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Problem Solving Strategies Students Use When Solving PROBLEM SOLVING STRATEGIES STUDENTS USE WHEN SOLVING COMBINATORIAL PROBLEMS by GARY YUEN A THESIS SUBMITTED iN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Mathematics Education) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2008 © Gary Yuen, 2008 ABSTRACT This research is a case study that examines the strategies that three grade 11 students use to manoeuvre through a series of three combinatorial problems. Grade 11 students were chosen as participants because they have had no formal training in solving this class of math problems. Data includes video recordings of each participant’s problem solving sessions along with each participant’s written work. Through analysis of this data, several themes related to problem solving strategies were identified. First, students tend to rely on algebraic representation and methods as they approach a problem. Second, students use the term “guess and check” to describe any strategy where the steps to a solution are not clearly defined. Thirdly, as students negotiate problems, they tend to search for patterns that will streamline their methods. Fourthly, students approach complicated problems by breaking up the problem into smaller parts. Finally, students who verify their work throughout the problems solving process tend to experience more success than those who do not. From these findings, I suggest that mathematics teachers need to ensure that they are not over-emphasizing algebraic strategies in the classroom. In addition, students need to be given the opportunity to explore various solution strategies to a given problem. Finally, students should be taught how to verify their work, and be encouraged to perform this step throughout the problem solving process. 11 TABLE.OF CONTENTS Abstract ii Table of Contents iii List of Figures v Acknowledgements vi Dedication vii CHAPTER 1 INTRODUCTION 1 1.1 The Problem Solving Dilemma 2 1.2 Resolving the Dilemma 3 1.3 Combinatorial Problems 5 1.4 The Research Question 6 CHAPTER 2 LITERATURE REVIEW 7 2.1 Problem Solving Models 7 2.1.1 Pólya’sHowToSolvelt 7 2.1.2 The Incorporation of Metacognitive Aspects 9 2.1.3 Other Cognitive and Metacognitive Problem Solving Models 11 2.1.4 Summary 14 2.2 Mathematical Problem Solving 15 2.2.1 The Effect of Uncertainty 15 2.2.2 TheEffectofBeliefs 16 2.2.3 The Effects of Ability and Prior Knowledge 17 2.2.4 The Effect of Multiple Strategies 19 2.2.5 The Use of Algebraic Methods 21 2.2.6 Summary 22 2.3 Combinatorics 23 2.3.1 The Classification of Combinatorial Problems 24 2.3.2 Strategies in Solving Combinatorial Problems 25 2.3.3 Obstacles to Successfully Solving Combinatorial Problems 29 2.3.4 Summary 30 2.4 Research Methods 31 CHAPTER 3 METHODOLOGY 34 3.1 The Participants 34 111 3.2 The Tasks .37 3.3 The Procedure 39 3.4 Data Analysis 41 CHAPTER 4 OBSERVATIONS, FINDINGS AND ANALYSIS 43 4.1 Jacqueline 43 4.1 .1 Jacqueline’s Attempt at Problem I 43 4.1.2 Jacqueline’s Attempt at Problem 2 47 4.1.3 Jacqueline’s Attempt at Problem 3 50 4.1.4 An Analysis of Jacqueline’s Approach 55 4.2 Andrea 58 4.2.1 Andrea’s Attempt at Problem 1 59 4.2.2 Andrea’s Attempt at Problem 2 62 4.2.3 Andrea’s Attempt at Problem 3 64 4.2.4 An Analysis of Andrea’s Approach 68 4.3 Tanja 71 4.3.1 Tanja’s Attempt at Problem 1 71 4.3.2 Tanja’s Attempt at Problem 2 75 4.3.3 Tanja’s Attempt at Problem 3 79 4.3.4 An Analysis of Tanja’s Approach 81 CHAPTER 5 DISCUSSION AND CONCLUSION 86 5.1 Problem Solving Strategies and Trends 86 5.1.1 Using Algebraic Representations and Methods 87 5.1.2 Guessing and Checking 88 5.1.3 Looking for Patterns 90 5.1.4 Breaking Up the Problem 92 5.1.5 Verifying 94 5.2 The Big Picture 96 5.3 Implications for Teaching 98 5.4 Future Directions 100 References 105 Appendix A: Copy of UBC Research Ethics Board’s Certificate of Approval 110 Appendix B: Parent Consent Form 111 iv LIST OF FIGURES Fig. 1: Jacqueline’s written work for problem 1 (page 1). 44 Fig. 2: Jacqueline’s written work for problem 1 (page 2) 45 Fig. 3: Jacqueline’s written work for problem 2 48 Fig. 4: Jacqueline’s written work for problem 3 51 Fig. 5: Andrea’s written work for problem 1 (page 1) 60 Fig. 6: Andrea’s written work for problem 1 (page 2) 61 Fig. 7: Andrea’s written work for problem 2 63 Fig. 8: Andrea’s written work for problem 3 66 Fig. 9: Tanja’s written work for problem 1 (page 1) 72 Fig. 10: Tanja’s written work for problem 1 (page 2) 73 Fig. 11: Tanja’s written work for problem 2 76 Fig. 12: Tanja’s two sets of diagonal zeroes 77 Fig. + 13: Tanja’s six permutations of”0 1 + 3” 77 Fig. 14: Tanja’s written work for problem 3 80 V ACKNOWLEDGEMENTS I would like to thank everyone who, directly or indirectly, made this Master’s Degree possible. First, I would like to thank the members of my committee for taking the time to review my thesis and for providing valuable feedback. Above all, I would like to thank Ann Anderson for believing in me and not giving up on me, and for all of her support and encouragement. This thesis would not have been possible without her. I would also like to thank Karen, Paul, Jana, and the other graduate students at the University of British Columbia whom I had the pleasure to work with and learn from. Many of you inspired me, and I consider some of you to be life-long friends. At the school level, I would like to thank my colleagues, who were also a great source of support and encouragement. In particular, I would like to thank Les for listening those many times when I needed an ear. In addition, I would like to thank the three participants in this study for cheerfully volunteering their time to take part. Finally, I would like to thank my friends and family for their encouragement and constant reminders. It was all appreciated! vi This thesis is dedicated to the little ones: Topher, Daniel, and Angie. vii Chapter 1 INTRODUCTION It is generally agreed upon by mathematics education researchers and educators that problem solving is a critical component of any successful mathematics program, and is central to mathematics education. The National Council of Teachers of Mathematics [NCTM] (2000) suggest that “problem solving is an integral part of all mathematics learning, and so should not be an isolated part of the mathematics program” (p. 52). In addition, they state that “by learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom” (NCTM, 2000, p. 52). With this in mind, current trends in mathematics education have moved away from traditional practices where the focus was on understanding mathematical knowledge and practicing mathematical procedures and algorithms. In this traditional model, problem solving was often treated as nothing more than an application or an afterthought. In contrast, the role of problem solving has been significantly elevated in mathematics classrooms today. Mathematics instruction has shifted to a model where problem solving plays a much more integral and prominent role. The rest of this chapter outlines my interest in the area of problem solving in mathematics education, my perspective on this topic, and the research question I explored in the current study. 1 1.1 The Problem Solving Dilemma As a secondary school mathematics teacher, frustrated students frequently confront me with the question, “Why do we have to know this?” During my early stages as a teacher, I was not sure how to respond to this question. Initially, I consulted with experienced teaching colleagues. Some of these colleagues suggested that the mathematics currently taught in our classrooms would be useful in the workplace. I was not convinced by this rationale. Although some specific mathematical topics that were taught may be relevant in some specialized fields, it was hard to argue that the wide breadth of mathematical content taught in the secondary curriculum would be relevant to anyone not specializing in mathematics or the physical sciences. With this in mind, it was difficult to explain to a student with aspirations of becoming a chef, for example, that learning to factor a quadratic expression was somehow important to their future career. Other colleagues I consulted argued that the mathematics we taught was useful for problem solving. I was somewhat sceptical about this justification as well, because these teachers frequently supported their explanation by referring to the standard textbook word problems that appear periodically throughout secondary school mathematics textbooks. The problem here was twofold. First, these problems generally consisted of a contrived and unrealistic, sometimes nonsensical, situation where the one required pathway to the solution involved the utilization of a specific mathematical process that was presented in a preceding section of the textbook. Furthermore, the word problems were often grouped and presented as a separate section, giving the impression that there is some sort of divide between a mathematical concept and its application. Second, and more importantly, most students were taught to solve these problems by applying a pre-determined “problem solving” procedure.
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