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Geometric transformations in middle school mathematics textbooks

ProQuest Dissertations and Theses, 2011
Dissertation
Author: Barbara Zorin
Abstract:
This study analyzed treatment of geometric transformations in presently available middle grades (6, 7, 8) student mathematics textbooks. Fourteen textbooks from four widely used textbook series were evaluated: two mainline publisher series, Pearson (Prentice Hall) and Glencoe (Math Connects); one National Science Foundation (NSF) funded curriculum project textbook series, Connected Mathematics 2; and one non-NSF funded curriculum project, the University of Chicago School Mathematics Project (UCSMP). A framework was developed to distinguish the characteristics in the treatment of geometric transformations and to determine the potential opportunity to learn transformation concepts as measured by textbook physical characteristics, lesson narratives, and analysis of student exercises with level of cognitive demand. Results indicated no consistency found in order, frequency, or location of transformation topics within textbooks by publisher or grade level. The structure of transformation lessons in three series (Prentice Hall, Glencoe, and UCSMP) was similar, with transformation lesson content at a simplified level and student low level of cognitive demand in transformation tasks. The types of exercises found predominately focused on students applying content studied in the narrative of lessons. The typical problems and issues experienced by students when working with transformations, as identified in the literature, received little support or attention in the lessons. The types of tasks that seem to embody the ideals in the process standards, such as working a problem backwards, were found on few occurrences across all textbooks examined. The level of cognitive demand required for student exercises predominately occurred in the Lower-Level , and Lower-Middle categories. Research indicates approximately the last fourth of textbook pages are not likely to be studied during a school year; hence topics located in the final fourth of textbook pages might not provide students the opportunity to experience geometric transformations in that year. This was found to be the case in some of the textbooks examined, therefore students might not have the opportunity to study geometric transformations during some middle grades, as was the case for the Glencoe (6, 7), and the UCSMP (6) textbooks, or possibly during their entire middle grades career as was found with the Prentice Hall (6, 7, Prealgebra) textbook series.

Table of Contents List of Tables _______________________________________________________ viii

List of Figures _________________________________________________________ x

Abstract _____________________________________________________________ xii

Chapter 1: Introduction and Rationale for the Study ___________________________ 1

Opportunity to Learn and Levels of Cognitive Demand __________________ 6

Statement of the Problem _________________________________________ 10

The Purpose of the Study _________________________________________ 11

Research Questions ______________________________________________ 12

Significance of the Study _________________________________________ 13

Conceptual Issues and Definitions __________________________________ 15

Chapter 2: Literature Review ____________________________________________ 21

Literature Selection ________________________________________ 21

The Curriculum and the Textbook __________________________________ 22

Types of Curriculum _______________________________________ 22

The Mathematics Textbook and the Curriculum _________________ 23

The Textbook and its Use in the Classroom _____________________ 25

Curriculum Analysis _______________________________________ 28

Related Textbook Content Analysis _________________________________ 29

Types of Textbook Content Analysis __________________________ 30

Curriculum Content Analysis for Textbook Selection _____________ 33

ii

Curriculum Content Analysis for Comparison to International Tests _______________________________________ 35

Content Analysis on Textbook Presentations and Student Expectations __________________________________________________ 38

Analyses of Levels of Cognitive Demand Required in Student Exercises ____________________________________________________ 43

Research on Transformation Tasks and Common Student Errors __________ 43

Transformations __________________________________________ 44

Issues Students Experience with Transformation Concepts _________________________________________ 46

Translations __________________________________ 46

Reflections __________________________________ 47

Rotations ____________________________________ 48

Dilations ____________________________________ 50

Composite Transformations _____________________ 51

Conceptual Framework for Content Analysis of Geometric Transformations _______________________________________________ 52

Summary of Literature Review _____________________________________ 53

Chapter 3: Research Design and Methodology 54

Research Questions 54

Sample 55

Development of the Coding Instrument for Analysis of Transformations 59

Global Content Analysis Conceptual Framework ______________________ 61

Sample Application of the Coding Instrument 69

iii Reliability Measures 71

Summary of Research Design and Methodology 75

Chapter 4: Findings ____________________________________________________ 77

Research Questions _____________________________________________ 77

Analysis Procedures _____________________________________________ 78

Organization of the Chapter _______________________________________ 79

Physical Characteristics of Transformation Lessons in Each Series _________________________________________________ 80

Location of Pages Related to Transformations __________________ 80

Relative Position of Transformation Lessons ____________________ 83

Lesson Pages Related to Each Type of Transformation ____________ 86

Characteristics and Structure of Transformation Lessons ________________ 88

Components of Transformation Lessons _______________________ 88

Characteristics of Transformation Constructs in Each Textbook Series _________________________________________ 91

Prentice Hall Textbook Series _________________________ 91

Symmetry, Line of Symmetry, and Reflection _______ 91

Translations __________________________________ 92

Rotations ____________________________________ 93

Dilations ____________________________________ 94

Glencoe Textbook Series _____________________________ 94

Symmetry ___________________________________ 95

Reflection and Translations _____________________ 95

Rotations ____________________________________ 96

iv

Dilations ____________________________________ 97

Connected Mathematics 2 Textbook Series _______________ 97

Symmetry and Line of Symmetry _________________ 98

Reflections, Translations, and Rotations ___________ 98

Dilations ____________________________________ 99

UCSMP Textbook Series _____________________________ 99

Symmetry and Reflections ______________________ 99

Translations _________________________________ 100

Rotations ___________________________________ 101

Dilations ___________________________________ 101

Summary of Textbook Series _________________________ 102

Number of Transformation Tasks __________________________________ 102

Number of Tasks in Each Series _____________________________ 103

Number of Each Type of Transformation Task in Student Exercises _______________________________________ 105

Characteristics of the Transformation Tasks in the Student Exercises _____________________________________________ 110

Translations _______________________________________ 111

Reflections _______________________________________ 113

Rotations _________________________________________ 118

Dilations _________________________________________ 120

Composite Transformations __________________________ 122

Student Exercises Analyzed by the Characteristics of Performance Expectations ___________________________ 124

v Suggestions for Instructional Aids and Real-World Connections ___________________________________________ 126

Student Exercises Summarized by Textbook Series ______________ 129

Prentice Hall ______________________________________ 129

Glencoe __________________________________________ 129

Connected Mathematics 2 ____________________________ 130

UCSMP __________________________________________ 130

Level of Cognitive Demand Expected by Students in the Transformation Exercises ______________________________________ 130

Summary of Findings ___________________________________________ 134 “Where” the Content of Transformation Lessons are Located in the Textbooks _________________________________ 135

“What is Included in the Transformation Lessons of each Textbook Series ________________________________________ 137

“How” Transformation Exercises are Presented in the Lessons _______________________________________________ 138

Level of Cognitive Demand Required by Student Exercises _______ 141

Chapter 5: Summary and Conclusions ____________________________________ 145

Overview of the Study __________________________________________ 145

Research Questions _____________________________________________ 146

Purpose of the Study ____________________________________________ 147

Summary of Results ____________________________________________ 148

Opportunity to Learn Transformation Concepts in the Prentice Hall Textbook Series _____________________________ 149

Opportunity to Learn Transformation Concepts in the Glencoe Textbook Series _________________________________ 151

Opportunity to Learn Transformation Concepts in the

vi Connected Mathematics 2 Textbook Series ___________________ 153

Opportunity to learn transformation concepts in the University of Chicago School Mathematics Project Textbook series ________________________________________ 156

Discussion ____________________________________________________ 158

Limitations of the Study _________________________________________ 163

Significance of the Study ________________________________________ 165

Implications for Future Research __________________________________ 168

References __________________________________________________________ 174

Appendices _________________________________________________________ 212

Appendix A: Pilot Study ________________________________________ 213

Appendix B: Composite Transformation Sample Conversions and Properties List ___________________________________ 233

Appendix C: Properties of Geometric Transformations Expected to be Present in Lessons ___________________________ 235

Appendix D: Aspects of Transformations and Student Issues __________ 236

Appendix E: Examples of Student Performance Expectations in Exercises ______________________________________ 238

Appendix F: Coding Instrument __________________________________ 243

Appendix G: Instrument Codes for Recording Characteristics of Student Exercises ________________________________ 248

Appendix H: Transformation Type Sub-grouped Categories and Exercises _______________________________________ 250

Appendix I: Examples of Tasks Characterized by Levels of Cognitive Demand in Exercises _____________________ 256

Appendix J: Background for Content Analysis and Related Research Studies _________________________________ 259

vii

Appendix K: Transformation Topic Sub-grouped Categories and Examples _______________________________________ 262

viii

List of Tables Table 1 Levels of Cognitive Demand for Mathematical Tasks 44

Table 2 Textbooks Selected for Analysis with Labels Used for This Study 57

Table 3 Three Stages of Data Collection and Coding Procedures 66

Table 4 Terminology for Transformation Concepts 67

Table 5 Reliability Measures by Textbook Series 75

Table 6 Pages Containing Geometric Transformations in the Four Textbook Series 81

Table 7 Geometric Transformations Lessons/Pages in Textbooks 84

Table 8 Number of Pages of Narrative and Exercises by Transformation Type 87

Table 9 Number and Percent of Each Transformation Type to the Total Number of Transformation Tasks in Each Textbook 109

Table 10 Percent of Each Type of Translation Task to the Total Number of Translation Tasks in Each Textbook 112

Table 11 Percent of Each Type of Reflection Task to Total Number of Reflection Tasks in Each Textbook 115

Table 12 Percent of Each Type of Rotation Task to Total Number of Rotation Tasks in Each Textbook 119

Table 13 Percent of Each Type of Dilation Task to Total Number of Dilation Tasks in Each Textbook 122

Table 14 Number of Composite Transformation Exercises in Each Textbook Series 123

Table 15 Number of Suggestions for the Use of Manipulatives, Technology, and Real-World Connections to Mathematics Concepts 128

ix Table 16 Percent of Each Level of Cognitive Demand Required by Student Exercises on Transformations in Each Textbook and Textbook Series 132

Table 17 Transformation Page Number Average and Standard Deviation in each Textbook Series 136

x

List of Figures Figure 1. Global Content Analysis Conceptual Framework 62

Figure 2. Conceptual Framework: Content Analysis of Two Dimensional Geometric Transformation Lessons in Middle Grades Textbooks 64

Figure 3. Example 1 - Sample of Student Exercise for Framework Coding 69

Figure 4. Example 2 - Sample of Student Exercise for Framework Coding 69

Figure 5. Example 3 - Sample of Student Exercise for Framework Coding 70

Figure 6. Example 4 - Sample of Student Exercise for Framework Coding 70

Figure 7. Placement of Transformation Topics in Textbooks by Percent of Pages Covered Prior to Lessons 85

Figure 8. Rotation Example 93

Figure 9. Number of Transformation Tasks in Each Series by Grade Level 104

Figure 10. Number of Each Transformation Type in Each Textbook by Series 106

Figure 11. Total Number of Transformation Exercises in Each Textbook Series 108

Figure 12. Example of General Translation Exercise 112

Figure 13. Summary of Translation Exercises in the Middle School Textbook Series 114

Figure 14. Example of Reflection Exercise - Rf over x 116

Figure 15. Example of Reflection Exercise - Rfo (over/onto preimage) 116

xi Figure 16. Summary of Reflection Exercises in the Middle School Textbook Series 117

Figure 17. Summary of Rotation Exercises in the Middle School Textbook Series 121

Figure 18. Summary of Dilation Exercises in the Middle School Textbook Series 123

Figure 19. Sample Composite Transformation Student Exercises 124

Figure 20. Analysis by Number of Type of Performance Expectations in the Transformation Exercises in the Textbook Series 125

Figure 21. Example of Exercise with Real-World Relevance without Connections 126

Figure 22. Example of Dilation Exercise with Real-World Connections 127

Figure 23. Example of Dilation with Real-World Connections 127

Figure 24. Level of Cognitive Demand Required by Students on Transformation Exercises in Each Textbook 133

Figure 25. Total Number of the Four Transformation Exercises in Each Textbook Series 141

Figure 26. Percent of Levels of Cognitive Demand in Student Exercises in each Textbook Series 143

xii

Abstract This study analyzed treatment of geometric transformations in presently available middle grades (6, 7, 8) student mathematics textbooks. Fourteen textbooks from four widely used textbook series were evaluated: two mainline publisher series, Pearson (Prentice Hall) and Glencoe (Math Connects); one National Science Foundation (NSF) funded curriculum project textbook series, Connected Mathematics 2; and one non-NSF funded curriculum project, the University of Chicago School Mathematics Project (UCSMP). A framework was developed to distinguish the characteristics in the treatment of geometric transformations and to determine the potential opportunity to learn transformation concepts as measured by textbook physical characteristics, lesson narratives, and analysis of student exercises with level of cognitive demand. Results indicated no consistency found in order, frequency, or location of transformation topics within textbooks by publisher or grade level. The structure of transformation lessons in three series (Prentice Hall, Glencoe, and UCSMP) was similar, with transformation lesson content at a simplified level and student low level of cognitive demand in transformation tasks. The types of exercises found predominately focused on students applying content studied in the narrative of lessons. The typical problems and issues experienced by students when working with transformations, as identified in the literature, received little support or attention in the lessons. The types of tasks that seem to embody the ideals in the process standards, such

xiii as working a problem backwards, were found on few occurrences across all textbooks examined. The level of cognitive demand required for student exercises predominately occurred in the Lower-Level, and Lower-Middle categories. Research indicates approximately the last fourth of textbook pages are not likely to be studied during a school year; hence topics located in the final fourth of textbook pages might not provide students the opportunity to experience geometric transformations in that year. This was found to be the case in some of the textbooks examined, therefore students might not have the opportunity to study geometric transformations during some middle grades, as was the case for the Glencoe (6, 7), and the UCSMP (6) textbooks, or possibly during their entire middle grades career as was found with the Prentice Hall (6, 7, Prealgebra) textbook series.

1

Chapter 1: Introduction and Rationale for the Study The branch of mathematics that has the closest relationship to the world around us, as well as the space in which we live is geometry (Clements & Samara, 2007; Leitzel, 1991; National Council of Teachers of Mathematics (NCTM), 1989). Furthermore, geometry is a vehicle by which we develop an understanding of space that is necessary for comprehending, interpreting, and appreciating our inherently geometric world (NCTM, 1989). Spatial geometry provides us with the knowledge to understand (Leitzel, 1991) and interpret our physical environment (Clements, 1998; NCTM, 1992); this knowledge provides us with intellectual instruments to sort, classify, draw (NCTM, 1992), use measurements, read maps, plan routes (NCTM, 2000), create works of art (Clements, Battista, Sarama & Swaminathan, 1997; NCTM, 2000), design plans, and build models (NCTM, 1992). Spatial geometry also provides us with the knowledge necessary for engineering (NCTM, 2000) and building (Clements, Battista, Sarama & Swaminathan, 1997), in addition to the aptitude to develop logical thinking abilities, creatively solve problems (NCTM, 1992), and design advanced technological settings and computer animations (Clements et al, 1997; Yates, 1988). Additionally, spatial geometry helps us understand and strengthen other areas of mathematics as well as provides us with the tools necessary for the study of other subjects (Boulter & Kirby, 1994). Spatial geometry includes the contemporary study of form, shape, size, pattern, and design. Spatial reasoning concentrates on the mental representation and manipulation

2 of spatial objects. Geometry is described by Clements and Battista (1992) and Usiskin (1987) as having four conceptual aspects. The first conceptual aspect is visualization, depiction, and construction; this conception focuses on visualization, sequence of patterns, and physical drawings. The second aspect is the study of the physical situations presented in the real world that direct the learner to geometric concepts, as a carpenter squaring a framing wall with the use of the Pythagorean Theorem. The third aspect provides representations for the non-physical or non-visual, as with the use of the number line to represent real numbers. The fourth aspect is a representation of the mathematical system with its logical organization, justifications, and proofs. The first three conceptual aspects of geometry necessitate the use of spatial sense, which can be learned and reinforced during the study of geometric transformations. The study of transformations supports the interpretation and description of our physical environment as well as provides us with a valuable tool in problem solving in many areas of mathematics and in real world situations (NCTM, 2000). The study of geometric transformations begins with the student‟s journey into the understanding of visualization, mental manipulation, and spatial orientation with regard to figures and objects. Through the study of transformations, Clements and Battista (1992) and Leitzel (1991) assert that students develop spatial visualization and the ability to mentally transform two dimensional images. Two dimensional transformations are an important topic for all students to study and the recommendation is that all middle grades students study transformations (NCTM, 1989, 2000, 2006). The study of geometry with transformations has enhanced geometry to a dynamic level by providing the student with a powerful problem-solving tool (NCTM, 1989).

3 Spatial reasoning and spatial visualization through transformations help us build and manipulate mental representations of two dimensional objects (NCTM, 2000). Students need to investigate shapes, including their components, attributes, and transformations. Additionally, students need to have the opportunity to engage in systematic explorations with two dimensional figures including representations of their physical motion (Clements, Battista, Sarama, & Swaminathan, 1997). Geometric transformations, for middle school students, are composed of five basic concepts: translations (slides), reflections (flips or mirror images), rotations (turns), dilations (size changes), and the composite transformation of two or more of the first three (Wesslen & Fernandez, 2005). Transformation concepts provide background knowledge to develop new perspectives in visualization skills to illuminate the concepts of congruence and similarity in the development of spatial sense (NCTM, 1989). Spatial reasoning, including spatial orientation and spatial visualization, is an aptitude that directly relates to an individual‟s mathematical ability (Brown & Wheatley, 1989; Clements & Sarama, 2007). It also directly influences success in subsequent geometry coursework and general mathematics achievement, which, in turn, directly affects the student‟s future career options (Ma & Wilkins, 2007; NCTM, 1989). Research suggests that students should have a functioning knowledge of geometric transformations by the end of eighth grade in order to be successful in higher level mathematics studies (Carraher & Schlieman, 2007; Flanders, 1987; Ina-Wilkins, 2007; Ladson-Billings, 1998; Knuth, Stephens, McNeil, & Alibali, 2006; National Assessment of Educational Progress (NAEP), 2004; NCTM, 2000; National Research Council (NRC), 1998). However, the academic performance of United States students in

4 geometry, and more specifically in spatial reasoning, is particularly low (Battista, 2007; Silver, 1998; Sowder, Wearne, Martin, & Strutchens, 2004). Because of long standing concerns about student achievement, recommendations by major national mathematics and professional educational organizations, such as the NCTM, the National Commission on Excellence in Education, and the NRC, call for essential alterations in school mathematics curricula, instruction, teaching, and assessment (NCTM, 1989, 1991, 1992, 1995, 2000, 2006; NRC, 1998). In particular, the NCTM published three milestone documents which developed mathematics curriculum standards for grades K - 12 that focused on school mathematics reform. The Curriculum and Evaluations Standards (NCTM, 1989) includes a vision for the teaching and learning of school mathematics, including a vision of mathematical literacy. This document also includes recommendations for the study of transformations of geometric figures to enhance the development of spatial sense for all students. The document‟s recommendations suggest that students should have an opportunity to study two dimensional figures through visualization and exploration of transformations. NCTM revised and updated the Standards with its publication of the Principles and Standards for School Mathematics (PSSM) (NCTM, 2000). This document extends the previous recommendations by providing clarification and elaboration on the curricula described, as well as specifically identifying expectations for each grade band: preK-2, 3- 5, 6-8 and 9-12. PSSM offers specific content guidelines for all students, and examples for teaching, as well as specific principles and features to assist students in attaining high quality mathematics understanding. The expectations for students are delineated in each of the mathematical strands. For example, in the PK - 2 grade band, PSSM recommends

5 that students should be able to recognize symmetry and geometric transformations of figures with the use of manipulatives; in grades 3-5, students should be able to predict and describe the results of geometric transformations and recognize line and rotational symmetry. In the 6 - 8 grade band, PSSM recommends that students should apply transformations; describe size, positions and orientations of geometric shapes under slides, flips, turns, and scaling; identify the center of rotation and line of symmetry; and examine similarity and congruence of these figures. The Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (Focal Points) (NCTM, 2006) further extended the recommended standards and delineated a coherent progression of concepts and expectations for students with descriptions of the most significant content for curriculum focus within each grade level from pre-kindergarten through grade eight. Focal Points extends the mathematics ideals set forth in the PSSM by targeting curriculum content and by providing resources that support the development of a coherent curriculum (Fennell, 2006). The Focal Points document reinforces the need for students to discuss their thinking, to use multiple representations that bring out mathematical connections, and to use problem solving in the process of learning. Of these milestone documents, PSSM (2000) offers the most specific and delineated recommendations for school mathematics content. Sufficient time has passed since the publication of PSSM to expect to observe substantial alignment to the recommended content in published textbooks. The NCTM (1989) stated that they expected the standards to be reflected in textbook content and that the standards should also be used as criteria for analyzing textbook content.

6 The PSSM (NCTM, 2000) can only be put into practice when its recommendations can be implemented. Hiebert and Grouws (2007) emphasize that the most important factor in student achievement is opportunity to learn, and one criterion for student opportunity to learn is the expectation that the prescribed curriculum standards be reflected within textbook contents (NCTM, 1989). The textbook is an influential factor on student learning (Begle, 1973; Grouws et al., 2004; Schmidt et al., 2001; Valverde et al., 2002), and it represents a variable that can be easily manipulated. On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (National Research Council, 2004) suggests that curriculum evaluation should begin with content analyses. Confrey (2006) affirms that content analysis is a critical element in the link between standards and the effectiveness of the curriculum. Textbook content analysis typically focuses on specific characteristics of the textbooks‟ content. Of the various characteristics analyzed, opportunity to learn and levels of cognitive demand are frequently used as measurements of the potential effectiveness of the reviewed materials. Both the characteristics, opportunity to learn, and levels of cognitive demand, are discussed in the next section. Opportunity to Learn and Levels of Cognitive Demand Tornroos (2005) describes the intended curriculum as the goals and objectives that are set down in curriculum documents; the curriculum documents most frequently used in the classroom are textbooks. An important contributing factor in learning outcomes is the opportunity to learn (OTL) based on textbook content (Tornroos, 2005). Tornroos found a high correlation between an item level analysis and student performance on the Third International Mathematics and Science Study (1999) and

7 suggested that content analysis of textbooks would be valuable when looking for justification for different student achievement in mathematics. Schmidt (2002) suggested that differences in student opportunity to learn did not suddenly appear in the eighth grade level, but rather in earlier grades, and that differences in curriculum diversity, to a large degree, cost student achievement exceedingly. Tarr, Reys, Barker, and Billstein (2006) report that it is crucial to identify and select textbooks that present critical features of mathematics that support student learning and assist teachers in helping students to learn. Tarr et al. describe the critical features of providing support, focus, and direction in the mathematics textbook and they call for the analysis of content emphasis within a textbook and across the span of textbooks within a series. Opportunity to learn can be studied in various ways as indicated above, and OTL can have a variety of meanings. Although Tornroos and Schmidt considered the relationship of OTL to test performance, Floden (2002) determined the opportunity to learn by the emphasis a topic receives in the written materials in the form of textbooks since they are the form used by the student. This study takes a somewhat broader view and considers opportunity to learn not only by the amount of emphasis a mathematical concept receives in student textbooks but also by the nature of lesson presentations, types of tasks presented for student activity, and the level of cognitive demand required by students to complete tasks. The NCTM set forth ideals for mathematics with recommendations for the teaching and learning of worthwhile tasks, including expectations that students will develop problem solving skills and critical thinking abilities. The PSSM (NCTM, 2000) document describes the necessity for learning mathematics content through meaningful

8 activities that focus on the Process Standards: problem solving, reasoning and proof, communications, connections, and representations. The mathematical tasks that students experience are central to learning because “tasks convey messages about what mathematics is and what doing mathematics entails” (NCTM, 1991, p. 24). Tasks need to provide an opportunity for the student to be active (Henningsen & Stein, 1997) and provoke thought and reasoning in complex and meaningful ways as categorized by Stein and Smith (1998). The results reported in Stein and Lane (1996) suggest, that in order for students to develop the capacity to think, reason, and problem solve in mathematics, it is important to start with high-level, cognitively complex tasks. Some of the high-level cognitive demand tasks include: exploring patterns (Henningsen & Stein, 1997) thinking and reasoning in flexible ways (Henningsen & Stein, 1997; Silver & Stein, 1996) communicating and explaining mathematical ideas (Henningsen & Stein, 1997; Silver & Stein, 1996) conjecturing, generalizing, and justifying strategies while making conclusions (Henningsen & Stein, 1997, Silver & Stein, 1996) interpreting and framing mathematical problems (Silver & Stein, 1996) making connections to construct and develop understanding (Silver & Stein, 1996; Stein & Smith 1998). A major finding of Stein and Lane (1996) and Smith and Stein (1998) was that the largest learning gains on mathematics assessments were from students who were engaged

9 in tasks with high levels of cognitive demand. Thus, the key to improving the performance of students was to engage them in more cognitively demanding activities (Boston & Smith, 2009) and hence provide the foundation for mathematical learning (Henningsen & Stein, 1997; Stein & Smith, 1998). Different types of tasks require higher levels of cognitive demands through active reasoning processes and the higher level demand tasks require students to think conceptually while providing a different set of opportunities for student cognition (Stein & Smith, 1998). Hence, students need to have the opportunity to learn worthwhile mathematical concepts, and be immersed in their mathematical studies with cognitively demanding tasks. NCTM (1989) stated “let it be understood that we hold no illusions of immediate reform” (p. 255), but they held the vision of having classroom materials, such as textbooks, produced so that standards would be aligned and in-depth learning take place. Yet, since the initial publication of the Standards, little has been done to analyze textbook contents. Because students do not learn what they are not taught (Tornroos, 2005), it is essential to examine the extent to which mathematical topics are presented in textbooks. Clements (1998) indicates it is essential to examine the extent to which middle school mathematics textbooks attend to the development of the concept of transformations in available instruction and in mathematics research. If there is a barrier to students in “opportunity to learn” which prevents them from attaining the full benefits from the Standards, educators need to address what can be done to eliminate the barriers; one way to know if a problem exists due to the lack of included content is to analyze the content of textbooks. With the inception of this study a pilot investigation was enacted to analyze the

10 extent and treatment of geometric transformations lessons in two middle grades textbooks to discern if sufficient differences in the curricula were present (Appendix A). The results suggested that an analysis of a larger variety of textbooks was a worthwhile endeavor, and hence this study was implemented. Statement of the Problem Research indicates that students have difficulties in understanding the concepts and variations in performing transformations (Clements & Battista, 1998; Clements, Battista, & Sarama, 1998; Clements & Burns, 2000; Clements, Battista, Sarama, & Swaminathan, 1996; Kieran, 1986; Magina & Hoyles, 1997; Mitchelmore, 1998; Olson, Zenigami & Okzaki, 2008; Rollick, 2009; Soon, 1989). Given recommendations from the mathematics education community about the inclusion of transformations in the middle grades curriculum, we might expect to observe the concepts in published textbooks; hence, there is a need to analyze contents. However, few examinations of the contents within textbooks have been found with respect to the alignment or development of mathematics concepts with current recommendations (Mesa, 2004), and none have been found to focus on the analysis of presentations and opportunity to learn for the study of geometric transformations. Because textbooks are the prime source of curriculum materials on which the student can depend for written instruction (Begle, 1973; Grouws et al., 2004; Schmidt et al., 2001; Valverde et al., 2002), the nature of the treatment of these concepts needs to be examined to insure that students are provided appropriate opportunities to learn. As a result there emerges a need to analyze the treatment of geometric transformations in middle school mathematics textbooks. This study examined the nature and treatment of

11 geometric transformations through the analysis of published middle grades textbooks in use in the United States. The textbooks chosen included publisher generated textbooks, curriculum project-developed textbooks, and National Science Foundation (NSF) funded curriculum materials; it was assumed that these textbook types would likely present the concepts differently. The lesson concepts were analyzed in terms of content of the narrative, examples offered for student study, number and types of student exercises, and the level of cognitive demand expected by student exercises. Additionally, this investigation addressed the possible changes of focus in the progression of content from grade six through grade eight. The Purpose of the Study This study had three foci: 1) to analyze the characteristics and nature of geometric transformation lessons in middle grades textbooks to determine the extent to which these textbooks provide students the potential opportunity to learn transformations as recommended in the curriculum standards; 2) to describe the content of geometric transformation lessons to identify the components of those lessons, including how they are sequenced within a series of textbooks from grades 6 through grade 8 and across different publishers; 3) to determine if student exercises included with the transformation lessons facilitate student achievement by the inclusion of processes that encourage conceptual understanding with performance expectations. Four types of middle school transformations were examined: the three rigid transformations and their composites (translations, reflections, and rotations), where rigid refers to the preimage figure and resulting image figure being congruent; and dilation where figures are either enlarged or shrunk. The sections of student exercises that follow

Full document contains 282 pages
Abstract: This study analyzed treatment of geometric transformations in presently available middle grades (6, 7, 8) student mathematics textbooks. Fourteen textbooks from four widely used textbook series were evaluated: two mainline publisher series, Pearson (Prentice Hall) and Glencoe (Math Connects); one National Science Foundation (NSF) funded curriculum project textbook series, Connected Mathematics 2; and one non-NSF funded curriculum project, the University of Chicago School Mathematics Project (UCSMP). A framework was developed to distinguish the characteristics in the treatment of geometric transformations and to determine the potential opportunity to learn transformation concepts as measured by textbook physical characteristics, lesson narratives, and analysis of student exercises with level of cognitive demand. Results indicated no consistency found in order, frequency, or location of transformation topics within textbooks by publisher or grade level. The structure of transformation lessons in three series (Prentice Hall, Glencoe, and UCSMP) was similar, with transformation lesson content at a simplified level and student low level of cognitive demand in transformation tasks. The types of exercises found predominately focused on students applying content studied in the narrative of lessons. The typical problems and issues experienced by students when working with transformations, as identified in the literature, received little support or attention in the lessons. The types of tasks that seem to embody the ideals in the process standards, such as working a problem backwards, were found on few occurrences across all textbooks examined. The level of cognitive demand required for student exercises predominately occurred in the Lower-Level , and Lower-Middle categories. Research indicates approximately the last fourth of textbook pages are not likely to be studied during a school year; hence topics located in the final fourth of textbook pages might not provide students the opportunity to experience geometric transformations in that year. This was found to be the case in some of the textbooks examined, therefore students might not have the opportunity to study geometric transformations during some middle grades, as was the case for the Glencoe (6, 7), and the UCSMP (6) textbooks, or possibly during their entire middle grades career as was found with the Prentice Hall (6, 7, Prealgebra) textbook series.