EDUCATIVE SUPPORTS FOR TEACHERS IN MIDDLE SCHOOL MATHEMATICS
CURRICULUM MATERIALS: WHAT IS OFFERED AND HOW IS IT EXPRESSED?
By
Lorraine Marie Males

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILSOPHY
Mathematics Education
2011

ABSTRACT
EDUCATIVE SUPPORTS FOR TEACHERS IN MIDDLE SCHOOL MATHEMATICS
CURRICULUM MATERIALS: WHAT IS OFFERED AND HOW IS IT EXPRESSED?
By
Lorraine Marie Males
Teaching can have a substantial impact on student learning (Darling-Hammond, 1999).
However, teaching excellence depends on many factors, including the need for high quality
teachers and their continued education, and high quality materials (Cohen, Raudenbush, & Ball,
2002; Putnam & Borko, 2000). This learning includes learning to plan and enact lessons that are
appropriate for all students, which requires learning to interpret and understand student thinking
and learning instructional routines and practices that will enable them to use student thinking
productively. As we enter into the era of the Common Core State Standards for Mathematics this
learning is even more critical, as the standards may require teachers to not only learn to
understand and unpack the standards themselves, but may also require teachers to learn new
content and learn to teach in different ways (Lappan, McCallum, Kepner, 2010).
Due to the complex nature of teaching and the myriad of demands placed on teachers,
mathematics educators need to consider all possible venues for teacher learning. In this paper, I
discuss my examination of the opportunities for teacher learning embedded within written
curriculum materials. Research indicates that teachers can and do learn from curriculum
materials. Curriculum materials, particularly educative ones, emerge as a potential source for
opportunities for teacher learning in ways that set them apart from more traditional professional
development, which is often criticized for being decontextualized, contrived, short-term,
fragmented, discontinuous, and disconnected (Ball & Cohen, 1999; Little, 1994; Lord, 1994;

Wilson & Berne, 1999). Educative curriculum materials are materials for Grades K-12 that are
―intended to promote teacher learning in addition to students‘ ‖ (Davis & Krajcik, 2005, p. 3).
I investigated the opportunities to learn embedded in four middle school curricular series
in the areas on introduction to variables and geometric transformations, by examining the content
and its expression in the teachers' guides. I developed and used two analytical frameworks; one
to code the content support derived from work in science education (Beyer et al., 2009) and a
second framework to describe the expression of text developed by Morgan (1996) and
augmented by Herbel-Esienmann (2007).
My results indicated that all four curricular series included opportunities for teacher
learning (mostly related to Pedagogical Content Support for Practices and Curricular
Knowledge, depending on the curriculum) in both the variable and the transformations units, but
these opportunities were quite minimal and focused heavily on particular types of supports. This
lack of support was particularly true for Rationale Guidance for teachers. In addition to the
content support, my analysis of aspects of voice indicated that although these four series
provided opportunities for teacher learning, they also may hinder teachers' learning by speaking
"through" teachers rather than "to" teachers (Remillard, 2000), as evidenced by the ways in
which personal pronouns were used and the frequencies of imperatives and modal verbs. I
discuss implications for curriculum development, teacher education, and research.

Copyright by
LORRAINE MARIE MALES
2011

DEDICATION
I dedicate this dissertation to my husband Josh,
a fellow mathematics teacher, who was there for me
throughout this journey and never made me feel
like I was making the wrong decision
and
to all the curriculum developers out there,
particularly those who have "pushed the envelope"
when it came to what and how students should learn
mathematics - you have a hard job. I thank you for
making my work as a mathematics educator worth it

v

ACKNOWLEDGEMENTS

Although this dissertation is my work, it is the product of my thinking and countless
conversations with my colleagues, including faculty, graduate students, and teachers. Without
these people I would not have been able to produce this dissertation or complete my program. It
is the conversations that I had with these people that have helped me to think deeply about
mathematics education and strive to understand and communicate my ideas.
First and foremost, thank you to my family. I want to thank my husband Josh, a fellow
mathematics teacher, for helping me through these four and a half years. Thank you for
reminding me about the Michigan State program, for moving (once again) so that I could pursue
this degree, for never doubting me, and for listening to me, since I know that can be a hard job
sometimes. Thank you also to my parents, Kevin and Patricia Burke, who have supported me in
all that I do.
Second, I want to thank my committee members, Jack Smith, Glenda Lappan, Beth
Herbel-Eisenmann, and Kristen Bieda. Jack, not only were you the chair of my dissertation
committee, but I have also been lucky to have you as an advisor and colleague. I thank you for
all of the opportunities you have given me, and for the many things I have learned about
research, academic life, and about being a good friend and colleague. I will treasure all of these
things as I embark on a new journey and I hope to put them to good use as I try to be as good of
an advisor to others as you have been to me. Glenda, I thank you for the many opportunities you
have given me and the entire mathematics education community. Your generosity has allowed
many people in our community to do the work they do. Beth, I would not be at this point in my
academic career had it not been for our shared work. Thanks for pushing me to think about
discourse in ways I never thought imaginable. Kristen, thanks for being a wonderful inspiration
vi

to those of us in the program. You have not only supported me in becoming a better researcher,
but I have thoroughly enjoyed my time working with you in the secondary mathematics teacher
education program. Finally, thanks for feeding my cupcake obsession. I guess since the PhD
worked out, I won‘t be opening a cupcake shop in East Lansing.
Finally, I want to thank the entire MSU community and the Program in Mathematics
Education, particularly Lisa Keller and Margaret Iding. Using Jack‘s phrase, I came to MSU ―to
get smarter‖ about a few things and can say now that I got smarter about more than just a few!
Although I am bound to forget someone, I wanted to name a few groups or people in the
community that have made my experiences quite memorable.
A huge thank you goes out to the entire STEM team, a project that not only helped me
develop the skills and tools for organizing my data for my dissertation, but that also served as a
support system for me throughout my time at Michigan State. A special thank you goes out to
Funda Gonulates, Kosze Lee, and Aaron Mosier. I will cherish all of the memories, both joyful
and stressful, that we shared as we worked together on this project and I look forward to
continuing to work with you all.
Thank you to my seventh floor colleagues, especially Glenda Lappan, Betty Phillips, and
Judith Miller. I had the pleasure to work with you as part of the Center for the Study of
Mathematics Curriculum. You have taken very good care of me over the past few years and I
would not be where I am without you and the intellectual and emotional support you provided.
Thank you to all of the graduate students that have helped push my writing, served as
inspirations, helped me in doing dissertation coding, or have been supportive friends, especially
Aaron Brakoniecki, Jillian Cavanna, Dan Clark, Leslie Dietiker, Nic Gilbertson, Aladar Horvath,
Sarah Kasten, Jill Newton, Sam Otten, Joanne Philhower, Shannon Sweeny, and Sasha Wang.
vii

TABLE OF CONTENTS
List of TABLES .............................................................................................................................. x
List of FIGURES .......................................................................................................................... xii
CHAPTER 1: INTRODUCTION ................................................................................................... 1
Overview of Study ...................................................................................................................... 3
Organization of this Dissertation ................................................................................................ 3
CHAPTER 2: THEORETICAL BACKGROUND ........................................................................ 6
Review of the Literature ............................................................................................................. 6
Meanings of curriculum .......................................................................................................... 6
Research on curriculum .......................................................................................................... 9
Research focused on teacher learning from curriculum use. ................................................ 16
Towards a Framework for Studying the Educative Nature of Curriculum ............................... 19
The role of the teacher in enacting the curriculum. .............................................................. 19
Role of curriculum materials in teacher learning. ................................................................. 23
Theoretical Framework ............................................................................................................. 24
Content Supports................................................................................................................... 25
Expression. ............................................................................................................................ 28
Focus of Study and Research Questions ................................................................................... 31
CHAPTER 3: METHOD .............................................................................................................. 33
Sample....................................................................................................................................... 33
Choice of curricular series. ................................................................................................... 33
Choice of mathematical content............................................................................................ 35
Units. ..................................................................................................................................... 40
Analytic Frameworks ................................................................................................................ 40
A framework for analyzing the content supports of teachers‘ guides. ................................. 42
A framework for analyzing expression. ................................................................................ 52
Procedures ................................................................................................................................. 57
Pilot analysis. ........................................................................................................................ 57
Coding Process...................................................................................................................... 57
Inter-rater reliability .............................................................................................................. 59
Analysis..................................................................................................................................... 60
CHAPTER 4: Results –DIFFERENCES IN STRUTURE AND UNIT CONTENT ................... 61
Structural Differences ............................................................................................................... 61
Embeddedness of Supports ....................................................................................................... 63
Unit Details ............................................................................................................................... 65
CHAPTER 5: RESULTS – CONTENT SUPPORTS AND EXPRESSION ............................... 68
Content Supports....................................................................................................................... 71
Types of Content Support. .................................................................................................... 71
viii

Frequency of Supports by Knowledge Type. ....................................................................... 72
Location of Content Supports. .............................................................................................. 74
Some differences within units across curricula. ................................................................... 75
Aspects of Voice ....................................................................................................................... 78
Personal Pronouns. ................................................................................................................ 79
"You"-Forms. ........................................................................................................................ 83
CHAPTER 6: DISCUSSION ........................................................................................................ 94
Relative Frequency of Supports and their Location ................................................................. 94
Why are certain supports more prevalent than others? ......................................................... 95
A note about Subject Matter Content Knowledge. ............................................................... 97
Differences in frequencies across curricula. ......................................................................... 98
Cognitive demand of curriculum materials........................................................................... 99
Design principles and structure. ............................................................................................ 99
Constructing Relationships: Teacher as Agent ....................................................................... 103
Opportunities to Learn in Variable and Transformations Units ............................................. 106
Content. ............................................................................................................................... 106
Expression. .......................................................................................................................... 107
Opportunities to Learn ............................................................................................................ 110
Contributions to the Field ....................................................................................................... 111
Implications............................................................................................................................. 112
Curriculum development. ................................................................................................... 112
Teacher education. .............................................................................................................. 114
Research. ............................................................................................................................. 116
Limitations .............................................................................................................................. 118
Future Directions for Research and Design ............................................................................ 119
Examining opportunities for teacher learning in written materials at other grade levels. .. 119
Examining teacher learning from opportunities embedded in written materials. ............... 120
Designing electronic educative curriculum materials and researching their use. ............... 121
APPENDICES ............................................................................................................................ 123
REFERENCES ........................................................................................................................... 138

ix

LIST OF TABLES

Table 1. Algebra Standards for Grades 6-8 (NCTM, 2000) ......................................................... 37

Table 2. Frequency of Topics within the Algebra Standards for Grades 6-8 from the Common
Core State Standards for Mathematics .......................................................................................... 37
Table 3. Geometry and Measurement Standards for Grades 6-8 (National Council of Teachers of
Mathematics, 2000)....................................................................................................................... 39
Table 4. Frequency of Topics within the Geometry Standards for Grades 6-8 from the Common
Core State Standards for Mathematics .......................................................................................... 40
Table 5. Units and Total Pages per Unit for the Curricular Series ............................................... 40
Table 6. Content Coding Scheme ................................................................................................. 42
Table 7. Examples of Pedagogical Content Knowledge for Topics Codes .................................. 47
Table 8. Examples of Pedagogical Content Knowledge for Topics Code .................................... 49
Table 9. Examples of Mathematics Curricular Knowledge Codes ............................................... 52
Table 10. Percentages of Agreement for content by Unit and Curriculum .................................. 59
Table 11. Percentages of Agreement for Personal Pronouns and "You"-forms by Unit and
Curriculum .................................................................................................................................... 60
Table 12. Unit Details by Curricular Series .................................................................................. 65
Table 13. Percentages of Enactment and Rationale Guidance by Unit and Curriculum .............. 72
Table 14. Percentages of Content Supports by Location .............................................................. 74
Table 15. Frequencies of Personal Pronouns by Unit and Curriculum......................................... 79
Table 16. Uses of ―We‖ by Unit and Curriculum ......................................................................... 82

x

Table 17. Frequencies of Implicit and Explicit Use of "You" by Unit and Curriculum ............... 83
Table 18. Percentages of Imperatives Directing Teachers and Students by Unit and Curriculum86
Table 19. Commonly Used Imperatives by Unit and Curriculum ................................................ 86
Table 20. Percentages of Inclusive and Exclusive Common Imperatives by Unit and Curriculum
....................................................................................................................................................... 88
Table 21. Frequencies of "You"-forms in the Variable Units by Curriculum .............................. 89
Table 22. Frequencies of "You"-forms in the Transformations Units by Curriculum ................. 89
Table 23. Finite Verbal Operators (Halliday, 1985, p. 116) ......................................................... 90
Table 24. Frequencies of Modal Verbs by Unit and Curriculum ................................................. 91
Table 25. Percentages of the Verbs "Want" and "Wish" Used with the Modal Verb "May" by
Unit and Curriculum ..................................................................................................................... 93
Table 26. Percentages of Each Knowledge Type by Curriculum ................................................. 95
Table 27. Process Standards form Principles and Standards for School Mathematics (NCTM,
2000) ............................................................................................................................................. 96
Table 28. Ratios of Codes per Sentence by Unit and Curriculum .............................................. 106
Table 29. Descriptions of Studies of Written Curriculum Materials .......................................... 124

xi

LIST OF FIGURES

Figure 1. The Temporal Phases of Curriculum Use (Stein, Remillard, & Smith, 2007, p. 322) .... 8
Figure 2. Points of View on Discourse: From Social Activity and From Grammar (recreated from
Martin & Rose, 2007, p. 5) ........................................................................................................... 30
Figure 3: Structure of the Units in this Study ............................................................................... 62
Figure 4. Examples of Pages from Embedded and Non-Embedded Teachers' Guides ................ 63
Figure 5. Percentages of Content Supports by Unit and Curriculum ........................................... 73
Figure 6. Percentages of Implicit and Explicit "You"-forms ........................................................ 84
Figure 7. Percentage of Sentences in the Imperative Form by Unit and Curriculum ................... 85
Figure 8. Scales for Negotiating Demands for Services and Giving Information (Martin & Rose,
2008, p. 53) ................................................................................................................................... 90
Figure 9. Percentages of "Low," "Median," and "High" Modal Verbs Used ............................... 92
Figure 10. Examples of Pages from Embedded and Non-Embedded Teachers' Guides ............ 100
Figure 11. Current Research and Future Needs .......................................................................... 121

xii

CHAPTER 1: INTRODUCTION

Teaching can have a substantial impact on student learning (Darling-Hammond, 1999).
However, teaching excellence depends on many factors, including the need for high quality
teachers and their continued education, and high quality materials (Cohen, Raudenbush, & Ball,
2002; Putnam & Borko, 2000). In the last two decades there have been many efforts to reform
mathematics teaching, but for the pedagogical change to be realized in the way envisioned by
these reformers, there is the need for substantial teacher learning beyond prospective teacher
education (Borko & Putnam, 1996; Remillard, 2000; Schneider & Krajcik, 2002). Teachers need
not only to know subject matter, but also must know subject matter in ways that allow them to
teach this subject matter to students in effective ways. This learning includes learning to plan and
enact lessons that are appropriate for all students, which requires being able to interpret and
understand student thinking and develop instructional routines and practices that will enable
them to use student thinking productively. As we enter into the era of the Common Core State
Standards for Mathematics (National Governors Association Center for Best Practices & Council
of Chief State School Officers, 2010) this learning is even more critical, as the standards may
require teachers to not only learn to understand and unpack the standards themselves, but may
also require teachers to learn new content and learn to teach in different ways (Lappan,
McCallum, Kepner, 2010).
Due to the complex nature of teaching and the myriad of demands placed on teachers,
such as having to teach for understanding and fluency across domains and topics while also
trying to make sure students meet the standards and raise their test scores, mathematics educators
need to consider all possible venues for teacher learning. In this paper, I discuss my examination
1

of the opportunities for teacher learning embedded within written curriculum materials. Research
indicates that teachers can and do learn from curriculum materials (Choppin, 2008; Collopy,
2003; Drake & Sherin, 2009; Lloyd, 2008a; 2008b; Remillard, 2000; Remillard & Bryans, 2004;
Schneider, Krajcik, & Blumenfeld, 2005; Schneider & Krajcik, 2002; Schneider, 2006; Van
Zoest & Bohl, 2002). Curriculum materials, particularly educative ones, emerge as a potential
source for opportunities for teacher learning in ways that set them apart from more traditional
professional development, which is often criticized for being decontextualized, contrived, shortterm, fragmented, discontinuous, and disconnected (Ball & Cohen, 1999; Little, 1994; Lord,
1994; Wilson & Berne, 1999). Educative curriculum materials are materials for Grades K-12
that are ―intended to promote teacher learning in addition to students‘ ‖ (Davis & Krajcik, 2005,
p. 3). Ball and Cohen (1996) advocated for the development of such materials, because unlike
other frameworks or mechanisms for guiding curriculum and engaging reform practices,
curriculum materials are used on a daily basis, affording curriculum materials a ―uniquely
intimate connection to teaching‖ (p. 6). Curriculum materials are an ideal source for
opportunities for teacher learning, particularly learning that has the potential to impact practice,
because curriculum materials are situated in the practice of teaching. Moreover, curriculum
materials often serve as a dominant source of curricular knowledge (Grouws, Smith, & Sztajn,
2004; Jackson, 1968; Schmidt, et al., 1996). Teachers use student texts and teacher‘s guides in
their daily planning and in the enactment of lessons.
Despite the plethora of research on teachers use of curriculum materials in mathematics
education, we know little about how the structure and features of written curriculum materials
impact teacher learning (Stein, Remillard, & Smith, 2007).

2

Overview of Study
As a starting point to investigating the impact of curriculum materials for teacher
learning, in this study, I examined the opportunities to learn for teachers in written middle school
mathematics curriculum materials by examining what supports were available and how these
supports were expressed. Specifically, I examined opportunities to learn by investigating the
content and the expression of this content (i.e., how curriculum authors speak to teachers in the
written text) in four middle school curriculum teachers' guides. By examining both the content
supports available and how curriculum authors speak to teachers, I will describe the
opportunities available for teacher learning in four middle school curriculum materials teachers‘
guides in the areas of introduction to variable and geometric transformations.
Organization of this Dissertation
This dissertation is organized into five chapters. This first chapter is meant to provide an
introduction to the issues surrounding teacher learning and motivate my study of written
curriculum materials.
In Chapter 2, I present the theoretical background of my study. I do this by presenting
both a review of the literature and the conceptual framing for my study. I situate this study within
the larger context of studies on curriculum, focusing specifically on studies of written curriculum
materials and those of the enacted curriculum that have focused on teachers use of and learning
from using curriculum materials. I then present my theoretical framework, beginning with a
discussion of the role of the teacher in enacting curriculum. I then discuss issues related to
teacher learning and the role of curriculum materials in teacher learning, specifically discussing
content and expression in written curriculum materials. Finally, at the end of this chapter, I
describe the focus of my study and present my research questions.
3

In Chapter 3, I describe my method. I begin by describing my choices of curricular series
and mathematical content. I then describe my analytical frameworks and how these came to be. I
begin first by introducing and describing the coding scheme I used to identify and capture the
content supports for Subject Matter Content Knowledge, Pedagogical Content Knowledge for
Topics, Pedagogical Content Knowledge for Practices, and Mathematics Curricular Knowledge.
I then describe my second framework that captures aspects of the voice of the materials. I
describe the major pieces of this framework; personal pronouns and ―you‖-forms, which include
imperatives and modality. I then describe my coding procedures and analysis. Finally, I end this
chapter with my percentages for inter-rater reliability and how I calculated these percentages.
In Chapters 4 and 5, I present my results. First, in chapter 4, I describe the structure of the
teachers' guides and details on each of the units. In chapter 5, I present my results related to
content, followed by my results for aspects of voice. For my content analysis I begin by
providing details about each of the units including the total number of sentences and codes. I
then provide frequencies for the types and locations of supports and discuss the content supports
that appeared frequently and infrequently. I discuss differences across curriculum and units
where appropriate. Following my content results, I begin the results related to aspects of voice
with person pronoun use. I discuss the frequency of pronouns and the ways in which these
pronouns were used. I then discuss "you"-forms and give special attention the most common
"you"-forms, imperatives and "you" + modal verbs. As with the content results, I discuss
differences in voice across curriculum and units where appropriate.
I found that, although teachers‘ guides provided some content support, this support often
lacked Rationale Guidance and was focused heavily on particular types of supports, including
those related to Pedagogical Content Knowledge for Practices and Curricular Knowledge. In

4

addition, although most support was located at the Lesson Level, I found that a significant
amount of support, particularly for CMP and MiC, was located at the Unit Level. This may have
implications for whether teachers use the supports. In addition to the content support, my
analysis of voice indicated that, although some curriculum authors chose to include personal
pronouns such as ―we‖ and ―you‖ in the written text, it is not clear that these pronouns served to
construct a collegial relationship with teachers. Curriculum authors were not speaking to
teachers. Instead, curriculum authors often chose to command teachers to perform actions as
evidenced by their use of imperatives.
Finally, in Chapter 6, I present my discussion. I return to my research questions and
discuss what my study indicates about teachers‘ opportunities to learn from middle school
curriculum materials. I begin by summarizing my results and then describing the relative
frequencies of content supports, how curriculum authors spoke to teachers, and how the
opportunities differed across the variable and transformations units. In addition, I provide my
study‘s contribution to the field and discuss limitations and future directions for my work.

5

CHAPTER 2: THEORETICAL BACKGROUND

In this chapter, I first present a review of the relevant literature, focusing specifically on
studies of written curriculum materials and those of the enacted curriculum that have focused on
teachers use of and learning from using curriculum materials. I then describe my theoretical
framework by discussing the critical role of the teacher in shaping the mathematics curriculum in
classrooms and the role of curriculum materials in teacher learning.
Review of the Literature
I begin my review of the literature by first discussing curriculum broadly, including
issues of terminology and meaning. Second, I focus on curriculum research in mathematics
education (and from science education, where appropriate) in the United States. In this section, I
briefly address trends in curriculum research and then focus specifically on a) research on the
written curriculum and b) research involving curriculum enactment and the interaction between
teachers and curriculum materials, focusing specifically on studies that attempt to describe
teacher learning from using curriculum materials.
Meanings of curriculum. First, as a way to frame the sections to come, it is necessary to
describe what I mean by "curriculum" and "curriculum materials" (or textbooks) and to describe
the ways in which educators have come to talk about curriculum. I discuss these issues before
describing the relevant literature. First, with the terminology issue, when I looked at the research
on ―curriculum,‖ not just in mathematics education, but in other disciplines as well, I saw that it
was diverse and encompassed the work of many scholars in many fields. Jackson (1992) pointed
out, that research on curriculum is "confused." Today, this confusion stems partly from what the
term curriculum might be taken to mean. To some curriculum, stemming from the Latin word
6

meaning race-course, includes the set of experiences that students undergo (Dewey, 1902)
throughout the course of their lives, whereas, to others, curriculum is synonymous with textbook.
More recently, particularly since the implementation of The No Child Left Behind Act, the term
curriculum is used to refer to state frameworks and policy documents. While, like textbooks,
these frameworks and policy documents may in fact be a part of what constitutes the curriculum,
these artifacts are not the curriculum. According to Bobbitt (1918), ―as applied to education, it
[curriculum] is that series of things which children and youth must do and experience by way of
developing abilities to do things well that makeup the affairs of adult life; and to be in all
respects what adults should be‖ (p. 42).
With this in mind, it is essential to have another term, rather than ―curriculum‖ to
describe the materials used by teachers and students to enact the mathematics curriculum in
classrooms. I have chosen to use, as others (Stein, Remillard, Smith, 2007) have as well, the
terms curriculum materials, and teachers' guides. I use curriculum materials to refer to the main
text that students and teachers have as part of a larger curriculum program. I specifically use
teacher's guide to signify that I mean the teacher's text. Although some educators use the term
curriculum materials to describe a particular kind of text, texts that focus more on the pedagogy
of teaching (Stein, Remillard, & Smith, 2007), I do not make this distinction at this point.
The issues of the meaning of "curriculum" become even more critical when we wish to
research and describe curriculum. When someone researches curriculum, what are they actually
researching? Not only do we have terminology issues, we have conceptualization issues.
Curriculum theorist have been using a number of different terms to describe the differences
between curriculum as outlined in some form of text or as enacted in classrooms. For example
Doyle (1992) used "formal curriculum," whereas Gehrke, Knapp, & Sirotnik (1992) used

7

"planned curriculum" to describe outlined goals, whether these be in policy documents,
textbooks or created by teachers. Yet, others have used terms such as "intended curriculum" to
describe this same thing. To describe the curriculum as it unfolds in classrooms, researchers have
used terms such as "implemented curriculum" (Snyder, Bolin, & Zumwalt, 1992) or "enacted
curriculum" (Gehrke et al., 1992; Stein, Remillard, & Smith). Based on years of research, Stein,
Remillard, and Smith (2007) proposed that curriculum use ―unfolds in a series of temporal
phases‖ (p. 321) as presented in Figure 1.
Figure 1. The Temporal Phases of Curriculum Use (Stein, Remillard, & Smith, 2007, p. 322)

Written
Curriculum

Intended
Curriculum

Enacted
Curriculum
Student
Learning








Explanations for Transformations
Teacher beliefs and knowledge
Teachers‘ orientations toward curriculum
Teachers‘ professional identity
Teacher professional communities
Organizational and policy contexts
Classroom structures and norms

This unfolding begins with the printed page (the written curriculum), progresses to the teachers'
plans for instruction (the intended curriculum), and ends with the actual implementation of

8

curricular-based tasks in the classroom (the enacted curriculum)" (p. 321), which hopefully
produce some form of student learning. It is within these phases, beginning with the written
curriculum, that transformations between written, intended, and enacted curriculum can occur.
Research on curriculum. In this section, I describe the literature pertaining to
curriculum. I begin by discussing the general trends in curriculum research and then focus on
research on written curriculum and research that has attended to issues of teachers‘ use of
curriculum materials. I focus primarily on what was studied. I provide specific information about
the findings of these studies when appropriate; I provide details of the results of these studies
when the results are related to the framing of my study.
Overview of curriculum research. Curriculum research in mathematics education in the
United States has focused on a) the content of curriculum materials, b) the engagement between
teachers and curriculum materials, c) the enactment of curricula in classrooms, and d) student
learning from curriculum materials (Stein, Remillard, Smith, 2007). In this review, I primarily
focus on the first three categories because the content of materials, the interaction between
teachers and materials, and curriculum enactment, provide valuable insight for this study in
which I propose and use a framework for studying teachers' opportunities to learn from using
curriculum materials. Particularly with the introduction of the Standards-based NSF-funded
curricula, there has been a focus on the content of textbooks. These studies have primarily
focused on what mathematical content is present in curriculum materials, rather than on how it is
presented. However, a few studies have aimed to describe the latter as well. In addition, a few
studies have focused primarily on what support, both mathematical and pedagogical, is available
for teachers to enact lessons. I discuss studies of this nature in the next section on written
materials. This section will be followed by my discussion of studies that describe the nature of

9

the interaction between teachers and curriculum materials and the enactment, with a focus on
what and how teachers learn from materials. I include research from science education as well,
because science researchers have also focused on curriculum and the teacher-curriculum
relationship.
Research on written curriculum. Although in some respects the written materials
influence all areas of curriculum research, the features of the written materials themselves have
gotten little attention. According to Stein, Remillard, and Smith (2007), written curriculum has
the potential to influence transformations between the written, intended, and enacted curriculum,
and deserves our attention. The question they pose is "How do characteristics or features of the
particular curriculum influence how a teacher uses it?" (p. 356). Although some curriculum
research has focused on the written materials themselves, particularly studies investigating the
content of curriculum materials, this is still an underdeveloped area of research. For a list of
these studies, including a summary of a) the number of curricular series analyzed, b) the level
(i.e., elementary, middle, high, college), c) what materials were analyzed (i.e., student‘s
textbook, teacher‘s guide), d) the focus of the analysis(i.e., content, sequence, voice), and e) what
countries‘ curriculum materials were analyzed, see Appendix A. Most studies focused primarily
on the content of the student textbook and many included the examination of multiple curricular
series, either textbooks from the same country (n = 23) or textbooks from different countries (n =
16); however, some studies (n=6) focused more centrally on the teacher materials. I first describe
the studies that focused more, or equally on student materials, including those that focused on the
mathematical content, the expression of content, and the alignment of content with standards
documents. I then discuss studies that focused more centrally on teacher materials.

10

Content of curriculum materials. Some studies focused broadly on the topics or content
present (or not) and its sequence in curriculum materials at both the elementary (Hook, Bishop,
& Hook, 2007) and secondary level (Flanders,1987). Larger-scale studies, such as the TIMSS
(Schmidt, McKnight, Valverde, Houang, & Wiley, 1997; Schmidt, Houang, & Cogan, 2002;
Valverde, Bianchi, Wolfe, Schmidt, & Houang, 2002) also focused on the content present (or
not) when examining curriculum guides and/or textbooks from 45 countries.
Studies that focused on particular content, did so in various strands; eight studies focused
on number and operations, three studies on algebra, five on geometry and measurement, and one
on statistics and probability. Whereas some studies focused more broadly on a strand, such as
Star, Herbel-Eisenmann, & Smith III (2000)'s report of algebra in both the first edition of the
Connected Mathematics Project and Forrester's (1990) Algebra I textbook, published by
Addison-Wesley, most focused more narrowly on specific mathematical content. These foci
included area measurement (Smith, Gonulates, Males, & Mosier, in preparation), arithmetic
average (Cai, Lo, & Watanabe, 2002), bar models (Hoven & Garelick, 2007), concatenations
(Lee & Messner, 2000), the distributive property (Ding & Li, 2010), estimation of linear
measurements (Chang, Males, Mosier, & Gonulates, 2011), fractions (Charalambous, Delaney,
Hsu, & Mesa, 2010; Watanabe, 2003; Yang, Reys, & Wu, 2010), functions (Mesa, 2004),
integers (Li, 2000), length measurement (Lee & Smith, 2011; Smith, Dietiker, Lee, Males, &
Mosier, in preparation), patterning (Olsen, 2010), probability (Jones, 2004), and transformational
geometry (Nissen, 2000).
In addition to studies that focused primarily on content, some focused on the integration
of certain aspects in a text such as the processes or tools involved including problem-solving
(Fan & Zhu, 2007), proof (Stylianides, 2005), and calculator use (Chval & Hicks, 2009).

11

Expression of content in curriculum materials. Some studies have described more than
just the mathematical content of the text, but have also described the ways in which this content
was expressed, or have attempted to describe how readers may interpret the text. These studies
may begin to help in answering the question posed by Stein, Remillard, and Smith (2007), as to
how the features of a textbook influence how a student or teacher uses it. Herbel-Eisenmann
(2007) investigated whether the text of one unit of the Connected Mathematics Project was
aligned with the ideas espoused in the NCTM Standards by investigating the voice of the text.
Drawing on Morgan (1996) and Halliday‘s (1985) Systemic Functional Linguistics, HerbelEisenmann investigated linguistic forms used in the text. Other studies have also tried to describe
the ways in which content is expressed, such as the work currently underway by the
Strengthening Tomorrow’s Education in Measurement project (Lee & Smith, 2011; Smith et al.,
in preparation) who have analyzed opportunities to learn spatial measurement in written
curriculum materials by examining both what is present in the text (content) and how this content
is present (textual forms). In addition to these studies, a recent study (Weinberg & Wiesner,
2011) attempted to provide a framework for describing a readers‘ response when reading written
curriculum materials. Drawing on ideas from reader-oriented theory, Weinberg and Wiesner
(2011) describe features of a calculus textbook that may impact ways in which students read
textbooks.
Alignment with standards. Some studies have investigated curriculum materials with
respect to the NCTM Standards. Nissen (2000), who specifically compared transformational
geometry tasks in four elementary, three middle, and one integrated high school textbook to the
NCTM Standards, found that whereas elementary and middle school textbooks met the
requirements of the NCTM Standards, high school textbooks did not. Martin, Hunt, Lannin,

12

Leonard Jr, Marshall, and Wares (2001), compared all the National Science Foundation-funded
high school curricula (Mathematics: Modeling Our World, Core Plus Mathematics, Interactive
Mathematics Program, MATH Connections, and SIMMS Integrated Mathematics) to the NCTM
Standards and identified features of the textbook that showed how each series aligned with the
Standards.
Focus on Teacher Materials. Whereas most of the studies above focused primarily on
describing student textbooks or opportunities for student learning (and in some cases using the
teacher‘s guides to aid in this analysis), a handful of studies (n = 7) focused primarily on, or
attended equally, to teacher materials in their analysis. These studies included investigations of
elementary and middle school textbooks and the analysis of textbooks within and outside the
United States.
In three studies on teacher materials, researchers examined curriculum materials used
outside of the United States. First, Netwon & Netwon (2006) studied 18 teachers‘ guides from
England. Specifically, these texts were investigated for their opportunities for teachers to teach
with attention to reason. This study indicated that the teachers' guides did not draw teachers‘
attention to discourse or attention to reasoning, but instead focused on computational skills.
Second, in two of these studies, researchers compared teachers‘ guides used in the United
States with those used in other countries. Li (2004) studied the opportunities in the teacher‘s
guide of Scott-Foresman Addison Wesley‘s Mathematics , University of Chicago School
Mathematics Project's Everyday Mathematics and People‘s Education Press Mathematics, used
in China, whereas, Watanabe‘s (2001) study compared two U.S. texts, Investigations and Math
in My World to two Japanese texts, Shintei sansuu, and Shinhes atarashi Jansu. These two
studies indicated that the teachers' guides used in the United States differed from those used in

13

China and Japan. Both studies described teachers' guides in the United States as lacking attention
to the mathematical big ideas, compared to the teachers' guides from these other countries. In
addition, Li (2004) indicated that the Chinese text included specific focus on conceptual content
and pedagogical approaches, including providing rationales. Watanabe (2001) pointed out that
the Japanese texts he analyzed were actually much more prescriptive than the American texts;
however, the text "appeared to encourage teachers to study the materials more carefully and
polish their daily lessons" (Watanabe, 2001, p. 200). The Japanese teachers' guides aimed to
help teachers learn mathematics and mathematics teaching by including discussions of the
mathematics, rationales for the organization of the materials, and samples of instructional plans
and student responses. Watanabe did not find these same features in the American teacher
materials.
Other studies that focused on teacher materials used in the United States included those
that focused on one text (Castro, 2006; Stylianides, 2007) and those that examined multiple texts
(Kim, Achubang, Lewis, Hoe, Reinke, & Remillard, 2010; Stein & Kim, 2009). The former
studies both investigated the teacher materials of the Connected Mathematics Project. Castro‘s
(2006) study focused on the teacher‘s guide as a resource in planning lessons, whereas
Stylianides (2007) specifically investigated the guidance provided in the teacher‘s guide for
teaching proof. These studies indicated that more opportunities may be needed for teachers to
learn from using the teacher's guide.
The latter studies have examined more than one text in order to describe the variation in
terms of teacher learning opportunities. Drawing on Ball and Cohen (1996) and on the heuristics
outlined by science educators (Davis & Krajcik, 2005), Stein and Kim (2009) investigated the
―educativeness‖ of two elementary curriculum materials, University of Chicago School

14

Mathematics Project's (2004) Everyday Mathematics and TERC‘s (1998) Investigations in
Number, Data, and Space. Using the ideas related to the cognitive demand of the tasks in each
curriculum, Stein and Kim (2009) found that the support material in each curriculum was aligned
with how cognitively demanding the tasks were for students and for teachers to enact. Stein and
Kim (2009) focused on whether the curriculum materials made the developers‘ rationales for
instructional tasks and pathways transparent and if the curriculum helped teachers to anticipate
how students might approach these tasks. They found that these two curricula differed in terms
of the demand for teacher learning and the opportunities provided for teacher learning. In their
analysis of over 40 lessons of each curricula across grades 1-5, they found that Everyday
Mathematics lessons tended to tell teachers what to do, but not why they were doing it, whereas
the Investigations lessons were judged to be 80% transparent with respect to why a particular
instructional activity was designed and how it represented worthwhile mathematics. Stein and
Kim (2009) also found that while the authors of Everyday Mathematics provided less demanding
tasks, they also provided fewer opportunities for teachers to learn about possible student
responses. The demanding tasks in Investigations were accompanied by supports that included
student responses, student work, examples of possible student difficulties, students‘ strategies for
solving problems, and explanations for how students might make sense of content.
In addition to this published work, a cross-university group is currently working on
investigating multiple elementary teachers‘ guides to describe the attempt to make ―visible the
mathematical and pedagogical features teachers encounter when reading curriculum resources in
order to guide research on teaching and curriculum materials and improved materials design‖
(Kim, Achubang, Lewis, Hoe, Reinke, & Remillard, 2010, p. 38).

15

Research focused on teacher learning from curriculum use. In both mathematics and
science education, curriculum researchers have investigated teacher learning from the use of
curriculum materials (Choppin, 2008; Collopy, 2003; Drake & Sherin, 2009; Lloyd, 2008a;
2008b; Remillard, 2000; Remillard & Bryans, 2004; Schneider, Krajcik, & Blumenfeld, 2005;
Schneider & Krajcik, 2002; Schneider, 2006; Van Zoest & Bohl, 2002). This research has
indicated that elementary and secondary teachers can and do learn from using curriculum
materials. In mathematics, this learning has been shown to be mediated by a variety of factors
including those that teachers bring with them, such as teachers' prior knowledge, identity, beliefs,
and orientations towards mathematics and towards curriculum materials (Collopy, 2003) and
those that happen as a result of using a set of curriculum materials. The latter included engaging
in the "reading process," or the constructive and dynamic process of making meaning through
engaging with the written text in a textbook, and also engaging with students and tasks
(Remillard, 2000; Remillard, 2005). In addition, struggling with new mathematics content (Van
Zoest & Bohl, 2002) and enacting lessons multiple times (Choppin, 2008; Drake & Sherin; 2009)
has enabled teachers to learn both about mathematics and pedagogy. Van Zoest and Bohl (2002)
found that the unfamiliar mathematical content in the Core Plus curriculum materials forced an
intern and mentor teacher to work through problems and have discussions about mathematics,
resulting in the development of mathematics subject matter knowledge. Choppin (2008) and
Drake and Sherin (2009) found that as teachers taught lesson multiple times they developed a
sense of where the curriculum was going and were able to draw on previous experiences
enacting lessons to improve them. These repeated enactments helped teachers to develop
curricular knowledge that enabled them to become more skilled enactors. Teacher learning was
also impacted by factors embedded in the written materials themselves, such as the structure of

16

the text, including how concise the text was (Lloyd, 2008a) and where support was located (i.e.,
at the lesson level or unit level) (Schneider, Krajcik, & Blumenfeld, 2005; Schneider & Krajcik,
2002; Schneider, 2006).
Learning from educative curriculum. Although research has rarely focused on how the
features of written mathematics curriculum impact teacher learning, there has been some focus
on this in science education. There has been a line of research that specifically investigated how
features of educative curriculum materials (i.e., materials designed for teacher learning)
supported teacher learning. Drawing on Shulman (1986), Ball and Cohen (1996), and Davis and
Krajcik‘s (2005) design heuristics (or the precursors to these heuristics), the educative
curriculum materials incorporated five principles: a) addressing content knowledge and
pedagogical content knowledge, b) situating teacher learning by meshing the content of the
support to lessons for students, c) linking different knowledge areas within lessons, d) making
knowledge accessible to teachers by including short scenarios in the language of teachers or
students involved in the lesson to illustrate or model the intended practice when possible, and e)
addressing immediate needs for understanding as teachers plan lessons that will be enacted
within a short time.
Three studies (Schneider, Krajcik, & Blumenfeld, 2005; Schneider & Krajcik, 2002;
Schneider, 2006) examining the use of these educative materials indicated that teachers used the
lesson-specific features, particularly when planning (Schneider & Krajcik, 2002). These included
examples of student work such as student-generated graphs and notes about possible student
misconceptions, strategies, and representations. According to Schneider and Krajcik (2002),
while there was evidence that teachers read the content explanations for the teacher that went
beyond what was needed by the students, observations indicated that the use of this content

17

knowledge was more effective in practice when it was related to the work of students, something
that was attended to in the lesson-specific features. One teacher indicated that she learned
physics content ―from reading the notes about student misconceptions because she held some of
those same misconceptions herself‖ (p. 238).
Schneider (2006) studied one science teacher, Ms. Shirley, over the course of two years
teaching five units, all with different levels of educative support. Schneider found that Ms.
Shirley used the materials purposefully to guide her teaching. At the beginning Ms. Shirley had
difficulty guiding student thinking, particularly during group and whole-class discussion. Over
time she improved in her ability to facilitate whole-class discussions, but still struggled doing
this when students worked in groups. Ms. Shirley‘s ability to facilitate thoughtful discussions
happened particularly when the curriculum materials included questions and provided possible
discussion scenarios. Ms. Shirley began teaching these units trying to ―fit the new lessons into
her previous teaching framework‖ (Schneider, 2006, p. 677). In the second year she began to
think about teaching within a framework intended by the materials and began to question
components of the materials. As she gained experience with the materials Ms. Shirley required
―finer grained support‖ (p. 677). She also grew in her attention to student thinking, moving from
being worried she might impart student misconceptions to thinking about how she could plan to
support students‘ developing ideas. Features of the materials, such as the scenarios and
questions, provided scaffolding for Ms. Shirley‘s efforts with discussion and enactments were
most thoughtful when features included explicit attention to how a lesson fit in with the overall
goals of the unit.
In all of these studies, lesson-specific educative features, such as descriptions of student
talk, or explanations of possible misconceptions and where they may arise, had the greatest

18

impact on teachers‘ learning to enact inquiry teaching in science, contributing to teachers‘
pedagogical content knowledge. These lesson-specific features have also been linked to the
improvement of content knowledge as reading about student misconceptions has resulted in
teachers confronting their own misconceptions about science.
Towards a Framework for Studying the Educative Nature of Curriculum
Although some may believe that student achievement is independent of the teacher, some
of the world‘s most respected educators remind us that teachers and teaching matter (DarlingHammond, 1999). It is not just getting the right textbook into the hands of students, but it is
about a combination of intricate factors, many of which rely on a student‘s teacher. In this
section I describe my framework. I begin by discussing the role of the teacher in shaping the
curriculum and how this role has been perceived in the American mathematics curriculum. I start
here because it is first necessary to recognize the role of the teacher in shaping the curriculum. If
one does not recognize the teacher's critical role, developing an educative curriculum and a
framework for analyzing such curriculum will make little sense. Second, I discuss teacher
learning, and finally I describe the role of curriculum materials in fostering teacher learning.
The role of the teacher in enacting the curriculum. The teacher plays an active role in
designing and enacting the curriculum in their classroom. Furthermore, the curriculum is a guide
not only for students, but for teachers. This notion of the curriculum being ―for teachers‖ is not
new. Dewey (1902) argued that the curriculum was in fact for the teacher, not the child.
Its [the curriculum‘s] primary value, its primary indication, is for the teacher, not for the
child. It says to the teacher: Such and such are the capacities, the fulfillments, in truth and
beauty and behavior, open to these children. Now see to it that day by day the conditions

19

are such that their own activities move inevitably in this direction, toward such
culmination of themselves. (p. 39).
Dewey saw the role of the teacher as instrumental in shaping the conditions of the
classroom in order for students to learn. He did not conceptualize the curriculum as something
for students, but instead saw curriculum as a guide for teachers, a guide that helped teachers see
the possibilities inherent in the culture. Using this curriculum, teachers could then design
activities and an environment that would help students meet their potential, which according to
Dewey, included being productive citizens in a democratic society.
However, over a century later, the role of teachers in the construction of curriculum was
not recognized by curriculum developers. In the last half of the 20th century, while curriculum
materials have exerted great influence over the mathematics teachers and their instruction
(Jackson, 1968; Schmidt, et al., 1996), curriculum materials have done little to engage teachers
in the active construction of the mathematics in their own classrooms. Mathematics curriculum
materials have merely been a source of problems and explanations for students (Stein, Remillard,
& Smith, 2007). Teachers relied heavily on textbooks as their primary source for what to teach
(Grouws, et al., 2004), but the "how" and the "why" were virtually absent from curriculum
materials. In mathematics, the textbook has had great influence over instruction, exerting so
much control that the instruction of many teachers was merely aimed at giving the student the
prerequisite knowledge to be able to complete the problems in the textbook (Jackson, 1968). This
long history of reliance on curriculum materials, according to Remillard (2005), is due to the
ways in which mathematics learning is viewed (e.g., math is learned by memorizing), the nature
of the content itself (e.g., a stable body of knowledge), and teachers‘ knowledge of the content
for teaching (e.g., lack of knowledge). For example, the same elementary teachers who adapted

20

textbook suggestions in language arts tended to stick to the textbook in mathematics (Sosniak &
Stodolsky, 1993). Despite this reliance on curriculum materials in shaping instruction,
curriculum developers underestimated the role of teachers when designing texts.
In the 1950‘s and 60‘s curriculum materials were a big part of the reform in mathematics
sometimes labeled ―New Math,‖ garnering this label from one particular set of curriculum
materials developed by the School Mathematics Study Group (Stein, Remillard, & Smith, 2007).
In this wave of reform, subject matter experts were solicited to design curriculum materials to
facilitate the development of scientifically productive citizens. Despite the unfamiliarity teachers
had with the content and sequence of the mathematics in these newly developed materials, these
materials were developed, like the texts before them, with the students as the intended audience.
Although some developers recognized the role of the teacher in shaping the curriculum and
attended to teacher learning (S. Senk, personal communication, April 2009), according to Cohen
and Barnes (1993) these developers continued to underestimate the influence of the teacher and
overestimate the educative power of the texts themselves. Curriculum materials from this era, in
general, were not designed to communicate with teachers, but were designed to be ―teacher
proof.‖
This discounting of the role of teachers in enacting curriculum did not go unnoticed.
Looking back on his work as head of the Woods Hole Conference, a conference commissioned
by the National Academies of Sciences to engage the leading subject matter specialists in
examining the American curriculum, Bruner (1977) argued, like Dewey, that curriculum is more
for teachers than students.
A curriculum is more for teachers than it is for pupils. If it cannot change, move, perturb,
inform teachers, it will have no effect on those whom they teach. It must first and

21

foremost be a curriculum for teachers. If it has any effect on pupils, it will have it by
virtue of having an effect on teachers. The doctrine that a well-wrought curriculum is a
way of ―teacher-proofing‖ a body of knowledge in order to get it to the student
uncontaminated is nonsense. (p. xv)
Bruner admits to brooding over this issue of curriculum since the first edition of this report in
1960. Like Dewey, in this preface, Bruner conceptualized curriculum as something for teachers.
Bruner spoke out against the notion of a teacher-proof curriculum, one that was so popular in the
50‘s and 60‘s, stating that this idea was nonsense. In order for curriculum to have an effect on
students, it must have an effect on teachers; it must change, move, perturb and inform teachers.
Despite Bruner‘s claims, it was not until the 1990‘s that developers began to recognize
the critical role of the teacher and write materials not only with students, but also with the
teacher in mind. The reformers of this new movement cast these curriculum materials as agents
of change hoping that they would help to transform instruction from a focus on procedural skills
to a focus on conceptual understanding that emphasized communication, reasoning and problemsolving (Senk & Thompson, 2003). This movement, heavily supported by The National Science
Foundation (NSF), was focused on developing materials that were aligned to the National
Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School
Mathematics (1989). One of the greatest differences in these new Standards-based materials and
the traditional texts or those developed in the ―New Math‖ era, was the attention developers gave
to the role of the teacher. The Standards-based materials acknowledged, some more explicitly
than others, the role of the teacher in shaping the enacted curriculum. Unlike materials in the
past, curriculum developers attempted to attend more to the teacher and issues of teacher
learning. For example, in writing about the design of The Connected Mathematics Project

22

Lappan, Phillips, & Fey (2007) describe their desire to write with teachers in mind, "It is for
teachers as well as students. The materials were written to support teachers‘ learning of both
mathematical content and pedagogical strategies. The teachers‘ guides include extensive help
with mathematics, pedagogy, and assessment‖ (p. 68). Although some curriculum developers
began to consider the role of the teacher more in their design of materials, others did not. In
addition, for those developers who considered the teacher, it is not clear that the types of learning
opportunities they provided were sufficient. In some cases teachers were still positioned as
passive receivers of the curriculum (Paris, 1993). In the next section I describe the issues related
to teacher learning to gain insight into the type of learning opportunities needed by teachers.
Role of curriculum materials in teacher learning. As research indicates, teachers can
and do learn from using curriculum materials (Choppin, 2008; Collopy, 2003; Drake & Sherin,
2009; Lloyd, 2008a; 2008b; Remillard, 2000; Remillard & Bryans, 2004; Schneider, Krajcik, &
Blumenfeld, 2005; Schneider & Krajcik, 2002; Schneider, 2006; Van Zoest & Bohl, 2002).
Although empirical work has rarely investigated the features of written curriculum materials and
how these features promote learning, the research on teachers‘ use of materials and what and
how they learn from using materials indicates that the written materials may play a role in this
learning. According to Beyer et al. (2009), curriculum materials seem like a fruitful option for
professional development as they are connected to teachers‘ everyday practice (Putnam &
Borko, 2000) and provide ongoing support (Collopy, 2003). In addition, according to Schneider
& Krajcik (2002), due to their wide-spread use, curriculum materials have a greater possibility of
effecting large-scale reform compared to other efforts professional development efforts aimed at
state and district level. Since curriculum materials are widely used and are intimately connected
to what teachers do on a daily basis, they have a greater chance of impacting practice than efforts

23

that are geared towards the needs of an entire district, as these are often, by necessity, less
connected to particular teachers‘ practices.
In order to serve as a source of professional development, curriculum materials need to
be more than just textbooks for students. Curriculum materials must ―speak to teachers, not
merely through them‖ (Remillard, 2000, p. 347). Rather than being merely a set of directions for
the teacher to follow (Remillard, 2000) or a source for problems to assign (Jackson, 1968), in
order for curriculum materials to be educative, the materials must communicate with teachers
directly about mathematics content, pedagogical practices, and curriculum. The authors of
materials that communicate in this way recognize the influence teachers have in enacting
mathematics curriculum with their students. Rather than treating teachers as passive receivers of
the curriculum (Paris, 1993), authors of educative curriculum position teachers as legitimate
readers of the curriculum (Remillard, 2000). This means that curriculum materials need to not
only have content that is appropriate for teachers to engage with the teaching of mathematics, but
also that materials need to written in ways that include the teacher in the text and position the
teacher as a legitimate reader and enactor of the mathematics curriculum. If the language choices
made by curriculum developers do not include teachers in the text, the available content may do
little to engage the teachers around important issues of subject matter, pedagogy, or curriculum.
Theoretical Framework
In this study I drew from two theoretical traditions. First, I draw on cognitive theories of
knowledge that underlie the construction of the content supports and guidance available in
teachers‘ guides. I specifically drew on the work of Shulman who described three types of
knowledge (subject matter content knowledge, pedagogical content knowledge, and curricular
knowledge) needed for teaching. Second, I drew on sociocultural theory, which focuses on social

24

and cultural practices as ways of knowing (not knowledge) and particularly those that examine
discourse practices as ways of making shared meaning in dialogue. Specifically, I drew on a
theory of language in social context, Systemic Functional Linguistics (Halliday, 1985), which
proposes that by examining grammar and drawing from social theory one can say something
about why clauses make the meanings they do. Since I am analyzing written text, I drew from
these two traditions because they provide a way for me to describe what (i.e., content supports) is
in teachers‘ guides and how it is expressed (i.e., aspects of voice). In the next sections I describe
these theories in greater detail and characterize how they contribute to my analysis of content
support and its expression.
Content Supports. My analysis of the content supports of teachers‘ guides required
investigating the types of knowledge and guidance available. In the next two sections I describe
Shulman‘s (1986) theory of knowledge for teaching and Beyer et al.‘s (2009) descriptions of the
types of guidance needed for teaching.
Knowledge. Learning to teach is complex. To develop expertise in teaching mathematics
one must have many types of knowledge and be able to integrate these in ways that help one
productively promote students‘ learning of mathematics. This requires more than just knowing
subject matter. Teachers need to know subject matter in ways that are appropriate for teaching.
In addition to Subject Matter Content Knowledge, teachers must have what Shulman (1986)
called Pedagogical Content Knowledge and Curricular Knowledge. In essence, teachers require
a specialized type of knowledge of their discipline, knowledge that allows them to teach, not just
know their subject matter.
Shulman's (1986) framework distinguished between three types of content knowledge:
Subject Matter Content Knowledge, Pedagogical Content Knowledge, and Curricular

25

Knowledge, all of which place special emphasis on the content knowledge needed for teaching.
First, Subject Matter Content Knowledge involves having a robust understanding of the subject
matter. According to Shulman (1986), ―the ways of discussing the content structure of
knowledge differ‖ (p. 9) in different disciplines. However, regardless of the discipline, it is
essential that teachers go beyond just the facts or concepts and have an understanding of the
structures of their respective subject matter. Teachers need understandings of their subject matter
that goes beyond the ―mere subject matter major‖ (p. 9). They must be able to understand that
something is so and also why something is so. For example, teachers need to know not only how
to compute 16  3 , but also know why the computation works. This requires knowing more than
just a procedure, but also knowing something about place value and recognizing that if we added
he 3 to the 1, we would in fact be adding 16 and 30 rather than 16 and 3. Second, Pedagogical
Content Knowledge in its most simplified terms can be described as subject matter knowledge
for teaching. This involves knowing more than subject matter, but knowing subject matter in a
way that is related to the teaching of this subject matter. This includes knowing the most useful
forms of representing content in ways that allow for its comprehensibility by others, knowing
students and when and how students may excel or struggle, and knowing strategies for working
with students‘ ideas. For example, this would include knowing some of the most common
mistakes students make. An example of this type of knowledge would include knowing that
when students simplify rationale expressions they often ―cancel out‖ identical terms found in the
numerator and denominator even when these terms are not common factors, such as canceling
the x ‘s in

x2  2
. Finally, Curricular Knowledge is knowledge about the range of programs for
x

the teaching of subject matter, the instructional materials available, and the knowledge related to
making decisions about the fruitfulness of using particular materials in particular situations. At
26

the time his work was published Shulman indicated that teacher education programs were remiss
in addressing curricular knowledge. More recently, drawing on Shulman‘s distinction, Choppin
(2008) defined curricular content knowledge as ―the knowledge of how a particular set of
curriculum materials functions to engage students in a particular context‖ (p. 288).
Some researchers, such as Ball and colleagues, have come to call this type of knowledge
for mathematics teachers Mathematical Knowledge for Teaching (MKT) to indicate that the
knowledge described in their framework is a specialized kind of knowledge specifically for
mathematics teachers. Ball's framework was built using Shulman‘s notion of pedagogical content
knowledge as the basis (Ball, Thames, & Phelps, 2008).
Guidance. Unlike materials that merely described what to teach, educative curricula go
beyond this and provide opportunities for teacher learning through two types of support:
1

Enactment Guidance and Rationale Guidance (Beyer, Delgado, Davis, & Krajcik, 2009). These
are supports for knowing what to do and how to do it, and also for understanding why particular
mathematical or pedagogical approaches are appropriate. Support in the form of Enactment
Guidance includes more than just knowing what to teach, but also knowing how to teach it.
According to Lappan and Phillips (2009), ―The ‗what to teach‘ and the ‗how to teach it‘ are
inextricably linked‖ (p. 9). Enactment Guidance includes detailed explanations for how to teach
mathematics. For example, guidance of this type might include a sample of a class discussion in
which the teacher asks specific questions to elicit students‘ justification for their reasoning or to
evaluate the reasoning of their classmates. Such examples provide support (but not explicit
1

Beyer et al. (2009) used the term Implementation for the guidance given to teachers about
what and how to teach, rather than Enactment, however I chose to use the word Enactment as it
acknowledges more the role of the teacher as a legitimate reader of curriculum materials and an
active participant in constructing the classroom curriculum with students (Remillard, 2005),
rather than as a passive implementer.
27

models) for how teachers might pose questions in related context to elicit similar student
responses.
The second type of support, Rationale Guidance, enables teachers to know why particular
mathematical or pedagogical approaches are appropriate and therefore opens up a dialogue
between the authors and the teacher, where teachers have the opportunity to engage with the
underlying conceptual ideas the curriculum is based on. Supports such as this allow teachers to
make sense of their curriculum materials and develop what Drake and Sherin (2009) call
―curriculum vision‖ or a sense of where the curriculum materials are going and an understanding
of the ―particular kinds of learning and teaching practices described in the curriculum materials‖
(p. 324). Teachers have many dimensions to deal with and in the midst of trying to decide how to
address student thinking and lead discussions of the mathematical content they either ―adapt,
omit, or augment materials‖ (Ball & Cohen, 1996, p. 7). Rationales provide teachers with the
opportunity to develop a curricular vision so they can adapt, omit, or augment materials
appropriately. Teachers need to know the importance of particular content or approaches and
how this content connects to previous and future content in order to make informed decisions
about what they may want to adapt, omit, or augment. An example of such support might include
a discussion of why having students create multiple representations for a particular situation (i.e.,
table, graph, equation) is important by describing how the facility between representations will
help students develop a stronger concept of linearity.
Expression. Equally important to investigating what was in the teachers‘ guides, was
investigating how it was expressed. Language influences "the production, maintenance, and
change of social relations of power" (Fairclough, 1989, p. 1). Therefore, curriculum authors‘
language choices, even subconscious ones, can set up particular types of relationships with

28

teachers, which potentially impact the role of curriculum materials for teacher learning. I do not
claim, nor does Fairclough (1989), that power is only a matter of language. I recognize that
issues of power and authority in teaching are embedded in more than just the written text, as they
are also involved in the social interactions and social institutions that teachers participate in each
and every day. However, when conceptualizing the role of written curriculum materials in
teacher learning, it seems clear that the linguistic features impact the reader in some way,
whether minimally or extraordinarily I do not know. In addition, I suspect this differs from
teacher to teacher, but nonetheless, this impact is a result of the words used in written materials.
If curriculum materials are to play a role in teacher learning, written curriculum materials must
be written to speak to teachers.
Halliday‘s (1985) Systemic Functional Linguistics (SFL) is a theory of language in social
context. In SFL, language is defined as systemic; it is a network of interrelated choices for
making meaning. Language is also functional; it fulfills various functions or purposes. According
to Martin and Rose (2009), despite the complexities of language in social context, the basic
principles of SFL make language issues relatively simple to manage. SFL is based on two
perspectives. First, there are relevant levels of language (called strata) and relationships between
these strata (realizations). Social contexts are realized as texts which are realized as a sequence
of clauses. See Figure 2 from Martin and Rose (2007). Discourse is situated within grammar and
social activity because discourse uses "the tools of the grammarians to identify the roles of
wordings in passages of text, and employs the work of social theorists to explain why they make
the meanings they do" (Martin & Rose, 2007, p. 4). This means that by looking at the grammar
of a written text, one can say something about the social activity within which that text appears.

29

Figure 2. Points of View on Discourse: From Social Activity and From Grammar (recreated from
Martin & Rose, 2007, p. 5)

social activity

discourse

grammar

Second, in SFL there are three general functions of language (called metafunctions).
These metafunctions include the interpersonal (i.e., enacting relationships), the ideational (i.e.,
representing experiences), and the textual (i.e., organizing discourse in meaningful ways). For
my study, I drew specifically on the framework developed by Morgan (1998) who examined
students' mathematical texts. Morgan‘s overall goal was to develop ways of communicating with
students so they could take control over their own writing and be assessed positively and she did
this by describing the ways in which student text may be interpreted by readers. Morgan‘s (1998)
framework was augmented by Herbel-Eisenmann (2007) to describe the voice of the student
edition of one unit from Connected Mathematics Project in order to illuminate the construction
of roles of the authors and readers. Herbel-Eisenmann‘s aim was to
―see whether the authors of the unit achieved the ideological goal (i.e., the intended
curriculum) put forth by the NCTM‘s Standards (1991) to shift the locus of authority

30

away from the teacher and the textbook and toward student mathematical reasoning and
justification. (p. 344).
Herbel-Eisenmann (2007) found that there may have been a mismatch between the goals set
forth my NCTM and conventional textbook forms.
To describe relationships, one can examine both the interpersonal and ideational
metafunctions, as these metafunctions describe how relationships are enacted and who is
involved in doing what processes. In my study, I was concerned primarily with the roles and
relationships that were constructed between curriculum authors and teachers and therefore, used
the interpersonal metafunction because this is the function used to enact relationships.
Focus of Study and Research Questions
The focus of this study is to describe teachers‘ opportunities to learn mathematics subject
matter, pedagogical practices related to mathematics topics and practices, and mathematics
curricular knowledge in existing middle school mathematics curriculum materials related to
introduction to variable and geometric transformations. In particular, I examine the opportunities
for teacher learning by investigating the content and its expression in teachers‘ guides.
Specifically, I address the following research questions:
1. What is the relative frequency of educative content supports in middle school
mathematics curriculum materials for teachers‘ Subject Matter Content Knowledge,
Pedagogical Content Knowledge for Mathematics Topics, Pedagogical Content
Knowledge for Mathematical Practices, and Mathematics Curricular Knowledge and
where are these supports located?

31

2. How do middle school mathematics curriculum materials speak to teachers (i.e., what
are some of the language choices they make) through the written text in the teachers‘
guides?
3. How does opportunity to learn (content and aspects of voice) differ for introduction to
variable and geometric transformations?
To address my first research question, I coded for content supports and their location. I
examined the teachers‘ guides of each unit and coded for educative content supports that
addressed Subject Matter Content Knowledge, Pedagogical Content Knowledge for Mathematics
Topics, Pedagogical Content Knowledge for Mathematical Practices, and Mathematics
Curricular Knowledge. I then determined the relative frequency of each of these categories of
supports and their location. To answer the second research question, since I analyzed a written
text, I coded for linguistic features in the text. Specifically, I coded for personal pronouns and
―you‖-forms as these forms indicate how authors are constructing relationships with teachers
through the written text. Finally, to address my last research question, I searched for differences
in the results from my content and expression analysis for the variable and transformations units.
In the next chapter, I describe my method for these analyses.

32

CHAPTER 3: METHOD

In this chapter, I describe the methodology I used in my study. I begin by describing my
sample, including my choices of curricular series and mathematical content. I then describe my
initial analysis which includes describing in detail my analytic frameworks. I begin first by
introducing and describing the coding scheme I used to identify and capture the content supports
for Subject Matter Content Knowledge, Pedagogical Content Knowledge, and Curricular
Knowledge. I then describe my second framework that captures aspects of the voice of the
materials. I describe the major pieces of this framework: a) personal pronouns and b) ―you‖forms, which include imperatives and modality. I then describe my coding procedures and
additional analyses. Finally, I end this chapter with my percentages for inter-rater reliability and
how I calculated these percentages.
Sample
Choice of curricular series. In this study, I examined the opportunities for teacher
learning provided in four middle school curricular series. Specifically, I analyzed a total of eight
units; two each from four curricular series, one pertaining to the introduction to variable and the
other pertaining to geometric transformations. I describe my reasons for choosing this
mathematical content in the next section. It is important to note that middle schools in the United
States vary considerably in terms of the types of mathematics courses offered. According to
Dossey, Halvorsen, and McCrone (2008), the patterns and usage of textbooks in the United
States for grades 6-8 are difficult to summarize because of the mixing of basal textbooks and the
addition of supplemental algebraic content. In some middle schools, course options include
Algebra I and therefore, textbooks might include Algebra textbooks typically used in high
33

school. For this study, I chose not to analyze textbooks such as these. I analyzed curriculum
materials designed specifically for students in grades 6 – 8.
Since it would be impossible to analyze all middle school mathematics curriculum
materials, I mindfully choose four series using the following criteria: a) market share in the
United States and b) varied design principles. By varied design principles, I mean I included
curriculum materials that are categorized as "Standards-based," or developed to be aligned with
the recommendations set forth in the National Council of Teachers of Mathematics' (1989)
Curriculum and Evaluation Standards for School Mathematics (Senk & Thompson, 2003), and
those that are not, called "traditional" or "conventional" (Stein, Remillard, & Smith, 2007)
curriculum materials. Although many curriculum materials claim to be standards-based and in
fact are, as textbooks are often written to align with some set of standards, I distinguish between
those written to align with the NCTM standards by using a capital "S." Therefore, I do not label
curriculum not written specifically to align with the 1989 NCTM standards as ―Standardsbased." The Standards-based materials had a common set of design specifications. These
curriculum materials included an expanded set of content, a de-emphasis of paper-and-pencil
skills, and focus on "students' active construction of communication about solutions to
challenging problems" (Stein, Remillard, & Smith, 2007, p. 320). I also purposefully chose
multiple curricular series within the Standards-based category because we know little about the
differences between curricula in the same category. In other words, although some mathematics
educators may talk about "Standards-based curricula" as if they were one thing, not all Standardbased curricula are the same. The National Research Council (2004) argues that this
categorization, in fact, masks some of the differences within categories difficult to see. In
addition, I used information available about the design of materials (Hirsch, 2007) to ensure that

34

the inclusion of each curricular series would provide additional information about the nature of
opportunities for teacher learning. For example, if two curricular series were designed with the
same guiding principles or with the same structure (e.g., problem-centered, spiraling) it might be
less likely that I would obtain something different regarding the nature of teachers‘ opportunities
to learn from including both of these curricular series. I chose the Connected Mathematics
Project 2 (Lappan, Fey, Fitzgerald, Friel, Phillips, 2006) [CMP], Glencoes‘s Math connects:
Concepts, skills, and problem solving (Michigan Edition) (Day, Frey, Howard, Hutchens,
Luchin, McClain, et al., 2009) [Glencoe], Mathematics in Context (Wisconsin Center for
Education & the Freudenthal Institute, 2010) [MiC], and UCSMP‘s Transition Mathematics
(Viktora, Cheung, Highstone, Capuzzi, Heeres, Metcalf, et al., 2008) [UCSMP]. These curricular
series were intended to be used with students in Grades 6 – 8, and in all but one case the series
contained three books. For UCSMP, there were only two books in the series; however
documentation from the publisher indicated that these two books were intended to be used with
students in Grades 5 – 9, depending on student readiness at each grade level.
Choice of mathematical content. In my initial examination of teachers‘ guides it was
apparent that for each curricular series (and others series not included in the final analysis), the
structure and features of the teachers‘ guides repeated for each unit or chapter. For example, the
Connected Mathematics Project’s Teacher‘s Guide provided a section entitled ―Mathematical
Background‖ and this section had the same structure for all units. Since the structure and features
were repeated throughout the texts at both the unit and lesson level, it was not necessary to
analyze every page of every textbook. Therefore, I mindfully chose units to analyze; one algebra
unit and one geometry unit, because algebra and geometry make up the dominant focus in the
high school curriculum (Schmidt, McKnight, Valverde, Houang, & Wiley, 1997), something that

35

students will encounter after middle school. Specifically, I analyzed units related to the
introduction to variable and geometric transformations. I choose these particular topics because
a) these topics are seen as important in the middle school mathematics curriculum as evidenced
by the number of standards addressing them in standards documents (National Council of
Teachers of Mathematics, 2000; Common Core Standards Initiative, 2010) and b) research
indicates that these topics are typically problematic for middle school students or teachers
(Clements, 2003; Kieran, 2007).
Introduction to variables. Algebra is the focus of much of the high school curriculum
(most students take at least two algebra courses and it is also integral in precalculus and calculus
courses as well) and has become increasingly integrated into and focused on in the middle
grades, as it has been named as the gateway to higher mathematics (National Mathematics
Advisory Panel, 2008). Investigating standards documents such as the NCTM Standards and the
Common Core Standards, I found that in the middle grades there is an emphasis placed on
variables, expressions, and equations, particularly related to linear expressions. Of the nine
individual standards outlined by NCTM in algebra for grades 6-8, four of these involved
symbolic algebra, including coming up with symbolic rules, developing an understanding of
variables, and using symbolic algebra to represent and solve problems, including recognizing and
generating equivalent expressions. See Table 1.
In the Common Core State Standards, this emphasis is even clearer. Across grades 6 – 8,
more than 65% of the algebra-related standards involve variables and working with expressions
and equations.

36

Table 1. Algebra Standards for Grades 6-8 (National Council of Teachers of Mathematics, 2000)
Instructional programs from
prekindergarten through grade
12 should enable all students to—
Understand patterns, relations, and
functions

Represent and analyze
mathematical situations and
structures using algebraic symbols

Use mathematical models to
represent and understand
quantitative relationships
Analyze change in various contexts

In grades 6–8 all students should—
 represent, analyze, and generalize a variety of
patterns with tables, graphs, words, and, when
possible, symbolic rules;
 relate and compare different forms of
representation for a relationship;
 identify functions as linear or nonlinear and contrast
their properties from tables, graphs, or equations.
 develop an initial conceptual understanding of
different uses of variables;
 explore relationships between symbolic expressions
and graphs of lines, paying particular attention to
the meaning of intercept and slope;
 use symbolic algebra to represent situations and to
solve problems, especially those that involve linear
relationships;
 recognize and generate equivalent forms for simple
algebraic expressions and solve linear equations
 model and solve contextualized problems using
various representations, such as graphs, tables, and
equations.
 use graphs to analyze the nature of changes in
quantities in linear relationships.

Specifically, 17 out of the 26 algebra-related standards focus on variables, expressions, and
equations in some way. See Table 2 for frequencies of the Common Core Standards in Grade 6 –
8 pertaining to various algebraic topics. (Note: Some standards address more than one of these
topics so the numbers will not add up to 26).
Table 2. Frequency of Topics within the Algebra Standards for Grades 6-8 from the Common
Core State Standards for Mathematics
Topic
Expressions
Equations
Inequalities
Variable/Unknown
Systems of Equations

Grade 6
8
4
6
8

Number of Standards
Grade 7
Grade 8
1
2
2
6
1
5
1
2
3
37

Total
11
12
12
11
3

Table 2 cont‘d
Linear Relationships
Functions
Graphing/Coordinate
Plane

1

2

11
5
4

5

14
5
10

Since it was not possible to analyze all content relations to variables, expressions, and
equations, I chose to focus specifically on the introduction to variables. Developing an
understanding of variables is critical to understanding and being able to use algebra and
understanding the concept of variable is difficult for students and teachers (Kieran, 2007). I also
recognize that there are different conceptions or approaches to variable (e.g., Arcavi, 1994;
Usiskin, 1988). Regardless of the conception of variable espoused by curriculum authors (e.g.,
variable as ―unknown‖ or ―true variable‖), and maybe more importantly because there are
various conceptions of algebra and variable, it seemed like an especially important topic to study.
Geometric transformations. In addition to algebra, geometry was named as being most
missing from U.S. curriculum compared to other countries in the TIMSS (Schmidt et al., 1996)
and more recently curricula have begun to put more emphasis on geometry to fill these gaps. I
chose to focus on geometric transformations. In her review of the history of the geometry
curriculum in the United States, Sinclair (2008) indicated that since the latter part of the
twentieth century, geometry, which once had lost its place in the curriculum, has struggled to
regain its place and has done this by adhering less to the strict Euclidean traditions. These lessstrict Euclidean traditions include the transformational approach to geometry, and according to
Sinclair transformations have recently figured prominently in middle school mathematics texts.
This focus on transformations seemed evident in the NCTM Standards where transformations
was included as a focal topic. See Table 3.

38

Table 3. Geometry and Measurement Standards for Grades 6-8 (National Council of Teachers of
Mathematics, 2000)
Instructional programs from
prekindergarten through
grade 12 should enable all
students to—
Analyze characteristics and
properties of two- and threedimensional geometric shapes
and develop mathematical
arguments about geometric
relationships

Specify locations and describe
spatial relationships using
coordinate geometry and other
representational systems
Apply transformations and use
symmetry to analyze
mathematical situations

Use visualization, spatial
reasoning, and geometric
modeling to solve problems

In grades 6–8 all students should—

 precisely describe, classify, and understand relationships
among types of two- and three-dimensional objects
using their defining properties;
 understand relationships among the angles, side lengths,
perimeters, areas, and volumes of similar objects;
 create and critique inductive and deductive arguments
concerning geometric ideas and relationships, such as
congruence, similarity, and the Pythagorean
relationship.
 use coordinate geometry to represent and examine the
properties of geometric shapes;
 use coordinate geometry to examine special geometric
shapes, such as regular polygons or those with pairs of
parallel or perpendicular sides.
 describe sizes, positions, and orientations of shapes
under informal transformations such as flips, turns,
slides, and scaling;
 examine the congruence, similarity, and line or
rotational symmetry of objects using transformations.
 draw geometric objects with specified properties, such
as side lengths or angle measures;
 use two-dimensional representations of threedimensional objects to visualize and solve problems
such as those involving surface area and volume;
 use visual tools such as networks to represent and solve
problems;
 use geometric models to represent and explain numerical
and algebraic relationships;
 recognize and apply geometric ideas and relationships in
areas outside the mathematics classroom, such as art,
science, and everyday life.

In addition, geometric transformations received considerable attention in the Grade 8
Common Core Standards (see Table 4) and continues to be focused on in the high school
standards as well. Furthermore, research indicated that students struggle with the geometry of
motion (Clements, 2003).
39

Table 4. Frequency of Topics within the Geometry Standards for Grades 6-8 from the Common
Core State Standards for Mathematics
Topic
Area
Surface Area
Volume
Length/Circumference
Scale Drawings
Angles
Relationships between
2-D & 3D Figures
Constructions
Coordinate Geometry
Similarity
Congruence
Transformations
Pythagorean Theorem

Number of Standards
Grade 6
1
1
1

1

Grade 7
3
1
1
2
1
1
1

Grade 8

1

1

1
1
2
2
4
2

Total
4
2
3
2
1
2
2
1
1
2
1
4
2

Units. Once I chose this mathematical content, I used the authors‘ descriptions of the
units to choose units that covered this content. The four curricular series, the units analyzed, and
the total number of pages for each unit can be found in Table 5.
Table 5. Units and Total Pages per Unit for the Curricular Series
Curricular Series
Connected Mathematics
Project 2 (CMP)
Math Connects (Glencoe)

Variable units
Variables and Patterns (72)

Algebra: Number Patterns and
Functions (77)
Mathematics in Context (MiC) Comparing Quantities (81)
Transition Mathematics
(UCSMP)

Using Variables (62)

Transformations units
Kaleidoscopes, Hubcaps,
and Mirrors (88)
Geometry: Polygons (92)
Triangles and Beyond
(121)
Some Important
Geometry Ideas (72)

Analytic Frameworks
Although often viewed as a subjective scheme, curriculum materials can be viewed as an
objectively given structure (Otte, 1983). Analysis can be focused not only on the interaction
between the reader and the text (subjective), but also on the structure of the written text itself and

40

the potential of this text to construct relationships between authors and readers (HerbelEisenmann, 2007). According to Remillard (2000), if texts are to be truly educative, authors must
speak to teachers rather than through teachers. The way authors speak is construed in the written
text by the content and language choices authors make and these choices, although not
necessarily conscious ones, between alternative structures and content affect ―the meaning that
listeners or readers may construct from the utterances‖ (Morgan, 1998, p. 79). Therefore, choices
of content and language in teachers‘ guides, impact the opportunities for teachers to make sense
of and learn from the written text.
To analyze the content and expression in the eight middle school teachers‘' guides, I
developed two frameworks. The first of these frameworks aimed to describe the supports for the
development of Subject Matter Knowledge, Pedagogical Content Knowledge for Topics,
Pedagogical Content Knowledge for Practices, and Curricular Knowledge in the teachers'
guides. I adapted a framework developed by Beyer et al. (2009) that included the types of
knowledge and guidance needed by teachers. Since this was developed for coding science
textbooks I adapted the codes to be appropriate for coding mathematics textbooks and I also
included codes to describe curriculum knowledge, which was not part of Beyer et al.‘s (2009)
framework. The second framework I developed was aimed at describing some aspects of the
language of the written text in the teacher's guides. I drew on Herbel-Eisenmann's (2007) voice
framework which was developed using ideas from Morgan (1996) and Halliday (1985).
Although research on voice more generally has been used to describe who is talking, HerbelEisenmann argues that voice can be used to investigate ―how speakers shift positions, identities,
and alignments toward the words they speak as well as toward one another‖ (p. 347). In the next
sections I describe both of these frameworks in detail.

41

A framework for analyzing the content supports of teachers’ guides. To analyze the
content of the teachers' guides, I developed a coding scheme (see Table 6), adapted from Beyer,
Delgado, Davis, & Krajcik (2009), based on the design heuristics outlined by Davis & Krajcik
(2005) and proposals put forth by Ball and Cohen (1996). In addition, this scheme included
codes derived from the literature on teacher learning from curriculum use (e.g., Choppin, 2008;
Collopy, 2003; Drake & Sherin, 2009; Lloyd, 2008a; 2008b; Remillard, 2000; Remillard &
Bryans, 2004; Schneider, Krajcik, & Blumenfeld, 2005; Schneider & Krajcik, 2002; Schneider,
2006; Van Zoest & Bohl, 2002). This scheme consisted of nine categories, which included codes
for both Enactment and Rationale Guidance. These categories were further grouped into four
domains: Subject Matter Content Knowledge (SMK), Pedagogical Content Knowledge for Topics
(PCK-T), Pedagogical Content Knowledge for Practices (PCK-P), and Curricular Knowledge
(CK).
Table 6. Content Coding Scheme
I. Mathematics Subject Matter Content Knowledge
Support Teachers in the Development of Subject Matter Knowledge
Content: Present mathematical content information [Con]
II. Pedagogical Content Knowledge for Mathematics Topics
Support Teachers in Engaging Students with Topic-Specific Mathematics
Enactment Guidance for:
Experiences: Engaging students in particular experiences. [E-Experiences]
Possible Student Pitfalls: Potential pitfalls with specific mathematical experiences
(including describing where students may have difficulty or potential misconceptions) [EPitfalls]
Engaging in Problematic Experiences: Engaging students in experiences that may
potentially be problematic [E-Problem]
Activity Sequences: Engaging in productive activity sequences [E-Sequence]
Rationale Guidance for:
Experiences: Why particular experiences are appropriate, including why a particular
problem or activity is useful in a students‘ mathematical development.[R-Experiences]
Activity Sequences: Why particular activity sequences are appropriate or useful. [RSequence]

42

Table 6 (cont'd)
Support Teachers in Using Mathematical Instructional Representations
Enactment Guidance for:
Representations: Using representations (verbal, equations, tables, graphs, pictures)
with students. [E-Representations]
Rationale Guidance for:
Representations: Why particular representations (verbal, equations, tables,
graphs, pictures) are appropriate including the advantages and disadvantages
for using particular representations. [R-Representations]
Support Teachers in Using Mathematical Tools
Enactment Guidance for:
Tools: Using particular tools (e.g., rulers, compasses, calculator)with students. [ETools]
Rationale Guidance for:
Tools: Why particular tools are appropriate including the advantages and
disadvantages for using particular tools [R-Tools]
Support Teachers in Anticipating and Using Students’ Mathematical Ideas
Enactment Guidance for:
Anticipating Student Ideas: Anticipate student ideas within a topic [I-Anticipate]
Using Student Ideas: Using students‘ ideas during the enactment of the lesson [IUse]
III. Pedagogical Content Knowledge for Mathematics Practices
Support Teachers in Engaging Students in Questions
Enactment Guidance for:
Using Questions: Using questions with students (Suggested answers provided) [IQuestionsA]
Using Questions: Using questions with students (Support for answers not provided)
[I-QuestionsNA]
Helping Students Ask/Answer Questions: Having students ask/answer their own
questions [I-Ask]
Rationale Guidance for:
Questions: Why particular questions or series of questions are appropriate or useful
[I-Ask]
Support Teachers in Engaging Students in Justification, Reasoning, and Proof
Enactment Guidance for:
Justification, Reasoning, & Proof: Using approaches for engaging students in
justification, reasoning and proof [E-Proof]
Rationale Guidance for:
Justification, Reasoning, & Proof: Why approaches to engaging in justification,
reasoning and proof are appropriate or useful [R-Proof]

43

Table 6 (cont'd)
Support Teachers in Developing Students' Mathematical Vocabulary/Terminology
Enactment Guidance for:
Developing Mathematical Terminology: Developing students‘ mathematical
terminology (including vocabulary and notation) and recognizing colloquial words
that may help students understand context or promote/hinder students conceptions
[E-Terminology]
Rationale Guidance for:
Developing Mathematical Terminology: Why developing mathematical vocabulary
or terminology is important and why particular strategies are appropriate or useful
[R-Terminology]
Support Teachers in Engaging Students in Appropriate Participation Structures
Enactment Guidance for:
Participation Structures: Using appropriate participation structures (e.g., individual,
small-group, whole-class) [E-Participation]
Rationale Guidance for:
Participation Structures: Why particular participation structures are appropriate or
useful [R-Participation]
IV. Mathematics Curricular Content Knowledge
Support Teachers in the development of Curricular Knowledge
Enactment Guidance for:
Curricular Overview: Using unit, section, or lesson content in the intended ways (by
providing a decription of what will unfold in the unit, section, or lesson) [EOverview]
Curricular Features: Using features of the curriculum [E-Features]
Curricular Storyline: Connecting previous and future mathematics content, including
within and across disciplines, courses, units, sections, and lessons [E-Storyline]
Curricular Goals: Using the goals of the unit, section or lesson (including knowing
what the goals are and how to acheive them) [E-Goals]
Rationale Guidance for:
Curricular Philosophy: curricular philosophy. [R-Philosophy]
Curricular Features: Why curricular features are appropriate or useful (including
explaining these features) [R-Features]
Curricular Storyline: Why particular pieces of content or processes are appropriate
in the storyline [R-Storyline]
Curricular Goals: Why particular goals are apporpriate or useful. [R-Goals]
Beyer et al.'s (2009) coding scheme was developed specifically for coding biology
textbooks. Due to the frameworks intimate connection to the teaching of science, I needed to
make adjustments in order to use it to code mathematics curriculum materials. That said, the
categories pertaining to the types of knowledge (i.e., Subject- Matter Content Knowledge,
Pedagogical Content Knowledge for Topics and Practices) and the types of guidance (Enactment
44

and Rationale) still seemed appropriate and served as organizing categories. Prior to generating
my codes, I felt that this framework lacked attention to the very thing that it was being used to
analyze, the curriculum and more specifically, curriculum materials. In addition to SubjectMatter Content Knowledge, and Pedagogical Content Knowledge, Shulman (1986) also
described a need for Curricular Knowledge. Furthermore, when writing about this in 1986,
Shulman indicated that this was the knowledge that teacher education was the most remiss in
attending to. Therefore, including codes for describing supports for the development of
Curricular Knowledge was something I decided I would do before generating my codes.
Generating codes. To generate codes I drew primarily on two sources: a) research
literature, including theoretical pieces (Ball and Cohen, 1996; Davis & Krajcik, 2005; Shulman,
1986), and the research on mathematics and science teacher learning and curriculum use,
including drawing heavily form the structure used by Beyer et al. (2009) and b) the curriculum
materials themselves and my knowledge of a variety of materials (from my years of teaching and
doing research). I used the latter as I developed my scheme in order to gauge the reasonableness
of my codes. I did not want to have codes in my scheme that were so far removed from what
could be expressed in written materials, or that were too specific to any one content or type of
problem. To aid me in making these decisions, in addition to my own knowledge of materials, I
also examined curricular series not included in my final study (i.e., Prentice Hall‘s Mathematics,
Math Thematics) and other units addressing different strands from the curriculum materials I
chose for my study (i.e., units focusing on number and data analysis and statistics). In the next
few sections, I describe the development of my codes organized by the four knowledge types. I
provide representative examples for each of the codes, drawing from either the variable or
geometric transformations unit from each curriculum. Since in this section my aim is to provide

45

examples of these codes, and not to compare the units, I include only one example for each code
from one of the units, not both. For some examples I provide more than one sentence, as context
may play a factor in understanding why I assigned particular codes. In these instances each of the
sentences received that code. If I provided three sentences in my example for Enactment
Guidance for Anticipating Student Thinking, all three sentences received this code.
Subject Matter Content Knowledge. The code that seemed the most obvious to include in
my scheme concerned subject matter. Drawing on Beyer et al. (2009), I considered this support
an ―other educational feature‖ (p. 983), rather than a support providing Enactment or Rationale
Guidance, although I recognize that having subject matter knowledge does impact the enactment
of a lesson. An example of a support is a sentence that provides information about the subject
matter, such as “The shape of a graph of a pattern of change over time shows the rate of
change in the dependent variable as time passes” (CMP, Variables & Patterns, p. 5).
Pedagogical Content Knowledge for Topics. Drawing from Beyer et al. (2009), I
included codes for Enactment Guidance for Experiences, Pitfalls, Activity Sequences,
Representations, Anticipating Student Thinking, and Using Student Thinking and Rationale
Guidance for Experiences, Activity Sequences, and Representations. These codes seemed
appropriate for mathematics teachers as well as science teachers. In addition to these codes, I
added three more. The first was Enactment Guidance Engaging Students in Problematic
Experiences. My pilot analysis revealed instances that went beyond identifying potential pitfall,
but provided information for teachers as to what to do when students found experiences difficult.
The other two codes that seemed appropriate were Enactment and Rationale Guidance for Using
Tools. In mathematics, tools such as calculators, rulers, and compasses, to name a few, are
integral to engaging in mathematical activity. It seemed necessary, and appropriate that teachers

46

be provided with both Enactment and Rationale Guidance for Using Tools with their students.
Table 7 includes examples of these codes.
Table 7. Examples of Pedagogical Content Knowledge for Topics Codes
Enactment
Guidance

Codes
Experiences

Examples
"Have students determined the height of trees or other objects
by using similar triangles" (Glencoe, Geometry: Polygons, p.
540a).
Possible
"Watch for students who are so ruled by the words that they
Student Pitfalls lose sight of their context. For example, students who add
whenever they see the word total in the problem" 'In the year
2000, the population of one town was 1,900 and in another
town was 2,100. What was the total population of the towns?'"
(USCMP, Using Variables, p. 76-77).
Engaging
"If students are having difficulty, you might remind them that
Students in
parallel arrows having the same length will produce the same
Problematic
change" (MiC, Comparing Quantities, p. 13).
Experiences
Activity
Sequences

Representation

Tools

Anticipating
Student Ideas
Using Student
Ideas

Rationale

Experiences

―After students work on problem 8, you may wish to discuss
how exchanging relates to the combination chart to remind
students what the numbers in the combination chart stands for‖
(MiC, Comparing Quantities, p. 18).
―In the case of tables, one has to decide which values of the
independent variable should be represented in the table to give
most informative results‖ (CMP, Variables & Patterns, p. 3).
"To draw a point at a certain distance from point A can be
done using a centimeter ruler" (MiC, Triangles and Beyond, p.
8B).
―Some students may trade for some corn, while others may
trade until they have only corn‖ (MiC, Comparing Quantities,
p. 5T).
"Some will talk about drawing lines from the center of the
design to corresponding points of two adjacent parts of the
pattern, thus making an angle that can be measured with an
angle ruler or a protractor. Help students to understand that
each design has a particular angle of rotation regardless of
which two adjacent parts are chosen" (CMP, Kaleidoscopes,
Hubcaps, and Mirrors, p. 24).
―We ask students to graph solutions to inequalities because
such graphs convey a lot of information‖ (UCSMP, Using
Variables, p. 115).

47

Table 7 (cont‘d)

Activity
Sequences
Representations

Tools

"We recommend going through Questions 1-10 in order. They
will take students through the key points of the lesson‖
(UCSMP, Some Important Geometry Ideas, p. 410).
"Unlike the combination chart, notebook notation can be used
to solve problems involving combinations of more than two
kinds of items. In addition, notebook notation can be used to
record the new combinations. Students should come to
appreciate the advantages of notebook notation over guessand-check strategies and combination
Charts" (MiC, Comparing Quantities, p. 22B).
"This tool [calculator] allows students to look at many
examples quickly and helps them observe patterns and make
conjectures about functions" (CMP, Variables and Patterns, p.
11).

Pedagogical Content Knowledge for Practices. The Beyer et al. (2009) framework did
not contain a category for practices, but included a category for "Scientific Inquiry." I was not
sure that I could translate scientific inquiry into one practice for mathematics teachers, but saw
this as support for enacting and understanding the practices or processes relevant to mathematics
teaching and learning. This would encompass some of what the Beyer et al. (20090) framework
included for helping science teachers enact inquiry teaching, but would include other codes
specific to mathematical teaching. From the Beyer et al. (2009) framework I included Enactment
Guidance for Using Questions and Helping Students Ask/Answer Their Own Questions and
Rationale Guidance for Questions. In addition to these codes, I generated six more codes; one
Enactment and one Rationale Guidance Support for Engaging in Justification, Reasoning, &
Proof, Developing Mathematical Terminology, and Participation Structures. The first two
seemed to be critical as these are practices or processes that are integral to the teaching and
learning of mathematics. Justification, reasoning, and proving are integral to the ways in which
mathematics is done and communicated. Proving is often difficult for students and teachers
(Harel & Sowder, 2007), particularly at the middle school level where students have difficulties
48

moving beyond examples-based reasoning (Knuth, Choppin, & Bieda, 2009) and teachers
themselves often have a limited understanding of proof (Chazan, 1993; Knuth, 2002). The
second addition, related to supports for developing mathematical terminology, seemed important
because communicating about mathematics requires the use of mathematical terminology. Some
may argue, like Lemke (1990) did about science that learning mathematics is learning to "talk
mathematics." Finally, I included Enactment and Rationale Guidance for Participation
Structures, as teachers are increasingly expected to have students interact in a variety for ways
(i.e., individual, pairs, small-group, whole-class) in order to actively engage students in
mathematical roles (Franke, Kazemi, & Battey; 2007; O‘Connor & Michaels, 1996) and these
structures are becoming integral to helping ELL students as well (Empson, Turner, Dominguez,
& Maldonado, 2006). Table 8 includes examples of these codes.
Table 8. Examples of Pedagogical Content Knowledge for Topics Code
Enactment
Guidance

Codes
Using Questions
(suggested answers
provided)
Using Questions
(suggested answers
not provided)
Helping Students
Ask/Answer
Questions:

Justification,
Reasoning, & Proof

Examples
―Ask: On which side of your reflection does your
left arm appear? [answer provided] right side‖
(Glencoe, Geometry: Polygons, p. 58).
―Ask them to tell you how they would solve it‖
(Glencoe, Algebra: Number Patterns and
functions, p. 40).
―Through their work in this and other geometry
units, students learn important questions to ask
themselves about any situation that involves the
principles explored, such as: How can I use
symmetry to describe the shapes and properties of
figures in a design or
a problem?‖ (CMP, Kaleidoscopes, Hubcaps, and
Mirrors, p. 2).
―Then you can show students an example of a
deduction. Begin with a
as the sum of the
measures of the angles around a point. Then it
follows that
is the sum of the measures of
the angles on one side of the line through that
point. Then it follows….‖ (UCSMP, Some
Important Geometry Ideas, p. 387).
49

Table 8 (cont'd)

Developing
Mathematical
Terminology

Participation
Structures

Rationale
Guidance

Questions

Justification,
Reasoning, & Proof

Developing
Mathematical
Terminology

Participation
Structures

―Help students understand that the term evaluate
an expression involves two steps. The first is to
replace the variables with their values. The
second step is to perform the indicated arithmetic
operations (UCMP, Using Variables, p. 85).
―Have students work Problem 1.5 individually or
in pairs. If they work with a partner, make sure
each student makes a table and a graph‖ (CMP,
Variables and Patterns, p. 33).
―These are important questions to discuss because
they utilize properties of congruence‖ (UCSMP,
Some Important Geometry Ideas, p. 399).
―It is likely that students will discover the angle
congruence theorems that are usually taught and
proved in high school geometry– Side-Side-Side,
Side-Angle-Side, and Angle-Side-Angle. This
engagement with the ideas in an informal way
will help make the more proof oriented approach
of high school geometry
more understandable‖ (CMP, Kaleidoscopes,
Hubcaps, and Mirrors, p. 8).
―The ideas of consecutive sides and consecutive
angles apply to all polygons not just
quadrilaterals. So this vocabulary is quite useful‖
(UCSMP, Some Important Ideas in Geometry, p.
402).
―Let the students work alone for a minute or two,
and then move them into small groups of 2 to 4.
This gives each student time to think individually
before discussing ideas with a group‖ (CMP,
Variables and Patterns, p. 55).

Mathematics Curricular Content Knowledge. As I previously stated, the Beyer et al.
(2009) framework did not include codes that pertained to supports for the development of
curricular knowledge. However, I included this knowledge because I was studying curriculum
materials and Shulman (1986) described this knowledge as the knowledge teacher educators
were the most remiss in teaching. Drawing on Shulman (1986), Ball and Cohen (1996), and the
research on curriculum use, I generated a total of eight codes, four Enactment (i.e, overview,
features, storyline, goals) and four Rationale Guidance supports (i.e., philosophy, features,
50

storyline, goals). Related to the idea that teachers should understand the resources that they use
(Shulman, 1986), I created codes for Enactment and Rationale Guidance for Curricular Features
and Rationale Guidance for Curriculum Philosophy. In order for teachers to understand their
materials they must have knowledge of its features, including not only what the features are, but
why these features were included in the textbook and an understanding of the philosophy of the
materials. In order to capture ideas that related to helping teachers to develop ―curriculum
vision‖ or a sense of where the materials were going and how particular activities it into this
(Drake & Sherin, 2009), I generated codes for Enactment and Rationale Guidance for Goals and
Curricular Storyline. Shulman (2009) advocated for this as a type of knowledge as well and
called it ―lateral‖ curriculum knowledge‖ (p. 10). In order to understand where a curriculum is
going, it is necessary to understand the mathematical goals and to have a sense of what came
before and after particular topics and lessons. Finally, the need for the Enactment Guidance for
Curricular Overview arose during my pilot analysis. I found text in all the curriculum materials
that described for teachers what the lesson or unit was about. At first I questioned whether this
was in fact support for the teacher. I decided that this text provided teachers with an opportunity
to learn about the activities that would unfold in a particular unit or lesson and so therefore, I
wanted to capture this in my analysis. Table 9 includes examples of each of these codes. For
Rationale Guidance for Goals I do not include an example from the curriculum materials
because I did not find this in any of the curricula. Although the curriculum materials all provided
goals for units and lessons, there was no support for understanding why these goals were chosen
by curriculum authors as appropriate.

51

Table 9. Examples of Mathematics Curricular Knowledge Codes
Enactment
Guidance

Rationale
Guidance

Codes
Examples
Curricular Overview ―Students solve a more complex exchange problem
involving the strength of horses, elephants, and
oxen‖ (MiC, Comparing Quantities, p. 3T).
Curricular Features
―The Differentiated Homework Options provide
suggestions for the exercises that are appropriate
for basic, core, or advanced students‖ (Glencoe,
Algebra: Number Patterns and Functions, p. 27).
Curricular Storyline ―For the teacher Students have encountered a
variety of patterns in the tables and graphs of
related variables. Many have been linear increasing
or decreasing patterns. Some have…. The details of
relating particular graph shapes to particular
function types and symbolic expressions will
develop over the next several years. (CMP,
Variables and Patterns, p. 104 ).
Curricular Goals
―The goal is for students to understand the use of
algebra as a tool to solve problems that arise in the
real world or in the world of mathematics, where
symbolic representations can be temporarily freed
of meaning to bring a deeper understanding of the
problem‖ (MiC, Comparing Quantities, p. xx).
Curricular
"Connected Mathematics was developed with the
Philosophy
belief that calculators should always be available
and that students should learn when their use is
appropriate" (CMP, Variable and Patterns, p. 11).
Curricular Features
―The Reading Math features throughout the chapter
help students understand the mathematical meaning
by relating them to everyday use‖ (Glencoe,
Geometry: Polygons, p. 508B).
Curricular Storyline ―Rotations are critical in the study of congruence
and tessellations used in this chapter to develop
properties of angles‖ (UCSMP, Important
Geometry Ideas, p. 374).
Curricular Goals
―These goals are important and/or appropriate
because….‖

A framework for analyzing expression. The ways in which text is expressed in the
teachers' guides indicates how curriculum authors are speaking to teachers. To investigate this
expression, I drew on a framework developed by Morgan (1988) and augmented by HerbelEisenmann (2007), who called hers a framework for analyzing the voice of the textbook.
52

Although research on voice, more generally, has been focused on examining who is speaking,
researchers argue that an examination of voice helps to identify ―how speakers shift positions,
identities, and alignments toward the words they speak as well as toward one another‖ (HerbelEisenmann, 2007, p. 347). To analyze voice, Herbel-Eisenmann (2007) used Halliday's (1985)
SFL and described both the interpersonal and ideational functions since these functions describe
the construction of personal relationships and who is speaking. According to Morgan (1998),
although it is possible to separate the grammatical features related to the ideational and
interpersonal functions, it is difficult to separate the effects that these forms have on readers.
That said, in my analysis, I drew most heavily on the interpersonal metafunction of language as
this function most closely describes the ways in which personal relationships are constructed. To
examine the construction of interpersonal relationships between curriculum authors and teachers,
I examined the use of personal pronouns and ―you”-forms, particularly focusing on imperatives
and ―you”+modal verbs because these were the most prevalent forms. In the next sections I
describe each of these grammatical features.
Personal pronouns. The first linguistic form I examined was personal pronouns because
this is one of the most explicit ways in which relationships are expressed between authors of a
text and their readers. First-person pronouns such as ―I‖ and ―we‖ indicate the authors direct
involvement in the activity of the text (Morgan, 1996). According to Herbel-Eisenmann (2007)
the use of the pronoun ―we‖ could indicate the authors speaking with the authority of the
discipline of mathematics, or it may be used to involve the readers in the mathematics.
Researchers such as Pimm (1987) and Rowland (1999) have alluded to the ideological aspects
embedded in the use of "we", such as how using "we" assumes the reader‘s agreement with the
author. In my analysis, I coded all of the uses of "we."

53

The second-person pronoun ―you‖ directly involves the reader, and is of particular
interest in my study because the readers of curriculum materials are the teachers using the
materials. In the next section I describe in more detail my analysis of "you." In addition, the
general absence of first-person pronouns, according to Morgan (1998), not only affects the
picture of the nature of mathematics, but also has the potential to distance the author from the
reader. Therefore, not finding personal pronouns in my analysis would also provide interesting
evidence about the ways in which authors are communicating with teachers through the written
text.
“You”-forms. Grammatically, the pronoun "you" serves to bring the reader into the text.
Since the reader of curriculum materials are teachers, I chose to examine the use of this pronoun
more fully to determine how it was used by curriculum authors. Although I captured all forms of
"you" in the text (i.e., verb+"you", modal verb+"you" "you"+verb, modal verb+"you",
imperatives), I specifically focus on imperatives and modality since these were the most
common. In the imperative form "you" is implicit, whereas when used with a modal verb, "you"
is explicit. I describe imperatives and modality below.
Imperatives. Imperatives are characteristic of academic mathematics texts (Morgan,
1998) and according to Herbel-Eisenmann (2007), imperatives ―implicitly address the reader and
involve him or her in the in the construction of mathematics‖ (p. 349) whether they want to be
implicated or not (Pimm, 1987). According to Morgan (1998) the use of imperatives marks an
author‘s membership in the mathematical community and is to be expected in mathematical
academic discourse. However, in the school context, Morgan questions the use of imperatives as
there may be tensions between pupils and teachers as pupils and teachers strive to both show that
they have familiarity with mathematical language, but at the same time explain their thinking.

54

The same may be the case between curriculum authors and teachers. This tension may arise as
curriculum authors both need to exhibit their knowledge and authority, but at the same time
explain their thinking.
According to Martin & Rose (2007), while there are different moods for realizing a
command (i.e., declarative, interrogative, imperative) and these all produce the same result, these
different grammatical moods differ in the kinds of social relationships they incur. Unlike other
forms, the imperative assumes a position of authority and does not open up the possibilities of
negotiation. Particularly useful is Rotman‘s (1988) distinction between two types of imperatives:
exclusive and inclusive. According to Rotman (1988), although "'the speaker of a clause which
has chosen the imperative has selected himself the role of controller and for his hearer the role of
controlled" (p. 8), different types of imperatives can include or exclude the hearer or reader.
Exclusive imperatives, position the reader as a ―scribbler,‖ or someone who must perform some
action. Some examples include ―Multiply,‖ ―Add,‖ and ―Bisect the angle.‖ Inclusive
imperatives, on the other hand, address the reader as a ―thinker‖ and these include imperatives
such as ―consider‖ and ―suppose.‖ These imperatives establish a ―commonality between speaker
and hearer‖ (p. 9). Although the reader is not positioned solely as either a "scribbler" or a
"thinker," the choice of imperatives foregrounds one positioning over the other.
Modality. Modality refers to ―the degree of likelihood, probability, weight or authority
the speaker attaches to the utterance‖ (Hodge & Kress, 1993, p. 9 as cited in Morgan, 1996).
There are two forms of modality, relational modality and expressive modality. Relational
modality describes the authority of one participant in relation to another, whereas expressive
modality describes the author's authority with respect to truth or probability (Fairclough, 2001).

55

In text, modality is established by the use of modal auxiliary verbs such as ―may,"
"must," or "should," or with hedges (Rowland, 1995). Hedges, or "linguistic pointers to
uncertainty" (Rowland, 1995, p. 328) are words that modify the certainty of an utterance. As
hedges and modal verbs get used together the strength of convictions of these utterances are
modified. For example, "I think you might want to explain..." is less certain than "You might
want to explain..." which is less certain than using the imperative "Explain..." According to
Herbel-Eisenmann, Kristmanson, and Wagner (2011), ―mathematics discourse that features
modality can be associated with a fallibilist philosophy of mathematics‖ (p. 146). They propose
that this underpins certain ways of teaching mathematics, including ways that involve students
working on problems without being given solution methods in advance. The same may be said
about the text in teachers' guides. Teachers' guides that do not feature modality may indicate that
the methods of teaching are completely known and certain, and therefore, this may impact
teachers‘ opportunities to engage around the ideas presented, hindering the potential for teacher
learning. On the other hand, text that features modality may serve to open up a dialogue between
curriculum authors and teachers, and teachers may feel empowered to make decisions about how
to enact the curriculum. This openness may promote a more equitable relationship between
authors and teachers, as teachers are being asked to take a greater role in making decisions about
how to enact the curriculum in their classrooms. For example, there is a grammatical difference
between "You might want to have students work in groups," as opposed to "Have students work
in groups." These differences, although subtle to some readers, may be interpreted quite
differently by teachers.
According to Fairclough (2001), there is an overlap between the modal verbs that
express permission and obligation (i.e., relational) and those that express certainty or possibility

56

(i.e. expressive). For example, the modal verb "may" is used to express both permission and
possibility. For example, in the expression, "You may multiply by four," it is possible that "may"
indicates permission being granted by the speaker for the listener to multiply by four. However,
it is also possible that "may" indicates the fact that it is mathematically possible to multiply by
four. Therefore, in my study, I captured both types of modality as issues of permission and
obligation seemed intimately related to how curriculum authors constructed relationships with
the teachers reading their materials.
Procedures
Pilot analysis. After the initial development of my analytic frameworks, I used these
frameworks to code teachers' guides from units that were not part of the final study. These units
included a variety of mathematical content including algebra, geometry, and statistics. I refined
my coding to account for additional aspects that were not initially part of my scheme. For
example, the original Content Coding Scheme included a code for Enactment Guidance for
Questions, but did not differentiate whether suggested answers were provided to teachers in the
text. Therefore, before beginning coding of the included units I broke this code into
QuestionsNA and QuestionsA to indicate whether suggested answers were provided in the text
or not, since Ball and Cohen (1996) indicated that providing suggested answers is more
supportive than the typical "Answers will Vary" (p. 7) or providing no answers at all.
Throughout the coding of the final corpus, I continually revisited the definitions of my codes and
either added codes or refined existing codes. When codes were added late into the analysis I
revisited all previously coded pages to see if there were instances of these codes.
Coding Process. I coded each page of the teachers‘ guide of each of the variables and
transformations units using both frameworks. The only exception to this page-by-page analysis

57

was in CMP, which included a condensed version of their teacher material for each lesson called
―At a Glance.‖ I did not code any of the ―At a Glance‖ pages as these included the same content
as the non-condensed teacher support. My unit of analysis for my content coding was the
sentence and for my expression coding was the clause. For my content coding I examined each
sentence and assigned one or more content codes, if applicable, and also coded the location of
the content support (i.e., unit, section, lesson). Sentences were coded for multiple supports if it
was warranted. For example, in order to help teachers use a particular question with students, the
text may also provide teachers with support for anticipating students' thinking, such as in the
following excerpt from Mathematics in Context: ―Some students may solve this problem without
using squares. They reason: 25 + 64 = 89, so the white square must have more than 89 tiles. So a
square of 100 tiles is possible‖ (MiC, Triangles and Beyond, p. 26T). This excerpt, found at the
lesson level, in "Comment About the Solutions," a section which included text related to the
questions in the student edition of the textbook, was coded for both Enactment Guidance for
Questions (suggested answers provided) and Anticipating Student Thinking because these three
sentences provided teachers with support in working with their students on a particular question,
and provided teachers with some ideas as to what their students might say. This excerpt received
three ―E-QueA-L‖ codes and three ―E-Ant-L‖ codes.
I also examined each clause coded all personal pronouns and "you"-forms and kept track
of the location as well. I entered all codes into a spreadsheet for ease of calculating frequencies
and percentages across all units and curriculum materials. This resulted in eight separate Excel
workbook files (one per unit), each containing a sheet for all pages in my sample within each
unit and two additional sheets that summed codes from all sheets in the workbook.

58

Inter-rater reliability. I used three additional coders to check the reliability of the
coding process. The content coding was done by one coder, a mathematics education graduate
student with curriculum analysis experience, who coded a random sample of 10% of the corpus,
stratified by unit, meaning that he coded pages from all eight units. Percent agreement was
calculated at the sentence level; agreement was achieved only when a sentence was coded in the
exact same way by both coders. For example, for the content coding if a sentence received two
codes from one coder and only one code was assigned by the second coder (even if this one code
was in agreement with one of the codes assigned by the first coder), agreement was not reached.
An agreement of at least 85% was reached for each unit. See Table 10 for the percentages of
agreement for each unit by curriculum.
Table 10. Percentages of Agreement for content by Unit and Curriculum
Curricula
CMP
Glencoe
MiC
UCSMP

Variable
85
88
85
91

Transformations
86
91
85
95

For the coding related to the aspects of voice, I had two coders (both mathematics
education graduate students) code personal pronouns and ―you‖-forms. One coder coded 10% of
the algebra units and the other, 10% of the geometry units. I calculated a separate reliability
percentage for personal pronouns, imperatives, and modal verbs used with the pronoun ―you.‖
Agreement was reached only when both coders had identified the same personal pronouns,
imperatives, and modal verbs used with the pronoun ―you‖ in each sentence within the text. For
example, if I coded a sentence as an imperative and the second coder did not, this was not
counted as agreement. The same was true if we did not code the same personal pronouns and
modal verbs in a sentence. See Table 11 for the percentages of agreement for personal pronouns,
imperatives, and modal verbs.
59

Table 11. Percentages of Agreement for Personal Pronouns and "You"-forms by Unit and
Curriculum
Curricula
CMP
Glencoe
MiC
UCSMP

Personal
Pronouns
98
100
100
99

Variable
Imperatives
99
98
99
96

Modal
Verbs
100
100
100
100

Personal
Pronouns
98
100
99
100

Transformations
Imperatives
97
97
96
96

Modal
Verbs
100
100
100
100

Analysis
Once my data was coded, I used the summary sheets in each workbook to determine the
relative frequencies in each unit of : a) each individual content code, b) each content category
(i.e., Subject Matter Content Knowledge, Pedagogical Content Knowledge for Topics,
Pedagogical Content Knowledge for Practices, Mathematics Curricular Knowledge), c) personal
pronouns, d) "you"-forms, e) imperatives (including frequencies of exclusive and inclusive
imperatives) , and f) modal verbs. I used these relative frequencies to search for themes within
units across curriculum materials and across the variable and transformations units.
In addition, for the voice analysis, I summed the frequencies of personal pronouns by
location. I then did a deeper analysis of the types of imperatives, including determining the
frequencies of imperatives that directed teachers versus those that directed students through the
teacher. Finally, since the modal verbs "may" and "might" were so pervasive in all four
curriculum materials, I investigated this further by determining the most frequent collocates (i.e.,
the words written directly after "may" or "might") in order to examine the types of processes that
teachers were asked to engaged in.

60

CHAPTER 4: RESULTS –DIFFERENCES IN STRUTURE AND UNIT CONTENT

Before outlining the main results of my study, I describe the curriculum materials and the
units in this chapter. This description provides an overall context for understanding the
curriculum materials that differed in their structure. CMP and MiC had separate teachers‘ guides
for each unit, whereas Glencoe and UCSMP each had a two-volume teachers‘ guide with each
volume having half of the teacher support for that course. These curricular series also differed in
terms of the ways in which support was broken up, with CMP and MiC having material
separated into three levels (i.e., Unit, Section, Lesson) and Glencoe and UCMP separated into
two (i.e., Unit, Section, Lesson). Finally, the curriculum materials varied in the "embeddedness‖
of their education supports (Beyer et al., 2009). Embedded supports are integrated within the
directions and content for enacting activities found in the student text, rather than being close to
but separate from student activities. CMP was the only teachers' guide that offered support in this
way.
In addition to impacting the amount of content support, these structural differences may
also impact the types of supports and the ways in which these were expressed. For example, the
unit and two-volume structures may impact the total number of pages possible, therefore
impacting the number of content supports and expression. With fewer pages there is less room
for support and this may also require authors to be terser in their expression. In the next sections,
I describe each of these structural differences.
Structural Differences
The curriculum materials varied in terms of the overall structures for their teachers'
guides. First, CMP and MiC had a separate teacher's guide for each unit, whereas Glencoe and

61

UCSMP each had a two-volume teacher's guide. Each of these volumes contained half of the
units for that course. Second, the ways in which the content in the teachers' guides was
distributed varied. Just as a curriculum is broken into units, these units are broken up into smaller
chunks of content. See Figure 3 for the structure of the units in this study.
Figure 3. Structure of the Units in this Study
Unit
Section

Did not exist in
Glencoe and UCSMP

Lesson

Although each curriculum used its own terminology to refer to the different levels of
support, I will use Unit, Section and Lesson throughout this chapter. CMP and MiC was
structured into three levels (i.e., Unit, Section, Lesson). Unit was the largest chunk of content
(encompassing the Section and Lesson Levels) and Lesson being the smallest, the chunk to be
covered in a typical class period. CMP had four sections (called Investigations) in their variable
unit and five in their transformations unit with each section having between two and four lessons
(called Problems) and MiC had five sections in their variable unit and six sections in their
transformations unit (called Sections), with each section having between three and four lessons
indicated by a specific topic name (e.g., Combining Transformations: Using Models to
Understand Transformations and Relationships Among Types of Two-Dimensional Objects).
Mathematics content in Glencoe and UCSMP was broken up into only two levels (i.e., Unit,
Lesson), and similar to CMP and MiC, Unit encompassed Lesson. Including chapter reviews,

62

Glencoe and UCSMP had a total of 10 and 9 lessons in their variable units, respectively, and 11
and 10 lessons in their transformations units, respectively.
Embeddedness of Supports
The curriculum materials also varied in terms of the "embeddedness of education
supports" (Beyer et al., 2009). The embedded teachers‘ guide pages integrated supports into the
student activities, whereas the non-embedded teachers‘ guides included supports close to, but
separated from student activities. These non-embedded teacher guide pages included a reduced
student page, which accounted for more than half of the space on the page. The text intended for
teachers was wrapped around the reduced student page. For an example of pages from an
embedded and non-embedded teacher‘s guide see Figure 4. The shaded portions indicate the part
of the page containing text intended for the teacher.
Figure 4. Examples of Pages from Embedded and Non-Embedded Teachers' Guides

Embedded Teachers' Guide Page

Non-embedded Teacher's Guide Page

Reduced Student Page
Reduced Student

Page

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CMP was the only teachers' guide that provided embedded supports. Glencoe, MiC, and
UCMP were not embedded, but were formatted as teacher editions of the student text with wraparound text for the teacher. These guides had text close to, but separated from the activities in the
student text, and support was often found in sections marked with specific headings (e..g, ―Error
Alert!,‖ ―Reaching All Learners,‖ ―About the Mathematics‖). CMP supports were integrated
within the directions and content for enacting activities found in the student text. For example,
―Summarize 1.4 Question A raises the issue of connecting points on a graph—an idea
introduced in Problem 1.3. Throughout this unit, students are asked to think about
whether connecting the points on a graph makes sense. In this case, it does because the
distance changes continuously over the 7.5-hour period. Students should understand that
connecting points can help them see patterns of change more easily and make predictions
or estimates about values between plotted points. However, they should also recognize
that, connecting points with straight segments shows a constant rate of change between
points, which may not accurately reflect the real situation‖ (CMP, Variables and Patterns,
p.29).
This CMP example shows how the text included supports within the narrative, rather than
in specially marked sections. This excerpt was intended to help teachers enact the Summarize, or
final part, of the lesson and provided support for helping teachers understand the curricular
storyline, details about the particular problem students were engaged with in the lesson, and
anticipate student thinking.

64

Unit Details
Although roughly the same amount of content, the texts differed in terms of the number
of lessons and pages in each unit. Table 12 provides the names of the units, the topics covered,
and the total number of lessons and pages for each curriculum.
Table 12. Unit Details by Curricular Series
Introduction to Variable
Curricular
Unit and
Topics
Series
Lesson, Page, &
Sentence Totals
Connected Variables and
variables,
Mathematics Patterns
representations
Project 2
(tables, graphs,
(CMP)
14 lessons
words, symbols)
72 pages
of relationships
2,019 sentences

Math
Connects
(Glencoe)

Algebra:
Number
Patterns and
Functions
11 lessons
50 pages
631 sentences

Mathematics Comparing
in Context Quantities
(MiC)
15 lessons
81 pages
782 sentences
Transition Using Variables
Mathematics
(UCSMP) 8 lessons
62 pages
688 sentences

Geometric Transformations
Unit and
Topics
Lesson, Page, &
Sentence Totals
Kaleidoscopes,
transformations,
Hubcaps, and
symmetry,
Mirrors
congruence
(including a focus
18 lessons
on triangle
88 pages
congruence)
2,249 sentences

variables,
equations, prime
factors, powers,
exponents, order
of operations,
functions,
formulas
(including area)

Geometry:
Polygons

variables,
equations,
systems of
equations
(informal solving)

Triangles and
Beyond

variables,
expressions,
formulas,
graphing
inequalities,
open sentences,
Pythagorean
Theorem
65

12 lessons
92 pages
782 sentences

transformations,
angles,
quadrilaterals,
polygons,
tessellations,
similar figures,
displaying data in
circle graphs
transformations,
congruence;
constructions

18 lessons
121 pages
1,258 sentences
Some Important
Geometry Ideas
10 lessons
72 pages
721 sentences

transformations,
symmetry,
tessellations,
angles and lines,
parallelograms,
distance,
Triangle-Sum
Property

Although the number of lessons, pages, and sentences varied, each covered relatively the
same content for introducing variables and geometric transformations. Both of Glencoe‘s units
included a lesson devoted to statistics content and this was not connected to the variable or
transformations content. Since the main goal of this study was to examine opportunities for
teacher learning and introduction to variable and geometric transformations served as a context
for this examination, I took this small difference to be acceptable. Each curriculum defined and
attended to the concept of variable and defined and attended to the basic transformations (i.e.,
translations, reflections, and rotations) in these units.
For each curriculum, the transformations units had more sentences in the teachers‘ guides
than the variable unit. This may make sense as all transformations units had more lessons and
pages in the teachers‘ guides than corresponding variable units. It is important to keep in mind
that the number of pages in the teachers' guides is related to the number of student lessons in
each of these units. In other words, if there are more student lessons in the unit with geometric
transformations then it would make sense that there would be more pages in the teachers‘ guides
for that unit and therefore, more sentences. I mention this because throughout this dissertation I
will often use percentages. Therefore, knowing that the number of sentences in the units differed
is important.
Other differences, such as the total number of pages and embeddedness of supports may
have implications for the amount of support available in each unit. The transformations units all
had more pages and sentences than the variable units. Therefore, one would expect to find more
support in the transformations units. In addition, the embeddedness of supports in CMP might
also impact the amount of support since each of these pages contained more space for text
intended for the teacher.

66

In the next two chapters, I discuss the results for content and expression and I return to
the issues of structure and how the differences in structure may impact the content and its
expression.

67

CHAPTER 5: RESULTS – CONTENT SUPPORTS AND EXPRESSION

My aim in this study was to describe teachers‘ opportunities to learn mathematics subject
matter, pedagogy, and to learn about mathematics curriculum from middle school mathematics
curriculum materials. Characterizing the content and expression of the text in the teachers‘
guides provided me with insight into not only what was in the teachers‘ guide, but also how text
was expressed to teachers in the written materials. Because my coding scheme generated a
diverse set of numerical results (frequencies of different kinds of content supports and
dimensions of their expression), I open this chapter by stating my clearest and most significant
findings. The balance of the chapter provides support for each of these major claims. Although I
could present the frequencies of all content supports and dimensions of their expression, this did
not seem warranted since frequencies were often quite low. In addition, reporting such a diverse
set of frequencies did not seem helpful in providing a description of the opportunities available
in these curricula.
First and foremost, my content coding revealed very little Rationale Guidance,
accounting for no more than 6% of the support in any curricular series. This was true for both the
variables and transformations units. Second, the supports I found most often addressed
Pedagogical Content Knowledge for Practices; these accounted for over 37% of the support in
three of the four curricula (CMP, Glencoe, and UCSMP). MiC, on the other hand, focused more
heavily on Curricular Knowledge and had an almost even split between this knowledge and PCK
for Topics or PCK for Practices. Subject Matter Content Knowledge was attended to least across
all curricula and units. When it was the focus, two curricula (CMP and MiC) located these
supports at the Unit level of organization. Finally, although most of the content support was

68

located at the Lesson level, a substantial amount of support, particularly for CMP and MiC, was
located at the Unit level. This may be a problem because previous research has shown that
teachers are more likely to attend to supports located the Lesson level.
In addition, there were some noticeable content support differences between curricula and
units. As previously stated, MiC attended more to Curricular Knowledge and PCK for Topics,
whereas CMP, Glencoe, and UCMP focused most prominently on PCK for Practices. Within the
variables units, CMP attended more to Enactment Guidance for Representations, than other
curricula, providing support for teachers to help students interpret, create, and use tables, graphs,
equations, and verbal representations. Glencoe attended more to Enactment Guidance for
Curricular Features by providing information about the various features of their curriculum,
including post-it notes that described what teachers would find in features such as Tips for New
Teachers and Error Alert!. MiC and UCSMP focused more on Enactment Guidance for
Curricular Overviews by providing descriptions of what each of their lessons would entail, such
as
―Students find out what triangle, when rotated, can make an octagon and what triangle
can make a hexagon. They investigate whether translation, rotation, or reflection can be
used to make a 10-gon. Then they investigate whether or not they would get the same
result if they started with a triangle that is not isosceles.(MiC, Triangles and Beyond, p.
50T)
In their transformations units, CMP and UCSMP attended more to Subject Matter
Knowledge than other curricula, whereas UCMSP attended more to Enactment Guidance for
Terminology. Although all curricula contained a large percentage of support devoted to
Enactment Guidance for Questions (suggested answers provided), this support was more

69

prominent in CMP. In addition, CMP focused more on Enactment Guidance for Proof and
Reasoning. As they did in their variables unit, Glencoe devoted much more support to Enactment
Guidance for Engaging Students in Potentially Problematic Experiences in their transformations
units, and MiC included a higher percentage of Enactment Guidance for Curricular Overviews.
Differences in content supports were also noticeable between units for all or most
curricula. The variables units frequently included Enactment Guidance for Representations,
whereas the transformations units included Enactment Guidance for Tools, and Justification,
Reasoning and Proof Approaches. Also, the Glencoe variables unit frequently included
Enactment Guidance for Curricular Features; however this was not true for their
transformations unit.
With respect to the linguistic expression, I first found that the first person plural pronoun
―we‖ was used only by CMP and UCSMP, whereas the second-person pronoun ―you‖ was used
in all curricular series and was by far the most common pronoun. The use of these pronouns is
important since they involve curriculum authors and teachers in the text. Second, when I looked
more carefully at the use of ―you,‖ I found that the most common ―you‖-forms were imperatives
(commands to do something), a form in which the reader is implicitly implicated, and ―you‖ +
modal verb, a form in which the modal verb functions to provide the reader some freedom to
choose what to do. Imperatives were much more common than all other ―you‖-forms, and at
least 7% of the sentences in each curricular series were in the imperative form. Finally, a variety
of modal verbs were used, but modal verbs ―may‖ and ―might,‖ verbs that carry a low level of
obligation (Halliday, 1985), were the most common across all units and curricula. Modal verbs
that carry a low level of obligation provide teachers with more freedom to choose to do (or not to
do) what curriculum authors stated.

70

There were noticeable differences in the use of pronouns and ―you‖-forms between
curricula, but not between units. In addition to the appearance of we‖ only in CMP and UCSMP,
Glencoe used imperatives much more frequently in their transformations unit than other
curricula. Although the use of imperatives was more common in the transformations units for all
curricula except for CMP, this difference was small. Finally, although the most common modals
were similar across curriculum materials with ―may‖ and ―might‖ being the most pervasive,
CMP and MiC, were the only curricula to use the high modal verb ―need.‖
In the next sections I provide descriptive statistics to substantiate these results. I begin
first with the results related to content, including the knowledge addressed, the types of supports
and the location of these supports. I then turn to issues of expression. I present the results related
to pronouns and then ―you‖-forms. I discuss any differences I found between curricula or units
where appropriate.
Content Supports
In this section I describe the findings related to content supports. I begin by describing
the types of guidance provided, the knowledge addressed, and where the content support was
located within the teachers‘ guides. I then discuss differences in content supports between
curricula and units.
Types of Content Support. For all curricula and units, content supports more often
provided Enactment Guidance, or described what to teach and how to teach it. There was very
little Rationale Guidance. Table 13 shows the percentages of each type of guidance. Since I did
not code Subject Matter Content Knowledge as either Enactment or Rationale Guidance these
percentages exclude these subject matter knowledge codes.

71

Table 13. Percentages of Enactment and Rationale Guidance by Unit and Curriculum
Variable
Curricula
CMP
Glencoe
MiC
UCSMP

Enactment
Guidance
97
99
94
94

Transformations
Enactment
Rationale
Guidance
Guidance
97
3
99
1
98
2
97
3

Rationale
Guidance
3
1
6
6

Enactment Guidance was much more prevalent than Rationale Guidance, which occurred
for no more than 6% of any curriculum This result is consistent with research in science
education (Beyer et. al, 2009), and even more startling because 20% of supports in their Biology
textbooks were found to provided Rationale Guidance. CMP and Glencoe were consistent in
their distribution across the two units. MiC and UCSMP included a higher percentage of
Rationale Guidance in their variable unit than in their transformations units, however this
difference was modest (4% and 3% respectively). In all four curricula most of the Rationale
Guidance occurred in the PCK for Topics category. The most frequent specific supports were
Rationale Guidance for Experiences, which was common for both the variable and the
transformations units, Representations which was more prevalent in the variable units, and
Tools, which was more prevalent in the transformations units. Also, common in other knowledge
categories was Rationale Guidance for Questions, Philosophy, and Storyline.
Frequency of Supports by Knowledge Type. Subject Matter Knowledge, Pedagogical
Content Knowledge for Topics and Practices, and Curricular Knowledge were addressed by all
four curricula in both units. Figure 5 shows the percentages of support for the four types of
knowledge for both the variables and transformations units in each curriculum.

72

Figure 5. Percentages of Content Supports by Unit and Curriculum

Subject Matter Knowledge
Pedagogical Content Knowledge for Topics
Pedagogical Content Knowledge for Practices
Curricular Knowledge
70%
60%
50%
40%
30%

20%
10%
0%
CMP

Glencoe

MiC

UCSMP

Variable

CMP

Glencoe

MiC

UCSMP

Transformations

The most prevalent type of content support for three of four curricula was PCK for
Practices, accounting for at least 37% of all support in CMP, Glencoe, and UCSMP. These
supports included mostly those designed to help teachers engage students in mathematical
practices such as questioning, reasoning and proving, and using and terminology. MiC, on the
other hand, split its attention evenly between PCK for Practices or PCK for Topics and
Curricular Knowledge. The most prevalent Curricular Knowledge supports included those
related to developing an understanding of the curricular features, and storyline, and provided a
curricular overview, or description of what a particular lesson was about, which accounted for
21% of all content supports in MiC for both units. The least prevalent content support across all
materials was in the domain of Subject Matter Knowledge, which accounted for no more than
14% for any curriculum, although in UCSMP's transformations unit support for Subject Matter
73

Knowledge was quite close to the percentage of support for PCK for Topics. It should be noted
that this category had only one element (Content Knowledge) and therefore, it might make sense
that it accounted for less support than the other categories which included far more elements. In
future sections I will discuss the frequency of particular types of support (including Content
Knowledge).
Location of Content Supports. Most content supports were located at the Lesson Level.
However, particularly for CMP and MiC, a significant amount of support, sometimes nearly 50%
of the total content support, was located at the Unit level. Table 14 shows the percentages of
supports at the Unit, Section, and Lesson levels for each curriculum.
Table 14. Percentages of Content Supports by Location
Curricula
CMP
Glencoe
MiC
UCSMP

Introduction to Variable
Unit
Section
Lesson
14
28
58
15
N/A
85
24
14
62
19
N/A
81

Geometric Transformations
Unit
Section
Lesson
48
5
47
12
N/A
88
16
11
72
18
N/A
82

For the variables units each curriculum included most of their supports at the Lesson
Level. For the transformations units three of four curricula included a majority of their support at
the Lesson Level, accounting for more than 70% of the support in Glencoe, MiC, and UCSMP.
CMP had a relatively even split between support that appeared at the Unit and Lesson Levels,
with 48% and 47% respectively. These results for CMP‘s transformations unit are important
because research indicates that the location of educative supports impacts whether teachers use
supports. For example, studies that investigated teachers‘ use of educative curriculum
(Schneider, Krajcik, & Blumenfeld, 2005; Schneider & Krajcik, 2002; Schneider, 2006)
indicated that teachers used and learned from the support located at the lesson-level, rather than
support located in other sections of the textbook and other books.
74

A deeper look into the location of supports indicated that supports for Curricular
Knowledge often appeared at the Unit Level. Both CMP and MiC units and the UCSMP
transformation unit all contained more Curricular Knowledge support at the Unit Level. Since
many of these supports, such as Rationale for Curricular Philosophy and Enactment and
Rationale Guidance for the Curricular Storyline involved content from the entire unit, finding
this support at the Unit Level made sense. Supports for other knowledge types always appeared
more often at the Lesson Level, except in CMP where in addition to Curricular Knowledge,
Subject Matter Knowledge appeared more often at the Unit Level and PCK for Practices either
appeared more often at the Unit Level or was split equally between the Section and Lesson
Levels. It is important to note that support may also appear at the curriculum level (e.g.,
Enactment guides, teachers' guides front matter), however, since this study focused on two units
in each curriculum, I do not discuss the support at the curricular level, but acknowledge that this
is another location, further removed from the lesson level, for potential teacher support.
Some differences within units across curricula. Although, generally, the most
prevalent content supports were the same for the units across curriculum materials, with
Enactment Guidance for Using Questions (suggested answers provided) one of the most
common supports, there were also some noticeable differences. First, I describe the differences
within units, starting with the variables units. I then describe differences in supports between the
two units.
Differences in variables units by curriculum. CMP included a much higher percentage
of support for Enactment Guidance for Representations in their variables unit than other
curriculum materials, accounting for 29% of support. In this CMP unit, one of the goals was to
"Observe relationships between two quantitative variables as shown in a table, graph, or equation

75

and describe how the relationship can be seen in each of the other forms of representation"
(CMP, Variables and Patterns, p. 2). Glencoe, on the other hand, focused more than other
materials on Enactment Guidance for Potentially Problematic Experiences and Curricular
Features in their variable unit. Glencoe included at the beginning of each lesson a section
entitled "Options for Differentiated Instruction." Although, this section included alternative
options for students at or above grade level, most of the supports (n = 73) were geared towards
helping teachers engage struggling students or students with special needs in experiences that
might be problematic for them. Glencoe also included more Enactment Guidance for Curricular
Features in their variable unit than other curriculum materials. I discuss Glencoe's' inclusion of
this support more in the section where I discuss differences across units. MiC, and to some extent
UCSMP, included substantial percentages of Enactment Guidance for Curricular Overviews,
accounting for 21% and 14%, respectively, of their total support.
Differences in transformations units by curriculum. In the transformations unit, there
were differences in the attention to Subject Matter Knowledge in the curriculum materials.
UCSMP and CMP both devoted over 17% of support to Subject Matter Knowledge. In addition,
UCSMP devoted 15% to Enactment Guidance for Terminology, more than other curriculum
materials, and often this support appeared on the same page as support for Subject Matter
Knowledge. Of their 77 instances of Enactment Guidance for Terminology, 46 appeared on the
same page and 23 on the page before or after Subject Matter Knowledge support. The close
proximity of these supports suggest that UCSMP authors may see the use of appropriate
terminology as related to learning subject matter. CMP contained a larger percentage of
Enactment Guidance for Questions (suggested answers not provided) and Enactment Guidance
for Proof and Reasoning than other curriculum materials. CMP included more suggested

76

questions and instances in which teachers were asked to have their students justify and explain
their reasoning and begin to establish the notion of proof more than other curriculum materials.
Finally, similar to the variables unit, Glencoe devoted much more support in their
transformations units than other curricula, to Enactment Guidance for Engaging Students in
Potentially Problematic Experiences than other curricula, accounting for 17% of its support, and
MiC included a higher percentage of Enactment Guidance for Curricular Overviews, accounting
for 21% of its total support.
Some overall differences across units. For the variables units, 13 content supports
occurred more than 10 times in at least three units, whereas 14 content supports occurred
frequently in at least three transformations units. The variables units frequently included
Enactment Guidance for Representations; however this was not frequently included in the
transformation unit for any curriculum. Instead, transformations units included Enactment
Guidance for Tools, and Justification, Reasoning and Proof Approaches. Although learning to
use representations, tools and engage in justification, proof, and reasoning are integral to the
study of mathematics and require support for teachers in any content domain, it may seem
appropriate that these supports were more prevalent in their respective units. In the variables
units, particularly CMP‘s, which devoted 29% of its support to guidance for using
representations, there was a larger focus than in the transformations unit on using different
representations (i.e., situation, table, graph, equation). This is not to say that Enactment
Guidance for Representations was completely absent from transformations units. In these units
teachers were given guidance for using or having students use representations such as sketches or
three-dimensional representations. The transformations units involved constructing and
transforming geometric shapes and investigating symmetry, therefore incurring the use of tools

77

such as rulers and compasses. In addition, these units, particularly CMP and UCSMP introduced
the ideas of using transformations to prove relationships such as similarity and congruence and
therefore provided more support for teachers to engage their students in reasoning and proof.
Another difference between units was specific to Glencoe. As I mentioned earlier,
Glencoe's variable unit had a high percentage of Enactment Guidance for Curricular Features;
however this was not true for their transformations unit. A possible explanation for this is that
the variable unit was the first chapter in the teacher's guide, whereas transformations was
covered at the end of the book in chapter 10. The Enactment Guidance for Curricular Features
supports, provided in the form of post-it notes, alerted teachers to the features of the curriculum
materials, such as the following: "Web addresses, or URLs, are provided to online assets such as
Personal Tutor, Extra Examples, Self-Check Quizzes, and Concepts in Motion" (Glencoe,
Algebra: Number Patterns and Functions, p. 37). A cursory examination of other units indicated
that these post-it note supports located in chapter 1 were absent in other units, not just the
transformations unit. The authors may expect teachers to remember the information provided in
Chapter 1 while reading other chapters.
Aspects of Voice
In this section, I describe the results regarding aspects of voice. I begin with findings
related to personal pronoun use in the four curricula, particularly focusing on the use of "we" and
"you." I begin with personal pronouns because the use of these words is one of the most obvious
ways to see how curriculum authors are speaking to teachers, as personal pronouns explicitly
express interpersonal relationships. I follow this with a more detailed look at how the pronoun
"you" is used by discussing "you"-forms, specifically the use of imperatives and modal verbs, as

78

these were the most prevalent forms and also illuminate the roles that are being constructed
between curriculum authors and teachers.
Personal Pronouns. The first linguistic form I examined was personal pronouns because
these are used to express relationships between authors of a text and their readers. First-person
pronouns such as ―I‖ and ―we‖ indicate the authors; direct involvement in the activity of the text
(Morgan, 1996). According to Herbel-Eisenmann (2007) the use of the pronoun ―we‖ could
indicate the authors speaking with the authority of the discipline of mathematics, or it may be
used to involve the readers in the mathematics. Researchers such as Pimm (1987) and Rowland
(1999) have alluded to the ideological aspects embedded in the use of we, such as how using we
assumes the readers agreement with the author. The second-person pronoun ―you‖ directly
involves the reader, and is of particular interest in my study because the readers of curriculum
materials are the teachers using the materials.
In addition, the general absence of first-person pronouns, according to Morgan (1998),
not only affects the picture of the nature of mathematics, but also has the potential to distance the
author from the reader.
Personal pronouns were used by all four curricular series. Table 15 shows the frequencies
of each of these pronouns by unit and curriculum.
Table 15. Frequencies of Personal Pronouns by Unit and Curriculum
Curricula
CMP
Glencoe
MiC
UCSMP

I
0
0
0
0

Variables
We
38
0
0
20

You
52
19
61
33

I
0
0
0
0

79

Transformations
We
102
0
3
17

You
42
25
129
38

As expected, I did not find any instances of the pronoun "I." As these texts were not
written by one person, this makes sense. Unlike the pronoun "I" which indicates a single voice,
the pronoun "we" indicates multiple voices. This pronoun seemed more likely to be found;
however, I only found "we" in CMP and UCSMP. "We" was completely absent from Glencoe
and almost completely absent from MiC, which had only three instances in the transformations
unit; two of which occurred within sample solutions.
By far the most common pronoun was the second-person pronoun "you," which occurred
at least 19 times in each unit and was very frequent in MiC's transformations unit which included
129 instances of "you." As a reminder, these teachers' guides did not contain the same number of
sentences. It would make sense to have more pronouns in teachers‘ guides where there were
more sentences.
Locations of personal pronouns. Generally, the personal pronouns ―we‖ and ―you‖ were
used in a variety of sections. The pronoun ―we‖ was used by CMP and UCMP within the
narrative sections that described how to enact lessons, such as "Notes on Lesson" in UCSMP,
and in CMP within the Launch, Explore, and Summarize sections, or within sections devoted
most prominently to mathematical content support such as "Background" in UCSMP and
"Mathematics of the Unit" in CMP. It was common for "we" to be used repeatedly in parts of the
text where explicit steps were given for solving a problem, particularly in CMP. In the next
section, I describe the uses and locations of ―we‖ in more detail.
Where the pronoun ―we‖ was found frequently in sections in which mathematical subject
matter was presented in CMP and UCSMP, the pronoun ―you‖ was not very frequent. CMP did
use the pronoun ―you‖ in sections devoted solely to the mathematical subject matter, such as
Mathematics in this Unit, but more often in the sections for the teacher that were devoted to the

80

enacting of individual lessons (called Problems), such as Launch, and Summarize, and in the
Possible Answers to the Mathematical Reflections which occurred at the end of the unit. This is
not to say that ―you‖ was not used when mathematical subject matter was discussed, but it was
just not frequent in sections devoted solely to discussions of subject matter. Similarly, for the
other curriculum materials ―you‖ was used most frequently in sections that were devoted to
helping teachers enact lessons.
Uses of we. "We" can be a grammatical device for bringing the authors directly into the
text. However, Pimm (1987) pointed out that often, particularly in classroom conversations, the
referent for "we" is not clear or explicit. The question he posed is ―To what community is the
teacher appealing when using the word we?‖ (p. 65). The same question is appropriate when
examining written text. When the referent is unclear, "we" may be used as a way of asserting
one‘s authority as a speaker for the discipline of mathematics (Herbel-Eisenmann, 2007) or
including oneself in the mathematics classroom community. In either case, "we" may "have to do
with both power and dominance, and attempts to enrol the (at least tacit) acquiescence of the
reader in what is being expressed" (Pimm, 1987, p. 68). These different uses and functions make
it necessary to look more closely at the instances of "we." Table 16 shows the frequencies of
―we‖ in which I determined that the referent was clear and those for which the referent was
unclear in the curriculum materials that contained this personal pronoun, CMP and UCSMP. In
the instances in which the referent was clear, "we" was used to refer to the group of authors of
the curriculum materials, or at least the authors of that particular teacher's guide (e.g., ―We start
with tables because many students are more comfortable working with tables than with graphs‖
(CMP, Variables and patterns, p. 87).). Unclear instances were uses of ―we‖ for which I could
not determine the referent. These instances went beyond the particular curriculum materials and

81

decisions made solely by the authors of these teachers‘ guides, such as marking mathematical
definitions or working through mathematical deductions It was difficult to tell who the ―we‖ was
in these cases.
Table 16. Uses of ―We‖ by Unit and Curriculum
Curricula
CMP
UCSMP

Function
Clear referent
Unclear referent
Clear referent
Unclear referent

Variable
9
25
20
0

Transformations
14
89
10
7

UCSMP contained more instances of clear than unclear referents. There were no unclear
referents in their variable unit, and fewer unclear referents in their transformations unit than clear
referents. The clear instances included the authors directly in the text and served three functions:
a) as a way of making suggestions or recommendations for teachers (e.g., "We recommend
that...," "We suggest that...," "We encourage..."), b) as a way of stating assumptions or choices
they made without rationale (e.g., "We introduce...," "We ask students...," "We assume..."), and
c) as a way of stating assumptions or explaining choices with rationale ("We start with tables
because...," "We have purposely chosen...because,"). CMP often chose to mark these instances
with text that indicated that they were providing special information for the teachers (e.g.,
"Note," "For the teacher").
I also found instances in which the referent for "we" was unclear. These types were most
often located at the Lesson or Unit Level when descriptions of solutions or mathematical subject
matter content were provided. The instances were of two grammatical forms, the first being
instances in which "we" appeared in a statement and the second in which "we" appeared in a
question. Examples of the former, included phrases such as, "We say 'non-trivial'..." (UCSMP,
Some Important Geometry Ideas, p. 375) or "Thus we know that the gap triangle is congruent to
the original triangle?" (CMP, Kaleidoscopes, Hubcaps, and Mirrors, p. 105). Examples of the
82

latter included questions such as, "Can we be sure that last transformation exactly filled the
gap?" (CMP, Kaleidoscopes, Hubcaps, and Mirrors, p. 105) or "Which variable should we put on
each axis so we can best see the “story” the graph tells? (CMP, Variables and Patterns, p. 21). I
found questions such as these only in CMP. Although these appeared throughout both CMP units
in various locations, I saw a combination of the two forms, "we" in questions and statements,
more often in the excerpts in which there were explicit steps given for solving a problem.
"You"-Forms. "You"-forms used in the teachers‘ guides indicated how curriculum
2

authors included teachers in the construction of their mathematics teaching . "You" was used in
two ways, implicitly in the imperative form and explicitly with the personal pronoun "you."
Table 17 shows the frequencies of explicit and implicit ―you‖-forms.
Table 17. Frequencies of Implicit and Explicit Use of "You" by Unit and Curriculum
Curricula
CMP
Glencoe
MiC
UCSMP

Variables
Implicit
Explicit (other
(imperative)
"you"-forms)
250
52
188
19
117
61
72
33

Transformations
Implicit
Explicit (other
(imperative)
"you"-forms)
154
42
150
25
115
129
51
38

Each curriculum used the imperative form much more frequently than explicitly using the
personal pronoun "you" when addressing the teacher, with the exception of MiC. See Figure 6
for the percentage of each form by unit and curriculum.

2

As a reminder, I did not analyze student pages of the textbook, even when these pages were
found in the teachers' guides. Therefore, when I report details about particular words used, these
details pertain only to the use of these words in the teachers' guides. These words may also have
been used on student pages of the text. In addition, I did not include any text that was a copy of
the student textbook within the narrative for the teacher.
83

Figure 6. Percentages of Implicit and Explicit "You"-forms

Implicit

Explicit

100%
80%
60%
40%
20%
0%
CMP

Glencoe

MiC

UCSMP

CMP

Glencoe

Variables

MiC

UCSMP

Tranformations

In all units and curricula more than 57% of the uses of "you" were expressed as
imperatives, except for MiC's transformations unit which had only 47% of "you"-forms
expressed as imperatives. In their transformations unit when the teacher was addressed more
often the word "you" was explicitly used. Glencoe had over 91% and 86% in the imperative form
in their variables and transformations units, respectively. In the next two sections I describe the
use of imperatives and explicit "you"-forms in more detail.
Imperative Form. An imperative expresses a command. A command positions the
addressee (i.e., the controlled) as the one to carry out the service and the speaker (i.e., the
controller) as the authority in the situation. Grammatically, this form does not require a subject,
but the pronoun "you" is implied. Figure 7 indicates the percentages of total sentences that were
in the imperative form.

84

Figure 7. Percentage of Sentences in the Imperative Form by Unit and Curriculum

Variable

Transformations

30%
25%
20%
15%
10%
5%
0%
CMP

Glencoe

MiC

UCSMP

Each curriculum had at least 7% of the sentences in each unit in the imperative from. The
Glencoe transformations unit had by far the greatest proportion of sentences in the imperative
form with 25%. Each curriculum had a greater percentage of sentences in the imperative from in
their transformations unit, except in CMP where 5% more of the sentences in the variable unit
were in the imperative form than in the transformation unit.
Types of imperatives. The types of imperatives that were used included those that directed
teachers to perform certain actions themselves (e.g., "Ask...," "Discuss...," "Remind..."), and
those that directed students through teachers (e.g., "Have students share...," "Have students
create...," "Have students discuss...") or to ensure that students engaged in particular actions or
processes (e.g., "Make sure students understand...," "Be sure students can explain..."). Table 18
illustrates the percentages of imperatives that directed teachers and students through teachers by
unit and curriculum.

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Table 18. Percentages of Imperatives Directing Teachers and Students by Unit and Curriculum
Variables
Curricula

Directed teacher

CMP

Transformations
Directed teacher

74

Directed
students through
the teacher
26

88

Directed
students through
the teacher
12

Glencoe

71

29

76

24

MiC

49

51

82

18

UCSMP

66

34

86

14

Only one unit, MiC's variable unit, contained more imperatives that directed students
through the teacher than directed the teacher to perform actions themselves. It is necessary to
look more closely at the specific imperatives used to see exactly what teachers were being
commanded to do. Table 19 shows the frequencies of the common imperatives (those with a
frequency of at least five), used in each of the units by curriculum. See Appendix B for the list
of all imperatives.
Table 19. Commonly Used Imperatives by Unit and Curriculum
Curricula

CMP

Glencoe

Variable
Imperative
Ask
Have students share
Use
Encourage
Discuss
Tell
Let
Explain
Remind
Choose
Use
Ask
Remind
Tell
Create
Write
Explain

Frequency
46
15
15
12
10
10
9
7
6
5
42
29
11
9
8
6
5
86

Transformations
Imperative
Frequency
Ask
30
Encourage
11
Have students work
9
Use
8
Have students share
7
Remind
7
Call
6
Discuss
5
Arrange
5
Help
5
Use
33
Ask
17
Remind
12
Have students write
10
Make
9
Encourage
7
Check
5

Table 19 (cont‘d)
Suggest

MiC

UCSMP

5

Ask
Have students share
Point out
Encourage
Provide
Be sure students
understand
Discuss
Ask
Give

Draw
Point out
Ask
Discuss
Encourage
Remind
Add
Have students discuss

16
10
8
7
7
6

Ask
Point out
Emphasize

5
15
7

5
5
26
23
12
6
6
5

13
7
5

CMP, Glencoe, and MiC had a greater variety of imperatives that appeared at least five
times in both units than UCMP. UCSMP‘s use is less varied as they used only two or three
different imperatives (i.e, ―Ask,‖ ―Give,‖ ―Point out,‖ ―Emphasize‖) more than five times. Most
of the commonly used imperatives directed teachers to perform certain actions; however, all
curricula, except UCMP, included imperatives that directed students through teachers at least
five times in one of their units. I describe these two types of imperatives in the next two sections.
Imperatives that directed teachers. According to Rotmann (1988), exclusive imperatives,
such as tell, remind, and give, position the reader as a ―scribbler,‖ or someone who must perform
some action, whereas inclusive imperatives, such as consider, think, and explain, address the
reader as a ―thinker.‖ The latter types of imperatives establish a ―commonality between speaker
and hearer‖ (p. 9). Although one is not positioned as only a "scribbler" or a "thinker," by
choosing certain words, one can foreground these roles. Table 20 shows the percentage of
inclusive and exclusive imperatives by unit and curriculum.

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Table 20. Percentages of Inclusive and Exclusive Common Imperatives by Unit and Curriculum
Curricula
CMP
Glencoe
MiC
UCSMP

Variable
Inclusive
Exclusive
28
72
31
69
43
57
0
100

Transformations
Inclusive
Exclusive
27
73
17
83
48
52
0
100

The commonly used imperatives were most frequently exclusive imperatives rather than
inclusive imperatives. The most common exclusive imperative across all curriculum materials
was "Ask," except for in the Glencoe transformations unit in which "Use" was the most common,
followed by "Ask." These two imperatives accounted for at least 35% of the commonly used
imperatives in each curriculum. MiC and UCMP did not frequently use the imperative "Use,"
instead MiC's second and third most common imperatives were "Discuss" and "Encourage"
whereas UCMP's included "Give," "Point Out" and "Emphasize."
Imperatives that directed students through teachers. The most common imperatives that
directed students through the teacher included "Have students share," "Have students work,"
"Have students write," "Have students discuss," and "Be sure students understand." Although
imperatives of this type still directed teachers, these imperatives positioned the teacher
differently than those that directly commanded teachers. Rather than being positioned as
controlled, in these instances, teachers were positioned as "instigators" (M. Schleppegrell,
personal communication, October 28, 2011). This grammatical form assumes that a teacher has
power over his or her students to make students do what he or she wants. Yet, since these
instances were in the imperative form, and therefore commanded teachers to direct students to do
certain things, a more apt interpretation would be that this positioned the teacher as an instigator
or agent to make students do what the curriculum authors want. So, while these instances may
have placed teachers in a more powerful position than other imperatives, this positioning was
88

quite subtle and narrow, as teacher may have power in these instances, but only over their
students (not themselves).There was no indication in these instances that teachers had a choice as
to what to make students do or even whether to make students perform these actions or not.
Explicit "You"- Forms. I now turn to describe the explicit "you"-forms I found in each
curriculum. Instances in which "you" was explicitly used fell into the following four forms: a)
verb + "you", such as "... tell you" or "...give you", b) modal verb + "you," such as "may need
you..." or "might want you..." c) "you" + verb, such as ―You have labeled...‖ or ―You fill...,‖ and
d) "You" + modal verb, such as ―You might want...‖ or ―You should explain....‖ See Tables 21
and 22 for the frequencies of these you-forms in each unit by curriculum.
Table 21. Frequencies of "You"-forms in the Variable Units by Curriculum
Curricula

You + verb

CMP
Glencoe
MiC
UCSMP

6
0
5
10

You + modal
verb
55
19
57
23

Verb + you
2
0
2
0

Modal verb +
you
0
0
1
0

Table 22. Frequencies of "You"-forms in the Transformations Units by Curriculum
Curricula

You + verb

CMP
Glencoe
MiC
UCSMP

2
3
3
3

You + modal
verb
47
23
126
35

Verb + you
0
0
0
1

Modal verb +
you
0
0
0
0

The most pervasive "you"-form was "you" + modal verb. Other forms were quite
uncommon across both units in all curriculum materials, although compared to other forms
UCSMP did have a significant number of "you" +verb forms in their variable unit.
"You"+modal verb. Modal verbs are grammatical tools that allow the authors to provide
readers with more freedom . Different modal verbs produce different levels of obligation or

89

probability. According to Martin & Rose (2008), modal verbs can be placed on a scale that
describes "how obliged" you are to act or "how probable" a statement is. See Figure 8.
Figure 8. Scales for Negotiating Demands for Services and Giving Information
(Martin & Rose, 2008, p. 53)
Demands for services scale
do it

Giving information scale

positive

it is

you must do it

it must be

you should do it

it should be

you could do it

positive

it might be

don't do it

negative

it isn't

negative

Demands for services can be negotiated on the left scale. This describes how likely one is
to act. On the poles "do it" and "don't do it" indicate no alternatives, or complete obligation to do
something or not; however, as one moves away from the poles there are varying levels of
obligation. The same is true on the information scale. According to Halliday (1985), these
varying levels of obligation or truth can be described as "low", "median", or "high." See Table 23
for the classification of modal operators. For example, ―You must explain‖ is more obligatory
than ―You should explain,‖ which is more obligatory than ―You may explain,‖ which is more
obligatory than the imperative ―Explain.‖
Table 23. Finite Verbal Operators (Halliday, 1985, p. 116)
Polarity
positive
negative

Low
can, may, could,
might
needn't, doesn't/didn't
+ need to, have to

Levels of Obligations
Median
will, would, should,
is/was to
won't, wouldn't
shouldn't, (isn't/wasn't
to)

90

High
must, ought to, need,
has/had to
mustn't, oughtn't to,
can't, couldn't (mayn't,
mightn't, hasn't/hadn't
to)

Although each curriculum used modal verbs, there was not much variety. Most of the
modal verbs fell into Halliday‘s (1985) low category. Table 24 indicates the frequencies of the
modal verbs that appeared in each unit by curriculum.

might

Should

will

need

Total

Transformations

may

Variables

Curricula

could

Units

can

Table 24. Frequencies of Modal Verbs by Unit and Curriculum

CMP
Glencoe
MiC
UCSMP
Total
CMP
Glencoe
MiC
UCSMP
Total

9
2
3
1
15
3
3
6
2
14

0
0
0
1
1
2
0
16
2
20

25
16
44
4
89
15
18
83
4
120

6
1
6
15
28
19
2
20
19
60

1
0
0
2
3
0
0
0
5
5

3
0
0
0
3
1
0
1
3
5

11
0
4
0
15
7
0
0
0
7

55
19
57
23
154
47
23
126
35
231

The most pervasive modal verbs used by each curriculum were the modal verbs "may"
and "might." "May" was always used more frequently than "might," except in UCMP, in which
"might" was over three times as frequent as "may" in both units and in the CMP transformations
unit, in which it was only slightly more frequent. Together these two modal verbs accounted for
at least 56% of the modal verbs used in each curriculum, and for Glencoe and MiC accounted for
over 80%. CMP had higher proportions of the modal ―need‖ than other curricula, whereas
UCMP had higher proportions of ―should.‖ In addition to the modals ―may‖ and ―might‖ other
low modals such as ―can‖ and ―could‖ and high modals, such as ―should‖ and ―need‖ were used,
but in lower frequencies. Figure 9 indicates the percentage of low, median, and high modal verbs
used in each unit by each curriculum.

91

Figure 9. Percentages of "Low," "Median," and "High" Modal Verbs Used

Low

Median

High

100%
80%
60%
40%

20%
0%
CMP

Glencoe

MiC

UCSMP

Variables

CMP

Glencoe

MiC

UCSMP

Transformations

In each curriculum low modal verbs (i.e., can, could, may, might), or verbs that required
the lowest level of obligation or probability of occurrence, were most common. In Glencoe, these
were the only modals used. All other curriculum materials included median modal verbs (i.e.,
will, should), whereas CMP was the only curriculum to use a high modal verb (i.e., need). Since
modal verbs were quite common, it seemed necessary to examine the verbs associated with these
modals operators, particularly for the most common modals, ―may‖ and ―might.‖ The most
commonly used verbs with the modal ―may‖ were ―want‖ and ―wish,‖ (e.g., ―You may want
students to explain…,‖ ‖ You may want to discuss…,‖ ―You may wish to ask…‖), accounting
for at least 25% of all verbs used. See Table 25. ―Might,‖ on the other hand, was most often used
with ―ask,‖ (e.g., ―You might ask them to …,‖ ―You might ask some questions such as…,‖ ―You
might ask if a triangle…‖), however, not nearly as often as ―may‖ was paired with ―want‖ or
―wish.‖
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Table 25. Percentages of the Verbs "Want" and "Wish" Used with the Modal Verb "May" by
Unit and Curriculum
Curricula
CMP
Glencoe
MiC
UCSMP

Variables
want
wish
26
29
0
88
24
0
21
32

Transformations
want
wish
41
0
5
70
42
7
9
9

I discuss possible issues related to using the words "want" and "wish" with modal verbs
in the discussion.

93

CHAPTER 6: DISCUSSION

In this chapter, I return to the focus of my study, teachers' opportunities to learn from
middle school mathematics curriculum materials. Specifically, I address my research questions.
As a reminder, I aimed to answer the following questions:
1. What is the relative frequency of educative content supports in middle school
mathematics curriculum materials for teachers‘ Subject Matter Content Knowledge,
Pedagogical Content Knowledge for Mathematics Topics, Pedagogical Content
Knowledge for Mathematical Practices, and Mathematics Curricular Knowledge and
where are these supports located?
2. How do middle school mathematics curriculum materials speak to teachers (i.e., what
are some of the language choices they make) through the written text in the teachers‘
guides?
3. How does opportunity to learn (content and aspects of voice) differ for introduction to
variable and geometric transformations?
In the sections that follow, I address each of these research questions, followed by what
these results indicated about teachers' opportunities to learn from middle school curriculum
materials. I follow this with a discussion of the contributions of my study to the field and
implications for curriculum development, teacher education, and research. I then discuss
limitations of the current study and finally, I end with future directions for research.
Relative Frequency of Supports and their Location
In this section I remind the reader of the overall relative frequency of content supports
and discuss possible reasons for these frequencies. In addition, I discuss differences in the four
curricular series that my analysis illuminated.
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As described in my results section, the most frequent content supports were Pedagogical
Content Knowledge for Practices and Curricular Knowledge. See Table 26 for percentages for
each knowledge type by unit and curriculum.
Table 26. Percentages of Each Knowledge Type by Curriculum
Curricula

SMK

CMP
Glencoe
MiC
UCSMP

4
4
5
7

Variable
PCKPCKTopics Practices
38
49
29
37
32
27
20
51

CK

SMK

10
30
37
22

7
6
7
14

Transformations
PCKPCKTopics Practices
12
70
33
42
24
34
16
51

CK
11
20
35
19

In all curricula, save MiC, the largest percentage of supports was supports for
Pedagogical Content Knowledge for Practices. MiC, on the other hand was different. It
contained mostly support for Curricular Knowledge, followed by Pedagogical Content
Knowledge for Topics or Pedagogical Content Knowledge for Practices. In addition, across all
curricula, Subject Matter Content Knowledge was least supported, accounting for no more than
7% of supports, except for in UCSMP‘s transformations unit, in which it accounted for 14% of
the content supports.
Why are certain supports more prevalent than others? What might explain the focus
on PCK for Practices and Curricular Knowledge? First, as three of these curricula were
Standards-Based, it might not seem surprising that there is a focus on Practices. The NCTM
Standards included more than just a list of content expectations; it included ―Process Standards‖
or a set of expectations that were about the mathematical processes students should engage in.
These included Problem Solving, Reasoning and Proof, Communication, Connections, and
Representations. See Table 27 for descriptions of the NCTM Process Standards.

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Table 27. Process Standards form Principles and Standards for School Mathematics (NCTM,
2000)
Process Standard
Problem Solving

Reasoning and
Proof

Communication

Connections

Representation

Description
Instructional programs from prekindergarten through grade 12
should enable all students to—
 Build new mathematical knowledge through problem solving
 Solve problems that arise in mathematics and in other contexts
 Apply and adapt a variety of appropriate strategies to solve problems
 Monitor and reflect on the process of mathematical problem solving
Instructional programs from prekindergarten through grade 12 should
enable all students to—
 Recognize reasoning and proof as fundamental aspects of
mathematics
 Make and investigate mathematical conjectures
 Develop and evaluate mathematical arguments and proofs
 Select and use various types of reasoning and methods of proof
Instructional programs from prekindergarten through grade 12 should
enable all students to—
 Organize and consolidate their mathematical thinking through
communication
 Communicate their mathematical thinking coherently and clearly to
peers, teachers, and others
 Analyze and evaluate the mathematical thinking and strategies of
others;
 Use the language of mathematics to express mathematical ideas
precisely.
Instructional programs from prekindergarten through grade 12 should
enable all students to—
 Recognize and use connections among mathematical ideas
 Understand how mathematical ideas interconnect and build on one
another to produce a coherent whole
 Recognize and apply mathematics in contexts outside of
mathematics
Instructional programs from prekindergarten through grade 12 should
enable all students to—
 Create and use representations to organize, record, and communicate
mathematical ideas
 Select, apply, and translate among mathematical representations to
solve problems
 Use representations to model and interpret physical, social, and
mathematical phenomena

CMP, MiC, and UCSMP, although originally designed to align with the 1989 NCTM
Standards, were all also revised and were intended to align with the NCTM 2000 Standards as
96

well. I argue that although Glencoe was not originally designed as a Standards-based curriculum
that its developers also attempted to align with the NCTM Standards, as evidenced by Standards
alignment charts in the teachers‘ guides that lists which lessons are aligned to which specific
NCTM Standards. Therefore, the curriculum I analyzed were designed, in part, to provide
students experiences with these processes, many of which were included in my coding scheme as
part of PCK for Practices. The most obvious of these is the Reasoning and Proof standard, as I
had a code within PCK for Practices for both Enactment and Rational Guidance for Reasoning
and Proof. However, my PCK-Practice codes, such as Enactment and Rationale Guidance for
Terminology and Participation Structures, also seem to fit within NCTM‘s Communication
Standard.
The same may be true for why Curricular Knowledge was prevalent in MiC, as the codes
I included within this category include much of what NCTM‘s Connections Standard
encompasses, such as understanding connections between mathematical ideas and how these
build on each other and other disciplines. In my scheme these fall within the Curricular
Knowledge category, as I see these supports are ones that help teachers understand how their
particular curriculum is making connections within and across disciplines.
A note about Subject Matter Content Knowledge. Although I know that the numbers
indicate that there was Subject Matter Content Knowledge I felt it necessary to draw readers
attention to this because although the percentages were quite small for each curriculum, there
were quite a few sentences devoted to subject matter content knowledge, especially in CMP,
MiC, and UCSMP. This knowledge was dwarfed by other categories of knowledge. The focus on
Subject Matter Content Knowledge may once again be attributed to the ways in which these
curricula were designed, particularly in the case of CMP and MiC. These two curriculum

97

projects, and later UCSMP which received funding in the second wave, were funded by the
National Science Foundation and at the time were qualitatively different than all other
curriculum materials on the market. These materials contained different mathematical content
and a different approach than traditional textbooks (Bradley, 2007, p. x) and therefore these
materials required a deeper understanding of mathematics than some of the materials middle
school teachers may have used previously.
Differences in frequencies across curricula. My study is not meant to be a horse race. I
am not trying to determine the best existing curriculum or the one that provides the most or best
supports. I am trying to describe the opportunities that currently exist so that we can learn from
these and improve curriculum materials. That said, there were some differences in the total
number of supports and in the relative frequencies of supports.
First, in terms of total number of content supports, CMP had a much larger number of
supports for both units, having at least 150% more codes in both units than others, and often
having close to or over three times as many codes. However, when taking into account total
sentences, CMP is only slightly higher than the other four curricula in the variable unit with 1.3
codes per sentence, while Glenoce, MiC, and UCSMP, have 1.1, 1.2, and 1.3 codes per sentence,
respectively. In the transformations unit, although CMP has more total supports than the other
curricula, it only averaged 1.0 code per sentences which is the same as both Glencoe and
UCSMP. For this unit, MiC actually had a larger codes per sentence ratio of 1.1. For both units,
CMP and MiC had equal or higher codes per sentence. Possible reasons for this might be related
to the demand of the tasks in the curriculum and the design principles or structure of these
curriculum materials. I discuss these issues in the next few sections.

98

Cognitive demand of curriculum materials. As described above, CMP and MiC were
two of the original NSF-funded curricula. The NSF-funded curricula were qualitatively different
than other curricula and these materials placed "a greater cognitive demand on students and
teachers" which made "the implementation of these materials difficult" (Bradley, 2007, p. x).
Although UCSMP received funding from the NSF in the second wave, the UCSMP curriculum
materials, although different from the traditional commercially-developed materials in arguably
more subtle ways, did not seem as radically different as curricula such as CMP and MiC. The
increased cognitive demand described by Bradley (2007) may be a contributing factor to the
increased number of supports in CMP and MiC. Since these curricula are more demanding for
teachers to teach with, teachers require more support. This is similar to what Stein and Kim
(2009) proposed. That said, I did not do an analysis of the cognitive demand of the four
curricular series I analyzed in this study. I merely propose this as a possible contributing factor
based on Bradley‘s (2007) statement that the NSF-funded texts were more cognitively
demanding for students and teachers than traditional curriculum materials.
Design principles and structure. Another factor that might describe the differences in
the curricula is the design principles and structure of the materials, particularly when it comes to
particular support codes, such as Enactment Guidance for Questions.
Structure of teachers' guides. Overall structure of the teachers' guides may impact the
total number of content supports. By structure I mean the ways in which the teachers' guides are
packaged and organized. CMP and MiC teachers' guides are, like their student books, standalone guides for each unit, whereas Glencoe and UCSMP use a two-volume structure; each
volume contains half of the teacher materials for that course. Although all curriculum developers
are plagued by publishing demands to keep their materials as few pages as possible, the two-

99

volume structure might require this even more as the volumes include support for multiple units.
An increased number of pages would result in a larger physical book, something that could
potentially discourage teachers from wanting to carry it.
In addition, the "embeddedness" of the teacher‘s guide might also play a factor. CMP was
the only embedded guide, a guide that included all teacher content within the narrative of the
lesson, rather than a teacher edition of the student text with wrap-around information for the
teacher. The wrap-around teachers guides might, by the nature of their structure include less
room for teacher support. Figure 10 provides a representation of each of these types of teachers'
guides. The shaded area indicates the space in which teacher support text is located.
Figure 10. Examples of Pages from Embedded and Non-Embedded Teachers' Guides

Embedded Teachers' Guide Page

Non-embedded Teacher's Guide Page

Reduced Student
Reduced Student Page
Page

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The CMP teachers' guide embedded the instructions to the teacher in a narrative that
followed the unfolding of the lessons and provided detailed descriptions for enacting the major
lesson pieces (i.e., Launch, Explore, and Summarize). Glencoe, MiC, and UCSMP, on the other
hand, provided support text around the student page. Unlike the CMP structure, this structure
leaves much less space for teacher text. For example, each Glencoe page was a 10 x 12 in. page,
which provided 120 sq. inches of space, however since a reduced student page was included, this
leaves only 53.5 sq inches or about 45% of the page for teacher text. This was typical for the
non-embedded teachers' guides. All text on each CMP teacher's guide page was 100% teacher
material. It is not clear whether curriculum developers have control over the structure of their
teacher' guides. It could be that the publisher of the materials makes these decisions and the
curriculum developers have little say in the matter. That said, it is an important issue for
developers to think about. If curriculum authors were not constrained by the structure, or even if
they were, thinking about ways in which they could include the types and amount of support
necessary is critical in improving teachers' guides.
Design principles. In addition to the overall structure of the teachers' guides, the design
principles of a curriculum may impact the types and frequencies of supports. First, if the
curriculum was designed specifically to attend to teacher learning this would have an impact on
the content supports. For example, CMP was specifically designed to be a curriculum for
teachers as well as students. "The materials were written to support teachers' learning of both
mathematical content and pedagogical strategies" (Lappan, Phillips, & Fey, 2007, p. 68).
Although each curriculum created a guide for the teacher these differed and these differences
may be attributed to the focus that the curriculum developers put on the teachers' guide and how
much this focus involved teacher learning as opposed to directions for enacting or implementing

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the curriculum. Whereas I can say that CMP stated teacher learning was a guiding principle in
the design of their materials, I cannot say this about the other curricula. This does not mean that
these curricula did not attend to teacher learning in the design of their materials, but this has not
either been described by the developers as a focus, or at least, I could not find evidence of this.
Design principles may also impact the types and frequencies of particular supports. All
curricula are designed with some set of principles. It would make sense that a curriculum's
teacher support materials would be geared towards supporting teachers with the types of content,
activities, and processes that teachers were expected to engage their students in. As I point out in
the results, this might explain why CMP had such a large number of content supports for
Enactment Guidance for Questions. CMP is "problem-centered" and the mathematics is
"embedded in interesting problems to promote deeper engagement and learning for students"
(Lappan, Phillips, & Fey, 2007, p. 68). When setting up a problem it seems natural to have
questions and since students are expected to engage with these problems through questioning, it
seems natural that the teacher support would warrant a focus on questions. This might also
explain why CMP was the only curriculum to provide support for Enactment Guidance for
Helping Students Ask Their Own Questions. Another instance in the data that indicated this
possibility is the large number of Enactment Guidance for Representations in CMP's variable
unit. Variables and Patterns places a special emphasis on different representations as evidenced
by one of the goals of the unit, "Observe relationships between two quantitative variables as
shown in a table, graph, or equation and describe how the relationship can be seen in each of the
other forms of representation" (CMP, Variables and Patterns, p. 2) and this may be related to the
large number of supports for teachers.

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Constructing Relationships: Teacher as Agent
In this section, I describe issues related to the curriculum developers‘ construction of
relationships with teachers. Specifically, I address how the curriculum authors spoke to teachers
in the text and what this indicates about the relationships and positioning of the teachers reading
these materials.
My results indicated that the curriculum authors did not speak "to" teachers, but instead
this study showed that often curriculum authors speak "through" teachers. This was evidence by
the large percentage of imperatives and the lack of Rationale Guidance. Although each
curriculum developed a guide for teachers, the content and the language used in these guides
may indicate that the curriculum authors, like writers in the early and mid- 1900's,
underestimated the role of the teacher. Furthermore, some of the linguistic forms, whether used
consciously or not, not only underestimated the teachers' role, but positioned the teacher in a
diminished role, the role of agent doing the bidding of the more knowledgeable curriculum
authors.
First, the large number of imperatives indicated that the curriculum authors often chose
to command teachers to perform certain actions and in some cases directed students through
commanding teachers. In addition, these imperatives often foregrounded the teacher‘s position as
that of "scribbler," merely performing the actions the curriculum authors desired. Although the
use of imperatives can be expected in academic discourse (Morgan, 1996), Morgan questions
their use in the school context. Embedded in this linguistic form are issues related to power and
authority. Although Morgan studied the written text of students, I argue that the same type of
concern is warranted when investigating text in the teachers' guides. An imperative, or
command, in the teachers' guides is a grammatical feature that allows for the curriculum authors

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to demand teachers to carry out a service and positions the curriculum authors as the authority in
the situation, the one (or in this case, the group) telling teachers what to do. The relationship
constructed here has implications for agency. Based on the thoughts of philosopher and
mathematician Charles Sanders Peirce, Rotman (1988) distinguished between two types of
mathematical agency:
"the one who imagines (what Peirce simply calls the ‗self‘ who conducts a reflective
observation), which we shall call the Mathematician, and the one who is imagined (the
skeleton diagram and surrogate of this self), which we shall call the Agent. In terms of the
distinction between imperatives, it is the Mathematician who carries out inclusive
demands to ‗consider‘ and ‗define‘ certain worlds and to ‗prove‘ theorems in relation to
these, and it is his Agent who executes the actions within such fabricated worlds, such as
‗count‘, ‗integrate‘, and so on, demanded by exclusive imperatives" (p. 11/106).
This distinction described by Rotman (1988), lends itself to describing the imperatives I saw in
the teachers' guides. Where Rotman wrote about mathematician and agent, in my study, it is the
curriculum authors that demand, and the teachers using the curriculum materials that execute the
actions.
In addition to imperatives, each curriculum made use of modal verbs to open up the
dialogue and provide the possibilities of negation. Modal verbs such as "may" and "might"
provided room for teachers to decide whether to enact suggestions posed by curriculum authors.
However, there was still a somewhat significant use of verbs such as "should" and "need."
Although these verbs provide some room for negotiation, they embed a higher degree of
obligation for teachers to act than the modals "may" and "might."

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A more subtle indication of the teachers positioning as agent is illustrated by the lack of
Rationale Guidance included in all the curriculum materials. Enactment Guidance, guidance that
supported teachers in what and how to enact lessons, was much more prevalent than Rationale
Guidance. Rationale guidance, which accounted for no more that 6% of the guidance in any unit,
enables teachers to understand why particular mathematical, pedagogical, or curricular
approaches are appropriate and creates a dialogue between authors and teachers. Supports such
as this can allow teachers to understand the reasoning of the authors of the materials and develop
what Drake and Sherin (2009) call ―curriculum vision‖ or a sense of where the curriculum
materials are going and an understanding of the ―particular kinds of learning and teaching
practices described in the curriculum materials‖ (p. 324). Furthermore, Rationale Guidance
supports would allow teachers to understand the choices made by curriculum authors and could
potentially help curriculum developers forge a different relationship with teachers; a more
reciprocal relationship, one in which teachers and curriculum developers work together to
construct the curriculum.
Another interesting finding from this study was that when curriculum authors opened up
a space for teachers by using modal verbs, such as ―may‖ and ―might,‖ curriculum authors
frequently used the verbs ―want‖ and ―wish.‖ It is possible that the use of the modal verbs "may"
or "might" allowed curriculum authors not to force wants and wishes on teachers. On the other
hand, the use of these words together may be a more subtle way for curriculum authors to
influence what their teacher do. Rather than telling a teacher to do something, or telling a teacher
he or she may do something, getting a teacher to want to do something might in fact be more
influential because it indicates a personal desire. Although the choice of these words may be an
unconscious one by curriculum authors, it is nonetheless a choice.

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Opportunities to Learn in Variable and Transformations Units
Content. When I began this dissertation one thought I had was that the supports for the
introduction to variables units and the geometric transformations units would differ. More
specifically, I thought that there would be more content supports in the variables units than in the
geometric transformations units because as a teacher I was always able to find more algebra
support, whether this be in the form of professional development sessions, or resources from my
district or the Internet. In addition, there is a greater emphasis on algebra than geometry since
success in algebra is seen an indicator for success in higher mathematics (National Mathematics
Advisory Panel, 2008). My results indicated that although the algebra units did have more
content support, the difference was not as large as I expected. Three of four curricular series
actually had a higher frequency of content supports for their geometric transformations unit,
CMP was the only exception. This might make sense since for each curriculum there were more
sentences in the transformations units. However, when I calculated a code per sentence ratio, the
variable and transformations units were quite close, with the variable units being only slightly
higher. See Table 28. This means that for each curriculum there were more supports in the
variables units per unit of text than in the transformations units, but the difference was minimal.
The transformations units may have included more sentences and supports because geometric
transformations is a relatively new topic in middle school (Sinclair, 2008). Curriculum authors
may have felt the need to include more teacher support for this potentially unfamiliar topic.
Table 28. Ratios of Codes per Sentence by Unit and Curriculum
Curricula
CMP
Glencoe
MiC
UCSMP

Variable
Codes per Sentence
1.3
1.1
1.2
1.1

Transformations
Codes per Sentence
1.0
1.0
1.1
1.0
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As described in the results section, although for both units the most prevalent supports
were the same, there were some differences in the content supports for the two units. The
transformations unit had a greater span of content supports than the variable unit; the
transformations units contained more types of support than the variable units. In addition, there
were differences in frequent content supports. Within the PCK for Mathematics Topics category,
the variables units contained more instances and a larger percentage of Enactment Guidance for
Representations, whereas the transformations unit included more Enactment Guidance for Tools.
The transformation units also included in the PCK for Practices category, more instances of
Enactment Guidance for Approaches to Reasoning and Proof. There were no noticeable
differences in the frequencies of other content supports between the variable and transformations
units.
Expression. Similarly, for the aspects of voice I analyzed the differences were quite
subtle. In the next two sections I describe the differences in personal pronoun use and "you"forms.
Differences in personal pronoun use by unit. As expected, since there were more
sentences in the transformations units, generally there was a larger frequency of personal
pronouns. The personal pronoun "I" was not found in either unit. The personal pronouns "we"
and "you" were more frequent in the transformations units except for the use of "you" in CMP,
which was more frequent in the variables you.
"We" was used differently in the units by CMP and UCSMP. In the transformations unit
the referent for "we" was more often unclear. In these instances I was unable to determine who
―we‖ referred to, unlike clear instances in which I could determine that ―we‖ referred to the
curriculum authors. Of all of the uses of "we" in CMP, 74% were unclear in their variables unit

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and 86% were unclear in their transformations unit. Similarly for UCSMP, there were no unclear
referents for "we" in the variable unit, whereas 41% were unclear in their transformations unit.
Why was this lack of clarity more common in the transformations unit? One possible explanation
for this is that most of the instances in which the referent for "we" was unclear was when the
authors provided details for a particular method or solution in a way that explicated the type of
thinking that one might engage in, and these were more common, particularly for CMP, in
transformations units. In these instances, "we" was used quite frequently and not in a way that
clearly referred to the curriculum authors or a particular plural group.
Why is it that transformations units included more explanations such as these that
detailed a particular method or solution? Two possible reasons come to mind. First, it could be
that due to geometric transformations being perceived as "new" content, that curriculum authors
thought it necessary to provide more teacher support. According to Sinclair (2008), up until the
mid 1960s, in the U.S., geometric transformations were deemed too hard, even for high school
students. Therefore, the inclusion of this new, hard content may have prompted curriculum
developers to provide more detailed (often step-by-step) methods or solutions and model the
types of thinking that performing transformations requires.
Another potential explanation for the differences in the two units, particularly related to
the use of different language, is authorship. It is likely that units, particularly in the Standardsbased curriculum materials, were written by different authors, or at least were primarily
supervised by different authors. This might be one reason why the transformations units and
variables units differ. Some authors may choose to use personal pronouns in particular ways,
including not using them all. The possibility of different authors may have also impacted the

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overall structure of the text including the inclusion of examples, such as the ones we see in the
transformations units that provide detailed explanations.
Differences in "you"-forms by unit. In general, implicit ―you‖-forms (i.e., imperatives)
were much more common across both units, except in MiC‘s transformations unit which had
more explicit than implicit ―you‖-forms. The ratios of implicit to explicit forms was higher in all
cases for the variables units, meaning that the differences between the total number of each of
these forms was greater relative to the total ―you‖-forms used in the variable units.
Imperatives. The use of imperatives in each curriculum was fairly consistent across units.
Across both units, generally there was a greater relative frequency of imperatives in the
transformations unit, with the Glencoe unit containing over 25% of the sentences in their unit in
the imperative form. CMP was the only curriculum to have more imperatives in their variable
unit. The locations and the types (i.e., inclusion, exclusive) of these imperatives were also fairly
consistent across units. However, there was a greater difference in common imperatives in the
transformations units between the percentage of imperatives that directed the teacher and those
that directed students through the teacher.
Explicit “you”-forms. The relative frequencies of explicit ―you‖-forms were virtually the
same across units. The most pervasive was ―you‖ + modal verb in both units. I also did not find
much of a difference in the types of modal verbs used in each unit. Across both units the
common modals were the same, low modals ―may‖ and ―might,‖ with CMP and MiC using the
median modal ―should‖ and the high modal ―need‖ and all curriculum materials including the
low modals ―can‖ and ―could,‖ but at much lower frequencies than ―may‖ and ―might.‖

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Opportunities to Learn
What do these results indicate about teachers‘ opportunities to learn about mathematics,
pedagogy, and curriculum, from middle school curriculum materials? Although each curriculum
provided access to some content supports, this access might not be sufficient. Over 15 of 31
content supports were infrequent or unobserved across at least three of the curricular series.
These absent supports were often Rationale Guidance supports, which not only impacted access
to content, but also diminished the ways in which authors spoke to teachers. When curriculum
authors discuss their rationale they open up a space in which teachers can engage with them
around the underlying principles on which the curriculum is designed and allows for teachers to
develop ―curriculum vision‖ (Drake & Sherin, 2009). Generally, this space was not provided.
Furthermore, the expression of the text, or the choice of words used by authors, as evidence by
the aspects of voice I analyzed, often positioned teachers as agents doing the bidding of
curriculum authors. This curriculum author – agent relationship may hinder teachers learning
from the text as it is possible to see the teachers‘ guide as merely a set of directions to enact the
curriculum.
As my discussion suggests, it is difficult to describe opportunities to learn in a diverse set
of curriculum materials. I propose, like Stein and Kim (2009), that the types of support are
intimately connected to the type of curriculum materials being analyzed. That said, regardless of
the structure or philosophy of a set of curriculum materials, my study indicated that the content
in these four series could be improved. All four series did not provide adequate access to content,
particularly Rationale Guidance and used language that may hinder teachers' opportunities to
engage with and learn from the text.

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Contributions to the Field
This analysis is quite timely. Although professional development for some time has been
deemed as a necessary piece in improving teaching, as we enter into the era of the Common Core
Standards (National Governors Association Center for Best Practices & Council of Chief State
School Officers, 2010), mathematics teacher educators and curriculum developers have already
recognized that this need is even more critical (Lappan, McCallum, & Kepner, 2010). Teachers
are going to be required to teach topics for which they have little or no background, such as
geometric transformations, and may be required to drastically change the ways in which they
approach particular topics or think more deeply about the ways in which they teach mathematics
to incorporate the practices outlined in the Common Core Standards. Research indicates that
teaching and curriculum matter and as the words of Bruner remind us, "A curriculum is more for
teachers than it is for pupils. If it cannot change, move, perturb, inform teachers, it will have no
effect on those whom they teach." As we enter into this new era, this idea from 35 years ago
carries an important message. For any "curriculum", whether it is a set of written materials, or an
outline (such as The Common Core Standards) for the types of things that should be taught, it
must affect change in teachers for it to affect students.
My dissertation illustrates that these four curricula series can do more to move and
inform teachers. I have not only provided the field with an analysis of the opportunities for the
kinds of opportunities in four popular middle school mathematics curricular series, but I have
also provided researchers with a framework for conducting similar analyses in other
mathematical content areas. My study has implications for curriculum development, teacher
education, and research. In the next section, I discuss these implications.

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Implications
Curriculum development. Curriculum developers have an extremely difficult task. I
recognize the pressures placed on curriculum authors from publishers to make their students‘ and
teachers' guides as short as possible. This pressure is not something that curriculum researchers
can ignore. I also recognize that what ends up in the final version of a textbook may not be
identical to what a curriculum development team intended. Nonetheless, this work has some
implications for curriculum developers. First and foremost, I hope that my study raises
awareness about how the choice of content and its expression might impact teachers'
opportunities to learn and that this encourages curriculum authors to critically examine their own
texts. Before curriculum developers can exert change, they need to see that this change is
necessary. I hope that this examination of text will enable curriculum developers to be more
intentional, particularly in when it comes to discussing their rationale and their choice of words.
In the next sections I outline a few recommendations for curriculum developers.
According to Remillard, for teachers to learn from their curriculum materials authors
need to speak to teachers. This is something that all four curricular series I analyzed could do
more of, particularly Glencoe, which addressed teachers through imperatives more than any
other curriculum series (at least 25% of sentences were in imperative form in each unit). In
addition, each curriculum suffered from the lack of Rationale Guidance. Including more
guidance of this type would help to include teachers more in the construction of their
mathematics teaching by opening up a dialogue with them and helping them understand why
particular approaches or content is appropriate. This could help in allowing teachers to develop
curriculum vision (Drake & Sherin, 2009) and enable teachers to become more skilled and
informed enactors of the curriculum. An area that curriculum authors may want to focus their

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attention on is providing more Rationale Guidance for Goals. This support was completely
absent from all curriculum materials. Including information related to why particular goals are
appropriate and why the curriculum authors chose particular goals for particular lessons may
allow teachers to more fully understand the goals of the unit or lesson and how these fit together
across the entire curricular series. This, in turn, may allow teachers to help their students achieve
these goals more effectively.
Another area of attention for curriculum authors is in providing more Implementation
Guidance for Anticipating and Using Students Thinking productively. If we expect teachers,
which often the more reform-oriented curricula do, to engage in discussions with their students
about the mathematics, more support is needed in anticipating and using student thinking (Star &
Strickland, 2008). This is particularly important for new teachers who have less experience, as
anticipating what students might do and say can be a challenge when you have limited
experience to draw from.
Finally, curriculum developers may want to think more strategically about where support
is located. For example, if all the subject matter support is at the unit level and teachers are not
looking at this prior to or during the lesson, these valuable supports may not be providing the
opportunities for learning that curriculum authors intended them to.
As I mentioned earlier, I recognize that curriculum developers are under pressure to make
their teachers' guides even shorter, often requiring them to be terse and make tough decisions
about what to include and what to exclude. With the increased availability and use of technology,
however, these constraints may become a thing of the past to some extent. I hope that the
findings from my study can influence the thinking related to the types of supports to include in
digital textbooks. For example, some curricula already have websites for professional

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development that include videos of lessons. With digital curriculum materials, images of
classroom interactions (Remillard, 2000), such as transcripts from classroom discussions (similar
to Dialogue Boxes in Investigations) or video could be embedded right into the lesson
guidelines. I make no claims that writing this kind of teacher's guide is easy, but that it may help
to eliminate some of the constraints that are placed on curriculum authors when developing a
paper teachers' guide. As a mathematics educator I find the possibilities inherent in this
technology quite liberating and potentially fruitful, but also think that as a field we need to think
carefully about the ways in which we use this technology. For example, if electronic curriculum
materials are just digital versions of paper teachers‘ guides, then these guides will suffer from
many of the same issues I described in my study. I discuss electronic curriculum materials more
in my section about future directions for design and research.
Teacher education. Although there have been many changes in the types of textbooks
that are available to teachers, it is still true that some prospective teachers develop the impression
that if they want "to be good teachers, they should avoid following textbooks and teachers‘
guides‖ (Ball & Feiman-Nemser, 1988) or more drastically, not just that they should not follow
them, but that they should completely throw them out all together and create all of their own
materials. This mentality, attributed to the need to be autonomous and to take control of every
aspect of their teaching especially their lesson planning (Ben-Peretz, 1990), seems unrealistic
and potentially problematic. Beginning teachers, while professionals in their own right, are not
curriculum developers, nor should they be positioned as such. The fear that teachers become
―text bound‖ has hindered some teacher education programs from addressing important issues
related to curriculum interpretation. It is within teacher education programs that prospective
teachers can begin to learn to read and interpret curriculum in meaningful and productive ways.

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Without this education, teachers may be doomed to ―remain ‗text bound,‘ using textbooks or
teacher guides because they ‗are there‘ without attempting adaptation or enrichment of existing
materials‖ (Ben-Peretz, 1990, p. 109). This study implies that, while teacher's guides are not
replete with opportunities for learning, opportunities do exist and that it is essential for teacher
educators to help prepare teachers that know how to use and learn from their textbooks.
First and foremost, this study implies that teacher educators need to help their students
learn to read their curriculum materials since rich opportunities for learning were not frequently
present. By reading, I mean the constructive process of meaning-making (Remillard, 2000). This
reading process will enhance the opportunities available because reading can incorporate more
than just what the supports present, such as thinking more deeply about student thinking. It takes
more to enact the curriculum than just reading the words on the page and if teacher educators do
not incorporate issues of written curriculum materials into their courses, teachers may remain
text-bound since they will not develop the skills necessary to use their teachers‘ guides. They
may not know where to find supports or how to adapt materials to meet the needs of their
students while still providing a coherent mathematical experience. This study provides valuable
insight into the types of opportunities that are available in four commonly used curricular series.
Although it is true that not all teachers will teach from these textbooks, teacher educators can use
these insights to help student teachers develop the skills necessary to use textbooks productively.
Remillard (2005) found that teacher learning was related to the activities that were central
in constructing the curriculum. These activities were wrapped up in the reading process which
involved reading the text, but also reading their students and the tasks they were engaged in. For
new teachers to be able to do this they will need assistance in learning how to read suggestions in
the text and make educated decisions about how to proceed in their classrooms based on what

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they know about their students (Kauffman, Johnson, Kardos, Liu, & Peske, 2002). This study
indicated that a large percentage of the sentences were commands in the imperative form. For
new teachers, particularly elementary and middle school teachers who may not feel empowered,
learning to adapt and change what their curriculum says in order to best meet their students needs
while still enacting a coherent and mathematically rich curriculum is difficult. Teacher education
programs can prepare students to do this by critically examining mathematics curriculum
materials. Assignments such as the ones given in mathematics methods courses (Lloyd & Behm,
2005; Lloyd & Pitt Bannister, 2010) where student were asked to compare lessons from two very
different textbooks or use a teachers' guide to plan and enact a lesson can not only help students
become more comfortable with the ways in which textbooks are designed, but these assignments
can also help prospective teachers develop a sense of and learn to critique different approaches
and content trajectories.
Another more simple, yet important implication of this study, is that opportunities for
teacher learning occur in various locations of the teacher's guide and teachers will need to
consult different locations to get this support. Directing prospective teachers to consult all parts
of whatever teacher's guide they use is important. For example, teacher educators can help
prospective teachers see that if they read the text at the unit level they may be able to gain insight
into how the individual lessons fit together to develop the big picture, or find the subject matter
support they needed.
Research. In my study I used use two frameworks for investigating opportunity to learn;
one framework that focused on content and one on aspects of voice. As described in the
background of my study, research on written curriculum materials often focuses more on content
and less on the expression of this content, with a few recent exceptions. My study indicated the

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importance of examining the expression of content in curriculum materials and the benefits of
using a Systemic Functional Linguistics framework to do this. First, if I only studied the content
of the teachers' guides I would not have gained insight into how curriculum authors speak to
teachers, a critical factor in how teacher supports may be experienced. For example, although
there were many supports for helping teachers use questions with students, my analysis indicated
that, in many of the instances authors chose to use imperatives, which indicated that teachers
were commanded to ask particular questions. The Systemic Functional Linguistics framework
allowed me to achieve a more nuanced description of the learning opportunities in the middle
school curriculum materials and illuminated the subtle shifts in the way language was used in
different units and curriculum materials. I argue that unless researchers look at how content is
expressed, we are not accurately describing opportunities to learn. Researchers should examine
more than just the content, whether this be in teachers' guides or student textbooks. In addition,
this means that researchers should employ other frameworks, such as ones that focus on forms of
expression, as these frameworks provide a means for critically examining text in ways that
previous frameworks that have focused on what is in a textbook do not. That said, it is equally
important when analyzing expression to consider what is expressed. Without examining the
content, the results tell us little about the social activity in which that expression occurs.
Therefore, in my study it was imperative to look at both the content supports and the expression
in order to be able to say something about the nature of the opportunities available for teachers in
the teacher‘s guides.
In addition researchers should consider how the philosophy and structure of curriculum
materials impact the types of supports provided in the text. Since I chose curriculum materials
that were designed using different guiding principles, these issues arose in my study as I tried to

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describe the difference I saw in the four curricular series. In my analysis I proposed that the
frequency and types of content supports and the language may be related to the philosophy and
structure of the curriculum itself. However, more analysis is needed in this area.
Limitations
No study is without limitations. In this section I discuss the limitations of my method.
First, this dissertation was written by one person. Although it is the product of my thinking and
countless conversations with colleagues (i.e., faculty, graduate students, teachers), the final
words in this document are my own. This represents my perspective. I acknowledge that others
using different frameworks (or developing different codes), or even the same frameworks, could
possibly bring different meaning to the data. Second, although I chose the four curricular series
in my study for particular reasons, I only analyzed four curricular series, all of which were
designed for middle school. As a field, researchers need to use methods such as the ones I used
in my study in order to make more nuanced claims about opportunities to learn. Finally, in my
study I did not attempt to make links to what or how teachers learn or experience curriculum.
This is an important aspect of this work, as more needs to be done in order to determine how the
opportunities to learn, particularly the content and its expression, in curriculum materials actually
promote teacher learning.
That said, despite these limitations, my study provided a way to describe a particular set
of curriculum materials, those that are widely used throughout the United States and vary in
design. In addition, my study provided a way to use two frameworks to obtain a nuanced
description of teachers‘ opportunities to learn from curriculum materials; two frameworks that
can be applied to other mathematics curriculum materials.

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Future Directions for Research and Design
I see many directions for further research and design, all of which I am interested in
pursuing myself. In this section, I describe three of these directions: a) examining curriculum
materials at other grade levels, b) focusing research on studies that combine the research on
curriculum use with the research in my study on opportunities in written curriculum materials,
and c) exploring ideas for designing and researching the use of electronic curriculum materials.
Examining opportunities for teacher learning in written materials at other grade
levels. In my study, I only investigated opportunities in middle school curriculum materials. I
chose middle school curricular materials for particular reasons, such as the unique position of
middle school teachers as those who bridge elementary and high school mathematics content and
that not all middle school teachers have had adequate preparation to teach middle school, but
have been former elementary or high school teachers (Hill, 2007). However, curriculum
materials at all levels need to be examined. Although I think that my frameworks are appropriate
for examining any mathematics curriculum materials, there may be level-specific issues that
researchers need to consider. This may be particularly important when interpreting what is
present in the materials. Most high school teachers are required to have college degrees in
mathematics, whereas elementary school teachers are not, but are often required to take more
child development courses and courses in other disciplines. These issues of prior coursework
may have implications for the types of opportunities needed. For example, subject matter content
knowledge may be more important for teachers who have had less exposure to mathematics
courses, whereas supports for connecting mathematics to other disciplines, may be needed by
those who have not had much exposure to the ideas of teaching literacy or science. If this is true,
then the opportunities needed in curriculum materials for different levels would be different.

119

Examining teacher learning from opportunities embedded in written materials. I see
my study as only the first step in investigating teacher learning from written curriculum
materials. Although research indicated that teachers can and do learn from using curriculum
materials, there has been minimal research that has looked at how the written curriculum impacts
this learning. In my study, I described the opportunities for teacher learning within four middle
school mathematics curricular series. However, I make no claims about whether or how teachers'
make use of these opportunities and furthermore, actually learn from these opportunities. Further
research must draw from both of these areas of research, the research on teacher learning from
curriculum use and the research on the opportunities in written materials (See Figure 11), to
investigate, whether or how teachers learn from the opportunities within written curriculum
materials. Specifically, as called for by Stein, Remillard, and Smith (2007), research should
investigate how the features or characteristics of particular curriculum materials influence how
teachers use it, and more specifically how teachers learn from the materials. My study can serve
as a starting point to address the following question: What and how do the features of curriculum
materials impact teacher learning? The detailed description I provided in my this study can serve
as the basis for studies of the enacted curriculum in which researchers observe teachers planning
and teaching lessons from the four curriculum I analyzed. It may be possible to make links
between the opportunities in these materials and what teachers are learning from using these
materials.

120

Figure 11. Current Research and Future Needs

Teacher
learning from
curriculum use

Opportunities
for teacher
learning in
written
curriculum
materials

Research needed at the intersection:
Studies that examine how teachers
learn from the opportunities in written
mathematics curriculum

Designing electronic educative curriculum materials and researching their use.
Although electronic curriculum materials have already been conceptualized and designed, most
of these materials have been merely digital versions of the old print curriculum materials. New
curriculum materials will likely be electronic (Center for the Study of Mathematics Curriculum,
2009). My study indicated that teachers' guides lacked many of the supports needed by teachers,
particularly those related to rationales and anticipating and using students thinking productively.
In the traditional print teachers' guides, including support such as this would require more
sentences, which would require more paper pages. Due to publishing demands, often curriculum
developers do not have the space for such supports as they cost more money. This is where
digital technologies provide a more publisher-friendly solution. First, there is not the same
pressure to keep the number of pages down when creating a digital book. Second, digital media
would allow for more than just sentences. Digital materials would allow elements such as
interactive applets, discussion boards, and videos. Rather than just having a discussion scenario,

121

a teacher could click on a video that illustrates a conversation in a real classroom around the task
that they are using that day.
Another advantage of electronic curriculum materials is that they can be made to be
customizable and adaptable. First, teachers could have more control over the activities they do in
their classroom by using a system that allowed them to easily choose from a menu of options,
somewhat like a more sophisticated version of the Choose your Own Adventure books for kids.
Brown (2009) indicates that materials such as these, like his Adaptive Instructional Materials
(Brown, Pellegrino, Goldman, Nacu, Julian, Tarnoff, et al., 2004), can help teachers develop
Pedagogical Design Capacity, the "capacity to perceive and mobilize existing resources in order
to craft instructional episodes (Brown, 2002; Brown & Edelson, 2003)" ( p. 29). Second,
electronic materials are also more easily adaptable. As researchers learn how teachers use
materials and what features promote learning, electronic curriculum materials can be adapted to
meet the needs of teachers easily. We live in a world in which people download updates to the
applications, or ―apps‖ on their phones on a daily basis (or in many cases, their apps
automatically update for them). The same could be true for curriculum materials. Designing
materials like this is no small feat; these materials would require both the mathematical
curriculum knowledge and the technical know-how, but I see this as a worthy direction for
design work. Designers, particularly those in the university system, could develop relationships
with those working in other departments such as computer science and engineering, or local or
national companies that specialize in technology, such as Google. I see this as an exciting avenue
for my own future as a researcher and possible designer.

122

APPENDICES

123

APPENDIX A

Table 29. Descriptions of Studies of Written Curriculum Materials
Authors

Number
of Series
Analyzed

Ashcraft, M. &
Christy, K.
(1995)

1

Cai, J., Lo, J. &
Watanabe, T.
(2002)

Resources
Examined
(i.e., student/
teacher)

Focus of Analysis

Countries

elementary

student

arithmetic facts
(simple addition
& multiplication)

U.S.

3

elementary/
middle

student

arithmetic
average

Asian
countries,
U.S.

Castro, A. M.
(2006)

1

middle

teacher

teachers' guide as
resource in
planning

U.S.

Chang, K.,
Males, L. M.,
Mosier, A., &
Gonulates, F.
(2011)

3

elementary

student and
teacher

estimation of
linear
measurement

U.S.

Charalambous,
C., Delaney, S.,
Hsu, H. &
Mesa, V. (2010)

5

elementary

student and
teacher

fractions
(addition and
subtraction) presentations of
content and
expectations
calculator use

Cyprus,
Ireland,
Taiwan

Chval, K. &
Hicks, S. (2009)

Level

elementary

124

U.S.

Table 29 (cont‘d)
Ding, M. & Li,
X. (2010)

3

elementary

student

distributive
property problem context,
type, and
variability

China,
U.S.

Fan, L. & Zhu,
Y. (2007)

9

high

student

problem-solving
procedures
(Polya)

China,
Singapore,
U.S.

Flanders (1987)

6

elementary/
middle

student

new content
versus review of
old

U.S.

HerbelEisenmann, B.
(2007)

1

middle

student

voice

U.S.

Hook, W.,
Bishop, W. &
Hook, J. (2007)

1

elementary

student

comparison to top
TIMSS
curriculum
materials

U.S.

Hoven, J. &
Garelick, B.
(2007)

1

elementary

student

bar models for
addition and
subtraction

Singapore

Huntley, M.
(2008).

2

middle

student

Framework for
analyzing
curriculum
materials

U.S.

125

Table 29 (cont‘d)
Jones, D.
(2004).

8

middle

student

probability historical (extent
and nature)

U.S.

Lee, K., &
Smith III, J. P.
(2011)

4

elementary

student and
teacher

length (content
and presentation
in text)

Singapore,
U.S.

Lee, M. &
Messner, S.
(2000)

10

middle/high

student

concatenations
and order of
operations instructional
emphasis

U.S.

Leinwand, S. &
Ginsburg, A.
(2007)

1

elementary

student

content and
organization
(broad)

Singapore

Li, J. (2004)

2

elementary

teacher

whole number
multiplication comparing
teachers' guides
US and China

China,
U.S.

Li, Y. (2000)

9

middle

student

integer addition
and subtraction

China,
U.S.

Martin, T.,
Hunt, C.,
Lannin, W.,
Leonard Jr., W.,
Marshall, G. &
Wares, A.
(2001)

5

high

student

Comparison to
NCTM Standards

U.S.

126

Table 29 (cont‘d)
Mesa, V. (2004)

24

middle

student

function describe the
practices
associated with
function notation

15
countries

Michalowicz,
K. & Howard,
A. (2003)

100s

elementary/
middle/high

student and
teacher

historical
development of
pedagogy and
content in text

Canada,
Mexico,
U.S.

Newton, D. &
Newton, L.
(2006)

18

Elementary

teacher

teachers' attention
to reason

England

Nissen, P.
(2000).

9

elementary/
middle/high

student

geometry/transfor
mational
geometry

U.S.

Olsen, T. (2010)

4

Middle

student

Patterning tasks articulated
learning
trajectories

U.S.

Pickreign, J. &
Capps, L.
(2000).

5

Elementary

student

comparing to
NCTM Standards
- geometry

U.S.

Schmidt, W.,
Houang, R. &
Cogan, L.
(2002)

45

elementary/
middle/high

student

content and
organization

45
countries

127

Table 29 (cont‘d)
Schmidt, W.H.,
McKnight,
C.C., Valverde,
G.A., Houang,
R.T. & Wiley,
D.E. (1997)
Smith III, J.P.,
Dietiker, L.,
Lee, K., Males,
L.M., & Mosier,
A. (in
preperation)
Smith III, J.P.,
Gonulates, F.,
Males, L.M., &
Mosier, A. (in
preperation)

45

elementary/
middle/high

student

content and
organization

45
countries

3

Elementary

student and
teacher

length (content
and presentation
in text)

U.S.

3

Elementary

student and
teacher

area (content and
presentation in
text)

U.S.

Star, J., HerbelEisenmann, B.
& Smith III
(2000)

2

middle/high

student

Algebra

U.S.

Stein & Kim
(2009)

2

Elementary

teacher

"educativeness"
of curriculum
materials supports for
teachers

U.S.

Stylianides, G.
(2005)

1

Middle

student and
teacher

proof
opportunities

U.S.

Stylianides, G.
(2007)

1

Middle

teacher

Opportunities for
teachers to learn
about proof

U.S.

128

Table 29 (cont‘d)
Valverde, G.,
Bianchi, L.,
Wolfe, R.,
Schmidt, W. &
Houang, R.
(2002)
Watanabe, T.
(2001)

45

elementary/
middle/high

student

content and
organization

45
countries

4

elementary

teacher

organization/cont
ent

Japan,
U.S.

Watanabe, T.
(2003).

4

elementary

student

fraction

Japan,
U.S.

Weinberg &
Weisner.
(2011).

1

college

students

framework for
analyzing the
reading of
textbooks

U.S.

Yang, D., Reys,
R. & Wu, L.
(2010)

3

elementary/
middle

student

fractions

Singapore,
Taiwan,
U.S.

129

APPENDIX B

List of all imperatives used by CMP in Variables and Patterns

Ask
Have students share
Use
Encourage
Discuss
Tell
Let
Explain
Remind
Choose
Allow
Emphasize
Have students work
Read
Continue
Describe
Have students look
Have students press
Make sure students understand
Move on
Be sure s. use
Check
Determine
Do not worry
Five
Help
Have students discuss
Have students put
Have students relate
Have students speculate
Look
Make
Point out
Repeat

Be sure students understand
Be sure use
Call
Collect
Compare
Conduct
Connect
Create
Display
Do not discuss
Elicit
Find
Go over
Go through
Highlight
Have students adjust
Have students construct
Have students draw
Have students examine
Have students explain
Have students follow
Have students graph
Have students make
Have students perform
Have students present
Have students record
Have students refer
Have students set up
Have students sketch
Have students summarize
Have students take
Have students talk
Inform
Introduce

130

Make sure students
discuss
Make sure students
mention
Make sure students points
are made
Make sure students can
talk
Make sure students
explain
Make sure you add
Pair
Put
Refer
Say
Suggest
Talk
Translate
Verify
Write

List of all imperatives used by CMP in Kaleidoscopes, Hubcaps, and Mirrors

Ask
Encourage
Have students work
Use
Have students share
Remind
Call
Discuss
Arrange
Help
Choose
Give
Draw
Direct
Explain
Review
Look
Read
Let
Challenge
Focus
Set
Distribute
Point out
Talk
Collect
Circulate
Have students demonstrate
Pose
Check
Put

131

List of all imperatives used by Glencoe in Algebra: Number Patterns and Functions

Use
Ask
Remind
Tell
Create
Write
Explain
Suggest
Check
Encourage
Have students complete
Have students create
Point out
Customize
Discuss
Give
Have students work in groups
Have students write
Post
Provide
Allow
Assign
Bring
Challenge
Consider
Display
Fill in
Guide
Have students add sticky notes
Have students compare
Have students count
Have students discuss
Have students draw
Have students exchange
Have students explain
Have students find
Have students look
Have students pronounce
Have students review

Have students simulate
Have students take notes
Include
Log on
Make sure student read
Make sure students fold
Make sure students understand
Monitor
Partner up
Present
Recommend
Refer
Repeat
Replace
Review
Roll
Search
See
Spot check
Start with
Stress
Urge
Watch

132

List of all imperatives used by Glencoe in Geometry: Polygons

Use
Ask
Remind
Have students write
Make
Encourage
Check
Draw
Point out
Have students complete
Have students tell
Have students draw
Tell
Display
Have students create
Have students use
Separate
Suggest
Demonstrate
Discuss
Explain
Have students exchange
Have students make
Have students measure
Have students update
Write

133

List of all imperatives used by MiC in Comparing Quantities

Ask
Have students share
Point out
Encourage
Provide
Be sure students understand
Discuss
Have parents review
Suggest
Copy
Have students give
Read
Allow
Be sure to discuss
Have students find
Have students try
Add
Be sure stud can explain
Begin
Challenge
Do not expect
Do not label one strategy
Do not try
Expect
Give
Have students act
Have students add
Have students begin
Have students circle
Have students create
Have students cross
Have students explain
Have students play
Have students put
Have students read
Have students use
Have students work
Leave
Let
Make

Make sure students are aware
Make sure students can find
Note
Reinforce
Remember
Remind
Say
See
Tell

134

List of all imperatives used by MiC in Triangles and Beyond

Ask
Discuss
Encourage
Remind
Add
Have students discuss
Have students reread
Be sure students know
Have students model
Point out
Have students explain
Let
Advise
Save
Use
Have students review
Involve
Show
Have students work
Have students share

135

List of all imperatives used by UCSMP in Using Variables

Ask
Give
Have students use
Point out
Encourage
Have students create
Have students work
Remind
Have students write
Emphasize
Administer
Have students look
Assure
Have students identify
Watch
Have students check
Help
ES evaluate
Look
Write
Compare
Relate
Call
Refer
Notice
Have students exchange
Have students draw
Display
Show
Have students record
Advise
Return
Assign

136

List of all imperatives used by UCSMP in Some Important Geometry Ideas

Ask
Point Out
Emphasize
Use
Be sure students include
Draw
Encourage
Explain
Extend
Advise
Have students trace
Administer
Have students solve
Assign

137

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138

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