# Fifth grade students' understanding of ratio and proportion in an engineering robotics program

ix TABLE OF CONTENTS

ACKNOWLEDGMENTS……………………………………………..…………vi TABLE OF CONTENTS..…………………………………………………......…ix LIST OF FIGURES……………….…………………………………………….xiv LIST OF TABLES……..……………………………………………………...xviii CHAPTER 1: THE STUDY………………………. …………………………..…1 Background……………………………………………………………..…2 Conceptual Underpinnings for the Study………..………………………..9 Statement of the Problem……………….………………………………..14 CHAPTER 2: REVIEW OF RELATED LITERATURE………………………..18 K-12 Engineering Education…………….……………………………….18 LEGO-robotics in support K-12 engineering education.…….......20 Student Development of Proportional Reasoning……………...……...…23 Students‘ proportional reasoning strategies.…………...………...24 Types of proportional reasoning problems..……………………..30 Assessment of Proportional Reasoning…..………………..………….....33 Video Research in the Learning Sciences ….……………………………35 Student Communities of Mathematical Inquiry……………………….....37 Summary………….…………………………………………………..….39 CHAPTER 3: RESEARCH DESIGN AND METHODOLOGY……………….41 Study Design Overview…………………………………………….……41

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Research Questions, Contexts and Conditions…………………………..46 Description of Setting and Population……...…………..…………….….50 Description of Study Participants……………………...……...…………51 Intervention Program Curricula…………………………………….........55 The non-engineering textbook-based intervention program….….56 The LEGO-based engineering robotics program…………..…….59 Data Collection...….…………………………………………………......69 Proportional reasoning assessment instruments……….…………72 Intra-mathematical proportional reasoning test (Intra-Prop)………………………………………………75 Extra-mathematical proportional reasoning test (Extra-Prop)....................................................................,.77 Engineering extra-mathematical proportional reasoning test (Engin-Prop)………………………..…………………...79 Definitions of engineering and mathematics test (Definitions)..…80 Classroom observational data……………….…...…………...,…81 Descriptive cases: LEGO-based engineering tasks...........86 Coding Criteria -Student Proportional Reasoning Strategies…,,,,,,,..,,….87 Example of proportional reasoning coding: level 0……………,,92 Example of proportional reasoning coding: level 1…………,,…93 Example of proportional reasoning coding: level 2…………..…94 Statistical Analysis Methods and Considerations……………………......96

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

xi CHAPTER 4: RESULTS OF ANALYSES…………….………………………100 Evaluating Research Question #1………………………………………100 Analysis of descriptive statistics………………………..……....101 Experimental group analysis……….…………..…..…..103 Control group analysis………………..…...……..……..104 Between & within comparisons measured by the Intra-Prop…..106 Between & within comparisons-measured by the Extra-Prop….108 Between & within comparisons-measured by the Engin-Prop…110 Evaluating Research Question #2…………….…….…………………..113 Proportional reasoning levels analysis- Intra-Prop……....…..…114 Proportional reasoning levels analysis- Extra-Prop……...……..117 Proportional reasoning levels analysis- Engin-Prop………...….121 Quantitative Analysis Summary………………………….…..……...…124 Classroom Observational Data……………………………..…………..126 The integrated teaching and learning classroom………………..126 Engineering team discussions- sample cases……...……………130 Mathematics and design actions: Cristian,Gaby and Adriana.............................................................................131 Collecting data: Cristian, Gaby and Adriana…….……..138 Velocity: Jorge, Nick and Jimena………………………141 Math & engineering: Cristian,Gaby and Adriana….…...144 Engineering design: Cristian,Gaby and Adriana…….....151 Learning about gears: Luci, Javier and Ramón…..….…158

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Designing with gears: Luci, Javier and Ramón……….. 165 Student verbal and written work- sample cases ….………….....170 Adriana -experimental group student…………….……. 172 Carlos -control group student…………………….…..…180 Proportional reasoning strategy levels-example cases………….187 CHAPTER 5: DISCUSSION & CONCLUSIONS...….………...……………..192 Discussion of Results of Analyses ………………………..…………....192 Impact of a Focused K-12 Engineering Education Program.……..……193 The Role that Contexts Play in Assessment Problems in Mathematics...195 The Process of Physically Distributed Learning in Engineering LEGO-robotics …………………………………………………………197 Teaching and Learning in an Integrated Manner…………………...…..200 Potential Impact on Student‘s Long Term Retention……………..……201 Limitations of the Study…………………………………...……………202 Implications of the Findings……………………………………..……..203 Future Research Recommendations………………………………….....204 BIBLIOGRAPHY…..…………………………………………………………..206 APPENDIX A...………………………………………………………..……….228 APPENDIX B……………………………………………………………….….232 APPENDIX C.………………………………………………………………….236 APPENDIX D.………………………………………………………………….241 APPENDIX E.………………………………………………………………….242

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

xiii APPENDIX F.………………………………………………………………….244 APPENDIX G.…………………………………………………………………246

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LIST OF FIGURES Figure 1. LEGO programmable brick (on top) with additional LEGO components ……………………………………………………………………………6 Figure 2. Ratio and Proportion in Intra-Mathematical and Extra-Mathematical Contexts………………………………………………………………………….14 Figure 3. Experimental and Control Group: Curriculum and Instruction……….16 Figure 4. Vergnaud's model for simple proportions (1983)……………………..28 Figure 5. Isomorphism of Measures Problems (Vergnaud, 1983)………………28 Figure 6. Students working in a videotaping-setup classroom…………………..45 Figure 7. Research Contexts and Conditions …………………………………...47 Figure 8. Data Collection Timeline……………………………………………..48 Figure 9. Classroom Mathematics Grade Distribution…………………………..53 Figure 10. Student-Reported Experience with LEGOs………………………….53 Figure 11. The Engineering Design Process (Massachusetts Department of Education,2006)………………………………………………………………....60 Figure 12. The 4-Step Fusion Engineering Design Process……………………..61 Figure 13. Systematic Networks (Roden, 1995)………………………………...82 Figure 14. Elementary Students' Integrated Design and Mathematics Actions in a Team Setting..……………………………………………………………………85 Figure 15. Example of use of Proportional Reasoning Level 0…………………93 Figure 16. Example of use of Proportional Reasoning Level…………………...94 Figure 17. Example of use of Proportional Reasoning Level 2………..……......95

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

xv Figure 18. Experimental Group Students Proportional Understanding over Time…………………………………………………………………………...103 Figure 19. Control Groups Students' Proportional Understanding over Time………………………………………………………………………...…..105 Figure 20. Mean Score on Intra-Prop for program groups across three time periods…………………………………………………………………………..108 Figure 21.Mean Scores on Extra-Prop for program groups across three time periods…………………………………………………………………………..109 Figure 22. Mean Scores on Engin-Prop for program groups across three time periods…………………………………………………………………………..111 Figure 23. Mean Level of Strategy-Use Scores (0-3) on Intra-Prop for program groups across three time periods………………………………………………..115 Figure 24. Mean Strategy Level Scores (0-3) on Extra-Prop for program groups across three time periods………………………………………………………..120 Figure 25. Mean Strategy Level Scores (0-3) on Engin-Prop for program groups across three time periods………………………………………………………..123 Figure 26. Four Categories of the Elementary Students' Mathematics and Design Actions………………………………………………………………………….131 Figure 27. Helping behaviors: Sharing…………………………………………132 Figure 28. Helping behaviors: Sharing…………………………………………133 Figure 29. Surviving mistakes behavior: Realizing mistake…………………...134 Figure 30. Surviving mistakes behavior: Fixing……………………………….135 Figure 31. Self-Focused Behaviors: Calculating……………………………….136

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Figure 32. Helping Behaviors: Sharing the Problem…………………………...137 Figure 33. Student Team A: How Fast? Data Sheet……………………………139 Figure 34. Student Team B: How Fast? Data Sheet……………………………142 Figure 35. Eastside Rodeo Delivery Warm-up Calculations: Student Work Sample…………………………………………………………………………..145 Figure 36. Eastside Rodeo Activity Worksheet………………………………..147 Figure 37. Eastside Rodeo Delivery Follow-up Questions…………………….150 Figure 38. LEGO Technic Kit Scale Drawing of the Truck……………………152 Figure 39. Students Building the LEGO Technic Kit Truck…………………...153 Figure 40. Adriana's Response to an Extra-Prop Ratio Scalar Problem at T1...156 Figure 41. Adriana's Response to an Extra-Prop Ratio Scalar Problem at T2…157 Figure 42. Gear Assembly Usedassembly used in the Texas Gears Activity….158 Figure 43. Sample student worksheet – Gear activity page 1………………….159 Figure 44. Sample student worksheet – Gear activity page 2………………….162 Figure 45. Sample student worksheet – Gear activity page 3………………….164 Figure 46. Sample student worksheet – Gear activity page 4………………….166 Figure 47. Luci showing her idea for adding gears…………………………….167 Figure 48. Javier discussing the assembly with gears………………………….167 Figure 49. Javier discussing the calculation of ratios……………………….…168 Figure 50. Ramón sharing a gear assembly idea………………………………169 Figure 51. Extra-Prop - Adriana‘s T1 work……………………………………175 Figure 52. Extra-Prop - Adriana‘s pre-assessment T1 work…………………...176

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

xvii Figure 53. Extra-Prop - Item #5 Adriana‘s T1 work…………………………...177 Figure 54. Extra-Prop - Item #5 Adriana‘s T2 post assessment………..………178 Figure 55. Extra-Prop - Item #8 Adriana‘s pre-intervention (T1) work………..178 Figure 56. Extra-Prop : Item #8 Adriana's post-intervention assessment (T2)...179 Figure 57. Extra-Prop - Item #1 Carlos‘ T1 assessment……………………….183 Figure 58. Extra-Prop - Item #1 Carlos‘s T2assessment work………………...184 Figure 59. Extra-Prop - Item #5 Carlos‘ T1 assessment……………………….185 Figure 60. Extra-Prop - Item #5 Carlos‘ T2-assessment……………………….186 Figure 61. Extra-Prop - Item #8 Carlos‘ T1 assessment……………………….187 Figure 62. Extra-Prop - Item #8 Carlos‘T2 assessment………………………..187 Figure 63. Control Student‘s Intra-Prop Item Response at 3 time points……...188 Figure 64. Adriana‘s Intra-Prop Item Response Across Three Time Periods…190

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LIST OF TABLES Table 1 Semantic Problem Types (From Lamon, 1993, p. 42)

..............................31 Table 2 Instruments Used in the Study

..................................................................49 Table 3 Representation of student factors in each group ..................................... 54 Table 4 TEKS and Activities in the Non-Engineering Textbook-based Intervention Program (Control Group)

..............................................................................56 Table 5 Mathematics and Engineering Robotics Instructional Goals for the LEGO-based engineering robotics program (Experimental Group)

.............61 Table 6 Activities in the LEGO-based Engineering Robotics Program (Experimental Group)

....................................................................................65 Table 7 Student Proportional Reasoning Data per Student in Experimental and Control Groups

...............................................................................................69 Table 8 Study Assessment Instruments Problem Types and Sources

.....................72 Table 9 Intra-Prop Test

.........................................................................................75 Table 10 Extra-Prop Test .......................................................................................76 Table 11 Engin-Prop Test

......................................................................................78 Table 12 Definitions Interview ...............................................................................79 Table 13 Coding of Assessment Data

....................................................................87 Table 14 Langrall and Swafford's Proportional Reasoning Scale (*modifications.)

……………………………………………………………………………………89 Table 15 Intra-Prop Descriptive Statistics - Strategy Rating by Student Groups . … …………………………………………………………………………………..113

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Table 16 Extra-Prop Descriptive Statistics - Strategy Rating by Student Groups …. …………………………………………………………………………………..118 Table 17 Engin-Prop Descriptive Statistics - Strategy Rating by Student Groups …. …………………………………………………………………………………..120

Running Head: Students‘ Understanding of Ratio and Proportion within Engineering Robotics

CHAPTER 1: THE STUDY The main goal for conducting this dissertation study was to offer students a mathematical learning experience that would allow the opportunity to explore their changes in understanding when solving mathematical problems of ratio and proportion in intra-mathematical and extra-mathematical contexts, and specifically, in the context of engineering robotics. The research questions for this dissertation were as follows: 1) How do students‘ test results compare for students learning ratio and proportion concepts within the LEGO-robotics integrated engineering and mathematics program versus when using a non-engineering textbook-based mathematics program? 2) How do students‘ proportional reasoning strategy levels compare for students learning ratio and proportion concepts within the LEGO-robotics integrated engineering and mathematics program versus when using a non-engineering textbook-based mathematics program? The following sections provide a brief background for the motivation in seeking to research the potential of K-12 engineering education as a context for the teaching and learning of mathematics. LEGO-robotics will be introduced as a specific application of K-12 engineering education, and some background on children‘s development of proportional reasoning will be presented.

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BACKGROUND An imperative in education research and reform is to study, understand, and then implement changes that will effectively address and impact the academic, social, and economic inequities that many students continue to experience. Students that are underserved academically, as defined by the Academic Pathways to Access and Student Success (2008), typically include low- income, underrepresented racial and ethnic minorities, and students of parents that have not attended college. These underserved children have inadequate access to quality educational experiences at home and this . This is echoed in their school experience, where they continue to receive inadequate access to quality teaching, challenging curricula, and relevant and engaging resources. This imperative motivated this study and was implemented with the cooperation of the target students. Underserved students require further support in the content area of mathematics. According to the 2005 National Assessment of Educational Progress (NAEP) results, the discrepancy in scores between White and African American eighth-grade students as well as those between White and Latino eighth-grade students remained over the last ten years (Loveless, 2008; NAEP, 2008). Mathematical literacy, in general, is considered to be a critical factor or gateway for students and linked to college readiness and success in higher education, career, and social stability (Allexsaht-Snider & Hart, 2001; Chazan, 2000; Moses

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

3 & Cobb, 2002). The United States Department of Education‘s final report of the National Mathematics Advisory Panel (2008) defined the three critical mathematical foundations for college readiness to include (a) fluency with whole numbers, (b) fluency with fractions, and (c) particular aspects of geometry and measurement. These content areas are certainly important, but are not enough for students to learn mathematical content to meet state content and process standards. The student must be prepared to apply this knowledge in multiple external situations, often through creative and personally meaningful understanding that best address mathematical problems in a variety of situations or contexts. This real world fluency and use of mathematics is referred to as mathematical literacy by the Programme for International Student Assessment (PISA) that was developed by the Organization for Economic and Co-operation Development (OECDP), an international team of researchers prominent in educational research and reform. Mathematical literacy is defined as: ―…the capacity to identify, understand and engage in mathematics, and to make well-founded judgments about the role that mathematics plays in an individual‘s current and future private life, occupational life, social life with peers and relatives, and life as a constructive, concerned and reflective citizen. (OECDP, 2003, p. 24)‖ Given this definition, and the stated importance of mathematics literacy, this dissertation study was designed with this imperative in mind so that educational research might be advanced while at the same time a direct learning benefit be

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provided to students in the content areas of mathematics as well as in engineering education. Engineering education, the teaching of engineering concepts and processes, has normally been reserved as a field of study for the college and university levels. However, in the last decade, preeminent engineering and technology professional groups in the U.S., such as the American Society for Engineering Education (ASEE), the National Academy of Engineering (NAE), the National Center for Engineering and Technology Education (NCETE), and the International Technology Education Association (ITEA) have made formal and significant policy and research coordination efforts to establish K-12 engineering education as an important part of school curriculum for young students. Various K-12 engineering education outreach efforts have focused on children‘s K-12 engineering education as a vehicle for developing technological literacy for all U.S. citizens through materials, curricula, and professional development for improved instruction. Research exploring the cognitive impact of K-12 engineering education suggests that such learning experiences can build creative design skills, foster the utilization of higher order thinking and problem solving skills (Amsel, Goodman, Savoi, & Clark, 1996; Fleer, 1999; Foster & Wright, 2001; Roden, 1995) and serve as a vehicle to effectively integrate and teach across content areas (Barlex & Pitt, 2000; Cross, 2007; Martínez Ortiz, 2004; Moundridou & Kaniglonou, 2008; Rogers & Portsmore, 2004). In this dissertation study K-12 engineering education was utilized as a curricular framework for the

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

5 integrated exploration of mathematics concepts. In addition, the use of robotics allowed opportunities and specific contexts for teaching and learning of ratio and proportion. K-12 engineering education , with an emphasis on the technology, can serve as a platform for providing inquiry-based, project-based learning with real- world contexts and hands-on approaches in collaborative problem solving. Papert (1980) investigated how technology could be used to help children learn mathematics differently. He developed a philosophical approach called ‗constructionism‘ and a supporting technological programming language called Logic Oriented Graphic Oriented (LOGO). Massachusetts Institute of Technology (MIT) researchers then collaborated with the LEGO company to design a control unit (the brick) embedded with computational power that could accommodate the mounting of traditional LEGO blocks to design and build controllable LEGO construction projects. This control brick, shown in Figure 1, runs basic programs written by students using a simple graphical interface on a personal computer.

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Figure 1. LEGO programmable brick (on top) with additional LEGO components

Together, the LEGO blocks and the programmable brick can empower children with the capability to design and interact with the physical world through their own insight and programming determination. The LEGO blocks can be used in a collaborative design environment that allow for team design and determination of variable setting that is further facilitated by the additional controllable options such as motor settings, and use of add-on sensors, and infrared communication devices (Resnick, Martin, Sargent, & Silverman, 1996). ). This exploration and development initiative led to the LEGO Mindstorms robotics kits that have been used in a wide variety of K-12 engineering education curricula and has been shown to serve as an effective application of K-12 engineering education that can impact students‘ improved learning through enhanced teaching of fundamental concepts in physics, measurement, and in mathematical problem solving. (Goldman, Eguchi, & Sklar, 2005; Mauch, 2001). LEGO-robotics has

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

7 also been used for students‘ learning of proportional reasoning (Norton, 2006; Polhemus, Olson, Jennings, Holmes & Rubenfeld, 2004). However, these studies have been problematic in that the focus of each study has been overly broad - aiming to improve problem solving in general or mathematics in general, versus specific concepts of mathematics. These studies have also been reliant primarily on qualitative data and have therefore been unable to quantify student learning. Norton (2004) utilized primarily qualitative descriptions of learning as a main research methodology to investigate his broad assertion that ―construction activities give the potential for students to achieve many learning outcomes in science, mathematics and technology‖ (p. 4). Norton later recognized that among the many potential mathematics and science learning objectives, LEGO-robotics seemed to have served to support student understanding of rates, ratios, and direct proportions. However, since this was not a main objective in of the study, the assessment instruments did not clearly reveal what students learned about proportional reasoning while engaged with LEGO-robotics. Another study by Polhemus, Olson, Jennings, Holmes, and Rubenfeld (2004) was not designed as a research study, rather as a report on a pedagogical model that integrates interactive robotics and multimedia for a middle school mathematics module on proportional reasoning. Here, students used both the construction and programming elements of LEGO-robotics. However, students did not receive were not directly instructed in without direct instruction of the mathematics content of ratio and proportion and researchers. Researchers

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concluded that direct instruction of mathematics concepts in the should have been made as part of the study to support and improve students‘ learning experience. This dissertation is based on the proposition that an engineering robotics context will offer students the opportunity for improving proportional reasoning through a deeper understanding of ratio and proportion through multiple representations. Furthermore, this learning in the context of engineering robotics may be more meaningful and long lasting for students than that offered by learning with a non-engineering, textbook-based mathematics learning curriculum. Multiple key elements highlighted in the studies and models above include (a) design of research as a carefully planned, mixed methods design approach that also includes a control group of participants, (b) focus on proportional reasoning as the specific content area of mathematics in which student learning may be affected, and (c) the study of mathematics strategies used when learning in the engineering context. This study and the supporting curriculum, written by the primary investigator of this study, were tailored to support learning in a school designated as high-need based on the population of students that it serves and their historically low performance in the annual statewide assessment program. The assessment program, called the Texas Assessment of Knowledge and Skills (TAKS), includes the four core academic areas (reading, writing, mathematics, and science.). According to the 2008 fifth grade student state assessment data for the four academic areas tested, the students at the designated study school scored 30%

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

9 lower than the average 5 th grade score in their same school district, and 40% lower than the regional average, The unique contribution of this dissertation study is the focus on capturing qualitative data of students‘ words and actions through videotape as they make sense of the mathematics of ratio and proportion as well as documenting and analyzing their understanding quantitatively, using some common assessment items (mathematical problems) found in the TAKS assessment. If this research can show that the K-12 an engineering education context can support the improved learning of important mathematical concepts such as proportional reasoning, it may point to K-12 Engineering as a curricular opportunity for strengthening mathematics learning while making learning engaging for students, regardless of their backgrounds. By situating mathematics in an engineering context, students could also gain other knowledge related to engineering and develop twenty first century skills such as collaboration, teamwork, technological literacy and effective communication.

CONCEPTUAL UNDERPINNINGS FOR THE STUDY The interconnected concepts of multiplication, division, fraction, ratio, and rational numbers often appear in experiential problem situations that students encounter in real world experiences. These concepts are also related mathematically to such an extent that taken all together, they may define a unique mathematical conceptual field. Vergnaud (1983) calls this the Multiplicative

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Conceptual Field (MCF) and defines it as ―a set of problems and situations for the treatment of which concepts, procedures, and representations of different but narrowly interconnected types are necessary‖ (p. 127). Vergnaud suggests that the MCF include concepts in multiplication and division, linear and bilinear functions, dimensional analysis, linear mapping, linear combinations of magnitudes as well as ratio, rate, fraction, and rational numbers. He maintains that these mathematical concepts do not exist in isolation but rather in a network of conceptual relations and problem situations. Similarly, student mathematical reasoning develops and is called upon to consider various mathematical problem solving situations. Proportional reasoning is one such kind of reasoning that students use when solving problems in the MCF. Research indicates that students‘ ability to reason proportionally is important for their own mathematical development and essential for learning advanced topics in mathematics (Behr, Harel, Lesh & Post, 1988). However, there is also some evidence that students perform better when encouraged to construct their own knowledge through collaborative problem solving activities involving proportion, than when they participate more passively in a traditional, textbook directed instruction (Ben-Chaim, Fey, Fitzgerald, Benedetto & Miller, 1998). Proportional reasoning is important to this dissertation study because the engineering robotics context include collaborative problem solving activities that directly call upon the use of proportional reasoning in collaborative problem

Students‘ Understanding of Ratio and Proportion within Engineering Robotics

11 solving activities. Proportional reasoning calls upon the understanding and application of skills within the MCF to make sense of mathematical relationships using ratios and equations involving ratios. Simply stated, a ratio is a comparison of two quantities, quantity a and quantity b, and a proportion is an equation stating when two ratios are equivalent. The two ratios are said to be proportional: ―For a mathematician, a proportion is a statement of equality of two ratios, i.e., a/b = c/d‖ (Tourniaire, 1984, p. 181). Mathematical problems of ratio and proportion can be analyzed by using the concepts of multiplication, division, fraction, and linear functions. Proportional reasoning involves the understanding of the relationship of two numbers (ratio) as the critical multiplicative comparator. Researchers propose that although proportional reasoning does not appear to be an automatically developed concept, students do encounter many opportunities for developing proportional reasoning in their daily lives as well as in the elementary and middle school classroom (Karplus, Pulos, & Stage, 1983; Lamon, 1993; Lo & WatanabeWatanbe, 1997; Noelting, 1980). Lesh, Post, and Behr (1988) have called proportional reasoning a watershed concept, a cornerstone of higher mathematics and the capstone of elementary concepts. Research on proportional reasoning in the elementary school (Tourniaire, 1986) has shown that children in grades 3, 4, and 5 have developed an idea of the concept of proportion but it is at first a fragmented ability that relies on the context of the problem. There remain gaps in the research as to the most effective methods for teaching proportional reasoning. Some evidence suggests that simply