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Proportionality in middleschool mathematics textbooks
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ABSTRACT: Some scholars have criticized the treatment of proportionality in middleschool textbooks, but these criticisms seem to be based on informal knowledge of the content of textbooks rather than on a detailed curriculum analysis. Thus, a curriculum analysis related to proportionality was needed. To investigate the treatment of proportionality in current middleschool textbooks, nine such books were analyzed. Sixth, seventh, and eighthgrade textbooks from three series were used: ConnectedMathematics2 (CMP), Glencoe's Math Connects, and the University of Chicago School Mathematics Project (UCSMP). Lessons with a focus on proportionality were selected from four content areas: algebra, data analysis/probability, geometry/measurement, and rational numbers. Within each lesson, tasks (activities, examples, and exercises) related to proportionality were coded along five dimensions: content area, problem type, solution strategy, presence or absence of a visual representation, and whether the task contained material regarding the characteristics of proportionality. For activities and exercises, the level of cognitive demand was also noted. Results indicate that proportionality is more of a focus in sixth and seventhgrade textbooks than in eighthgrade textbooks. The CMP and UCSMP series focused on algebra in eighth grade rather than proportionality. In all of the sixthgrade textbooks, and some of the seventh and eighthgrade books, proportionality was presented primarily through the rational number content area. Two problem types described in the research literature, ratio comparison and missing value, were extensively found. However, qualitative proportional problems were virtually absent from the textbooks in this study. Other problem types (alternate form and function rule), not described in the literature, were also found. Differences were found between the solution strategies suggested in the three textbook series. Formal proportions are used earlier and more frequently in the Math Connects series than in the other two. In the CMP series, students are more likely to use manipulatives. The Mathematical Task Framework (Stein, Smith, Henningsen, & Silver, 2000) was used to measure the level of cognitive demand. The level of cognitive demand differed among textbook series with the CMP series having the highest level of cognitive demand and the Math Connects series having the lowest.
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Proportionality i n Middle School Mathematics Textbooks by Gwendolyn Joy Johnson A dissertation su bmitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Secondary Education College of Educ ation University of South Florida Major Professor: Denisse R. Thompson Ph.D. Richard Austin, Ph.D. Helen Gerretson, Ph.D. Elizabeth Shauness y, Ph.D. Date of Approval: May 7, 2010 Keywords: algebra, cognitive demand curriculum, reasoning, rational numbers Copyright 2010 Gwendolyn Joy Johnson
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ACKNOWLEDGEMENTS I would like to thank the members of my dissertation committee: Dr. Denisse Thompson, Dr. Richard Austin, Dr. Helen Gerretson, and Dr. Elizabeth Shaunessy. They have devoted considerable time reading various drafts of the dissertation and have provided valuable guidance and constructive criticism In particular, Dr. Denisse Thompson has always found time to provide assistance, both with this dissertation and with a variety of other issues that arose over the past six years. I am gratefu l for her concern for the welfare of the doctoral students at the University of South Florida I would also like to thank my parents, Fred and Ruth Wilson. It was my step to apply to a doctoral program and th ey have both provided support and encouragement throughout the process. Knowing that they are always in my corner gives me peace of mind in a way that nothing else can. Finally, I am thankful for the doctoral students that I have met at the University of S outh Florida. They have commiserated with me, read drafts of manuscripts, and provided encouragement. In particular, I would like to thank James Dogbey for assistance with the check coding and reliability processes used in this dissertation.
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i TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ................. viii LIST OF FIGURES ................................ ................................ ................................ ................... x ABSTRACT ................................ ................................ ................................ ............................ x ii CHAPTER 1: INTRODUCTION ................................ ................................ .............................. 1 Statement of the Problem ................................ ................................ ............................... 3 Purpose of the Study ................................ ................................ ................................ ...... 5 Research Questions ................................ ................................ ................................ ........ 6 Funding by the National Science Foundation ................................ ................................ 7 Definition of Terms ................................ ................................ ................................ ........ 9 Additive and Multiplicative Reasoning ................................ ............................. 9 Fractions and Rational Numbers ................................ ................................ ...... 10 Measure Space ................................ ................................ ................................ 1 1 Proportionality ................................ ................................ ................................ 1 2 Proportional Reasoning ................................ ................................ .................... 1 3 Rates and Ratios ................................ ................................ ............................... 1 4 Task ................................ ................................ ................................ .................. 1 5 Significance of the Study ................................ ................................ ............................. 1 6 Summary ................................ ................................ ................................ ...................... 18 CHAPTER 2: LITERATURE REVIEW ................................ ................................ ................. 2 0 Content Areas ................................ ................................ ................................ ............... 2 1 Algebra ................................ ................................ ................................ ............. 2 1 Data Analysis and Probability ................................ ................................ ......... 2 2 Geometry and Measurement ................................ ................................ ............ 2 4 Rational Numbers ................................ ................................ ............................ 2 5 Summary of Content Areas ................................ ................................ .............. 2 8 Problem Types ................................ ................................ ................................ ............. 28 Missing Value Problems ................................ ................................ .................. 29 Ratio Comparison Problems ................................ ................................ ............ 3 0 Qualitative Problems ................................ ................................ ........................ 3 1 Summary of Problem Types ................................ ................................ ............ 32 Solution Strategies ................................ ................................ ................................ ....... 3 3 Missing Value Problems ................................ ................................ .................. 3 3 Building Up (Factor of Change) Strategy ................................ ............ 3 3 Unit Rate Strategy ................................ ................................ ................ 3 4 Proportion Strategy ................................ ................................ .............. 37 Ratio Comparison Problems ................................ ................................ ............ 39 Summary of Solution Strategies ................................ ................................ ...... 40
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ii Common Errors ................................ ................................ ................................ ............ 4 0 Inappropriate Application of Additive Reasoning ................................ ........... 4 1 Inappropriate Application of Proportional Reasoning ................................ ..... 4 2 Curriculum Analysis Methodology ................................ ................................ .............. 4 4 Textbook Selection ................................ ................................ .......................... 4 5 Funding by the National Science Foundation ................................ ...... 4 5 Selecting Textbooks ................................ ................................ ............. 4 6 Measuring t he Challenge Inherent in Tasks ................................ ..................... 4 7 Common Methods of Measuring Challenge ................................ ........ 48 The Mathematical Task Framework ................................ .................... 4 9 Common Features of Curriculum Analysis ................................ ..................... 5 1 Number of Pages or Lessons ................................ ................................ 5 2 Problem Types ................................ ................................ ..................... 5 3 Visual Representations ................................ ................................ ......... 5 3 Examples, E xercises, and Tasks ................................ .......................... 5 4 Example of Curriculum Analysis: Reasoning and Proof ..................... 5 4 Example of Curriculum Analysis: Probability ................................ ..... 5 6 Summary of Curriculum Analysis Methodology ................................ ............. 57 Summary of Literature Review ................................ ................................ .................... 5 8 Unanswered Questions ................................ ................................ ................................ 5 9 Summary of the Pilot Study ................................ ................................ ......................... 6 1 CHAPTER 3: METHODS ................................ ................................ ................................ ....... 6 3 Textbook Selection ................................ ................................ ................................ ...... 6 3 Selection Criteria ................................ ................................ ............................. 6 4 Connected Mathematics2 (CMP) ................................ ................................ ..... 6 5 Goals of the CMP curriculum ................................ .............................. 6 5 Format of the CMP curriculum ................................ ............................ 6 6 Glencoe Math Connects ................................ ................................ ................... 6 7 Goals of the Math Connects curriculum ................................ .............. 6 7 Format of the Math Connects curriculum ................................ ............ 6 7 University of Chicago School Mathematics Project (UCSMP) ....................... 6 8 Summary of Textbook Selection ................................ ................................ ..... 6 9 Lesson Selection ................................ ................................ ................................ .......... 6 9 Excluded Sections of Textbooks ................................ ................................ ...... 6 9 Lesson Selection Criteria ................................ ................................ ................. 7 1 Included Algebra L essons ................................ ................................ .... 7 1 Excluded Algebra L essons ................................ ................................ ... 7 1 Included Data Analysis L essons ................................ .......................... 7 2 Excluded Data Analysis L essons ................................ ......................... 7 2 Included Geometry L essons ................................ ................................ 7 2 Excluded Geometry L essons ................................ ................................ 7 3 Lessons on I ntegers ................................ ................................ .............. 7 4 Included Decimal L essons ................................ ................................ ... 7 4 Excluded Decimal L essons ................................ ................................ .. 7 5 Included Fraction L essons ................................ ................................ ... 7 5
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iii Excluded Fraction L essons ................................ ................................ .. 7 6 Included Percent L essons ................................ ................................ ..... 7 6 Excluded Percent L essons ................................ ................................ .... 7 6 Summary of Lesson Selection Criteria ................................ ............................ 7 7 Number of Lessons Included in the Study ................................ ....................... 7 9 Grade Levels of Lessons Included in the Study ................................ ............... 80 Task Selection and Counting ................................ ................................ ....................... 80 Algebra Tasks ................................ ................................ ................................ .. 8 1 Tasks Rela ted to Patterns and Sequences ................................ ............ 8 1 Tasks Related to Function Rules ................................ .......................... 8 1 Tasks Related to Graphing ................................ ................................ ... 8 2 Data Analysis Tasks ................................ ................................ ......................... 8 3 Geometry/Measurement Tasks ................................ ................................ ........ 8 4 Rati onal Number Tasks ................................ ................................ .................... 8 5 Counting Tasks ................................ ................................ ................................ 8 6 Connected Mathematics2 ( CMP ) ................................ ......................... 8 6 Glencoe Math Connects ................................ ................................ ....... 8 7 UCSMP ................................ ................................ ................................ 8 8 Summary of Task Selection and Counting ................................ ...................... 8 9 Number of Tasks Included in the Study ................................ .......................... 8 9 Grade Levels of Selected Tasks ................................ ................................ ....... 90 Framework for Curriculum Analysis ................................ ................................ ........... 9 1 Content Area ................................ ................................ ................................ .... 9 2 Problem Type ................................ ................................ ................................ ... 9 3 Missing Value ................................ ................................ ...................... 9 3 Ratio Comparison ................................ ................................ ................ 9 4 Qualitative ................................ ................................ ............................ 9 5 Alternate Form ................................ ................................ ..................... 9 6 Function Rule ................................ ................................ ....................... 9 6 Other ................................ ................................ ................................ .... 9 7 Solution Strategy ................................ ................................ .............................. 9 7 Building Up ................................ ................................ .......................... 9 8 Decimal ................................ ................................ ................................ 9 8 Manipulatives ................................ ................................ ....................... 9 8 Proportion ................................ ................................ ............................ 9 9 Unit rate ................................ ................................ ............................... 9 9 Other ................................ ................................ ................................ .. 100 No solution strategy ................................ ................................ ........... 100 Characteristics of Proportional Reasoning ................................ ..................... 100 Visual Representation ................................ ................................ .................... 1 0 1 Level of Cognitive Demand ................................ ................................ ........... 10 1 Memorization ................................ ................................ ..................... 10 2 Procedures Without Connections ................................ ....................... 10 2 Procedures With Connections ................................ ............................ 10 3 Doing Mathematics ................................ ................................ ............ 10 4 Summary of the Framework ................................ ................................ .......... 10 4
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iv Narrative Text ................................ ................................ ................................ 1 0 7 Reliability ................................ ................................ ................................ ................... 1 0 7 Reliability Procedures ................................ ................................ .................... 1 0 7 Lessons Coded by the Doctoral Student ................................ ............ 1 0 8 Reliability Training ................................ ................................ ............ 1 0 9 Reliability of Task Selection ................................ ................................ .......... 1 0 9 Reliability of Task Selection in CMP ................................ ................ 1 10 Reliability of Task Selection in Math Connects ................................ 1 10 Reliability of Task Selection in UCSMP ................................ ........... 11 1 Summary of Reliability of Task Selection ................................ ......... 11 2 Reliability of Coding ................................ ................................ ...................... 11 2 Reliability of Problem Type Coding ................................ .................. 1 1 3 Reliability of Level of Cognitive Demand Coding ............................ 1 1 4 Reliability for the CMP Textbooks ................................ ........ 1 1 5 Reliability for the Math Connects Grade 8 Textbook ............ 1 1 6 Reliability for the UCSMP Grade 8 Textbook ...................... 11 7 Reliability of Coding Over Time ................................ ....................... 1 1 7 Summary of Reliability of Coding ................................ ..................... 1 1 8 Summary of Research Methods ................................ ................................ ................. 1 1 9 CHAPTER 4: RESULTS ................................ ................................ ................................ ....... 1 2 0 Content Areas ................................ ................................ ................................ ............. 1 2 3 Content Areas and Textbook Series ................................ ............................... 1 2 3 Content Areas, Textbook Series, and Grade Levels ................................ ...... 1 2 4 Content Areas of Sixth Grade Textbooks ................................ .......... 1 2 4 Content Areas of Seventh Grade Textbooks ................................ ..... 1 2 6 Seventh grade CMP ................................ ............................... 1 2 7 Seventh grade Math Connects ................................ ............... 1 2 8 Seventh grade UCSMP ................................ .......................... 1 2 8 Content Areas of Eighth Grade Textbooks ................................ ........ 1 29 Eighth grade CMP ................................ ................................ 1 3 0 Eighth grade Math Connects ................................ ................. 1 3 0 Eighth grade UCSMP ................................ ............................ 1 3 0 Summary of Findings Related to Content Area ................................ ............. 1 3 1 Problem Type ................................ ................................ ................................ ............. 1 3 2 Problem Type and Conte nt Areas ................................ ................................ .. 1 3 2 Problem Type and Textbook Series ................................ ............................... 1 3 3 Problem Type and Grade Levels ................................ ................................ .... 1 3 4 Problem Type, Textbook Series, and Grade Levels ................................ ...... 1 3 6 Missing Value Tasks in UCSMP ................................ ................................ ... 1 3 7 Qualitative Tasks ................................ ................................ ........................... 1 3 8 Solution Strategy ................................ ................................ ................................ ........ 1 39 Solution Strategy and Grade Level ................................ ................................ 1 4 0 Solution Strategy, Textbook Series and Grade Level ................................ ... 1 4 1 The Proportion Strategy ................................ ................................ ................. 1 4 2 The Unit Rate Strategy ................................ ................................ ................... 1 4 4
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v Other Solution Strategies ................................ ................................ ............... 1 4 5 Level of Cognitive Demand ................................ ................................ ....................... 1 4 5 Level of Cognitive Demand and Content Area ................................ .............. 1 4 6 Level of Cognitiv e Demand and Textbook Series ................................ ......... 1 4 7 Cognitive Demand in CMP Textbooks ................................ .............. 1 4 8 Level of Cognitive Demand and Grade Level ................................ ............... 1 49 Visual Representat ion ................................ ................................ ................................ 1 5 0 Visual Repres entation and Textbook Series ................................ .................. 1 5 0 Visual Representation in CMP Textbooks ................................ ......... 15 1 Visual Re presentation and Grade Level ................................ ........................ 15 2 Characteristics of Proportionality ................................ ................................ .............. 1 5 3 Characteristics and Textbook Series ................................ .............................. 1 5 3 Characteristics in CMP Textbooks ................................ .................... 1 5 4 Characteristics in Math Connects Textbooks ................................ .... 1 5 7 Characteristics in UCSMP Textbooks ................................ ............... 1 5 8 Characteristics and Grade Level ................................ ................................ .... 1 59 Overview of Textbook Series ................................ ................................ .................... 1 6 0 Proportionality in CMP Textbooks ................................ ................................ 1 6 0 The Sixth grade CMP Textbook ................................ ........................ 1 6 1 The Seventh grade CMP Textbook ................................ ................... 16 2 T he Eighth grade CMP Textbook ................................ ...................... 1 6 3 Proportionality in Math Connects Textbooks ................................ ................ 1 6 4 The Sixth grade Math Connects Textbook ................................ ........ 1 6 4 The Seventh grade Math Connects Textbook ................................ .... 1 6 5 The Eighth grade Math Connects Textboo k ................................ ...... 1 6 6 Proportionality in UCSMP Textbooks ................................ ........................... 1 6 7 The Sixth grade UCSMP Textbook ................................ ................... 1 6 8 The Seventh grade UCSMP Textbook ................................ .............. 1 69 The Eighth grade UCSMP Textbook ................................ ................. 1 69 Summary of Results ................................ ................................ ................................ ... 1 7 0 CHAPTER 5: DISCUSSION ................................ ................................ ................................ 1 7 2 Grade Levels ................................ ................................ ................................ .............. 17 4 Sixth Grade ................................ ................................ ................................ .... 17 4 Seventh Grade ................................ ................................ ................................ 1 7 5 Eighth Grade ................................ ................................ ................................ .. 1 7 6 Content Areas ................................ ................................ ................................ ............. 1 7 7 Content Areas of Tasks in Sixth Grade Textbooks ................................ ....... 1 7 8 Content Areas of Tasks in Seventh Grade Textbooks ................................ ... 1 8 0 Content Areas of Tasks in Eighth Grade Textbooks ................................ ..... 1 8 1 Content Area Progression From Six th to Eighth Grade ................................ 1 8 1 Proportionality as a Connection Between Content Areas .............................. 1 8 2 Problem Types ................................ ................................ ................................ ........... 18 4 Alternate Form and Ratio Comparison ................................ .......................... 18 4 Missing Value ................................ ................................ ................................ 18 4 Function Rule ................................ ................................ ................................ 1 8 6
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vi Qualitative ................................ ................................ ................................ ...... 1 8 7 Differences Between Textbook Series ................................ ................................ ....... 1 8 7 Amount of R epetition Among Grade Levels ................................ ................. 1 8 7 Algebra in Eighth Grade Textbooks ................................ .............................. 1 8 8 Procedural Versus Conceptual Understanding ................................ .............. 1 89 Solution Strategies ................................ ................................ ............. 1 9 0 Level of Cognitive Demand ................................ ............................... 1 9 2 Ch aracteristics of Proportional Situations ................................ ......... 1 9 3 Conclusion ................................ ................................ ................................ ................. 1 9 5 Limitations ................................ ................................ ................................ ................. 19 7 Limited Sample Size ................................ ................................ ...................... 19 8 Limitations of the Framework ................................ ................................ ....... 19 8 Intended Versus Implemented Curriculum ................................ .................... 199 Recommendations ................................ ................................ ................................ ...... 20 0 Recommendations for Curriculum Developers ................................ ............. 20 0 Recommendations for Future Research ................................ ......................... 20 2 REFERENCES ................................ ................................ ................................ ...................... 20 4 APPEND IX A: PILOT STUDY ................................ ................................ ............................ 2 1 7 APPENDIX B: LESSONS INCLUDED IN THE STUDY ................................ ................... 2 5 2 A PPENDIX C : TRAINING MODULE ................................ ................................ ................. 2 6 2 A PPENDIX D : R ELIABILITY TEST ................................ ................................ .................. 2 6 4 ABOUT THE AUTHOR ................................ ................................ ............................. End Page
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vii LIST OF TABLES Table 1 Topics of Included and Excluded Lessons ................................ ....................... 7 8 Table 2 Number and Percent of Lessons Included in the Study From Each Textbook ................................ ................................ ........................ 7 9 Table 3 The Curriculum Analysis Framework ................................ ........................... 1 0 6 Table 4 Number and Percent of Lessons that were Double Coded ............................ 1 0 8 Table 5 Reliability of Task Selection by Textbook Series ................................ .......... 1 1 3 Table 6 Reliability of Problem Type Coding by Series and Grade ............................ 1 1 4 Table 7 Reliability of Cognitive Demand Coding by Series and Grade ..................... 1 1 5 Table 8 Number of Tasks Related to Proportionality by Series and Grade ................ 1 2 2 Table 9 Percentage of Tasks in Each Content Area by Series ................................ .... 1 2 4 Table 1 0 Percentage o f Tasks of Each Problem Type by Series ................................ .. 1 3 4 Table 1 1 Percentage of Tasks of Each Problem Type by Series and Grade ................. 1 3 7 Table 1 2 Number and Percentage of Tasks w ith a Solution Strategy ........................... 1 4 0 Table 13 Percentage of Tasks with Each Solution Strategy by Grade .......................... 14 1 Table 1 4 Percentage of Tasks with Each Solution Strategy by Textbook Series and Grade Level ................................ ................................ .................. 1 4 2 Table 1 5 Perc entage of Exercises at Each Level of Cognitive Demand by Content Area ................................ ................................ ................................ .. 1 4 7 Table 1 6 Perc entage of Ex ercises at Each Level of Demand by Textbook Series ................................ ................................ ............................. 1 4 8 Table 1 7 Visual Representation by Textbook Series and Grade ................................ .. 1 5 2 Table 1 8 Number and Percentage of Tasks that Point Out the Characteristics of Proportionality ................................ ................................ .. 1 5 4
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viii Table 1 9 Conceptual and Procedural Aspects of Textbook Series ............................... 19 4 Table A1 Lessons and Tasks Included in the Pilot Study ................................ .............. 22 0 Table A2 Percentages of Tasks Related to the Appropriateness of Proportional Reasoning ................................ ................................ .................. 24 2 Table A3 Percent of Examples Related to Proportionality in Each Content Area ................................ ................................ ................................ .. 24 4 Table A4 Percent of Exercises Related to Proportionality in Each Content Area ................................ ................................ ................................ .. 24 4 Table B1 Grade 6 CMP Investigations Included in the Study ................................ ....... 25 2 Table B2 Grade 7 CMP Investigations Included in the Study ................................ ....... 25 3 Table B3 Grade 8 CMP Inves tigations Included in the Study ................................ ....... 25 4 Table B4 Grade 6 Math Connects Lessons Included in the Study ................................ 25 5 Table B5 Grade 7 Math Connects Lessons Included in the Study ................................ 25 6 Table B6 Grade 8 Math Connects Lessons Included in the Study ................................ 25 7 Table B7 Grade 6 UCSMP Le ssons Included in the Study ................................ ........... 25 8 Table B8 Grade 7 UCSMP Lessons Included in the Study ................................ ........... 2 59 Table B9 Grade 8 UCSMP Lessons Included in the Study ................................ ........... 26 0 Table C1 Answer Key for Training Module ................................ ................................ .. 26 3 Table D1 Answer Key for Reliability Test ................................ ................................ .... 266
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ix LIST OF FIGURES Figure 1 Function Rule Exercise Not Related to Proportionality ............................ 8 2 Figure 2 Function Rule Exercise Related to Proportionality ................................ ... 8 2 Figure 3 Data Analysis Exercise Related to Proportionality ................................ .... 8 3 Figure 4 Ratio Example Not Related to Proportionality ................................ .......... 8 5 Figure 5 Percentage of Tasks Related to Proportionality at Each Grade Level ................................ ................................ ................................ 9 1 Figure 6 Task of Problem Type Ratio Comparison ................................ ................. 9 5 Figure 7 Exercise of Problem Type Qualitative ................................ ....................... 9 5 Figure 8 Exercise of Problem Type Function Rule ................................ .................. 9 6 Figure 9 Example with Proportion Solution Strategy ................................ .............. 9 9 Figure 10 Exercise of the Doing Mathematics Level of Demand ............................ 10 4 Figure 1 1 Percentage of Sixth Grade Tasks in Each Content Area ......................... 1 2 5 Figure 1 2 Percentage of Seventh Grade Tasks in Each Content Area ..................... 1 2 7 Figure 1 3 Percentage of Eighth Grade Tasks in Each Content Area ....................... 1 29 Figure 1 4 Percentage of Tasks in Each Content Area by Series .............................. 1 3 2 Figure 1 5 Percentage of Tasks of Types Alternate Form and Ratio Comparison ................................ ................................ .............................. 1 3 5 Figure 1 6 Percentage of Tasks of Types Function R ule and Missing Value ................................ ................................ ................................ ........ 1 3 6 Figure 1 7 Percentage of Tasks at Each Level of Cognitive Demand by Grade ................................ ................................ ................................ ... 1 49 Figure A1 Example of the Problem Type Alternate Form ................................ ........ 22 2
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x Figure A2 Example of the Problem Type Function Rule ................................ .......... 22 3 Figure A3 Example of the Problem Type Missing Value ................................ ......... 22 3 Figure A4 Example of the Problem Type Ratio Comparison ................................ ... 22 4 Figure A5 Example of the Solution Strategy Building Up ................................ ....... 22 5 Figure A6 Example of the Solution Strategy Manipulatives ................................ .... 22 6 Figure A7 Example of the Solution Strategy Proportion ................................ .......... 22 6 Figure A8 Example of the Solution Strategy Unit Rate ................................ ............ 22 7 Figure A9 Percent of Lessons Related to Proportionality in Each Content Area ................................ ................................ ............................ 2 29 Figure A10 Problem Type of Tasks in the Middle School Math Textbook ................ 23 3 Figure A11 Percent of Tas ks in Selected Lessons Related to Proportionality ............ 24 1 Figure A12 Percent of Exercises at Each Level of Cognitive Demand ...................... 2 49
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xi PROPORTIONAILITY IN MIDDLE SCHOOL MATHEMATICS TEXTBOOKS GWENDOLYN JOY JOHNSON ABSTRACT Some scholars have criticized the treatment of proportionality in middle school textbooks, but these criticisms seem to be based on informal knowledge of the content of textbooks rather than on a detailed curriculum analysis. Thus, a curriculum analysis related to proportionality was needed. To investigate the treatment of proportionality in current middle school textbooks, nine such books were analyzed. Sixth seventh and eighth grade textbooks from three series were used: ConnectedMathematics2 (CMP) Math Connects and t he University of Chicago School Mathematics Project (UCSMP) Lessons with a focus on proportionality were selected from four content areas: algebra, data analysis /probability geometry/measurement, and rational numbers. Within each lesson, tasks (activitie s, examples, and exercises) related to proportionality were coded along five dimensions: content area, problem type, solution strategy, presence or absence of a visual representation, and whether the task contained material regarding the characteristics of proportionality. For activities and exercises, the level of cognitive demand was also noted. Results indicate that proportionality is more of a focus in sixth and seventh grade textbooks than in eighth grade text books. The CMP and UCSMP series focused on algebra in eighth grade rather than proportionality. In all of the sixth grade textbooks, and some of the
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xii seventh and eighth grade books, p roportionality was presented primarily through the rational number content area. Two problem types desc ribed in the research literature, ratio comparison and missing value were extensively found However, q ualitative proportional problems were virtually absent from the textbooks in this study. O ther problem types ( alternate form and function rule ) not des cribed in the literature were also found D ifferences were f ound between the solution strategies suggested in the three textbook series. Formal proportions are used earlier and more frequently in the Math Connects series than in the other two. In the CMP series, students are more likely to use manipulatives The Mathematical Task Framework ( Stein, Smith, Henningsen, & Silver, 2000) was used to measure the level of cognitive demand. The level of cognitive demand differed among textbook series with the CMP series having the highest level of cognitive demand and the Math Connects series having the lowest.
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1 CHAPTER 1: INTRODUCTION For decades, there have been criticisms of the U.S. educational system and and science. Educators and researchers have recently come to believe that one way to improve mathematics education is to improve the curriculum ( Tarr et al. 2008 ; Valverde, Bianchi, Wolfe, Schmidt, & Houang, 2002 ) Ball and Cohen ( 1996) expressed the impo rtance of curriculum: Unlike frameworks, objectives, assessments, and other methods that seek to guide curriculum, instructional materials are concrete and daily. They are the stuff of culum schools. They ha Schmidt and colleagues (2001) agreed th at curriculum is important, stating is at the very center of intentional learning in schools, specifying content and directing Valverde et al. pointed out that curriculu m bridges the gap between content standards and classroom Mathematics education at all levels is important, but the middle grades may be ing this time, many students will solidify conceptions about themselves
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2 as learners of mathematics about their competence, their attitude, and their interest and motivation. These conceptions will influence how they approach the study of mathematics in l 2000, p 211). Although the middle school mathematics curriculum contains many important concepts, one of the most pervasive is that of proportionality. An u n derstanding of proportionality i s essential for comprehending high school and college level mathematics. Proportionality is also an area of mathematics with numerous applications to real life These applications include two or more purch asing options, as well as many other useful skills Proportionality is also closely related to many topics within the middle school mathematics curriculum such as algebra, fractions, measurement, percent, and rates and ratios Specifically proportionality is ideally suited to provide two types of connections for which the NCTM has called : connections between school and real life mathematics and connections between mathematical concepts (NCTM, 1989, 2000). Thus, the NCTM has stated that propo grades mathematics The development of proportional reasoning is considered one of the most important goals of the middle school mathematics curriculum. It represents a significant shift from additive reasoning, in which amounts are compared to multiplicative reasoning, in which percents are compared. Signifying the magnitude of this shift Lesh, concept.
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3 Similarly, Smith (2002) d escribed the importance and complexity of proportionality in this way : No area of elementary school mathematics is as mathematically rich, cognitively complicated, and difficult to teach as fractions, ratios, a nd proportio nality. These represent relationships between two discrete or continuous quantities, rather than p. 3). Statement of the Problem Proportionality is a topic with which many children and adults struggle. In fact, p. 637). P roportionality is related to many of the most difficult topics in the middle school mathematics curriculum such as equivalent fractions, long division, place value and percents, measurement con version, and ratio and rates (Lesh et al., 1988) Some of difficulty with proportionality is that it requires multiplicative reasoning, which is different from the additive reasoning learned in elementary school. Making this shift is so difficult that students often revert back to additiv e reasoning on hard problems, even when they are capable of multiplicative reasoning (Karplus, Pulos, & Stage, 19 8 3). Some of difficulty with proportionality is due to the complexity of the topic and the shift from additive to multiplicative reasoning but the way proportionality In most classes, textbooks are used on a daily basis (Grouws, Smith, & Sztajn, 2004), so it is r easonable
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4 to assume they affect student understanding. In fact, at least one study has indicated a proportionality (Ben Chaim, Fey, Fitzgerald, Benedetto, & Miller, 1998). Ho wever, although Ben Chaim et al. claimed that there are significant differences in the treatment of proportionality between the two curricula used in their study, these differences have not been described in detail. Researchers have criticized middle scho ol curricula for failing to help students recognize the differences between additive and multiplicative relationships, for placing too much emphasis on proce dural rather than conceptual understanding of proportionality, and for paying inadequate attention to proportionality (Cramer, Post, & Currier, 1993; Post, C ramer, Behr, Lesh, & Harel, 1993; Watson & Shaughness y, 2004). These criticisms seem to be proportionality in textbooks rather than on actual research. In fact, there is a remarkable lack of research on the treatment of proportionality in textbooks. A search of the Education Full Text and ERIC databases produced no documents that reported the results of a curriculum analysis related to proportio nality. Two studies contained curriculum analyses relevant to the current study. (1988) reported results from an informal look at proportionality in secondary textbooks, but the textbooks used were publi shed in the 1970s and 1980s and the investigation was not detailed. The American Association for the Advancement of Science ( AAAS ) conducted an evaluation of contemporary middle school textbooks, but did not focus on proportionality (AAAS, 2000) Therefore it appears that an analysis of the treatment of proportionality in middle school mathematics textbooks is needed.
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5 Purpose of the Study Th e purpose of the study was to describe how proportionality is treated in contemporary middle school mathematics text books. Six aspects of this treatment were investigated. First, grades conducted to determine whether proportionality is used this way in textbooks. The scaling, linear equatio ns, slope, relative Therefore, there was a need to investigate the extent to which textbooks highlight the proportionality aspect of various topics, such as algebra, measurement, percent, probability, ratios and rates, rational numbers, scale factors, and similar figures. Second, because researchers have suggested that students should be exposed to a wide range of proportional situations, one goal of the study was to determine the extent to which textbooks i nclude various types of proportional problems. Three types of proportional problems have been identified in the research literature (Cramer et al., 1993); these will be described in Chapter 2. Additionally, a pilot study conducted by the researcher reveale d that several types of proportional problems exist in textbooks that have not been identified in the research literature. Third, students often use additive reasoning in situations where multiplicative reason ing is required and vice versa. The result is that students seem not to know when to apply proportional reasoning. Therefore, there was a need to examine the extent to which
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6 textbooks point out differences between the two types of reasoning and the extent to which textbooks illustrate situations where each type is appropriate and inappropriate. Fourth r esearch findings have indicate d that some solution methods seem more n atural to students than others (Karplus et al., 1983) For example, the use of proportions and cross multiplication is traditionall y emphasized in textbooks, but is highly symbolic. Therefore, there was a need to analyze the solution strategies presented in textbooks Fifth because problems with higher levels of cognitive demand may lead to better conceptual understanding, it was imp ortant to note the level of cognitive demand ( Boston & Smith, 2009 ; Stein, Smith, Henningsen, & Silver, 2000 ) of tasks. Finally because visual representations may lead to conceptual understanding (e.g., Martinie & Bay Williams, 2003a), it was also importa nt to note the extent to which textb ooks use visual representations to express ideas related to proportionality. Research Questions The purpose of the study was to investigate the f ollowing six research questions. 1. To what extent is proportionality emphasiz ed in the treatment of various content areas within mathematics, such as algebra, data analysis/probability, geometry/ measurement, and rational numbers? How does this vary among grade levels and textbook series? 2. Among the tasks related to proportionality in middle school mathematics textbooks, which problem types ( e.g., missing value, ratio comparison, qualitative) are featured most and least often? How does this vary among mathematical content areas grade levels, and textbook series?
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7 3. Which solution strategies (e.g., building up, unit rate, proportion) to tasks related to proportionality are encouraged by middle school mathematics textbooks? How does this vary among grade levels and textbook series? 4. What level of cognitive demand (Boston & Smith, 2009 ; Stein et al., 2000) is exhibited by the proportional exercises in middle school textbooks? How does this vary among mathematical content areas, grade levels, and textbook series? 5. To what extent are visual representations used in middle school mathematics textbooks to illustrate concepts related to proportionality? How does this vary among gra de levels and textbook series? 6. To what extent are the characteristics of proportional situations pointed out in middle school mathematics textbooks? How does this var y among grade levels and textbook series? Funding by the National Science Foundation the twentieth century and culminated in A Nation at Risk : The I mperative for Educational R efo rm (National Commission on Excellence in Education 1983 ) al foundations of our society are presently being eroded 5). According to Senk and Thompson (2003) levels of achievement that are seen as inadequate result in calls for changes in the curriculum. Senk and Thompson stated the r ecommendations about what to teach in schools and how to teach it. Advocates of
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8 reform often attempt to influence classroom practice, and hence, student achievement, by In response to these concerns about mathematics achievement, the NCTM issued changes that were needed in mathematics education, including changes to the curriculum. Another effect of the concerns about mathematics achievemen t was the National Science Foundation (NSF) funding of curriculum development projects. According to Senk and Thompson (2003) comprehensive instructional materials for the elementary, midd le, and high schools consistent with the calls for change in the Curriculum and Evaluation Standards [NCTM, p 13 14). Five curriculum development projects funded by the NSF produced textbooks for middle school students (Senk & Thompson). Scholar s have described the differences between NSF funded and traditional curricula (e.g., Robinson, Robinson, & Maceli, 2000) According to Robinson et al., NSF past, both in con funded curricula are more likely than traditional curricula to use problem solving contexts and to portray mathematics as a unified discipline by making connections between topics. Researchers ha ve also investigated the effects of NSF funded curricula on student achievement ( Ridgway, Zawojewski, Hoover, & Lambdin, 2003; Tarr et al., 2008). A few detailed comparisons of NSF funded and traditional curricula have been conducted. For example, Hodges, Cady, and Collins (2008) examined the ways fractions are represented in two NSF funded and one traditional textbook. However, few in depth,
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9 rigorous curriculum analyses have been conducted to determine how NSF fund ed curricula cover various mathematical topics or how their treatment differs from that of traditional textbooks. Definition of Terms In this section, the meanings of some important terms are discussed. The terms defined in this section include the follow ing: additive reasoning multiplicative reasoning fraction rational number measure space proportionality proportional reasoning rate ratio and task Th is section includes not only a definition of each of these terms, but also a broader discussion o f their meanings and usage. Therefore, the sections that follow are lengthier than simple definitions would be. Additive and Multiplicative Reasoning Additive reasoning is one of the first types of mathematical reasoning learned by young children. It con sists of skills related to counting, adding, joining, subtracting, separating, and removing (Bright, Joyner, & Wallis, 2003; Lamon, 2007 ; Post et al., 1993 ). Multiplicative reasoning refers to reasoning about multiplication, division, linear functi ons, rat ios, rates, rational numbers shrinking, enlarging, scaling, duplicating, e xponentiating, and fair sharing ( Lamon, 2007 ). Behr, Harel, Post, & Lesh (1992) described the differences between additive and multiplicative reasoning in this way: As early as poss ible, children should be brought to understand tha t the change in 4 to get to 8 ( or the difference between 4 and 8) can be defined in two ways: additively (with an addition or subtraction rule) or multiplicatively (with a multiplication or division rule) ( p. 316).
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10 reasoning is in contrast to additive reasoning. Additive reasoning involves using counts for example, sums or differences of numbers as the critical factor in comparing quantities. Multiplicative or proportional reasoning involves using ratios as the critical There is some disagreement about the extent to which multiplicative reasoning should be taught as a natural extension of additive reasoning From a constructivist, student centered point of view, Steffe (1994) viewed multiplicative reasoning as an extension of additive reasoning and might become for c hildren, the operations would be constructed as modifications of their In contrast, Confrey (1994) believed that too much emphasis current appr She pointed out that although some uses of multiplication are related to addition, others, such as sharing and d ividing symmetrically, are not. Fractions and Rational Numbers Rational numbers are numbers that can be written in the form where a and b are integers and b is not equal to zero For example, 0.5, and 50% are all rational numbers. Fractions are numbers that are written in the form For example, is a both a fraction and a rational number; 0.5 and 50% are rational numbers but not fractions. Not all fractions are rational numbers. For ex ample, is not rational even though it is written in fractional form (Lamon, 2007). Thus, many numbers are both fractions and rational
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11 numbers, such as , and (Lamon). Other numbers are fractions but not rational numbers and still other numb ers are rational numbers but not written in fractional form. Further complicating the issue is that rates and ratios can also be written in fractional may not be rational. The fractional method of writing a number points out comparisons. For example, compares the numbers three and four and also compares to one. It is this comparison aspect of fractions that relates to proportionality. Researchers agree that all ration al numbers, including fractions, are closely related to proportionality. For example, proportions together in the same phrase simply because the terms are so close mathemati Measure Space The notion of measure spaces was developed by the French psychologist Gerard Vergnaud. Cramer et al. (1993) provided the following example of measure spaces U.S. dollars can be exchanged for 2 Brit ish pounds, then at this rate 21 U.S. dollars can are U.S. dollars and British pounds. T discuss two methods of solving prob lems: Between and Within strategies. Using a Between strategy, a person would compare the number of U.S. dollars to the number of British pounds. Using a Within strategy, a person would determine the ratio between 21 U.S. dollars and three U.S. dollars.
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12 Proportionality teria are the following: (a) the rate of change is positive and constant and (b) the y inter cept is zero ( Cramer et al., 1993; Lamon, 1999) Both of these characteristics are described below. In proportional situations, the rate of change is constant. In other words, a change of a certain amount in the independent variable produces a consistent c hange in the dependent variable. This implies that the slope is constant and that a graph of the relationship will feature a straight line. Lamon (1999) provided the following examples: In the midst of change, important relationships can remain constant. For example, whether you are mixing a quart of lemonade or a gallon, the ratio of scoopfuls of drink mix to pints of water is the same. If you buy grapes, although the amount you pay increases if you buy 3 pounds instead of 2 pounds, the rate $1.19 per pou nd remains constant (p. 187). In proportional situations, the y intercept is zero, meaning that a value of zero for the independent variable is associated with a value of zero for the dependent variable. Because the y intercept is zero, the familiar equati on for a line, simplifies to (Cramer et al. 1993; Lamon, 2007). It is important to note that the equation involves only multiplication, not addition. Thus, proportional relationships are always multiplicative in nature (Cramer et al.).
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13 Proportional Reasoning Lamon (1999) provided the following description of proportional reasoning: Proportional reasoning results after one has built up competence in a number of knowledge. proportional reasoning is to say that it is the ability to recognize, to explain, to th ink about, to make conjectures about, to graph, to transform, to compare, to make judgments about, to represent, or to symbolize relationships. (p. 5) Although proportional reasoning is difficult to define, there is no doubt it is research, proportional reasoning has been considered a p. 97). The term proportional reasoning is closely related to two other terms : proportionality and multiplicative reasoning According to Lamon (2007), the terms proportionality and proportional reasoning are often used interchangeably. However, there is a distinction between the two. Proportionality is a mathematical characteristic of some quantitative situations, whereas proportional reasoning is a type of reasoning and thus a cognitive function of the mind. Multiplicative reasoning and proportional reasoning are closely related to each other, but the distinction between them is not completely clear. In Principles and Standards for School Mathematics, the NCTM indicated that multiplicative reasoning is a
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14 prerequisite for or less advanced form of proportional reasoning. NCTM (2000) st ated, 5 provides foundational knowledge that can be built on as students move to an emphasis on proportional reasoning in the Thus, although proportional reasoning is closely relat ed to both proportionality and multiplicative reasoning there are distinctions between the terms. Rates and Ratios Rates and ratios are both comparisons of two quantities. Various authors have suggested different distinctions between rates and ratios So me scholars believe that a rate that a ratio compares quantities from the same measure space, such as 20 children in one class to 18 children in another (Lamon, 2007). O th er scholars have offered different distinctions between rates and ratios. A ccording to Kaput and West (1994), a ratio arises from a situation in which the variables are discrete, such as two pencils for 10 cents, 4 pencils for 20 cents, and so on. Conversely, a rate arises from a situation in which the variables are continuous, such as a mixture of paint or orange juice. Because no consensus has been reached, the definitions of rate and ratio have not been firmly established. Lamon (1999) stated the following: E veryday language and usage of rates and ratios is out of control. The media have long employed ratios and rates and the language appropriate to ratios and rates in many different ways, sometimes inconsistently, sometimes interchangeabl y Students are exposed to less than correct usage and terminology, and it is no easy task to reconcile precise mathematical ideas wi 165).
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15 Task According to Doyle (1979, 1983) earlier studies of curriculum focused on the curriculum in broad terms, such as the percentage of the school day devoted to mathematics, but, as a result of cognitive theories o f psychology attention to individual tasks grew Doyle (1979) defined a task as consisting of two elements: a goal and a set of ope rations necessary to achieve that goal. According to Doyle, in a learning task goal is to be able to display a capabil Doyle (1983) described four categories of academic tasks: memory tasks; procedural or routine tasks; c ompr ehension or understanding tasks; and opinion tasks. Recently, t he analysis of tasks has been conducted in teacher education (Stein et al., 2000) Stein et al. developed a task analysis framework with categories similar to those described by Doyle (1983). The attention to individual tasks rests on the Analysis of tasks has also been conducted in curriculum analysis ( Jones, 2004; Jones & Tarr 2007). Jones used the following definition of probability task exercise, or set of exercises in a textbook that has been written with the intent of focusing task was used in this study, but the author is interested in prop ortionality rather than probability.
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16 Significance of the Study standardized testing. Another strong determinant of what students learn is the curriculum (Hirsch, 2007). Both national and international studies have shown that t extbooks have a significant impact on the content that is covered in mathematics classes ( Grouws et al., 2004 ; Sch midt et al., 2001 ; Valverde et al., 2002 ). For example, a ccording to data obtained through the National Assessment of Educational Progress, in 2000, 72% of eighth (Grouws et al. ). A variety of education organizations have expressed the opinion that the mathematics curriculum is of vital importance ( AAAS 2000; N CTM 1989). In spite of the importance of the mathematics curriculum there have been few recent studies of textbook s c ontent. Most recent research related to curriculum ha s implementation of it or learning from it. While these studies provide valuable insights into the instruction that occurs in classrooms, there is also a need to delv e into the subje ct of what textbooks actually contain. The mathematics curriculum at all levels of education affects student learning, but the middle school curriculum, in particular, is in need of attention ( AAAS 2000). One of the primary goals of the middle school mat hematics of proportional reasoning. For example, in the Principles and Standards for School Mathematics thread that connects many of the mathema tics topics studied in grades 6
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17 2000, p. 217). In both that document and the Curriculum Focal Points (NCTM, 2006), the NCTM offered suggestions regarding what middle school students should learn about proportional reasoning. For example, the NCTM suggested that because proportions are highly symbolic, they may not be the most natural way for students to solve problems (NCTM, 2000). Therefore, students should develop other solution strategies in addition to proportions. However, because curriculum materials related to proportional reasoning have not been analyzed in depth, the degree to which NCTM suggestions are currently followed by curriculum developers is not clear. Furthermore, because suggestions offered by the NCTM regarding proportional reas oning are meant only to be general guidelines, curriculum developers have a great deal of latitude and may choose to present proportional reasoning in a wide variety of ways. For example, the percentage of questions of missing value and comparison types is left to the discretion of curriculum developers. Because of this discretion, it is likely that proportional reasoning is treated differently in various curricula. reasoning was a major goal of one of the curricula selected for this study ( Connected Mathematics ), and research indicates that students studying from this curriculum have a better understanding of proportionality than students u sing other curricula (Ben Chaim et al. 1998). However, little speci fic information regarding the differences of the treatment of proportionality in various curricula is available; thus, curriculum developers do not know which aspects of the Connected Mathematics curriculum are successful. Information regarding the treatment of proportional reasoning in textbooks could help curriculum developers make improvements to their treatment of the topic. For example, curriculum developers may inadvertently emphasize proportional reasoning in
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18 some con tent areas more than others limiting the extent to which proportionality serves as a connective thread as encouraged by the NCTM. Curriculum developers may also inadvertently place a higher emphasis on some problem types while neglecting others An in dep th analysis of their materials may help them recognize these imbalances. The information provided by this study could help states and school districts select textbooks that b e st meet the needs of their middle school students. Information about the strength s of textbooks could help teachers maximize the effects of those strengths and information about their weaknesses could help teachers make modifications to the curriculum when necessary. Summary Textbooks affect what is covered in mathematics classes and what students learn. Therefore, careful analyses of the contents of textbooks are important. The middle school curriculum, in particular, has been the subject of recent interest (e.g., AAAS, 2000). One of the primary goals at this level is the development of p roportional reasoning which has (Lesh et al., 1988) Because proportionality can be a challenging concept, textbook authors must do all they can to help students understand it. Researchers have identified three ma in types of proportional reasoning questions, which are described in the next chapter; it seems reasonable to assume that textbooks should contain all three types. Because the NCTM encourages middle school the proportionality inherent in algebra, measurement, rational numbers, and probability should be highligh ted. Because students often use additive reasoning in situations where proportional reasoning is called for, and vice versa, textbooks should help students
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19 understand the differences between the two types of reasoning and when each is required Researchers claim to have identified several weaknesses in the coverage of proportionality in middle school textbooks; h owe ver, these criticisms d o not seem to be based on a thorough curriculum analysis. This study provides the data necessary for a better understanding of the treatment of proportionality in three contemporary middle school textbook series.
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20 CHAPTER 2: LITER ATURE REVIEW In Chapter 3, a framework that was used to analyze the presentation of proportionality in middle school mathematics textbooks is presented. This framework was developed by the author based on research literature regarding the mathematical natu re of proportionality as well as research literature on how students develop proportional reasoning. The purpose s of this chapter are to discuss the literature o n which the framework is based, to discuss criticisms of the treatment of proportionality in textbooks, and to discuss research methods that have been used in other curriculum analyses. This presentation of the research literature and discussion of curriculum analysis methodology provide s the reade r with an understanding of why the study wa s necessary and the methodology that was used. This chapter is divided into eight sections. First, because the NCTM has stated grades mathematics appears is discussed. Second, the problem types that have been identified in the research literature are discussed. Third, because researchers have suggested that some soluti on strategies contribute more toward conceptual understanding than do others, a discussion of solution strategies is provided Fourth common errors and common misunder standings about proportionality will be discussed Fifth methods other researchers have used to analyze textbooks are discussed. In the sixth section, the liter ature on proportional reasoning is summarized In the seventh section, questions that remain unanswered by
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21 previous research are identified, and in the final section, a summary of the pilot study conducted by the researcher is provided. Together, these eight sections should prepare the reader to see connections between the research literature, the pilot study, and the frame work presented in Chapter 3. Content Areas Many middle school However, proportionality cannot be confined to a single lesson or even a single chapter. Rather, proportionality is a fundamental characteristic of mathematical relationships that appear s in sever al areas of mathematics. Because the NCTM encourages middle school educators to use proportionality as a connective thread that runs through the curriculum (NCTM, 2000), proportionality should be emphasized in the teaching of several different areas of mat hematics. Emphasizing the proportionality that is inherent in various content areas may help students see mathematics as a coherent discipline. These content areas and their connection to proportionality are the subject of this section. Algebra One of the content areas closely connected to proportionality is algebra (Cramer et al., 1993; Karplus et al., 1983 ; Martinie & Bay Williams, 2003b; NCTM, 2000; Seele y & Schielack, 2007). Some scholars view an understanding of proportionality as a prerequi site to success in algebra. For example, Post et al. stated that proportionality is of proportionality as a prerequisite to success in algebra, Cramer et al. (1993) saw an even tighter mathematical connection between algebra and proportionality. They
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2 2 relationships can be expressed through a rule with the form of y = mx Expre ssing a proportional relationship in the form y = mx highlights the connections to slope and graphing. Telese and Abete (2002) showed how a lesson related to nutrition could be used to teach middle school students about algebra and proportionality. Their s tudents found a linear relationship between grams of fat and calories from fat. One of the y = 9 x Examples like these show how linear relationships are closely related to multiplicat ion. Proportionality is usually highlighted in algebr a lessons in one of two ways: thro ugh attention on constant slope or through rate problems. An example of a rate problem is this exer cise from a sixth marathon is approximately 26 mile s. If Joshua ran the marathon in 4 hours at a constant rate, how far did he run per hour? Day et al., 2009a, p. 318 ). Data Analysis and Probability Data from the 1996 National Assessment of Educational Progress (NAEP) nderstanding of data analysis and probability wa s Shaughnessy, 2000, p. 266). At least th ree topics within the data analysis and probability standard can be connected to proportionality: arithmetic mean, misleading graphs, and probability. Watson and Shaughness proportional reasoning that was relate d to arithmetic mean. They presented students with
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23 assessment. On the basis of the graphs, students were asked to determine which class did better. In one pair of graphs, on e set of data represented a larger class than the other. To someone using additive reasoning, the larger class appeared to have done better, but to someone using proportional reasoning, the smaller class di d better. Watson and Shaughness y found that additi ve reasoning was common among elementary and middle school students and even extended into ninth grade. Proportional reasoning is also needed when analyzing graphs; Spence and Krizel (1994) explain why: The most commonly seen graphs are pie and bar char apparent dimensionality of graphical elements can be one, two, or three, even though the represented numerical quantities and proportions are unidi distortion (p. 1193) dimensional, but are often drawn in three dimensions. Modifications like these can mislead the reader; proportion al reasoning is required to correctly interpret these graphs. Another content area in which proportional reasoning is required is probability. For example, Watson and Shaughness fundamental to making connections b etween populations and samples drawn from those by comparing experimental and empirical data.
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24 Geometry and Measurement Within geometry and measurement, proportional reasoning is most evident in the middle grades students understand similar ity, which is closely related to their more published activities related to size changes and similar figures that are designed to promote or assess proportional reasoning (Che 2009; Frost & Dornoo, 2006; Moss & Caswell, 2004). Che states that she posts pictures of giant pencils on the walls of her Frost and Dornoo (2006) discussed some of the issues that arise when helping middle could have several different meanings. They expla ined how students can construct squares on geoboards and discover that doubling the lengths of the sides doubles the perimeter but not the area, a common misunderstanding mentioned in the Curriculum and Evaluation Standards (NCTM, 1989). In another example of proportionality related to size changes, Moss and Caswell (2004) had fifth and sixth grade students construct dolls whose measurements were in proportion to the average measurements of students in the class. They found that e of percents help them understand size changes. For
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25 Because pi is the ratio of the circumference of any circle to its diameter, reasoning abou t pi and circles is another area of geometry and measurement related to proportionality (NCTM, 2000). Converting between units of measure is another application of proportionality. Additionally, because perimeter is related proportionally to side lengths b ut area is not, textbooks often use the topics of area and perimeter to discuss the differences between proportional and nonproportional relationships. Rational Numbers There has been an enormous amount of literature published on rational numbers. However, not all of the literature on rational numbers is relevant to a study of proportionality. The purpose of this section is not to review all of the literature on rational numbers, but instead to demonstrate the connection between proportionality and rational numbers. Rational numbers are interpreted in different ways, depending on the context (Behr et al., 1983). Three of the interpretations of rational numbers are closely related to proportionality: fractions, decimals, and percent. Common in middle school textbooks are tasks in which students are asked to find equivalent fractions. Post et al. (1993) described the tight connection between equivalent fractions and proporti between the numerical procedures used in manipulating and finding equivalent fractions and numerical procedures necessary to solve missing value and numerical comparison problems. In both equivalent fraction tasks, students are given three numbers and asked to find a fourth. For games. How many of its next 50 games must the team win in order to maintain the ratio
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26 equivalent to that has a denominator of 50. This is an example of a missing value problem. In other eq uivalent fraction tasks, students are given only two numbers and asked to find the other two. For example, students could be asked to find a fraction equivalent to This is not a missing value problem as defined by Lamon (2007). In fact, this type of pr oblem is not explicitly discussed in the literature on proportional reasoning. In other fraction problems, students are asked to compare the size of two or more fractions. For example, a problem on the fourth administration of the NAEP asked students to id entify the larger of and About two thirds of the seventh grade students correctly identified as the larger fraction (Kouba, Carpenter, and Swafford, 1989). These problems are closely related to proportionality; if fact, they fit into one of the t hree types of proportional problems, the ratio comparison type. To solve the problem, students must not only view and as units but must also then compare the size of those units. Decimals are closely related to fractions. For example, 0.6 is equival ent to school students have traditionally had difficulty converting between decimals and fractions. For example, on the fourth administration of the NAEP, only 60% of the seventh graders tested could correctl y write as a decimal (Kouba et al., 1989). Although rational numbers are one of the content areas traditionally associated with proportional reasoning (Behr et al., 1992; Lamon, 2007), decimals are discussed in the research literature far less often t han are fractions. Another rational number concept closely connected to proportionality is percent (Dole, 2000; Lamon, 1994, 1999; Martinie & Bay Williams, 2003b; Moss & Caswell,
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27 2004; Parker & Leinhardt, 1995). In a comprehensive literature review of per cent Parker and Leinhardt pointed out that although percent appears to be a simple, straightforward cause of these multiple personalities of percent, middle school students seem to lack an understanding of percent and do relatively poorly on NAEP items involving the concept (Dole, 2000). The various interpretations of percent include the following: (a) p ercent as a number, (b) percent as an intensive quantity, (c) percent as a fraction or ratio, and (d) a percent functions of percent is the one most prominently featured i n textbooks. Parker and Leinhardt give the example of a political candidate who receives 35% of the votes; the whole set is the total number of votes and the subset is the number of votes received by the candidate. In addition to functioning as a fraction, a percent can also be a ratio. Parker and Leinhardt involving different sets, different attributes of the same set, or the change in a set over the example of a comparison between the number of students in School A to the number of students in School B and the example of a comparison between the length of a board and its width.
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28 Summary of Content Areas The NCTM has encouraged teachers of mathematics to point out connections between various mathematical topics and content areas. Because proportionality plays a major role in each of the content areas previously described, it provides a unique opportunity t o illustrate the connections the NCTM has recommended As Seeley and relationships is a particularly noteworthy example of how ideas from Number and Operations, Algebra, Geometry, Measurement and Data Analysis and Probability can this connective role, the proportionality inherent in various content areas must be made explicit to student s. Therefore, it seems reasonable to assert that proportionality should be Problem Types Many mathematical concepts have multiple meanings and are used in a variety of situations. For e xample, the concept of multiplication has a variety of meanings, including repeated addition, the area of a rectangle, or the number of combinations of two or more quantities. Similarly, proportionality is inherent in a wide variety of situations and vario us types of proportional problems require different types of reasoning and solution methods. Therefore, it is important to understand the various types of problems that require proportional reasoning. Researchers generally agree that there are three main types of proportional problems: missing value, ratio comparison, and qualitative ( Behr et al. 1992; Ben Chaim et al., 1998; Cramer et al., 1993; Singh, 2000), although some scholars (e.g., L amon
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29 2007 ) recognize two main types ( missing value and ratio comparison ) and consider qualitative problems to be a subtype within the se two main types Because missing value ra tio comparison and qualitative problems are the three main types of tasks that involve proportionality, each of these types is discussed in this section. Missing Value P roblems value problem provides three of the four values in the proportion Missing value problems can be either purely numeric al or can be stated in the form of a word problem. Although the above quotation by Lamon mentioned a proportion, there are other methods of solving missing value problems as discussed in a later section. Karplus et al. (1983 ) offer ed this example of a missing value In this example, the three known values are 175 kilometers, 3 hours, and 12 hours. The missing value is the number of miles traveled in 12 hours. Another example of a missing value problem is the well known Mr. Tall/Mr. Short problem (Karplus, Karplus, Formi sano, & Paulson, 1978 ) In this task, participants are shown two stick figures, one short and one tall. Each stick figure is accom panied by a stack of buttons showing his height. The height of Mr. Short is also given in paper clips and the task is to calculate the height of Mr. Tall in paper clips. Missing value problems can be very simple or quite complicated. Lo (2004) provides th is example that involves time, fractions, and hour to paint a wall that was 12 ft by 12 ft, how long will it take to paint another wall that is 15 ft McKenna and Harel (1990) pointed out that missing value pro blems
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30 can be worded in two different ways One way of wording a missing value problem is to have only one measure space in each sentence, as in Matthew bought 6 lb of candy and Kim bought 10 lb of the same kind. If Matthew paid $12, how much did Kim pay? The other way of wording a missing value problem is to mention both measure spaces in a (McKenna & Harel, p. 589). Ratio Comparison P roblems In a ratio comparison problem, four values are given. These values form two ratios. The task is to d etermi ne which ratio is greater or whether the two ratios are equivalent This type of problem has been referred to by several different names inc 2007 ), ; Singh, 2000 ). These authors appear to be referring to roughly the same type of problem, however, Lamon includes qualitative problems in this category whereas oth er authors distinguish between ratio comparison and qualitative problems. Karplus et al. (1983 ) offered this example of a ratio is driven 180 km in 3 hours. Car B is driven 400 km in 7 hours. Which car was drive n 0). Note that this problem is equivalent to asking students to compare the ratios 180 : 3 and 400 : 7. great soccer players. Pele scored 300 goals in 400 matches, while Maradona scored 400 goals i n 500 matches. Who had a better scoring record, Pele, Maradona, or do they have
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31 A nother example of this type of problem is Noelting s (1980 a) orange juice problems. E ach task featured two pitchers of orange juice with different n umbers of cups of water and orange juice concentrate. Each pitcher of juice contained some ratio of concentrate to water; the task was to compare the two ratios. As a final example, Karplus et al. (1983) showed a picture of 7 girls clustered around three p izzas and three boys problem is equivalent to asking students to compare the ratios 7 : 3 and 3 : 1. Qualitative Problems In a task that requires qualitative proportional reasoning, a numerical answer is not desired or possible. Some researchers distinguish between two types of qualitative proportional tasks: qualitative prediction tasks and qualitative comparison tasks (Cramer et al., 1993) In qu alitative prediction tasks students compare a past or present situation to a future one. Cramer et al. provided this example of a qualitative prediction Devan ran fewer laps in more time than she did yesterday, would her running speed be (a) faster, (b) slower, (c) exactly the same, (d) not enough information to tell (p. 166). with some friends. Today, you share fewer cookies with more fr iends. Will everyone get In qu alitative comparison tasks, students compare two past situations or two current situations. Cramer et al. (1993) provided this example of a qualitative comp arison One advantage of qualitative problems is that students cannot answer them u sing a
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32 memorized procedure. Therefore, some scholars believe they are a better assessment of a that cannot be obtained just by solving an equation. In contrast to [quantitative] tasks, qualitative of this type of thinking, qualitative prediction and comparison sho uld have a recognized Summary of Problem Types Three types of proportional problems have been discussed in the research literature: missing value ratio comparison and qualitative types knowledge, no researcher has specifically stated that all three types of problems should appear in middle school textbooks. However, scholars have stated their belief that proportional reasoning develops as students are exposed to a variety of proportional situations. Therefore, it seems reasonable to conclude that all three types of problems should appear in middle school textbooks.
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33 Solution Strategies In the previous section, the differences between missing value and ratio comparison pr oblems were described. Because these problem types are quite different, they are solved using different methods or strategies. Researchers have identified three ways students solve missing value problems and two ways they solve ratio comparison problems. M issing Value Problems Missing value p roportional problems generally require both multiplication and division as part of their solution process. This leads to three general solution methods: a) the multiplication can be performed first and the division seco nd, b) the division can be performed first and the multiplication second, or c) they can be performed more or less simultaneously. Several authors have described these three general methods ( Karplus et al., 1983 ; Noelting, 1980a, 1980 b) Building Up (Fact or of Change) Strategy Two similar strateg ies fit under the umbrella of the building u p strategy. The first, Tourniaire orange juice and water. One is made with 2 glasses of orange juice and 4 glasses of water. The other is made with 404). Tournaire stated that a I need 4 glasses of water for the This Scholars recognize that this approach is less sophist icated than other methods, but also that it can
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34 be successfully used by elementary school students and can provide a bridge to true proportional thinking (Lamon; Parker, 1999; Tournaire) Other versions of the building up strategy involve simple forms of multiplication such as doubling When t he solution strategy involves multiplication, it is sometimes referred to as t Cramer et al 1993) but is more often called the ( Heinz & Sterba Boatwright, 2008 ; K aput & West, 1994; Lamon, 1994). Cramer et al described this method: A student using the factor of change twice as many apples, then the cost will be multiplicative relationship within a measure space (p. 168). Because the building up strategy involves finding the multiplicative relations hip within a scribed by Karplus et al. (1983 ). Students who have not received instruction in solving proportional problems seem to naturally and successfully use the building up strat egy (Lamon, 1993 b ). Some scholars believe that success with the building up strategy does not demonstrate that a student has learned to reason proportionally. For example, Piaget referred to use of the building up Unit R ate Strategy Cramer et al. (1993) described the unit rate approach in this way: This approach is characterized by finding the multiplicative relationship between measure spaces. The unit rate is found through division. For example, if 3 apples cost 60 cen ts, find the cost of 6 apples. The cost for 1 apple is found by dividing:
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35 60 cents 3 apples = 20 cents per apple. This unit rate is the constant factor that relates apples and cost To find the cost of 6 apples, you simply multiply 6 apples by 20 cents per apple (p. 167). M any au thors, including Cramer et al. associate the unit rate strategy with situations in which a person is purchasing several units of the same item for a given price. In this case, the unit rate is the price per unit. Although often associated with situations involving purchases, the unit rate strategy can be used in a variety of situations. For example, Singh (2000) showed how the Mr. Tall/Mr. Short problem can be solved through a unit rate strategy. As another example of the unit r ate strategy in a non money situation, Telese and Abete (2002) said that if a three ounce serving of hamburger has 21 grams of fat, the unit rate is seven grams of fat per ounce of hamburger. As Cramer et al. (1993) pointed out, the unit rate strategy invo lves finding the multiplicative relationship between measure spaces. Thus, it is equivalent to the between al. (1983 ). This strategy or very similar strategies have been ca by L o (2004) a nd the 1981). S everal researchers have found that many students use the unit rate strategy and are generally successful with it ( Ben Chaim et al., 1998; Cramer et al., 1993; Post et al., 1993 ; Rupley, 1981 ) Ben Chaim et al. found that students using Connected Mathematics were much more likely than other students to use the unit rate approach on some problems and were very successful in its use. Rupley found that ninth grade students instructed in the use of the unit measure strategy had a success rate on proportion word problems that was similar to the success rate of the eleventh grade students in the study.
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36 Rupley also found that students invent this strategy on their own before receiving instruction in proport ionality and therefore suggested that thi s method be taught prior to other strategies. Although students seem to be successful when using the unit rate strategy this strategy also has disadvantages. For example, it is much easier to use when the first number in the problem is smaller than the second (Rupley, 1981). For example, consider p. 2). Rupley believed this is a fairly easy problem because the first step, using the unit rate method, is to divide 28 by seven. The pieces of pipe is 7 feet. The l would have involved dividing seven by 28. Thus, the size of the numbers can influence students invented the u nit rate method on their own; once students received instruction in this method, the order of the numbers did not significantly influence their success with the unit rate method. Another disadvantage of the u nit rate method is that once students have divi ded, they can use additive reasoning rather than multiplication to solve the problem. For example, in the problem above, once students have divided 28 by seven and determined that each piece of pipe is four feet long, they could add four five times rather than multiplying four and five. Therefore, Singh (200 0 ) cautioned that use of the unit rate method does not ensure that students are using multiplicative reasoning.
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37 Proportion Strategy A nother solution method involves the writing of a proportion After a n informal review of textbooks, Fisher ( 1988) reported that the proportion strategy is the most common strategy presented. Although t he proportion method is a powerful problem solving tool (Behr et al., 1983), it has advantages and disadvantages, which are described in the following paragraphs. An advantage of the proportion strategy is that it can be applied to all missing value problems with more or less equal ease. In other words, students are less affected by the specific numbers in the problem when th ey use proportions than they are when they use the unit rate method (Rupley, 1981). A disadvantage of the proportion strategy is that it is a procedure that some students perform mechanistically; they may arrive at correct answers but not develop proportio nal reasoning or und e rstand why the procedure works. F or example, Lamon (1999) wrote that when faced with a proportional tempted to use an equation of the form (p. 5). Heinz & Sterba Boatwright (2008) found that preservice elementary teachers could successfully set up proportions but had little conceptual understanding of them. Students have learned how to arrange the four quantities in a r, the students did not seem to be attending to the fact that a Because the use of proportions seems to not foster conceptual understanding, researchers have suggested the y be less heavily emph asized (Karplus et al., 1983 ).
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38 Proportions can be solved several different ways. Two of the most common are t and the cross multiply and divide algorithm. The fraction strategy is one method of solving a proportion It has been referred to as the cio & Bezuk, 1994 ). Cramer et al. stated that rate pairs are treated as fractions and that the multiplication rule for generating equivalent fractions is used. Cramer et al. provided this example: Another method of solving proportions is the c ross m ultiply a nd d ivide a lgorith m. This is the traditional method of solving proportions and is quite common in textbooks. However, when students are given a choice of what strategy to use, very few choose to cross multiply a nd divide (Karplus et al., 1983 ). Some educators have suggested that teaching students to use cross multiplication results i n lower success rates than teaching students to use the unit rate m ethod For example, Lamon (1993b) stated, multiply and divide algorithm for solving missing value proportion problems is used meaningfully by very few students in spite of the fact that it has been the main strategy used in textbooks for many years (p. 152). The NCTM s to be that cross multiplication should be taught along with other methods of solving proportion problems and that students should understand why it works rather than simply memorizing the procedure.
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39 Ratio Comparison Problems As described above, there is a considerable body of literature regarding solution strategies for missing value problems. There is far less research literature on solution strategies for ratio comparison problems. Lamon (1993a, b) and Noelting (1980a, 1980b) are two of the few rese archers who have focused on solution strategies for ratio comparison problem s. (2007) discussed a problem in whic h seven girls share three pizzas and three boys share Lamon stated that children us e there was always 1 pizza for 3 people [as there is for the boys], the first group [girls] Noelting (198 0 a, b) described two methods of comparing ratios; he called these the Between and Within strategies. Noelting asked participants to compare two pitchers of orange juice, each made with different amounts of orange juice concentrate and water. According to Noelting, the Within strategy was used by participants who compared the amounts of orange juice and wa ter within each recipe first and then compared these two ratios. The Between strategy was used by participants who compared the two amounts of orange juice and the two amounts of water. Lamon (1993a) also discussed Between and Within strategies. Using her pizza between strategy a option is to think about children and pizzas separately, in whic h case quantities would be
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40 related using a within strategy Between strategy compared two like Thus, there appears to be in consistency in the research literature in the use of the terms Between and Within Summary of Solution Strategies Solution strategies for missing value problems are more often discussed in research literature than are solution strategies for ratio compari son problems. The traditional method of solving a missing value problem is through the use of a proportion and the cross multiply and divide algorithm. However, this method is highly symbolic and not easily understood conceptually. Less symbolic methods, s uch as the building up and unit rate strategies are less efficient, but may be more easily understood by students. Common Errors S tudents have predictable error patterns when working with proportional situations. Understanding these common errors may enab le curriculum authors to design problems and activities that help students recognize and avoid them. One common error is that students often use additive reasoning when proportional reasoning is needed ( Br ight et al. 2003 ; Singh, 2000 ). Another common err or is that s tudents often attempt to apply proportional reasoning to situations in which it is not appropriate (Heinz & Sterba Boatwright, 2008; Van Dooren, De Bock, Hessels, Janssens, & Verschaffel, 2005; Van Dooren et al. 2009). Examples of these errors are provided in th e following paragraphs.
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41 Inappropriate Application of Additive Reasoning As described in Chapter 1, there is an important difference between additive and multiplicative (proportional) reasoning. Additive reaso ning is developed in early elementary school and many students have difficulty making the transition to multiplicative reasoning in late elementary or middle school. Thus, o ne mistake students make when working with proportional situations is to attempt to apply additive reasoning (Bright et al., 2003; Che, 2009; Karplus et al., 1983; Lamon, 2007; Parker, 1999; Singh, 2000) Post et al. (1993) pointed out that this error is an attempt to apply previous understandings to new situations; they stated: Two types of relationships exist between any two numbers: additive and multiplicative. Additive considerations based on variations of the counting theme (count up, count back, skip count up, skip count back) dominate mathematics prior to the introduction o f rational number concepts. We know that this additive baggage is difficult for children to modify when new content domains require multiplicative, rather than additive conceptualizations (p. 333). Bright et al. (2003) posed the following problem to 14 ei ghth grade students: A farmer has three fields. One is 185 feet by 245 feet, one is 75 feet by 114 feet, and one is 455 feet by 508 feet. If you were flying over these fields, which one would seem most square? Which one would seem least square? Explain yo ur answers (p. 167). Five of the 14 students used the difference between length and width as the determining factor for an error that reflects the use of additive reasoning
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42 u had planned to purchase 2 pounds of mixed nuts for 8 people, but now 10 people are coming. How state that because the number of people increased by two, the pounds of nuts should increase by two. However, as Lamon points out, adding two people and two pounds means that the people : pound ratio would be 1 : 1, which is not correct. Although most researchers view additive reasoning as an error when multiplicative reasoning is needed Parker (1999) and Tourniare (1986) pointed out that additive reasoning is a natural transition to multiplicative reasoning. Parker stated, an additive building up strategy to solve problems. The use of an additive strategy can be Parker showed how building up strategies can be based on either addition or multiplication, thereby providing scaffolding for t he transition between the two types of reasoning. Tournaire suggested that proportional reasoning can be taught in elementary school provided that instruction built s trategies are sometimes seen as an error in proportional situations, they can also be seen as a starting place for instruction about proportionality. Inappropriate Application of Proportional Reasoning According to the Curriculum Focal Points (NCTM, 2006), seventh grade students should be able to d istinguish between situations that are proportional and th ose that are not proportional. However, r esearch findings have shown that both students and teachers have difficulty distinguishing between p roportional and non proportional situations
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43 ( Heinz & Sterba Boatwright, 2008; Van Dooren et al., 2005 ; Van Dooren et al., 2009 ). A common example of the difference between proportional and non proportional situations is the doubling of the dimensions of a rectangle. If both the length and width of a rectangle are doubled, the perimeter is also doubled. This is because the relationship between length, width, and perimeter is proportional. However, if the length and width of a rectangle are doubled, the area is not doubled. This is because the relationship between length, width, and area is not proportional. As Van Dooren et al. (2005) pointed out, many of the situations children ls of sand to fill one bucket, so 12 handfuls to fill three buckets; one toy car has four wheels, so two the elementary school curriculum is centered around proporti onal situations. They stated: Much attention is paid to proportional relations because of their wide applicability and usefulness not only for understanding numerous everyday life situations but also many problems in mathematics and science. This begins al ready in Grade 2 Van Dooren et al. speculated that the prevalence of proportional situations in everyday life and school mathematics results in students incorrectly using proportional reasoning to you think it will sell altogether in January, February
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44 In some mathematical relationships, when the value of one variable increases, the value of another variable decreases. This is called inverse variation Thus, proportional reasoning cannot be used to correctly solve these problems. Fisher (19 8 8) asked teachers that many secondary education teachers attempt to apply proportional re asoning to inverse relationships and then fail to notice that their answers are unreasonable. Because students have difficulty determining whether a problem can be solved with proportional reasoning, it seems reasonable to state that textbooks should help students learn to determine which situations require proportio nal reasoning and which do not. The extent to which recent textbooks contain material intended to help students discriminate between proportional and nonproportional situations is not clear. In this section, errors that students often make when dealing with proportionality have been discussed In the next section, curriculum analysis methodology is discussed. Curriculum Analysis Methodology The previous sections suggest that a n analysis of middle school textbooks would be helpful in determining how proportionality is covered. Before researchers begin a curriculum analysis, they should understand the criticisms that have been aimed at current curricula, frameworks that have been u sed to analyze curricula, and methods that have been used in other curriculum analyses. Therefore, i n this section, the following three topics are discussed: (a) methods of selecting textbooks for analysis (b) methods of measuring the challenge inherent i n tasks and (c) common features of curriculum analysis.
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45 Textbook Selection conduct of a content analysis requires identifying either a set of standards against which a curriculum is State content standards provide a general outline of what should be taught at various grade levels, but do not provide details regarding problem types and solution strategies that should be present ed or other factors researchers may look for. Thus, rather than comparing curricula to standards, curricula are often compared to each other, as recommended by the NRC. Funding by the National Science Foundation In an attempt to improve student achievement in mathematics and science, the National Science Foundation provided funds to develop curricula that would offer creative alternatives to traditional curricula. Tarr et al. (2008) provide background on this occurrence: In response to the Curric ulum and Evaluation Standards for School Mathematics (NCTM, 1989 ) and in an effort to influence and strengthen the quality of U.S. mathematics textbooks, the National Science Foundation (NSF) has invested an estimated $93 million in K 12 mathematics curric ulum development efforts active engagement of students, a focus on problem solving, and attention to conne ctions within mathematical strands as well as to real life contexts (p. 248) Because curriculum development projects that received NSF funding attempted to develop curricula that differed from existing curricula, it is reasonable to expect that the
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46 conten t and methods of presentation in NSF funded textbooks differs from that of commercially generated textbooks. In fact, Robinson et al. (2000) stated that NSF funded co One textbook series, developed by the University of Chicago School Mathematics Project, is considered neither NSF supported nor commercially generated ( NRC 2004). Selecting Textbooks Researchers who analyze mathematics text books and their effects on achievement generally use two criteria for selecting textbooks: they attempt to select widely used series and both NSF funded and non NSF funded curricula ( Hodges et al. 2008 ; Johnson, Thompson, & Senk, 2010; Tarr et al., 2008). For example, Hodges et al. Math Connects Connected Mathematics and Math Thematics because they are NSF funded series. Similar, Ta rr et al. chose textbook funded textbooks. Tarr et al. used the Glencoe publication as the non NSF funded curriculum and Connected Mathematics as the alternative curric ulum. Although researchers attempt to select textbooks with significant market share, determining which textbooks are most widely used can be difficult. The 2000 Mathematics and Science Education Survey has been used to estimate market share (Jones, 2004; Tarr et al., 2008), but the data used in the survey is more than a decade old. Additionally, there has been reorganization among the publishers in the past decade, complicating the use of the 2000 Survey.
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47 Measuring the Challenge Inherent in Tasks One feature of textbooks some researchers have sought to analyze is the level of challenge inherent in tasks. In this section, the author will describe how researchers have attempted to measure the level of challenge. In the 19 9 0s, the Quantitative Unders tanding: Amplifying Student Achievement and Reasoning ( QUASAR ) project drew attention to the importance of the level of challenge offered to students through the problems on which they worked. The QUASAR ntional mathematics instruction has placed a heavier emphasis on memorization and imitation than on understanding, & Stein 1996, pp. 477 478). Silver and Stein udents to con struct meaning and/or to relate There is some evidence that, internationally, tasks in mathematics textbooks offer little challenge. Valverde et al. (2002) believe th is situation represent s a paradox; they stated student performance promo ted by textboo ks were remarkably basic and comparatively 138). Increasing the level of challenge offered by tasks in textbooks requires a method of describing the level of challenge. Researchers hav e used various methods to do so; these methods are described below.
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48 Common Methods of Measuring Challenge Yan and Lianghuo (2006) described several ways of measuring the challenge inherent in tasks. One way is to distinguish routine problems Yan and Lianghuo defined a non be resolved by merely applying a standard algorithm, formula, or procedure, which is However, most problems can be solved thr ough the use of some algorithm or procedure and determining which algorithms According to Yan and Lianghuo (2006), t he level of challenge offered by tasks can also be measured problems. However, this requires careful definitio ns of these words. Yan and Lianghuo described four sub other re are not clear. According to Yan and Lianghuo (2006), the level of challenge offered by tasks can also be measured by distinguishing between open ended and close ended proble ms ; ended problem is a problem with several or many correct answers. Correspondingly, a close ( Yan & Lianghuo p. 613 ). However, although open ended tasks often offer more c hallenge than close ended tasks, this is not always the case. Therefore, distinguishing between these two types of tasks is not sufficient when attempting to measure the level of challenge inherent in a task.
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49 The previous paragraphs illustrate that describing the chall enge inherent in tasks can be difficult and requires careful definitions. Therefore, researchers have frameworks that organize tasks into levels based on the cognitive demand required to complete the task. One such framework is the Mathematical Task Framew ork. The Mathematical Task Framework A tool that has been used to measure what is expected of students is the Mathematical Task Framework (Stein et al., 2000) which describes levels of cognitive 11). Smith et al. described four levels o f cognitive demand which are summarized in the following paragraphs. Stein et al (200 0 demand. These are M emorization tasks and P rocedures W ithout C onnections tasks. According to Stein et al. a M emorization learned facts, rules, formulae, or definitions OR committing facts, rules, formulae or M emorization task, students simply reproduce previously seen material. In order to classify an exercise as a M emorization task, the researcher would need to know to what content students had been exposed. For example, Stein et al. gives this example of a M emorization equivalents for the fr actions and M emorization task implies that students had previously memorized these facts. If students had never before seen the decimal and percent equivalents for these fractions, this would not be a M emorization task.
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50 The second level in the Mathematical Task Framework includes tasks that require procedures without connections to understanding, meaning, or concepts which is usually shortened to Procedures Without C onnections is either specifically called for or its use is evident bas ed on prior instruction, experience, category, a researcher would need to be familiar with the prior instruction. Stein et al. gives this example of a Procedures Without C onnections to this task in this category is appropriate only if students are familiar with that procedure. Stein et al (200 0 ) describ demand. These are P rocedures W ith C onnections tasks and Doing M athematics tasks. According to Stein et al. in a Procedures With C onnections Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding These types of ta sks often involve graphical representations. For example, asking students to represent on a 10 by 10 grid is a Procedures With C onnections task (Stein et al.). In order to complete the task, student s must reason that a 10 by 10 gri d contains 100 squares ; thus, would need to be converted to a fraction with 100 as the denominator. The highest level in the Mathemat ical Task Framework is titled D oing M athematics mathematical con Shade 6 small squares in a 4 10 rectangle. Using the rectangle,
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51 explain how to determine each of the following: a) the percent of area that is shaded, b ) 13). This is considered a Doing M athematics task because students are provided with little guidance regarding how to proceed. It is up to the student to acces s relevant knowledge. According to Jones (2004), classifying tasks according to their level of cognitive demand was more difficult than classifying tasks along other dimensions. Also, the reliability of this coding was lower than the reliability of coding along other dimensions. Jones compared his coding to that of two other raters; his coding of levels of cognitive 73% of the tasks. He also coded 60 tasks twice with a t ime delay between and found that his earlier codes matched the later codes 88% of the time. Common Features of Curriculum Analysis Studies of mathematics textbooks generally focus on a single content area, such as data analysis (Cai, Lo, & Watanabe, 2002), probability (Jones, 2004) or reasoni ng and proof (Johnson et al., 2010 ). Despite the varying content areas, researchers conducting curriculum analyses generally look for certain aspects of the treatment of their topic. These common features of curri culum analysis are discussed in the following paragraphs T hen, to familiarize the reader with curriculum analysis methods, two case studies in curriculum analysis are described Although the research projects discussed below do not relate to proportional reasoning, they do show how curriculum analys i s is conducted. Although the procedures used in curriculum analysis vary from study to study, there are some commonly used methods, which are described below.
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52 Number of Pages or L essons In some curriculum anal yses, researchers attem pt to determine the approximate percentage of a textbook that is devoted to a certain topic. This is often done by counting pages or lessons. For example, Jones (2004) counted both the number of pages and the number of lessons devote d to probability. He then calculated percentages of textbooks devoted to probability. This method allows researchers to compare textbooks from various time periods or various publishers. However, this method is not always feasible. Some mathematical topic s are spread throughout the book, making this type of estimate much more difficult. For example, Johnson et al. (2010 ) investigated the treatment of proof related reasoning. Rather than examine every page in the textbooks, they restricted their analysis to three topic areas that they thought were most likely to contain proof related reasoning. Because they did not examine every page in the textbooks, they could not arrive at an estimate of the percentage of pages in the textbook that dealt with proof relate d reasoning. Similarly, as an mathematics topics studied in grades 6 appear throughout middle school textbooks. Researchers could choose to examine every page in middle school textbooks, or they could choose to focus on sections that are likely to relate to proportionality, such as sections on algebra, rational numbers, similar figures, and probability. Making this latter choice would allow researchers to devote mo re attention to these sections but would preclude estimating the percentage of pages devoted to the subject.
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53 Problem T ypes When conducting curriculum analyses, researchers often note the various types of problems contained in textbooks. Problem types are often supported by research literature. For example, the literature on mathematical proof indicated that students often confuse specific examples of a property with a general proof or argument that applies to all instances of the property. For this reason Johnson et al. (2010 Similarly, Jones (2004) relied on previous research suggesting He counted the number of tasks that related to ea ch of the four constructs and then calculated the percentage of probability tasks that related to each of the four constructs. As with other curriculum analyses, research on the treatment of proportionality should include attention to problem types. B ecaus e the research literature on proportional reasoning suggests that there exists three problem types (missing value, comparison, and qualitative), a curriculum analysis on proportionality should include attention to these problem types. Visual R epresentation s symbols, drawings, and graphs help s middle school students reason about and understand mathematics mathematical ideas 2008). Specifically, research has shown that middle school students understand some representations of rational numbers bet ter than others (Martinie & Bay Williams, 2003a). R esearch findings indicate that the representat ions indicate that interpreting fractions and decimals as locations on a number line is more
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54 difficult for students than interpreting fractions as part of geometric regions (Ko uba Zawojewski, & Struchens, 1997). Examples, Exercises, and Tasks Researchers who conduct curriculum analyses make decisions about how to segment the curriculum into manageable pieces. Several researchers have defined blocks 004 ; Jones & Tarr, 2007; Valverde et al. 2002). Looking at consists of one example or one exercise. Other researchers have focused on statements of properties in th idual exercises in the exercise sets of textbooks (e.g., Johnson et al., 2010 ). In this was defined as a problem or question that is labeled as an example and that is solved or answered in th e was defined as a problem or question that appears in an exercise set and is not solved or answered Exercises may contain multiple parts, as explained in a later section. an activity, example, or exercise. Example of Curriculum Analysis: Reasoning and Proof Johnson et al. ( 2010 ) analyze d the treatment of proof related reasoning in high school text books. The researchers began by conducting a literature review to familiarize themselves with major developments in the field. This literature review affected the framework they developed. For example, research findings indicated that many students and tea chers do not understand the difference between an empirical argument based on specific examples of a phenomenon and a logical argument that applies to all cases of the
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55 phenomenon. The researchers later built into the framework a distinction between the two types of arguments. Once the literature review was complete, the researchers consulted standards documents published by the NCTM. The researchers noted what the standards documents said that students should be able to do. Because some expectations that ap pear in standards documents are difficult to define precisely, only some of these expectations expectations were likely to appear in textbooks in a measureable manner; t he standards bas ed expectations they felt could be defined in a precise and measureable way were built into the curriculum analysis framework. On the basis of these expectations, the ply to the examples and exercises in the textbooks. The researchers chose s ix textbooks series to analyze. They based the selection on their knowledge of which series are frequently used and their desire to compare commercial series to curriculum developme nt projects. They included a total of 20 textbooks. The researchers then developed a list of key words associated with reasoning and proof. They searched both the table of contents and ind ices for these key words. On the basis of this search, the researchers chose three content areas that seemed to contain the most material related to reasoning and proof. Rather than conducting a curriculum analysis on every lesson in each textbook, they fo cused on lessons in the three content areas they had identified.
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56 The researchers then examined the examples and exercises in the textboo ks and compared them to the categories they had developed. Because they found some examples and exercises that did not fit well into the existing categories, new categories were added As the researchers examined additional lessons, they continually reviewed the framework to ensure their categories adequately represented the types of reasoning presented in the examples and required in the exercises. The researchers found that developing a curriculum analysis framework to examine the reasoning and proof in high school textbooks required several revisions. The researchers often initially disagreed about which category an ex ample or exercise most closely resembled. In the end, however, they did develop a framework that could be used reliably. Example of Curriculum Analysis: Probability Jones (2004) developed a framework to analyze the treatment of probability in middle scho ol textbooks. Because he wished to compare contemporary textbooks to ol der ones, he chose two textbook series from each of four time periods. Within each time textbooks, he analyzed the textbooks for their treatment of probabili ty. One of the research questions posed by Jones (2004) related to the extent to which middle grades textbooks incorporate probability. He answered this question by counting the number of pages that contained probability tasks and comparing it to the num ber of pages in the textbook. This allowed him to state whether the percentage of pages devoted to probability has changed in the past few decades. He also identified probability
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57 ndependence, experimental probability, sample space, etc. and determined the percentage of analyzed tasks that related to each of the constructs. (p. 7). To answer this question, Jones noted which manipulatives were used, whether tasks were new or repetitions of previously encountered tasks, and the level of cognitive demand required by tasks. The third and final research question posed by Jones ( 2004) concerned the definitions, worked examples, and oral or written exercises. He also reported on lesson sequences, or the organization of lessons related to probability. Su mmary of Curriculum Analysis Methodology Although every curriculum analysis study utilizes different research methods, many have similarities. In selecting textbooks for their study, researchers often include both NSF Standards nd non NS F Some researchers attempt to measure the challenge inherent in tasks; one method that has been used (Jones, 2004) is the Mathematical Task Framework (Stein et al., 2000). In some curriculum analyses, page numbers devoted to a given topic are counted. This allows the researcher to state the percentage of the textbook devoted to the topic. Researchers conducting a curriculum analysis regarding a given topic, such as proof or probability, generally do not analyze the entire textbook bu t instead analyze the lessons most closely related to that topic.
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58 Summary of Literature Review important transition from elementary, additive ways of thinking to more advance d, multiplicative modes of thought. However, successfully completing this transition can be difficult Part of this difficulty is likely due to the complexity of this type of reasoning as it involves comparisons of ratios, which are themselves comparisons of quantities. Researchers have identified three problem types related to proportionality : missing value ratio comparison and qualitative In a missing value problem, students are given three out of four values that form a proportion and are asked to find the fourth. In a ratio comparison problem, students are asked to compare or order ratios or fractions. To solve a qualitative problem, students must think ab out which values are bigger or smaller without being given numerical quantities. Proportionality is a mathematical characteristic of a wide variety of situations and could, therefore, appear within textbooks in a variety of content areas. Proportionality could appear in lessons on algebra through discussions of patterns, slope, and y intercept. Proportionality is likely present in some lessons on probability, statistics, and data analysis. It could appear in lessons on geometry in the guise of similar figu res or in measurement in the form of measurement conversions. Proportionality is often closely associated with rational numbers (e.g., Behr et al., 1992; Lamon, 2007). Methods of solving missing value problems have been widely discussed and include the bu ilding up strategy, proportions and the unit rate method. Students seem to be able to discover the building up and unit rate strategies on their own; thus, they may provide a bridge to true proportional reasoning. The proportion strategy is more advanced
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59 and some researchers have suggested that students use it mechanistically without true understanding. Researchers have discussed errors that students commonly make. These include using additive reasoning when multiplicative reasoning is needed and vice ver sa. Elementary school mathematics focuses on additive reasoning, and some scholars believe Despite the large body of research on proportional reasoning and ration al numbers, there are still unanswered questions. Existing research has focused on understanding the mathematics inherent in these problems or understanding thinking about them. No rigorous analysis of the treatment of proportionality in textboo ks has been conducted. Unanswered Questions Because proportionality is difficult for many students it is important that its coverage in textbooks is the best that it can be. Because no recent curriculum analyses of the treatment of proportionality have be en conducted, it is unknown how textboo ks cover the topic. A curriculum analysis is needed. This research could help curriculum authors improve how they present proportionality to students and could help teachers improve their use of existing textbooks. W hen conducting an analysis of curriculum materials related to proportionality, researchers should look for several factors. First, because the NCTM has stated that proportionality is a connective thread that runs throughout the middle school curriculum, pr oportionality should be emphasized in several content areas including algebra, data analysis, geometry and measurement, percent, and probability. It is not necessary that
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60 proportionality be emphasized equally in each of these content areas, but it should b e mentioned in lessons in each content area. Second, three main types of problems have been identified ( missing value ratio comparison and qualitative ) and textbooks should contain all three. Again, it is not necessary that one third of the examples an d exercises deal with each type of problem, but each type of problem should be represented. Third, researchers should examine the solution strategies encouraged by textbooks. P roportion s have been the traditional method of answering proportional questions, but many researchers believe that this method is less intuitive than others. Researchers believe that the building up strategy can provide a scaffold from additive to multiplicative reasonin g and that the unit rate method seems more natural to students than setting up a proportion and that many students are more successful with the unit rate method (Cramer et al., 1993) A fourth factor that researchers should look for is attention to common errors. Many students use additive reasoning when multiplicative reasoning is required and vice versa. Educators have criticized textbooks for inadequately pointing out the differences between proportional and nonproportional situations ( Cramer et al., 19 93) However, these criticisms are more than a decade old. The extent to which current textbooks point out the differences between proportional and nonproportional situations in unknown. Fifth researchers should investigate the level of cognitive demand required by examples and exercises that involve proportionality. There are large differences between to foster different kinds of knowledge and skills. Finally, be cause students have more
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61 difficulty with some graphical representations of rational numbers than with others (Kouba et al., 1997), researchers should determine which types of graphical representations are used. Summary of the Pilot Study As part of the refinement of the framework for analyzing the treatment of proportionality in textbooks, the researcher applied the framework to two sixth grade textbooks, one from the Math Thematics series (Billstein & Williamson, 2008) and the other publi shed by McDougal Littell (Larson, Boswell, Kanold, & Stiff, 2004) A summary of the pilot study is provided below and details are located in Appendix A. Of the three problem types described in the literature review many examples of missing value and ratio comparison problems were found. Less than 1% of the tasks were of the qualitative problem type. The researcher found tasks related to proportionality that could not be classified as any of the three types found in the research literature. For example, put ting a fraction into lowest terms cannot be classified as any of the three problem types. The researcher found tasks related to proportionality in all of the content areas described in the literature review. More than half of the tasks were found in the ra tional number content area, reflecting the middle The Math Thematics textbook also emphasized proportionality in the geometry/ measurement content area whereas in the traditional textbook, less than 10% of the pro portional exercises were in this content area. The researcher found very few tasks that could help students understand the differences between proportional and non proportional situations. Textbooks could help
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62 students understand these differences by pointing out the characteristics of proportional situations: they have a constant rate of change and a y intercept of zero. In the two textbooks combined, less than 4% of the exercises related to proportionality poi nted out these characteristics.
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63 CHAPTER THREE: METHODS The purpose of this chapter is to describe the methods used in the study. Th e chapter contains six sections The first discusses how the textbooks used in the study were selected, the second dis cusses how lessons from those textbooks were selected, the third describes how tasks were selected and counted, the fourth describes the curriculum analysis framework, the fifth discusses reliability, and the sixth presents a summary of the methods used. T extbook Selection Sample selection occurred in three stages. First, of the wide variety of textbook series available for middle school students, three widely used series were selected. Second, from the dozens of lessons in each textbook, lessons that were most closely related to proportionality were selected for the study. Third, from each of the selected lessons, tasks (activities, examples, and exercises) related to proportionality were selected and coded. The textbook selection criteria are described in this section. Subsequent sections describe the lesson selection and task selection processes. T hree contemporary middle school textbook series were selected : Math Connects: Concepts, Skills, and Problem Solving published by Glencoe McGraw Hill, the third edition of books developed by the University of Chicago School Mathemat ics Project (UCSMP) which are titled Pre transition Mathematics Transition Mathematics and Algebra and the second edition of books developed by the Connected Mathematics Project (CM P) Thes e textbooks were selec ted because are all widely used. O ne is NSF
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64 funded ( CMP) and two are not ( UCSMP and Math Connects ). Details about selection criteria and i nformation about each textbook series is provided below. Selection Criteria Tw o criteria were used to select textbook series for the study. First, only textbooks from widely used series were selected. Textbook publishers consider market share data to be proprietary information; thus, this information was unavailable. However, other so urces suggested that the three series chosen are widely used ( Hodges et al., 2008; Huntley, 2008; Jones & Tarr, 2007; Lappan Phillips, & Fey, 2007; Ridgway, Zawojewski, Hoover, & Lambdin, 2003; Tarr et al., 2008 ). Second, because the researcher was inte rested in obtaining varying pictures of the treatment of proportionality, she chose series with different histories and goals. Curriculum projects that received funding through the National Science Foundation ( NSF ) have generally been heavily influenced by the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) and the Principles and Standards for School Mathematics (NCTM, 2000); thus, they are often called Standards based curricula ( Robinson et al., 2000) Standards based curricula teaching and learning that is qualitatively different from conventional practice in content, Thus, in order to obtain varying pictures of the treatment of pro portionality, the researcher chose one textbook series that was NSF funded and Standards based (CMP), one series that was not NSF funded ( Math Connects ), and one series that was not NSF funded but was designed as an alternative to traditional curricula (UC SMP).
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65 Connected Mathematics2 (CMP) One of the three textbook series chosen for this study wa s the CMP curriculum. The following paragraphs describe the goals and format of the CMP curriculum. Goals of the CMP C urriculum The authors of the CMP curriculum provide this background: The Connected Mathematics Project (CMP) was funded by the National Science Foundation between 1991 and 1997 to develop a mathematics curriculum for grades 6, 7, a nd 8. The result was Connected M athematics 1 (CMP 1), a complete mat hematics curriculum that helps students develop understanding of important concepts, skills, procedures, and ways of thinking and reasoning in number, geometry, measurement, algebra, probability, and statistics. In 2000, the National Science Foundation fun ded a revisi on of the CMP materials (Lappan et al., 2007, p. 67) Proportionality is a central focus of the CMP priority goal for the seventh grade Connected Mathematics curriculum is the development of proportional reason Ridgway et al., 2003, p. 213). The development of CMP and understand ing of rational numbers (Lappan et al. 2007). One would therefore expect the treatment of proportiona lity in these materials to be different from that of textbooks with different goals and whose development was not guided by this research. Although there is no published information regarding the market penetration of the second edition, the first edition of Connected Mathematics was widely used ( Hodges et al., 2008; Huntley, 2008; Lappan et al. 2007 ; Ridgway et al.).
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66 CMP is a problem centered curriculum (Lappan et al., 2007), meaning that the for students but generally do not explain how to solve problems. Because the focus is on student solution of problems CMP the study. Format of the CMP C urriculum The CMP curriculum consists of eight modules at each of the three grade levels The modules are usually purchased as separate small booklets. Thus, there is no single together and i t was this format that the researcher used during this study. Therefore in this study refer s to the set of modules for a given grade level. Four sixth grade modules were included in the study: Bits and Pieces I (Lappan et al., 2009a), B its and Pieces I I (Lappan et al., 2009b), Bits and Pieces I II (Lappan et al., 2009c), and How Likely Is It? (Lappan et al., 2009d). Six seventh grade modules were included: Variables and Patterns (Lappan et al., 2006a), Stretching and Shrinking (Lappan et al., 2006b), Comparing and Scaling (Lappan et al., 2006c), Moving Straight Ahead (Lappan et al., 2006d), Filling and Wrapping (Lappan et al., 2006e), and What Do You Expect? (Lappan et al., 2006f). Three eighth grade modules were included: Thinking with Mathematical Models (Lappan et al., 2006g), Say It with Symbols (Lappan et al., 2006h), and Samples and Populations (Lappan et al., 2006i). A list of the Investigations included from each of these modules is provided in Appendix B Each Inves tigation was counted as one lesson.
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67 Glencoe Math Connects school mat hematics were published in 2009 and are titled Math Connects : Concepts, Skills, and Problem Solving. The Math Connects th grade). Math Connects middle school mathematics publications which were titled Mathematics: Applications and Connections (published 1998 2001) and Mathemat ics: Applications and Concepts (published 2004 school textbooks are widely used (Hodges et al., 2008; Jones & Tarr, 2007; Tarr et al., 2008). Goals of the Math Connects C urriculum According to a Glencoe publication (Macmillan/McGra w Hill/Glencoe, 2009), the Math Connects Curriculum Focal Points (NCTM, 2006), process standards (NCTM, 2000), and research on problem solving, reasoning, representation, discourse, reading, and writing. According to the Glencoe publication, the Math Connects world applicat ions, hands on labs, direct instruction, writing exercises, higher order thinking and practice that (p. 1). Format of the Math Connects C urriculum Each of the three Math Connects books contain s five units. Each unit is divided into two or three chapters. Each of these chapters contains between seven and 10 lessons. Thus, each Math Connects textbook contains about 125 lessons. Each lesson contains about a page of narrative and several worked examples. T hus, t he Math Connects books
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68 contain more narrative and worked examples for students to read than do the CMP textbooks. extend another lesson. Each ch lessons generally consist of an activity, usually with multiple steps. University of Chicago School Mathematics Project (UCSMP) Thompson and Senk (2001) provided the following background regarding the UCSMP textbooks : The University of Chicago School Mathematics Project (UCSMP) is a research and curriculum development project that was begun in 1983, prior to the development of the NCTM Standards to the publication of the Standards, continued work has been influenced by the Standards movement and the discussions about reform that have resulted from this movement (p. 59) The middle school UCSMP books are title d Pre Transition Mathematics (sixth grade), Transition Mathematics ( seventh grade), and Algebra (eighth grade). The third edition of these textbooks, published by Wright Group/McGraw Hill in 200 8 and 200 9 was used in the study. These middle school textboo school textbooks and are meant to provide a transition to the high school curriculum (Usiskin, 2007). The U C S MP textbooks are buil t on four key program features: a wider scope of mathematical content than found in t raditional programs, an emphasis on real world applications, use of technology, and a focus on four dimensions of understanding: skills, properties, uses, and representations (McConnell et al., 2009).
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69 Some studies of middle school mathematics textbooks ha ve excluded algebra textbooks from their sample (Jones, 2004; Jones & Tarr, 2007). However, because UCSMP intends for their algebra textbook to be used by at grade level eighth grade students, it is considered a middle school textbook and was therefore inc luded in the study. Summary of Textbook Selection The information presented above illustrates that the three textbook series selected for this study have different histories, goals, and emphases on proportionality. These three textbook series were chosen because they were expected to provide different pictures of how proportionality is treated in middle school textbooks. Lesson Selection Once the three textbook series were selected, the researcher identified lessons most closely related to proportionality. These lessons were selected only from the main part of the textbooks. Excluded sections of the textbooks are described below. Excluded Sections of Textbooks To allow the researcher to focus on the main con tent of each textbook, c ertain sections of textbooks were excluded from the study. Thes e include d introductory material designed to prepare students for the main content of the textbook as well as glossaries, indices, and standardized test preparation sect ions that appear ed in the back of the textbooks The CMP the end of each unit. These sections are each approximately three pages long and were
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70 excluded from the study. Some units also ha ve one or two Unit Projects, each of which is about two pages long. The Unit Projects were excluded. T he Glencoe Math Connects in the were excluded from the study. Additionally, each chapter in the Glencoe Math Connects books has a one page review for checking prerequisite skills. These were also excluded from the study. A few of the lessons in each Math Connects Although some of these Investigations contain exercises related to proportionality, their focus is on problem solving strategies rather than proportionality and they were excluded from the study. Mid chapter quizzes, study guides, and practice tests were also excluded. The Glencoe Math Connects were excluded from the study. The Glencoe Math Connects books also have Student Handbooks at the back of the textbooks that include extra practice, preparation for standardized tests were excluded. Finally, most lessons in Math Connects textbooks begin with a short section one to eight questions that usually focus on lower level skills that are prerequisites to the lesson. These s ections were not analyzed. In t he UCSMP textbooks were were excluded from the study.
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71 Lesson Selection Criteria In this section, the criteria used to select lessons are described. The criteria are grouped by content area. Included Algebra L essons Patterns, sequences, and functions lead into the concept of slope; a constant slope is one of the hallmarks of proportional situations. Thus, lessons on patterns, sequences, and functions were included in the study. L essons on slope were also included in the study. Some lessons o like Lessons such as these were included in the study. Lessons on direct variation were also included. L essons that relate d to solving proportions were included. Lessons on using graphs and tables to express linear relationships were included as were lessons on the differences between linear and non linear functions. Lessons on writing equations to correspond to a data table o r verbal description were included in the study. Excluded Algebra L essons Lessons on negative numbers and absolute value were excluded from the study. Lessons on evaluating expressions and equivalent expressions were also excluded. Lesson s on domain and ra nge were excluded from the study as were lessons on expressions and equations. L essons on the commutative and distributive properties were excluded Lessons on order of operations, simplifying algebraic expressions and solving algebraic equations were also excluded unless they relate d to solving proportions L essons on systems of equations were also excluded.
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72 Included D ata Analysis L essons Lessons on circle graphs generally involve percent or proportions. For example, an example in the eighth grade Math Connects textbook states that 12 out of 30 countries won between 1 and 5 medals in the 2006 Winter Olympics. To make a circle graph, students must determine how many degrees (out of 360) correspond to 12 out of 30 countries. This could be solved throu gh the use of a proportion such as Therefore, lessons on circle graphs were included. Additionally, l essons on using data, ratios, or probability to make predictions were included in the study. Excluded Data Analysis L essons Lessons on bar grap hs, box plots, histograms, line plots, and stem and leaf graphs were excluded from the study Lessons on interpreting line graphs were excluded unless they contain ed a discussion of slope. Lessons on the use of tables as a problem solving strategy were ex c luded from the study. L essons on the parts of graphs, such as titles, scales, axes, and intervals were excluded. Probability and statistics are included in the data analysis standard. However lessons on basic probability including tree diagrams outcomes and sample spaces were excluded. Lessons on basic statistics, such as mean, median, mode, and variability were also excluded. Included G eometry L essons L essons on area, perimeter or circumference were included only if they contained material regarding how these quantities are affected when the side lengths or the radius is changed Similarly, lessons on volume or surface area were included only if they
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73 contained material regarding how these quanti ties are affected when side lengths are changed L essons on dilations, similar figures and scale drawings were also included. Converting from one unit of measurement to another requ ires multiplication or division; proportions and /or ratio tables are often used. Thus, lessons on measurement conversion were included. Excluded Geometry L essons Lessons on the coordinate plane and ordered pairs were excluded from the study Lessons on translations, reflections, and rotations were excluded. Lessons on types of angles (acute, right, and obtuse) were excluded from the study as were lessons on angle relationships (complementa ry, supplementary, and vertical ) Lessons on the sum of the angle s in polygons were excluded as were lessons on constructi ons with straightedge, compass, and protractor. Lessons designed to familiarize students with units of measurement were excluded unless they also involve d conversions between units. Lessons on elapsed time o r temperature were also excluded. Lessons on area of triangles and parallelograms usually focus on identifying the base and height and using formulas. Some lessons are intended to help students formulate or understand the origin of these formulas, but these lessons are not directly related to proportiona lity. Therefore, lessons on area of triangles and parallelograms were excluded from the study. Some lessons on congruent and similar figures focus on definitions of these terms and help students identify corresponding parts. Because these skills do not in volve
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74 proportionality, lessons on congruent and similar figures were not included unless they also involve d calculating lengths of sides of polygons. Lessons on I ntegers Although integers are rational numbers, they are less obviously comparisons between qu antities than are fractions or ratios. For example, the fraction compares the quantities 2 and 3. The integer 5 can be written as which compares 5 to 1, but the integer 5 is usually thought of as a single quantity rather than a ratio. Thus, most les sons on integers were excluded. In middle school textbooks lessons on integers generally involve computation with integers divisibility, order of operations, real life uses of integers, or rounding integers These lessons were excluded from the study. Included D ecimal L essons least to greatest or vice versa. The skills of c o mparing and ordering decimals are related to pr oportionality. For example, when comparing 2.78 and 2.81, a person could compare two mixed numbers, and or two fractions, and Because this involves ratio comparison, lessons on comparing and ordering decimals were included in the study. Rounding decimals also involves ratio comparison. For example, when rounding 99.96 to the nearest tenth, one compares 99.9, 99.96, and 100.0, or 99 99 and 100. Thus, lessons on rounding decimals were included Lessons on converting bet ween decimals and fractions were also included.
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75 Excluded D ecimal L essons on place value and various ways of writing decimals. For example, when explaining the meaning of 1.65, a textbook may point out that the one is in the ones place, the six is in the tenths place, and the five is in the hundredths place. Textbooks often use 10 by 10 grids, base ten blocks, place value charts, and money to explain the meaning of decimals. For example, 1.65 can be represented by one dollar bill, six dimes, and five pennies. These lessons have little to do with proportionality and were excluded. Lessons on adding, subtracting, multiplying, and div iding decimals were excluded. Lessons on estimating sums, differences, and quotients of decimals were excluded from the study. Although lessons on estimation involve rounding, which is included in the study, lessons on estimation usually assume students kn ow how to round and instead focus on computation procedures. Included Fraction L essons L ess ons on equivalent fractions, mixed numbers, improper fractions, and simplifying fractions were included in the study. L essons on comparing and ordering fractions wer e included. Rounding fractions involves ratio comparison. For example, rounding to the nearest half involves comparing 2, and Thus, lessons on rounding fractions were included. Ratios can be written as fractions and can be simplified as fractions. For example, the ratio 2:6 can be written and simplified as either 1:3 or Thus, lessons on ratios were included in the study. Lessons on writing and solving proportions were also included.
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76 Excluded Fraction L essons Lessons on understanding fractions, such as explanations of what the numerator and denominator represent were excluded from the study. Lessons on greatest common factor and least common multiple were excluded. Less ons on fraction computation were excluded from the study. Although adding and subtracting fractions involves finding equivalent fractions, which wa s included in the study, lessons on adding and subtracting fractions usually assume students are familiar wit h equivalent fractions and focus instead on procedures for computation. Lessons on estimating sums, differences, products, and quotients of fractions were also excluded. Although lessons on estimation involve rounding, which wa s included in the study, less ons on estimation usually assume students know how to round and instead focus on computation procedures. Included Percent L essons Lessons on converting between fractions decimals, and percents were included in the study Proportions can be used to find th e percent of a number. For example, to calculate 52% of 298, one could use the proportion Thus, lessons on finding the percent of a number were included in the study. To estimate 52% of 298, one could round 52% to 50% and use the proportion Thus, lessons on estimating with percents were also included. Excluded Percent L essons Many middle school textbooks include a lesson on using algebraic equations to solve percent problems. L essons in which students are taught to use an equation to solve percent problems focus on helping students put numbers in their proper places in the
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77 equation and on solving the equation. Because these skills are not related to proportionality, lessons on using algebraic equations to solve percent problems were e xcluded. Many middle school textbooks include a lesson on percent increase and percent a formula, Similarly, lessons on interest usually involve the formula Use of the s e formula s does not require proportional reasoning Therefore, lessons on percent of change or interest were excluded. Summary of Lesson Selection Cr iteria T able 1 summarizes the types of lessons that were included and excluded in each of the content areas.
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78 Table 1 Topics of Included and Excluded Lessons Content Area Included Topics Excluded Topics Algebra methods of solving proportions solving equations and systems constant slope (rate of change) the distributive property function rules order of operations Data Analysis circle graphs interpretation of graphs using probability to make predictions parts of graphs problem solving strategies Geometry area, perimeter and circumference the coordinate plane similar figures and scale drawings transformations measurement conversion types of angles Decimals comparing and ordering decimals representing decimals rounding decimals computation with decimals converting decimals to fractions Fractions equivalent fractions greatest common factor improper fractions, mixed numbers least common multiple comparing and ordering fractions computation with fractions Percents converting pe rcents to fractions equations and percent problems finding the percent of a number percent of change
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79 Number of Lessons Included in the Study Prior to beginning the study, the researcher examined each of the textbooks and developed a preliminary list of 200 lessons to be included in the study. During the study, closer examination of these lessons revealed that 35 of the lessons on the preliminary list did not focus on proportionality to an extent that would warrant inclusion in the study. Therefore, the sample included 165 lessons. Because Math Connects textbooks contain more lessons than textbooks from the other series, more Math Connects lessons were included in the study. Of the 165 lessons, 28 (17%) were from CMP, 91 (55%) were from Math Connects and 46 (28%) were from UCSMP. Table 2 presents the number and percent of lessons from each textbook. Table 2 Number and Percent of Lessons Included in the Study From Each Textbook C MP Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Number of Lessons Included 9 1 5 4 3 3 3 2 2 6 2 1 19 6 Number of Lessons in Book 34 34 35 133 125 122 106 105 108 Percent of Lessons Included 26 44 11 25 26 21 20 18 6 Of the 165 lessons, 35% were in the Algebra content standard. The rational number and geometry/measurement content standards were represented approximately equally, with each accounting for about 26% of the lessons included. Less than 13% of the lessons were in the Data Analysis/Probability content standard.
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80 Grade Levels of Lessons Included in the Study M ore lessons were coded from the sixth and seventh grade textbooks than from the eighth grade books. One reason could be that the eighth grade textbooks focus ed on algebra. This wa s especially true in the CMP and UCSMP series. In most of the sixth and seventh grade textbooks, more than 20% of the lessons were related to pr oportionality; in the eighth grade books, less than 20% of the lessons wer e related to proportionality. A larger number of lessons were coded from the Math Connects series than from the other two series. At the sixth and seventh grade levels, this reflects the fact that the Math Connects textbooks contain more lessons than the sixth and seventh grade books from the other series. Similar percentages of lessons were coded from each of the sixth and seventh grade books. For example, between 20 and 26% of the lessons in each sixth grade textbook were coded. At the eighth grade lev el, a higher percentage of lessons were coded from the Math Connects textbook than from the other series. This reflects the fact that the eighth grade CMP and UCSMP textbooks contain more of a focus on algebra (and less of a focus on proportionality) than does the eighth grade Math Connects book. Task Selection and Counting Three types of tasks were included in the study: activities intended to be completed in class, examples with worked out solutions, and exercises intended to be completed at home. Each of the tasks in the selected lessons were read and considered, but only tasks related to proportionality were coded according to the curriculum analysis framework. The following sections describe how tasks were selected from each of the lessons included in the study.
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81 Algebra Tasks Three types of algebra tasks involve proporti onality. S ome tasks related to patterns and sequences, some related to function rules, and some tasks related to graphing involve proportionality. However, not all tasks related to these topics involve proportionality. Details are provided below. Tasks Rel ated to Patterns and Sequences Tasks related to patterns typically show a visual pattern and ask the student to draw the next item in the pattern or predict how many blocks or tiles would be used to make either the next item or an item further along in the pattern. Sequences consist of numbers rather than visual patterns, but the task is generally the same to predict the next number or a number further along in the sequence. Sequences that were presented as lists of numbers were not coded. An example of a task that was not coded states, (Day et al., 20 09 a p. 27). Some tasks related to patterns and sequences ask students to write a function rule. These tasks are directly related to the concept of slope and were coded. Tasks R elated to Function R ules Tasks related to function rules typically have one of two goals. Some tasks ask students to create a function table based on a given function rule. These tasks require students to substitute numbers for a variable and compute with these numbers. Because these tasks require only computation, they were not cod ed. An example of an exercise that was not coded appears in Figure 1.
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82 Figure 1 Function rule exercise not related to proportionality Other tasks require students to do the opposite, to identify a function rule that match es a given table. These tasks require students to notice a pattern and to consider rate of change. Therefore, tasks that require students to identify a function rule to match a given table were coded. An example of an exercise that was coded appears in Figur e 2 Figure 2 Function rule e xercise related to proportionality Tasks Related to G raphing Tasks related to graphing positive and negative integers on a number line were not coded. Tasks that focus ed on graphing ordered pairs were not coded. Tasks related to line graphs were coded only if a discussion of slope or rate of change was present. Find the rule for each function table. x 2 4 5 10 8 16 (Day et al., 2009a, p. 51) Copy and complete each function table. Input ( x ) Output ( x + 3) 0 2 4 (Day et al., 2009a, p. 51)
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83 Da ta Analysis Tasks Two topics related to data analysis and probability require proportional reasoning: (a) the construction of circle graphs and (b) the use of circle graphs or probability to make predictions. Tasks related to the construction of circle graphs were coded, as in the exercise in Figure 3. Figure 3 Data analysis exercise related to proportionality Tasks related to the use of circle graphs to make a prediction were also coded. For example, an exercise that was coded in the pilot study read types of hits by Cal Ripkin, Jr., in one season. In Exercises 12 15, predict the number of graph show ed that Ripkin hit 124 singles in one season; to complete the exercise, students must solve a missing value problem and compute the number of singles he would be likely to hit in four seasons. Tasks in which students were asked simply to compare sizes of the sections of a circle graph were not coded. Tasks related to bar graphs, line plots and stem and leaf plots were not coded. Tasks in which students used probability to make predictions were coded A random sur vey of people at a mall shows that 22 prefer to take a family trip by car, 18 prefer to travel by plane, and 4 prefer to travel by bus. If 500 people are surveyed, how many should say they prefer to travel by Day et al., 2009c, p. 656 ). and 5 ride in a car. a. How many sectors should a circle graph of this information contain? b. Determine the measure of the centra l angle for each sector. c. Complete a circle graph for the data (McConnell et al., 2009, p. 297).
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84 Geometry/Measurement Tasks Tasks in which students were asked to convert between units of measurement were coded. For example, a student could convert four yards to feet by solving the proportion However, tasks in which students are asked to add or subtract measurements were not coded. Although adding and subtracting measurements often requires measurement conversion, the focus of addition and subtraction problems is generally not on the conversion. One way textbooks hi nt at when proportional reasoning is appropriate is to point out that the relationship between side lengths and perimeter is proportional but the relationships between side lengths and area and volume are not proportional. Tasks that discuss ed what happens when side lengths or radii are doubled and tripled were coded. For example, in the pilot study, the following exercise was al., 2004, p 495). Area and perimeter tasks were coded only if they were directly related to proportionality in this way. Tasks in which students simply calculate area or perimeter were not coded. The same reasoning applie d to lessons and tasks on surface area and vo lume: they were only coded if students are asked to consider the consequences of changing side lengths. For example, in the pilot study, the following exercise was coded: 2004, p. 616). Tasks on size changes were coded, as in this seventh model of a car is
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85 Rational Number Tasks T ask s that involve d combining two numbers to form a fraction or ratio were considered to not involve proportionality. The example in Figure 4 was considered not to involve proportionality. Figure 4 Ratio e xample not related to proportionality Lessons that involve d conversion between improper fractions and mixed numbers were analyzed, but some of the examples and exercises in these lessons did involve proportional ity For example, if the examples in such a lesson instruct ed students to convert an improper fraction into a mixed number by dividing the numerator by the denominator, proportio nal reasoning wa s not involved. Lessons that involve d finding the percent of a number were analyzed, but the examples in them were coded only if a proportion strategy wa s used to solve them. Many textbooks instruct students to change percents into decimals or common fractions in order to solve percent problems. For example, one can find 75% of 60 by multiplying by 60. This does not require proportional reasoning. Therefore, examples that suggest this method of solving percent problems were not coded. However, students may solve exercises in these sections with the use of a proport ion, such as Therefore, most of the exercises in lessons on finding the percent of a number were coded. Exercises that instruct students to use the formula to find simple interest were not coded. Example 1 Writing a Ratio in Different Ways In the orchestra shown above, 8 of the 35 instruments are violas. The ratio of the number of violas to the total number of instruments, can be written as as 8 : 35, or as 8 to 35 (Larson et al., 2004, p. 374).
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86 Counting Tasks Researchers who condu ct curriculum analyses must make decisions concerning methods by which tasks are counted. One issue that researchers face is that the narrative section s of many lessons contain problems for students to work or questions for them to answer. Whether these ta sks should be included in the analysis of the narrative sections or the analysis of the exercise sets is a decision researchers must make. Another issue is that exercises typically contain several parts, which are often labeled a, b, c, etc. Some researche rs count these as separate exercises and others do not. D ecisions regarding these questions for the current study are described below. Connected Mathematics2 ( CMP ) CMP textbooks are different from most middle school textbooks in several ways. For instance, the CMP textbooks contain few if any are intended to be co mpleted in class and were, therefore, coded as activities. The tasks in homework sets were coded as exercises. In the Investigations, each Problem contains parts that are lettered (typically A through C). Some of the lettered parts contain numbered subpar ts (typically 1 and 2). In the Investigations, the numbered subparts are often quite different from each other. For example, in a sixth grade problem, students are shown a floor plan of a bumper car facility. Subpart 1 asks students to calculate the area a nd perimeter of the floor plan. In subpart 2, students are asked whether they think the area or perimeter is a better
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87 descriptor of the size of the facility. These subparts are quite different from each other. Thus, in the study, each numbered subpart was counted as a separate exercise. In addition to the Problems in the Investigations, CMP textbooks contain exercises in three types of sets: Applications, Connections, and Extensions. Based on the numbering of the exercises, it seemed that th e s e sections were intended to comprise one exercise set. Thus, no distinction was made between exercises in each of the three sections within each exercise set. Exercises in the exercise sets were numbered and often contained several lettered parts. These lettered part s are usually closely related to each other. Thus, each numbered exercise was counted as one exercise even if it contained several lettered subparts. Glencoe Math Connects Examples in Math Connects textbooks are clearly labeled and numbered. They contain p narrative o f the lesson; therefore, these examples were analyzed. Exercises in Math Connects textbooks are clearly numbered and generally do no t have multiple parts. Exercise that the five sections were intended to comprise one exercise set. Additionally, the two seemed similar to each other. The other three sections typically contained small numbers
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88 of exercises. Thus, the five sections within each exercise set were combined and analyzed as one exercise set. Math Connects textbooks do not have examples and exercises that are labeled and numbered like the examples and exercises in the regular lessons. However, some of these lessons contain material related to proportionality, so the se lessons were analyzed. Each of these lessons contains one or were considered to be activities. UCSMP Examples in the narrative sections of UCSMP textbooks are clea rly labeled. Some the two was made. Most narrative sections contain a short question problems in the narrative sections of the Glencoe Math Connects textbooks and were, therefore, not analyzed. Some narrative sections also co ntain a few questions called Exercises in UCSMP textbooks are clearly numbered and often have two, three, or four lettered parts. As in the ConnectedMathematics2 textbooks, lettered parts are usually close ly related to each other. For example, in the Pre Transition Mathematics
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89 several lettered subparts. Exercise Based on the numbering of the problems, it seems that the sections are intended to comprise one exercise set. Thus, no distinction was made between exercises in the four sections within each UCSMP exercise set. Summary of Task Selection and Counting Many of the tasks in the selected lessons were related to proportionality, but many others were not. Only the tasks related to proportionality were coded. To determine which tasks would be coded, the researcher used the task selection criteria previously descri bed. In each of the four content areas, detailed task selection criteria have been described. Researchers who conduct curriculum analysis must also decide how to count tasks. The diversity of numbering systems used in textbooks can make counting tasks dif ficult. In this study, methods of counting tasks varied between series and between sections of textbooks, as previously described. Number of Tasks Included in the Study In all, 4,563 tasks were coded. This number includes 421 activities, 358 examples, and 3,784 exercises. Sixth and seventh grade textbooks contained more tasks related to proportionality than did eighth grade books. The CMP and UCSMP textbooks contained a similar number of tas ks related to proportionality (965 and 832, respectively). The Math Connects textbooks contained a far greater number of tasks related to proportionality
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90 (2,766). Because the total number of tasks in each textbook is unknown, it is not possible to report t he percentage of tasks from each book that was related to proportionality. Grade Levels of Selected Tasks The sixth and seventh grade textbooks in the sample contained more tasks related to proportionality than did the eighth grade textbooks. Across series, 34% of the tasks coded were from sixth grade textbooks, 42% were from seventh grade textbooks, and 24% were from eighth grade textbooks. The percentage of tasks at each grade level varied by series, as explained below. In the CMP and UCSMP series, a far greater number of tasks were coded from the sixth and seventh grade textbooks than from the eighth grade book. In the Math Connects series, there were slightly fewer proportionality related tasks in the eighth grade textbook than in the sixth and seventh grade books, but the decrease between seventh and eighth grades was smaller in the Math Connects series than in the other two. Figure 5 shows the percentage of tasks from each grade level for each textbook series. Figure 5 shows that, in all three series, the seventh grade textbook contained more tasks related to proportionality than either the sixth or the eighth grade textbook. In the CMP and UCSMP series, the focus on proportionality drops off sharply between seventh and eighth grades. The Math Connects series had the smallest difference between the grade levels.
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91 Figure 5 Percentage of tasks related to prop ortionality at each grade level Note: 1,628 sixth grade, 1,898 seventh grade, and 1,037 eighth grade tasks were coded. Framework for Cur riculum Analysis T o analyze the activities, examples and exercises in each textbook, the researcher use d a framework that she developed based on research literature. The framework was tested and modified based on a pilot study (Appendix A ) The framework involves analyzing tasks along several dimensions. Each dimension involve s several categories. In the paragraphs below, the researcher describe s the dimensions along which tasks were coded and the categories that were used. Then, examples of tasks that fit into each of the categories are provided. The framework differ s from that used in other studies in two ways. First, i n some curriculum analyses, only activities and exercises were analyzed (Jones, 2004; Jones & Tarr, 2007). Howeve r, this practice ignores the examples in the narrative portion of textbooks. Second, i n other curriculum analyses, different frameworks were used to 0 10 20 30 40 50 60 Gr. 6 Gr. 7 Gr. 8 Percent CMP Math Connects UCSMP
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92 examine the narrative and the exercise sets (e.g., Johnson et al., 2010 ) However, in this study the frame work used to analyze the narrative and the exercise sets was the same, with one exception This exception is that the level of cognitive demand was coded for the exercises, but not the examples, for the reasons stated in a later section Thus, e ach example and activity was analyzed along five dimensions: content area, problem type, solution strategy, characteristics of proportionality, and the presence or absence of a visual representation. E ach exercise was analyzed along six dimensions: the five listed ab ove and the level of cognitive demand. Content Area Proportionality appears in many areas of mathematics. Thus, the content area of each task was noted. Each task was coded as primarily relating to algebra, data analysis/probability, geometry/measurement, or rational numbers. Determining the content area of tasks was generally clear and straightforward. However, coding some tasks according to content area presented minor difficulties. For example, it was difficult to decide whether tasks related to ratios and/or rates should be coded as algebra or rational number To decide whether a task should be coded as algebra or rational number the resea rcher attempted to determine which content area was the focus on the lesson. For example, in the seventh grade CMP Investigation his could have been coded as a rational number task (because the answer is a decimal) but the previous two pages of the textbook discussed linear equations and asked students to write an equation to match the graph of a line. Because these are algebra topi cs, the rate exercise was coded as an algebra task.
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93 Because proportions are used in all four content areas, some tasks related to proportions were difficult to code according to content area. For example, in the seventh x that mak es the fractions equivalent: argue that, because the task contains a variable, it should be coded as algebra Others may argue that the module focused on geometry and measurement. However, because th e task specifically mentioned fractions, it was coded as rational number Problem Type The literature review suggested the existence of three problem types: missing value ratio comparison and qualitative The pilot study indicated that tasks of these pro blem types can indeed be identified in middle school mathematics textbooks. However, there are many tasks in middle school mathematics textbooks that do not fit neatly into one of these types. Th us, six codes for problem type were used: the three from the literature and alternate form function rule and other Each of these six types is discussed below. M issing V alue In a missing value task, three value s are given and students are asked to find the fourth. Missing value tasks are found in every content area within mathematics. Missing value tasks often appear as rate or ratio problems; many of these were coded as algebra The following is an example of a seventh grade missing value task that was coded as alg ebra
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94 In the data analysis/probability content area, missing value tasks often appear when students are asked to use data to ma ke a prediction as in this seventh grade fish. Suppose 150 of the 500 fish in his net are salmon. How many salmon do you predict 2006f, p. 45). In the geometry/measurement content area, missing value tasks ar e related to similar figures, measurement conversion, or scale models, as in this seventh grade cale model is 5 inches wide, how w Many missing value tasks are in the rational number content area. Some rational numb er tasks that were coded with the problem type missing value involved equivalent fractions, as in this sixth with a number so the fractions are equivalent. missing value tasks involve d finding the percent of a number. For example, calculating 75% of 120 could be accomplished by solving the proportion Therefore, tasks in which students find the percent of a number were coded with the missing value problem type. Ratio C omparison In a ratio comparison problem, students are given two fractions or rati os and are asked whether they are equivalent or which is greater. Most ratio comparison problems are related to rational numbers. For example, students can be shown two fractions and asked whether t he fractions a re equivalent as in this sixth grade exerci whether the statement is correct or incorrect
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95 Because decimals are mathematically equivalent to fractions, tasks in which students are asked to compare the size of decimals were also coded, as in Figure 6 Figure 6 Task of problem type ratio comparison Tasks that involved rounding fractions or mixed numbers were also coded as ratio comparison, as in this sixth using envelopes that are inches by inches. To the nearest half inch, how large ca n This was coded as ratio comparison because it involves comparing to Ratio comparison problems also appear in lessons on geometry. For example, if students are shown two polygons that have measurements marked and are asked whether the polygons are similar, students must use the measurements to make ratios and then compare those ratios. Q ua litative In qualitative tasks, n o numbers are prov ided. Thus, students must think about relationships rather than quantities, as in the seventh grade exercise in Figure 7 Figure 7 Exercise of problem type q ualitative For each pair of numbers, find another number that is between them. 1. 0.8 and 0.85 2. 0.72 and 0.73 3. 1.2 and 1.205 4. 0.0213 and 0.0214 (Lappan et al., 2009a, p. 40). In which situation will the rate increase? Give an example to explain your reasoning. a. x increases, y is unchanged b. x is unchanged, y increases (Day et al., 2009b, p. 292)
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96 Alternate F orm Alternate form tasks involve converting between decimals, fractions, and percents or putting fractions and ratios into lowest terms. For example, a textbook could ask students to write the decimal 2.04 as a mixed number in simplest form. Alternate form tasks are similar to missing va lue tasks; the difference is that in an alternate form task, students are given two numbers and are asked to compute two other numbers. In a missing value task, students are given three out of the four numbers in a proportion and are asked to find the four th. Tasks that require students to convert a division problem into a fraction or mixed number were not coded. For example, a task could ask students to convert into a mixed number. If the task had asked students to convert into a mixed number, it would be coded as an alternate form task However, when students are asked to think about division problems like they are often asked to think about divisors, dividends, and re mainders rather than fractions. Function R ule In a function rule task, st udents write an expression, equation or description of a function based on data or a table as in the sixth grade exercise in Figure 8 Figure 8 Exercise of problem type function rule Find the rule for each function table. x 6 3 22 11 34 17 (Day et al., 2009a, p. 51)
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97 In some function rule tasks, the relationship between variables was not proportional or even linear. Tasks in which the relationship was not proportional were coded because they encouraged students to think about ra te of change which may help them understand the characteristics of proportional situations. Other Any task that did not fit neatly into the five categories previously described was coded other For example, in a sixth grade exercise, students were given the quantities 45 2009 p. 403). This exercise is related to proportionality because it involves the rate 22.5 miles per hour, but the exercise does not fit into any of the other problem type categories. Thus, it was coded with the problem type other Solution Strategy Most of the worked examples that appear in textbooks are solved using a specific solution strategy. Some activities and exercises suggest that students use a specific solution strategy. T he solution strategy suggested by each task was noted. The research l iterature suggested the existence of three solution strategies: the building up proportion and unit rate strategies. The pilot study suggested the need for two additional codes for solution strategy: decimal and manipulative The codes that were used for the solution strategy dimensions are described below.
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98 Building U p The building up strategy can be based on either addition, in which a quantity is repeatedly added, or multiplication, in which doubling or tripling is used. Construction of a function tab le is one method of using the building up strategy, as in this seventh grade function table that shows the total number of songs downloaded after 1, 2, 3, and 4 months. The This task is related to proportionality because a proportion could be used to find the number of songs at the end of each month, as in It was coded with the building up strategy because t he function table encourages students to repeatedly add eight. Decimal The pilot study (Appendix A) established a need for a code for the decimal solution strategy. In a task coded with the decimal solution strategy, students are encouraged to convert fractions or percents into decimals, as in this sixth grade exercise: , and 2009, p. 113). Manipulatives The pilot study (Appendix A) established a need for a code for the manipulative solution strategy. S ome textbooks instruct students to draw pictures and/or use manipulatives such as fraction strips, to help them answer questions. Thus, manipulatives was used as a code for the solution strategy dimension.
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99 Proportion Proportions are one of the solution strategies discus sed in the research literature. The example in Figure 9 demonstrates the use of the p roportion strategy. Figure 9 Example with p roportion solution strategy Unit R ate A task was coded as unit rate only if students were instructed to use a unit rate strategy to solve a problem. Tasks that ask students to state the unit rate or to convert a rate into a unit rate were not coded because in these situations, the unit rate wa s not used as a solution strategy. For example, in the pilot study a sixth grade example instructed 2004 p. 379). This exercise was not coded with the unit rate strategy because students were instructed to simply comp ute the unit rate An adult African elephant can be 30 feet long and 11 feet high at the shoulder. Estimate the length of a baby elephant that is 3 feet high at the shoulder. Solution Compare lengths on the adult with the corresponding lengths on the baby. Set up a proportion by forming two equal ratios. Let x be the length of the baby. Since the elephants are similar, the ratios are equal. We estimate that the baby elephant i s slightly over 8 feet long (Brown et al., 2009, p. 310).
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100 Other Tasks in which a solution strategy wa s suggested that d id not fit neatly into one of the above categories were coded as other Many of the tasks coded with the solution strategy other were also coded with the problem type function rule Because the function rule problem type has not been discussed in the research literature, solution strategies for this type of problem have also not been discussed. Some tasks related to converting between fractions, decimals, and percents were also coded with the solution strategy other No Solution S trategy Tasks in which no solution strategy wa s suggested were coded as n o solution strategy Most worked example s offered a solution strategy but m ost activities and exercises simply posed a question or problem and did not offer a solution strategy. Thus, most activities and exercises were coded as n o solution strategy Because the CMP series did not contain worked examples, almost all of the CMP tas ks were coded as n o solution strategy Characteristics of Proportional Reasoning Many students have difficulty identifying situations in which proportional reasoning is appropriate (Singh, 2000; Van Dooren et al., 2005, 2009; Watson & Shaughnessy 2004). T herefore, whether or not each task point ed out whether proportional reasoning is appropriate was noted. There are two methods textbook authors use to help students recognize when proportional reasoning is appropriate. One is by pointing out the characteris tics of proportional situations. These characteristics are that the rate of change is constant and the y intercept is zero. The other is by considering the
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101 effects of doubling (or tripling or quadrupling) one of the variables. In proportional relationships if one quantity doubles, the other does also. One way textbook authors help students recognize situations in which proportional reasoning is or is not appropriate is by discussing enlargements of rectangles. When a rectangle is enlarged, the perimeter in creases by the same percentage as the side lengths, but the area does not. This is because perimeter is related proportionally to side lengths whereas area is not. The same reasoning applies to the volu me of three dimensional objects. S tudents may incorrec tly think that doubling the radius of a cylinder doubles the volume. Visual Representation The researcher noted whether each task was accompanied by a visual real world contexts, pictures, written language, manipulatives, and symbols (Hodges et al., 2008), the focus in this study was on visual representations, such as charts, diagrams, graphs, illustrations, and pictures. Pictures that were merely decorative and not inte nded representation. When a visual representation was present, the researcher entered a short verbal description of the representation into the coding sheet. Level of Cognitive Demand T he level of cognitive demand of exercises was noted. Consistent with Jones (2004) and Stein et al. (2000), each exercise was coded as M emorization P rocedures Without C onnections P rocedures With C onnections or D o ing M athematics As was explained in Chapter Two, classifying exercises into these categories requires knowing
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102 what procedures students have been taught. Exercises were classified into these categories by comparing them to the examples in the lesson. For example, if an exercise was similar to an example in the narrative, the exercise wa s likely to be coded M emorization or Procedures Without C onnections It would be much more difficult to classify examples in the narrative according to their level of cognitive demand because doing so would r equire knowing what procedures students ha d been taught in previous lessons. Therefore, the researcher classified activities and exercises, but not examples, according to their level of cognitive demand. M emorization Acc ording to Stein et al. (2000), M emorization tasks involve either reproducing previously learned facts, rules, formulae, or definitions OR committing facts, rules, ] are not ambiguous such tasks involve exact reproduction of previous ly seen material was coded as M emorization if it require d no computation and if the answer could be easily obtained by reading the narrative section of the lesson. For example, tasks in which students were asked for decimal equi valents of common fractions and percents were coded as M emorization P rocedures W ithout C onnections The second level of task in the framework designed by Stein et al. (2000) is called procedure s without connections to understanding, meaning, or concepts w hich they shorten to P rocedure s Without C onnections. These procedure is either specifically called for or its use is evident based on prior instruction, experience, or placement of the task [They] require limited cogniti ve demand for
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103 successful completion Stein et al., p. 16). For example, tasks in which students were asked to solve a straightforward missing value task such as were coded as P rocedures W ithout C onnections More complex tasks were also coded as P rocedures Without C onnections if students had been shown via a worked example exactly how to complete the task. For example, in an eighth fast runner can run a half 2009, p. 267). Students are then instructed to find the average rate in miles per minute. This was coded as P rocedures Without C onnections because an example illustrated virtually the same skill required in the exercise. P rocedures W ith C onnections According to Stein et al. (2000), P rocedure s With C onnections Stein et al. also state that Procedures With C onnections multiple ways (e.g., visual For example, an exercise from the pilot study Three blue rhombuses make one yellow hexagon. Use seven blue rhombuses to make yellow hexagons. Use division t o write as a mixed number (Billstein & Williamson, 2008, p. 44). The exercise was coded as Procedures With C onnections for two reasons. First, manipulatives (pattern blocks) were required. Second, use of the pattern blocks was intended to help students understand why the procedure for converting fractions to mixed numbers works. In a Procedures Without C onnections task, students apply a procedure and may have no understanding why it works; a task designed
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104 to help students develop a deeper understanding of procedures was coded as Procedures With C onnections Tasks in which students were required to explain their reasoning were usually coded as Procedures With C onnections Doing M athematics According to Stein et al. (2000), Doing M athematics complex and nonalgorithmic thinking (i.e., there is not a predictable, well rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked 16). Tasks coded as Doing M athematics often had multiple parts and required an mathematical topics. For example, the exercise in Figure 1 0 points out that multiplicative reasoning can be used in purchasing decisions as well as in tasks rel ated to similar figures. Figure 1 0 Exercise of the Doing M athematics level of demand While shopping for sneakers, Juan finds two pairs he likes. One pair costs $55 and the other costs $165. He makes the following statements about the prices. a. Are both of his statements accurate? b. How are the comparison methods Juan uses similar to the methods you use to compare the sizes and shapes of similar figures? c. Which method is more appropriate for comparing the size and shape of an enlarged or reduced figure to the original? Explain. (Lappan et al., 2006b, p. 16)
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105 Summary of the Framework In summary examples were coded along five dimensions: content area, problem type, characteristics of proportionality solution strategy, and visual representation. Activities and exercises were coded along six dimensions: the five previously listed and the level of cognitive deman d. Table 3 presents the codes associated with each of these dimensions.
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106 Table 3 The Curriculum Analysis Framework Dimension Categories Content Area Algebra Data Analysis/Probability Geometry/Measurement Rational Numbers Problem Type Alternate Form Function Rule Missing Value Ratio Comparison Qualitative Other Characteristics Yes No Solution Strategy Building Up Decimal Manipulatives Proportion Unit Rate Other No Strategy Visual Representation Yes No Level of Cognitive Demand* M emorization Procedures without connections Procedures with connections ________________________ D oing mathematics *The level of cognitive demand was noted for activities and exercises, but not for examples.
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107 Narrative Text The basic unit of analysis was the task defined as tive text, intended for Because the number of narrative passages that explicitly discuss proportionality wa s small, they were not analyzed quantitatively. Instead, the researcher note d where they occur red and their content. Reliability This section is divided into three parts. In the first, the procedures used to monitor reliability are described. In the second, the reliability of the task selection is described. In the final part of this section, the reliability of the coding is discussed. Reliability Procedures To ens u re reliability of the coding, the researcher employ ed a check coding method in which t he researcher was assisted by a fellow doctoral student in mathematics education. The doctoral student had completed all coursework, passed comprehensive exams, and was w orking on his own dissertation. The doctoral student had read several procedures of the study. The researcher and doctoral student met 13 times during the months in which the coding took place (November and Decem ber of 2009 and January of 2010 ) Each meeting lasted one to two hours. To ensure that the researcher and doctoral student shared a common understanding of the framework, during the first six meetings, they discussed
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108 any tasks on which there was disagreement. After the sixth meeting, the researcher felt that there was a common understanding of the framework; thereafter, tasks were not discussed individually. Th e doctoral student coded seve ral lessons from each textbook. In all, he coded 26 of the 165 lessons included in the study (15%). The doctoral student recorded his coding data in a spreadsheet that he emailed to the researcher after each meeting. The researcher also kept handwritten no tes regarding the substance of each meeting and transferred these notes to a typed Reliability Log within a day or two of each meeting. Lessons Coded by the Doctoral Student The researcher used a purposeful selection process to choose the lessons for the doctoral student to code. In selecting lessons to be coded by the doctoral student, the researcher used several criteria. First, at least 10% of the lessons from each textbook were coded. Second, lessons from all of the content areas were selected. In most textbooks, lessons from two or three different content areas were selected. Table 4 presents the number of lessons from each textbook included in the study and the number and percent of lessons that were coded by the doctoral student. Table 4 Number and Percent of Lessons that were Double Coded CM P Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Number of Lessons Double Coded 3 2 1 6 4 3 4 2 1 Percent of Lessons Double Coded 33 13 25 18 13 12 19 11 17 Total Lessons Coded 9 1 5 4 33 3 2 2 6 2 1 1 9 6
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109 Reliability Training The researcher and doctoral student began meeting on November 6, 2009 to code lessons for the study. During the month of November, the researcher and the doctoral student met five times and coded nine sixth grade lessons. Although reliability percentages had not been calculated at that point, the researcher had some concerns about the level of reliability After consulting the Chair of the dissertation committee, the researcher developed a Training Module consisting of 1 3 exercises from the textbooks in the study. The Training Module can be found in Appendix C Working together, the researcher and doctoral student coded the exercises in the Training Module. Once the Training Module was completed, a 1 3 item Reliability Test was administered. The Reliability Test can be found in Appendix D O n the Reliability T est, the researcher and doctoral student disagreed on one code for content area, one code for problem type, and two for level of cognitive demand. Thus, on the reliability test, the reliability for content area and problem type was 92% and the reliabilit y for cognitive demand was 8 5 %. Re liability regarding content area and solution strategy were both 100%. Reliability of Task Selection Reliability of task selection refers to the extent to which the researcher and doctoral student selected the same tasks to code; i.e., considered the same tasks to involve proportionality. For each textbook, two percentages associated with this type of reliability can be calculated: a) the percent of tasks coded by the doctoral student that were also coded by the researcher and b) the percent of tasks code d by the researcher that were also coded by the doctoral student The following paragraphs describe both types of reliability for each textbook
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110 Reliability of Task Selection in CMP Three lessons were selected from the sixth grade CMP textbook for check coding. From these lessons, 123 tasks were coded by both the researcher and doctoral student, 10 were coded only by the researcher, and 1 6 were coded only by the doctoral student. Two lessons were selected from the seventh grade CMP textbook for check co ding. From these lessons, 51 tasks were coded by both the researcher and doctoral student, nine were coded only by the researcher, and six were coded only by the doctoral student. One lesson was selected from the eighth grade CMP textbook for check codin g. From this lesson, 37 tasks were coded by both the researcher and doctoral student, nine were coded only by the researcher, and three were coded only by the doctoral student. Combining the data from the three CMP textbooks, 211 tasks were coded by both the researcher and doctoral student, 28 were coded only by the researcher, and 25 were coded only by the doctoral student. Of the 239 tasks coded by the researcher, 211 were also coded by the doctoral student. Thus, 8 8 % of the tasks coded by the researcher were als o coded by the doctoral student. Of the 236 tasks coded by the doctoral student, 211 were also coded by the researcher. Thus, 89% of the tasks coded by the doctoral student were also coded by the researcher. Reliability of Task Selection in Math Connects Six lessons were selected from the sixth grade Math Connects textbook for check coding. From these lessons, 111 tasks were coded by both the researcher and doctoral student, eight were coded only by the researcher, and one was coded only by the doctoral student.
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111 Four lessons were selected from the seventh grade Math Connects textbook for check coding. From these lessons, 99 tasks were coded by both the researcher and doctoral student, none were coded only by the researcher and three were coded only by the doctoral student. Three lessons were selected from the eighth grade Math Connects textbook for check coding. From these lessons, 86 tasks were coded by both the researcher and doctoral student, four were coded only by t he researcher, and none were coded only by the doctoral student. Combining the data from the three Math Connects textbooks, 2 9 6 tasks were coded by both the researcher and doctoral student, 12 were coded only by the researcher, and four were coded only b y the doctoral student. Of the 308 tasks coded by the researcher, 296 were also coded by the doctoral student. Thus, 96% of the tasks coded by the researcher were also coded by the doctoral student. Of the 300 tasks coded by the doctoral student, 296 were also coded by the researcher. Thus, 99% of the tasks coded by the doctoral student were also coded by the researcher. Reliability of Task Selection in UCSMP Four lessons were selected from the sixth grade UCSMP textbook for check coding. From these lesson s, 57 tasks were coded by both the researcher and doctoral student, eight were coded only by the researcher, and one was coded only by the doctoral student. Two lessons were selected from the seventh grade UCSMP textbook for check coding. From these less ons, 37 tasks were coded by both the researcher and doctoral
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112 student, five were coded only by the researcher, and none were coded only by the doctoral student. One lesson was selected from the eighth grade UCSMP textbook for check coding. From this lesso n 12 tasks were coded by both the researcher and doctoral student, t hree were coded only by the researcher, and none were coded only by the doctoral student. Combining the data from the three UCSMP textbooks, 106 tasks were coded by both the researcher and doctoral student, 16 were coded only by the researcher, and one was coded only by the doctoral student. Of the 122 tasks coded by the researcher, 106 were also coded by the doctoral student. Thus, 87% of the tasks cod ed by the researcher were also coded by the doctoral student. Of the 107 tasks coded by the doctoral student, 106 were also coded by the researcher. Thus, 99% of the tasks coded by the doctoral student were also coded by the researcher. Summary of Reliability of Task Selection Table 5 summarizes the number of exercises coded by the researcher and the doctoral student for ea ch series. O f the two types of reliability of task selection, the more important is the percent of tasks coded by the doctoral s tudent that were also coded by the researcher. A high percentage indicates that the researcher did not miss many tasks able 5
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113 Table 5 Reliability of Task Selection by Textbook Series CMP Math Connects UCSMP Number of Tasks Coded by Both 211 2 9 6 106 Number of Tasks Code d Only by the Researcher 28 12 16 Number of Tasks Coded Only by Doctoral St udent 25 4 1 Reliability of Task Selection (Percent)* 89 99 99 *Note: For example, for CMP, the doctoral student coded 236 exercises. 211 of these were also coded by the researcher. 211 236 = 0.894 0.894 was rounded to 89 % Reliability of C oding Each example related to proportionality was coded along five dimensions: content area, problem type, solution strategy, characteristics of proportionality, and the presence or absence of a visual representation. Each activity and each exercise was coded along these five dimensions and also the level of cognitive demand. For four of these dimensions, the codes to be assigned were obvious and the researcher and the doctoral student agreed on almost every item. These dimensions were content area, solut ion strategy, characteristics of proportionality, and the presence or absence of a visual representation. For two dimensions, the necessary codes were less obvious and the researcher and doctoral student disagreed more frequently. These dimensions were pro blem type and the level of cognitive demand. Thus, reliability related to these dimensions is discussed in this section. Reliability of Problem Type Coding Codes for problem type included alternate form function rule missing value ratio comparison qualitative and other Over all textbooks and series, the problem type
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114 codes matched for 540 of the 613 tasks coded by both researchers (88%). As indicated in Table 6, r eliability regarding problem type ranged from a low of 75% in the CMP sixth grade book to a high of 100% in the Math Connects sixth grade book. Reliability of the problem type coding was lowest in the CMP series (7 8 %) and high in both Math Connects and UCSMP (94% and 92%, respectively ). Table 6 Reliability of Problem Type Coding by Series and Grade CMP Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Number Tasks w/ Agreement 92 42 31 111 94 72 54 33 11 Total Number Tasks Coded 12 3 51 37 11 1 9 9 86 5 7 37 12 Percent Agreement 75 82 84 100 95 84 9 5 89 92 *Refers to the number of tasks coded by both the researcher and the doctoral student Most of the difficulty with coding the problem type dimension related t o the code of other It was often difficult to tell whether to code a task as other or to assign a specific problem type code. Many tasks were mathematically equivalent to a missing value or ratio comparison task but had cosmetic features that made them appear different. For example, in an eighth grade Math Connects exercise, students were given coordinates of the three vertices of a triangle and were asked to find the vertices after a dilation with a given scale factor and then graph the original and ne w triangle. The researcher coded this with the problem type other because it did not neatly fit into the categories and the doctoral student coded it with the problem type missing value because it involve d finding missing values.
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115 Reliability of Level of Co gnitive Demand Coding The Mathematical Task Framework (Stein et al., 2000) was used to classify each activity and each exercise by level of cognitive demand. As indicated in Table 7 reliability regarding level of cognitive demand of the exercises ranged from a low of 5 2 % in the eighth grade Math Connects textbook to a high of 90% in the seventh grade CMP textbook. Over all textbooks and series, the reliability regarding cognitive demand was 7 2 %. Reliability was lowest in the CMP series (65%) and highest i n the UCSMP series (76%) Most of the difficulty with coding the cognitive demand dimension related to the two middle levels (the codes of Procedures Without Connections and Procedures With Connections ). Table 7 Reliability of Cognitive Demand Coding by Series and Grade CMP Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Number Tasks w/ Agreement 52 27 17 89 66 39 37 27 6 Total Number Tasks Coded 98 30 20 1 03 86 75 44 37 10 Percent Agreement 53 90 8 5 86 7 7 52 84 73 6 0 *Refers to the number of activities plus the number of exercises Jones (2004) also used the Mathematical Task Framework (Stein et al., 2000) to measure the level of cognitive demand of tasks found in textbooks. He also found that the level of cognitive demand was, in some cases, difficult to de termine. He reported reliabilities of 73% to 88%. He also reported the percentage of tasks on which the two rater s were within one level of cognitive demand. Applying this strategy to the current
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116 study would result in a reliability of 100%. On each task for which the two codes disagreed, the difference was only one level of cognitive demand. Reliability for CMP Textb ooks Cognitive demand was difficult to code in the CMP textbooks because the series contains no labeled examples, making it difficult to determine what procedures students had learned. For example, in an exerc ise in the sixth grade textbook, students must find a fraction between and The researcher coded this exercise as Procedures Without Connections because there are simple procedures that can be followed to solve the problem. The doctoral student coded the exercise as Procedures With Connections b ecause the procedures for solving this type of problem had not been presented to students. Reliability for t he Math Connects Grade 8 Textbook In the Math Connects Grade 8 t extbook 75 exercises were coded by both the researcher and the doctoral student. I nitially, the two sets of codes were identical for 29 of these exercises, resulting in an initial reliability of 39%. To investigate why the reliability was low, the researcher looked at the tasks on which the two raters disagreed. the level of cognitive demand for 10 exercises. In these exercises, students were given a set of x and y values, asked to determine whether the linear function was an example of direct va riation, and asked to state the constant of variation. The research er originally coded these exercises as Procedures With Connections because they required several steps. However, examples in the narrative clearly showed students how to solve problems of t his type. Therefore, the codes were changed to Procedures With out Connections
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117 because the use of a procedure was evident based on prior instruction and because the exercises required no explanation. Correcting this error meant that the two sets of codes we re identical for 39 of the 75 exercises, resulting in a reliability for the Math Connects Grade 8 t extbook of 52%. Reliability of the cognitive demand dimension was low in another lesson from amples show students how to obtain measurements from a diagram and set up and solve a proportion to obtain a measurement that might be difficult to obtain directly, such as the distance across a lake. Fifteen exercises follow the examples and are very simi lar to the examples. Because the procedure for solving the problems was clearly illustrated in the examples, the researcher coded all 15 exercises as Procedures Without Connections The doctoral student believed that the presence of visual representations and word problems elevated the level of cognitive demand to Procedures With Connections Thus, the two raters disagreed on all 15 exercises. presence of visual representations and word problems is said to increase the depth of knowledge. Be cause this is not consistent with the Mathematical Task Framework (Stein et al., 2000), the Reliability for t he UCSMP Grade 8 Textbook Reliability of the coding of the cognitive demand dimension was also low in the UCSMP Algebra textbook ( 6 0%). However, only one lesson was coded by the doctoral student and only 10 tasks from this lesson were coded by both the researcher and the doctoral student. Had more tasks been double coded, reliability may have been higher.
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118 Reliabili ty of Coding Over Time To determine whether reliability increased, decreased, or remained constant over the three months that coding occurred, the 26 lessons that were coded by the doctoral student were separated into three sets. One set consisted of nine lessons that were coded by the doctoral student between November 18, 2009 and January 2, 2010. A second set consisted of 12 lessons that were coded by the doctoral student between January 2, 2010 and January 10, 2010. The third set, with five lessons, was coded by the doctoral student between January 10, 2010 and January 18, 2010. Reliability was calculated for each of the three sets. For lessons that were discussed on more than one occasion, the last date was used to classify the lesson into one of the th ree sets. For the first set of lessons, the reliability of the Problem Type coding was 89%, for the second set it was 84% and for the third set it was 86%. Thus, it appears that the reliability of coding the problem type dimension did not significantly ch ange over the three months in which coding took place. Summary of Reliability of Coding Coding of four of the six dimensions presented no difficulties. However, coding the problem type and level of cognitive demand dimensions was, in some cases, more diff icult. Reliability of problem type codes was above 90% in five of the nine textbooks. Overall, reliability of problem type codes was 8 8 % and was lowest in the CMP series. Coding the level of demand was the most challenging. Reliability of these codes range d from 5 2 % to 90% and was 7 2 % overall.
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119 Summary of Research Methods In this chapter, the researcher described the methods that were used to select a sample of textbooks and lessons, described how examples and exercises would be counted and grouped, described the framework that was used to categorize examples and exercises, and discussed reliability measures that were employed Three contemporary middle school textbooks series were selected for the study: Connected Mathematics2 Math Connect s and books from the University of Chicago School Mathematics Project. These series were selected because all are widely used and because their treatment of proportionality wa s expected to vary in significant ways. Lessons from those textbooks were select ed on the likelihood of their inclusion of material related to proportionality. The selected lessons include d most, but not all of the material related to proportionality in the textbooks. Every task in the selected lessons was analyzed, but only tasks th at involve d proportionality were coded. Th re e types of tasks were analyzed: activities meant to be completed in class, examples that contain ed worked out solution s, and exercises meant to be completed as homework Examples related to proportionality were c oded along five dimensions: content area, problem type, solution strategy, visual representation, and whether the example highlights whether proportional reasoning is appro priate in the given situation. Activities and e xercises were coded along six dimensi ons: the five listed above and the level of cognitive demand. A pilot study was conducted w ith two sixth grade textbook s. T he pilot study demonstrated the reliability with which the framework can be applied an d the usefulness of the results. The pilot stud y and its findings are described in Appendix A
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120 C HAPTER 4: RESULTS The purpose of this study was to investigate the nature of the treatment of proportionality in contemporary middle school textbooks. Specifically, the study was designed to answer the following six questions: 1. To what extent is proportionality emphasized in the treatment of various content areas within mathematics, such as algebra, data analysis/probability, geometry/ measurement, and rational numbers? How does this vary a mong grade levels and textbook series? 2. Among the tasks related to proportionality in middle school mathematics textbooks, which problem types (e.g., missing value, ratio comparison, qualitative) are featured most and least often? How does this vary among m athematical content areas, grade levels, and textbook series? 3. Which solution strategies (e.g., building up, unit rate, proportion) to tasks related to proportionality are encouraged by middle school mathematics textbooks? How does this vary among grade lev els and textbook series? 4. What level of cognitive demand (Boston & Smith, 2009; Stein et al., 2000) is exhibited by the proportional exercises in middle school textbooks? How does this vary among mathematical content areas, grade levels, and textbook series ? 5. To what extent are visual representations used in middle school mathematics textbooks to illustrate concepts related to proportionality? How does this vary among grade levels and textbook series?
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121 6. To what extent are the characteristics of proportional sit uations pointed out in middle school mathematics textbooks? How does this vary among grade levels and textbook series? T o answer these questions, t hree contemporary, widely used textbooks series Math C onnects and the sixth seventh and eighth grade textbooks from the University of Chicago School Mathematics Project (UCSMP) The unit of analysis was the task Three types of tasks were analyzed: activities intended to be completed in class, examples wi th worked out solutions, and exercises intended to be completed as homework. The researcher analyzed 165 lessons that contained 4,563 tasks related to proportionality. Information on the number and percent of lessons included in the study fro m each textboo k was provided in Chapt er 3. I n most books, between 11% and 26% of the lessons in the textbook were included in the study. A higher percentage of lessons (44%) from the seventh grade CMP textbook was y. More lessons and tasks from the Math Connects textbooks were included than from the other two series. This does not necessarily mean that Math Connects textbooks have more of a focus on proportionality than do the other two series. Rather, it is a refle ction of the fact that Math Connects textbooks have more overall lessons and tasks than do the other two series. Similarly, more exercises were coded from the CMP series than from the UCSMP series This reflects the fact that CMP exercise sets tend to cont ain a larger number of exercises than do UCSMP textbooks. CMP exercise sets typically contain about 40 exercises whereas UCSMP sets typically contain about 20 exercises.
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122 The problems in the Investigations in the CMP textbooks were considered to be activit ies because they are intended to be done in class. Exercise (homework) sets in all textbooks were considered to be exercises. Overall, 9% of the tasks were in class activities, 8% were examples, and 83% were exercises. In the Math Connects and UCSMP textbo oks, about 3% of the tasks were activities, about 10% were examples, and about 87% were exercises. In the CMP textbooks, 34% of the tasks were problems in the Investigations and 65% were exercises found in the homework sets. The CMP textbooks did not have labeled examples. Table 8 shows the number of activities, examples, and exercises coded from each series and each textbook Table 8 Number of Tasks Related to Proportionality by Series and Grade CMP Math Connects UCSMP 6 th 7 th 8 th total 6 th 7 th 8 th total 6 th 7 th 8 th total Activities 103 192 34 329 32 26 9 67 13 9 3 25 Examples 0 0 0 0 81 82 82 245 48 49 16 113 Exercises 234 336 66 636 833 891 730 2454 284 313 97 694 Total Tasks 337 528 100 965 946 999 821 2766 345 371 116 832
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123 Content Areas The first research question related to the content areas within mathematics through which proportionality is presented. Overall, 53% of th e tasks related to proportionality were in the rational number content area 21% were in g eometry /measurement 20% were in algebra and 6% were in d ata analysis/p robability. In general, the percentage of activities, examples, and exercises in each content area were similar to the percentage of tasks in each content area. For example, 20% of the tasks were related to algebra and approximately 20% of the activities, examples, and exercises were related to algebra. There were two exceptions to this rule. I n the ge ometry/measurement content area, there were more activities and fewer examples and exercis es; in the rational number content area there were fewer activities and more examples and exercises The content areas of tasks related to proportionality varied by textbook series and by grade level, as explained in the following sections. Content Areas and Textbook Series In all three textbook series, a higher percenta ge of the tasks related to proportionality were from the rational number content area than any other. In all three series, between 4 8 % and 5 6 % of the tasks related to proportionality were from the rational number content area. In the Math Connects and UCSM P series, the percentage s of tasks in the geometry/measurement and algebra content areas were approximately equal. In the CMP series, more tasks related to proportionality were in geometry/measurement than in the algebra content area. In all three series, few tasks related to proportionality were from the data analysis and probability content area. In all
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124 three series, between 4% and 8% of the tasks related to proportionality were from the data analysis and probability content area. This information is deta iled in Table 9 Table 9 Percentage of Tasks in Each Content Area by Series Algebra Data Analysis Geometry Rational Number CMP ( n = 965) 15.7 7.9 26.1 50.2 Math Connects ( n = 2,766) 19.4 5.6 19.1 55.9 UCSMP ( n = 832 ) 24.8 4.1 22.4 48.3 Content Areas, Textbook Series and Grade Levels In all of the sixth grad e textbooks, a higher percentage of the tasks related to proportionality were in the rational number content area than any other The Math Connects and UCSMP textbooks continued the focus on rational numbers through seventh grade. The Math Connects textbooks continued the focus on rational numbers through eighth grade as well, but the focus in the eighth grade CMP and UCSMP textbooks shifted to algeb ra. Content Areas of Sixth Grade Textbooks Combining the data from all three series, 66% of the sixth grade tasks related to proportionality were in the rational number content area. In each of the three sixth grade books, more than 55 % of the tasks were in the rational number content area. As indicated in Figure 1 1 t he sixth grade CMP textbook had a higher percentage of tasks related to rational numbers than the other two sixth grade textbooks Other than the focus on rational numbers, there was little c ontent similarity between sixth grade textbooks. The
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125 sixth grade CMP textbook contained tasks in the data analysis and probability content area but virtually no tasks related to proportionality in the algebra and geometry/measurement content areas. T he Mat h Connects and UCSMP sixth grade books had more tasks related to proportionality in the algebra and geometry content areas than in data analysis Figure 1 1 Percentage of Sixth Grade Tasks in Each Content Area In the s ixth grade CMP curriculum, a higher percentage of tasks related to proportionality were in the rational number content area than in any other. The sixth grade CMP curriculum contained eight modules, three of which focus ed on rational numbers. In the first module on ratio nal numbers students compare and order fractions and decimals and convert among fractions, decimals, and percents. In the second, students learn to estimate and compute with fractions. The third module related to 0 10 20 30 40 50 60 70 80 90 100 CMP Math Connects UCSMP Percent Algebra Data Analysis Geometry Rational Number
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126 rational numbers cover ed computation with decimals and algorithms for solving percent problems. In the s ixth grade Math Connects textbook, a higher percentage of tasks related to proportionality were in the rational number content area than in any other. Of the 12 chapters in the textbook, five f ocused on rational numbers. The sixth grade Math Connects lesson were determined to be in the rational number content area. As in the other two sixth grade textbooks in the UCSMP textbook, a higher percentage of tasks related to proportionality were in the rational number content area than in any other. Of the 13 chapters in the textbook, two focused on rational numbers Multiplication rational numbers The sixth grade UCSMP textbook had a higher percentage of tasks related to geometry than did the other two sixth grade textbooks. The UC SMP textbook use d rulers and measurement as a way to help students understand fractions, mixed numbers, and decimals. Content Areas of Seventh Grade Textbooks As indicated in Figure 1 2 i n the Math Connects and UCSMP series, the seventh grade book continued to cover proportionality primarily through the rational number content area In the seventh grade Math Connects and UCSMP textbooks, more than 55% of the proportional tasks were in the rational number con tent area. Unlike the other two series, in the seventh grade CMP textbook, a higher percentage of tasks related to proportionality were in the g eometry/ m easurement content standard than any other.
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127 Figure 1 2 Percentage of Seventh Grade Tasks in Each Content Area Seventh g rade CMP In the seventh grade CMP curriculum, more tasks related to proportionality were in the geometry/measurement content area than in any other. The seventh grade CMP curriculum contain ed eight modules, two of which focused on sions of rectangular prisms and b p. 4). related proportionally to the linear dimensions. For ex ample, a task asks students to tripled, quadrupled, a p. 68). 0 10 20 30 40 50 60 70 80 90 100 CMP Math Connects UCSMP Percent Algebra Data Analysis Geometry Rational Number
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128 Seventh grade Math Connects In t he seventh grade Math Connects book a higher pe rcentage of tasks related to proportionality were in the rational number content area than in any other content area. Specifically, 58% of the tasks related to proportionality were found in the rational number content area. Of the 12 chapters in the textbo ok, only three focus ed specifically ) None of the lessons from As ex plained in C hapter 3, tasks related to rates and ratios could be coded as either algebra or rational number. The seventh grade Math Connects book contained a determined to be in the rational number content area. Thus, the seve nth grade Math Connects book contained a large percentage of tasks related to rational numbers because a large number of rational number tasks related to proportionality were found in three Seventh grade UCSMP. As in the Math Connects book, the seventh grade UCSMP textbook featured a higher percentage of tasks related to proportionality in the rational number content area than in any other content area. Specifi cally, 55% of the tasks related to proportionality were found in the rational number content area. Of the 12 chapters in the textbook, only two focused specifically on rational numbers These two chapters
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129 contained the vast majority of the tasks related to proportionality. Of the 313 exercises in this textbook, 221 were in one of these two chapters that focused on rational numbers. Content Areas of Eighth Grade Textbooks As indicated in Figure 1 3 i n the eighth grade CMP and UCSMP textbooks, more of the proportional tasks were in the algebra content standard than any other, with more than 70% of the proportional exercises in the algebra standard. This is likely due to the fact that the CMP and UCSMP textbooks focus on algebra. In the eighth grade Math Connects textbook, a higher percentage of the tasks related to proportionality were in the rational number content standard than any other. Figure 1 3 Percentage of Eighth Grade Tasks in Each Conten t Area All of the eighth textbook had a lesson devoted to inverse relationships; many of the tasks in this lesson were related to proportionality. 0 10 20 30 40 50 60 70 80 90 100 CMP Math Connects UCSMP Percent Algebra Data Analysis Geometry Rational Number
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130 Eighth g rade CMP Four Investigations in the eighth grade CMP textbook were The Investigation ssed equations and graphs of direct variation and inverse x and y is a direct variation if it can be expressed as where k g p. elationship between two non zero variables, x and y is an inverse variation if or In the eighth grade CMP textbook, more tasks Investigatio n Eighth grade Math Connects The Math Connects book ha d a lesson on direct variation which inform ed their relationship is called a direct variation The constant ratio is called the consta nt of variation c p. 487). The direct variation relationship wa s described in words, with a graph, and with the equations and However, many of the Math Connects lessons relate d to rational numbers rather than algebra; one chapter focuse d on rational numbers and another on percent. A wide variety of other topics related to proportionality are also covered, such as ratios, rates, rate of change, proportions, similar figures, dilat ions, sequences, circle graphs, probability, and statistics. Eighth grade UCSMP In the eighth grade UCSMP text, in a lesson title d A special case of a linear equation occurs when the y intercept is at the origin. Then the y intercept is 0 and becomes This means that y is a constant multiple of x y is a constant multiple of x it is sai d that y varies
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131 directly as x T his situation is called direct variation (Brow n et al., 2009, pp. 352 353). Other lessons in the UCSMP textbook focused on rates, ratios, and similar figures. More Summary of Findings Related to Content Area As indicated in Figure 1 4 in all three textbook series, a higher percentage of the tasks related to proportionality were in the rational number content area than in any other. This may reflect the fact that the framework included rational number tasks that other scho lars might not consider to be closely related to proportionality, such as finding a percent of a number. In all three textbook series, a lower percentage of the tasks related to proportionality were in the data analysis and probability content area than in any other. This is perhaps not surprising given the emphasis on algebra, geometry, and rational numbers in the Curriculum Focal Points for sixth, seventh, and eighth grades (NCTM, 2006) Figure 1 4 also shows differences among the three series. The CMP se ries featured a higher percentage of tasks related to proportionality in the d ata a nalysis /probability and g eometry /measurement content areas than did the other two series. The Math Connects series featured a higher percentage of tasks related to proportionality in the Rational Number content area than did the other two series. The UCSMP series featured a higher percentage of tasks related to proportionality in the a lgebra content area than di d the other two series.
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132 Figure 1 4 Percentage of t asks in e ach c ontent a rea by s eries Problem Type The second research que stion concerned the problem types of tasks related to proportionality. Three problem types were identified in the research literat ure: missing value ratio comparison and qualitative Two others were identified in the course of a pilot study: alternate form and function rule (See Appendix D.) Overall, 39% of the tasks were missing value 24% were alternate form 20% were ratio comp arison 6% were function rule less than 1% were qualitative and 11% were coded as other Large differences in the problem types of tasks existed among content areas, textbook series and grade levels, as explained in the following sections. Problem Type and Content Areas Problem types were related to content area. For example, the alternate form problem type is closely related to the rational number content area since alternate form tasks involve finding equivalent fractions or converting between fractio ns, decimals, and 0 10 20 30 40 50 60 70 80 90 100 CMP Math Connects UCSMP Percent Algebra Data Analysis Geometry Rational Number
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13 3 percents. The ratio comparison problem type is also related to rational numbers as ratio comparison is involved in comparing the magnitude of fractions, decimals, percents, and ratios. The function rule problem type is naturally associate d with the algebra content area. The missing value problem type is related to a variety of topics such as equivalent fractions, rates and ratios, similar figures, and probability. Thus, the missing value problem type is related to all of the content areas. Problem Type and Textbook Series As indicated in Table 10, in the CMP series, missing value and ratio comparison tasks were common as were tasks coded with the problem type other The relatively high percentage of tasks coded as ratio comparison may refl ect the sixth focus on rational numbers; many of the tasks in this textbook asked students to compare fractions. The relatively high percentage of tasks coded as other reflects the fact that many CMP tasks are fairly involved, contain mult iple parts, and are nonroutine. In the Math Connects series, alternate form and missing value tasks were common. The relatively high percentage of tasks coded as alternate form may reflect the Math Connects focus on rational numbers; many of the tasks in t his series asked students to find equivalent fractions or to convert between fractions, decimals, and percents. In the UCSMP series, missing value tasks were much more common than any other problem type. M issing value tasks were more common in the UCSMP series than in any other series. In the UCSMP series, missing value tasks were especially comm on in the eighth grade textbook w h ere they appeared in lessons on rates, ratios, proportions, and similar figures. Function r ule problems were rare in the UCSMP series.
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134 Table 1 0 Percentage of Tasks of Each Problem Type by Series AF FR MV RC Other CMP ( n = 965) 1 4 10 26 28 21 Math Connects ( n = 2,766) 29 6 40 18 7 UCSMP ( n = 832 ) 22 <1 49 15 14 Note: AF means alternate form FR means function rule MV means missing value and RC means ratio comparison Problem Type and Grade Levels All of the problem types either steadily increased in frequency from sixth to seventh to eighth grade or st eadily decreased in frequency The two problem types associated with rational numbers, alternate form and ratio comparison decreased as the grade levels increased. The other two problem types, function rule and missing value increased as the grade levels increa sed. The percentage of tasks of the alternate form and ratio comparison problem types decreased as the grade level increased As illustrated in Figure 1 5 t he alternate form problem type decreased from 33% in sixth grade to 22% in seventh grade and 15% in eighth grade. The ratio comparison problem type decreased from 3 1 % in sixth grade to 15 % in seventh grade and 1 2 % in eighth grade.
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135 Figure 1 5 Percentage of tasks of types alternate form and ratio comparison Note: 1,628 sixth grade, 1,898 seventh grade, and 1,037 eighth grade tasks were coded. T he percentage of tasks of the function rule and missing value problem types increased with grade level. As illustrated in Figure 1 6 the function rule problem type increased from 4% in sixth grade to 6% in seventh grade and 7% in eighth grade. The function rule problem type is, naturally, associated with the algebra content area. Thus, the fact that its frequency increased with grade level seems reasonable. As illustrated in Figure 1 6 the missing value problem ty pe increased from 29% in sixth grade to 43% in seventh grade and 46% in eighth grade. The missing value problem type appeared in all four content areas; thus, it seems reasonable that it was common at all three grade levels. The missing value problem type was especially common in the eighth grade UCSMP textbook, in which 71% of the tasks were coded as missing value 0 5 10 15 20 25 30 35 40 45 50 Gr 6 Gr 7 Gr 8 Alternate Form Ratio Comparison
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136 Figure 1 6 Percentage of tasks of types function rule and missing value Note: 1,628 sixth grade, 1,898 seventh grade, and 1,037 eighth grade tasks were coded. Problem Type, Textbook Series and Grade Levels As indicated in Table 11, i n the CMP textbooks, the most frequent problem type shifted from ratio comparison in sixth grade to missing value in seventh grade and function rul e in eighth grade. This may reflect the sixth grade focus on rational numbers and the eighth grade focus on algebra. CMP seventh and eighth grade textbooks, perhaps indicating that the CMP tasks were difficul t to classify. In the Math Connects textbooks, the most frequent problem type shifted from alternate form in sixth grade to missing value in seventh and eighth grades. In the UCSMP textbooks, the most frequent problem type was missing value throughout all three grades. The missing value problem type in UCSMP textbooks is discussed in the following section. 0 5 10 15 20 25 30 35 40 45 50 Gr 6 Gr 7 Gr 8 Function Rule Missing Value
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137 Table 1 1 Percentage of Tasks of Each Problem Type by Series and Grade C MP Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Alternate Form 32 5 0 37 30 19 24 25 4 Function Rule 0 10 40 8 5 4 0 1 0 Missing Value 14 37 14 27 47 46 49 43 71 Ratio Comparison 52 15 10 26 15 14 22 13 3 Other 0 33 33 2 3 17 6 19 22 Number of Tasks 337 528 100 946 999 821 345 371 116 Missing Value Tasks in UCSMP Missing value tasks were common in most of the text books in the study and particularly in the UCSMP textbooks. The UCSMP sixth grade textbook contained two chapters ( number of missing value related to percentages, most of which were coded as missing value For example, one exercise instructed students to calculate 36% of 50. This was coded as missing value because it could be sol ved with the proportion The chapter proportion, such as in 6 seconds. If she continues working at this rate, how many words can she type in 3
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138 Missing value tasks appeared in the seventh grade UCSMP textbook in many of the same ways that they did in the sixth grade book: through percentages and les sons grade textbook also grade textbook. The seventh grade textbook contain ed chapters that point out when multipl chapters contained large numbers of missing value tasks. Almost all of the missing value tasks in the eighth grade UCSMP textbook chapter were related to rates, ratios, proportions, and similar figures. All of these lessons conta ined large numbers of missing value tasks. For example, all of the 13 exercises that missing value Qualitative Tasks but re would her running speed be (a) faster, (b) slower, (c) exactly the same, or (d) not enou gh Although qualitative tasks have been discu ssed in the research literature, they were virtually absent from the textbooks in this study. Two percent of the tasks related to proportionality in the eighth grade CMP textbook w ere coded as qualitative ; in all other texts, less than one percent of the tasks related to proportionality were coded as qualitative The tasks in the eighth grade CMP textbook
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139 that were coded as qualitative t in the equation is held constant. What happens to the distance d as the rate r (Lappan et al., 2006g, p. 60). Solution Strategy The third research question concerned the solution strat egies suggested for solving tasks related to proportionality. The codes for solution strategy included the building up strategy in which one of the quantities is doubled or tripled, the decimal strategy in which a fraction or percent is converted to a deci mal, the manipulatives strategy in which students are instructed to use manipulatives or pictures, the proportion strategy, and the unit rate strategy. Most of the tasks included in the study did not suggest a solution strategy. Across all series and grad e levels, solution strategies were suggested in 14% of the tasks. Solution strategies were suggested more often in examples than in exercises. The frequency with which solution strategies were suggested varied by series and by grade level. In CMP texts, s olution strategies were suggested in 10% of the tasks. In Math Connects texts, solution strategies were suggested in 14% of the tasks. In UCSMP texts, solution strategies were suggested in 22% of the tasks. In the sixth grade texts, solution strategies wer e suggested in 15% of the tasks. In the seventh grade texts, solution strategies were suggested in 13% of the tasks. In the eighth grade texts, solution strategies were suggested in 17% of the tasks. As indicated in Table 12, tasks in UCSMP textbooks were more likely than tasks from other series to suggest a solution strategy.
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140 Table 12 Number and Percentage of Tasks With a Solution Strategy CMP Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Number of Tasks with SS* 43 50 0 103 126 151 94 67 24 Percent of Tasks with SS 13 9 0 11 13 18 27 18 21 Number of Tasks in Book 337 528 100 946 999 821 345 371 116 Solution Strategy and Grade Level One might expect to see the strategies that are less formal and more likely to foster conceptual understanding at lower grade levels and more symbolic methods at higher grade levels. Previous research suggested that students find t he building up method to be more natural than the proportion strategy (Lamon, 1999; Parker, 1999; Tournaire, 1986). Therefore, one might expect to see the building up strategy in sixth grade textbooks. However, as indicated in Table 1 3 the building up str ategy was relatively rare at all three grade levels. The manipulative strategy, in which students are encouraged to use manipulatives or pictures, is less symbolic and less formal than other methods; one might expect to find the manipulative strategy more often at lower grade levels. This was the case in the textbooks in this study; the manipulative strategy decreased in frequency as the grade level increased. In contrast the proportion strategy is quite symbolic; one might therefore expect to find it more frequently at higher grade levels. This was the case in the textbooks
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141 in the study; although it was about equally common in sixth and seventh grades, it was much more common in the eighth grade textbooks. Table 1 3 Percentage of Tasks W ith Each Solution Strategy by Grade Grade 6 Grade 7 Grade 8 Building Up 5 7 0 Decimal 6 9 8 Manipulative 26 21 8 Proportion 29 28 52 Other 27 25 30 n* 240 243 175 *For this table, n was the number of tasks in which a solution strategy was suggested. Solution Strategy, Textbook Series, and Grade Level The frequency with which specific solution strategies were suggested varied by textbook series. As indicated in Table 1 4 the CMP texts tended to suggest the manipulativ e strategy whereas the Math Connects and UCSMP texts tended to suggest the proportion strategy. In the Math Connects and UCSMP texts, about 30% of the tasks were coded with the solution strategy other suggesting that the solution strategies in these serie s did not fit neatly into the framework. In none of the eighth grade CMP tasks that were coded was a solution strategy suggested. The building up strategy, which has been identified in the research literature as a useful tool for transitioning to true
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142 prop ortional reasoning, was found in the seventh grade CMP textbook and in the sixth grade Math Connects textbook, but was rare in the other seven texts. Table 14 Percentage of Tasks W ith Each Solution Strategy by Textbook Series and Grade Level CMP Math Connects UCSMP 6 th 7 th 8 th overall 6 th 7 th 8 th overall 6 th 7 th 8 th overall Building Up 0 22 0 12 10 3 0 4 3 2 0 2 Decimal 7 4 0 5 0 6 9 6 13 18 4 14 Manipulative 70 42 0 55 16 21 9 15 18 5 0 11 Proportion 2 26 0 15 41 27 52 41 28 22 25 32 Other 7 4 0 5 34 31 29 31 29 28 38 30 n* 43 50 0 93 103 126 151 380 94 67 24 185 *For this table, n was the number of tasks in which a solution strat egy was suggested. The Proportion Strategy Proportions are a common and traditional method of solving proportional problems; thus, the proportion strategy is of particular interest. As shown in Table 1 4 the frequency with which proportions were used varied co nsiderably with textbook series. The percentage of tasks in which proportions were used varied most in the sixth grade textbooks. Proportions were common in the sixth grade Math Connects textbook, with 41% of the solution strategies featuring a pr oportion. Proportions were less common in the sixth grade UCSMP textbook, with 28% of the solution strategies featuring a
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143 proportion. Proportions were rare in the sixth grade CMP book, with only 2% of the solution strategies featuring a proportion. The following is an illustration of a type of task that can be present ed either through the use of a proportion or through the use of a visual aid. Common in the sixth grade books were tasks in which students found fractions equivalent to a given fraction, such as converting to The sixth grade Math Connects textbook contains a two page exercises in which students are instructed to use counters to name fractions equivalent to given fractions. The next lesson in the Math Connects textbook is intended to help students use proportions to find equivalent fractions. The five page l esson contains four examples and 49 exercises. Twenty four proportions appear in the lesson. Thus, although equivalent fractions are presented first through the use of manipulatives, proportions seem to be the preferred solution strategy. By way of contras t, the sixth is also intended, in part, to help students find equivalent fractions. This is accomplished primarily through the use of pictures of manipulatives (fraction strips) and number line s. No complete proportions appear in the in class portion of the Inves tigation, although in six tasks students are asked to insert one of the signs <, >, or = between two fractions. The exercise set of the investigation contains four complete proportions a nd in 12 tasks students are asked to insert one of the signs <, >, or = between two fractions. In writing or with pictures, how compares to 9a, p. 29). Thus,
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144 although proportions appear in the CMP textbook, manipulatives and visual representations are encouraged. The Unit Rate Strategy All of the textbook series in this study contained material about the definition of a unit rate and procedur es for calculating a unit rate. However, very rarely was the unit rate method suggested as a means for completing a task. As a reminder to the reader, a task was coded with the unit rate strategy only if a unit rate was used as a means for solving a larger problem The following is an example of an exercise coded with the unit rate 20 cm, and 25 cm in diameter. The cakes are the same height and cost $8, $16, and $20, respectively. Which size gives the most cake per dollar? Which size gives the least cake McConnell et al., 2009, p. 536). Tasks in which students were instructed to simply compute a unit rate were not coded. For example, an exercise that read because the unit rate was not used to solve a larger problem. In five of the nine textbooks in the study, in no task was the unit rate strategy explicit ly suggested. In all of the textbooks in the study, the unit rate strategy was explicitly suggested in less than 1% of the tasks. However, for many tasks, the unit rate method may have been applied or assumed. For example, in a seventh grade Math Connects unit rates are used. The exercise set following these examples contains 46 exercises similar to the examples. In might be reasonable to assume that students would follow the exa mples and use the unit rate strategy for most of the exercises. However, in none of the
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145 exercises was the unit rate strategy explicitly suggested. Thus, the exercises were coded as no solution strategy Other Solution Strategies About 30% of the tasks in the Math Connect s and UCSMP textbook series were coded with the solution strategy other This suggests that the framework used to code solution strategies was not adequate to describe many of the solution strategies suggested in the textbooks. Virtually al l of the tasks coded with the solution strategy other were examples rather than exercises, as exercises generally did not suggest any solution strategy. In some of the examples coded as other students were instructed to compare decimals by lining up the decimal points and noticing the location of the digits that differed. Other examples coded as other instructed students to round decimals using a similar place value procedure. Other examples coded as other instructed students to compare fractions by using a common denominator. Level of Cognitive Demand The fourth research question concerned the level of cognitive demand (Stein et al., 2000) of tasks related to proportionality. According to th e framework designed by Stein and colleagues Memorization facts, rules, formulae or definitions OR committing facts, rules, formulae or definitions to Procedures Without C onnections Use of the procedure is either specifically called for or its use is evident based on prior Procedures With Connections some degree of cognitive effort. Although general
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146 Tasks at the highest level, Doing Mathematics O ne of the ways to determine the level of cognitive demand of a task is to notice whether students had recently seen a worked out example similar to the task. Thus, the researcher coded the level of cognitive demand only of exercises, not of activities and examples. Coding the level of cognitive demand was especially difficult in the CMP series which has no labeled examples. Far l ess than 1% of the exercises coded were at the lowest level of cognitive demand, the Memorization level. This was true in all thr ee textbook series. Overall, 68% of the tasks were Procedures Without Connections 25% of the tasks were Procedures With Connections and 7% of the tasks were Doing Mathematics The level of cognitive demand varied by textbook series and by grade level, as explained in the following sections. Level of Cognitive Demand and Content Area The level of cognitive demand varied according to the content area of the exercise. As indicated in Table 15, most of the exercises in the rational number content area were coded as Procedures Without C onnections Thus, e xercises in the rational number content area had the lowest levels of cognitive demand. E xercises in the data analysis and probability content area had the highest levels of cognitive demand.
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147 Table 1 5 Percentage of Exercises at Each Level of Cognitive Demand by Content Area Algebra Data Analysis Geometry Rat. Number Procedures Without Connections 57 38 56 79 Procedures With Connections 34 47 29 20 Doing Mathematics 9 16 15 2 Total Number of Exercises 734 212 765 2062 Level of Cognitive Demand and Textbook Series As indicated in Table 16, t he level of cognitive demand in the Math Connects and UCSMP textbooks was similar, with about 70% of the tasks being at the level Procedures Without Connections about 25% of the tasks being at the level Procedures With Connections and about 5% of the tas ks at the level Doing Mathematics The level of cognitive demand in the CMP textbooks was much higher, with 43% of the tasks being at the level Procedures Without Connections 36% of the tasks being at the level Procedures With Connections and 21% of the tasks at the level Doing Mathematics
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148 Table 1 6 Percentage of Exercises at Each Level of Demand by Textbook Series CMP Math Connects UCSMP Procedures Without Connections 43 75 65 Procedures With Connections 36 22 29 Doing Mathematics 21 3 7 Total Number of Exercises 636 2454 694 Cognitive Demand in CMP Textbooks As indicated in Table 1 6 the level of cognitive demand of tasks related to proportionality was higher in the CMP series than in the o ther two textbook series. M any CMP tasks ask students to create or design a mathematical or real life situation. This involves nonalgorithmic thinking, which is one of the hallmarks of a Doing Mathematics task (Stein et al., 2000). For example, a sixth gra de CMP exercise related to probability person game. Explain why your game is fair. Make a spinner and a set of rules for a two person game that is not fair. Explain why your game is not fai an et al., 200 9d p. 47). Additionally, many of the CMP exercises ask students to make and justify conjectures and to use drawings and examples in their response. For example, a seventh grade CMP task on similar s similar to rectangle B and to rectangle C. Can you conclude that rectangle B is similar to rectangle C? Explain. Use drawings and examples b p. 54).
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149 Level of Cognitive Demand and Grade Level The level of cognitive demand increased across the three grade levels. As illustrated in Figure 1 7 the percentage of tasks at the level Procedures Without Connections decreased through the grade levels. Specifically, the percentage of tasks at the level P rocedures Without Connections decreased from 73% in sixth grade to 68% in seventh grade and 59% in eighth grade. The percentage of tasks at the higher levels of cognitive demand increased between sixth and eighth grades. Figure 1 7 Percentage of tasks at each level of cognitive demand by grade The increase in cognitive demand at higher grade levels was most noticeable in the CMP textbooks. In the CMP textbooks, the percentage of tasks at the Procedures Without Connections level decreased steadily from 55% in sixth grade to 42% in seventh grade and 11% in eighth grade. The percentage of tasks at the Doing Mathematics level increased steadily from 4% in sixth grade to 28% in seventh grade and 40% in eighth 0 10 20 30 40 50 60 70 80 Gr. 6 Gr. 7 Gr. 8 Procedures Without Connections Procedures With Connections Doing Mathematics
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150 grade. Similar trends were seen in the Math Conne cts and UCSMP series, but in these series, the difference between sixth and eighth grades was smaller. Visual Representation The fifth research question concerned the visual representations that appear in textbooks in tasks related to proportionality. Ove rall, 28% of the tasks related to proportionality involved a visual representation such as a table, graph, or picture of manipulatives. This percentage varied with textbook series and grade level, as explained in the sections below. Activities were more li kely than examples or exercises to have a visual representation; 52% of the activities had one compared to 33% of the examples and 25% of the exercises. The two most common visual representations were tables and similar figures. Tables were used for a vari ety of purposes, such as showing the equivalence of various fractions, decimals, and percents, or for showing a relationship between an x and y variable. Visual Representation and Textbook Series Visual representations were more common in the CMP textbooks than in the other series. In the CMP textbooks, 45% of the tasks were accompanied by a visual representation. This number was 26% for the Math Connects textbooks, and 1 7 % for UCSMP. In the CMP se ries, tasks in Investigations and exercise sets were about equally likely to contain a visual representation. In the Math Connects and UCSMP series, activities were more likely to contain visual representations than were examples and exercises. For example in the UCSMP series, 76% of the activities included a visual representation compared with 15% of the examples and 15% of the exercises.
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151 Visual Representations in CMP Te x tbooks Because visual representations were more common in CMP textbooks than in the other two series, the researcher paid special attention to the visual representations in CMP textbooks. The sixth grade CMP textbook focuses on rational numbers; thus most of the visual representations are related to rational numbers. Many of the tasks in the sixth decimals and fractions. Other tasks in the same module refer to a table of data and ask students to use the data to calculate percentages. The sixth grade cur riculum also contains include a picture of a spinner. The seventh d to similar figures. Many of these tasks include a visual representation of similar figures. Another seventh fractions, decimals, and perc ents to compare quantities. The data are often presented in a table. The eighth grade CMP textbook focuses on algebra and thus contains visual representations of functions, such as function tables and graphs. More than half of the tasks coded from the eig hth grade CMP textbook contained a visual representation, most of which were either a table or a graph. Quite a few of the eighth grade tasks contain both a table and a graph. Tables were used for two distinct purposes. Some tables contained data from a re al life situation. Other tables were function tables that showed a relationship between x and y variables.
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152 Visual Representation and Grade Level T he percentage of tasks with a visual representation increased through the grade levels, with 3 5 % of the sixth grade tasks, 60% of the seventh grade tasks, and 67% of the eighth grade tasks accompani ed by a visual representation. As indicated in Table 17, i n the CMP and Math Connects series, the frequency of visual representations increased with grade level. For e xample, in the CMP series, 32% of the sixth grade tasks, 49% of the seventh grade tasks, and 63% of the eighth grade tasks were accompanied by a visual representation. In the UCSMP series, the percentage of tasks with a visual representation increased betw een sixth and seventh grades and then decreased slightly in eighth grade. Table 1 7 Visual Representation by Textbook Series and Grade CMP Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Number of Tasks with VR* 109 253 70 204 235 278 32 82 2 3 Percent of Tasks with VR 32 4 9 63 22 24 34 9 22 20 Number of Tasks in Book 337 528 100 946 999 821 345 371 116 The specific visual representations that were used also varied with grade level. Number lines and 10 by 10 grids were used to convey concepts related to rational numbers and were, thus, common in sixth grade textbooks. Similar figures were common in seventh gr ade textbooks, and tables and graphs were common in eighth grade texts.
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153 Characteristics of Proportional Situations The sixth research question related to the characteristics of proportional situations and the extent to which tasks in textbooks point out when proportional reasoning is appropriate. Overall, 9% of the tasks related to proportionality pointed out the characteristics of proportional situations or whether proportional reasoning was appropriate in a given situation. Activities were more likely t o cover this content than examples and exercises. Nineteen percent of the activities pointed out the characteristics of proportional situations as did 7% of the examples and 8 % of the exercises. Characteristics and Textbook Series CMP textbooks contained a higher percentage of tasks that pointed out the characteristics of proportional situations than did the Math Connects and UCSMP series. Seventeen percent of the CMP tasks covered this content as did 7% of the Math Connects tasks and 9% of the UCSMP tasks As indicated in Table 1 8 in most of the textbooks, less than 20% of the tasks pointed out the characteristics of proportionality. In this respect, the eighth grade CMP textbook was much different from the others in the study; 62% of the tasks in this te xtbook pointed out the characteristics of proportionality or discussed the appropriateness of proportionality
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154 Table 1 8 Number and Percentage of Tasks that Point Out the Characteristics of Proportionality CMP Math Connects UCSMP 6 th 7 th 8 th 6 th 7 th 8 th 6 th 7 th 8 th Number of Tasks with Char* 3 96 62 34 30 127 15 56 3 Percent of Tasks with Char 1 18 62 4 3 15 4 15 3 Number of Tasks in Book 337 528 100 946 999 821 345 371 116 Characteristics of Proportionality in CMP Textbooks In the sixth grade CMP textbook, there are a few tasks related to the differences between additive and multiplicative reasoning when percents are introduced. For example, s tudents are provided with this information: During a recent year, Yao Ming made 301 out of 371 free throw attempts and raw numbers to tell who was better at free throws. But in sports, the announcers give these raw numbers as percents (Lappan et al., 200 9a p. 56). Students then use a visual representation to estimate the percent of shots each player made. Students are Lappan et al., p. 56). S tudents using additive reasoning would reason that 451 is greater than 301 and agree with Alisha. Students using multiplicative reasoning would realize that fractions or percents are nece ssary to compare the players.
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155 Both the seventh and eighth grade CMP textbooks contained a relatively large percentage of tasks that pointed out the characteristics of proportionality. In general, the seventh grade textbook accomplishes this through geometry/measurement tasks and the eighth grade textbook accomplishes this through algebra tasks The seventh grade CMP textbook contained many tasks that prompt students to consider what happens to the side lengths, perimeter, and area of a shape that undergoes a size change. These tasks could help students realize that the side lengths and perimeter increase by the same percentage as the size change, but that the area does not. This could help students understand that the relationship between size change factor and perimeter is proportional, but the relationship between size cha nge factor and area is not proportional. The seventh grade CMP textbook contains quite a few tasks that point out that a figu re enlarged by a scale factor will have its perimeter increased by a factor of s its area increased by and its volume incr eased by The seventh grade CMP textbook also has several tasks that explicitly do you decide when to compare numbers using rates, ratios, or percents rather than by To point out the differences between additive and multiplicative reasoning, the seventh which students are asked a series of questions designed to illustrate the differences between additive and multiplicative reasoning, such as the following: There are 300 students in East Middle School. To plan transportation services for the new West Mid dle School, the school system surveyed East students. The
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156 compare the number of students in Mr. Archer the number who are walkers? Which seems to be the best comparison statement? (Lappan et al., 2006 c p. 65). Ideally, students would state both that five more students ride than walk (additive reasoning) and that 57% of the students ride the bus (multiplicative reasoning.) However, because little guidance is provided, teachers should ensure that students come up with both answers. The eighth grade CMP textbook contained many tasks that prompt students to compare linear to no n linear relationships. These were coded as pointing out the characteristics of proportional situations since linearity is one of the characteristics of those situations. However, many of the tasks in the eighth grade CMP textbook that were coded as pointi ng out the characteristics of proportional situations were not closely students are shown 18 equations and asked which represent functions that are linear, exponential, and quadratic. Although distinguishing between linear and non linear relationships may be an important part of distinguishing between proportional and non proportional situations, it is possible that other researchers would not have coded this and similar tas ks as pointing out the characteristics of proportional situations. Thus, although the researcher found that 62 % of the eighth grade CMP tasks cover this content, this may be an over estimate. In the eighth tasks ask students to write an equation of the form to match a given data
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157 table or graph. At least one of these graphs contains both a line with a y intercept of zero and a line with a positive y intercept. This would be a logical place to ask students to consider the effects of a positive y intercept, but students are not asked to do so. Characteristics of Proportionality in Math Connects Textbooks All three of the Math Co nnects textbook s contain material designed to help students distinguish between proportional and nonproportional situations. However, in some cases, this material is presented in an abstract, symbolic manner that seems to encourage the use of procedures ra ther than foster conceptual understanding. For example, the sixth grade textbook states the following: To verify a proportion, you can use cross products. If the product of the means equals the product of the extremes then the two ratios form a proportion In a proportion, the top left and bottom right numbers are the extremes. The top right and bottom left numbers are the means. (Day et al., 2009a, p. 333) Although this explanation seems abstract and procedural, a subsequent exercise encourages multiple s olution strategies; it read s: w o large pizzas for $15 and four large pizzas for $28. Describe and use three different ways to determine if The seventh grade Math Connec ts textbook shows that one serving of milk contains 300 milligrams of calcium and that four servings of milk contain 1,200 milligrams of calcium. The textbook uses the proportion to show that the relationship is proportional. The exercise set for this lesson consists of 12 exercises in which students are given two ratios and asked whe ther
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158 313). This method of comparing ratios is mathematically correct and w ill help students distinguish between proportional and non proportional relationships, but is likely to lead only to procedural understanding and will likely not help students understand the differences between additive and multiplicative reasoning. In th e eighth grade textbook, two examples are provided in which data is taken from a ratio table and ratios are formed and compared. However, in the exercise set, no ratio tables are provided and in some cases would be impossible to construct due to a lack of information. Characteristics of Proportionality in UCSM P Textbooks The sixth and seventh grade UCSMP textbook s explicitly address the issue of recognizing situations in which proport ional reasoning is required. However, the number of tasks devoted to the topic is small. For example, in the sixth grade textbook, in a make sense to use proportions al., 2009, p. 496). Neither of the two examples in the lesson is related to the appropriateness of proportional reasoning. Thus, if students complete and understand these exercises, they will have c onsidered the issue. However, since students generally do not complete or understand every exercise in a textbook, the issue could easily be missed. The seventh grade UCSMP textbook also devotes a small amount of space to the issue. The textbook states:
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159 There are two parts to proportional thinking: (1) the sense to recognize situations in which setting up a proportion is a way to find an answer and (2) the ability to get or estimate an answer to a proportion without solving an equation. Some people believ e that proportional thinking is one of the most important kinds of thinking you can have in mathematics (Viktora et al., 2008, p. 596). An in class activity in this same lesson presents students with four different situations and asks which of the four ca lls for proportional thinking. Few tasks in the eighth grade textbook are designed to help students distinguish proportional from non proportional situations. A few exercises are designed to help students realize that the ratio of the areas of similar fig ures is not proportional to the size change factor and several exercises help students distinguish between linear and non linear functions. Characteristics and Grade Level Overall, the percentage of tasks that pointed out the characteristics of proportion al situations increased with grade level. Three percent of the sixth grade tasks covered this content as did 10% of the seventh grade tasks and 1 9 % of the eighth grade tasks. However, this varied with textbook series. In the CMP texts, the percentage of ta sks that pointed out the characteristics of proportional situations increased steadily with grade level from 1% in sixth grade to 1 8 % in seventh grade to 62 % in eighth grade. In the Math Connects series, the percentage of tasks that pointed out the charact eristics of proportional situations was higher in eighth grade than in either the sixth or seventh grades. In the UCSMP series, the percentage of tasks that pointed out the characteristics of proportional situations was highe st in seventh grade.
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160 According to the Curriculum Focal Points (NCTM, 2006), seventh grade In the textbook series in this study, contrasting proportional and in verse relationships was covered in eighth grade rather than seventh. The developers of these series will need to decide whether they wish to comply with the recommendations of the Curriculum Focal Points and move their comparisons of proportional and inver se relationships from eighth to seventh grade Overview of Textbook Series In this section, an overview of the treatment of proportionality in each textbook series is provided. This should help the reader understand the material related to proportionalit y that students studying from each series experience as they move through middle school. Proportionality in CMP Textbooks Each of the textbooks in the CMP series covers proportionality differently; in each grade level, students are presented with a different view of proportionality. All of the CMP textbooks use visual representations and hands on activities to foster conceptu al understanding, but the content areas through which proportionality is presented vary among grade levels. At the sixth grade level, proportionality is presented through rational numbers and probability. At the seventh grade level, proportionality is pres ented through all of the content areas. At the eighth grade level, it is presented through algebra. Proportions were used less often in the CMP textbooks than in the other series. Some important topics, such as the definition of direct variation, appeared only in exercise sets and were not explicitly addressed in the portion of the textbook that would be covered in
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161 class. Thus, students who study from CMP textbooks for three years have the opportunity to see proportionality as a theme that connects content areas. However, these students will have had less practice with formal proportions than students who study from other textbook series. T he S ixth grade CMP T extbook In the sixth grade CMP textbook, there is a focus on helping students develop a conceptual understanding of fractions. Students use fraction strips, number lines, and visual representations of fractions to develop understandings of equivalent fractions and the relative size of various fractions. Proportions are used to compare fractions after st udents have had opportunities to develop conceptu al understandings of fractions. Similarly, information related to decimals is first presented visually, through the use of hund redths grids and number lines. As previously described, t he differences between additive and multiplicative reasoning are hinted at when percents are introduced. In addition to the focus on rational numbers, the sixth grade CMP book also presents proportionality through probability; one of the eight modules is related to probability and includes three lessons that involve proportionality. In the lessons on probability, students first conduct hands on probability experiments A large number of tasks ask students to predict and analyze the data sets that would result from various spinne rs. In summary, the sixth grade CMP textbook presents proportionality through rational numbers and probability. The visual representations and hands on activities that are featured in this book are likely to help students develop conceptual understanding of rational numbers and probability. In the sixth grade CMP book, there are almost no tasks
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162 related to proportionality in the algebra and geometry content areas. Thus, there are no lessons on similar figures, rates, ratios, or slope, which are common in mo st middle school textbooks. Proportions were used in the sixth grade CMP textbook only to compare fractions. T he Seventh grade CMP T extbook Six seventh grade CMP modules were included in the study. Two related primarily to algebra, two to geometry/measurement, one to rational numbers, and one to data analysis/probability. However, in each of the algebra modules, only one investigation related to proportionality. In one of the geometry/measurement modules, Wrapping: Three Dimens related to proportionality. Thus, of the 15 investigations related to proportionality, seven involved geometry and measurement. ion titled dimensional solids. This was the only textbook in the study to contain such a lesson. The investigation pointed out that when a solid figure is enlarged by a scale factor, the relation ship between the dimensions and the scale factor is proportional, but the relationship between the surface area and scale factor is not proportional and the relationship between the volume and the scale factor is not proportional. Although one lesson in t he seventh grade book was devoted to the use of proportions, in other lessons, students were instructed to use tables, graphs, and equations
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163 to find missing values rather than proportions. Thus, as in the sixth grade book, proportions were used less often than one might expect. T he Eighth grade CMP T extbook Only three of the eight modules in the eighth grade curriculum featured a focus on proportionality and were included in the study. Thus, one could say that proportionality is less of a focus in the eighth grade book than in the sixth and seventh grade CMP textbooks. Of the eight modules in the eighth grade CMP curriculum, five focus on algebra, two on geometry, and one on data analysis. T he geometry covered in the textbook is primarily the Pythagor ean Theorem and transformations, which are not closely related to proportionality. Two of the algebra modules as well as the data analysis module contained material related to proportionality and were included in the study. In the data analysis module, onl y one investigation contained material related to proportionality and was included in the study. Thus, in the eighth grade CMP textbook, proportionality i s presented primarily through the algebra content area. Because of the focus on algebra, there were mo re tasks of the function rule problem type than any other. Because none of the modules focus on rational numbers, none of the tasks were coded with the alternate form problem type. Many of the tasks in the algebra modules ask students to compare linear rel ationships to inverse or other nonlinear relationships. These tasks were included in the study because linearity is one of the characteristics of proportional situations. Many of the tasks in the algebra modules ask students to translate between tables, gr aphs, and equations. The definition of inverse variation and the inverse variation equations are
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164 highlighted, but the definition of direct variation and the direct variation equation appear only in one exercise in a homework set. Proportionality in Math Co nnects Textbooks Whereas in the CMP series, proportionality was presented differently at each grade level, in the Math Connects series, students see a similar view of proportionality at each grade level. All three of the Math Connects textbooks cover a gr eat deal of material. Thus, in all three textbooks, proportionality is covered through all four content areas. Although each textbook contains tasks related to proportionality in all four content areas, more tasks are related to rational numbers than any o ther content area T he sixth and seventh grade Math Connects textbooks seem repetitive; much of the material related to proportionality that is covered in sixth grade is also covere d in the seventh grade textbook. For example, b oth textbooks contain several chapters on decimals, fractions, and percent. Although the eighth grade textbook contains five chapters on algebra, the book as a whole has much less of a focus on algebra than do the eighth grade CMP and UCSMP textbooks. The algebra that does appe ar in the Math Connects book is at a lower level than in the eighth grade CMP and UCSMP textbooks. The Sixth grade Math Connects Textbook Proportionality in the sixth grade Math Connects textbook is presented primarily through rational numbers ; 62% of the tasks related to proportionality were coded as rational number Three of the 12 chapters focus on fractions, decimals, and/or percents More of the tasks are of the problem alternate form than any other problem type. This indicates that many tasks in the sixth grade book ask students to convert between fractions, decimals, and percents.
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165 The lessons on rational numbers contain a few visual representations, but far fewer than were found in the sixth grade CMP textbook. The lessons on rational numbers contain large numbers of tasks of the cognitive level of demand Procedures Without Connections These tasks seem repetitive standard form, and expanded form of decimals. The sixth grade textbook also presents proportionality through algebra; 18% of the tasks related to proportionality were coded as algebra Two of the 12 chapters focus on algebra. One of the chapters on algebra contain which includes numerous tasks of the function rule problem type. The sixth grade Math Connects textbook has much more material related to formal proportions than does the sixth grade CMP textbook. The textbook contai ns a presented as a way of writing ratios in simplest form. Most of the solution strategies that are suggested involve proportions. However, in very few of these tasks is c ross multiplication used to solve the proportion. The Seventh grade Math Connects Textbook Proportionality in the seventh grade Math Connects textbook is presented primarily through rational numbers and, to a lesser extent, geometry. Three of the 12 chapters focus on fractions, decimals, and/or percents. Several of the seventh grade lessons on rational numbers seem similar to lessons in the six th grade textbook. For
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166 contain lessons on converting between fractions, decimals, and percents and both textbooks cover computation with fractions The seventh grade c information that was covered in the sixth This chapter, in both books, involves algebra. The chapter in the seventh Three of the 12 chapters focus on geometry. There is one lesson in the seventh grade Math Connects book on similar figures, but there is far less material on similar figures in this book than in th e seventh grade CMP textbook. Although there are several lessons on three dimensional solids, there is not a focus on changing the dimensions of these solids as there was in the seventh grade CMP textbook. The Eighth grade Math Connects Textbook Proportio nality in the eighth grade Math Connects textbook is presented primarily through rational numbers. Two of the 12 chapters focus on fractions, decimals, and/or percents. As in the sixth and seventh grade books, the eighth grade book contains material relat ed to converting among fractions, decimals, and percents comparing the magnitudes of rational numbers and computation with rational numbers. Each of the rational number lessons contain a large number of exercises which are mostly of a low level of cognit ive demand. For example, in the eighth Procedures without Connections
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167 Proportionality in the eighth grade Math Connects textbook is also presented through the algebra content area. Three of the 12 chapters focus on algebra. One of these two variable quantities is constant, their relationship is called a direct variation. The 9c p. 487). This lesson explicitly address the differences between proportional linear functions and nonproportional linear functions. Proportionality in UCSMP Textbooks A s in the CMP series, in the UCSMP series, proportionality is more prominent in the sixth and seventh grade textbooks than in eighth grade. This is likely due to the fact that the eighth grade CMP and UCSMP textbooks focus on algebra. In other ways, the UC SMP series resembles the Math Connects series more than the CMP series. For example, the sixth grade Math Connects and UCSMP series both contain a lesson on proportions, which are virtually absent from the sixth grade CMP textbook. In the UCSMP textbooks, rates, ratios and proportions arise from division. For example, the seventh grade textbook contains a chapter titled Division which includes four lessons on rates, ratios, and proportions. The UCSMP textbooks number of words by a certain number of minutes and that units such as these can be written as fractions, as in Th is connection between division and proportionality was much less visible in the other two series.
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168 The Sixth grade UCSMP Textbook Pro portionality in the sixth grade UCSMP textbook is presented primarily through rational numbers. T wo of the 12 chapters focus on fractions, decimals, and/or percents and other chapters also contain lessons on rational numbers Fractions are first presented with a geometric context that was not present in the other two series; the fraction is compared to the midpoint of a segment and rule r s are used to provide a visual represe ntation of other fractions. Decimals are first presented through the use of number lines instead of the hundredths grids that were used in the other two series. Rulers and number lines are shown frequently in the lessons on rational numbers, but few other visual representations are used. As in the Math Connects series, there is a sixth grade lesson on ratio and proportion. As in the Math Connects series, proportions are first introduced as a way to put a ratio in lowest terms. As in the Math Connects serie s, a connection is made between rate tables and proportions. The UCSMP lesson on proportions contains several exercises that check whether students understand the definition of a proportion; in the Math Connects to write and solve proportions. In the sixth grade UCSMP textbook, ratios and proportions ar ise from division. The sixth grade book presents the Rate Model for Division and the Ratio Comparison When two quantities with different kinds of units are divided, the quotient is a rate McConnell et al., 200 9 p. 401 ). The Ratio a and b are quantities with the same units, then compares a to b al., 2009, p. 462).
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169 The Seventh grade UCSMP Textbook As in the sixth grade book, proportionality in the seventh grade UCSMP textbook is presented primarily through rational numbers. Although only one of the 12 chapters focuses on fractions, decimals, and percents, five of the lessons in this chapter were included in the study Thus, five of the 19 lessons coded from the seventh grade book were related to fractions, decimals, or percents. Chapters on multiplication and division contained three lessons relat ed to rational numbers. Thus, eight of the 19 lessons coded from the seventh grade book were related to rational numbers. As in the sixth grade book, there are few visual representation s of rational numbers. M ost of the representations that are present are number lines, not the fraction strips and hundredths grids that are present in other series. As in the sixth grade textbook, ratios and proportions arise from division. The seventh grade book reiterates the Rate Model for Division and the Ratio Comparison Model for Division that were covered in the sixth grade textbook The Ratio Comparison Model for division is presented identically in both textbooks, but the Rate Model for Division is presented more algebraically in the seventh grade a and b are quantities with different units, then is the amount of quantity a per amount of quantity b The Eighth grade UCSMP Textbook As in the CMP series, the eighth grade UCSMP textbook presents proportionality through algebra. In fact, a chapter in the eighth grade UCSMP textbook is titled grade book is a repeti tion of material in the sixth and seventh grade books. For example, the sixth and
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170 seventh are covered again in the eighth grade textbook. The eighth grade UCSMP textbook co ntains less material on the differences between linear and non linear functions than do the other two series. The eighth grade UCSMP book has lessons on exponential functions, including a lesson comparing linear increase to exponential growth, but there is less of a focus on this than in the other two series. The UCSMP book also does not cover inverse variation, which is covered in the Intercept Equations for Lines Summary of Results The purpose of this study was to investigate how the treatment of proportionality in textbooks varies by grade level and textbook series. Results indicate that the amount of attention paid to proportionality varies by grade level; sixth and seventh grade textbooks contain considerably more tasks related to proportionality than do eighth grade textbooks. Results also indicate that the treatment of proportionality in textbooks varies by textbook series. Tasks related to proportionality in CMP textbooks have higher levels of cognitive demand and are more likely to contain a visual representation. Proportionality is a characteristic of mathematical relationships that appear in a variety of topics within mathematics, such as algebra, data analysis, geometry, and rational numbers. Results of this study indicate that more of the tasks related to proportionality are also related to rational numbers than to any other content area. This likely occurs because lessons on rational numbers typically have large numbers of relatively simple practice problems. In the Math Connects series, this focus on rational
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171 numbers was present at all three grade levels, whereas in the other two series, the eighth grade textbook focused on algebra. Other findings of this study indicate tha t the qualitative problem type is virtually absent from the textbooks in this study as were the building up and unit rate solution strategies. Textbooks contain material on unit rate, but typically have students simply compute the unit rate rather than use it as a solution strategy.
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172 CHAPTER 5: DISCUSSION The purpose of this study was to investigate the treatment of proportionality in three contemporary, widely used middle school textbook series: Connected Mathematics2 (CMP), Math Connects and the sixth seventh and eighth grade textbooks from the University of Chicago School Mathematics Project ( UCSMP ) The three series were chosen because their treatment of proportionality was expected to be quite different. Results of the study indicate that this was indeed the case. In each of the nine textbooks, t he researcher analyzed lessons in which proportionality was a focus, such as lessons on rational numbers, rates, ratios, proportions, similar figures, and certain probability le ssons. The researcher compared the treatment of proportionality across grade levels to see whether there is a logical progression as students move from sixth to Principles and Standards for School Mathematics (N CTM, 2000), proportionality can be used to connect various content areas in mathematics. The researcher noted the content area of each task related to proportionality in order to determine the extent to which proportionality is indeed used in this manner. S everal conclusions based on the analysis of the nine textbooks in this study can be made Three of these conclusions seem particularly salient and are elaborated upon in the following sections. First, a clear finding of this study is that more of the tas ks related to proportionality were in the rational number content area than any other. This was particularly true in the sixth and seventh grade textbooks. The Principles and Standards
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173 for School Mathematics state that middle op a deep understanding of rational number concepts, become proficient in rational number computation and estimation, and learn to think flexibly about relationships among frac p. 212). However, more recent recomm endations in the Curriculum Focal Points (NCTM, 2006) state that fractions and decimals should be studied earlier, in third and fourth grades. If one follows the newer recommendations, a focus on fractions and decimals in middle school may be inappropriate Partly because of the focus on rational numbers in sixth and seventh grades, there were more tasks related to proportionality in the sixth and se venth grade textbooks than in the eighth grade textbooks The sixth and seventh grade CMP and UCSMP textboo ks had a much heavier focus on proportionality than did the eighth grade textbooks in the series. This may be appropriate, given the recommendations in the Curric ulum Focal Points (NCTM, 2006) which indicate that eighth grade students should study a variet y of algebra topics. Proportionality should be studied in eighth grade, but should be studied through an algebra lens, which was present in some, but not all of the eighth grade textbooks in the study. A third salient finding of this study is that real d ifferences exist in the presentation of proportionality in the three series in this study. Differences can be seen in the content area of the proportional tasks, the solution strategies encouraged by the textbooks the level of cognitive demand of the task s, and the ways in which textbooks point out the characteristic s of proportionality and the differences between additive and multiplicative reasoning Although student understanding was not measured in this study, it seems
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174 likely that many of the Math Conn ects lessons would encourage procedural understanding and that the CMP series is designed to foster conceptual understanding Justification of this claim appears in a later section. Grade Levels One of the findings of this study is that, in the textbook series included in this study, proportionality is emphasized in sixth and seventh grades more than in eighth grade. This finding applies to the CMP and UCSMP series more than to the Math Connects s eries. This finding reflects the fact that the eighth grade CMP and UCSMP textbooks focus more on algebra than other content areas Sixth Grade According to the Curriculum Focal Points (NCTM, 200 6 ) sixth grade students should connect rates and ratios to multiplication and division The example given is the of 12 items by first dividing $3.75 by 5 to find out how much one item costs and then multiplying the cost of The Curriculum Focal Points suggest that sixth grade students develop this reasoning by using a multiplication table or simple drawings. Noticeably absent from this recommendation is any mention of formal proportions or cross multiplication Because one of the three sixth grade f ocal p oints directly relates to proportionality, one would expect proportionality to be a focus of sixth grade textbooks. This was the case in the textbooks in the study. A fairly large number of lessons and tasks were coded from each sixth grade te xtbook. T he sixth grade lessons and tasks that related to proportionality may have been too focused on rational numbers, but at least proportionality was present and common in sixth grade textbooks.
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175 Seventh Grade According to the Curriculum Focal Points (NCTM, 200 6 ) seventh grade students should develop an understanding of proportionality, including similarity. All three of the seventh grade textbooks in the study had at least one lesson on similar f igures. T he seventh grade CMP textbook had five Investigations related to similar figures. The seventh grade Math Connects included tions in The Curriculum Focal Points also state that seventh grade students should: use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase proportional relationships and identify the unit rate as the slope of the related line. inverse proportionality (NCTM, 2006, p. 19). Thus one would expect proportionality to be a focus of seventh grade textbooks. This was the case in the textbooks in the study. In each of the three series, more tasks were coded from the seventh grade textbook than from either of the other two grade levels. In the CMP series, there were more tasks related to proportionali ty in the seventh grade book than in the sixth and eighth grade books combine d Furthermore, one would expect to see seventh grade tasks related to algebra, geometry, and rational numbers. This also was the case with the textbooks in the study. All three seventh grade textbooks had
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176 relatively large percentages of proportional tasks related to each of these three content areas. Eighth Grade One of the eighth grade f ocal points mentions proportionality but seems more focused on other algebraic concepts. The f ocal p oint states that eighth grade students should recognize a proportion ( y / x = k or y = kx ) as a special case of a linear equation of the form y = mx + b understanding that the constant of proportionality, k is the slope and Curriculum Focal Point s also describes a wide variety of other algebraic concepts to be studied that are not closely related to proportionality. Addition ally, the other two eighth grade f ocal p oints are not directly related to proportionality. Thus, one might expect to see proportionality less of a focus in eighth grade than in sixth and seventh grades. This was the case in the textbooks in the study. In e ach of the three series, fewer tasks were coded from the eighth grade textbook than from the other two grade level s. T he number of tasks coded in the eighth grade CMP and UCSMP textbooks is sharply lower than the number of task s related to proportionality in the seventh grade textbook. The difference is much less dramatic in the Math Connects series. T hus, t he CMP and UCSMP se ries are consistent with the recommendations in the Curriculum Focal Points in that they focus on proportionality in sixth and sevent h grades and on algebra in eighth grade. However, for this strategy to work, students must understand proportionality by th e time they leave seventh grade since there is less material related to it in the eighth grade textbooks. In contrast, the Math Conne cts series has only slightly less material in the eighth grade book related to proportionality than in
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177 the sixth and seventh grade textbooks. This provides the advantage of review for eighth grade students who have not mastered proportionality, but has th e disadvantage of a lesser emphasis on algebra in the eighth grade Math Connects textbook. Content Areas Overall, 55% of the tasks related to proportionality were in the rational number content standard, 20% were in the algebra standard, 22% were in g eometry /measurement and 6% were in d ata analysis/p robability. The large percentage of tasks related to rational numbers is likely a reflection of two factors. First, rational numbers have historically constituted a large part of the middle school mathemat ics curriculum. Although the Curriculum Focal Points (NCTM, 2006) states that an understanding of rational numbers should be developed in third and fourth grades, the textbooks examined in this study contain many tasks aimed at helping students understand and translate among fractions, decimals, and percents. Thus, the large percentage of tasks related to rational numbers in part reflects a perhaps outdated focus on helping middle school students understand and translate between various forms of rational nu mbers. The large percentage of tasks related to rational numbers is also a reflection of the framework used in this study The researcher intentionally adopted a broad view of proportionality which includes some tasks that other scholars may not classify a s directly related to proportionality. Many of these tasks are related to rational numbers. For example, many tasks in the sixth grade textbooks asked students to compare the size of two or more decimals. Some scholars may not recognize this as proportiona l reasoning, but these tasks were included in the study. Thus, the large percentage of tasks related to rational numbers in part reflects the framework.
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178 The low per centage of tasks related to d ata a nalysis is likely a reflection of the overall focus on thi s standard in the middle school curriculum. Of the nine focal points in the middle school curriculum, only one is directly related to data analysis. This focal point states that students should use descriptive statistics to compare data sets (NCTM, 2006). Because mean, median, and mode were not included in this study, it is not surprising that only a small percentage of tasks in this study related to data analysis One of the connections to the seventh grade focal points states that students should use prop ortions to make predictions and use percentages in relation to circle graphs. These types of tasks were found in the textbooks in the study and were included in the study. Content Areas of Tasks in Sixth Grade Textbooks All of the sixth grade textbooks in this study present proportionality primarily through the rational number content area. Specifically, 87% of the tasks in the CMP textbook, 62% of the tasks in the Math Connects textbook, and 57% of the tasks in the UCSMP textbook were in the rational numb er content area. As previously mentioned, this could be due, in part, to the nature of the framework. However, it is clear that all of the sixth grade textbooks in the study contain a large number of tasks that require students to compare the size of fract ions or decimals, to convert between fractions, decimals, and percents, or to find percentages of various numbers. Many scholars may not consider these activities to involve proportional reasoning, but because of the broad definition of proportionality use d in this study, the se topics were included in the framework. Two characteristics of the sixth grade textbooks resulted in this tendency to represent proportionality through rational numbers. First, most of the lessons related to
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179 proportionality were als o related to rational numbers. For example, in the sixth grade CMP textbook, nine Investigations were analyzed; six of these focused on rational numbers and the other three were in the data analysis and probability content area. Second, lessons related to rational numbers had a greater number of tasks coded than did lessons in other content areas. For example, in the CMP book, in the Investigation sixth grade textbooks contain a large number of lessons related to rational number s, but also many of the tasks in the lessons fit into the framework for identifying proportional tasks. The Curriculum Focal Points (NCTM, 2006) state that students should develop an understanding of fractions and decimals in third and fourth grades. The content standards of most states indicate that students should develop an understanding of equivalent fractions in fourt h or fifth grade, but that equivalence of fractions, decimals, and percents is studied in sixth and seventh grades (Reys et al., 2006). Thus, whether the sixth grade focus on rational numbers is appropriate is not clear as mathematics educators are not in agreement regarding the placement of rational numbers Geometry and measurement tasks involving proportionality were also common in the sixth grade Math Connects and UCSMP textbooks. In the Math Connects textbook, many of these tasks had to do with conver ting units of measurement. The Math Connects textbook contains a large number of exe rcises similar to 6 yd = ____ ft. The UCSMP textbook contains fewer exercises of this nature.
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180 Content Areas of Tasks in Seventh Grade Textbooks The Curriculum Focal Points (NCTM, 2006) suggest that proportionality in seventh grade should be presented through a variety of content areas. This document state s that seventh graders should solve a wide variety of percent problems, learn about similar figures, distinguish pr oportional relationships from other relationships, including inverse, and use proportions to solve data analysis and probability problems. In all of the seventh grade textbooks, proportionality was presented through all four content areas. All of the seven th grade textbooks contained at least one lesson on similar figures. However, content designed to help students distinguish proportional relationships from other relationships was minimal. For example, the seventh grade Math Connects textbook had a one pag and no discussion of when proportional reasoning is appropriate. A noticeable difference between the seventh grade textbooks was the focus on geometry in CMP and rational numbers in Math Connects and UCSMP. A focus o n rational numbers could be appropriate in seventh grade, provid ed that percents rates, and ratios constitute the bulk of that focus, rather than decimals or fractions, which should be studied by students in third and fourth grades, according to the Curri culum Focal Points (NCTM 2006 ). The framework used in this study was not designed to distinguish between the various types of rational numbers, so percentages cannot be reported, but the titles of the lessons provide some insight into their content. The r ational number tasks in the seventh grade UCSMP textbook appear in lessons on a variety of topics including fractions, decimals, percents, rates, and ratios. However, in the seventh grade Math Connects textbook, many of the rational number tasks appear in lessons on simplifying
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181 fractions, converting between fractions and decimals, and comparing and ordering fractions and decimals, topics that, according to the Curriculum Focal Points should have been studied prior to seventh grade. Content Areas of Tasks in Eighth Grade Textbooks The Curriculum Focal Points (NCTM, 2006) suggest that proportionality in eighth grade should be presented primarily through algebra. This was the case in the eighth grade CMP textbook, in which three of the four Investigations related to proportionality were also related to algebra. The UCSMP textbook also focused on algebra; five of the six lessons related to proportionality were also related to algebra. The Math Connects textbook, however, contained a plethora of topics. Twenty six lessons related to proportionality, covering ratios, rates, rate of change, proportions, similar figures, dilations, indirect measurement, solid figures, s equences, slope, direct variation, graphs of functions, circle graphs, probability, and statistics. Although some of these lessons related to algebra, they may be overshadowed by the mountain of other content. Content Area Progression From Sixth To Eighth Grade Given the nature of the mathematics curriculum as described in the Curriculum Focal Points (NCTM, 2006), one would expect to see a progression from a focus on rational numbers in late elementary and early middle school to algebra in late middle scho ol. This progression is clear in the CMP textbooks. T he CMP textbooks present proportionality primarily through rational numbers in sixth grade, geometry in seventh grade, and algebra in eighth grade. One could perhaps argue that the transition from ration al numbers to algebra in the CMP series is too extreme as none of the 342 tasks related to proportionality in the sixth grade book were in the algebra content area and
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182 only the seventh grade book had a significant percentage of tasks related to proportiona lity in the geometry content area The progression from rational numbers to algebra is less dramatic but still present in the UCSMP books. In the UCSMP series, a high percentage of sixth and seventh grade tasks are related to rational numbers but almost n one are in the eighth grade Algebra text. Not surprisingly, most of the proportional tasks in the eighth grade Algebra text are in the algebra content standard. The progression is less clear in the Math Connects textbooks. The percentage of tasks in the ra tional number content area decreases little from sixth to eighth grade, the percentage of tasks in the geometry and data analyses content areas remain relatively constant, and the percentage of proportion related tasks in the algebra standard increases onl y slightly over the three years. Research cannot determine whether proportionality should be presented through a different content area in sixth grade than it is in eighth grade. Research findings can describe differences between textbooks and the results of this study indicate that, among tasks related to proportionality, there is a progression from rational numbers to algebra in the CMP and UCSMP series. Proportionality as a Connection Between Content Areas Principles and Standards for School Mathematics suggest that proportionality can be used as a theme to integrate various topics. The Principles and Standards state the following: Curricular focus and integration are also evident in the proposed emphasis on proportionality as an integrative theme in the middle grades mathematics
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183 program. Facility with proportionality develops through work in many areas of the curriculum, including ratio and proportion, percent, similarity, scaling, linear equations, slope, relative frequency histograms, and p robability (NCTM, 200 0 p. 212). To what degree do the textbooks in this study use proportionality as a connective theme? Because this was not one of the research questions of this study, the study was not designed specifically to answer this question. How ever, based on the analysis of the textbooks, it appears that the connections between content areas results from the capacity of prop ortions to solve problems in diverse situations. L essons that focus on proportions generally incorporate several content ar eas. Algebra is usually involved in these lessons as students learn the cross multiplication procedure. Lessons that focus on proportions often include ratio and rate problems as well as geometry problems related to maps or similar figures. Thus, lessons o n proportions are one way textbooks use proportionality to connect content areas. Most textbook series, including Math Connects and UCSMP, separate content into lessons that each focus on a narrow range of content. For example, the seventh grade Math Conne cts textbook has separate lessons on ratios, rates, rate of change, solving proportions, scale drawings, similar figures, and probability. This may lead students to believe that the topics have nothing in common. In contrast to this, the CMP seventh grade brings together tasks related to percent, proportions, similar figures, scale factors, indirect measurement, rates, ratios, ratio comparison, probability, and sampling in a single investig ation. Having tasks from various content areas in close proximity may help students recognize connections between them.
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184 Problem Types There is little discussion in the existing research literature regarding the difficulty of various problem types or the g rade levels at which each type should be studied. However, based on this study, some generalizations can be made. Alternate F orm and Ratio C omparison Alternate form and ratio comparison problem types tend to accompany a focus on rational numbers. For exam ple, the alternate form problem type occurs primarily when students are asked to simplify fractions or convert between fractions, decimals, and percents. Similarly, in the textbooks analyzed, the ratio comparison problem type appears primarily when student s were asked to compare fractions or decimals or to place fractions and decimals on number lines. According to the Curriculum Focal Points (NCTM, 2006), these skills should be studied in third and fourth grades. Therefore, one might hope to see the alterna te form and ratio comparison problem types common in third to sixth grade textbooks, and less common in seventh and eighth grade textbooks. In fact, in the books in this study, the alternate form and ratio comparison problem types did in fact decrease be tween sixth and eighth grades. This could be seen as an encouraging sign as it may indicate that seventh and eighth graders are assumed to have mastered skills like simplifying fractions and converting between decimals, fractions, and percents. Missing Value The missing value problem type is often associated with rates and ratios as in this example from the seventh grade Math Connects b p. 287). According to the C urriculum Focal Points, sixth
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185 missing value problem type is also associated with similar figures and proportions. According to the Curriculum Focal Points, seventh might expect the missing value problem type to be common in sixth and seventh grade textbooks and less common in eighth grade textbooks. However, findings of the study indicate that the missing value problem type is more common in seventh and eighth grade textbooks than in sixth grade books. As explained above, more missing value probl ems were found in eighth grade books than one might expect given the recommendations of the Curriculum Focal Points The missing value problem type is particularly common in the eighth grade Math Connects textbook, in which 46% of the proportional tasks ar e missing value and in the eighth grade UCSMP textbook, in which 71% of the proportional tasks are missing value In the eighth grade UCSMP textbook, the majority of the missing value tasks were found in lessons on rates, ratios, proportions, and similar figures. Given that the Curriculum Focal Points places these topics in sixth and seventh grades, one might wonder whether these lessons would be more appropriate in the sixth and seventh grade textbooks. In the eighth grade Math Connects textbook, a wide variety of lessons contain missing value tasks including lessons on proportions, similar figures, indirect measurement, scale drawings, and percents. As with the eighth grade UCSMP series, one might wonder whether these lessons would be more appropriate i n the sixth and seventh grade textbooks.
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186 Function Rule The function rule problem type has not been described in the research literature on proportionality, but was identified by the researcher during the course of a pilot study (Appendix A ) In a typical function rule task, students are shown a function table and asked to write the function rule. In sixth grade textbooks, this rule is generally written as an expression, such as 4 x In eighth grade texts, this rule is generally an equation, such as Although there was some variety in the types of tasks code d with the function rule problem type, most seem to be related either to the sixth grade f ocal p oint on algebra uch as 3 x = y grade f ocal p oint on algebra n rule tasks should be fairly common at more than one grade level. However, this was not the case in the textbooks studied. Function rule tasks were fairly rare, with less than 6% of the tasks being of this type. In the CMP texts, they were common in seven th and eighth grade texts, but virtually absent from the sixth grade text. In the Math Connects series, they were present but rare at all three grade levels. Function rule tasks were virtually absent from all three UCSMP texts. When functions are written as equations, the slope and y intercept are clear. Thus, function rule problems may help students distinguish between proportional and nonproportional relationships. Because the function rule problem type has not been described in the research literature, more discussion of this problem type is necessary. If
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187 educators decide that the function rule problem type is an important part of the middle school mathematics curriculum, textbook authors will need to give more attention to it. Qualitative Many researc hers have mentioned qualitative proportional tasks as an important part of proportional reasoning (e.g., Ben Chaim et al., 1998; Lamon, 2007 ). One might expect, therefore, to see qualitative proportion tasks in middle school textbooks. However, they were v irtually absent from the textbooks in this study. Educators should decide whether qualitative proportion tasks are important for middle school students to encounter; if they are, textbook authors will need to pay them more attention. Differences Between Te xtbook Series The three textbook series were chosen because the researcher expected their treatment of proportionality to differ In fact, the three textbook series differ in at least three important ways: (a) the amount of repetition of content among the three grade levels in each series, (b) the degree of emphasis on algebra in the eighth grade textbook, and (c) the degree to which each series seems designed to foster conceptual understanding. The purpose of this discussion is not meant to offer an opini on on the quality of each series but instead to describe the differences between the series. Amount of Repetition Among Grade Levels One of the findings of this study is that content related to rational numbers and proportionality is often repeated in six th seventh and eighth grade textbooks. For example, in the Math Connects series, all three textbooks contain lessons on rates, ratios, and solving proportions. The content in these lessons is very similar. In all three textbooks, the lesson on ratios an
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188 The exercises on rates and ratios in the sixth grade book include more visual representations, but the content covered is virtually identical in all three books; in all three b ooks, students are asked to write ratios in simplest form and to compare ratios. The sixth and seventh grade UCSMP textbooks also contain repetition related to io Comparison Model for Divis i The examples and exercises in these lessons are somewhat different; the sixth grade lessons focus on helping students understand the connection between division, rates, and ratios and the eighth grade textbook contains mo re tasks with variables. The sixth and seventh grade UCSMP the definition of a proportion, how to state a proportion in words, and how to determine whethe r a propo rtion is true or false. A content analysis such as this cannot state how much repetition is desirable. To determine the amount of review needed by students would require a study involving measures of student understanding. However, the similarity of the l essons on rates, ratios, and proportions in the Math Connects series seems excessive; if a lesson is effective in helping students learn material, it seems unnecessary to repeat virtually the same lesson in sixth, seventh, and eighth grades. Algebra in the Eighth Grade Textbooks Textbook authors cannot emphasize all content to the same degree; they must make choices regarding the content on which to focus at each grade level. The results of this study indicate that authors of eighth grade textbooks choose b etween a focus on
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189 algebra or proportionality. I n t he CMP and UCSMP series, a much larger number of tasks were coded in the sixth and seventh grade books than in the eighth grade book. This suggests that the CMP and UCSMP series emphasize proportionality i n sixth and seventh grades Not only do these series have more tasks related to proportionality in sixth and seventh grades, but also, the percentage of tasks related to rational numbers plummets in eighth grade in both series. The topics covered in the CM P and UCSMP eighth grade books clearly indicate that algebra is the focus of these books This suggests that the authors of the CMP and UCSMP textbooks expect students to have developed proportional reasoning and an understanding of rational numbers by the end of seventh grade, which allows them time to study algebra in eighth grade. By contrast, the Math Connects series continues to feature a focus on both rational numbers and proportionality through eighth grade As discussed in a previous section, the Curriculum Focal Points (NCTM, 2006) seem to suggest that eighth grade textbooks should focus on many aspects of algebra, of which proportionality is only one part. Thus, given the recommendations, the focus on algebra in the eighth grade CMP and UCSMP series seems more appropriate than the focus on rational numbers and proportionality in the eighth grade Math Connects textbook, particularly when much of the content related to proportionality in the eighth grade Math Connects textbook is a repetition of material in the sixth and seventh grade textbooks. Procedural Versus Conceptual Understanding Another difference between the series in this study is that the intent of CMP textbooks seems to be to foster a conceptual understanding of rational numbers an d proportionality whereas many of the lessons in the Math Connects series seem likely to
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190 promote procedural understanding of these topics Student understanding was not measured in this study; therefore, it is not possible to state definitively whether a s eries promotes procedural or conceptual understanding However, a close look at the tasks and lessons of these textbooks clearly suggests that they are likely to result in different types of understanding of proportionality Three of the dimensions coded i n this investigation help justify the claim that the CMP series is more likely than the Math Connects series to foster conceptual understanding of proportionality. These dimensions are the solution strategies encouraged, the level of cognitive demand, and the way textbooks distinguish between proportional and non proportional relationships. The UCSMP textbooks in some ways resemble the CMP textbooks. For example, both the CMP and UCSMP textbooks encourage conceptual understanding of the distinctions betwee n proportional and non proportional situations. In other ways, the UCSMP textbooks resemble the Math Connects books. For example, in both the Math Connects and UCSMP textbooks, more than 30% of the tasks that suggest a solution strategy encourage the use of a proportion. Also, the level of cognitive demand of the tasks in the UCSMP textbooks is closer to that of the Math Connects books than the CMP series, a t least for the topic of proportionality. Solution Strategies According to the researc h literature, informal solution strategies such as the unit rate and building up methods may be easier for children to understand than the symbolic and procedural use of proportions and cross multiplication (Cramer et al., 1993). One might expect therefore, that proportionality be introduced through solution strategies that foster conceptual understanding, such as through the use of pictures and manipulatives.
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191 D ifference s exist in the solution strategies encouraged by the three textbooks in this study. The CMP textbooks feature a large percentage of tasks that encourage students to use manipulatives and/or pictures and a small percentage of tasks that encourage the propo rtion strategy. Thus, the CMP series seems to encourage a conceptual understanding of proportionality. The Math Connects textbooks feature a large percentage of tasks that encourage students to use the proportion strategy and thus seem to encourage a proce dural understanding of proportionality In the Math Connects textbooks, the proportion strategy was more common than any other at all three grade levels. This use of symbolism before proportional rea soning has been fully developed may encourage only proced ural understanding and prevent students from understanding proportionality more conceptually. In the UCSMP textbooks, the proportion strategy is also common at all three grade levels, but the decimal and manipulative strategies are also common, as is encou ragement to use technology. In the CMP textbooks, encouragement to use manipulatives or pictures is much more common than the proportion strategy. Thus, the CMP textbooks seem to lean strongly toward conceptual rather than procedural understanding. In fact the CMP textbooks could be criticized for inadequate instruction in procedures; some might argue that the lack of encouragement for the proportion and cross multiplication strategies leaves students without procedures to eff iciently solve routine problem s related to proportionality.
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192 Level of Cognitive Demand Helping support the claim that the CMP textbooks are more likely to foster conceptual understanding is the ir level of cognitive demand. Almost none of the tasks coded were at the Memorization level; thus, the lowest level of demand was the Procedures Without Connections In this type of task, a student is asked to follow a simple procedure which ha s likely been demonstrated For example, the following is a task from the sixth grade CMP textboo k that was coded as Procedures Without Connections 9a p. 23). In the CMP textbooks, 43% of the coded tasks were at this level compared to 65% of the UCSMP tasks and 75% of the Math Conne cts tasks. The highest level of demand is Doing Mathematics According to Stein et al. (2000), Doing M athematics complex and nonalgorithmic thinking (i.e., there is not a predictable, well rehearsed approach or pathway explicitly suggested b y the task, task instructions, or a worked out In the CMP textbooks, 21% of the coded tasks were at this level compared to 7% of the UCSMP tasks and 3% of the Math Connects tasks. Thus, it is clear that the CMP textbooks had a higher leve l of demand than either of the other two series. This is not necessarily a benefit of the CMP textbooks; one could perhaps argue that the level of demand is so high that it might frustrate students and their teachers. Also, higher level tasks take much mor e time to complete, leaving less time for routine tasks that allow students to practice procedures. However, if level of cognitive demand is associated with conceptual understanding, then it seems likely that the CMP textbooks are more likely than the othe r two series to foster conceptual understanding, at least with respect to the topic of proportionality.
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193 Characteristics of Proportional Situations Textbooks have been criticized for years for failing to help students understand when proportional reasoning is appropriate. For example, Cramer et al. (1993) stated, so they can determine when it is appro 169). Recent studies have found that children who have studied proportionality in school apply proportions where they are not appropriate (e.g., Van Dooren et al., 2009). Although researchers seem to re cognize this as a problem, there has been little discussion about how to help children understand the characteristics of proportionality or know when to apply proportional reasoning. Some of the textbooks in this study make an attempt in this direction, a s explained in Chapter 4. However, the number of tasks in each textbook that address the characteristics of proportionality is very small; if a teacher choose s to skip a single lesson, students could never see this material. Additionally, the Math Connects textbooks address certain issue s in only a very procedural way that may not help students develop true proportional reasoning or understand the difference between additive and multiplicative reasoning. Thus, the researcher concludes that although the CMP and UCSMP series have made steps toward helping students understand when proportional reasoning is called for, more attention to this issue is needed. Based on the solution strategies encouraged, the level of cognitive demand, and the way textbooks instruct students to distinguish between proportional and non proportional situations, the researcher believes that the CMP series is likely to promote
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194 concep tual understanding of proportionality and that the Math Connects series is likely to promote procedural understanding of proportionality. The researcher does not claim that one type of understanding is better than the other, just that the textbook series are likely to promote different types of understanding As shown in Table 1 9 the UCSMP series in some ways resembles the Math Connects textbooks and in other ways resembles the CMP series. The classifications in Table 1 9 are somewhat arbitrary; for Chapter 4 indicate that there are substantial differences between the series. Thus, although the classifications have not been defined, the researcher neverthe less believes Table 1 9 illustrates important differences between the series. Table 1 9 Conceptual and Procedural Aspects of Textbook Series CMP Math Connects UCSMP Common Solution Strategies Building Up Propo rtion Proportion Manipulative Decimal Level of Cognitive Demand High Low/Moderate Moderate Proportional versus Nonprop.* Conceptual Procedural Conceptual *Refers to the method by which students are taught to identify proportional situations
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195 Conclusion Textbooks are influential in mathematics classes, affecting the content that is covered and the ways in which that content is presented. As Ball and Cohen (1996) system and are likely to be an effective way of influencing what happens in mathematics classes. Different curricula have been shown to have dif fering effects on student achievement ( Ben Chaim et al. presentation of material seems clear. f proportional reasoning is widely recognized as one of the most important goals of the middle school mathematics curriculum (e.g., Curcio & Bezuk, 1994) This study has shown that there are some similarities between middle f proportionality and also important differences. The series in this study were similar in the fact that all of the sixth grade textbooks focused on helping students understand rational numbers, despite the fact that the Curriculum Focal Points states that this understanding should be achieved in third and fourth grades (NCTM, 2006) The differences between textbook series are striking. For example, proportions are common in the Math Connects sixth grade textbook, but almost absent from the CMP sixth grade book. Another difference is that the presentation of proportionality in the Math Connects textbooks is clearly more procedural than in the other two series, and the CMP series is clearly more conceptual, with its focus on manipulatives and pictures. A thi rd difference is that the level of cognitive demand of the CMP series seems higher than
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196 that of the other two series, with over 20% of the CMP tasks classified at the highest level of cognitive demand. The results of this study show a disconnect between t he research literature on proportionality and a sample of middle school textbooks. The research literature indicates the existence of three problem types : missing value ratio comparison and qualitative Not only were these three problem types insufficien t to describe many of the tasks in textbooks, but the qualitative type is virtually absent from textbooks. The research literature also indicates that students have difficulty distinguishing between proportional and non proportional relationships. Although all of the seventh grade textbooks in the study made an attempt to cover this topic, it was often explicitly addressed in one lesson, and in the case of Math Connects was addressed in a very procedural manner. Finally, the research literature suggests that the building up and unit rate strategies may help students transition toward proportional reasoning before learning the symbolism of proportions and cross multiplic ation. However, the building up and unit rate strategies were virtually absent from most of the textbooks in the study. One would hope to see evidence of progression and development as students move from sixth to seventh to eighth grades. Some evidence of such a progression was present. In all three series, the focus on rational numbers decreased through the three years. The two problem types associated with rational numbers, alternate form and ratio comparison also decreased through the three years. Alon g with the decreased emphasis on rational numbers, one would expect to see an increased focus on algebra in eighth grade. This occurred in the CMP and UCSMP series, but much less so in the Math
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197 Connects series. Progression was also evident in the level of cognitive demand, which increase d from sixth to seventh to eighth grade in all three series. In conclusion, some encouraging signs are present, such as the intertwining of algebra and proportionality that was seen in the eighth grade CMP and UCSMP textboo ks. Perhaps in response to recent literature, all of the series attempt to help students understand the difference between proportional and non proportional situations. In some areas and in some textbooks, improvement is necessary, such as the procedural m anner in which the seventh grade Math Connects textbook explains how to compare ratios to determine if a situation is proportional. Curriculum content analyses such as the one reported here cannot answer certain students devel op an understanding of fractions and decimals in third and fourth grades, as recommended by the Curriculum Focal Points (NCTM, 200 6 ), or in the sixth and seventh grades, as the material is covered in the textbooks in ot answer the question about when students should learn certain topics, but curriculum content analyses can identify areas in which textbooks differ from recommendations, as this study has done. Limitations of the Study The study has several limitations. The first is that only three textbook series were studied and these series may not be representative of all the middle school textbooks used reliability and validity of th e framework used to analyze the textbooks. The third limitation is that textbooks are not the only influence on the mathematical content presented to students. Each of these limitations is described in more detail below.
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198 Limited Sample Size To allow the r esearcher to study the selected texts in depth and devote sufficient attention to each, only three textbook series were used. The results may not apply to other textbook series. An attempt was made to choose textbooks that are widely used; however, because market share data are not available, the percentages of U.S. students using the selected series are not known. The researcher also attempted to select textbooks that were authored under different educational philosophies, thereby capturing some of the div ersity among middle school textbooks. However, many middle school textbooks series are used in the United States, and three series cannot capture all of the diversity. Thus, the limited sample size allow ed an in depth analysis of the selected textbooks, bu t may not represent the curriculum studied by many middle school students. Also, the validity of the researcher ed on her ability to select lessons that present an accurate picture of the coverage of proportionality in each textbook. Many curriculum analyses restrict themselves to only certain sections of textbooks (e.g., Joh nson et al., 2010; Jones, 2004) and this study did the same. The extent to which these selections provide an accurate portrait of the entire book affect s the validity of the conclusions. Limitations of the Framework Another limitation of the study relates to the reliability and validity of the framework that was used to code each task The reliability of the framework refers to the extent to which a rese archer unrelated to this study would code tasks in the same way the researcher would. To ensure reliability, another doctoral student coded 16% of the lessons analyzed by the researcher. Two types of reliability were calculated. One
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199 reflected the extent to which the researcher and graduate chose the same tasks to code and the other type reflected the extent to which they assigned the same codes. Reliability percentages were reported in Chapter 4. The validity of the framework refers to the accuracy with whi ch the framework measures important features of proportional examples and exercises in middle school mathematics textbooks. To ensure validity of the framework, the researcher conducted a thorough literature review on the aspects of proportionality that ar e taught in middle school and based the framework on the research literature. The aspects of proportionality that the research community has deemed important were incorporated into the framework. The researcher acknowledges that not all content related to proportionality was examined. For example, long division is related to proportionality (Lesh et al., 1988), but was not included in the study. However, the definition of proportionality incorporated in the framework is quite broad and captures the vast maj ority of topics related to proportionality. Intended Versus Implemented Curriculum Scholars interested in curriculum have distinguished between the intended and the implemented curriculum, describing the intended curriculum as the set of goals set forth in standards and policy statements and the implemented curriculum as what actually is covered in classrooms (Schmidt et al., 2001; Valverde et al., 2002). Valverde et al. offered this explanation: The inclusion of a learning goal in the intended curriculum d oes not guarantee that it will be covered. Including an intention as a goal does not guarantee that the
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200 opportunity to attain that goal will actually be provided in classrooms but does greatly increase the probability that it will (p. 8). In this study, percentages were used in an attempt to describe the relative emphasis on various content areas, problem types, and solution strategies. However, when tasks are implemented, it is likely that they do not all receive equal attention. Because only student texts were studied, the researcher was unable to account for activities th were not included in the study. A study of stu dent texts is considered to be a study of the intended curriculum whereas a study of what actually occurs in classrooms would be a study of implemented curriculum. Also, the implementation of textbooks in classes is affected by teacher knowledge, beliefs, and many other factors (Tarr et al., 2008). Therefore, textbooks are far from the only influence on student achievement. Although the significance of this study rests on the assumption that textbooks have some influence on student achievement, the research er acknowledges that other influences also have an impact on student achievement. Recommendations Two types of recommendations are offered below: recommendations for curriculum developers and recommendations for future research. Recommendations for Curric ulum Developers The researcher has t wo recommendations for curriculum developers. First, curriculum developers should consider which content related to proportionality should appear at more than one grade level; content should not be repeated unless there is a
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201 purpose behind the repetition. In some of the textbook series in this study, repetition appear s in two different forms. One, particularly in the Math Connects series, proportional ity i s presented primarily through rational numbers at all three grade levels. It is not clear whether a conceptual understanding of fractions and decimals should be achieved in third and fourth grades, as the Curriculum Focal Points (NCTM, 200 6 ) suggest or at the sixth grade level, as the CMP series suggests, but the repetition of rational number content in sixth, seventh, and eighth grades is likely unnecessary. The second form of repetition occurs when lessons that a re very similar to each other a re pla ce d at multiple grade levels. When similar lessons appear at more than one grade level, there should be a logical progression and development through the grade levels rather than mere repetition. As discussed in an earlier section, in the Math Connects and UCSMP series, multiple textbooks in each series contain lessons on ratios, rates, and proportions. In some cases, the presentation of the content d oes not vary significantly from grade level to grade level. Second, curriculum developers should include more material designed to help students understand the differences between additive and multiplicative reasoning This material should foster more than just procedural understanding. Most of the textbooks in the study contain material designed to help stud ents understand these differences. Some of the textbooks contain visual representations that may help students develop conceptual understanding of the difference between additive and multiplicative reasoning. An sixth grade CMP However, in most books, only a few exercises explicitly
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202 address the issue of additive versus multiplicative reasoning. In some textbooks, these exercises appear in unit reviews which may or may not be assigned by the teacher. Recommendations for Future Research The researcher has identified three areas in need of additional research : a framework for measuring cognitive demand, proportional reasoning problem types, and from informal to more symbolic solution strategies. In this study and others (e.g., Jones, 2004), the framework developed by Stein et al. (2000) was used to measure cognitive demand. However, this framework was originally developed for the context of teac her education rather than curriculum analysis. Not surprisingly, then, its re liability was in some instances lower than would be desired, both in this study and others (e.g., Jones). The level of cognitive demand is an important aspect of tasks and mathema tics education researchers need a reliable way to measure it. The development of a framework, or the refinement of an existing one, would be a valuable contribution. Researchers seem to have accepted without question that there are two or three types of p roportional reasoning problems: ratio comparison, missing value, and qualitative problems. Ratio comparison and missing value problems are indeed common in middle school textbooks. However, not only are qualitative problems virtually absent from textbooks, but there are additional problem types that have not been described in the research literature. This study has provided examples of these additional types and also connected various problem types to content areas. Further investigation of the problem type s actually found in textbooks would be beneficial.
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203 A few studies have indicated that certain solution strategies may seem more natural to students than others. For example, the unit rate strategy is often used by children who have not had instruction in p roportionality (Lamon, 1993). Additional research is needed to discover how informal strategies can help students transition to proportions and cross multiplication. Some of the textbooks in this study used an equivalent fractions method of solving proport ions before introducing cross multiplication; whether the equivalent fractions method is an effective transition between informal and more symbolic reasoning is unknown.
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204 R EFERENCES American Association for the Advancement of Science. (2000). Middle grades mathematics textbooks: A benchmarks based evaluation Washington DC: Author. Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is or might be the role of curriculum materials in teacher learning and instructional reform? Educational R esearcher, 25 (9), p. 6 8, 14. Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), H andbook of research on mathematics teaching and learning (pp. 296 333). New York: Macmillan. Behr, M. J., Les h, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91 126). New York: Academic Press. Ben Chaim, D., Fey, J. T., Fitzger ald, W. M., Benedetto, C. & Mi ller, J. (1998). Proportional reasoning among 7 th grade students with different curricular experiences. Educational Studies in Mathematics, 36 247 273. Billstein R., & Williamson J. (2008). Math Thematics Book 1 Evanston, IL: McDougal Littell. Boston, M. D. & Smith, M. S. (2009). Transforming secondary mathematics teaching: classrooms. Journal for Research in Mathematics Education, 40 119 156.
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205 Bright, G. W., Joyner, J. M., & Wal lis, C. (2003). Assessing proportional thinking. Mathematics T eaching i n the Middle S chool, 9 166 172. Brown, S. A., Breunlin, R. J., Wiltjer, M. H., Degner, K. M., Eddins, S. K., & Edwards, M. T., et al. (2009). University of Chicago School M athematics P roject: A lgebra (3 rd ed.) Chicago: Wright Group/McGraw Hill Cai, J., Lo, J. J., & Watanabe, T. (2002). Intended treatments of arithmetic average in U.S. and Asian school mathematics textbooks. School Science and Mathematics, 102 391 404. Che, S. M. (200 9). Giant pencils: Developing proportional reasoning. Mathematics Teaching in the Middle School, 14 404 408. Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In G. Harel & J. Confre y (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 29 3 330) Albany, NY: State University of New York Press. Cramer, K., Post, T., & Currier, S (1993). Learning and teaching ratio and proportion: Research implications. In D. T. Owens (Ed.) Research ideas for the classroom: Middle grades mathematics (pp.159 178). New York: Macmillan. Curcio, F. R., & Bezuk, N. S. (1994). Curriculum and evaluation standards for school mathematics addenda series, grades 5 8: U nderstanding rational numbers and proportions Reston, VA: National Council of Teachers of Mathematics. Day, R., Frey, P., Howard, A. C., Hutchens, D. A., Luchin, B., McClain, K., et al. (2009a). Math connects: Concepts, skills, and problem solving Course 1 Columbus, OH: Glencoe/McGraw Hill.
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206 Day, R., Frey, P., Howard, A. C., Hutchens, D. A., Luchin, B., McClain, K., et al. (2009 b ). Math connects: Concepts, skills, and problem solving Course 2 Columbus, OH: Glencoe/McGraw Hill. Day, R., Frey, P., Howard, A C., Hutchens, D. A., Luchin, B., McClain, K., et al. (2009 c ). Math connects: Concepts, skills, and problem solving Course 3 Columbus, OH: Glencoe/McGraw Hill. Dole, S (2000). Promoting percent as a proportion in eighth grade mathematics. School Science and Mathematics, 100 380 389. Doyle, W. (1979). The tasks of teaching and learning in classrooms : R&D report no. 4103 Austin, TX: Research and Development Center for Teacher Education, The University of Texas at Austin. Doyle W. (1983). Academic work. Review of Educational Research, 53 159 199. Fisher, L. C. (1988). Strategies used by secondary mathematics teachers to solve proportion problems. Journal for Research in Mathematics Education, 19 157 168. Frost, J. H., & Dornoo, M. D. (2006). Similar sha pes and ratios. Mathematics Teaching in the Middle School, 12 222 224. Grouws, D. A., Smith, M. S., & Sztajn, P. (2004). The preparation and teaching practices of United States mathematics teachers: Grades 4 and 8. In P. Kloosterman & F. K. Lester (Eds.), Results and interpretations of the 1990 2000 mathematics assessments of the National Assessment of Educational Progress (pp. 221 267). Reston, VA: National Council of Teachers of Mathematics.
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207 Hirsch, C. R. (2007). Curriculum materials matter. In C. R. Hir sch (Ed.), Perspectives on the design and development of school mathematics curricula (pp. 1 5). Reston, VA: National Council of Teachers of Mathematics. Heinz K., & Sterba Boatwright, B. (2008). The when and why of using proportions. Mathematics Teacher, 101 528 533. Hodges, T. E., Cady, J. & Collins R. L (2008). Fraction representation: The not so common denominator among textbooks. Mathematics Teaching in the Middle School, 14 78 84. Huntley, M. A (2008). A framework for analyzing differences across mathematics curricula. NCSM Journal, Fall, 2008 10 17. Johnson, G. J., Thompson, D. R., & Senk, S. L. ( 2010 ). A f ramework f or investigating proof related r eason ing in high school mathematics t extbooks Mathematics Teache r, 103 410 418. Jones, D. L. (2004). Probability in middle grades mathematics textbooks: An examination of historical trends, 1957 2004 Doctoral dissertation, University of Missouri. Jones, D L., & Tarr, J. E. (2007). An examination of the levels of cognitive demand required by p robability tasks in middle grades mathematics textbooks. Statistics Education Research Journal, 6 (2), 4 27. Kaput, J., & West, M. M. (1994). Missing value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Con frey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235 287). Albany, New York: State University of New York Press.
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208 Karplus, R., Karplus, E., Formisano, M., and Paulsen, A (1978). Proportional reasoning and control of variables in seven countries. In J. Lochhead and J. Clement (Eds.), Cognitive process instruction: Research on teaching thinking skills (pp. 47 104). Philadelphia, PN: The Franklin Institute Press. Karplus, R., Pulos, S., & Stage, E. K. (1983). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 45 91). New York: Academic Press. Kouba, V. L., Carpenter, T. P., & Swafford, J. O. (1989). Number and operations. In M. M. L indquist (Ed.), Results from the fourth mathematics assessment of the N ational Assessment of Educational Progress (pp. 64 93). Reston, VA: National Council of Teachers of Mathematics. Kouba, V. L., Zawojewski, J. S., & Struchens, M. E. (1997). What do stud ents know about numbers and operations? In P. A. Kenney and E. A. Silver (Eds.), Results from the sixth mathematics assessment of the N ational Assessment of Educational Progress (pp. 87 140). Reston, VA: National Council of Teachers of Mathematics. Lamon S. J. (1993 a tacognitive processes. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 131 156). Hillsdale, NJ: Lawrence Erlbaum Associates. Lamon S. J. Jo u r nal for Research in Mathematics Education, 24 41 61.
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209 Lamon, S. J. (1994). Ratio and proportion: Cognitive foundations in un itizing and norming. In G. Harel and J. Confrey ( Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89 122). Albany, NY: State University of New York Press. Lamon, S. J. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructio nal strategies for teachers Mahwah, NJ: Lawrence Erlbaum Associates. Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teac hing and learning (pp. 629 667). Charlotte, NC: Information Age. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006a). Connected m athematics2: Variables and patterns Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006b). Connected m athematics2: Stretching and shrinking Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006c). Connected m athematics 2: Comparing and scaling Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006d). Connected m athematics2: Moving straight ahead Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Philli ps, E. D. (2006e). Connected m athematics2: Filling and wrapping Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006f). Connected m athematics2: What do you expect ? Boston: Pearson.
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210 Lappan, G., Fey, J. T., Fitzg erald, W. M., Friel, S. N., & Phillips, E. D. (2006g). Connected m athematics2: Thinking with mathematical models Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006h). Connected m athematics2: Say it with symbo ls Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006i). Connected m athematics2: Samples and populations Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Frie l S. N., & Phillips, E. D. (2009a ). C onnected m athematics2: Bits and p ieces I Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2009b). Connected m athematics2: Bits and p ieces II Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2009c). Connected m athematics2: Bits and p ieces III Boston: Pearson. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2009d). Connected m athematics2: How l ikely i s i t ? Boston: Pearson. Lappan, G., Phillips, E. D., & Fey, J. T. (2007). The case of Connected Mathematics. In C. R. Hirsch (Ed.), Perspectives on the design and development of school mathematics curricula (pp. 67 80). Reston, VA: National Council of Teachers of Mathematics. Lar son, R., Boswell, L., Kanold, T., & Stiff, L. ( 2004 ). McDougal Littell middle school math: Course 1 Boston: McDougal Littell. Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In M. Behr & J. Hiebert (Eds.), Number concepts and operations in the middle grades (pp. 93 118). Reston, VA: National Council of Teachers of Mathematics.
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211 for a missing value proportion task. Proceedings of the 28 th conference of the International Group for the Psychology of Mathematics Education Volume 3 (pp. 265 272). Martinie, S. L., & Bay Williams, J. M. (2003a) understanding of decimal fractions using multiple representations. Mathematics Teach ing in the Middle School, 8 244 247. Martinie, S. L & Bay Williams, J. M. (2003 b ). Using literature to engage students in proportional reasoning. Mathematics Teaching in the Middle School, 9 142 148. McConnell, J. W., Feldman, C. H., Heeres, D., Kallemeyn, E., Ortiz, E., Winningham, N., et al. (2009). The University of Chicago School Mathematics Project pre transition mathematics (3 rd ed.). Chicago: Wright Group/McGraw Hill. McKenna, C., & Harel, G. (1990). The effect of order and coordination of the problem quantities on the difficulty of missing value problems. International Journal of Mathematical Education in Science and Technology, 21 589 593. Macmillan/McGraw Hill/Glencoe. (2009). Pre development research: The research base for preK 12 mathe matics. Columbus, OH: Author. Moss J., & Caswell, B (2004). Building percent dolls: Connecting linear measurement to learning ratio and proportion. Mathematics Teaching in the Middle School, 10 68 74. National Commission on Excellence in Education. (198 3). A n ation at r isk : The imperative for educational reform Washington, DC: U.S. Government Printing Office.
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212 National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics Reston, VA: Author. National Coun cil of Teachers of Mathematics. (2000). Principles and standards for school mathematics Reston, VA: Author. National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics Reston, VA: Author. N ational Research Council. (2004). On evaluating curricular effectiveness: Judging the quality of K 12 mathematics evaluations. Washington, D. C.: National Academies Press. Noelting, G (1980a). The development of proportional reasoning and the ratio concept. Part 1 Differentiation of stages. Educational Studies in Mathematics, 11, 217 253. Noelting, G (1980b). The development of proportional reasoning and the ratio concept. Part 2 Problem structure at successive stages: Problem solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11, 331 363. Parker, M reasoning activities for future teachers. Mathematics Teaching in the Middle School, 4 286 289. Parke r, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65 421 481. Post, T. R., C ramer, K. A., Behr, M., Lesh, R., and Harel, G (1993). In T. P. Carpenter, E. Fennema, & T. A. Romberg. (Eds.), Rational numbers: An integration of research (pp. 327 362). Hillsdale, NJ: Lawrence Erlbaum Associates.
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213 Reys, B. J., Di n gman, S., Olson, T., Sutter, A, Teuscher, D., & Chval, K. (2006). Analysis of K 8 number and operation grade level learning expectations In B. J. Reys (Ed.), The intended mathematics curriculum as represented in state level curriculum standards: Consensus or confusion ? (pp. 15 58). Charlotte, NC: Information Age Publishing. Ridgway, J. E., Zawojewski, J. S., Hoover, M. N., & Lambdin, D. V (2003). Student attainment in the Connected Mathematics curriculum. In S. L. Senk & D. R. Thompson (Eds. ) Standards based school mathematics curricula: What are they? What do students learn? (pp.193 224). Mahwah, NJ: Lawrence Erlbaum. Robinson, E. E., R obinson, M. F., & Maceli, J. C. ( 2000). In M. J. Burke & F. R. Curcio (Eds.), Learning mathematics for a new century (pp. 112 126). Reston, VA: National Council of Teachers of Mathematics. Rupley, W. H. (1981). The e ffects of numerical characteristics on the difficulty of proportional problems Doctoral dissertation, University of California, Berkeley. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E., Cogan, L. S., & Wolfe, R. G. (2001). Why schools matter: A cross national comparison of curriculum and learning San Francisco, CA: Jossey Bass. Seeley, C., & Schielack, J. F. (2007). A look at the development of ratios, rates, and proportionality. M athematics T eaching in the M iddle S chool 13 140 142. Senk, S. L., & Thompson, D. R. (200 3). School mathematics curricula: Recommendations and issues In S. L. Senk and D. R. Thompson (Eds.), Standards based school mathematics curricula: What are they? What do students learn? (pp. 3 27). Mahwah, NJ: Lawrence Erlbaum Associates.
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214 Silver, E. A., Urban Education, 30 476 521. Singh, P. (200 0 ). Understanding the concepts of proportion and ratio constructed by two grade si x students. Educational Studies in Mathematics, 43 271 292. B. Litwiller and G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 3 17). Reston, VA: National Council of Teachers of Mathematics. Child Development, 65 1193 1213. (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3 39) Albany, NY: State University of New York Press. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A ( 2000) Implementing standards based mathematics instruction: A casebook for professional development New York: Teachers College Press. Tarr, J. E., Reys, R. E., Reys, B. J., Chavez, O., Shih, J., & Osterlind, S. J. (2008). The impact of middle grades mathematics curricula and the classroom learning env ironment on student achievement. Journal for Research in Mathematics Education, 39 247 280. Telese, J. A. & Abete, J. (2002). Diet, ratios, proportions: A healthy mix. Mathematics Teaching in the Middle School 8 8 13.
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215 Thompson, D. R., & Senk, S. L. (2001). The effects of curriculum on achievement in second year algebra: The example of the University of Chicago School Mathematics project. Journal for Research in Mathematics Education, 32 58 84. Tourniaire, F. (1986). Proportions in eleme ntary school. Educational Studies in Mathematics, 17 410 412. Usiskin, Z. (2007).The case of the University of Chicago School Mathematics Project secondary component. In C. R. Hirsch (Ed.), Perspectives on the design and development of school mathematic s curricula (pp. 173 182). Reston, VA: National Council of Teachers of Mathematics. Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R., T. (2002). According to the book: Using TIMSS to investigate the translation of policy into pra ctice through the world of textbooks Boston: Kluwer Academic Publishers. Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralizati on. Cognition and Instruction, 23 57 86. Van Dooren, W., De Bock, D., Evers, M., & Verschaffel, L use of proportionality on missing value problems: How numbers may change solutions. Journal for Research in Mathematics Education, 40 187 211. Viktora, S. S., Cheung, E., Highstone, V., Capuzzi, C., Heeres, D. Metcalf, N. et al. (2008). The University of Chicago School Mathematics Project transition mathematics (3 rd ed.). Chicago: Wright Group/McGraw Hill.
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216 Watson, J. M. & Shaugh nessy, J. M (2004). Proportional reasoning: Lessons from research in data and chance. Mathematics Teaching in the Middle School, 10 104 109. Yan, Z. & Lianghuo, F. (2006). Focus on the representation of problem types in intended curriculum: A comparison of selected mathematics textbooks from mainland China and the United States. International Journal of Science and Mathematics Education, 4 609 626. Zawojewski, J. S., & Sha ughnessy, J. M. (2000). Data and chance. In E. A. Silver & P. A. Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp.235 268). Reston, VA: National Council of Teachers of Mathematics.
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217 Appendix A : Pilot Study
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218 PROPORTIONALITY IN TWO SIXTH GRADE MATHEMATICS TEXTBOOKS : A PILOT STUDY In 1989, the NCTM published Curriculum and Evaluation Standards for School Mathematics which, among other things, described thirteen curriculum standards. One of include the investigation of mathematical connections so that students can see mathematics published Principles and Standards for School Mathematics which proposed certain mathematical topics, including proportionality, be used to help students investigate connections. NCTM (2000) st proposed emphasis on proportionality as an integrative theme in the middle grades In order for proportionality to function effectively as an integrative theme, st udents must be able to reason proportionally. However, researchers agree that proportional reasoning is a difficult skill to acquire. For example, Karplus, Pulos, and Stage (1983) found that about 30% of the sixth and eighth grade students studied never u sed proportional reasoning and Lamon (2007) estimated that about 90% of adults do not reason proportionally. Researchers believe that part of the difficulty lies in the complexity inherent in this type of reasoning, but also that curriculum resources do no t present the topic as well as they could (e.g., Cramer, Post, & Currier, 1993).
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219 To determine how proportionality is treated in middle school mathematics textbooks, the researcher designed a curriculum analysis framework. In this section, the researcher d escribes a pilot study that was undertaken to test the framework. proportionality, the researcher used several variations of it to analyze the treatment of proportionality in two sixth grade textbooks: Middle School Math Course 1 (Larson, Boswell, Kanold, & Stiff, 2004) and Math Thematics Book 1 (Billstein & Williamson, 2008). Although these books are both published by McDougal Littell, they represent different educational philosophies. Middle School Math was not developed with funding by the Nation al Science Foundation (NSF) whereas Math Thematics is based on the standards and principles of the NCTM and was developed through funding by the NSF. Methods In the following paragraphs, the researcher will describe how she selected lessons to be included in the pilot study and how she decided which tasks would be coded using the framework. Then, a description of the framework is provided. Lesson Selection The researcher selected lessons from each textbook that seemed the most likely to involve proportion al reasoning. These lessons were drawn from all content areas, including algebra, data analysis/probability, geometry/measurement, and rational numbers. Within the algebra standard, lessons on function rules, rate of change, slope, and solving proportions were selected for the study. Within the data analysis and probability standard, lessons on circle graphs and using probability to make predictions were selected. Within the geometry and measurement standards, lessons on area,
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220 measurement conversions, simil ar figures, and volume were selected. Within the number and operations standard, lessons on equivalent fractions and lessons on converting between decimals, fractions and percents were selected, but lessons on fraction computation were excluded. Of the 93 lessons in the Middle School Math textbook, 17 (18%) were selected and analyzed. The majority of these lessons were in the rational number content area. Of the 43 sections in the Math Thematics textbook, 15 (35%) were selected and analyzed. The largest per centage of these lessons related to geometry and measurement. Task Selection The researcher analyzed every example and every exercise in the selected lessons. However, not all examples and exercises in these lessons involved proportionality. Only tasks tha t did involve proportionality were coded. A task was defined as a labeled were not included. In the Middle School Math textbook, 485 tasks were coded and from the Mat h Thematics textbook, 827 tasks were coded. This information is summarized in Table A1. Table A1 Lessons and Tasks Included in the Pilot Study Lessons Tasks Number Percent Number Percent Total Included Included Total* Included Included Math Thematics 43 17 40 1389 827 60 Middle School Math 93 17 18 704 485 69 *The total number of tasks refers to the number of tasks within the selected lessons.
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221 The percentage of lessons coded in the Middle School Math textbook was far less than the percentage of lessons coded in the Math Thematics textbook. This is due to the fact that the Middle School Math textbook includes four ch apters (27 lessons) on computation with rational numbers. Because these lessons were excluded from the study, a smaller percentage of lessons was coded. The Framework The framework was based on research literature that identified various problem types and solution strategies (e.g., Cramer et al., 1993). Because research has suggested that students have difficulty distinguishing between proportional and nonproportional situations, the framework notes whether tasks in textbooks point out the characteristics of proportional situations. Because the NCTM has stated that proportionality could be used to connect various mathematical topics, the framework looks for proportionality in several different content areas. Each example was coded along five dimensions: pro blem type, content area, solution strategy suggested, whether a visual representation was present, and whether the example pointed out the characteristics of proportional situations. Each exercise was coded along six dimensions: the five listed above and a lso the level of cognitive demand (Stein et al., 2000). Problem Types Each task related to proportionality was classified into one of the following problem types: alternate form function rule missing value ratio comparison qualitative and other
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222 Alternate form problem type. In tasks of this problem type, students are asked to put a fraction, decimal, or percent into another form. Figure A1 shows an example of this type. Figure A1 Example of problem type alternate form Although the problem type alternate form was not identified in the research literature, problems of this type involve proportionality and appear in middle school textbooks. Therefore, the problem type alternate form is a necessary part of the framework. These task s are similar to missing value tasks, but in alternate form tasks students must find two values, not one. Function rule problem type. In tasks of this problem type, students are asked to find a rule that relates members of one set to members of a second s et. For example, in Write two fractions that are equivalent to Multiply the numerator and denominator by 2. Multiply the numerator and denominator by 3. Answer The fractions and are equivalent to (Larson et al., 2004, p. 228).
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223 Figure A2 Exercise of the problem type function rule Although this problem type was not identified in the research literature on proportional reasoning, problems of this type are related to proportionality and appear in middle school textbooks. Therefore, the pr oblem type function rule is a necessary part of the framework. Missing v alue problem t ype In a missing value task, students are given three out of four numbers that form a proportion and are asked to find the fourth. This was one of the problem types ide ntified in the research literature as an important part of proportional reasoning. Figure A3 Example of the problem type missing value a. Copy and complete the table. Term number 1 2 3 4 ? ? ? ? Term 12 24 36 48 ? ? ? ? b. How are the term numbers and terms related? c. Write an equation for the rule for the sequence. Use t for the term and n for the term number. d. Use the equation to find the 30 th term in the sequence. Solve the proportion Solution Write the cross products. They are equal. Write the related division equation. Divide. Answer The solution is 45 (Larson et al., 2004, p. 385).
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224 Ratio comparison problem t ype In a ratio comparison problem, students are given two ratios and are asked which is larger or if the ratios are equal. This was one of the problem types identified in the research literature as an important part of proportional reasoning (e.g., Lamon, 2007). Figure A4 shows an example of this type. Figure A4 Example of the problem type ratio comparison Qualitative problem type In the research literature, questions like the following were identified as qualitative yesterday, would her running speed be (a) faster, (b) slower, (c) exactly the same, (d) not enough information to t no tasks resembling this one was found in either of the two sixth grade textbooks examined in this pilot study. Solution Strategies In middle school mathematics textbooks, most examples are accompanied by a suggested solution strategy. Some exercises also suggest a solution method to students, although most do not. Each task related to proportionality was classified into one of the Football Allen completes of his pass attempts. Mike completes 7 out of every 10 pass attempts. Who has the better record? Solution Write each ratio as a decimal. Then compare the decimals. Allen: Mike: 7 out of 10 = Answer Because 0.75 > 0.7, Allen has the better record. (Larson et al., 2004, p. 375)
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225 following solution strategies: building up decimals pro portion unit rate other and none The proportion solution strategy was sub divided into two categories: proportion: cross multiplication and proportion: multiply or divide B uilding up strategy. Figure A5 shows an example of the building up strategy. The example in the textbook featured a picture of a canoe with measurements indicating correspond to 1 inch, 2 i nches, and 3 inches on the scale drawing. How long is the actual canoe? Figure A5 Example of the solution strategy building up Decimals. Tasks in some textbooks instruct students to compare fractions by converting them to decimals. The example in Figure A4 was coded with the solution strategy decimals Manipulatives Some tasks suggest that students use drawings or manipulatives to compare fractions and mixed numbers. These tasks were coded with the manipulatives Solution Make a table. The scale on the drawing is 1 in : 5 ft. Each inch on the drawing represents 5 feet on the canoe. Scale drawing length (inches) Length 5 Actual canoe lengt h (feet) 1 1 5 5 2 2 5 10 3 3 5 15 Answer The actual canoe is 15 feet long. (Larson et al., 2004, p. 68)
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226 solution strategy. The exercise in Figure A6 was coded with the solution strategy manipulatives (Billstein & Williamson, 2008, p. 42). Figure A6 Exercise of the solution strategy manipulatives Proportion The example in Figure A7 demonstrates the use of the proportion strategy. Figure A7 Exercise of the solution strategy proportion. Unit rate One way to solve missing value problems is through the use of a unit rate. Using the method, students conver t the given rate to one that has a denominator of one. For example, to solve the problem in Figure A8 students would divide 3,000 by 60 and 180 by 15. Find the missing term in the proportion (Billstein & Williamson, 2004, p. 401)
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227 Figure A 8 Exercise of the solution strategy unit rate A task was coded as unit rate only if it suggested that students use a unit rate strategy to solve a problem. Tasks that ask students to state the unit rate or to convert a rate into a unit rate were not coded with the unit rate solution strategy because in these situations, the unit rate is not used as a solution strategy. For example, in a sixth grade average rate of s mentioned, this exercise was coded as no solution strategy because the unit rate in this example is not used as a solution strategy. Other Tasks in which a solution strategy is suggested that does not fit neatly into one of the above categories were cod ed as other For example, consider comparison of fractions, such as determining whether or is larger. This could be done by converting both fractions to decimals, which would be coded with the solution strategy decimals It could also be accomplished by using a common denominator, i.e., converting both fractions to eighteenths. In the pilot study, the researcher found few tasks in which students were asked to compare fractions by using common denominators. Therefore, the framework does not include a c ode for it. On average, a Ruby throated Hummingbird beats its wings about 3000 times in 60 seconds. A Giant Humm ingbird beats its wings an average of about 180 times in 15 seconds. Which bird beats its wings faster? (1 Find the unit rate for the Ruby throated Hummingbird. (2 Find the unit rate for the Giant Hummingbird. (3 Compare the unit rates (Larson et al., 200 4, p. 381).
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228 T asks in which multiple solution strategies were suggested were also coded as other For example, an exercise in a sixth 2008, p. 292). Although in some curriculum analyses, tasks that fit into several categories receive multiple codes, the percent of tasks in which multiple solution strategies were offered was far less than 1%. Thus, in the pilot study, it was not problem atic to code these tasks as other When the framework is applied to other series, it may be necessary to assign multiple codes to tasks in which multiple sol ution strategies are suggested depend ing on the number of these tasks that is encountered. Results In this section, results related to the following three topics are reported: a) the content area of lessons related to proportionality, b ) proportionality in the Middle School Math textbook, and c) proportionality in the Math Thematics textbook. Compariso ns between the two textbooks are made in the Discussion section. Content Area of Lessons Related to Proportionality As indicated in Figure A9, in the Middle School Math textbook, more lessons related to proportionality were in the Number and Operations standard than any other. In the Math Thematics textbook, more lessons related to proportionality were in the Geometry/Measurement standard than any other. Specifically, i n the Middle School Math textbook, the researcher found 17 lessons related to proportionality: seven (41%) in the Number and Operations standard, four (24%) in the algebra standard, four (24%) in the Geometry/Measurement standard, and two (12%) in the data analysis standard. In the Math Thematics textbook, the researcher found 17 lessons related to proportionality: five
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229 (29%) in the Number and Operations standard, three (18%) in the Algebra standard, eight (47%) in the Geometry/Measurement standard, and one (6%) in the Data Analysis/Probability standard. Figure A 9 Percent of l essons r elated to p roportionality in e ach c ontent a rea Proportionality in the Middle School Math Textbook In this section, the treatment of proportionality in the Middle School Ma th t extbook is described. First, the examples related to proportionality from this book are discussed. In a later section, the exercises related to proportionality are described. Examples in the Middle School Math Textbook In the lessons selected for the study from the Middle School Math textbook, the researcher found 54 examples, 41 (76%) of which were related to proportionality. As explained below, very few of the examples pointed out the characteristics of proportionality Rational number was the most common content area, followed by 0 5 10 15 20 25 30 35 40 45 50 Algebra Data Analysis Geometry Number Middle School Math MathThematics
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230 geometry/measurement A lternate form and missing value were the most common problem types. The solution strategy most commonly suggested was the proportion strategy. Proportions were more ofte n solved through multiplication or division rather than cross products. Characteristics of p roportionality in Middle School Math examples. Of the 41 examples related to proportionality in the Middle School Math textbook, three (7%) contained material that could help students realize that proportional reasoning is examples were designed to p oint out that area is not related proportionally to the lengths of the sides of a rectangle or to the radius of a circle. The two examples from the lesson used as a scale model for a mural. One example asked students to write a ratio comparing the perimeter of the drawing to the perimeter of the actual mural. The answer provided The o ther example asked students to write a ratio comparing the area of the drawing to hat the relationship between the side lengths of a rectangle and its perimeter is proportional whereas the relationship between the side lengths of a rectangle and its area is not proportional.
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231 Content a rea of Middle School Math e xamples Of the 41 examp les related to proportionality in the Middle School Math textbook, 24 (59%) related to rational numbers, eight (20%) to geometry/ measurement, six (15%) to algebra, and three (7%) to data analysis /probability Problem t ype of Middle School Math e xamples Of the 41 examples related to proportionality in the Middle School Math textbook, 16 (39%) were of the type alternate form 11 (27%) of the type missing value seven (17%) of the type ratio comparison three (7%) of the type function rule three (7%) of the type other and one (2%) of the qualitative problem type. All 41 examples clearly fit into one of the six problem type categories. Solution s trategy of Middle School Math e xamples Of the 41 examples related to proportionality in the Middle School Math textbook, 18 (44%) suggested the proportion strategy, three (7%) suggested the decimals strategy, two (5%) suggested the building up strategy, and 18 (44%) were coded as other None of the examples in the Middle School Math textbook suggested solving a pr oblem through the use of manipulatives or the unit rate strategy. Of the 18 examples that suggested the proportion strategy, 14 (78%) suggested multiplying or dividing both the numerator and denominator by the same number. Of the 18 examples that suggested the proportion strategy, 4 (22%) suggested using cross products. Visual r epresentation s of Middle School Math e xamples An example was coded as having a visual representation if it included a chart, diagram, graph, or picture that could help a student understand the mathematical concepts. Pictures that seemed purely
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232 decorative were not coded. Of the 41 examples related to p roportionality in the Middle School Math textbook, 12 (29%) were accompanied by a visual representation. Exercises in the Middle School Math Textbook In the selected lessons of the Middle School Math textbook, the researcher found 650 exercises, 447 (69%) of which were related to proportionality. Exercises resembled the examples in several ways: few pointed out the characteristics of proportionality, rational number was the most common content area, and few were of the qualitative problem type. Characteristics of p roportionality in Middle School Math exercises. Of the 447 exercises related to proportionality in the Middle School Math textbook, 10 (2%) contained material that could help students realize that proportional reasoning is appropriate i n some situations but not others. In all of these exercises, the differences between proportional and non proportional situations were only hinted at rather than stated explicitly. These exercises were spread throughout the book and related to a variety of content areas. Content a rea of Middle School Math exercises. Of the 447 exercises related to proportionality, 344 (77%) were in the rational number content area 41 (9%) were in geometry/measurement 36 (8%) were in algebra and 26 (6%) were in data anal ysis /probability Thus, in the Middle School Math textbook, the majority of both the examples and the exercises related to proportionality were in the rational number content area. This emphasis on proportionality in the rational number content area wa s ev en more pronounced in the exercises than it is in the examples.
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233 Level of c ognitive d emand of Middle School Math exercises. Of the 447 exercises related to proportionality in the Middle School Math textbook 10 (2%) were at the level M emorization 362 (81%) were at the level Procedures Without C onnections 14 (3%) were at the level Procedures With C onnections and 11 (2%) were at the level Doing M athematics It is not possible to compare exercises to examples because the level of cognitive demand of examples was not coded. Problem t ype of Middle School Math e xercises Of the 447 exercises related to proportionality in the Middle School Math textbook 165 (37%) were of the type missing value 138 (31%) were of the type alternate form 84 (19%) were of the type ratio comparison 32 (7%) were of the other problem type, 18 (4%) were of the function rule type, and 10 (2%) were of the qualitative problem type. Figure A 10 shows a comparison of the problem type of examples and exercises. The figure indicat es that, in the Middle School Math text book, problem types were distributed in approximately the same way in examples and exercises. Problem types emphasized in examples were also emphasized in exercises. Figure A 10 Problem t ype of t asks in the Middle School Math t extbook 0 5 10 15 20 25 30 35 40 45 Examples Exercises
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234 In the research literature, questions like the following were identified as qualitative running speed be (a) faster, (b) slower, (c) exactly the same, (d) not enough information to this in the sixth grade Middle School Math textbook. Nine exercises were coded as qualitative All of these involved examining a circle or bar graph and estimating the fraction of the population in a given category. Solution s trategy of Middle School Math exercises. Of the 447 exercises related to proportionality in the Middle School Math textbook, 401 (90%) did not suggest a solution strat egy, 40 (9%) suggested the proportion strategy, five (1%) were coded as other and one (0.2%) suggested the unit rate strategy. None of the exercises in the Middle School Math textbook suggested the building up decimals or manipulatives strategies. Prop ortionality in the Math Thematics Textbook In this section, the treatment of proportionality in the Middle School Math t extbook is described. First, the examples related to proportionality from this book are discussed. In a later section, the exercises rel ated to proportionality are described. Examples in the Math Thematics Textbook In the 17 lessons selected for the study from the Math Thematics textbook, the researcher found 62 examples, 36 (58%) of which were related to proportionality. Examples in the Math Thematics textbook were similar to examples in the Middle School Math textbook in that few pointed out the characteristics of proportionality, most were related to rational numbers, and few were of the qualitative problem type.
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235 Characteristics of pro portionality in the Math Thematics examples. Of the 36 examples related to proportionality in the Math Thematics textbook, only one contained material that could help students realize that proportional reasoning is appropriate in some situations but not ot that showed why the ratio 10 : 4 is equivalent to the ratio 5 : 2. This example was coded as pointing out w hether proportional reasoning is appropriate because it emphasized that every five cashews are paired with two pretzels, thus emphasizing the constant ratio. Because a constant ratio (or rate of change) is one of the characteristics of proportional situati ons, this example was considered to point out whether proportional reasoning was appropriate. Content a rea of the Math Thematics e xamples Of the 36 examples related to proportionality in the Math Thematics textbook, 23 (64%) were in the rational number content area seven (19%) were in algebra five (14%) were in geometry/ measurement and one (3%) was in data analysis /probability Thus, in both the Middle School Math and Math Thematics textbooks, more than 50% of the examples related to proportionality were in the rational number content area. Problem t ype of the Math Thematics e xamples Of the 36 examples related to proportionality in the Math Thematics textbook, 12 (33%) were of the type alternate form 11 (31%) were of the type ratio comparison eight (22%) were of the type missing value three (8%) were of the type function rule one (3%) was of the qualitative problem type, and one (3%) was of the type other
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236 Solution s trategy of the Math Thematics e xamples Of the 36 examples related to pro portionality in the Math Thematics textbook, 18 (50%) were coded as other six (17%) did not suggest a strategy, five (14%) suggested manipulatives five (14%) suggested the proportion strategy, one (3%) suggested the decimals strategy, and one (3%) sugges ted the cross products strategy. None of the examples in the Math Thematics textbook suggested the building up or unit rate strategies. A large number of examples were coded with the solution strategy other for a variety of reasons. One reason is that the examples in the Math Thematics textbook tended to be longer and contain more verbal explanation than examples in the Middle School Math textbook. Examples in the Math Thematics textbook typically contained several mathematical ideas and were thus harder to classify by solution strategy. In six of the examples, no solution strategy was suggested. Visual r epresentation in the Math Thematics examples. Of the 36 examples related to proportionality in the Math T hematics textbook, 12 (33%) were accompanied by a visual representation. This is slightly greater than the percent of examples in the Middle School Math textbook that were accompanied by a visual representation. In the Math Thematics textbook, many of the examples that included a visual representation were examples that encouraged the use of manipulatives; a picture of the manipulatives was often included to illustrate their use. Exercises in the Math Thematics Textbook The Math Thematics textbooks separat e exercises into three types of exercise sets. of exercises were coded separa tely. In the selected lessons, there were 149 exercises
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237 related to proportionality in the Explorations, 323 in the Practice and Application sets, and 220 in the Extra Skills pages, for a total of 692 exercises related to proportionality. Because these sect ions contained a total of 1,327 exercises, 52% of the exercises in the lessons selected from the Math Thematics t extbook were related to proportionality. Characteristics of Proportionality the Math Thematics exercises. Of the 692 exercises related to prop ortionality in the Math Thematics textbook, 32 (5%) contained material that could help students realize that proportional reasoning is appropriate in some situations but not others. These exercises included 11 of the 149 exercises in the Exploration sectio ns, 13 of the 323 exercises in the Practice and Application sets, and 9 of the 220 exercises in the Extra Skills pages. Thus, the percentage of exercises that emphasized the characteristics of proportional situations was higher in the Exploration sections than in the other two types of sections. Some of the exercises that pointed out the characteristics of proportional situations involved collecting data, making a scatterplot, drawing a line of best fit, and using this line to make predictions. These were c oded as pointing out the characteristics of proportional situations because the line of best fit had a constant slope, which could help students understand that, in proportional situations, the ratio of the variables is constant. Content a rea of the Math Thematics e xercises Of the 692 exercises related to proportionality in the Math Thematics textbook, 430 (62%) were in the rational number content area, 183 (26%) were in geometry/ measurement 64 (9%) were in algebra and 15 (2%) were in data analysis / probability Thus, in both the examples and exercises in the Math Thematics textbook, rational number was the most common content area.
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238 Examples were more likely than exercises to involve algebra and exercises were more likely to involve geometry and measu rement. Problem t ype of the Math Thematics e xercises Of the 692 exercises related to proportionality in the Math Thematics textbook, 260 (38%) were of the type missing value 195 (28%) were of the type alternate form 179 (26%) were of the type ratio com parison 19 (3%) were of the function rule type, three (0.4%) were of the qualitative problem type, and 36 (4%) were of the other problem type. As in the Middle School Math textbook, problem types were distributed in approximately the same way in examples and exercises. In the research literature, questions like the following were identified as qualitative running speed be (a) faster, (b) slower, (c) exactly the same, (d) not enough information Middle School Math textbook, no tasks in the Math Thematics book resembled the qualitative tasks described in the literature. Four tasks in the Math Thematics book were coded as qualitative Three of these involved estimating the fraction of a circle graph represented by a given sector and one required students to examine an arrangement of rectangles and triangles and estimate the fraction of the figure represented by each shape. Although fr actions are quantitative, students needed to use visual estimation skills rather than computation to arrive at these answers; thus, the tasks were coded as qualitative In the Math Thematics textbook, the percentages of examples and exercises of each prob lem type were similar. Approximately one third of the tasks were of the type alternate form approximately one third were of the type missing value and slightly less
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239 than one third were of the type ratio comparison In both the examples and exercises, fu nction rule qualitative and other tasks were uncommon. Level of c ognitive d emand of Math Thematics exercises. Of the 692 exercises related to proportionality, none were at the M emorization level, 516 (75%) were at the level Procedures w ithout C onnection s 141 (20%) were at the level Procedures w ith C onnections and 35 (5%) were at the level Doing M athematics Exercises in the Explorations sections were more likely to be at the level Procedures with C onnections than exercises in the other two types of exe rcise sets. Problems in the Extra Skills sets were almost exclusively at the level Procedures without C onnections Solution s trategy of the Math Thematics e xercises Of the 692 exercises related to proportionality in the Math Thematics textbook, 563 (81%) did not suggest a solution strategy, 58 (8%) were coded as other 28 (4%) suggested using manipulatives, 24 (3%) suggested the proportion strategy, 14 (2%) suggested using decimals three (0.4%) suggested the building up strategy, and two (0.3%) suggested the unit rate strategy. Discussion This discussion section is divided into six parts: a) the prevalence of proportionality, b) the extent to which textbooks discuss the appropriateness of proportional reasoning, c) content areas, d) p roblem types, e) solution strategies, and f ) level of cognitive demand. Prevalence of Proportionality The extent of each textbook related to proportionality can be analyzed in two different ways: (a) the percentage of lessons related to proportionality and (b) the percentage of tasks related to proportionality.
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240 Percentage of Lessons Related to Proportionality The percentage of lessons related to proportionality may indicate how integrated proportionality is into the curriculu m. A high percentage of lessons related to proportionality may indicate that it is covered throughout the school year whereas a small percentage of lessons related to proportionality indicates that it is concentrated in a few lessons or chapters. Eighteen percent of the lessons in the Middle School Math textbook and 35% of the lessons in the Math Thematics textbook were related to proportionality. Thus, the percentage of lessons in the Math Thematics textbook related to proportionality was almost twice as h igh as the percentage in the Middle School Math textbook. This indicates that proportionality is spread throughout the Math Thematics textbook and is more concentrated in the Middle School Math textbook. NCTM (2000) suggested that as an integrative theme in the middle grades mathematics to appear in a significant percentage of lessons. The results of this pilot study may suggest that proportion ality is used to illustrate connections between mathematical concepts in the Math Thematics textbook more than in the Middle School Math textbook. Percentage of Tasks Related to Proportionality As indicated in Figure A11, 53% of the tasks in selected lessons in the Math Thematics textbook and 69% of the tasks within selected lessons in the Middle School Math textbook were related to proportionality. Thus, although the Math Thematics textbook had a much higher percentage of les sons related to proportionality than did the Middle School Math textbook, the percentage of tasks related to proportionality within
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241 the selected lessons was lower in the Math Thematics textbook. This is consistent with the finding discussed above, that pro portionality is more spread out in the Math Thematics textbook and more concentrated in the Middle School Math textbook. Figure A 1 1 Percent of t asks in s elected l essons r elated to p roportionality Calculating the percentage of tasks within an entire b ook that relate to proportionality would require counting the number of tasks in an entire book, which was not done in this study. However, this percentage can be estimated by multiplying the percent of tasks coded in the selected lessons by the percent of lessons selected. For example, in the Middle School Math textbook, 18% of the lessons and 69% of the tasks within them were coded. One can multiply these probabilities to arrive at an estimate that 12% of the tasks in the Middle School Math are related to proportionality. Using this method, an estimated 19% of the tasks in the Math Thematics textbook are related to proportionality. 0 10 20 30 40 50 60 70 80 Exercises Examples Total Middle School Math Math Thematics
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242 Characteristics of Proportional Reasoning Research findings have indicated that both students and teachers have difficulty recognizing situations in which proportional reasoning is or is not appropriate (Cramer et al., 1993; Van Dooren, De Bock, Hessels, Janssens, & Verschaffel, 2005). Given that many students and teachers attempt to apply proportional reasoning to situations in which it is not appropriate, it might be beneficial for tasks in middle school mathematics textbooks to point out the characteristics of proportional and non proportional s ituations. The researcher has identified three problems with the way textbooks address this issue. First, too few tasks point out the characteristics of proportional situations. Second, in the Middle School Math textbook, there was little diversity in the tasks that did point out the characteristics of proportional situations. Third, tasks that did point out the characteristics of proportional situations did so in only subtle ways; there was little explicit discussion of these characteristics. Too F ew T asks P oint O ut the C haracteristics of P roportional S ituations As illustrated in Table A 2 only small percentages of the tasks related to proportional reasoning seem to be designed to help students recognize the characteristics of proportionality, which may he lp them recognize situations in which proportional reasoning is or is not appropriate. Table A 2 Percentages of Tasks Related to the Appropriateness of Proportional Reasoning Examples Exercises Middle School Math 7 2 Math Them atics 3 5
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243 In the two textbooks combined, only 42 of the 1,139 exercises (3.7%) related to proportionality pointed out these characteristics. Thus, there are likely too few tasks that point out the differences between proportional and non proportional situations. There is L ittle D iversity in the T asks In the Middle School Math textbook there was little diversity in the examples and exercises that did point out the ch aracteristics of proportional situations. For example, in the Middle School Math textbook, all three examples related to this topic involved area of rectangles or circles. There was more diversity in the tasks in the Math Thematics textbook. At least one e xample in the Math Thematics textbook used a diagram to illustrate the meaning of a constant ratio and some exercises involved using a line of best fit to point out that the relationship between the variables was constant. There are F ew E xplicit D iscussio ns of the C haracteristics of P roportional S ituations T asks were coded as pointing out the appropriateness or inappropriateness of proportional reasoning even if the issue was touched on very subtly. Students and teachers would have to be very perceptive to recognize some of these tasks as pointing out the characteristics of proportional situations. Very rarely was an explicit discussion of these characteristics seen in either textbook. Content Areas In both textbooks, most of the examples and most of the exercises related to proportionality were also related to rational numbers. This likely reflects both the the middle school curriculum.
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244 Content Area of E xamples As illustrated in Table A 3 the content area of examples in the Math Thematics textbook was similar to the content area of examples in the Middle School Math textbook. Table A 3 Percent of Examples Related to Proportionality in Each Content Standard Content Area Algebra Data Analysis Geometry Rational Numbers Middle School Math 15 7 20 59 Math Thematics 19 3 14 64 Content Area of E x ercises As illustrated by Table A 4 most of the proportional exercises in the Middle School Math textbook were related to rational numbers. The Math Thematics textbook had many proportional exercises that were related to rational numbers but also featured a considerable number of proportional exercises that were related to the geometry/ measureme nt content area Table A 4 Percent of Exercises R elated to P roportionality in Each Content Standard Content Area Algebra Data Analysis Geometry Rational Numbers Middle School Math 8 6 9 77 Math Thematics 9 2 26 62
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245 The large percentage of examples and exercises that were related to rational any scholars would not consider conversion between decimals, fractions, and percents to involve proportional reasoning. Because the researcher did include this and similar concepts, the percentage of proportional exercises related to rational numbers may be overstated. Problem Types Problem T ype of E xamples The problem types of the examples in the two textbooks were somewhat similar. The Math Thematics textbook contained a smaller percentage of examples of the type alternate form and a larger percentage of examples of the type ratio comparison but the percen tage of examples of the other problem types did not vary greatly between the two textbooks. Problem T ype of Exercises T he problem types of the exercises in the two textbooks were also similar. The Math Thematics textbook contained a larger percentage of e xercises of the type ratio comparison but the percentage of examples of the other problem types did not vary greatly between the two textbooks. Alternate Form Problem Type In the Middle School Math textbook, almost half of the examples and almost one thi rd of the exercises were of the alternate form problem type. This type of example involves converting between decimals, fractions, or percents, or finding equivalent fractions. Tasks of the type alternate form are useful in helping students acquire skills
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246 related to rational numbers, but tasks of this type relate almost exclusively to rational numbers and thus do not point out the usefulness of proportional reasoning in algebra, data analysis, geometry, or measurement. Furthermore, this problem type rarely points out why proportional reasoning is appropriate in some situations and not others and rarely includes a visual aid. Thus, although this type of example is important, having almost half of the examples related to proportionality of the type alternate f orm may over emphasize this problem type. Qualitative Problem T ype Although qualitative was one of the problem types identified in the research literature as an important part of proportional reasoning (e.g., Lamon, 2007), the middle school textbooks exam ined contained few examples or exercises of this type. Although no guidelines have been set for the percentage of proportional reasoning examples that should be of each problem type, given that qualitative problems are one of the major components of propor tional reasoning, a textbook in which less than 3% of the proportional reasoning examples are of the qualitative type may not provide students sufficient opportunity to view examples of this type. Solution Strategies The result s of the pilot study le d to t hree conclusions. One, in some sixth grade textbooks, the building up and unit rate strategies are rarely suggested. Two, the solution strategies suggested in examples varies between series. Three, there exist solution strategies that are suggested by some textbooks that have not been described in the research literature.
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247 The Building Up and Unit Rate Strategies Two solution strategies recognized in the literature as fostering conceptual understanding of proportionality are the building up strategy (Heinz & Sterba Boatwright, 2008; Parker, 1999) and the unit rate strategy (Ben Chaim, Fey, Fitzgerald, Benedetto, & Miller, 1998). These strategies seem natural to students, who often use them before they have received instruction in proportionality. Thus, one m ight expect that they would appear in sixth grade textbooks and that more formal, algebraic methods, such as proportions and cross products, might appear in seventh and eighth grade textbooks. Based on the two textbooks studied, this is not the case. The building up and unit rate strategies were virtually absent from the textbooks studied. By not including these solution strategies in sixth grade textbooks, authors may be missing an opportunity before transitioning students into formal, algebraic approaches to solving proportional problems. The Solution S tr ategies S uggested V aries Between Series The Math Thematics textbook contained examples that suggested the use of manipulatives and the Middl e School Math textbook did not. In the Math Thematics textbook, the examples that suggested the use of manipulatives were generally of problem type alternate form In these examples, students were shown how to use manipulatives to find equivalent fraction s or to convert improper fraction to a mixed number. The manipulatives used were generally pattern blocks. The Middle School Math textbook contained a much higher percentage of examples in which a proportion strategy was suggested. When proportions were used in the first half of the Middle School Math book, they were solved by multiplying or
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248 dividing the numerator and denominator by the same number. Once a section on cross products had been covered, they were typically used to solve proportions in the examples. Solution S trategies That Have Not Been D escribed Several solution strategies are well known, such as the cross multiply a nd divide method of solving proportions. Other solution strategies have been described in the research literature. However, in this study, a large percentage of the solution strategies were classified as other because they have not been recognized as solut ion strategies. To some extent, this reflects the inclusion in the study of problem types that are not recognized in the research literature. For example, many of the tasks coded with the solution strategy other were coded with the problem type alternate f orm or function rule which are not recognized in the research literature. However, some tasks of well known problem types were also solved by solution strategies that have not been discussed in the literature. Levels of Cognitive Demand Textbooks based on the principles of the Curriculum and Evaluation Standards for School Mathematics (NCTM, 2000) are generally believed to have higher levels of cognitive demand than traditional textbooks. For example, Senk and Thompson (2003) stated that Standards based ma textbooks studied was similar and most of the exercises in both the Middle School Math and Math Thematics textb ook were at the level Procedures without C onnections Given the need for students to practice skills, perhaps this is to be expected.
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249 As indicated in Figure A1 2 in both the Middle School Math and the Math Thematics series, most exercises were at the level Procedures without Connections The percentage of exercises at this level in the Middle School Math textbook was consistent with the findings of Jones (2004) who found that 75% of the exercises related to probabilit y in a Middle School Math textbook are at this level. However, the percentage of exercises at this level in the Math Thematics estimate; he repo rted that 40% of the tasks in a Math Thematics textbook were at this l evel whereas the current study places this percent at 75%. The Math Thematics textbooks may be unusual among NSF funded series in that Math Thematics books exclusively at t he level procedures without connections Other Math Thematics textbooks that do not contain these sets of extra practice problems may have higher cognitive demand. Figure A1 2 Percent of exercises at each level of cognitive demand 0 10 20 30 40 50 60 70 80 90 Middle School Math Math Thematics
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250 Conclusion This pilot study demonstrated that an analysis of middle school textbooks can because only two textbooks were used in this pilot study and because only sixth grade textbooks wer e analyzed, a more complete study should be conducted to determine whether the results hold true in a larger sample size and for other grade levels. Because one of the problem types and some of the solution strategies discussed in the literature were virtu ally absent from the two textbooks analyzed, the issues of problem types and solution strategies should be further addressed. Finally, because the findings regarding the level of cognitive demand of tasks in the Math Thematics textbook differed dramaticall y from that reported by Jones (2004), the level of cognitive demand of tasks should be further investigated.
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251 A PPENDIX B: L ESSONS INCLUDED IN THE STUDY
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252 Table B 1 Grade 6 CMP Investigations Included in the Study Module Investigation Bits and Pieces I Fundraising Fractions Bits and Pieces I Sharing and Comparing with Fractions Bits and Pieces I Moving Between Fractions and Decimals Bits and Pieces I Wo rking with Percents Bits and Pieces II Estimating with Fractions Bits and Pieces III Decimals More or Less! How Likely Is It? A First Look at Chance How Likely Is It? Experimental and Theoretical Probability How Likely Is It? Ma king Decisions with P robability
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253 Table B 2 Grade 7 CMP Investigations Included in the Study Module Investigation Variables and Patterns Rules and Equations Stretching and Shrinking Enlarging and Reducing Shapes Stretching and Shrinking Similar Figures Stretching and Shrinking Similar Polygons Stretching and Shrinking Similarity and Ratios Stretching and Shrinking Using Similar Triangles and Ratios Comparing and Scaling Making Comparisons Comparing and Scaling Comparing Ratios, Percents, and Fraction s Comparing and Scaling Comparing and Scaling Rates Comparing and Scaling Making Sense of Proportions Moving Straight Ahead Walking Rates Filling and Wrapping Prisms and Cylinders Filling and Wrapping Scaling Boxes What Do You Expect? Analyzing Situations Using an Area Model What Do You Expect? Expected Value
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254 Table B 3 Grade 8 CMP Investigations Included in the Study Module Investigation Thinking with Mathematical Models Linear Models and Equations Thinking with Mathematical Models Inverse Variation Say It With Symbols Looking Back at Functions Samples and Populations Choosing a Sample from a Population
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255 Table B 4 Grade 6 Math Connects Lessons Included in the Study Chapter Lesson Algebra: Number Patterns and Functions Function Machines Algebra: Number Patterns and Functions Algebra: Functions Algebra: Number Patterns and Func tions Algebra Lab: Writing Formulas Operations with Decimals Comparing and Ordering Decimals Operations with Decimals Rounding Decimals Fractions and Decimals Math Lab: Equivalent Fractions Fractions and Decimals Simplifying Fractions Fractions and Decimals Mixed Numbers Fractions and Decimals Comparing and Ordering Fractions Fractions and Decimals Writing Decimals as Fractions Fractions and Decimals Writing Fractions as Decimals Operations with Fractions Math Lab: Rounding Fractions Operations with Fractions Rounding Fractions and Mixed Numbers Ratio, Proportion, and Functions Ratios and Rates Ratio, Proportion, and Functions Math Lab: Ratios and Tangrams Ratio, Proportion, and Functions Ratio Tables Ratio, Proportion, and Functions Proportions Ratio, Proportion, and Functions Algebra: Solving Proportions Ratio, Prop ortion, and Functions Sequences and Expressions Ratio, Proportion, and Functions Proportions and Equations Ratio, Proportion, and Functions Graphing Proportional Relationships Percent and Probability Percents and Fractions P ercent and Probability Circle Graphs Percent and Probability Percents and Decimals Percent and Probability Experimental and Theoretical Probability Percent and Probability Making Predictions Percent and Probability Estimating with Percents Systems of Measurement Length in the Customary System Systems of Measurement Capacity & Weight in the Customary System Systems of Measure ment Changing Metric Units Geometry: Angles and Polygons Similar and Congruent Figures Perimeter, Area, and Volume Measurement Lab: Area and Perimeter Perimeter, Area, and Volume Measurem ent Lab: Circumference
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256 Table B 5 Grade 7 Math Connects Lessons Included in the Study Chapter Lesson Introduction to Algebra and Functions Algebra: Arithmetic Sequences Introduction to Algebra and Functions Algebra: Equations and Functions Linear Equations and Functions Measurement: Perimeter and Area Linear Equations and Functions Functions and Graphs Fractions, Decimals, and Percents Simplifying Fractions Fractions, Decimals, and Percents Fractions and Decimals Fractions, Decimals, and Percents Fractions and Percents Fractions, Decimals, and Percents Percents and Decimals Fractions, Decimals, and Percents Comparing Rational Numbers Ratios a nd Proportions Ratios Ratios and Proportions Rates Ratios and Proportions Rate of Change and Slope Ratios and Proportions Changing Customary Units Ratios and Proportions Changing Metric Units Ratios and Proportions Algebra: Solving Proportions Ra tios and Proportions Math Lab: Inverse Proportionality Ratios and Proportions Scale Drawings Ratios and Proportions Spreadsheet Lab: Scale Drawings Ratios and Proportions Fractions, Decimals, and Percents Applying Percents Math Lab: Percent of a Number Applying Percents Percent of a Number Applying Percents The Percent Proportion Applying Percents Estimating with Percents Statistics: Analyzing Data Spreadsheet Lab: Circle Graphs Statistics: Analyzing Data Using Data to Predict Statistics: Analyzing Data Using Sampling to Predict Probability Theoretical & Experimental Probability Geometry: Polygons Display Data in a Circle Graph Geometry: Polygons Similar Figures Two and Three Dimensional Figures Circumference of Circles Two and Three Dimensio nal Figures Graphing Geometric Relationships Geometry and Measurement Measurement Lab: Changes in Scale
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257 Table B 6 Grade 8 Math Connects Lessons Included in the Study Chapter Lesson Algebra : Rational Numbers Rational Numbers Algebra: Rational Numbers Comparing & Ordering Rational Numbers Proportions and Similarity Ratios and Rates Proportions and Similarity Proportional and Nonproportional Relationships Proportions and Similarity Rate of Change Proportions and Similarity Constant Rate of Change Proportions and Similarity Solving Proportions Proportions and Similarity Similar Polygons Proportions and Similarity Dilations Proportions and Similarity Indirect Measurement Proportions a nd Similarity Scale Drawings and Models Percent Ratios and Percents Percent Comparing Fractions, Decimals, and Percents Percent Algebra: The Percent Proportion Percent Finding Percents Mentally Percent Percents and Estimation Measurement: Area and Vol ume Circumference and Area of Circles Measurement: Area and Volume Explore Similar Solids Measurement: Area and Volume Similar Solids Algebra: Linear Functions Sequences Algebra: Linear Functions Slope Algebra: Linear Functions Direct Variation Alge bra: Linear Functions Geometry Lab: Slope Triangles Statistics Circle Graphs Probability Experimental and Theoretical Probability Probability Using Sampling to Predict
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258 Table B 7 Grade 6 UCSMP Lessons Included in the Study Chapter Lesson Some Uses of Integers and Fractions Measuring Length Some Uses of Integers and Fractions Mixed Numbers and Mixed Units Some Uses of Integers and Fractions Equal Fractions Some Uses of Integers and Fractions Negative Fractions and Mixed Numbers Some Uses of Decimals and Percents Decimals for Numbers between Whole Numbers Some Uses of Decimals and Percents Decimals and the Metric System Some Uses of Decimals and Percents Converting Fracti ons to Decimals Some Uses of Decimals and Percents Decimal and Fraction Equivalents Some Uses of Decimals and Percents Percent and Circle Graphs Some Uses of Decimals and Percents Comparing Fractions, Decimals, and Percents Statistics and Displays Circle Graphs Using Multiplication Calculating Percents in Your Head Using Multiplication Using the Percent of a Quantity Using Division The Rate Model for Division Ratio and Proportion The Ratio Comparison Model for Division Ratio and Proportion Proportions Rati o and Proportion Solving Proportions Ratio and Proportion Proportions in Pictures and Maps Area and Volume Area and Operations of Arithmetic Area and Volume The Circumference and Area of a Circle Area and Volume Surface Area and Volume of a Cube
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259 Table B 8 Grade 7 UCSMP Lessons Included in the Study Chapter Lesson Representing Numbers Decimals for Numbers between Integers Representing Numbers Equal Fractions Representing Numbers Fraction Decimal Equivalence Representing Numbers Fractions, Decimals, and Percents Representing Numbers Using Percents Multiplication in Geometry The Area Model for Multiplication Multiplication in Geometry Circles Multiplication in Geometry The Size Change Model for Mul tiplication Multiplication in Algebra The Rate Factor Model for Multiplication Multiplication in Algebra Combining Percents Patterns Leading to Division The Rate Model for Division Patterns Leading to Division The Ratio Comparison Model for Division Patte rns Leading to Division Proportions Patterns Leading to Division Proportional Thinking Patterns Leading to Division Proportions in Similar Figures Geometry in Space Volume of Prisms and Cylinders Geometry in Space How Changing Dimensions Affects Area Geome try in Space How Changing Dimensions Affects Volume Statistics and Variability Representing Categorical Data
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260 Table B 9 Grade 8 UCSMP Lessons Included in the Study Chapter Lesson Division and Proportions in Algebra Rates Division and Proportions in Algebra Multiplying and Dividing Rates Division and Proportions in Algebra Ratios Division and Proportions in Algebra Proportions Division and Proportions in Algebra Similar Figures Slopes and Lines Rate of Change
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261 A PPENDIX C: T RAINING MODULE AND ANSWER KEY
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262 TRAINING MODULE Work with the researcher to c ode the content area, problem type, level of cognitive demand, and solution strategy of the following exercises. 1. Marissa is 5 feet 6 inches tall. How tall is she in inches? 2. In the first five basketball games, Jamil made 9 out of 12 free throw attempts. Find the probability of Jamil making his next free throw attempt. 3. Write as a percent. 4. Insert >, < or = to make a true statemen t. 0.86 5. Express as a unit rate: 153 points in 18 games 6. Explain how to write a rate as a unit rate. 7. Which measure is greater? Or are the measures the same? Explain. One square yard or one square foot 8. Students at Neilson Middle School are asked if they prefer watching television or listening to the radio. Of 150 students, 100 prefer television and 50 prefer radio. Decide whether this statement is accurate: At Neilson Middle School, 1/3 of the students p refer radio to television. 9. A car can average 140 miles on 5 gallons of gasoline. Write an equation for the distance d in miles the car can travel on g gallons of gas. 10. Jasmin and Janelle are driving to Cincinnati. They think they can average 60 miles per hour for the 310 miles. At this rate, how long will it take them to get to Cincinnati? a. Let t be the time (in hours) it will take them. Write an equation involving t that can answer the question. b. Solve your equation. Check the solution in the original equation. c. Answer the question with a sentence. 11. There are 640 acres in a square mile. On August 20 and 21, 1910, what may have been the largest forest fire in American history burned three million acres of timberland in northern Idaho and western Montana. A bout how many square miles is this? 12. Multiple Choice In t minutes, a copy machine made n copies. At this rate, how many copies per second does the machine make? A B C D 13. The Talkalot cell phone comp any sells a pay as you go phone with 700 minutes for $70. a. What is the rate per minute? b. What is the rate per hour?
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263 Table C1 Answer Key for Training Module Question Content Problem Solution Cognitive Visual Charact. of Number Area Type Strategy Demand ** Repres. Proportionality 1 GM MV No ne Low: Proc. None No 2 DAP MV None Low: Proc. None No 3 RN MV None Low: Proc. None No 4 RN RC None Low: Proc. None No 5 ALG MV None Low: Proc. None No 6 ALG Other None High: Proc. None No 7 GM Other None High: Proc. None Yes 8 DAP RC None High: Proc. None No 9 ALG FR None High: Proc. None No 10 ALG FR None High: Proc. None No 11 GM MV None L ow: Proc. None No 12 ALG Other None High: Proc. None No 13 ALG MV None Low: Proc. None No *DAP = data analysis/probability GM = geometry/measurement RN = rational number **Low: Proc = Procedures Without Connect. High: Proc = Procedures With Connect.
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264 A PPENDIX D: R ELIABILITY TEST AND ANSWER KEY
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265 RELIABILITY TEST On your own, c ode the content area, problem type, level of cognitive demand, and solution strategy of the following exercises. 1. Write as a decimal. 2. In the first five basketball games, Jamil made 9 out of 12 free throw attempts. Suppose Jamil attempts 40 free throws throughout the season. About how many free throws will you expect him to make? Justify your reasoning. 3. 720 cm = ____ m 4. Order this set of numbers from least to greatest: , 22%, 0.3, 0.02 5. The Statue of Liberty stands about 305 feet tall, including the pedestal. Christina wants to build a model of the statue that is at least 20 inches tall. Find the s cale if the model is 20 inches tall. 6. Find 26% of 360. 7. Using a sales tax of 6%, find the total cost for a $2.00 magazine. 8. Rewrite using a denominator of 10. Then, write a decimal for each fraction. 9. Find the value of x that makes the fractions equivalent. 10. A school photograph measures 12 centimeters by 20 centimeters. The class officers want to enlarge the photo to fit on a large poster. Can the original photo be enlarged to 60 centimeters by 90 centimeters? 11. What fraction of an hour is 3 minutes? 12. algebraic expression. 13. A 5 pound bag of dog food costs $4.89. A 12 pound bag costs $9.5 9. Which is the better buy?
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266 Table D 1 A nswer Key for Reliability Test Question Content Problem Solution Cognitive Visual Charact. of Number Area Type Strategy Demand Repres. Proportionality 1 RN AF None Low: Proc. None No 2 DAP MV None High : Proc. None No 3 GM MV None Low: Proc. None No 4 RN RC None Low: Proc. None No 5 GM MV None High : Proc. None No 6 RN MV None Low : Proc. None No 7 RN MV None Low : Proc. None No 8 RN MV & AF None Low : Proc. None No 9 RN MV None Low : Proc. None No 10 GM RC None High: Proc. None No 11 RN MV None Low: Proc. None No 12 ALG FR None Low : Proc. None No 13 ALG RC None Low: Proc. None No *D AP = data analysis/probability GM = geometry/measurement RN = rational number **Low: Proc = Procedures Without Connect. High: Proc = Procedures With Connect.
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ABOUT THE AUTHOR Gwendolyn Johnson received a bachelors degree in Secondary Mathematics Education from Bowling Green State University in 1993 and a masters degree in Business Administration from the same institution in 2000. She taught high school in Toledo, Ohio and Ann Arbor, Michigan from 2000 to 2004. In 2004, she entered the doctoral program in Mathematics Education at the University of South Florida in Tampa. From 2004 to 2010, she taught mathematics methods courses for undergraduate and graduate pre service teachers. She was also employed by ITT Technical Institute in 2009 and 2010 where she taught mathematics content courses.
