just look different but they're the same - but they're equivalent.
T: So if] took this piece [points to shaded fourth in the center third] and
1 moved it over to here [points next to the shadedfourth in the leftmost
third], would it be the same size as this one sixth [points to shaded one-

sixth in the bottom model]?

Ss: Yes.

176

(77) T: Okay. So either way would be okay?

(78) C: Um hmm.
This discussion in the enacted curriculum compares the two approaches suggested in the
written curriculum and the class concludes that the answers are equivalent. It could be
argued that although the enacted curriculum never uses a number line model, the methods
of partitioning number lines advocated for in the written curriculum are present in
conjunction with linear area models in the enacted curriculum. In addition, the simple
presence of number line models is not enough to accomplish the goals of objectification.
If learning mathematics is a change in participation in mathematical discourse, it is the

ways in which the number line is talked about that is the key to this development. In
. . . . . 1
order to rerfy fractions, it is necessary to encapsulate “1” and “6” into the number “-6— ”

and although number lines can facilitate this process, it is also possible to use number
lines in ways in which the talk is of parts and pieces instead of numbers.

Examples 133-146 certainly highlight the question of the affordances and
limitations of certain types of iconic mediators and prompt the question, “Which iconic
mediators best accomplish the goals set forth in the written curriculum for this
Investigation?” It is not the purpose of this project to definitively answer this question,
rather to discuss the visual mediators present in the written and enacted curricula;
however, its importance seems indisputable.

Symbolic Mediators
Seventy-six percent and 58% of the visual mediators in the written and enacted

curricula respectively are symbolic mediators (i.e., number sentences). In many cases,

177

the symbolic mediators are constructed from the iconic mediators. In these cases,
students are encouraged first to model the situation and then to write the symbolic version

from that model. An example of such use from the written curriculum is provided here:

Example 14 7 (WC).

 

2
4!

l .
For 2 3 x the area model could be as shown

 

§\\
\\
S\\

 

 

 

 

 

 

 

 

 

 

 

QIN

 

 

 

 

 

 

 

 

 

 

 

In Example 147, the symbolic mediator is completed only after the situation has been
modeled with an iconic mediator. The written curriculum also suggests the use of
symbolic mediators to help students see patterns. For example:

Example 148 (WC).
“It is helpful to write the number sentence by the models. By doing this, students

typically notice that you can multiply the numerators and denominators to find the
product. ” (T G, p. 61 )

I78

In Example 148, the symbolic mediator is used to illustrate the traditional algorithm for
fraction multiplication. That is, seeing several symbolic mediators, students may identify
the pattern of multiplying numerators and denominators.

Similar uses of symbolic visual mediators are seen in the enacted curriculum. An
example of deriving the symbolic mediator from the iconic mediator is provided here:

Example 149 (EC - Student Work).

9136;? it . 5t

 

 

 

 

 

 

 

m Durables sob "‘- Gem

: at)" 3(3 3r: El
7: e: a (hire): .1? firm {£39396 3’5 +6 ,6
3M 399‘” ”It. to '15 an} act 5/6.

 

 

 

l 1
First, the rectangular area model is constructed showing “%of 2” and “-3- x -2- ,” then the

. 1 1 5 . . .
model 18 used to construct the final number sentence “-3— x 25 = g .” It is interesting here

66 99

to note the use of “of” and “x” in this diagram. The relationship between “of” and x is

discussed in other sections (e.g., pp. 50-54 and 118-119); however, the transition between

179

. . . 3 . 3 . . .
a quasr-symbolic mediator (e.g., “201 2 = g ") and a standard symbolic mediator

. 3 1 3 .. . . . . , .
(“ — x 2 = g ) is common in the enacted curriculum. In fact, the teacher often writes “X”

4
on top of “of” on student work to emphasize the relationship (see Example 17 on p. 53).
Symbolic mediators are also used to identify the traditional algorithm for fraction
multiplication in the enacted curriculum. In fact, it is exactly after the teacher has
changed several quasi-symbolic mediators to standard symbolic mediators on Day 2 that

a student notices the pattern and suggests the traditional algorithm.

Example 150 (EC).
(1) T: Instead of saying nine tenths of a sixth, 1 can also say nine tenths times
a sixth [writes “X” on top of “of" in “%of g- = 6%. ” Go ahead. Trevor.

( 2) S: I think it's times the numerator and numerator and then times the
denominator and denominator.

Examples 147-150 indicate similar uses of symbolic mediators in the written and enacted
curriculum even though they are relatively more common in the written curriculum.
Summary
The question addressed in this summary is, “What does an investigation of the
visual mediators in the written and enacted curricula allow us to see?” That is, “What do
we know now about the relationship between the written and enacted curricula that we

did not know before?”49 Recall that visual mediators are artifacts created specially for

 

49 The statements in this summary and throughout this analysis are made with several caveats. First, this
analysis compares the written text with an enactment of the written text (one of infinitely many possible
enactments); this should be kept in mind when reading these statements as some results may be attributed
to this difference in curricular form. Second, similarity (and difference) here is through my eyes only.
That is, another person (e.g., a teacher, a textbook author) using their own lens may see things quite
differently. Finally, the evidence for my claims is gleaned from five days in the written and enacted
curricula. That is, none of my statements can be generalized either to the written curriculum as a whole or

180

the sake of mathematical communication. Table 16 summarizes the findings from the

investigation of visual mediators for the purposes of examination for overall conclusions.

Table 16.

Summary of “ Visual Mediators " Analysis

 

Written Curriculum

Enacted Curriculum

 

Analysis of
References to
Visual
Mediators
(i.e., word

analysis)

26% of references to visual
mediators are “general” references
94% of references to visual
mediators refer to iconic mediators
(remaining 6% refer to symbolic

mediators)

The most common general word
referencing iconic mediator is

“model” (64%)

14% of references to visual
mediators are “general” references
81% of references to visual
mediators refer to iconic mediators
(18% refer to symbolic mediators
and 1% to concrete mediators)
The one reference to concrete
mediators in the enacted
curriculum is “blocks”

The most common general word
referencing iconic mediator is

“drawing” (45%)

Neither curricula use “representation” as a label for an iconic mediator

The most common function of references to iconic mediators

are “constructing” and “mediational flexibility”

(frequencies range from 29% to 35%)

 

 

the enacted curriculum as a whole. Rather, my statements highlight insights gained through the use of this
framework regarding the relationship between the written and enacted curriculum on these five days that
may be of interest to teachers, curriculum developers, and mathematics education researchers.

181

Table 16 (cont’d)

 

Written Curriculum Enacted Curriculum

 

Analysis of
References to
Visual
Mediators
(i.e., word

analysis)

Analysis of
Visual
Mediators
(i.e., visual
mediator

analysis)

Rectangular area models are the most common
iconic mediators in both curricula
28% of iconic mediators 0% of iconic mediators
are number line models are number line models
The most common function of references to
symbolic mediators is “writing”
Virtually the same set of verbs associated with
visual mediators is present in both curricula
Both curricula include references that relate one iconic mediator to
another, one symbolic mediator to another,
and an iconic mediator to a symbolic mediator
The primary use of visual mediators in both curricula
is the construction of endorsed narratives
58% of the visual mediators
76% of the visual mediators are
are symbolic mediators
symbolic mediators (remaining
(41% refer to iconic mediators and
24% are iconic mediators)
1% to concrete mediators)
----- The one use of concrete mediators
in the enacted curriculum

involve the use of blocks

Both curricula include rectangular area models and linear area models

 

182

Table 16 (cont‘d)

 

Written Curriculum Enacted Curriculum

 

Analysis of Rectangular area models are the most common type of iconic mediator

Visual in both curricula (53% and 60% respectively)
Mediators Includes contextual pictures
(i.e., visual and number lines .....
mediator In both curricula, instructions regarding the use of rectangular
analysis) area models are quite detailed early in the week

and much briefer later in the week
Both curricula use “coloring” and vertical and horizontal lines
(for “cutting” and “dividing” with rectangular area models,
dividing the rectangle into A x B pieces where A and B
are the denominators of the two fractions
----- Students use alternative cutting
methods in addition to dividing the
rectangle into A x B pieces where
A and B are the denominators
of the two fractions
Both curricula use “coloring” and vertical lines for partitioning
the linear area models, dividing the model into A x B pieces
where A and B are the denominators of the two fractions;

an alternative partitioning approach is used as well

 

183

Table 16 (cont‘d)

 

 

 

Written Curriculum Enacted Curriculum
Analysis of
Use of the number line . -----
Visual
includes partitioning in ways
Mediators
similar to the linear area model;
(i.e., visual
however, coloring is not used
mediator
In both curricula, symbolic mediators are constructed using information
analysis)
gained from constructing iconic mediators
Both curricula include both “quasi-symbolic” mediators (e. g.,
“20f —1— = —3- ”) and “standard symbolic” mediators (e.g., “2 x l = 2”)
4 2 8 4 2 8
Note. Dashes (i.e., “ ----- “) indicate that the category is not applicable to the designated curriculum.

Table 16 and the more detailed analysis included within the chapter highlight many
discursive similarities and differences between the written and enacted curricula with
regard to visual mediators. The problem with such a table is that some of the richness
and interpretation present within the text of the chapter is lost. However, the summary
provided within the table allows a look at the data as a whole.

It seems that there are far more similarities between the written and enacted
curricula in this analysis (presented in Table 16) than are present in the analysis of
mathematical words (presented in Table 10 on p. 124). From the information presented
in Table 16, I conclude that the use of visual mediators in the written and enacted
curricula is similar. Of particular importance is that both curricula address the goals of

mediational flexibility and fluency with symbolic mediators in similar ways. That is,

184

students are introduced to a variety of visual mediators, expected to recognize the
relationships between them, and transition from one to another.

One notable exception to this overall similarity is the lack of number lines in the
enacted curriculum.50 This is particularly important for the purposes of objectification
because the number line affords opportunities to reify fractions (i.e., discuss them as
numbers). Area models (whether rectangular or linear) do not necessarily afford these
same Opportunities for reification because the discourse if more likely to include words
such as “part” or “piece” and phrases such as “X of Y” which do less to promote
fractions as numbers. This greater opportunity for reification in the written curriculum
supports similar earlier conclusions in Chapters 4 (Goals of the Written and Enacted

Curricula) and 5 (Mathematical Words in the Written and Enacted Curricula).

 

5° This difference is highlighted in Table 16.

185

CHAPTER 7: ENDORSED NARRATIVES
IN THE WRITTEN AND ENACTED CURRICULA

Endorsed narratives are the third mathematical feature under consideration in this
investigation. A narrative is defined as “any text, spoken or written, that is framed as a
description of objects, or of relations between objects or activities with or by objects, and
that is subject to endorsement or rejection, that is, being labeled true or false” (Sfard,
2008, p. 176). A “narrative” that has been established as true within some mathematical
discourse community has been “endorsed.” Endorsed narratives in formal mathematical

discourse include, but are not limited to, axioms, definitions, and theorems. Narratives

can be classified as object-level (e. g., “%x—;— = g”) or meta-level (e.g., “multiply the

numerator of the first fraction by the numerator of the second fraction to get the
numerator of their product”).

There is a wide variety of ways in which narratives can be endorsed depending on
the mathematical context. On one end of the continuum, professional mathematicians
adhere to very specific rules regarding the endorsement of narratives. That is,
mathematicians produce fully endorsed narratives using formal deduction. This takes the
form of a proof containing a series of endorsed narratives sequentially linked through
strict deductive processes.5 I Induction and abduction are often used by mathematicians
as heuristics for investigating mathematics, but neither by itself can serve to endorse a
narrative in scholarly mathematics. In contrast, empirical evidence is often used to
endorse narratives in colloquial mathematics. For example, children might use counting

marbles as a procedure to “prove” that they have more marbles than their playmates. One

 

5' This is the tradition in mathematics; however, there are current debates regarding what counts as proof in
mathematics. For example. can computer-generated data serve as proof?

186

of the goals of school mathematics is to move students from the everyday use of
empirical or quasi-empirical evidence to more formal mathematical methods of narrative
endorsement (Sfard, 2008).

Much of the discourse in mathematics involves (1) recalling, (2) constructing, and
(3) substantiating narratives. One can imagine a mathematics classroom in which
students are asked to recall basic mathematical facts such as “3 x 6 = 18,” construct the

number sentence “45 + 86 = 131,” and substantiate the narrative, “The probability of

choosing a jack from a standard deck of cards is 1i .” As illustrated here, recalling

involves summoning a previously endorsed narrative, constructing results in a newly
endorsed narrative, and substantiating addresses the ways in which decisions are made
about whether to endorse a narrative (i.e., decide that it is true) (Sfard, 2008, p. 291).
Clearly the nature of these processes varies both quantitatively and qualitatively
depending on many factors including the age and grade level of the students. For
example, many first grade students may construct the narrative “2 + 7 = 9,” whereas a
fourth grade student may recall the same narrative. Similarly, fourth grade students and
eleventh grade students may have different methods by which to substantiate the
narrative. Narratives and their endorsement play a key role in mathematics. That is,
using previously substantiated narratives to construct new narratives is the work of
professional mathematicians.

Given this, it seems appropriate to examine recalling, constructing, and
substantiating narratives in the comparison of the mathematical features of the written

and enacted curricula. This section first introduces the types of narratives present in the

187

written and enacted curricula and then describes the ways in which recalling.
constructing. and substantiating play out in relation to these narratives.
Types of Narratives

Only those narratives directly related to fraction multiplication are considered in
this analysis. That is, the narrative is either an example of fraction multiplication (object-
level narrative) or addresses the derivation or magnitude of the product of fraction
multiplication (meta-level narrative). As with the mathematical features discussed
earlier, it is necessary to explicate the analytic decisions made for the process of

identifying the endorsed narratives in the written and enacted curricula. In the case of the

. . . . . 2 1
written curriculum, many narratives are opened in the Student Guide (e. g., “—3— x — ”) and

closed in the Teacher Guide (e.g., Egg”)? In these cases, the opening and closure of

the narratives were conjoined (e. g., “3 x l = i ”) and included in the analysis. This

3 5 15
decision is consistent with the conceptualization of the written curriculum as the union of
the Teacher’s Guide and the Student Guide.

Given the multi-modal nature (i.e., it contains both spoken and written forms of
communication) of the enacted curriculum and the general complexity of spoken
language, several analytic decisions applicable to the enacted curriculum are described
here.53 First, many narratives are included multiple times in close proximity to one

another within the enacted curriculum. This repetition takes several forms, including

 

52 Throughout this analysis, “opening” will refer to the “factor x factor” component of the narrative (e.g.,
2 99

_ )_

15

53 The written form ofthe enacted curriculum includes the teacher and students writing on paper, posters,
transparencies, and the whiteboard.

2 l . .
“ 3 x g ”) and “closure” wrll refer to the “= product” component of the narrative (e.g., “=

188

saying the narrative as well as writing the narrative, saying the narrative more than once
in adjacent utterances, and writing the narrative in two forms, hereafter referred to as

1 3

3 3
x =—” and“ uasi-s mbolic form” e. .,“——of—=—”. In
8 ) q y ( g 4 2 8 )

All»
NI—

“symbolic form” (e.g., “

the first two cases (i.e., saying and writing, saying more than once in adjacent utterances),
the narrative was counted only once in this analysis. In the case of saying and writing,
the written form was included because this best served the comparison with the written
curriculum. It can be argued that these repetitions are important and indicate emphasis
on particular narratives; however, the major purpose of this analysis is to compare the
written and enacted curricula and such redundancy is not a feature of the written
curriculum. Therefore, the exclusion of these redundancies is appropriate for this
comparative study. In the final case (i.e., writing the narrative in symbolic form and
quasi-symbolic form), both forms were counted because both forms appear in the written
and enacted curricula.

Several times in the enacted curriculum, narratives are proposed in which the

opening (‘éof é”) is the same, but the closure of the narrative is different (e.g.,

9? 6‘

of

6630f
3

01A

”). Note that in these examples, one of the narratives is closed

wlN
NI"

-3
69

Nl—t

with “ = ” and the other is closed with “=

O\IN
ONIA

.” In some of these cases (as in this one),

the narratives are non-equivalent (i.e., one of them is mathematically incorrect). In other

99 “_2_0f_1_ = i 5,). In these caSeS,

3212

of

WIN

3
6 9

. . 1
cases, the narratives are equivalent (e.g., “ 3

189

whether equivalent or not, the proposed narratives are included in the analysis. These
instances of “conflicting narratives” will be elaborated later in this section.

134 and 108 narratives in the written and enacted curricula respectively are
directly related to fraction multiplication. Here, I examine these narratives in several
ways to provide a more elaborated picture of the narratives. Figure 23 summarizes the

relative frequencies of meta—level and object-level narratives in the written and enacted

curricula.
Written Curriculum Enacted Curriculum
Meta— Meta-
level level ,
21% 23% /

   

or): t-
Object- 1‘”
1e e1 level
V
770/
79% °

Figure 23. Types of narratives in the written and enacted curricula

The relative frequencies of object-level and meta-level narratives related to fraction
multiplication in the written and enacted curricula are very similar. In both cases, object-
level narratives make up more than 75% of the narratives. This may indicate that
students are expected, in both curricula, to multiply fractions more than to talk about
fraction multiplication. Next, I describe the nature of each of these types of narratives in

greater detail.

190

Object-level Narratives
All object-level narratives in both the written and enacted curricula involve the
multiplication of two numbers, at least one of which is a fraction or includes a fraction
(i.e., a fraction or a mixed number). Table 17 summarizes the relative frequencies of the
types of numbers involved in these narratives in both curricula.
Table 17.

Relative Frequencies of the Types of Numbers involved in Object-level Narratives in the
Written and Enacted Curricula

 

 

 

Written Enacted
Type Sample Narrative Curriculum Curriculum
Fraction x Fraction “ 1 x 2 _ 2 ,, 50% 69%
3 3 9
Fraction x Whole Number “ix 2 = 1,, 14% 6%
Fraction x Mixed Number “1x 2 l _ _5_ ,, 15% 8%
3 2 6

Mixed Number x Whole Number , 2 ,, 3% 1%
‘ l—x 12 = 20

Mixed Number x Mixed Number “ 21x 1 l _ 21_1,, 18% 16%
2 6 12

. . 1 2 5 2 1 5
Note. Order ofthe numbers was not consrdered (i.e., “2 3x; = 1;” and “ 3x 2 3 = I; ” were treated as

the same type).

“Fraction x Fraction” object-level narratives are proportionally more common in the
enacted curriculum than in the written curriculum (69% compared to 50%). Three other
types (“Fraction x Whole Number,” “Fraction x Mixed Number,” and “Mixed Number x
Mixed Number”) are represented in approximately 15% of the object-level narratives in

the written curriculum. Only one other type (“Mixed Number x Mixed Number”)

191

receives this much attention in the enacted curriculum. Figures 24 and 25 indicate the
way in which these object-level narratives are distributed across the five days in the
written and enacted curricula. Figure 24 presents the distribution of the “Fraction x
Fraction” type and Figure 25 presents the distribution of the “Other” types (all types

except “Fraction x Fraction”).

 

 

 

>.

O

G

a l

g. I Written

E Curriculum
a; C! Enacted

“g Curriculum ‘
3
a:

 

Day 1 Day 2 Day 3 Day 4 Day 5

 

 

 

Figure 24. Distribution of “Fraction x Fraction” type object-level narratives across the

five days of Investigation 3 in the written and enacted curricula.

192

 

g» 50 y
5 —
i g. 40 I Written 1
2 Curriculumli
‘ In 30 l
2 ll Enacted
I 3g 20 ’Curriculumi
T»
a:

 

0
i Day 1 Day 2 Day 3 Day 4 Day 5

L,_ ,~, if, ._./,E _7

Figure 25. Distribution of “Other” type (i.e., all types except Fraction x Fraction) object-
level narratives across the five days of Investigation 3 in the written and enacted
curricula.
Figures 24 and 25 indicate that in both the written and enacted curricula, “Fraction x
Fraction” narratives are more common early in the Investigation and the other types
become more prevalent later in the Investigation. Most notable, however, is the distinct
separation of “Fraction x Fraction” from the other types in the enacted curriculum. In
fact, the other types are completely absent in the enacted curriculum during the first three
days and the “Fraction x Fraction” type is virtually absent during the last two days of the
Investigation. When taken together, Table 17 and Figures 24 and 25 indicate that both
the types of object-level narratives and their distribution across the Investigation differ
between the written and enacted curricula.

In order to further compare the types of object-level narratives in the written and
enacted curricula, I classified the narratives into “simple” and “complex” categories. The
simple object-level narratives include three forms: (1) “A x B = C,” (2) “A of B = C,”

and (3) “A x B ~C,” where A and B are fractions, mixed numbers, or whole numbers,

193

and where no more than one of A or B is a whole number. The complex object-level
narratives include any forms other than these three basic forms. In both the written and
enacted curricula, the complex forms present in the curricula illustrate the Distributive
Property and converting a mixed number into an equivalent improper fraction in order to
facilitate multiplication. Figure 26 summarizes the relative frequencies of simple and

complex object-level narratives in the written and enacted curricula.

Written Curriculum Enacted Curriculum
Complex
Complex 10%

13% ,.

   

/"

   
 

/

 

Simple Simple
87% 90%
Figure 26. Types of object-level narratives in the written and enacted curricula.
The relative frequencies of simple and complex object-level narratives in the written and
enacted curricula are very similar; nearly 90% of the object-level narratives in both
curricula are simple.
Table 18 summarizes the relative frequencies of the three simple forms of object-

level narratives present in the written and enacted curricula.

I94

Table 18.

Relative Frequencies 9’ three Forms of Simple Object-level Narratives

 

 

Form Written Curriculum Enacted Curriculum
“A x B = C” 47% 45%
“A ofB = C” 16% 41%
“A x B as C” 34% 13%

 

Table 18 indicates that the form “A x B = C” is represented in similar proportions in both
the written and enacted curricula, in both cases accounting for nearly half of the simple
object-level narratives. However, “A x B as C” is more than twice as common in the
written curriculum than in the enacted curriculum and conversely “A of B = C” is more
than twice as common in the enacted curriculum than in the written curriculum. Both of
these findings support conclusions make in earlier chapters. The greater relative
frequency of simple object-level estimation narratives supports earlier findings that
suggest that a greater emphasis on estimation is present in the written curriculum than in
the enacted curriculum. The greater relative frequency of simple object-level narratives
containing “of” in the enacted curriculum supports earlier findings that concluded that “x
of y” is more common in the enacted curriculum. This discrepancy is important because
estimation narratives have the potential to reify fractions while “X of Y” do not provide
this opportunity. Figures 27, 28, and 29 summarize the distribution of the simple object-
level narratives, A x B = C, A of B = C, and A x B ~C, respectively across the five days

included in this analysis.

195

 

I Written Curriculrnn
El Enacted Curriculum

Relative Frequency

 

Day 1 Day 2 Day 3 Day 4 Day 5

 

 

 

Figure 27. Distribution of “A x B = C” across the five days of Investigation 3 in the

written and enacted curricula.

 

I Written Curriculum
El Enacted Curriculum

Relative Frequency

 

Day 1 Day 2 Day 3 Day 4 Day 5

 

 

 

Figure 28. Distribution of “A of B = C” across the five days of Investigation 3 in the

written and enacted curricula.

I96

Relative Frequency

 

. ‘._:‘ ”f IWritten Curriculum
L j ,7 D Enacted Curriculum

Day 1 Day 2 Day 3 Day 4 Day 5

 

 

 

Figure 29. Distribution of “A x B ~C” across the five days of Investigation 3 in the
written and enacted curricula.
Figures 27, 28, and 29 indicate interesting features of the distribution of each of the basic
forms of simple object-level narratives. First, narratives of the “A x B = C” form are
most common in Days 2 and 5 in the written curriculum and Days 2 and 3 in the enacted
curriculum. It is striking that this is the only form present on Day 5 of the written
curriculum. Second, narratives of the “A of B = C” form are more common in the early
days of the Investigation in both the written and enacted curriculum; however, this
tendency is much more pronounced in the written curriculum than in the enacted
curriculum where such forms are present on each day. Finally, narratives of the form “A
x B ~ C” are present on four days in the written curriculum, but only on two days in the
enacted curriculum. Together, Table 18 and Figures 27, 28, and 29 indicate that the
written and enacted curricula vary in both the proportion of the three forms of simple
object-level narratives and their distribution across the five days in this analysis.

As mentioned previously, 13% and 10% of the object-level narratives in the

written and enacted curricula respectively are “complex.” At least 75% (79% in the

197

written curriculum and 75% in the enacted curriculum) of these complex object-level
narratives in both the written and enacted curricula illustrate the distributive property and
approximately 25% (21% in the written curriculum and 25% in the enacted curriculum)
demonstrate the conversion of a mixed number to. an equivalent improper fraction for the
purposes of calculating the product.

An example of the “distributive property” form of object-level narratives from the
written curriculum is provided. here:

Example 151 (WC).

1 1 I 1 1 1
“2.. 10—= 2— 10 + 2— — =24— TC, .74
3x 2 (3x 1 (312) 2 f P 1

An example from the same category, the same problem as well, in the enacted curriculum

is demonstrated by the following poster presented on Day 5:

Example 152 (EC - Student Work).

 

Y} Q‘lgx \Oi’i
2 ‘13 x 9‘23}:
2hxhtt

(if '

 

 

 

 

 

 

Examples 151 and 152 illustrate the same complex object-level narrative in the written
and enacted curricula. Examples from the written and enacted curriculum of the second
form of “complex” object-level narratives, converting a mixed number to an improper
fraction for the purposes of finding the product, are provided here. First, from the written

curriculum:

198

Example 153 (WC).

1 3 5 3 7 _
“‘2—x—=—x—=1—‘ TO, .71
2 4 2 4 s ( p )

The following poster and associated explanation provide an example from Day 5 of this

category in the enacted curriculum:

Example 154 (EC - Student Work).

@813" 1012‘ 31%

”2......—
”WWI-7" :

c
1" ‘.... “Zr /)(
rice-4 /9 a 2 .

 

 

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:15 Q)
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5'13

 

at
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S: “Okay. I did two thirds times ten halves [points to “2% " and “IO-:— ”] and

then I changed two thirds [points to “2::— ”] to seven thirds and ten and a half

to twenty one seconds, and then three times two is six and then seven times
twenty one is a hundred forty seven and I showed that here [points to 21 x 7 =
147]. So I did six, which is my denominator, divided by one hundred forty
seven [points to 14 7 + 6 = 24R3], which is my numerator, and then 1 got

twentyfour and three sixths [points to “24% ”j. ”

In this explanation, the student describes the conversion of the mixed numbers to
improper fractions followed by the use of the stande algorithm for fraction

multiplication, albeit while displaying some confusion regarding reading mixed numbers

199

(e.g., saying "two thirds while pornting to “2—3-"). This language confusron, however,

does not seem to impact the student’s ability to complete the process of fraction
multiplication correctly. The relative frequencies(see Figure 26 and discussion on p.
195) and Examples 151-154 indicate that the written and enacted curricula are similar,
both quantitatively and qualitatively, with respect to “complex” object-level narratives.
Meta-level Narratives

As mentioned previously, meta-level narratives represent 21% and 23% of the
narratives in the written and enacted curricula respectively. These narratives were
classified into two types in the written curriculum, those that describe an algorithm for
fraction multiplication (hereafter “algorithm” meta-level narratives) and those that
describe the relationship between the magnitude of the factors and the product (hereafter
“factor-product relationship” meta-level narratives). A third type of meta-level narrative
present in the enacted curriculum describes the commutativeproperty as it relates to
fraction multiplication (hereafter “commutative” meta-level narrative). Table 19
summarizes the relative frequency of these types of meta-level narratives in the written

and enacted curricula.

200

Table 19.
Relative Frequencies of Types of Meta-level Narratives in the Written and Enacted

(,‘urricula

 

Type of Meta-level

 

 

Narrative Written Curriculum Enacted Curriculum
“Algorithm” 50% 84%
“F actor-Product 50% 8%
Relationship”
“Commutative” ----- 8%
Note. Dashes (i.e., “ ----- “) indicate that the category was not present in the designated curriculum.

Table 19 indicates that the enacted curriculum contains substantially more “algorithm”
meta-level narratives than the written curriculum, and conversely the written curriculum
contains substantially more “factor-product relationship” meta-level narratives than the
enacted curriculum.

“Algorithm” meta-level narratives were categorized into those that describe the

standard algorithm for multiplying fractions (i.e., x = g) and those that describe

3 _c_
b d
other algorithms, including procedures used for mixed numbers and whole numbers. In
the written curriculum, 57% are of the first type and 43% are of the second type. The
corresponding frequencies in the enacted curriculrun are 48% and 52%. An example of
each type of “algorithm” meta-level narrative from both the written and enacted
curriculum is provided here (“standard” followed by “other”):

Example 155 (WC).

“If you multiply the numerators and multiply the denominators, you will get the
product of the fractions. ” (T G, p. 70)

201

Example 156 (EC).

S: “What I realized is that when you plus it you are, uh, finding, like, common
denominators, and then when you’re, uh, timesing it, you can just, uh, you're
timesing one of the numerator by the, the numerator by the numerator, and
then you're timesing the denominator by the denominator. ” (Day 2)

Example 15 7 (WC).

“ When the denominator of the first number is equal to the numerator of the
second number, just use the first numberfor the numerator and the second
number for the denominator. ” (T G, p. 6 7)

Example 158 (EC).

T: “Oh my gosh. Now wouldn't that be nice if we could multiply any fraction by
a whole number just by doing that [points to where she has just put 1 under 20

in “—x— = — = 10 ” on the whiteboard] and we could still use our

algorithm? ” (Day 5)
Example 157 from the written curriculum describes an algorithm for a special case of
fraction multiplication (i.e., when the denominator of the first number is equal to the
numerator of the second number), whereas Example 158 illustrates an algorithm for
multiplying a fraction by a whole number. Note that Examples 155-157 suggest the
reification of fractions because they involve a procedure for "multiplying fractions, which
implies that fractions are numbers. Example 158 goes a step further and explicitly refers
to fractions as “numbers.”

“Factor-Product Relationship” meta-level narratives make up 50% of the meta-
level narratives in the written curriculum, but only 8% in the enacted curriculum. An
example from the written curriculum is provided here:

Example 159 (WC).

“When you multiply a fraction by a fraction, the product is less than each of the
two fractions you multiplied. ” (SG, p. 35)”

 

54 This “Factor-Product Relationship” meta-level narrative was stated as a conjecture by Libby, a fictitious
student in the written curriculum.

202

Similarly, the following excerpt from the enacted curriculum addresses the relationship

between the magnitude of the factors and the product in fraction multiplication:

Example 160 (EC).

T: Well, that 's why 1 said it 's sometimes hard to think about multiplying with

fractions. Because we 're not getting like seven groups of five where we ’re
getting this bigger number. W e 're taking a part of something that 's already a
part. So your, what's going to happen to your answer if you’re taking a part
of something that 's already only a part? Is your answer going to get bigger,
or is your answer going to get smaller?

S: Smaller. (Day 2)

The large discrepancy between the relative frequencies of “factor-product relationship”

meta-level narratives in the written and enacted curricula (50% compared to 8%) supports

earlier findings that the relationship between the magnitude of the factors and product of

fraction multiplication receives more emphasis in the written curriculum than in the

enacted curriculum. As with estimation object-level narratives, this is important because

discussions of this relationship promote reification of fractions.

Finally, 8% of the meta-level narratives in the enacted curriculum address the

Commutative Property. An example is illustrated in the following excerpt from Day 3:

Example 161 (EC).

(1)
(2)

( 3)
( 4)

( 5)

SI: So it could be either way.

T: So it could be either way [points to “ x

klN
MIN
MIN
klN.
OoIN

written on the transparency]?
S2: One fourth of one half because that ’s really what you took.

T: Okay, you're right. That 's really what he took. But what point is
Collin bringing out that's an important one about multiplication?

S3 : It doesn't matter which, which way -

203

( 6) T: Do you remember what you call that property? Remember, we talked
about with addition?

( 7) S] : Commutative.
(8) T: You're good. Excellent. Excellent.
Overall, Table 19 and Examples 155-161 indicate that although the relative frequency of
meta-level narratives in the written and enacted curricula is quite similar (approximately
20-25%) and qualitatively many of the sample meta-level narratives are similar, the
proportion of the meta-level narratives representing the three types varies greatly.
Recalling narratives
Students are expected, both in the written and enacted curricula, to recall
previously endorsed narratives. For example, Problem 3.1 in the written curriculum
states “In this investigation, you will relate what you already know about multiplication
to situations using fractions” (SG, p. 32). This sentence is likely asking students to recall
previously endorsed narratives related to whole number multiplication because this has
been the students’ primary experience (if not their entire experience) with multiplication.
Similarly, in the enacted curricula (specifically early on Day 1) the class recalls narratives
regarding whole number multiplication:
Example 162 (EC).
(1) T: So what if I had something like this (writes “12 x 13 ” on the board),
twelve times thirteen? How would you think of this problem right here?
Grace, how would you think of this problem?
(2) S1: As twelve groups of thirteen.
(3 ) T: Twelve groups of thirteen. How many of you think about it like that?
(Some students raise their hands). Okay? Can you tell me the answer to
this problem right here (points to “8 x 3 ” written on the board), eight

groups of three or eight plus eight plus eight? What 's the answer,
anybody?

204

( 4) Ss: T wenty-four.
In this example, the class is asked to recall the endorsed narrative “8 x 3 = 24.” In
addition, students are asked to recall that 24 is 8 groups of 3 or 8 + 8 + 8. Recalling
endorsed narratives, however, does not explicitly take place during this Investigation,
either in the written or enacted curricula, very often. It can be argued that every time
students calculate the product of fractions using the standard algorithm they recall their
basic multiplication facts; however, because these facts are not the focus of this
investigation, recalling will take a backseat to constructing and substantiating narratives
for the remainder of this chapter.

Constructing Narratives

Recall that constructing results in a new narrative that is subject to rejection or
endorsement. Before examining the relationship between the written and enacted
curricula in terms of the construction of narratives, it is important to note that the way in
which a narrative is constructed proved to be more difficult to glean from the written
curriculum than from the enacted curriculum where the construction is often captured in
the classroom discourse. Statements such as Example 155 (see p. 202) are the exception
to this. In this case, the algorithm is a prescription for constructing narratives. Some
general statements made in the written curriculum provide subtle clues regarding the
ways in which the authors are envisioning narrative construction in the classroom. For
example:

Example 163 (WC).

“Students develop strategies for multiplying mixed numbers built on their
previous work. " (TG, p. 71)

205

Example 164 (WC).
“Use what you know about equivalence and multiplying/fractions tofirst estimate,
and then determine, thefollowing products. ” (SG, p. 38)

Example 165 (WC).
“Allow students to find ways to make sense of the problem using the model. ” ( T G,

p. 60)
Example 163 indicates that students should use strategies deveIOped “in their previous
work” to construct narratives for multiplying mixed numbers. Example 164 again
suggests using previous knowledge, here about equivalence and estimation, to construct
narratives. Finally, Example 165 advocates for the use of “models” to construct
narratives. These types of statements along with specific directions given in the
Teacher’s Guide and Student Guide provide evidence regarding the methods of narrative
construction in the written curriculum. Because the method of construction in both
curricula is largely dependent upon the type of narrative, this section will be organized
around the classification of object-level and meta-level narratives.

Construction of Object-Level Narratives

The construction of object-level narratives varies depending whether the
narratives come from contextual or non-contextual problems.55 For example, consider
the following contextual question (i.e., Question B-3 from Problem 3.2) in the written
curriculum:

Example 166 (WC)
Blaine drives the machine that points stripes along the highway. He plans to

paint a stripe that is .120 of a mile long. He is g of the way done when he runs

out of paint. How long is the stripe he painted? (SG, p. 35)

 

55 Recall that “contextual” here is used for Questions that are associated with a “real life” situation.

206

. . _ . . 2 9 ,. .
In this Question, the opening of the narrative (i.e., “—3— x16 ‘) rs constructed from the

context of the problem and the closure of the narrative (e.g., “=;—:”) may be constructed

using a variety of strategies. In contrast, consider this non-contextual Question (Question
A-1.d. from Problem 3.2) in the written curriculum:

Example 16 7 (WC):

“Use your algorithm to multiply fix; " (SG, p. 3 9)
The Question provides the opening of the narrative and requires only the construction of

the closure of the narrative (e.g., “:8 ‘). In this case. the question specrfies the use of an

algorithm for this construction. Given the importance of whether the narrative comes
from a contextual Question or a non-contextual Question, this section begins with a
comparison of the relative frequencies of narratives associated with contextual and non-

contextual questions in the written and enacted curricula (see Figure 30).

  

Written Curriculum Enacted Curriculum
Context
21%
Non- /
Context [ Context

47% \\ 53%

Non- "/

Context
79%

Figure 30. Relative frequencies of narratives associated with contextual and non-

contextual Questions in the written and enacted curricula.

207

Figure 30 indicates that less than one-fourth of the narratives in the written curriculum
are associated with contextual Questions compared to approximately half of the
narratives in the enacted curriculum. Figures 31 and 32 illustrate the distribution of the
narratives associated with contextual and non-contextual Questions respectively across

the five days.

 

60

I Written Curriculum
C1 Enacted Curriculum

Relative Frequency
8

_|
O

 

20 I
,I- I

Day 1 Day 2 Day 3 Day 4 Day 5

 

 

 

Figure 31. Distribution of narratives associated with contextual Questions across the five

days of Investigation 3 in the written and enacted curricula.

208

 

I Written Curricullnn
ELElagted Cru'riculuLn

Relative Frequency

 

Day 1 Day 2 Day 3 Day 4 Day 5

 

 

 

Figure 32. Distribution of narratives associated with non-contextual Questions across the
five days of Investigation 3 in the written and enacted curricula.
Figures 31 and 32 highlight several interesting features of the distribution. For example,
the relative frequency of narratives associated with contextual Questions in the written
curriculum increases steadily over Days 1-3, reaches a maximum of 50% on Day 3, and
then disappears completely. That is, there are no contextual narratives presented in the
written curriculum on Days 4 and 5. Narratives associated with contextual Questions are
present on all days in the enacted curriculum and are actually most prevalent on Day 5.
The narratives associated with non-contextual Questions are represented quite
consistently in the written curriculum, approximately 20% each day with a slight dip on
Day 3. It is also notable that more than 50% of the non-contextual narratives in the
enacted curriculum occur on Day 2. Some of these differences may be attributable to the
fact that Problems 3.4 and 3.5 were not enacted in the classroom during these five days.

The possibility remains; however, that the students may have had more experience with

209

contextual narratives and less experience with non-contextual narratives proportionally
than was intended by the authors.
Sixty-eight percent and 73% of the contextual object-level narratives in the

written and enacted curricula respectively include both context and iconic mediators in

. . . . 1 3 ,,
their construction, the opening of the narrative (e.g., “-2—of g ) constructed from the

. . 3 .
context of the Question and the closure of the narrative (e. g., “=1—6—”) constructed usrng

an iconic mediator representing the Question.56 Two examples are provided here in
which the written curriculum (speaking to the student in Example 168 and the teacher in

Example 169) encourages the use of iconic mediators for constructing narratives.

Example 168 (WC).
“Remember, to make sense of a situation you can draw a model or change a
fraction to an equivalent form. ” (SG, p. 32)
Example 169 (WC).
“Help them connect what is happening in the drawing to what is happening in the
problem. ” (T G, p. 61)
The use of iconic mediators for completing the construction of object-level narratives in
contextual questions is also encouraged in the enacted curriculum. This is evidenced by
questions asked by the teacher, such as: “Are you guys following his drawing?” on Day
1 and “But what is your drawing showing?” on Day 2. The following excerpt from Day
1 contains a discussion in the enacted curriculum in which a narrative is constructed from

a contextual question during a whole class discussion. The poster presented by a group

of students is also included here.

 

5" 1 will not provide details here regarding the ways in which the models were used; model use is detailed in
the “Visual Mediators” section of this chapter.

210

Example I 70 (EC - Student Work).

 

 

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#1.

 

 

 

 

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U

 

 

 

 

(l)

(2)

( 3)

(4)
(5)
(6)
(7)
(8)
(9)
(10)

T: Okay. Can we go over there to number 1? This is ofl that extra sheet
of problems. "Jennifer wants to buy some brownies. When she gets there
the pan is half full. She buys two thirds of that pan. " So talk about what
you guys did.

S 1 : Um, well, there, we knew that she, there was, it was half fitll, so we
split it into half [waves hand down middle of picture ], and then we, she
bought two thirds of it, so we shaded, we made thirds first [points to each
line dividing the rectangle into thirds], and then we shaded in two [points
to shaded two thirds], and then, um, she, it said what fraction of the pan
she bought, and she would buy two sixths of the pan because if you split
all these into thirds [points to picture] you'll get two sixths.

T: So can somebody say in words what she bought? She bought what?
What fraction of what fraction? Anybody. What'd she buy?

Ss: Two sixths.

T: But that's the answer. But what did she buy?
S? : Two

T: Two

S ?: Two thirds

T: Two thirds of what?

S?: Ofa half

211

(11)

(12)

(13)

(14)
(15)
(16)

(17)

T: Can you write that down right by your problem. She bought two
thirds of one half Can you write that next to it [SI writes “2 ” on
whiteboard next to poster]? You might have to write it underneath [SI
erases “2 ” on whiteboard]. I don 't know if you can squeeze that in there.
Underneath your drawing. Two thirds of what?

S? : One half

T: So two thirds of one half [S1 writes “gof % ” underneath picture],

and, and what was two thirds of a half? How much of that pan, you guys?
S?: Two thirds
T: What answer did they get?

S? : Two sixths

<3le

T: Could you write "equals one. equals two sixths" [SI writes “=

’9] so

we can look at that for a minute?

The length of Example 170 illustrates the complexity of constructing narratives in a

. 2 1 2 ,, . .
classroom. Here the class constructed “Eof 2 = g usrng a contextual Question and an

iconic mediator.

The other common method of construction of object-level narratives occurs in the

enacted curriculum. These cases, comprising the construction of 23% of the objective-

level narratives, are those in which the contextual obj ect—level narrative is constructed

using a previously-constructed narrative. The student work included in Example 170

. . . . 2
represents this type. The narrative on the original poster was “30f

. . . 2
an “X” was placed over the “of,” constructing a new narrative “3 x

.” On Day 2,

MI—
axlrx)

O\IN

” (see Figure

MI—

20 on p. 146). Construction of a new object-level narrative from a previously-

212

constructed narrative does not occur in the written curriculum; however, the use of
previously-constructed narratives to construct new meta-level narratives is common in
both the written and enacted curricula. This type of narrative construction is very
important as this is the way that narratives are constructed in formal mathematics. That
is, previously proven statements are used in the construction of new statements. More
examples of this type will be discussed in the next section, Construction of Meta-Level
Narratives.

As discussed earlier, in non-contextual object-level narratives, the opening of the
narrative is provided in the written curriculum (and therefore also accessed in the enacted
curriculum if the question is completed by the class). The closure of the construction of
these narratives occurs in two primary ways in both the written and enacted curricula.
The first method uses an iconic mediator in the same way described previously in the
construction of contextual object-level narratives (see description on p. 207). The only
difference is whether or not the opening is given (non-contextual) or needs to be derived
from a contextual situation. This method of construction is used in 21% and 59% of the
non-contextual object-level narratives in the written and enacted curricula respectively.

The second method is the use of estimation strategies to complete the construction
of object-level narratives when the opening of the narrative is given. This estimation
method is used in 33% and 23% of the non-contextual object-level narratives in the
written and enacted curricula respectively. An example of this method occurs in both the
written and enacted curricula in association with the “Getting Ready” in Problem 3.3

(SG, p. 36):

213

Example I 71 (WC).

 

 

Getting Ready for Problomgb

Estimate each product to the nearest whole number (1. 2. 3. . . .).

l 9 1 9 I 4 I 11
2X2m 15x27?) 25x7 3_X2E
Will the actual product be greater than or: less than your whole number

estimate?

 

 

 

 

The following excerpt from the written curriculum provides suggestions for the use of
estimation strategies for narrative construction with reference to the first and second of
the four fraction multiplication questions here:

Example 1 72 (WC).
“There are diflerent ways to approach estimation. For example, with the first

problem in the Getting Ready, ix 21, students might use a benchmark strategy.
2 1 0
Since 2.190 is almost 3, a reasonable estimate would be 5 of 3 or] 1. In the

second problem, 1 fix 2796, students might use a combination of benchmarking

and distribution. ” (T G, p. 71 )
The first question (i.e., “%x 2% ”) referenced here is enacted in a similar way in the

classroom on Day 4. That is, the same method of narrative construction (i.e., estimation)
is used. An excerpt from this discussion is included here:
Example I 73 (EC).
(1) T: We 're just going to estimate how much this might be before we try to
figure out some sort of a strategy. Could you help me with an estimate? If
I said I was going to get one half of two and nine tenths, Graham, how

could we think about estimating that answer?

(2) SI: Well, nine tenths. that's close to a whole, so, one. Yeah, so

214

(3 ) T: So what, Graham?
(4) SI : Nine tenths is close to a whole.
( 5) T: Okay.

(6) SI : So you could round that up into a whole and then two would be a
three.

( 7) T: So you would call this [points to “ 279-0— ”] about three?

(8) S1 : Yeah.

( 9) T: And then can you think about what it would be to be half of three?

(10) SI : Um

(11) T: What's half of three?

(12) S1: One anda half
Examples 172-173 included here illustrate several interesting points regarding the
construction of object-level narratives in the written and enacted curricula. The use of
iconic mediators in the construction of contextual object-level narratives is common in
both the written and enacted curricula (68% and 70% respectively). The frequency of use
of iconic mediators in the construction of non-contextual object—level narratives,
however, is quite different, 21% of the constructions in the written curriculum and 59%
of the constructions in the enacted curriculrun. In addition to using iconic mediators,
33% of the contextual object-level narratives in the written curriculum and 23% in the
enacted curricula are constructed using estimation strategies. Furthermore, 23% of the
contextual object-level narratives in the enacted curriculum are constructed using

previously-constructed narratives. This greater emphasis on estimation in the written

curriculum supports earlier findings. Finally, if all iconic mediator use for the

construction of object-level narratives (both contextual and non-contextual) is combined,
the relative frequencies for the written and enacted curriculum are strikingly different,
31% and 66% respectively. This seems to indicate a greater variety of construction
techniques in the written curriculum.
Construction of Meta-Level Narratives

Recall that there are two types of meta-level narratives in the written curriculum
(“algorithm” and “factor-product relationship”). The enacted curriculum includes these
two types as well as a third type (“commutative”). The vast majority of “algorithm”
meta-level narratives are constructed using previously-constructed narratives, 79% and
90% in the written and enacted curricula respectively. That is, it is expected that students
will reason inductively and generalize from a series of fraction multiplication object-level
narratives to a general pattern or algorithm. Evidence of this method of construction in
the written curriculum is provided here in the form of a suggestion from the Teacher’s
Guide:

Example I 74 (WC).

“It is helpful to write the number sentence by the models. By doing this, students
typically notice that you can multiply the numerators and denominators to find the

product. ” (T G, p. 61)
On Day 2 in the enacted curriculum, the teacher returns to a series of six posters and
changes the form of the narratives from “of” narratives to “X” narratives (see Example 17
on p. 53). The associated exchange is included here:
Example I 75 (EC).
T: Okay. W e 're going to come back and look at all these, 'cause you know what
I 'm going to do? I'm going to change that word "of" to a mathematic symbol.
[Changes “of” to “X” on six posters] What you guys have done here, 1 love

the way that you're all looking at me, I can tell you're listening. Thank you
very much. When I take a part of another part, I can write that as a

216

multiplication problem. Instead of saying nine tenths of a sixth. I can also say
nine tenths times a sixth.

S1: 1 think it's times the numerator and numerator and then times the
denominator and denominator.

In fact, this exchange demonstrates the desired response from the written curriculum, that
the student, when shown a series of these previously-constructed object-level narratives
may construct a meta-level “algorithm” narrative.

In the written curriculum, 71% of the “factor-product relationship” meta-level
narratives are constructed by “exploration.” Remember that these narratives address the
relationship between the magnitude of the factors and product of a fraction multiplication
object—level narrative. The written curriculum suggests that students explore object—level
narratives in order to construct meta-level narratives about these relationships. For
example, the following excerpt appears in the written curriculum:

Example I 76 (WC).

“ You might want to explore what happens when one factor is less than 1 and the

other is greater than I. ” (T G, p. 70)
This “exploration” method of construction is not very different than that of using
previously-constructed narratives. The subtle distinction is that the previously-
constructed narratives to be considered are not specified in the curriculum. That is, it is
left to the students and teacher to decide which object-level narratives to construct and
“explore.”

All of the “factor-product relationship” and “commutative” meta-level narratives
in the enacted curriculum are constructed using previously-constructed narratives (only

four such narratives are constructed in the enacted curriculum). On Day 1, the teacher

points to a series of posters containing object-level narratives and states:

217

Example I 77 (EC).
T: “You know what. I'm noticing that every time. I'm noticing, that, ooh, I got a

little two sixths, or a third. I got three eighths, I got two twenty oneths. My
answer got so little. ”
She is likely using these products of fraction multiplication to begin to encourage
students to think about the magnitude of the product when multiplying two fractions less
than one.

Overall, both qualitatively and quantitatively, the construction of meta-level
narratives in the written and enacted curricula is quite similar. Nearly all of the meta-
level narratives are constructed by noticing patterns in previously-constructed narratives.
This result is in keeping with earlier findings that indicated similarities between the
written and enacted curricula with regard to meta-level narratives.

Substantiating Narratives

As stated previously, one of the primary purposes of school mathematics is to
change the ways in which children substantiate (and therefore endorse) narratives,
moving them from the use of empirical or quasi-empirical evidence to more formal
mathematical justification. Given that proof is the gold standard of substantiation in
scholarly mathematics, I began my investigation by searching for “prove” and its
derivatives (e. g., proving) in the written and enacted curricula. It occurs four times in the
written curriculum and six times in the enacted curriculum. All four uses in the written
curriculum refer to the use of iconic mediators for proving. Two examples are given

here:

Example I 78 (WC).
“Draw a picture to prove that your calculation makes sense "(SG, p. 38)

218

Example I 79 (WC).

“ Who can use their model to prove that the answer )3? is sensible? ” (TG, p. 76)

Four of the six uses of “prove” and its derivatives in the enacted curriculum are similar in
nature to the Examples 178 and 179, one such use is illustrated in the following excerpt
(including the corresponding student poster from Day 3):

Example 180 (EC - Student Work).

 

_.-—0
q

mi

 

 

 

 

 

 

L l

T: Go ahead. Can you prove that this is one sixth?

 

 

S: Because, if you, just like on the one big bar that we had, if you just end up
splitting that one, it ends up giving you six pieces, and so, like, if we use those
big bars, it would just show you like you had one sixth.

One of the other two instances of “prove” occurs on Day 2 in the enacted curriculum. In

this case, using the standard algorithm for fraction multiplication for proving is

suggested. The student in this excerpt has just finished constructing the narrative

to 99
__x_ = _

as shown in the poster included here:

219

Example 181 (EC - Student Work).

 

 

 

 

 

(1)

(2)
(3)

( 4)
(5)
( 6)

S1: And there ’sfour pieces in each of the fifths and we colored in three
pieces in each of those fifths. That 's what we did.

T: Okay.

SI : Another way to prove it would be three times two equals six and then

five
T: Right. But do we know that that for sure works?

Ss: No.

T: 1 uess that’s what we'ret in to l ure out with our drawin s.
g ’3’ g g g

In this excerpt, the student suggests another method of proving (i.e., using the algorithm),

but the teacher points out that the class has not yet fully endorsed the algorithm narrative,

suggesting that it is unavailable for substantiation at this point. The final use of “prove”

is a statement that allows student choice of substantiation method. It occurs on Day 4 in

the enacted curriculum:

220

Example 182 (EC).
T: “Okay, alrighty. I know some of you are really thinking hard about this. Now
I'm going to push you to think even harder, okay? I’m going to give you a
couple of situations, and what you're going to do at your table is you’re going
to talk about is the answer to the problem going to be more or less than one
whole pan of brownies. That 's all you have to do is talk about it. However
you want to prove it is up to your table. ”
These instances of “prove” provide a glimpse into the methods of substantiation in the
written and enacted curricula. An analysis of these methods will be the focus of the
remainder of this section.
I found the line between narrative construction and substantiation to be blurry in
the analysis of the written and enacted curriculum. The methods of narrative construction

are in most cases also used for narrative substantiation. For example, the previous

section indicated that iconic mediators are often used to construct narratives (in particular

99

the “ = is constructed from a contextual

\OIUi

., . . 2 5
portion of the narrative) once 3x6

problem. It is also true that iconic mediators are used to substantiate narratives. That is,

-1
12

X

all.»

if a student questions a narrative (e. g., “ ”), the same iconic mediator used to

Cali--

construct the narrative may also be used to substantiate the narrative. Therefore, for the
purposes of these analyses, 1 am conceptualizing these constructs temporally. That is,
constructing is conceived as resulting in a narrative and any discussion that occurs about
the narrative following its construction is conceived as substantiation. Therefore, this
analysis will be framed using the narrative categories described in Types of Narratives

and consider any discussions that occur after the narratives are presented or stated. In the

221

written curriculum, substantiation is also discussed in terms of how students are expected
to support their narratives once they have been constructed.

As might be expected given the findings summarized in Constructing Narratives
and the blurry line described in the previous paragraph, the primary methods for
substantiating narratives are quite predictable. More than 80% of the object-level and
meta-level narratives (88% and 81% in the written and enacted curricula respectively) are
substantiated using either iconic mediators or estimation strategies. In the written
curriculum, the 88% breaks down into 47% and 41% for iconic mediators and estimation
strategies respectively. As one might expect given the previous discussions regarding the
lesser emphasis on estimation in the enacted curriculum, the breakdown is quite different,
68% and 13% for iconic mediators and estimation strategies respectively.

Although individual narratives are substantiated using one method (often the same
method used to construct it), another interesting phenomenon is present in both the
written and enacted curricula. In the written curriculum, students are encouraged to
substantiate 25% of the narratives with a method beyond the particular method of
substantiation associated with the narrative. The method involves using the production of
the same or similar narratives using several strategies as further substantiation of the
narratives. For example, in Problem 3.3, students are asked to estimate an answer, create
a “model,” and write a number sentence for a given contextual problem. Problem 3.4
asks students to use these same three strategies in a non-contextual problem. The
agreement of these three methods (i.e., estimating, “modeling,” and writing a number
sentence) is used to further substantiate the narratives. That is, if you do the same

Question in three different ways and get the same or similar (in the case of estimation)

222

answers, there is a good chance that you are correct and the narrative can be endorsed.
There are statements in the written curriculum that provide additional support for this
phenomenon:

Example 183 (WC).

“Help them use estimation to decide if their models and computations are

reasonable. ” (T G, p. 66)

Example 184 (WC).
“ You can also estimate to see if your answer makes sense. ” (SG, p. 32)

“Agreement” between related narratives as substantiation is also present in the enacted

curriculum. For example, a discussion (Example 185) ensues after the incomplete

narrative “ 2% x g ” is displayed using the overhead projector on Day 4. First, a student

uses the Commutative Property to rearrange the problem and construct the remainder of

the narrative.
Example 185 (EC).

(1) T: What if I have something like this? What if I have two and a half
groups that are four sevenths [shows “ 33x; ”], or maybe 1 want to think

about it like four sevenths of a group that two and a half How are we
going to think about, how are you going to think about estimating
something like, just estimating?

(2) S?: I think it

(3 ) T: Connor, can you see okay? How, how can we think about estimating
that one? Katie, you have any ideas about how to estimate that one?

(4) S1: Well, um, I think it might be, um, well, 'cause if it was four and four

sevenths, um, times two and one half it would be like four sevenths of two
and a half

(5) T: Okay, so you're thinking about it like this [writes ";x 2%” ?

( 6) SI : Yeah. So

223

(7)
(8)

(9)

(10)
(11)

(12)

T: Okay.
Sl : And/our sevenths is about halfso 1 think it ’d be like two.
T: So you ’re thinking of half of two and a half[ crosses out “57,: ” and

writes “

s].

MIN

S1: Yeah.
T: So what would a half of two and a half be?

S]: Oh, wait. It would be. um, one and one fourth.

. . . . . 4 1 1 ,, .
The resulting narrative from this conversation 18 “—x 2— ~ 1—. The conversation then

turns to a debate regarding the Commutative Property and whether or not they are

allowed to rearrange the factors when constructing narratives involving mixed numbers.

The class decides to construct the original given narrative to see if the answers are the

same, thus demonstrating the strategy of “agreement.” The conversation continues:

Example I 86 (EC).

(1)

(2)

( 3)

(4)
(5)

T: Hmmm. Let's think about this for a second. This is two and a half

groups of about how much is this [points to “; "j?
Ss: Half

T: Okay [crosses out “; ” and writes “5 "j. So let's think about this

for a second. If I have two and a half halves, how much do I have? Okay.

How about if I get rid of this [covers the “—21—- ” in the “2-3 ”]? Can you

figure out two halves?

S?: You have about

Ss: A whole. A whole.

224

(6)
(7)
(39

($9

(ICU
(11)
(12)

T: What 's two halves?
Ss: A whole.

T: One whole [holds up index finger]. Could I get a half of a half

1 1 1
[gestures between the ”:2— in 23and the “2 ” on its own]?

S? : It'd be the same.

S? : It would be zero fourths.

T: Careful. W hat’s a half of a half?
S? : One fourth.

l

. . . 1 4 .
This conversation eventually constructs the narrative “ 23x 7 ~ 12 ”. After a bit more

discussion, a student suggests another yet another way to think about it:

Example 187 (EC).

(1) S1: To make it less confusing, maybe if you changed the two wholes into
halves, like, for, five halves '

(2) T: Oh

(3) SI : So maybe

( 4) T: Would that help us?

(5) S? : No.

( 6) T: What if we called this [points to “ Zéx; ” ] five halves of four
sevenths [writes “ gx; ”]? But could you multiply that using our
algorithm?

( 7) Ss: Yeah.

225

(8) T: Let's do itjus't to see what we get. Could we do five halves offour

, 5 4 ,, , ,
sevenths [writes “ —2-x—7— ]? What would we end up With?

( 9) S? : Fourteen twentieths

(1 1)) S? : Twenty fourteenths

(11) T: Actually, twentyfourteenths [writes “=j—3 ”], right? And what is that

the same as?

(12) S: One whole and sixfourteenths [writes “I 76; ”].

Example 187 constructs the narrative “i—x; =53— =1 % Examples 185-187

individually suggest various methods of narrative substantiation; however, taken as a
whole they suggest the method of “agreement.” That is, the class seems to believe that if
they can make these narratives the same or similar (in the case of estimation), the
agreement serves to substantiate (at least to some extent) the narrative. The excerpt that
follows illustrates the conversation in which the class discusses whether these narratives

(“fl—x 21 ~11,” “21xfl ~11,” and “Exilz—Q: 1%”) are close enough to serve as

7 2 4 2 7 4 2 7 14
substantiation for one another:
Example 188 (EC).

(1) T: [S that about - now these are estimates. Is that close to this estimate

. l . .
[pornts to 11m each estimate]?

(2) Ss: Yes.
(3) T: Or no?

(4) S 1 : Not really.

226

(5 ) T: No?

( 6) Ss: Yeah, yes.

( 7) T: Isn't that about -

(8) $2: 'Cause you’ve got to divide -

( 9) T: Almost one and a half and that 's one and afourth. Nope. W e ’re not
even close? Oh, okay. What do you think?

(10) S3 : I think we are pretty close, because, first of all, it's not yet a half
(11) T: Okay.

(12) S3 : So, um, I think we '11 only be like afewfourteenths ofl and that 's not
very big of a chunk because -

(1 3) T: Okay.
(14) S4: F ourteenths are very, are kind of small, small pieces.
A closely related type of “agreement” substantiation occurs when students produce

equivalent narratives using slightly different iconic mediators. For example, the

. . 1 2 . . .
incomplete narrative “40f 3— ” was wrltten on the whiteboard on Day 3. F 1rst, a student

shaded two-thirds of the bar (indicated here by light grey shading):

 

l l 1

Then the student divides the two-thirds into fourths and shades one of them (dark grey

shading). The student also cuts the unshaded third in half:

 

 

.” Another student suggests the

. . . . l 2 1
These actrvrtles construct the narrative “40f -3— = g

following to complete the same narrative “211-of % .” He begins in the same way as the

227

first diagram presented above (i.e., shading two-thirds of the bar). He then divides each

of the thirds into fourths and shades one fourth of each third (indicated in dark grey):

 

 

   

This method establishes the narrative "iof 5:: = 123 .” The following discussion

established their equivalence in the class:
Example 189 (EC).
(1) T: Are those different, two twelfths and one sixth?
(2) S I : No. They 're not different.
(3 ) T: They're not different?
(4) S1: They just look different but they're the same - but they're equivalent.
(5 ) T: So if I took this piece [points to one of the small dark gray pieces in
the third diagram] and I moved it over to here [points to the other small

dark gray piece], would it be the same size as this one sixth [points to the
dark gray piece in the second diagram]?

( 6) Ss: Yes.

( 7) T: Okay. So either way would be okay?

(8) Ss: Um hmm.
In the same way mentioned previously, these equivalent narratives substantiate one
another by the very fact of their equivalence. That is, two different methods with the
same or equivalent answers further substantiate a narrative.

A final version of the “agreement” substantiation of narratives in the enacted
curriculum involves the “agreement” of the class. That is, it seems that consensus or
convincing your classmates serves as a method of narrative substantiation. The examples

provided here illustrate the teacher questioning students regarding their agreement:

228

Example 190 (EC).
T: “How many of you agree with Trevor that 1 ate two .sixths? How many of you
disagree with him? ” (Day 1)

Example 191 (EC).
T: “Does everybody agree with her about that? " (Day 1)

Example 192 (EC).
T: “Do you guys agree with that? " (Day 2)

In addition, statements like the following further indicate the use of consensus as a
method of substantiation:

Example 193 (EC).
T: “Talk to them [your classmates]. They're the ones you have to convince. ”
WWO

Example 194 (EC).
T: “You need to convince your classmates it 's less than a whole. ” (Day 2)

Although this method of substantiation does not occur in quite the same way in the
written curriculum, there are three instances (SG, pp. 35, 37, and 38) in which fictitious
students are created that either disagree with one another or are presenting a strategy to
the “live” students in the classroom for their appraisal. One such example from the
written curriculum is included here:

Example I 95 (WC).

 

Yuri and Paula are trying to find the following product.

2 1
23 X :4-
Yuri says that if he rewrites 2%. he can use what he knows about multiplying
fractions. He writes:

8 l
s X 3
Paula asks. “Can you do that? Are those two problems the same?“

What do you think about Yuri‘s idea? Are the two multiplication problems
equivalent?

 

 

 

229

The creation of these fictitious students and asking for agreement or disagreement from
the “live” students in the written curriculum may have been intended to serve many
purposes (e. g., It may be easier for “live” students to disagree with “virtual” students than
their own peers). However, the creation of these fictitious students and the questions
associated with them seem to indicate the possibility that consensus can serve as a form
of substantiation in the written curriculum as well.
Summary

The question addressed in this summary is, “What does an investigation of the
endorsed narratives in the written and enacted curricula allow us to see?” That is, “What
do we know now about the relationship between the written and enacted curricula that we
did not know before?”57 Recall that a narrative is “any text, spoken or written, that is
framed as a description of objects, or of relations between objects or activities with or by
objects, and that is subject to endorsement or rejection, that is, being labeled true or false”
(Sfard, 2008, p. 176). Table 20 summarizes the findings from the investigation of

endorsed narratives for the purposes of examination for overall conclusions.

 

57 The statements in this summary and throughout this analysis are made with several caveats. First, this
analysis compares the written text with an enactment of the written text (one of infinitely many possible
enactments); this should be kept in mind when reading these statements as some results may be attributed
to this difference in curricular form. Second, similarity (and difference) here is through my eyes only.
That is, another person (e.g., a teacher, a textbook author) using their own lens may see things quite
differently. Finally, the evidence for my claims is gleaned from five days in the written and enacted
curricula. That is, none of my statements can be generalized either to the written curriculum as a whole or
the enacted curriculum as a whole. Rather, my statements highlight insights gained through the use of this
framework regarding the relationship between the written and enacted curriculum on these five days that
may be of interest to teachers, curriculum developers, and mathematics education researchers.

230

Table 20.

Summary of “Endorsed Narratives ” Analysis

 

 

Written Curriculum Enacted Curriculum
Types of Includes 138 narratives Includes 108 narratives
Narratives Approximately three-fourths of the narratives

in both curricula are object—level narratives
(the remainder are meta-level narratives
50% of the object-level 69% of the object-level
narratives are “Fraction x narratives are “Fraction x

Fraction" (other combinations Fraction” (othereombinations

that represent at least 10% that represent at least 10%
include “Fraction x Whole include “Mixed Number x

Number,” “Fraction x Mixed Mixed Number”)
Number,” and “Mixed I
Number x Mixed Number”)
“Fraction x Fraction” narratives occur early in the week
in both curricula

The other types of obj ect-level The other types of object-level

narratives (not “Fraction x narratives (not “Fraction x
Fraction”) are present Fraction”) are present
on all five days on Days 4 and 5

Nearly 90% of all object-level narratives are “simple”

(the remainder are “complex”)

 

231

Table 20 (cont’d)

 

Written Curriculum Enacted Curriculum

Types of Nearly half of all simple object-level narratives are of the form
Narratives “A x B = C,” where A and B are fractions, whole number, or

mixed numbers (A and B are not both whole numbers)

The remainder of
the simple object-level
narratives are primarily

of the form “A x B ~ C”

Simple object-level narratives
of the form “A x B = C” are
most common on Days 2 and 5
Simple object-level narratives
of the form “A of B = C” are
most common on Days 1 and 2
Simple object-level narratives
of the form “A x B ~ C”

occurs on Days 1-4

The remainder of the
simple object+level narratives
are primarily of the form
“A of B = C”

(i.e., quasirsymbolic form)
Simple object-level narratives
of the form “A x B = C” are
most common on Days 2 and 3
Simple object-level narratives
of the form “A of B = C”
occur across all five days
Simple object-leVel narratives
ofthe form “A x B ~ C”

occurs on Days 2 and 4

 

232

Table 20 (cont’d)

 

 

Written Curriculum Enacted Curriculum
Types of Approximately 75% of the complex object-level narratives in
Narratives both curricula address the Distributive Property (the remainder

address the conversion of a mixed number to an improper

fraction for the purposes of fraction multiplication)

50% of the meta-level 84% of the meta-level
narratives represent the narratives represent the
“algorithm” type and 50% “algorithm” type (8%

represent the “factor-product represent the “factor-product
relationship” type relationship” type and 8%
represent the “commutative”
, type)
Approximately half of the “algorithm” meta-level narratives
address the traditional algorithm for fraction multiplication

and the other half address other algorithms

Constructing Object-level narratives Object-level narratives
Narratives associated with contextual associated with non-contextual
Questions represent 21% Questions represent 53%
of the narratives of the narratives

 

233

Table 20 (cont’d)

 

 

Written Curriculum Enacted Curriculum
Constructing Object-level narratives Object-level narratives
Narratives associated with contextual associated with contextual
Questions occur on Days 1-3 Questions occur

across all five days

The construction of object-level narratives varies depending

on whether it is associated with a contextual
or non-contextual Question
Approximately 70% of the contextual object-level narratives are
constructed using both context and iconic mediators
----- 23% of the contextual
object-level narratives are
constructed from previously-

constructed narratives

21% of the non-contextual 59% of the non-contextual
object-level narratives object-level narratives
are completed using are completed using
iconic mediators iconic mediators

33% of the non-contextual 23% of the non-contextual

object-level narratives object-level narratives
are completed using are completed using
estimation strategies estimation strategies

 

234

Table 20 (cont'd)

 

 

Written Curriculum Enacted Curriculum
Constructing 31% of all object-level - 66% of all object-level
Narratives narratives are constructed narratives are constructed
using iconic mediators using iconic mediators

Nearly all “algorithm” meta-level narratives are constructed
using previously-constructed narratives

(79% and 90% respectively)

71% of the “factor-product The “factor-product
relationship” meta-level relationship” meta-level
narratives are constructed narratives are constructed
using “exploration” using previously-

constructed narratives
----- The “commutative” meta-level
narratives are constructed
using previously-
constructed narratives
Substantiating Nearly all uses of “prove” refer to proving
Narratives using iconic mediators

47% and 41% of narratives are 68% and 13% of narratives are

substantiated using iconic substantiated using iconic
mediators and estimation mediators and estimation
strategies respectively strategies respectively

 

235

Table 20 (cont’d)

 

 

Written Curriculum Enacted Curriculum
Substantiating Both curricula use “agreement” between strategies
Narratives for substantiating narratives

Both curricula use “consensus” for substantiating narratives

 

Note. Dashes (i.e., “ ----- “) indicate that the category is not applicable to the designated curriculum.

Table 20 and the more detailed analysis included within the chapter highlight many
discursive similarities and differences between the written and enacted curricula with
regard to endorsed narratives. The problem with such a table is that some of the richness
and interpretation present within the text of the chapter is lost. However, the summary
provided within the table allows a look at the data as a whole.

The comparison between the written and enacted curricula in Table 20 seems to
indicate several substantial differences. First, several points seem to indicate that the
enacted curriculum contains more experiences with object-level narratives addressing
fractions and fewer experiences with object-level narratives including combinations of
fractions, mixed numbers, and whole numbers. Similar results emerged from the
mathematical words analysis (see Chapter 5). This difference in experiences with a
broader range of numbers in the written curriculum may further limit students’
opportunities to reify fractions because operations with other types of numbers (e.g.,
mixed numbers, whole numbers) reinforces the use of fractions as numbers.

Table 20 also includes evidence of more experiences with narratives addressing
estimation (as a type of simple object-level narrative and for constructing and

substantiating narratives) and the relationship between the magnitude of the factors and

236

the product in fraction multiplication in the written curriculum than in the enacted
curriculum. This supports similar results from Chapter 4 (Goals of the Written and
Enacted Curricula), Chapter 5 (Mathematical Words in the Written and Enacted
Curricula), and Chapter 6 (Visual Mediators in the Written and Enacted Curricula). In
addition, the table indicates that the enacted curriculum provides more experiences with
quasi-symbolic narratives (supports conclusions in Chapter 4 and 5) and narratives
associated with contextual Questions and iconic mediators.58 That is, experiences with
estimation and relating the magnitude of the factors and product seem more likely to lead

to fraction objectification than quasi-symbolic narratives (e.g., “%of % = g”) and the use

of iconic mediators because the former use fractions and numbers and the latter use
fractions as descriptors or parts. This analysis lends additional support to the conclusion
that objectification is supported more substantially in the written curriculum than in the

enacted curriculum.

 

58 These differences are highlighted in Table 20.

237

CHAPTER 8: MATHEMATICAL ROUTINES
IN THE WRITTEN AND ENACTED CURRICULA

Routines are the fourth and final mathematical feature to be examined in the
written and enacted curricula. Routines are “well-defined repetitive patterns in
interlocutors’ actions, characteristic of a given discourse” (Sfard, 2007, p. 574). In this
case, the discourse is mathematical in nature and the context is a sixth grade classroom.
It is expected that the mathematical routines found in classrooms will differ to a greater
or lesser extent from routines used by professional mathematicians depending on the
grade level of the students. The preceding three chapters include analyses of routines in
conjunction with the other mathematical features under investigation. The ubiquitous
nature of routines makes this inevitable. In particular, thus far I have examined the
routines associated with word use (e.g., emphasizing “of” as a proxy for fraction
multiplication), the routine ways in which visual mediators are used (e. g., partitioning
area and number line models), and the routines of narrative construction and
substantiation (e.g., constructing meta-level narratives from previously endorsed object-
level narratives). Therefore, these routines will not be further explicated in this chapter
except as they naturally arise in the discussion.

Questioning is a widely used mathematical routine in both the written and enacted
curricula (and arguably in scholarly mathematics) which has only been mentioned in
passing thus far. Danielson (2005) noted the pervasiveness of questioning in both the
written and enacted curricula in his study of the introduction to linearity in Connected
Mathematics. In fact, the Connected Mathematics Teacher’s Guides contain a section
entitled “Suggested Questions” for each Problem in the Investigation. Given the fact that

“learning mathematics” is conceptualized here as changing discourse practices, the

238

questions included in the written and enacted curricula are worthy of examination. That
is, “What discursive practices are elicited through questions in the written and enacted
curriculum?” The discussion here is limited to questions that address the processes and
products of fraction multiplication. '

I compared the ranking of common question Words (Who, What, When, Where,
Why, and How) in the written and enacted curricula. Such a comparison is somewhat
problematic because the written curriculum contains 837 distinct words and the enacted

curriculum contains 1293 distinct words. To account for this difference, each word

 

ranking m the written curriculum was multiplled by to “scale it up” for comparison

purposes. Table 21 provides the ranking for the common question words in both
curricula. For example, “What” is the 31S‘ most common word in the written curriculum
(after being scaled up) and the 13th most common word in the enacted curriculum (of
1293 distinct words).

Table 21.

Rankings of Common Question W ords in the Written and Enacted Curricula

Question Word Written Curriculum Enacted Curriculum

What 31 13
How 46 23
When 59 123
Why 120 97
Where 141 l 3 8
Who 334 299

 

239

Several notable features are illustrated in Table 21. First, each of the curricula has three
question words in the top 100 words in the text. “What” and “How” are included in this
top tier in both curricula. However, the third question word in the written curriculum is
“When” whereas the third in the enacted curriculum is “Why.” Aside from this
exception, the order of the remaining question word rankings is the same between the two
texts. In addition, with the exception of “When,” all six words rank higher in the enacted
curriculum than in the written curriculum. This may be a result of the differential nature
of the written and enacted curricula or it may indicate a greater emphasis on questioning
in the enacted curriculum than in the written curriculum. It should be noted that the
context in which these words are used was not considered in this analysis (i.e., whether
they were used in questions or embedded in statements), only the presence of the words
themselves.

The written text of Investigation 3 includes a total of 110 suggested questions (an
average of 22 questions per day). These questions are locatedin the Student Guide and
Teacher’s Guide, which include 22 and 88 questions respectively. In the Teacher’s
Guide, questions are included both in the “Suggested Questions” section mentioned
earlier, as well as throughout the text. Usually the questions have a question format;
however, several questions are mentioned as suggestions in statement form. Both types
are included in this analysis. An example of the latter form follows:

Example 196 (WC).
“Ask students to explain how each of these is different from the fraction
multiplication problems they have solved already. ” (T G, p. 71)

Given the complexity of spoken language in the enacted curriculum, several

analytic decisions were made regarding which questions to include in the analysis. First,

240

only questions related to mathematics were included. For example, “Did you watch
American Idol last night?” was not included. Second, several questions in the enacted
curriculum are rhetorical in nature (e.g., “Can I ask you something about
multiplication?”). Questions of this type were excluded from the analysis. Third, it is
common in the enacted curriculum to make a statement into a question by adding “okay”
or “right” on the end. For example, “There are three pieces, right?” is for all intents and
purposes a statement with a request for confirmation. Therefore, questions of this type
were also excluded from the analysis. Finally, as mentioned in previous sections, it is
very common to repeat utterances (e. g., questions) in spoken language. Therefore, any
question repeated in the same turn or in any two consecutive turns by the same person
was only included once in the analysis. In Example 197 from Day 4, the teacher asks the
same question more than once within a turn.

Example I 97 (EC).

T: “Okay. I tell you what. Let's look at one more and let's see if we can just
figure out what it's saying. Not necessarily how to solve it, but what is it
saying if I have three and a fourth times two and eleven twelfths? What
does that mean? What does that mean?”

Here, “what is it saying if I have three and a fourth times two and eleven twelfths?”
“What does that mean?” and “What does that mean?” are all asking students for the
meaning of mixed number multiplication. Therefore, only the first question from this
utterance is included in the analysis. Similarly, the following excerpt from Day 2
provides an example in which the same question is asked in consecutive turns.

Example 198 (EC).

SI : First I would divide it into fifths and then, if I have this and then we would

split these two pieces into thirds, and then two thirds of that would be, like,
um, these parts.

24]

T: Where 's your one third?

S] .' I messed up.

T: Where '5 your one third? Can you show us, okay.

The teacher asks the question “Where’s your one third” in consecutive turns here.
Therefore, only the first of these questions was included in the analysis. Using these
guidelines in the analysis of the enacted curriculum, 579 questions were identified over
the course of the five days (an average of 116 questions per day).

To examine the relationship between the questions in the written and enacted
curricula, the remainder of this chapter will present a series of analyses of the 110
questions in the written curriculum and the 579 questions in the enacted curriculum.
These analyses include: (1) Leading Words of Questions, (2) Elicited Answers to
Questions, (3) Mathematical Processes addressed by Questions, and (4) Miscellaneous
Questions.

Leading Words of Questions

For this analysis, the first word of each question in the written and enacted
curricula was noted. In several cases in both curricula the question was refrained in order
to put the “question word” first. For example, in “For Question A, how did you first
mark your brownie pan?” (TG, p. 61), “For Question A” was ignored and the first word
was recorded as “How.” Similarly, “And then” was ignored and “Did” was recorded as
the first word in the following question from Day 5: “And then did she just use our
algorithm, multiply the numerators and multiply the denominators?” Figure 33
summarizes the relative frequencies of the first words of questions that are present in at

least 10% of either the written or enacted curriculum or both.

242

 

 

I Written
Curriculum

D Enacted
Curriculum

 

 

 

 

Relative Frequency

 

What How Do/Does/ Am/ls/Are/ Can/Could Why
Did Was/Were

 

 

 

Figure 33. Relative frequencies of leading words of questions in the written and enacted
curricula.

Interestingly, “What” and “How” are the most common in both curricula (each
representing between 17% and 30%). This supports the earlier findings from the ranking
in each curriculum. In both cases, however, the relative frequency in the written
curriculum is greater than in the enacted curriculum. This does not support the previous
finding in which both “What” and “How” are ranked higher in the enacted curriculum
and may suggest that the word uses in the enacted curriculum are not from direct
questions. This rationale also applies to “Why” because the Figure 33 indicates that it is
twice as common in the written curriculum as in the enacted curriculum. Remember that
its ranking is slightly higher in the enacted curriculum. The relative frequency of
“Do/Does/Did” is very similar between the two curricula (approximately 15%). Finally,
both “Am/Is/Are/Was/Were” and “Can/Could” are more common in the enacted

curriculum.

243

Here, “learning mathematics” is defined as changes in participation in
mathematical discourse. Given this, it seems that “open” questions (i.e., questions that
require more than l-2 word answers) would be preferable to “closed” questions, as they
would provide opportunities for extended mathematical discourse. “How” and “Why”
questions tend to be open, whereas the “Do/Does/Did,” “Am/ls/Are/Was/Were,” and
“Can/Could” families of question words tend to be closed. “What” questions are not
obviously open or closed and they appear in fairly equal frequencies in both curricula,
therefore they were not included in this calculation. That is, 34% and 23% of the
questions in the written and enacted curricula respectively are open questions. The next
section describes a closely related analysis.

Elicited Answers to Questions

The questions in the written and enacted curricula ask students either to say
something or to do something. Again, if learning mathematics is conceptualized as a
change in ways of participating in mathematical discourse, then what students are
expected to say and/or do is of critical importance. In particular, if students are expected
primarily to “do,” then it is questionable how much mathematics learning can take place.
Figure 34 indicates the proportion of questions in the curricula that expect students to

6‘say99 or £6d0.99

244

Written Curriculum Enacted Curriculum

Do Do
16% 14%

/’

   

   

/

     

,/

Say

say 86%

84%

Figure 3 4. Relative frequencies of questions asking student to “say” something versus
those asking students to “do” something.

Figure 34 indicates that questions expecting students to “say” and “do” are present in
similar frequencies in the written and enacted curricula. In both cases, questions asking
students to “say” are much more common than questions asking students to “do.”

The questions were classified into what the students are expected to say or do. In
making these decisions, I utilized the context of the question, including the answer given
in the enacted curriculum or the “possible response” in the written curriculum. When
expected to “say” something, several types of responses are elicited. These include
explanations, 1-2 word answers, yes or no, a number, a number sentence, or a question.
Table 22 provides examples from the written and enacted curricula of questions eliciting

each of these responses.

245

Table 22.

Sample Questions eliciting Particular Response Types in the Written and Enacted

Curricula

 

 

Response Written Curriculum Enacted Curriculum
Explanation “ Why does it make sense that the S: “ Why did you write twenty twos,
product [ of reciprocals ] is though? ” (Day 5)
always I? " (TG, p. 81)
1-2 Words “Would the estimate 01,2 % be an T: “Do you remember what you call
that pr0perty? ” (Day 3)
underestimate or an
overestimate? ” (TG, p. 76)
Yes/No “Do you think Takoda ’s strategy T: “So, is it fair to say that these
works? ” (SG, p. 38) two girls took each one of these
thirds and split them into seven
pieces? ” (Day I)
Number “ W hat fraction of a whole pan S: “So, how many thirds?” (Day 5)
does Mr. Williams buy? ” (SG, p.
33 )
Number ----- T: “Did, what, what was our number
Sentence sentence? ” (Day 5 )
Question ----- T: “A ny other questions for

Trevor? ” (Day I)

 

Note. Dashes (i.e., “ ----- “) indicate that the category is not present in the designated curriculum.

Note that no questions in the written curriculum elicit a number sentence or a question.
Figure 35 summarizes the relative frequency of each category of response in the written

and enacted curricula.

 

 

 

 

 

 

 

>~.

U

=

3 I Writte

a -"-
h (‘IIIILIIIIIIII
k-

2 U Enacted
'5 Curriculum
.2

0

n:

 

Explanation 1-2 Word Yes/No Number Number Question
Sentence

 

 

 

Figure 35. Relative frequencies of categories of elicited answers to questions in the
written and enacted curricula.
Figure 35 indicates that explanations are the most commonly elicited type of response in
the written curriculum (representing nearly 50% of all “saying” questions), whereas
Yes/No responses are slightly more common than explanations in the enacted curriculum
(40% compared to 31%). Questions eliciting a number occur in relatively similar
proportions in the written and enacted curricula, 22% and 24% respectively. Figure 35
provides a slightly different picture than in the analysis of leading words (see Figure 33)
because “explanation” questions are “open” and all other categories are “closed.” In this
analysis in which only “saying” questions are considered, the discrepancy between open
and closed questions is greater. In the previous analysis, the relative frequencies of open
questions in the written and enacted curricula were 34% and 23% respectively. In this

analysis, the corresponding frequencies are 49% and 31%.

247

When a question expects the students to “do” something, the expected actions

include calculating; constructing a symbolic mediator (e. g., a number sentence);

constructing, manipulating, or indicating an iconic mediator (e.g., a diagram); and

manipulating a concrete mediator (e. g., blocks). “Constructing” involves producing the

visual mediator, “manipulating” refers to changing the visual mediator in some way once

it has been constructed, and “indicating” entails pointing to something when asked a

question about the visual mediator. Table 23 provides examples of questions from each

category in the written and enacted curricula.

Table 23.

Sample Questions eliciting Particular Actions in the Written and Enacted Curricula

 

Action

Written Curriculum

Enacted Curriculum

 

Calculate

Construct

Symbolic Mediator

Construct

Iconic Mediator

“Ask groups to write the

number sentences for the

problems on their

transparency. ” (TG, p. 72)

“How would you represent

Ex? on a number line? ” (SG,

p. 34)

T: “But could you multiply that
using our algorithm? ” (Day 4)
T: “Can you write that

[number sentence] next to it? ”

(Day 1)

T: “Who can come up and
show what that would look like
on a long, skinny model?”

(Day 3)

 

248

Table 23 (cont’d)

 

Action Written Curriculum

Enacted Curriculum

 

Manipulate “ What could you do in your

Iconic Mediator drawing to make this

clearer? " (T G. p. 60)

Manipulate -----
Concrete Mediator
Indicate “Where do you see this [the
numerators] on the brownie

Iconic Mediator

pan drawing? ” (T G, p. 61 )

T: “Could you continue them
as if this brownie were here, or
this part of the goal was

there? ” (Day 4)

T: “How much - could you get
a half of them [blocks]? (Day
4)

T: “So where is one whole

ounce? ” (Day 5)

 

Note. Dashes (i.e., “ ----- “) indicate that the category is not present in the designated curriculum.

Table 23 indicates that the written curriculum does not contain any questions that ask

students to calculate or to manipulate a concrete mediator. Figure 36 summarizes the

relative frequencies of the types of “doing” responses to questions in the written and

enacted curricula.

249

 

 

I Written
Curriculum

L1 Enacted
Curriculum

 

 

Relative Frequency

 

‘2; \\° ‘0 \0 ‘0 \c’
\e’v 9°? x°°° 96‘ \°°
0% \%$ 6°C} $9 (Jo (9,0
c . a ‘0
oer” 0°” at” at? “‘6
0 w“

 

 

 

Figure 36. Relative frequencies of “doing” question responses in the written and enacted
curricula.

Figure 36 illuminates several differences between the “doing” question responses in the
written and enacted curricula. First, constructing iconic visual mediators is more than
twice as common as any other expected “doing” question response in the written
curriculum. In fact, half of all “doing” questions in the written curriculum expect this
action. In contrast, several categories have relatively similar frequencies in the enacted
curriculum. However, it should be noted that all three of the responses with the highest
relative frequencies in the enacted curriculum involve iconic visual mediators
(constructing, manipulating, and indicating). This is not to say, of course, that the
Questions in the written curriculum does not expect students to calculate, only that the
questions (i.e., sentences ending with “?”) in the written curriculum do not ask students to

calculate.59

 

59 Recall that Questions (with a upper-case “Q”) indicate specific exercises from the written curriculum.

250

Mathematical Processes Addressed by Questions

Most questions address, directly or indirectly, the construction or substantiation of
narratives. In particular, they address the mathematical processes involved in both of
these practices. The processes addressed by questions in either the written or enacted
curricula or both include: (1) estimation, (2) using the Commutative Property, (3)
decomposing numbers, (4) converting a mixed number to an improper fraction, (5)
modeling (including concrete, iconic, and symbolic mediators), and (6) using an
algorithm. There are also several questions that address mathematical processes more
generally. Table 24 includes sample questions from the written and enacted curricula
which exemplify questions addressing each process.
Table 24.
Sample Questions addressing Particular Mathematical Processes in the Written and

Enacted Curricula

 

Process Written Curriculum Enacted Curriculum

 

Estimation “ Who can explain how they T: “If! said I was going to get

estimated 1% x 2% ? "(TG, p. one halfof two and nine tenths,

Graham, how could we think

72)
about estimating that
answer? " (Day 4)
Commutative ----- T: “So can I switch my two
Property factors around, my two

numbers that I ’m multiplying.

around, or no? ” (Day 4)

 

25|

Table 24 (cont’d)

 

 

Process Written Curriculum Enacted Curriculum
Decomposing “W ould you want to use this T: “Could I do ten groups of
Numbers strategy [distributive property] two and a third and then a half
d
with the problem 3 g x 2 g ? ” ofa group oftwo an a
third? ”(Day 5)
(TG, p. 77)
Mixed Number —-+ “ What do you think about T: “Elliott, what do you think

Improper Fraction

Modeling

Algorithm

Yuri ’s idea [to rewrite a mixed
number as an improper
fraction before multiplying]? ”
(SG, p. 37)

“How does your drawing help
someone see the part of the
whole pan that is bought? ”

(T G, p. 60)

“What observations can you
make from Questions A and B
that help you write an
algorithm for multiplying

fractions? ” (SG, p. 35)

[about changing the mixed
number to an improper

fiaction]? ” (Day 5)

T: “So how far, what fraction
of a whole mile is this one little
piece right here [pointing to
part of model]? ” (Day 3)

T: “Do you think that these are
the steps that we should tape -

take? ” (Day 3)

 

252

Table 24 (cont’d)

 

Process Written Curriculum Enacted Curriculum
General “What method did you use to T: “Can you do this in a whole

solve the problem? ” (T G, p. day, so how could you figure

 

81) out what you do in a half of a
day? " (Day 5)
Note. Dashes (i.e., “ ----- “) indicate that the category is not present in the designated curriculum.

Note that, as discussed earlier, the commutative property is not addressed by any
questions in the written curriculum. Figure 37 summarizes the relative frequencies of the
mathematical processes addressed in the questions. Some questions address more than
one method; therefore, they were counted more than once (this type of question will be

further discussed later in the section).

 

#010)
000

N
O

 

 

Curriculum

  

Relative Frequency
(a)
O

 

10
o -
C 0 00 E
g a? a Ag 3" E. E
g gg 8. 3g 'g E 5
1;; E0 E EE 2 i” O
U '82
X9-
'55

 

 

 

Figure 3 7. Relative frequency of mathematical processes addressed in questions in the

written and enacted curricula.

253

Figure 37 indicates that more than 50% of the questions in the enacted curriculum
address modeling (recall that this modeling may involve concrete, iconic, or symbolic
visual mediators). In contrast, 25% of the questions in the written curriculum address
estimation and another 25% address modeling. Much fewer questions in the enacted
curriculum address estimation. This finding lends further support to the conjecture
proposed and supported earlier that estimation receives more attention in the written
curriculum than in the enacted curriculum. Questions addressing algorithms and general
methods also represent at least 10% of the questions in the written curriculum. The same
is true for decomposing numbers and algorithms in the enacted curriculum.

Of the questions that address more than one mathematical method, three
categories emerged, including those that address the relationship between (1) iconic
visual mediators, (2) symbolic visual mediators, (3) an iconic visual mediator and a
symbolic visual mediator, and (4) other methods (e.g., estimation). Examples are
provided in Table 25 representing each type in the written and enacted curricula.

Table 25.

Sample Questions addressing Comparison of Mathematical Methods in the Written and

 

 

Enacted Curricula
Relationship Written Curriculum Enacted Curriculum
Iconic—Iconic “Who can share a number line “Is that the same thing that

model? [asked after a brownie you guys saw Trevor do in the
pan model had been shared]? first drawing, or is this

(TG, p. 6 7) different?” (Day I)

 

254

Table 25 (cont’d)

 

Relationship

Symbolic-Symbolic

Iconic-Symbolic

Other

Written Curriculum

“What does this [“é—x 3 e
1
I 2 ] tell you about the

I 9
roduct or —x2—?” TG,
p f 2 10 (

p. 72)

“ What part of your drawing in
Question A shows the
denominator? ” (T G, p. 61)
“Does your exact answer seem
reasonable given the

estimate? ” (T G, p. 72)

Enacted Curriculum

“How is this problem

[” 30f? ”] like that problem

over there [“le—ofé ”]? ” (Day

2)

“But does your work over here
[number sentence] match this
drawing? ” (Day 2)

“ Without multiplying it [only
estimating it] can I know? ”

(Day 4)

 

Twelve percent of the questions in the written curriculum address the relationships

illustrated in Table 25 compared to 11% of the questions in the enacted curriculum.

These questions seem important for the goal of mediational flexibility (i.e., the ability to

transition between visual mediators).

Miscellaneous Questions

Questions that address the “answer” to a Question are very common in both the

written and enacted curricula. In fact, 45% of the questions in the written curriculum

address the answer to a Question compared to 23% of the questions in the enacted

curriculum. These questions take several forms including asking for the answer itself,

255

asking about the relative size of the answer, asking about the answer’s relationship to
another answer, etc. Examples given here provide a flavor of the variety of these
questions in the written and enacted curricula:

Example 199 (WC).
“Who can use their model to prove that the answer 123 sensible? " (TG, p. 76)

Example 200 (EC).
T: “Do you think that it 's onefourth or do you think it 's showing one sixth of the
whole bar? ” (Day 3)

Example 201 (WC).
“What fraction of a whole pan does Aunt Serena buy? ” (SG, p. 33)

Example 202 (EC).
T: “W hat fiaction of the candy bar did I eat? ” (Day 1)

Example 203 (WC).
“How did you come up with 2—3 ? " (TG, p. 71 )

Example 204 (EC).
T: “How did you decide three eighths? ” (Day 1)

Example 205 (WC).
“Does multiplication with fractions always lead to a product that is less than

each factor? (T G, p. 6 7)

Example 206 (EC).

T: “Is your answer going to get bigger, or is your answer going to get smaller? ”
(Day 1)

Qualitatively the questions in the written and enacted curricula seem quite similar;
however, quantitatively their relative frequencies are quite different (45% compared to
23%). This seems to indicate a greater focus in the enacted curriculum on process than
on product. I

Each of the following types of questions represents approximately 2-3% of all
questions in both curricula with one exception which is noted below. The first type asks

students about the meaning of fraction multiplication:

256

Example 207 (WC).
“What does it mean tofind éof i— ? ” (TG, p. 60)

Example 208 (EC).
T: “What does that mean again, one third’times one fourth? ” (Day 2)

The second type asks students if they agree with an answer or strategy that has been
suggested:

Example 209 (WC).
“Do you agree with this answer and the reasoning? ” ( T G, p. 72)

Example 210 (EC).
T: “Do you agree with that, Jacob, or no? ” (Day 2)

The third type asks students if a suggested answer or strategy makes sense:

Example 211 (WC).
“Do you think Paula’s strategy of rewriting the mixed numbers as fractions is
sensible? ” (T G, p. 80)

Example 212 (EC).
T: “Could you guys live with that, does that make sense? ” (Day 3)

Finally, the fourth type asks students whether they are following or understand an answer
or strategy (this type is present only in the enacted curriculum):

Example 213 (EC).
“Are you guysfollowing his drawing? (Day 1)

Example 214 (EC).
“Everybody understand what Katie did? ” (Day 3)

Examples 207-212 indicate that these question types, quite similar in nature, are

represented in both the written and enacted curricula (with the exception of the final

tyrae).

257

Summary
The question addressed in this summary is, “What does an investigation of the
mathematical routines (in this case, the routine of asking questions) in the written and
enacted curricula allow us to see?” That is, “What do we know now about the
relationship between the written and enacted curricula that we did not know before?”
Table 26 summarizes the findings from the investigation of mathematical routines for the

. . . 60
purposes of examination for overall conclusrons.

Table 26.

Summary of “Mathematical Routines " (i. e., Questionitg) Analysis

 

Written Curriculum Enacted Curriculum

 

Word Rankings
“What” and “How” are the first and second ranked question words

in both curricula

“When” is the third ranked “Why” is the third ranked

question word question word

 

 

(’0 The statements in this summary and throughout this analysis are made with several caveats. First, this
analysis compares the written text with an enactment of the written text (one of infinitely many possible
enactments); this should be kept in mind when reading these statements as some results may be attributed
to this difference in curricular form. Second, similarity (and difference) here is through my eyes only.
That is, another person (e.g., a teacher, a textbook author) using their own lens may see things quite
differently. Finally, the evidence for my claims is gleaned from five days in the written and enacted
curricula. That is, none of my statements can be generalized either to the written curriculum as a whole or
the enacted curriculum as a whole. Rather, my statements highlight insights gained through the use of this
framework regarding the relationship between the written and enacted curriculum on these five days that
may be of interest to teachers, curriculum developers, and mathematics education researchers.

258

Table 26 (cont’d)

 

 

Written Curriculum Enacted Curriculum
Leading Words Average 22 questions/day Average 116 questions/day
of Questions “What” and “How” are the most common first word of a question

in both curricula

10% of questions begin with 5% of questions begin with
“Why” “Why”
34% of the questions are open 23% of the questions are open
Elicited Approximately 85% of the questions ask students to “say” something
Answers to rather than to “do” something
Questions The most common elicited The most common elicited

answer to “saying” questions is answer to “saying” questions is
an “explanation” (49%) _ “yes/no” (40%)
Approximately 25% of the “saying” questions elicit
a “number” in both curricula
49% of the “saying” questions 31% of the “saying” questions
are open are open
Nearly all elicited responses to “doing” questions

involve visual mediators

 

259

Table 26 (cont’d)

 

 

Written Curriculum Enacted Curriculum
Mathematical 26% of the questions address 55% of the questions address
Processes “modeling” and 24% address “modeling” and 6% address
addressed by “estimation” “estimation”
Questions Approximately 10% of the questions address the relationship

between two or more mathematical processes
Miscellaneous 45% of the questions address 23% of the questions address
Questions the answer to a Question the answer to a Question
Both curricula include questions that ask for the meaning of fraction
multiplication, requests for agreement/disagreement,

and whether particular mathematics makes sense

 

Table 26 and the more detailed analysis included within the chapter highlight
many discursive similarities and differences between the questions included in the written
and enacted curricula. The problem with such a table is that some of the richness and
interpretation present within the text of this chapter is lost. However, the summary
provided allows a look at the data as a whole.

Table 26 indicates a discrepancy in the written and enacted curricula between the
quantity of questions asked in each. This table also highlights (in several places) that a
greater proportion of the questions in the written curriculum require an explanation (i.e.,
are open questions). If learning mathematics involves changing participation in
mathematical discourse, then the opportunities that “open” questions provide for students

to participate actively in mathematical discourse (i.e., beyond short answers) make a

260

difference in students’ learning. That is, the 4-stage model of word use (See Figure 4 on
p. 75) indicates the need to move beyond phrase-driven use and routine-driven use to
reach object-driven use. Open questions provide opportunities for students to use words
in a greater variety of ways, thus possibly facilitating objectification. In addition,
questions that require explanation allow teachers greater access to the ways in which
students are using words and therefore their stage of word use in particular situations.
Finally, the table provides additional evidence that estimation is under-represented in the
enacted curriculum compared to the written curriculum; this suggests that opportunities

to objectify fractions are under-represented as well.6|

 

6' These differences are highlighted in Table 26.

26l

CHAPTER 9: DISCUSSION

In this chapter, I return to the question that motivated this study and those that
guided my analysis. First, the question that served as the impetus for the study was:
“Which mathematics curricula are the most effective for promoting student learning?” It
can be argued that this study does little to answer the question because no
recommendation for the “right” curricula will be given within its pages. The contribution
of this study toward the answering of this question is more subtle. Efforts to answer this
question in the mathematics education research community have come largely in the form
of comparative curricular studies, including large quantitative studies, case studies, and
curricular analyses, among others. In these studies, two or more curricula are compared
using a variety of methods, and results often include the endorsement of a particular
curriculum.

A complaint in the field regarding these studies has been that the credit or blame
for the achievement (defined in myriad ways) of the students in the classrooms in which
these curricula are being used is assigned to these curricula with little or no knowledge of
the “fidelity of implementation” or “treatment integrity” of these curricula in the
classrooms (e. g., National Research Council, 2004; Senk & Thompson, 2003).62 In an
attempt .to address this concern, recent studies have conducted classroom observations
and documented “implementation” in a variety of ways, including textbook-use diaries
(in which students and teachers record the frequency with which they use their

textbooks), table-of-contents implementation records (in which teachers note the chapters

 

”2 “Fidelity of implementation” and “treatment integrity” are in quotation marks here because they are not
the language used in my study. Rather, I use “the relationship between the written and enacted curricula.”
They are, however, used in the literature within this conversation. Therefore, I use them here to position
my study in the literature.

262

completed in the textbook), and various observation protocols (addressing instructional
strategies, standards-based practices, etc.) (e.g., Post et al., 2008; Tarr et al., 2008).

In spite of these efforts, it still seemed that an in-depth focus on the way in which
particular mathematics was being presented in the classroom compared to the way it was
presented in the textbook was missing. I proposed this study to investigate the
relationship between the written and enacted curricula in the context of a particular
mathematical topic, fraction multiplication. I conducted this analysis using Sfard’s
Commognition framework (2008) in which the mathematical features of the curricula
(i.e., mathematical words, visual mediators, endorsed narratives, and mathematical
routines) are highlighted.63 The results of this study confirm the importance of
considering the mathematics when observing classrooms for “fidelity of implementation”
or “integrity of treatment” of the curriculum. That is, methods that do not address the
mathematics in the written and enacted curricula do not provide enough information to
make claims regarding the effectiveness of a particular cru'riculum.

The question guiding my study was: What does an investigation of the key
features of mathematical discourse, using the Commognition framework, in the written
and enacted curricula reveal? This question served to maintain a mathematical focus in
the study and as a reminder that the primary question was one of comparison between the
written and enacted curricula. I have addressed these questions in each chapter summary
with particular attention to the mathematical feature described there. In this discussion, I

address the question more broadly.

 

(’3 As earlier, I cite Sfard (2008) even though this framework has been elaborated in other publications,
because this book represents her most recent and comprehensive rationale for and description of this
theoretical framework.

263

I begin with a quick refresher course on Commognition to set up my later
comments. First, recall that Commognition defines “learning mathematics” as changing
participation in mathematical discourse. Second, the four key mathematical features
highlighted in Commognition (along with their definitions) are:

' Mathematical words: Words that signify mathematical objects or processes

° Visual mediators: Symbolic artifacts, created specially for the sake of
mathematical communication

° Endorsed narratives: Any text, spoken or written, which is framed as a
description of objects, of relationships between processes with or by objects,
and which is subject to endorsement or rejection, that is, to being labeled as
true or false

' Mathematical routines: Repetitive patterns characteristic of mathematical
discourse (Sfard, 2008)

It is important to note that these features interact with one other in a variety of ways. For
example, endorsed narratives contain mathematical words, mathematical routines are
apparent in the use of visual mediators, visual mediators are used in the construction of
endorsed narratives, etc. The most interesting part of the investigation was the way in
which the examination of each mathematical feature contributed to the richness of the
comparison between the written and enacted curricula.

Finally, objectification is a key element in the Commognition framework. I will
describe this part of the theory in some detail because it played a substantial role in my
analysis and much of my later comments are framed using objectification. Sfard (2008)
defines objectification as “a process in which a noun begins to be used as if it signifies an
extradiscursive, self-sustained entity (object), independent of human agency” (p. 412).

She describes the process as consisting of “two tightly related, but not inseparable sub-

264

processes: reification and alienation” (p. 412). She defines reification and alienation as
follows:

0 Reification: Replacement of talk about processes with talk about objects

° Alienation: Using discursive forms that present phenomena in an impersonal

way, as if they were occurring of themselves, without the participation of
human beings

The case of whole numbers illustrates the importance of objectification. A young child
talks about a number as a process, that is “three” is not an object to them; rather it is the
result of counting. This is demonstrated when you ask a child how many cookies she has
and she counts them. Wherever she finishes when the last cookie has been counted
(assuming she has mastered one-to-one correspondence) is her answer. A child will often
then begin using “three” as an adjective. Her answer when you ask her how many
cookies she has will be “three cookies.” It is later (sometimes much later) before a
number is objectified (i.e., takes on a life of its own).64 When a child has objectified
“three,” she is able to operate on it without counting (i.e., “three” as a process) or
associating it with cookies (i.e., “three” as an adjective). Instead, “three” is now an
object for her and she is able to talk about it in new ways (e. g., “three” is greater than
two). That is, it has a life of its own and does not depend on human agency for its object-
ness (no counting, no cookies). In this case, the change in discourse that counts as
learning is the transition from non-objectified ways of speaking to objectified ways of
speaking.

Objectification is not straight-forward to detect; however, there are clues in the

ways in which individuals speak that provide hints about how they are thinking. In the

 

(”Commognition does not support a Platonic view of mathematical objects. That is, objectification is not
meant to imply that mathematical objects actually exist and objectification represents their discovery.
Rather, they are proposed as theoretical constructs to facilitate communication.

example in the previous paragraph, “three is greater than two,” we see “three” used with
“is” and “greater than.” These are both clues that “three” has been objectified because
“is” is used with objects (e.g., The cat is fat). That is, “three” is a noun. In addition,
“greater than” is discourse used exclusively with numbers (i.e., the objectified use of
“three”). In the “adjective” stage of the use of “three,” we would be more likely to hear,
“Three cookies are more than two cookies.” Note here the use of “are” and “more than.”
In this case, “cookies” is the noun. Finally, “three is greater than two” is stated as a
mathematical fact and is not dependent upon human agency to make it so. These types of
hints were used throughout this analysis to detect the objectification of fractions and

other mathematical words.

The objectification of fractions (the mathematical topic under investigation here)

6

. . 5 . .
rs perhaps even more complex. A fraction (e.g., ‘ -6— ”) rs an “encapsulation” of two

whole numbers. Encapsulation is combining two objects (e.g., “5” and “6”) so that they

form a new object (i.e., a fraction). It is not a given that a student that has spent several
years studying fractions has objectified them (i.e., he now secs “2— ” as a number in its
own right). He may still see “five sixths” as five pieces (the pieces are sixths). If he has
objectified “g- ,” he is likely to say “five-sixths is If not, he is more likely to say

“five sixths are ...” I argue that this encapsulation (i.e., the objectification of fractions) is
one of many to come in mathematical discourse. As the number system in which

students are expected to operate grows, so does the expected level of encapsulation.

266

Down the road, we will expect these same students to treat “F ” as a number in its
own right and operate on it as such.

Given this background, I will summarize my analysis of the relationship between
the written and enacted curricula through the lens of objectification. Recall that
objectification has two requirements: reification and alienation. It is impossible to
address these processes completely separately, but I will begin by saying a few general
words about alienation in the written and enacted curricula.65

As stated previously, alienation is the depersonalization of mathematics in
general, and in this case, fractions in particular. Personalization is a key component in
both the written and enacted curricula. For example, the traditional algorithm for fraction
multiplication is noticed by a student in the class, Trevor, as he examines several
symbolic mediators (i.e., number sentences) constructed using an iconic mediator (i.e.,
diagram). Once he makes the suggestion, the algorithm is referred to as “Trevor’s Way,”
Trevor’s Idea,” and “Trevor’s Method” by both the teacher and students during the
remainder of the week. That is, the mathematics in this case is personalized. In the
written curriculum, fictitious students are used in several instances to suggest various
strategies (e. g., Distributive Property). This is in contrast to a more traditional
mathematics textbook that might state the algorithm or the Distributive Property as a rule

in a box at the beginning of the lesson. Another example of personalization in the

 

(’5 The statements in this summary and throughout this analysis are made with several caveats. First, this
analysis compares the written text with an enactment of the written text (one of infinitely many possible
enactments); this should be kept in mind when reading these statements as some results may be attributed
to this difference in curricular form. Second, similarity (and difference) here is through my eyes only.
That is. another person (e.g., a teacher. a textbook author) using their own lens may see things quite
differently. Finally, the evidence for my claims is gleaned from five days in the written and enacted
curricula. That is, none of my statements can be generalized either to the written curriculum as a whole or
the enacted curriculum as a whole. Rather, my statements highlight insights gained through the use of this
framework regarding the relationship between the written and enacted curriculum on these five days that
may be of interest to teachers, curriculum developers, and mathematics education researchers.

267

curricula is the use of consensus as a strategy for narrative substantiation (i.e., proving).
That is, agreement of the class often seems to be the standard for narrative endorsement
(i.e., deciding whether a mathematical statement is true or false). Again, in the written
curriculum, the questions posed to students regarding these fictitious students, is “Do you
agree?” It is not my conclusion that alienation never occurs in the written or enacted
curriculum. This is not the case; however, it is my claim that alienation of the
mathematics is not the norm in either curriculum. It is not my intention to convince the
reader whether or not personalization or depersonalization (i.e., alienation) is the “right”
answer for teachers or curriculum developers, only to raise the question for discussion
and to acknowledge that this choice likely affects how mathematics is learned.

The second component of objectification is reification. I argue here, and
throughout this analysis, that more opportunities for fraction reification are present in the
written curriculum than in the enacted curriculum. Several specific mathematical
features of the written curriculum, that receive less emphasis in the enacted curriculum,
have the potential to facilitate reification. First, four of the five major analyses in this
study (see Chapters 4, 5, 7, and 8) indicate that estimation strategies for fraction

multiplication feature more prominently in the written curriculum than in the enacted

curriculum. Estimation strategres promote rerficatron—type language (e.g. “—5- rs

approximately 1”) and therefore provide opportunities for students to use fractions in this
way (i.e., as numbers). Second, two of the five major analyses in this study (see Chapters
4 and 7) suggest that exploring the relationship between the magnitude of the factors and

the product of fraction multiplication receives more attention in the written curriculum

than in the enacted curriculum. Like estimation, an exploration of this relationship likely

268

encourages the reification of fractions because “factors” and “product” are words
associated with numbers.

Third, three of the five major analyses in this study (Chapters 4, 5, and 7) revealed
that the written curriculum provides more opportunities for fraction multiplication with
combinations of fractions (proper and improper), whole number, and mixed numbers.
That is, the majority of the fraction multiplication addressed in the enacted curriculum
involves the multiplication of two proper fractions. The extension of fraction

multiplication to include these combinations expands fraction use to move beyond “part

. 1 3 ,, . . . . .
of part" (e.g., “3x 4 ) which seems to be problematic for rerficatron because it does not
. “ 1 3 ,, . . . .
promote encapsulatron. For example, 3x 4 rs described, in both currrcula as

l 3 . . . . . . . . .
“30f -4— Iconic mediators are used to Illustrate the meamng of thrs multiplication and

the discussion turns out to be about “parts” and “pieces,” rather than about numbers.

Improper fractions are much more difficult to speak about in this way (i.e., ‘éx? ” does
not easily translate into “%of g ”), therefore they may promote the reification of
fractions. In addition, mixed numbers (e.g., “2% ”) are often known by students to be
between two numbers (e.g., “2%— is between 2 and 3”), therefore mixed number

. . . . . . , 2 ,,
multrplrcatron may lead to drscussrons usrng “23 as a number.

269

Finally, although the majority of the types of and uses of visual mediators are
quite similar in the written and enacted curricula (see Chapters, 4, 6, 7, and 8), the one
notable exception to this is the absence of number lines in the enacted curriculum. Of all
of the visual mediators proposed in the written curriculum, the number line is the one
with the most potential for reification of fractions because it is an iconic mediator created
specifically for “numbers” and it does not appear in the enacted curriculum. Although
the linear area model in the enacted curriculum is used in ways similar to a number line,
the discussions still primarily include words such as “parts” and “pieces” and do not
promote reifying fiactions (i.e., fractions as numbers).

The use of the Commognition framework for these analyses revealed many
discursive similarities and differences in the written and enacted curricula. An important
feature of mathematical discourse, objectification, was highlighted in the five primary
analyses (Chapters 4-8) and several mathematical differences emerged in the presentation
of the mathematics between the two curricula. Recall that objectification is made up of
two closely related processes, alienation and reification. I conclude that both the written
and enacted curricula are largely personalized (i.e., not alienated); however, experiences
that promote reification are more evident in the written curriculum than in the enacted

curriculum.

270

CHAPTER 10: CONCLUSION

This study contributes to the field of mathematics education in several ways. It
pilots an analytic method for investigating the relationship between the written and
enacted curricula (i.e., another way to think about “fidelity of implementation” or
“integrity of treatment”) that focuses on the mathematics present in the curricula. This is
important because we need methods that include mathematics when documenting
“implementation” for use in comparative studies. This study is a step in that direction.

In addition, this study illustrates the usefulness of the Commognition framework
for highlighting opportunities for objectification in the written and enacted curricula.
That is, by conducting a detailed analysis of the mathematical features, differences
between the mathematics in the written and enacted curricula emerged that may impact
the ways in which students speak about, and therefore learn mathematics. This study also
provides fodder for thought and discussion to mathematics educators interested in
fractions in general or fraction operations more specifically. For example, should
“fraction as number” take a backseat to “fraction as part-whole” while students are
learning fraction multiplication or is it possible to address these two uses of fractions
simultaneously? Finally, this study acknowledges students’ agency in the classroom by
conceptualizing the enacted curriculum as the union of the words and actions of the
teacher and the words and actions of the students. This is in contrast to many studies of
implementation that focus primarily on the words and actions of the teacher.

The primary limitations of this study have been stated throughout this dissertation,
but will be reiterated here briefly. First, comparing two forms of curricula (i.e., written

and enacted) is challenging. Hopefully, the analytic decisions made here regarding the

27l

challenges of comparing these two modes will be helpful to future researchers. Second,
the primary perspective represented here is mine.66 I acknowledge that another
individual using a different (or even the same) framework may see things quite
differently. Related to this, important mathematics may have been missed. That is, I
cannot claim that any of these analyses are comprehensive in scope. Finally and most
importantly, the results of this study cannot be generalized beyond the five days analyzed .
here. That includes generalization to the whole written curriculum or the whole enacted
curriculum. These results are much less about this particular curriculum and its use in
this particular classroom than about the questions the analysis allows us to ask and the
discussions it allows us to have, all toward the improvement of mathematics teaching and
learning. In addition, it is possible that reification of fractions is more present in the
enacted curriculum on days before or after the five days included in this study. Even if
this is true, questions addressing how and when opportunities for the reification of
fractions are most appropriate are still worthy of discussion.

This study leaves us with a long list of questions and ideas for important further
research. First, how would curriculum developers see these data? How would a teacher
see these data? As mentioned previously, I have presented my analysis and interpretation
of the data, but more perspectives would provide a more complete picture. For those
interested in fraction multiplication, an investigation using this lens into other curricula
(traditional or standards-based) or other classrooms (with teachers who have more/less
experience with the curriculum) would be a next step. This work could also be extended

into other mathematical topics. In addition, more discussions regarding the affordances

 

6" I say “primary” here because my perspective has certainly been affected by my interactions with others
and l have received input on my ideas from many individuals throughout this process.

272

and limitations of alienation are needed. What does personalization afford us? What
does depersonalization (i.e., alienation) afford us? What are the tradeoffs?

Finally, we need to develop instruments for textbook analyses and classroom
observations that are sensitive to the mathematics presented in the written and enacted
curricula in order to better address the issue of “fidelity of implementation” or “treatment
integrity.” The commognition framework provides one possible set of mathematical
features (i.e., mathematical words, visual mediators, endorsed narratives, and
mathematical routines) to highlight in these instruments. In this particular case, I focused
on objectification through the lens of these mathematical features, but in other cases,
different aspects of mathematics might be brought to the fore.

These discussions regarding whether mathematics curriculum and classrooms
promote objectification seem particularly important now given the move in many states
toward an increase in high school mathematics requirements. With more students taking
advanced mathematics, we need to focus more resources on increasing the chances for
students’ success in these challenging courses. Further investigation into both how
objectification serves students in these advanced mathematical settings and how to
facilitate objectification would be a useful research topic, because students who in the
past needed familiarity with only whole numbers, integers, and rational numbers are now
expected to operate with irrational and even complex numbers.

Additional questions for curriculum developers include: How can desired
discourse practices be made more explicit in the curriculum? Some of this has already
begun (e. g., “Suggested Questions” in Connected Mathematics), but what more can be

done? Can a balance be struck between personalization and depersonalization of the

273

mathematics? How can objectification be made more explicit in the curriculum? Is it
beneficial to do this? Finally, how can objectification be presented in ways that are
accessible and convincing to teachers? How can teachers best facilitate objectification at
various levels and in various contexts? i

In sum, this study provides opportunities for important conversations among
mathematics education researchers, curriculum developers, and teachers, in which
important and challenging mathematics is the focus. Such conversations, in this case
addressing fraction multiplication through the lens of the Commognition framework,
function to improve the work in all three domains, research, curriculum, and teaching, for

these three communities ultimately serve the same master, the mathematics learner.

274

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