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 . Table of Contents ii iii List of Figures v Acknowledgements vi Dedication CHAPTER 1 1.1 1.2 1.3 1.4 CHAPTER 2 2.1 2.2 2.3 2.4 CHAPTER 3 3.1 vii INTRODUCTION 1 The Problem Solving Dilemma Resolving the Dilemma Combinatorial Problems The Research Question 2 3 5 6 LITERATURE REVIEW 7 Problem Solving Models 2.1.1 Pólya’sHowToSolvelt 2.1.2 The Incorporation of Metacognitive Aspects 2.1.3 Other Cognitive and Metacognitive Problem Solving Models 2.1.4 Summary Mathematical Problem Solving 2.2.1 The Effect of Uncertainty 2.2.2 TheEffectofBeliefs 2.2.3 The Effects of Ability and Prior Knowledge 2.2.4 The Effect of Multiple Strategies 2.2.5 The Use of Algebraic Methods 2.2.6 Summary Combinatorics 2.3.1 The Classification of Combinatorial Problems 2.3.2 Strategies in Solving Combinatorial Problems 2.3.3 Obstacles to Successfully Solving Combinatorial Problems 2.3.4 Summary Research Methods METHODOLOGY 7 7 9 11 14 15 15 16 17 19 21 22 23 24 25 29 30 31 34 The Participants 34 111 3.2 3.3 3.4 CHAPTER 4 4.1 4.2 4.3 CHAPTER 5 5.1 5.2 5.3 5.4 The Tasks The Procedure Data Analysis .37 39 41 OBSERVATIONS, FINDINGS AND ANALYSIS Jacqueline 4.1 .1 Jacqueline’s Attempt at Problem I 4.1.2 Jacqueline’s Attempt at Problem 2 4.1.3 Jacqueline’s Attempt at Problem 3 4.1.4 An Analysis of Jacqueline’s Approach Andrea 4.2.1 Andrea’s Attempt at Problem 1 4.2.2 Andrea’s Attempt at Problem 2 4.2.3 Andrea’s Attempt at Problem 3 4.2.4 An Analysis of Andrea’s Approach Tanja 4.3.1 Tanja’s Attempt at Problem 1 4.3.2 Tanja’s Attempt at Problem 2 4.3.3 Tanja’s Attempt at Problem 3 4.3.4 An Analysis of Tanja’s Approach DISCUSSION AND CONCLUSION Problem Solving Strategies and Trends 5.1.1 Using Algebraic Representations and Methods 5.1.2 Guessing and Checking 5.1.3 Looking for Patterns 5.1.4 Breaking Up the Problem 5.1.5 Verifying The Big Picture Implications for Teaching Future Directions 43 43 43 47 50 55 58 59 62 64 68 71 71 75 79 81 86 86 87 88 90 92 94 96 98 100 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” Fig. 14: Tanja’s written work for problem 3 77 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. I, 2 too, recall a time early in my career when I taught these word problems by listing rigid stepby-step pathways to the solution, demonstrating my procedure through the working of several examples on the whiteboard, then having students practice my algorithm through a series of repetitive exercises. Of course, students were only presented with the one prescribed pathway that made use of one specific mathematical concept, very often an algebraic concept. Other solution methods were not only discouraged, but students were often penalized for using them. Through observation and reflection, I eventually came to the realization that the memorization of an inflexible algorithm is not good problem solving. Instead, it was a shortcut that removed critical thinking from the problem solving process and reduced the problem to an exercise in numerical and symbolic manipulation. Being able to solve problems in this manner was not a good rationale for learning mathematics. 1.2 Resolving the Dilemma If a mathematical problem was to be set up in some context, then good mathematical problems should ideally have some basis in a context that can be modelled. Students are more likely to be engaged in a problem if they feel that the problem is somehow relevant to their daily lives. More importantly, however, good mathematical problems must allow for students to explore various solution pathways. They should not be contrived in a way such that they function only as an exercise in one specific algorithm. Furthermore, teachers need to de-emphasize a “correct” solution. Instead, they should encourage creativity and critical thinking. The focus needs to be on the processes that lead to an answer, and not the answer itself. 3 So, “why do we have to know this?” To answer this question, I gave some thought to the content of our high school mathematics courses. Much of the mathematics that is taught in secondary schools involves following a logical, linear sequence of procedures to a desired result. This is evident when we consider some examples of what is regularly taught in secondary school mathematics classrooms today: solving equations, graphing functions, proving identities. All three of these activities, and many others like them, involve systematic procedures that will lead from a starting point to a desired goal. Almost all of the mathematics that is taught in secondary school classrooms can be viewed in this fashion. I believe that such structured exercises in linear logic model the organized thinking that is often required for successful problem solving. This is not to suggest that problem solving should become a structured exercise in linear logic. Instead, I believe these processes form the pool of techniques and strategies that students can draw upon when they encounter a mathematical problem solving situation. The critical feature that distinguishes problem solving, however, is that students should not be restricted to one or more prescribed solution methods. The choice in solution methods and strategies should be one that students experiment with and make on their own. I believe that mathematical problem solving, when done right, involves the modelling of logical thinking and reasoning processes that is essential for solving problems that are encountered in our daily lives. These problems can range from simple everyday problems like how to complete a series of errands within a set timeframe, to complex problems that may lead to significant advances in science and technology. Furthermore, the continued popularity of word puzzles and the recent popularity of Sudoku-type mathematical puzzles 4 seem to indicate that the act of solving a challenging problem has inherent value to many people. Along these lines, Pólya (1957) noted that: The space devoted by popular newspapers and magazines to crossword puzzles and other riddles seems to show that people spend some time in solving unpractical problems. Behind the desire to solve this or that problem that confers no material advantage, there may be a deeper curiosity, a desire to understand the ways and means, the motives and procedures of solution. (pp. vi-vii) However, for mathematical problem solving to be useful in this regard, the problems themselves must be open-ended in nature, as most everyday problems are. 1.3 Combinatorial Problems The mathematical field of combinatorics is especially conducive to good problem solving. A firm grasp of combinatorial problems (involving permutations, combinations, and the fundamental counting principle) is crucial for subsequent understanding of probability. Furthermore, these problems often involve concrete situations that make them less abstract than other branches of mathematics like algebra or geometry. Combinatorial problems can be of varying difficulty and are accessible to a wide range of ability and developmental levels. One advantage of this class of problems is that they can often be solved using a variety of methods, from simple counting techniques, to strategic use of lists and tables, to complex manipulation of formulas. With this in mind, a single combinatorial problem can be presented to students with a wide range of mathematical abilities, potentially resulting in a breadth of solution strategies. Although this can be said for many non-combinatorial problems as well, the relatively limited body of research into this class of problems makes 5 them especially conducive for examination. Furthermore, the intuitive nature of combinatorial problems often allows them to be solved without formal instruction. In fact, since formal instruction often equates to instruction in the use of formulas, students who have been taught a combinatorics unit tend to rely solely on the formulas when solving combinatorial problems and lose the creativity and critical thinking skills they had displayed prior to instruction (Batanero, Navarro-Pelayo & Godino, 1997). For this reason, participants selected for this study will have had no previous formal training in combinatorics. 1.4 The Research Question Each student understands and potentially utilizes any given mathematical concept in a unique manner. This certainly has an impact on how students approach mathematical problems in general, and combinatorial problems in particular. Since combinatorial problems can often be solved using a variety of solution pathways, it would be of interest to examine how students with no formal instruction in combinatorics approach such problems. Therefore, the goal of this research is to investigate the strategies that stand out as students at the secondary level attempt to solve a series of combinatorial problems. 6 Chapter 2 LITERATURE REVIEW A great deal of research exists that analyses problem solving from a myriad of perspectives. This chapter begins by discussing some of the cognitive and metacognitive models of problem solving in the literature. Next, current research in mathematical problem solving strategies in general and combinatorial problem solving strategies in particular are examined. Finally, several studies that relate to research methods that will be used in my study will be considered. 2.1 Problem Solving Models The current literature contains a number of models that describe the problem solving process. Many of these models describe problem solving using both cognitive and metacognitive components. In reviewing these various models, it seems that many of them are descendents of a model described by George Pólya (1957). 2.1.1 Pólya’s How To Solve It It seems that the formal exploration of problem solving in mathematics education began in 1945, with the publication of Polya’s How To Solve It. Pólya (1957) described four simple steps to problem solving: 1. Understanding the problem. 7 2. Devising a plan. 3. Carrying out the plan. 4. Looking back. He suggests that, within each of these four steps, teachers should not specifically direct students to the solution. Instead, teachers should use questions to “help the student discretely, unobtrusively” (Pólya, p. 1). Furthermore, Pólya states: a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. (p. v) Along the same lines as Pólya (1957), Schoenfeld (1992) noted that mathematics lessons today maintain the format where “students are showii a strategy..., given practice exercises using the strategy, given homework using the strategy, and tested on the strategy” (p. 354). He goes on to state that this model of mathematics instruction reduces mathematical strategies to a simple collection of mathematical tasks and students miss out on the goal of learning mathematics, which is to develop broad mathematical thinking skills. Despite the fact that recent research (Lovaszova & Gabriela, 2002; Mamona-Downs, 2002; Schoenfeld, 1992) consistently confirms what Pólya so progressively and prophetically stated some 60 years ago, some mathematics teachers continue to “misuse their opportunity” by de emphasizing the importance of problem solving or by teaching it as an exercise in algorithmic repetition. 8 2.1.2 The Incorporation of Metacognitive Aspects In 2007, Passmore reviewed Pólya’ s model for problem solving in light of current research. He found that the heuristic training, as discussed by Pólya, should be done in the context of particular problems rather than in general. In addition, Passmore suggested that metacognitive training needs to be incorporated into Pólya’s model in order for problem solvers to self-regulate and monitor the problem solving process. Flavell (1976) defined metacognition as “one’s knowledge concerning one’s own cognitive processes and products or anything related to them” (p. 232). The introduction of metacognitive aspects to Pólya’s model has been thoroughly discussed in a number of research studies. Garofalo and Lester conducted one such study in 1985. They developed a cognitive metacognitive framework for studying mathematical performance (as opposed to mathematical problem solving in particular) that is a direct descendent of Polya’s work. In this framework, Garofalo and Lester described four categories of activities involved in performing a mathematical task: orientation, organization, execution, and verification. The parallels between these four categories and the four steps to problem solving proposed by Pólya (1957) are clear. In addition to these “cognitive” categories, Garofalo and Lester identified three metacognitive components that allow for the knowledge and regulation of the cognitive components: 1. Person variables (the problem solver’s beliefs and characteristics). 2. Task variables (features of the problem solving task, such as content, context, structure, syntax, and process). 3. Strategy variables (the problem solver’s awareness of strategies in the cognitive components). 9 Garofalo and Lester state that, during problem solving, these three metacognitive components must be considered in conjunction with the cognitive components. Along the same lines, Schoenfeld (1992) discussed the evolution of the concept of self-regulation as an important complement to the cognitive aspects of problem solving. Another model of mathematical problem solving that included both cognitive and metacognitive components was developed by Montague and Applegate (1993). In their model, Montague and Applegate describe mathematical problem solving as the overlap between cognitive and metacognitive strategies and processes. In this model, they identified seven specific cognitive processes (read, paraphrase, visualize, hypothesise, estimate, compute, and check) that could be viewed as an interpretation of Pólya’s (1957) four steps to problem solving. In addition, they proposed three metacognitive processes (self-instruct, self-question, and self-monitor) that approached metacognition differently than Garofalo and Lester (1985), in that the metacognitive processes proposed by Montague and Applegate are more directly linked to the problem solver. Metacognition itself was a focus of study for several researchers. Biryukov (2004) studied the role of metacognition in mathematical problem solving by asking first and second year pedagogical college students to solve two combinatorial problems and then complete a questionnaire. The questions on the questionnaire were designed to determine subjects’ cognitive and metacognitive behaviour during their problem solving session. Biryukov concluded that there is a greater chance of problem solving success when subjects had metacognitive experiences and could apply them. Specifically, he determined that successful students constructed a schematic model of the problem condition and developed a solution 10 strategy prior to actually solving the problem. Based on his results, Biryukov suggested that mathematics teachers need to include metacognition as a focus of instruction. Schurter (2002) studied the role of metacognition in problem solving through comprehension monitoring, which he describes as a self-regulatory aspect of metacognition that involves “the awareness and control of one’s understanding or lack of understanding” (p. 22). In this study, the subjects, approximately 400 college students enrolled in a pre-college level mathematics course, were divided into three groups: a control group which received traditional instruction, a treatment group where instruction included an emphasis on the use of comprehension monitoring, and a second treatment group where instruction focused on the use of comprehension monitoring in conjunction with Polya’s four-step problem solving technique and heuristics. Schurter found that students in both treatment groups scored higher on their final exams than students in the control group, therefore concluding that comprehension monitoring plays a positive role in mathematical problem solving performance. Furthermore, Schurter stated that the increase in performance of the comprehension monitoring groups was primarily due to the use of self-questioning. Interestingly, Schurter also determined that the metacognitive techniques that lead to the improved performance in the treatment groups occurred subconsciously. 2.1.3 Other Cognitive and Metacognitive Problem Solving Models Numerous other models have been developed that describe mathematical problem solving using cognitive and metacognitive structures. For example, Lesh and Harel (2003) examined the cognitive processes that middle school students undergo during problem solving. They found that these students developed models or “conceptual tools” (p. 186) 11 during problem solving through an iterative process of testing and revising, and that the progression through these modelling cycles was comparable to the general stages of cognitive development described by Piaget and others. Like Lesh and Harel (2003), Pape and Wang (2003) also examined the way in which problem solvers formed cognitive models of their problems. In their study, Pape and Wang tIi examined the self-regulating behaviours of 6 and 7thgrade problem solvers. Pape and Wang suggested that successful problem solvers formed a mental model or representation of the problem elements and their relationships from the text of a problem. Meaning is constructed through “internal representations of the problem or. concrete or semi-concrete . . (i.e. pictures) external representations” (p. 419). They go on to state that good problem solvers read and analysed the problem, then followed a pathway to the solution using a variety of cognitive processes while monitoring the success of each step and making necessary adjustments to the mental model along the way. In their study, Pape and Wang classified the behaviours of their subjects using 14 categories of self-regulated learning strategies described by Zimmerman and Martinez-Pons (1988). These categories included cognitive strategies (organizing and transforming, seeking information, and rehearsing and memorizing), monitoring strategies (self-evaluating, goal-setting and planning, keeping records and monitoring, and reviewing records), environmental structuring (working in a quiet setting, rewards for progress), and help seeking strategies. They found that, although students realized the importance of “strategies associated with transformative behaviour” (p. 437) such as organizing, transforming, goal-setting and planning, most students did not exhibit these strategies during an actual problem solving activity. Pape and Wang concluded 12 that this was due to the students generally not knowing how to perform these strategies when solving mathematical word problems. Pugalee (2004) examined 9thgrade students’ written versus verbal descriptions of problem solving strategies and processes, and analysed them in terms of Garofalo and Lester’s (1985) metacognitive framework. In addition to verifying the relationship between the number of different strategies attempted and problem solving success, Pugalee also found that the majority of the students’ cognitive problem solving actions were execution behaviours, as opposed to orientation, organization, or verification behaviours. In examining the thought processes of problem solvers, Mamona-Downs (2002) distinguishes between conceptual knowledge, structural knowledge, and procedural knowledge. Conceptual knowledge, as it relates to problem solving, involved viewing a problem holistically and in context. Structural knowledge, on the other hand, involved a more in-depth analysis of the components and their interconnectedness, independent of context. This concept seems to relate to the idea of cognitive models described by Lesh and Harel (2003) and Pape and Wang (2003). The third type of knowledge proposed by Mamona-Downs, procedural knowledge, involved rote memorization of procedures that were not fully understood, and is clearly not useful in good problem solving strategies. Furthermore, Mamona-Downs defined a technique as a structured method that leads to a goal. She stated that techniques are not as general as heuristics, but not as mechanical or restrictive as prescribed algorithms. They lie somewhere in between, and are triggered by structural cues. Mamona-Downs argued that techniques are important in problem solving because they facilitate the act of retrieving pertinent information. Therefore, during problem 13 solving, students require both an awareness of techniques as well as an alertness in recognizing the cues that trigger different techniques. 2.1.4 Summary Current research in problem solving has come a long way since Polya’s (1957) heuristical steps to problem solving. Nonetheless, many of the current models of mathematical problem solving use Polya’s work as a foundation. One major progression in recent research has been the examination of metacognitive as well as cognitive aspects to problem solving (Garofalo & Lester, 1985; Pugalee, 2004; Schoenfeld, 1992; Schurter, 2002). The research clearly shows not only that metacognitive aspects play a role in problem solving, but that they are instrumental to problem solving success (Biryukov, 2004; Schurter, 2002). Some of the current research examines specific cognitive and metacognitive processes observed in students as they engaged in various problem solving exercises (Lesh & Hare!, 2003; Mamoma-Downs, 2002; Pape & Wang, 2003; Pugalee, 2004). From this research, it is clear that, although researchers seem to agree on a cognitive-metacognitive framework in describing the problem solving process, there is some disagreement on the specifics. For example, Pape and Wang suggest in their study that students do not engage in metacognitive activities because they have not been shown how. However, this is in contradiction to Schurter’s (2002) suggestion that metacognition during problem solving is a subconscious activity. It is difficult to conduct a research study that examines the strategies used in solving combinatorial problems alone, without touching upon the underlying cognitive and 14 metacognitive processes. The specific problem solving strategies that are observed occur due to a subject’s thought processes. In essence, an understanding of cognitive and metacognitive processes is required in order to try to understand why specific problem solving strategies were observed. 2.2 Mathematical Problem Solving Problem solving continues to be an extremely popular area of research in mathematics education. Current research into mathematical problem solving takes a wide variety of approaches and explores a great expanse of topics. Some specific examples include the role uncertainty plays as mathematical problems are negotiated, the role beliefs play in problem solving, the effect of ability and prior knowledge on problem solving, the use of multiple problem solving strategies, and the use of algebraic strategies. Each of these will be discussed in detail in this section. 2.2.1 The Effect of Uncertainty De Hoyos, Gray, & Simpson (2004) examined the role that uncertainty plays during the initial stages of problem solving. They described uncertainty as a context “in which students lack the knowledge and understanding necessary to know exactly what to do next in order to deal with the situation” (p. 258). De Hoyos et a!. concluded that there is a positive correlation between the amount of uncertainty during problem solving and the need to make arbitrary decisions. They go on to state that, while arbitrary decisions may lead to an “unsustainable course of action” (p. 260), they can also “bring unexpected knowledge and understanding and allow the student to open promising avenues” (p. 260). In addition, De 15 Hoyos et al. noted that students deal with uncertainty during problem solving by looking at simpler cases and examples in order to reduce complexity, to keep the situation manageable, and to help them gain an understanding of the problem. 2.2.2 The Effect of Beliefs Several studies examined the effect that students’ beliefs have on their problem solving performance. Mason (2003) used Kloosterman and Stage’s (1992) Indiana Mathematics Beliefs Scales to examine Italian high school students’ beliefs about problem solving. In particular, they focused on differences in grade and gender, possible relationships between mathematical beliefs and mathematical achievement, and reasons for mature and naïve beliefs. Mason found that, from the first year to the final year of high school, students become increasingly convinced that not all problems could be solved using routine procedures. However, she also found that, from the first year to the final year of high school, students’ belief in the usefulness of mathematics decreased. With respect to the variable of gender, Mason concluded that boys were more likely to simply memorize a procedure or algorithm whereas girls were more likely to believe in the importance of understanding why the procedure or algorithm worked. Mason also looked at the relationship between students’ mathematical beliefs and their corresponding mathematical achievement. Mason found that the best predictor of high achievement in mathematics was the student belief that they had the ability to solve time-consuming mathematics problems. This belief was followed by the belief that not all mathematics problems could be solved using step-by-step procedures, the belief that mathematics is useful in everyday life, and the belief that understanding 16 mathematical concepts is important. Interestingly, Mason found that belief in the value of effort to improve mathematical ability was not a predictor of mathematical achievement. The effect of student beliefs on problem solving was also studied by Lerch (2004). She worked with four subjects who were enrolled in a college level elementary algebra course, and examined the effect of “control decisions” and personal beliefs on how students worked with both familiar problems (i.e. problems that were related to content students were recently instructed upon) and unfamiliar problems (i.e. problems unrelated to recent course content). “Control”, as used by Lerch, referred to the decisions made by students during the problem solving process. Lerch observed that students stopped working due to lack of confidence and previous lack of success. In addition, students persisted with incorrect strategies when attempting to solve unfamiliar problems, believing that specific types of problems could be solved using specific strategies. Finally, Lerch found that more successful control decisions were made when students were solving familiar problems than when they were solving unfamiliar problems. Unfortunately, the use of “familiar problems” in this study sounds too much like the procedural textbook problems described earlier. As Resnick and Glaser (1976) stated, problem solving should ideally involve problems which possess some aspect of unfamiliarity, where the process of arriving at a solution involved making new connections between existing knowledge. 2.2.3 The Effects of Ability and Prior Knowledge The impact of student ability and prior knowledge on the subsequent use of problems solving strategies was a focus of several research studies. For example, Montague and Applegate (1993) organized their American middle school subjects into three groups based 17 on mathematical achievement (learning disabled, average achieving, and gifted students) and examined their problem solving skills through the use of think-aloud protocols. They observed that learning disabled and average achieving students were less strategic in their problem solving than gifted students. In addition, they found that students in the learning disabled group generally lacked strategies for problem representation. Consequently, these students approached problems differently than students in the other two groups. Finally, Montague and Applegate also noticed that students were more willing to persevere with a problem when it appeared easy to them, and gave up easily when a problem appeared to be difficult. Students’ intelligence and mathematical ability was also a variable in Meijer and Riemersma’ s (2002) study that investigated the effect of a computer-supported teaching programme on problem solving. Specifically, during a problem solving activity, st 1 grade students were given up to six hints by a computer, as the subjects required them. Meijer and Riemersma found that computer-supported heuristic assistance facilitated transfer across problems with similar contexts, but not to novel problems where the context was considerably different. Furthermore, Meijer and Riemersma found that this transfer was limited to students with high intelligence and mathematical abilities. Meijer and Riemersma go on to state that, due to this limitation in the experimental effect and the fact that teacher student interactions are more flexible than computer-student interactions, computer supported instruction should be used only to supplement current teaching techniques. Shotsberger (1993) used think-aloud protocols to study the use of cognitive strategies by th 6 grade students in problem solving. In this study, Shotsberger categorized specific behaviours of students during problem solving tasks according to Garofalo and Lester’s 18 (1985) four cognitive processes. He found that some specific problem solving strategies were more natural to students than others. However, these were not necessarily the most effective or efficient strategies. Furthermore, Shotsberger noticed that students’ problem solving behaviours were affected when they were told whether their solution was correct. Finally, Serafino and Cicchelli (2003) looked specifically at the effect of prior knowledge on problem solving and subsequent transfer. According to Serafino and Cicchelli, success in problem solving is measured by students’ ability to reach a solution as well as their ability to transfer their learning to an analogous problem of similar difficulty. In their study of fifty 5thgrade students, Serafino and Cicchelli concluded that students with high prior knowledge resulted in better problem solving scores and more transfer. 2.2.4 The Effect of Multiple Strategies Several researchers examined the effect on problem solving when participants used more than one strategy to solve a problem. One such study is Pugalee’s (2004) examination of 9thgrade students’ written versus verbal descriptions of problem solving strategies and processes, where he observed the relationship between the number of different strategies attempted by students and subsequent success in problem solving. Pugalee found a positive correlation between these two variables. This was similar to a result stated by Pape and Wang (2003), who concluded in their research that high achieving students used a greater variety of strategic behaviour than lower achieving students did. Furthermore, Pugalee determined that students who developed a global plan were more likely to solve problems successfully, and students generally did not verify the accuracy of their answers. Finally, 19 Pugalee observed that students who wrote descriptions of their thinking were more successful at solving problems than students who simply verbalised their thoughts. Lovaszova and Hvorecky (2002) stated that “teaching mathematics.. .means more than pure training of calculation techniques and skills — it is also about learning their interrelationships and in-depth understanding of the concepts” (p. 263). They suggested that students should be encouraged to attempt the same problem using different methods. In their study of future mathematics teachers in their final year of training, students were assigned a problem and given approximately a week to form a solution. Students then discussed and demonstrated their various solutions within a group. Finally, the discussion turned to the strengths and weaknesses of each solution and the relationships between the different solutions were examined. Through this conferencing, students saw and understood multiple solution methods to the problem, as well as the power and limitations of each solution. Herman’s (2007) research also involved multiple solution strategies during problem solving. Specifically, Herman presented algebraic problems to students in a first-year advanced algebra course and examined their preferred solution strategies. She found that, even after instruction in a variety of solution methods, students preferred symbol manipulation over graphing or the use of tables on a graphing calculator. In fact, some students viewed the use of a graphing calculator to produce solutions by graphing or using tables as “cheating”. Herman explained that students generally considered symbol manipulation to be more mathematically proper, and viewed graphical and tabular approaches using a calculator as secondary methods that are useful for checking the results they obtained by symbol manipulation. However, Herman concluded that those students who were capable of using multiple representations to solve algebraic problems were generally 20 better problem solvers. This finding is consistent with findings from studies by Pape & Wang (2003) and Pugalee (2004). 2.2.5 The Use of Algebraic Methods The use of algebraic strategies in problem solving was a focus of several studies. Among them, Gagatsis and Shiakalli (2004) looked at the “translation ability” (p. 645), or the ability to move from one form of representation to another, as it related to functions. The representations of functions used in this study were verbal statements, graphs, and algebraic expressions. Gagatsis and Shiakalli found that translation ability was one component which led to successful problem solving, but that translation ability alone was not the only factor. Furthermore, they found that students experienced less problem solving success when the graphical representation was involved. Mamona-Downs (2002) noted that it was difficult for students to exit an algebraic mode of manipulation once they entered it. She goes on to state that, unfortunately, it is also difficult to extract meaning or intuitive significance while working algebraically, so it is unlikely that knowledge will be retained in the long-term. Van Dooren, Verschaffel, and Onghena (2003) conducted a study that compared pre service elementary and secondary teachers in terms of their mathematical strategies in solving arithmetic and algebraic word problems. They found that secondary teachers tended to use algebraic strategies exclusively for both arithmetic and algebraic problems, even in cases where a simple arithmetic method would be more efficient. On the other hand, elementary school teachers were divided into two subgroups: Those who stubbornly stuck with arithmetic strategies for both types of problems, and consequently experienced limited success and ultimate frustration with algebraic problems, and those who demonstrated 21 flexibility by using arithmetic strategies with arithmetic problems and algebraic strategies with algebraic problems. However, Van Dooren et al. also found that secondary school teachers performed well on both types of problems whereas elementary teachers performed well on arithmetic problems, but generally performed poorly on algebraic problems. Some elementary teachers could not solve the algebraic problems because they were unfamiliar with algebraic methods. 2.2.6 Summary Numerous researchers have keyed in on general and specific strategies used by students during mathematical problem solving. This collection of research can be classified along various themes. For example, De Hoyos et al. (2004) studied the notion of uncertainty and its effects during problem solving. Several studies examined the role of student beliefs on problem solving (Lerch, 2004; Mason, 2003). Some of the the studies discussed look at the relationship between prior knowledge or mathematical ability and subsequent problem solving strategies and success (Meijer & Riemersma, 2002; Montague & Applewhite, 1993; Serafino & Cicchelli, 2003; Shotsberger, 1993). The use of multiple problem solving strategies is a fourth theme that was identified in current literature (Herman, 2007; Lovaszova & Hvorecky, 2002; Pugalee, 2004). A final theme is the use of algebraic methods during problem solving (Gagatsis & Shiakalli, 2004; Mamona-Downs, 2002; Van Dooren et al., 2003). In fact, regardless of the focus of the research, the majority of researchers tended to use algebraic problem solving and reasoning as a tool in their studies. In addition, most of the current research seems to consist of quantitative experiments conducted on relatively large groups of subjects. These studies, although informative, tended to treat the subjects as 22 homogeneous groups and reported results without taking individual differences within the groups into account. There seems to be a general lack of qualitative studies that take an indepth look at individual differences in students as they work through mathematical problem solving activities. 2.3 Combinatorics The volume of research relating to combinatorial problem solving is less substantial than that of other branches of mathematics (e.g. algebra). This may be a reflection of the relative lack of emphasis on combinatorics in the K- 12 curriculum. Along these lines, English (2005) noted that, “despite its importance in the mathematics curriculum, combinatorics continues to remain neglected, particularly at the elementary school level” (p. 121). English goes on to outline the important role of combinatorics in developing problem solving skills, student difficulties when working with combinatorial problems, some of the findings in current research, and ideas for increasing students’ access to combinatorics. English notes that “combinatorics provides the basis for meaningful problems to be solved in a variety of ways and with a variety of representational tools (including manipulatives)” (p. 121). Since this is the case, these problems allow students with minimal background knowledge or instruction in combinatorics to successfully reach a solution. In fact, English argued that “children should be given opportunities to explore combinatorial problem situations without direct instruction. The rich and meaningful contexts in which these problems can be couched means that children have sufficient resources to tackle the problems unassisted” (p. 131). From her review of current research, English concluded that elementary school children have the ability to develop combinatorial concepts when 23 presented with meaningful problems. Furthermore, combinatorial problems “facilitate the development of enumeration processes, as well as conjectures, generalisations, and systematic thinking” (p. 134). However, for this to happen, it is important that students “be given the freedom to use different representations and approaches, and that they be encouraged to explain and describe their actions” (p. 132). This section discusses one researcher’s classification of combinatorial problems, the strategies that participants employed in solving combinatorial problems, and potential obstacles to successfully solving combinatorial problems. 2.3.1 The Classification of Combinatorial Problems A classification of combinatorial reasoning was studied by Batanero, Navarro-Pelayo and Godino (1997). They developed an “implicit combinatorial model” which separates combinatorial problems into three groups: 1. Selection problems (select a sample of n elements from a set of m objects). 2. Distribution problems (distribute a set of n objects into m cells). 3. Partition problems (split a set of n objects into m subsets). In their study of 14 and 15-year olds, they found that, after instruction, most students preferred using a formula with selection problems. However, they continued to use listing with partition and distribution problems. Furthermore, some students who were successful in solving a selection problem encountered difficulties when the same problem was expressed in partition or distribution form. This is in line with Mamona-Downs’ (2002) observation that whether or not a student is successful in solving a problem can depend on how it is formulated. In their study, Batanero et al. identified fourteen specific errors that were 24 observed as students attempted to solve combinatorial problems. They found that conceptual errors (as opposed to procedural errors) were linked to four factors: 1. Students’ inability to interpret the problem statement correctly. 2. Specific errors relating to the partition model. 3. An increase in errors when repeated elements were introduced to permutation problems. 4. Difficulties in discriminating between combinations and permutations with repetitions. However, the focus of this research was the classification of combinatorial problems according to the implicit combinatorial model, not on the specific strategies students used to solve them. 2.3.2 Strategies in Solving Combinatorial Problems Several researchers looked at how students solved combinatorial problems. Sriraman (2003) examined the problem solving behaviour of nine 9 t h grade students in an advanced algebra class as they worked through a series of five increasingly difficult combinatorial problems over the course of three months. Students were assigned a new problem every other week, and asked to attempt each problem and to record their work through journal writing. Sriraman looked at the combinatorial problem solving behaviours in which high school students engaged, the differences in combinatorial problem solving behaviours of gifted and non-gifted students, and how gifted students abstracted and generalized mathematical concepts. He interpreted his findings using Lester’s (1985) model, a cognitive metacognitive framework identical to the one described by Garofalo and Lester (1985). 25 Sriraman found that gifted students spent considerable time understanding the problem, identifying assumptions, and devising a plan, compared to non-gifted students. Gifted students also worked their way up by beginning with simpler cases. During the execution phase, gifted students consistently performed correct procedures and monitored the accuracy of their work. Furthermore, gifted students tried to gain mathematical insight during the problem solving process. As a result, gifted students showed conceptual understanding and were able to state generalizations around the combinatorial problems that they worked on. Finally, Sriraman found that gifted students expressed positive feelings associated with the construction of new ideas. In Watson’s (1996) study, British pre-service teachers and pre-university secondary school students’ use of strategies was observed as they solved three combinatorial problems. The subjects in this study had formal instruction in combinatorics, including the use of combinatorial formulas. The focus of this study was on the use of three strategies: 1. Listing combinations and permutations. 2. Sub-dividing into cases. 3. Using a formula. Watson found that subdividing into cases was the most common strategy of the three studied. Watson observed that, when formulas were used, they were usually the correct ones. Furthermore, he noted that few students attempted to solve combinatorial problems by listing all possible cases, even when students were struggling with a particular problem. This is in contrast to research done by English (2005), and likely due to the fact that the subjects in this study were adults. Consequently, they were developmentally and intellectually much more advanced than the young children that English studied. Nonetheless, Watson noted that 26 listing is often a good place to start and may provide insight into more sophisticated solution strategies, especially with more challenging combinatorial problems. This is consistent with De Hoyos et al.’s (2004) observation that students dealt with cases of uncertainty during problem solving by looking at simpler cases in order to develop an understanding of the problem. Finally, Watson stated that, regardless of the method, the key to success in solving combinatorial problems was to work systematically. In another study by English (1991), children aged 4 to 9 years were individually presented with a series of increasingly difficult combinatorial problems (using manipulatives), and students’ problem solving strategies were observed. English noted a range of strategies, from random selection to systematic patterns. In addition, English concluded that children tended to adopt more efficient methods as they progressed through the series of tasks. Eizenberg and Zaslavsky (2004) looked specifically at the verification strategies employed by 14 undergraduate students as they solved a series of combinatorial problems. Although all of the students had formal instruction in solving combinatorial problems, all the students also claimed that they have never been formally taught how to verify their solutions to combinatorial problems. Eizenberg and Zaslavsky classified the verification behaviours of their students into five categories: 1. Reworking the solution. 2. Adding justifications to the solution. 3. Evaluating the reasonableness of the answer. 4. Modifying some components of the solution. 5. Using a different solution method and comparing answers. 27 Eizenberg and Zaslavsky found that many students were unable to come up with strategies for verifying their solution. Of those that could, simply reworking the problem was the verification method most commonly used. Unfortunately, Eizenberg and Zaslavsky deemed this method to be the least efficient of the five. They observed that evaluating the reasonableness of an answer was not frequently used. Eizenberg and Zaslavsky speculated that this may be due to estimation generally being difficult with combinatorial problems. They concluded that reworking the problem with a different method was the best way for students to detect errors in their solution. Glass & Maher (2004) looked at the role ofjustification in solving combinatorial problems. In particular, they suggest that students should be encouraged to justify their solutions when problem solving so that their thinking and logic is revealed. In their study, Glass & Maher asked high school, undergraduate, and graduate students to justify their solutions to a combinatorial problem. Through these justifications, they identified four categories of problem solving strategies: 1. Use of cases. 2. Use of inductive argument. 3. Eliminating incorrect solutions. 4. Use of formulas. In addition, several participants used some combination of these methods. Glass & Maher found that most high school students used cases to solve the problem successfully, confirming the results of Watson’s (1996) study, whereas undergraduate students used either cases or induction. The successful use of formulas was limited to a few graduate students. 28 2.3.3 Obstacles to Successfully Solving Combinatorial Problems In reviewing the literature, English (2005) noted some obstacles to successfully solving combinatorial problems that were encountered by young students. A major obstacle was the “sample-space misconception” (English, p. 129), where students were either unable to list all possibilities in a sample space or produced a sample space with duplicate entries. This difficulty subsequently led to difficulty with basic probability calculations. A second obstacle was the use of repeated addition when multiplication was clearly a more efficient choice or operation. These obstacles are understandable given that English tended to study the combinatorial problem solving behaviours of relatively young children. Since my study examined the combinatorial problem solving behaviours of much older students, it was of interest to see if these same behaviours were observed in the participants of my study. Hadar and Hadass (1981) also looked at obstacles to successfully solving combinatorial problems. In their study, they identified seven common obstacles to solving combinatorial problems. These obstacles are as follows: 1. Misinterpreting what the question is asking for. 2. Choosing inappropriate notation. 3. Not deconstructing the problem into a set of sub-problems. 4. Non-systematic methods of counting and/or solving. 5. Not applying constraints on one or more variables. 6. Not realizing the counting plan. 7. Not generalizing a specific solution. Although this list of obstacles is thorough, it is not without its drawbacks. First, it could be argued that some of these obstacles apply to all mathematical problem solving, and are not 29 specific to combinatorial problem solving. More importantly, Hadar and Hadass seem to have developed this list of obstacles with one specific linear method of solving particular combinatoric problems in mind. For example, not all combinatorial problems must be solved by “deconstructing the problem into a set of sub-problems”. This is an aspect of one specific approach to solving some combinatorial problems. Most combinatorial problems can be solved using a variety of strategies. Whether solving mathematical problems in general or combinatorial problems in particular, students should be allowed the freedom to explore various problem solving strategies and processes, and to choose one with which they are most comfortable. 2.3.4 Summary Since combinatorics is a branch of mathematics, having some exposure to research into mathematical problem solving in general is important if one wants to develop an indepth understanding of combinatorial problem solving. In contrast to the amount of research available on algebraic problem solving, there is relatively little research available on combinatorial problem solving. Most of the studies reviewed focus on specific aspects of combinatorial strategies used in solving problems (Eizenberg & Zaslavsky, 2004; English, 1991; Glass & Maher, 2004; Sriraman, 2003; Watson, 1996). Batanero et al. (1997) studied how combinatorial problems can be classified and how these classifications affect problem solving. Hadar & Hadass (1981) discussed pitfalls to combinatorial problem solving. Like studies into mathematical problem solving, many of these studies were quantitative in nature that looked at general trends of large groups of subjects rather than the specific behaviours of individual subjects. Furthermore, several of these studies used subjects that have had prior 30 instruction in combinatorics. In these cases, the use of combinatoric formulas can become a prominent strategy. Unfortunately, the use of formulas in combinatorics problem solving is an algorithmic exercise that generally minimizes understanding and critical analysis of the problem. The goal of this research is not to observe adolescent participants’ ability to follow algorithmic procedures. Instead, it is to observe their intuitive strategies as they manoeuvre through a series of combinatorial problems. Therefore, the focus of the current study was on participants whom have had no formal instruction in combinatorics. 2.4 Research Methods Many of the studies reviewed thus far use think-aloud and/or videotaping as methods of data collection. These same methods were the primary methods of data collection in this study. Although there are numerous research studies that employ videotaping and thinkaloud protocols as data collection tools, few studies have addressed the strengths and challenges of these methods themselves. One exception is a study conducted by Pine in 1996. Pine (1996) thoroughly analysed the use of video-recording in collecting research data. She notes that there are both advantages and disadvantages to working with videotaped data. During the process of videotaping, we make decisions that determine what data will be captured and what data will be lost. If data is captured on video, Pine suggested that we should work with the video directly rather than a transcript of the video. She noted that it is impossible to translate into words all the details, nuances and richness that can be observed in a piece of video. Whereas a transcript usually tells us little more than what is said, video gives many more details, including the context of the research and how things are expressed. 31 Furthermore, video can be returned to and viewed with a different purpose or focus. This is much harder to do with a transcript because of it’s one-dimensional nature. A downside to using videos is that they need to be sorted through in order to identify the relevant portions, a task that can be time consuming. For this reason, transcripts of the videotaped data are often easier to work with than the actual videos. Pine (1996) proposed two methods that can be used to determine what students are thinking during the videotaped data collection process: 1. Think-aloud (students are asked to verbalise their thought as much as possible during the data collection). 2. Stimulated recall (students do not necessarily verbalise during the videotaping, but they watch the videotaped data with the researcher afterwards and describe their thoughts during portions of the tape that the researcher identifies as relevant). With both of these methods, Pine noted that students’ beliefs might influence their verbalisations. In general, Pine seemed to prefer think-aloud over stimulated recall. However, the process is not perfect. First, since students generally do not naturally verbalise when they solve problems, asking them to verbalise during problem solving may alter students’ mental actions. Secondly, there may be mental actions that students cannot put into words. Therefore, Pine suggested that verbalisations might not accurately reflect the students’ mental actions. Montague and Applegate (1993) used think-aloud protocols to monitor students’ cognitive and metacognitive processes during problem solving. In using think-aloud protocols, they noted that students were only able to verbalise information in focal attention, 32 and not fast automatic processes that were not necessarily conscious. Montague and Applegate found that, with gifted students, verbalisations increased as they attempted problems that were more difficult. This was interpreted as a metacognitive act where the students were using verbalisation to consciously control and regulate cognitive processes and strategies. Informed by these researchers, I chose to videotape the participants in my study, after prompting them to verbalize their thoughts as they solved each problem in my presence. Details of the procedure used in the current study are provided in chapter 3. 33 Chapter 3 METHODOLOGY A qualitative approach was selected for this research because it would allow for a thorough examination of the problem solving strategies that individual students employed as they attempted to solve a combinatorial problem. In addition, a qualitative method would provide a nice complement to the abundant quantitative research into mathematical problem solving in general and into combinatorial problem solving in particular that currently exists. This chapter describes the participants selected for this study, the tasks that they were asked to complete, the procedures followed in this study, and how the findings were analysed. 3.1 The Participants It was anticipated that, for each of the participants involved, the thorough nature of this case study would yield a great deal of data and information regarding their problem solving strategies. Therefore, in order to keep this study manageable, the number of participants was limited to three. A major goal of this study was to examine students’ intuitive and instinctive combinatorial problem solving strategies when a novel problem solving situation is encountered. A novel problem solving situation was critical because this would eliminate the simple recall and application of previously learned heuristics, algorithms, or strategies that may occur should participants be familiar with a problem in this study. Furthermore, Batanero et al. (1997) noted that “the teaching of the formula for combinations. seems to disturb the intuitive empirical strategies for this type of problems” . . 34 (p. 183). With this in mind, students in grade 12 were not selected as participants because the topic of combinatorics is a unit that is taught in grade 12 mathematics courses in British Columbia. In addition, this unit is typically taught with a strong emphasis on the use of combinatorial formulas. Therefore, grade 12 students would very likely have had formal exposure to the class of problems used in this study in general, and on specific methods of solution in particular. Instead, students at the grade 11 level were selected because these students would hopefully have experienced and developed a wide range of mathematical problem solving strategies over the course of their schooling, while likely receiving no formal instruction in solving combinatorial problems. Students in lower grades were not selected because they would, in theory, not possess the repertoire of problem solving strategies that a grade 11 student should. The three participants themselves attended a suburban secondary school in the Lower Mainland of British Columbia where I work as a mathematics teacher. A mathematics teacher colleague of mine at this school was acquainted with this research study and agreed to recommend three students that he felt were suitable participants. Participants were selected in this manner because, with a sample size of three, the mathematical backgrounds and ability levels of the three student participants should not be so diverse that comparisons cannot be made. Furthermore, since think-aloud protocols were instrumental in this study, participants who were relatively fluent in English and who were generally comfortable verbalising in class were selected. Therefore, the mathematics teacher was included in the participant selection process because this process needed to include someone who knew the participants fairly well. I met with the three participants that the teacher recommended, described the research study and their potential role in it to them in detail, and then I asked if 35 they would like to volunteer to take part in the study. All three students agreed to participate in this study. For the purposes of this study, the pseudonyms Jacqueline, Andrea, and Tanja were used to represent the three volunteer participants. All three participants were female and came from similar socio-economic backgrounds. Jacqueline is a fourth-generation Canadian who came from an Irish and Scottish-English descent. She was born in Vancouver and grew up in the Lower Mainland, where she attended elementary school. Andrea is a firstgeneration Canadian who was born in Ghana to Bangladeshi parents. She attended elementary school in New York before moving to the Lower Mainland in grade 3, where she completed her elementary education. Tanja is a first-generation Canadian of Serbian descent, born in the former Yugoslavia. She moved to the Lower Mainland when she was six and attended elementary school there as well. Each of the three participants attended a different elementary school in the same suburban area, but all three attended the same secondary school from grade 8 onwards. At the time of the study, each of the three students was 16 years old and each was near the end of their grade 11 year. Furthermore, all three students were in the same Principles of Mathematics 11 class. Their mathematics teacher described them as generally hard working students with above average, but not exceptional, mathematical abilities. At the completion of the course, the teacher reported that Jacqueline achieved a final grade of 73%, Andrea achieved a final grade of 82%, and Tanja achieved a final grade of 88%. Andrea and Tanja proceeded to take Principles of Mathematics 12 in the subsequent school year whereas Jacqueline did not. 36 3.2 The Tasks Each participant was asked to solve three combinatorial problems. This class of problems was selected because it was felt that a wide variety of strategies could be used in solving these problems. This variety of possible solution strategies made these problems ideal for focussing on the students’ methods and thought processes leading to a possible answer, rather than on the answer itself. Furthermore, as English (2005) pointed out, these problems were intuitive in nature and could usually be solved successfully without formal instruction in this specific branch of mathematics. The problems themselves were selected from several books that were essentially compilations of various types of mathematical problems, and are as follows: Problem 1 (from Meyer & Sallee, 1983, p. 101) At Blackfoot School, Friday is pizza day, and all of the 60 students look forward to getting their own slice. Mrs. Richards was part way through cutting each of the ten pizzas when she discovered one of them had not been cooked. She and Mrs. Hendricks decided to cut some of the pizzas into seven pieces and cut some into eight pieces so that there would be enough pieces to go around. How many pizzas might have been cut into six pieces? Seven pieces? Eight pieces? Problem 2 (from Giblin & Porteous, 1990, p. 38) Davey Jones has forgotten the number of his locker. The combination is a five-digit number, each digit being one of the numbers 0, 1, 2, 3, 4, 5. All he can remember about the correct number is 37 • The first three digits add up to 4; • The fifth digit is greater than the fourth. Only knowing this information, how many combinations might be correct if repetitions of digits is allowed? Problem 3 (from Fisher, 1982, p. 59) A palindromic number is one that reads the same from left to right and from right to left. An example is 46764. How many palindromic numbers are there between 10 and 100,000? Using my experience teaching mathematics at the secondary level, I chose these three combinatorial problems because I felt that they were not so simple that a solution would be immediately obvious to a typical grade 11 student, but not so difficult that most grade 11 students would be unable to solve them. In addition, I believed that these three problems would be engaging to students at the grade 11 level. All three problems selected for this study followed the combinatorial theme but were different enough so that participants were not simply repeating the same problem three times. Furthermore, this slight variation across problems would reduce the effect a problem solution may have on subsequent ones. According to Batanero et al.’s (1997) implicit combinatorial model, problem 1 may be classified as a partition problem (i.e. a problem where solvers are asked to split a set of n objects into m subsets), whereas problems 2 and 3 may be classified as selection problems (i.e. problems where solvers are asked to select a sample of n elements from a set of m 38 objects). Finally, since all three problems involved combinatorics, there is the potential to observe the transferability of strategies from one problem to another. 3.3 The Procedure Participants were asked to participate in three separate individual problem solving sessions, where they were presented with a new problem during each session. All of these sessions took place after school, in a classroom in the participants’ school, and were scheduled to accommodate the participants’ after school activities. Since individual differences rather than the behaviours of a homogenous group was a focus of this study, it was important that observations be made on participants as individual problem solvers and not as members in a group setting (e.g. students within in a classroom setting). Furthermore, observations of individuals added a degree of focus to this study by eliminating the variable of inter-student interactions during the problem solving process. Therefore, during each session, students worked individually on a given problem as I videotaped and observed them. Due to the nature of the problems, it was anticipated that each problem solving session would last no more than 30 minutes. However, participants were not subjected to a strict time limit. As it turned out, no one problem solving session went beyond 30 minutes. The format of each problem solving session for each participant was identical. Participants were seated at a table with me seated to their left. Although this arrangement was not a conscious decision, it worked out well because it allowed me to easily follow each participant’s work on her paper as she worked through each problem. In addition, it diminished the distraction of my presence since I was off to the side and not directly in the participants’ line of vision. Participants were given a brief description of the format of the 39 problem solving session, reminded to verbalise their thoughts as much as possible, and asked if the interview procedure was clear. If there were no questions, they were then given one of the combinatorial problems. The three combinatorial problems were presented to each participant over three sessions in the same order (i.e. problem 1, followed by problem 2, then by problem 3). In this way, if an earlier problem should have some effect on the solution of a subsequent problem, the effect would be the same for each participant. Little consideration was given to the actual order of the three problems themselves. I felt that the three problems were similar in difficulty and there was no indication that any one of the problems should logically be presented to the participants before or after one of the other problems. The problems themselves were presented on a sheet of paper with space for recording calculations and solutions. Additional paper and a TI 83-Plus graphing calculator were also made available to the participants. Each participant’s completed written work for each problem is presented in chapter 4. My main role was to make observations as each participant worked through each respective problem. However, I prompted participants to verbalise their reasoning and thinking as they progressed through a problem. These prompts generally took the form of brief statements or questions, used as reminders when participants fell silent, if their words were indistinguishable, or if their actions or mathematical reasoning was unclear to me. Some examples of such prompts and questions are “Can you think of another method?” and “Can you explain what you did there?” I did not work with a predetermined set of prompts and questions. Instead, I made statements and posed questions that I thought were appropriate for each specific instance. I also answered questions that participants posed as they progressed through the problems. In most cases, participants tended to ask questions 40 when they felt unsure of a mathematical procedure or when they felt they needed some sort of clarification. I tried to be as neutral as possible in answering these questions, with responses like “What do you think?” Participants were videotaped during each session as they worked through the problem at hand. The video camera was set up on the table and trained on each participant’s paper as she worked through each problem. I started the recording after the participants indicated that they were ready, and stopped the recording once they indicated that they were finished. Although participants were often interested in whether or not they had solved each respective problem correctly, this information was not initially shared with the participants in order to reduce the possibility that the correct solution may inadvertently be shared with other participants. Furthermore, as Shotsberger (1993) observed, knowledge of whether a solution is correct may influence a participant’s problem solving behaviours. Therefore, participants were invited instead to see me regarding the accuracy of their solutions once the entire data collection process was completed. In addition, participants were asked not to discuss any of the problems with the other participants. 3.4 Data Analysis After the data collection was completed, the complete video-recordings of the nine problem solving sessions were viewed several times in conjunction with each participant’s written work during each respective session. This was done in order for general themes to emerge and for points of interest to present themselves. Once these themes and points of interest were determined, portions of each session that illustrated them were transcribed. During this data analysis phase, the focus was on the problem solving strategies used by the 41 participants. For example, particular attention was paid to the types of strategies used by each student as they approached a problem, how these strategies changed and evolved over the course of each problem solving session, and any evidence as to why these changes occurred. Although I did not approach this analysis with a pre-determined list of problem solving strategies, a clear group of strategies was formed through the observation and analysis of the participants’ problem solving sessions. In addition, the progression of each participant’s problem solving strategies over the three problems was observed and generalizations relating to these observations as well as the similarities and differences between each participant’s approach was made. 42 Chapter 4 OBSERVATIONS, FINDINGS, AND ANALYSIS The problem solving sessions were conducted over the course of two weeks. After all the data collection had been completed, video-recordings of each problem solving session were viewed several times and observations were made. This chapter presents a detailed case-bycase description of the problem solving processes of each participant as they progress through each of the three combinatorial problems, followed by an analysis of each participant’s approach to each problem and their strengths and weaknesses. Each participant’s completed written work for each problem is presented in this chapter. 4.1 Jacqueline In solving these problems, Jacqueline tends to approach each problem systematically and with great care. Consequently, Jacqueline experiences the most success, solving two of the three problems correctly. 4.1.1 Jacqueline’s Attempt at Problem 1 Jacqueline seems to have some initial difficulty in understanding the problem. After I clarify the problem for her, Jacqueline takes a few seconds to ponder, then states, “I don’t know how to do this algebraically.” She thinks a bit more and says, “...I’ll just do this by guessing and checking.” 43 Problem I At J3laekthot School. Friday is pizza day. and all of he 60 students look ibrward to getting their own slice. Mrs. Richards was part way through cutting each of the ten pizzas when she discovered one of them had not been cooked. She and Mrs. Hendricks decided to cut some of the pines into seven pieces and cut some into eight pieces so that there would be enough pieces to go Flow many’ pizzas might have been cut into six pieces? Seven pieces? Eight around —1 lucces 1 / CL ‘ x Q-C / ‘I 1: /24 4t: jg Ko C oiEC —- 2 /f A ,; /? / ,:‘* ‘p •,,4/’/ , 1$ / --— —— - / \_) — _\ — - —; 0 - - 4 - I - 4 (3(.) Figure 1. Jacqueline’s written work for problem 1 (page 1). 44 /7 >4 <.... 4 ‘4 7’7 ‘1) 7 / Figure 2. Jacqueline’s written work for problem 1 (page 2). 45 Jacqueline starts by subtracting 6 from 60 to arrive at 54, and quickly subtracts another 6 from 54 to arrive at 48. She then considers how she can make up the difference (i.e. 12) using 7 and 8. She recognizes that she cannot, and subtracts another 6 to arrive at 42 (see Figure 1). She repeats the process, taking more time to consider combinations of multiples of 7 and 8. Again, she realizes that no such combination exists, so she subtracts 6 again to arrive at 36. This time, Jacqueline quickly realizes that the difference of 24 can be made up by 3 times 8. Upon arriving at this solution (cutting six pizzas into 6 slices each and three into 8 slices each), she asks herself, “Is there more?” She continues with her procedure of subtracting 6 and considering how combinations of multiples of 7 and 8 can make up the difference from 60. When she subtracted 6 from 36 to arrive at 30, Jacqueline realizes that the difference of 30 could be made up by “2 times 15, which is 7 plus 8.” Thus, she writes out a second solution: cutting five pizzas into 6 slices each, two into 7 slices each, and two into 8 slices each. As Jacqueline continues with her procedure, she finds a third solution at 18: cutting three pizzas into 6 slices each and six into 7 slices each. Jacqueline seems to have developed an efficient strategy for solving this problem, and stops again at 12. Here, she calculates that the difference needed to make 60 is 48, and concludes that cutting two pizzas into 6 slices each and six pizzas into 8 slices each is a fourth solution. This solution is incorrect because it gives a total of eight pizzas, not nine. Therefore, in using her strategy, it seems that Jacqueline either has not been considering the condition that there must be nine pizzas (although all previous solutions met this condition), or has neglected this condition in this particular instance. Jacqueline subtracts another 6 to arrive at 6, and incorrectly calculates the difference to be 56. She then returns to the four solutions she has found so far (three correct and one 46 incorrect) and states, “I’m just looking to see if there is a pattern.” After some time, she declares, “I don’t see a pattern. in the way I’m doing it at least.” She goes back to the . . incorrect difference of 56, and comes up with two more solutions that are incorrect: cutting one pizza into 6 slices and seven into 8 slices each, and cutting one pizza into 6 slices and eight into 7 slices each. Suddenly she utters, “Oh, I made a mistake, its 54,” and proceeds to scribble out the last two incorrect solutions (see Figure 1). Having reached the end of her pattern of subtracting 6, she goes back and considers the incorrect solution of cutting two pizzas into 6 slices each and six pizzas into 8 slices each. She realizes that this solution involves eight pizzas, not nine, and scribbles it out as well. Having found mistakes with the last two iterations of her procedure, she goes back and reconsiders possible combinations of 7 and 8 that will make up the difference from 12 and 6, respectively, to 60. She spends considerable time thinking about the possibility of cutting one pizza into 6 slices. She sets up a table to calculate all the possible combinations of multiples of 7 and 8, to check if any add up to 54 (i.e. the difference between 6 and 60, see Figure 2). From her table, she decides that there is none. She ponders a bit more, and finally declares, “I think just the three solutions.” In the end, Jacqueline manages to find three of the four possible solutions. 4.1.2 Jacqueline’s Attempt at Problem 2 Jacqueline reads the problem and asks herself aloud, “I can do this algebraically?” She then attempts an algebraic solution, but quickly realizes that this will not take her far, as she utters slowly, “The first three digits.. .ifx is the first.. .no, this will be too many letters I think.. .1’ 11 just do it to show you.” She writes the equation “x +y + z = 4”, followed by the statement “a> b”, but immediately abandons any algebraic methods and begins anew. 47 Problem 2 s Jones h w Icugottun iL numbec ol his k eLi inc cornbinsoon tc atiseJjtJ nurnKr each digit being one of the numbers 0. L 2. 3, 1. 5 All he can remember about the correct nnmhcr is • The first three digtis acid up Ic) 4: • Inc ii hih digit is greater than the Jbnrth, Only knowing this inlorinanon. how ianv combinations mit bt he correct if repetitions of digits is ailowcd 4 ‘x > c& ED r( ( 9 Th * g42Z 0 344 ® — // 54 6 4 q5 :60 \225. o+2 I%: _ S) 45 / Figure 3. Jacqueline’s written work for problem 2. 48 “Okay.. .it’s a five digit number, but I have six to choose from. I’m probably just going to do a guess and check here [for the first three digits]. I know it can’t be 4 or 5. but . . it can be multiples of 1 or 2.” Jacqueline has decided that she will focus on the first three digits. She has also decided that none of the first three digits can be 4 (an erroneous assumption) or 5 (a correct assumption). Furthermore, she quickly and correctly concluded that the digits 1 and 2 could be used more than once in any one permutation of the first three digits. Jacqueline then begins listing permutations of digits that add to four, beginning with “0 + 4”. She immediately excludes this, because it consists of two digits, not three. She continues to list all the permutations of digits that add to four where the first digit is 0 (see Figure 3). This is systematic in that she has fixed the first digit, but the order in which she writes these five permutations suggests that she came up with the second and third digits in a somewhat arbitrary manner. After these five permutations are listed, Jacqueline considers possibilities for the fourth and fifth digits, for the first permutation of “0 + 3 + 1”. She reasons that, if the fourth digit is 0, then there are five possibilities for the fifth digit. If the fourth digit is 1, then there are four possibilities for the fifth digit. Continuing this line of reasoning, she concludes that, for the first permutation of “0 + 3 + 1”, there are a total of 15 possible permutations for the fourth and fifth digits. She then realizes that these 15 permutations for the fourth and fifth digits can be applied to each of the five first-three-digit permutations she has written, for a total of 75 different five digit permutations (all starting with the digit 0). Jacqueline then lists the first-three-digit permutations that start with 1, “to see if [she] can find a pattern.” She correctly arrives at four different permutations that begin with 1. Jacqueline multiplies 15 by 4 to get 60 permutations of the five digits that begin with 1. 49 Jacqueline then proceeds to list the three first-three-digit permutations that begin with 2, and quickly notices a pattern. She multiplies 15 by 3 to get 45 permutations of the five digits that begin with 2. She steps back and states, “If I continue the pattern here...”, and writes “5 + 3 + 2 + 1 = + 4 15”. She concludes that 15 is the total number of first-three-digit permutations, and she has done this without listing each of these permutations. She multiplies this 15 with the 15 different permutations for the fourth and fifth digits, to correctly arrive at a final answer of 225 permutations. 4.1.3 Jacqueline’s Attempt at Problem 3 After her initial reading of this problem, Jacqueline quickly decides, “There’s no way I can guess and check all that, so I have to find a pattern.” Since palindromic numbers are the same forwards and backwards, Jacqueline strategizes, “If I look at half.. .that’s half of the palindrome, so I only have to find three digits... so from one to three digits [to form the first half of a palindromic number]. and then just flop them over [to get the second half of the . . same palindromic number].” Jacqueline then lists the one-digit numbers from zero to nine, and notes that there are 10 digits (see Figure 4). She then quickly observes that there are 100 two-digit numbers (i.e. 00 to 99) and 1000 three-digit numbers (i.e. 000 to 999). Jacqueline then goes back to the question, and crosses out the list of one-digit numbers because the question is asking for numbers starting with 10, so one-digit numbers “wouldn’t count.” According to her initial strategy, this is an erroneous step because the list of 10 one-digit numbers actually represents the first half of 10 two-digit numbers. 50 Problem 3 A paliruirom& o,miher is one that reads the same from left to right and from tight to Jell. An example is 46164. 1 C? Low many palindromic numbers are there between 1001000? 2 H V 4t and ± (q) - it / i - 9q7 CQei C:iI 1 qci CI 00 0 (1. .J \%4 1 (:$ •:Le’?<.? / 3d4J ‘4 J iiui (41Z, : t4s fti.c w iOi Figure 4. Jacqueline’s written work for problem 3. 51 ‘1 Jacqueline then looks at her list of 100 two-digit numbers and eliminates the numbers 00 to 09, again reasoning that 10 is the lower limit of the numbers asked for in the problem. She then says, “10 is not a palindromic number so 11 is the smallest it can go.” It seems that Jacqueline has drifted from her strategy of dividing the numbers into two symmetrical portions. Jacqueline lists the two-digit palindromic numbers from 11 to 99, and quickly notes that there are nine of these numbers. She then moves on to three-digit numbers, starting with 101. Changing only the middle digit (i.e. 101, 111, 121, 131, ...), Jacqueline produces 10 three-digit palindromic numbers that start and end with 1. She repeats this process to produce 10 three-digit palindromic numbers that start and end with 2, and quickly realizes that the same can be done to produce corresponding lists for three-digit palindromic numbers that begin and end with the digits 3 through 9. However, Jacqueline incorrectly calculates the total number of three-digit palindromic numbers to be 100 (i.e. 10 x 10), instead of 90 (i.e. 9 x 10). Moving on to four-digit palindromic numbers, Jacqueline asks, “Can I do ‘0110’?” When told that she cannot, Jacqueline responds, “So it has to be at least starting with a ‘1 ‘.“ She then lists the four-digit palindromic numbers that begin and end with 1 (i.e. 1001, 1111, 1221, ..., 1991) and notes that there are ten. She quickly also lists the four-digit palindromic numbers that begin and end with 2 and notes that there are ten of these as well. Noticing the pattern, Jacqueline concludes that there are 10 four-digit palindromic numbers that begin and end with 3, ten that begin and end with 4, and so on. In calculating the total number of four digit palindromic numbers, Jacqueline makes another arithmetical error, somehow arriving at a total of 900 instead of 90. 52 Looking at five-digit numbers, Jacqueline starts with 10001 and says, “This one is going to be a little more difficult because I have to... [inaudible].. .the middle number now.” Beginning with 10001, she produces a sequence of 10 five-digit palindromic numbers by fixing the first, middle, and last digits and changing the second and fourth digits: 10001, 11011, 12021, 13031, ..., 19091. She looks at this list of numbers, ponders, and says, “But then all of those can also be done. each of [these ten numbers]. with ten separate ones in . . . . the middle.” Jacqueline recognizes that each of the ten numbers she has produced can have ten different middle-digits. She states, “So, I guess times 10, so that’s 100 for the ones.” Noticing a pattern to her model, Jacqueline continues, “And then twos, and threes, all the way to nine would be the same, so 900.” Jacqueline has quickly calculated the correct number of five-digit palindromic numbers. However, noticing that her result is identical to the number of four-digit palindromic numbers, Jacqueline then says, “[That’s] the same for four digits, so maybe it’s not right.” She thinks a bit and reticently decides that she probably did not make a mistake. However, doubting herself, she says, “Okay, I’ll go over [the fivedigit calculation] again.” Jacqueline then proceeds to review her five-digit number calculations, starting with 10001. “[The three digits in the middle are] all zeroes, but I’m only looking at the two outside digits [i.e. the second and fourth digits]. All the way to 9 is ten. Each of these can be substituted with ten numbers in the middle spot, so there are ten options for each of those.. .and also there are ten of those.. .so there should be 1000. I don’t know why I get 900.. .no, 100.. .900?” Verifying her work seems to have produced some confusion for Jacqueline. Leaving it for now, Jacqueline proceeds to look for six-digit palindromic numbers and quickly realizes that this is not necessary. She adds the number of two-, three-, four-, 53 and five-digit palindromic numbers she has found to arrive at a total of 1909. “That’s it,” she proclaims. Wanting to delve deeper, I decide to ask Jacqueline why she felt that it was erroneous for there to be 900 four-digit and 900 five-digit palindromic numbers. Jacqueline replies, “Because there’s more numbers to change around in five-digits, that there’d be more possibilities.” Trying to rationalize, she continues, “But I guess not because they’re basically the same thing...” She writes ‘12321’ and ‘123321’, circles the central threes in each number and thinks. Unable to clear up her confusion, she finally says, “There’s more possibilities.. .1 don’t know, I don’t know why that is. It seems wrong to me.. .there shouldn’t be the same amount because with five digits there are more possibilities.” Having reached an impasse, I provide Jacqueline with the following prompt: “If this [five-digit calculation] is right, maybe there’s something wrong with the [four-digit] one.” Jacqueline proceeds to check her four-digit calculation by writing “1001, 1111, 1221, ...“ She says, “Those two [middle digits] have to be the same because they’re palindromes, so I do ten of those.” Moving on, she determines that there are ten four-digit palindromic numbers that begin and end with 2, ten that begin and end with 3, and so on, up to 9. She then states, “But I can only go up to nine, because I can’t have zero, 0000 is not right. There’s nine of those.. .oh, 90!” Jacqueline changes the number of four-digit palindromic numbers from 900 to 90, and she changes her total from 1909 to 1099. Jacqueline asks, “Is that right? Did I make a mistake on the three digits too?” I reply with a neutral, “What do you think,” and Jacqueline decides to check her three-digit calculation again. She quickly lists the sequence “101, 111, 121, 131, ..., 191” and determines that this sequence consists of ten palindromic numbers. She then says, “Same 54 with twos, three, four, five, six, seven, eight, nine, but not zero. So I did make a mistake then. It should be 90.” Jacqueline then looks at her two-digit numbers and states, “Yeah, for sure those are the only ones.” Finally, Jacqueline adds her subtotals (without using her calculator) to arrive at the correct total of 1089 palindromic numbers. “Yeah, I think that’s it.” 4.1.4 An Analysis of Jacqueline’s Approach Of the three participants, Jacqueline appears to be the most deliberate in her problem solving. After she is presented with the first problem, she takes a few moments to read and understand the problem before attempting a solution. With this problem, Jacqueline initially, almost instinctively, contemplates an algebraic approach, but quickly realizes that an algebraic pathway to the solution would be difficult and opts for what she calls “guess and check” instead. Jacqueline’s approach to guess and check is very systematic. She uses information given in the problem to determine a starting point, and quickly develops a strategy for producing possible solutions to the problem. Guess and check is described by Johanning (2007) as a strategy where students “choose a value to represent an unknown quantity and. test or check the accuracy of the guessed value using relational reasoning. [If . . their guess does not lead to the targeted total, then] a new guess is made and the trial is repeated by applying the same relational reasoning, checking to see if the new guess will produce the desired total” (p. 123). Although Jacqueline refers to her approach as guess and check, it is more sophisticated than what Johanning has described. Jacqueline is not simply guessing at values of the solution (i.e. numbers of pizza) and checking these guesses. Instead, she seems to be guessing at a solution strategy. Based on information from the 55 problem, Jacqueline develops a strategy for producing possible solutions. She proceeds to use her strategy to produce possible solutions and subsequently checks each of her solutions to see if they satisfy the conditions set out in the problem. Interestingly, Jacqueline does not stop after coming across an initial correct solution. She pauses briefly, questions whether there are any more correct solutions, and forges ahead. Furthermore, she does not use her current solutions (whether they are correct or incorrect) to guide her to her next guess. Instead, she simple discards any that do not work and returns to her strategy to systematically produce the next solution. At several points, Jacqueline looks at the correct solutions her procedure has produced and looks for a pattern. When she does not see one, she returns to her strategy. As she progresses, Jacqueline seems to become more efficient and her procedure becomes more systematic. However, Jacqueline neglects to consider all relevant cases and, therefore, manages to find only three of the four possible solutions. In terms of Hadar and Hadass’ (1981) study, it can be said that she was unable to overcome the obstacle of implementing her counting plan. Jacqueline’s approach to problem 2 is similar to how she approached problem 1. After reading the problem, she initially considers an algebraic method. She very quickly realizes that an algebraic solution would be difficult. Nonetheless, she represents the problem using algebraic notation in order to organize her thoughts, it would seem. Again, Jacqueline refers to her strategy as “guess and check”. However, similar to her strategy in solving problem 1, she does not initially guess at the solution to the problem. Instead, she systematically develops a procedure for producing the solution. Interestingly, a traditional guess and check method would not work with this problem because a guess at the solution (i.e. the number of combinations) cannot be verified by simply substituting the guess back 56 into the problem. It is unclear whether Jacqueline realizes this. Using information from the problem, Jacqueline develops a strategy that she refines as she progresses. Jacqueline’s strategy here involves separating the problem into three steps. She uses a series of procedures to find the number of permutations of the first three digits, then a different set of procedures to find the number of permutations of the last two digits. She then combines the answers from these two steps to arrive at the final answer. As was the case with problem 1, she seems to be continuously searching for patterns and efficiencies as she works through the problem. In the case of this problem, Jacqueline’s description of her reasoning as she progresses through the problem as well as her written work indicate that her thoughts are well organized and she easily recognizes patterns that quickly lead to an efficient solution. After reading the third problem, Jacqueline proclaims, “There’s no way I can guess and check all that, so I have to find a pattern.” This statement indicates that Jacqueline has adopted guess and check as her solution method of choice, and seems to consider looking for a pattern as a secondary method that she can turn to when she decides that guess and check may not be viable in producing a solution. Firstly, it is interesting that Jacqueline did not begin by representing the problem using algebraic symbols or expressions. This is a departure from how she approached the first two problems. Furthermore, it should be noted that the use of patterns was integral to Jacqueline’s relative success in solving the first two problems. However, in both those cases, she referred to her method as “guess and check”. This may be because her strategy included guessing in order to close in on a procedure that produces a solution and because checking her solutions was an important step throughout her strategy. Although she refers to her strategy here as “finding a pattern” instead of “guess and check”, the steps she follows here are not significantly different from the steps she used 57 in solving the first two problems. In general, Jacqueline takes time to understand the problem and uses information given to her to develop a procedure that yields some solutions that fit the criteria given in the problem, as she checks her solutions along the way. Jacqueline’s strategy involves breaking up the problem by identifying the different cases implied in the problem, adjusting her strategy to accommodate these different cases, looking for patterns in the solutions that her methods produce, and synthesizing these patterns to produce a final answer. As was the case with the first two problems, Jacqueline’s frequent verification of her reasoning and procedures as well as her solutions allows her to catch errors and to arrive at the correct solution. Through her frequent verification of her strategy, Jacqueline manages to catch most (but not all) errors soon after she makes them. This seems to be one of Jacqueline’s strengths. In addition, Sriraman (2003) noted that monitoring the accuracy of one’s work on an ongoing basis is a sign of a good problem solver. However, Jacqueline generally seems to verify her work by reworking the solution. Although this is a valid method of verification and Eizenberg and Zaslavsky (2004) noted that it is the method most frequently observed, they also stated that there are more effective methods to verifying the solutions to a combinatorial problem. 4.2 Andrea Andrea seems to approach each problem with some mathematical preconceptions. She also has the most difficulty as a result. Andrea does not successfully solve any of the three problems (although she comes very close to solving problem 3 correctly). 58 4.2.1 Andrea’s Attempt at Problem 1 After reading the problem, Andrea states, “I know there’s a way to do this, I remember doing these problems last year.” Andrea then tries to recall “using a formula”, and mentions solving by “substitution”. After some thought, Andrea writes the equation “6a + 7b + 8c = 60” and proceeds with the substitution “b 9 — a” (an incorrect statement). It appears that Andrea is attempting to solve the problem algebraically by employing a linear system. Unfortunately, there is not sufficient information to solve this problem using this method. After reducing the initial equation to two variables through substitution, Andrea proceeds to simplify the expression through algebraic manipulation. What results is an equation involving the variables a and c (see Figure 5). She realizes that there is a problem with the number of variables, and states, “I’m not sure if this is going to work.” She also seems to understand the error in her initial substitution, as she says, “c has to be 9 minus a minus ‘something’ [b]. . .“ Using the erroneous result from her first series of algebraic manipulations, she tries again. This time, she arrives at a solution of”2a = —6”, and recognizes that this is incorrect because the number of pizzas cannot be negative. As it is evident that Andrea has reached an impasse and is getting frustrated, I ask her, “Is there another way?” Andrea thinks for a moment and continues struggling with algebraic equations. She goes over the work she has done, trying to find her error. Several minutes later, I question her again, “Can you think of another method?” Almost ignoring the question, Andrea starts anew with the equation “6a + 7b + series of erroneous algebraic manipulations (see Figure 6). 59 8c 60” and goes through another Problem 1 1 L ,\t l3lackihot School, Friday is pizza day, and all of he 61) students look forward to getting their own slice. Mrs. Richards was part way through cutting each of the ten pizzas when she discovered one nt them had not been cooked. She and Vii’s. Hendricks decided to cut some of the pizzas into seven pieces and cut some into eight pieces So that there would he enough pieces to go around. I low many pizzas mieht have been cut into six pieces? Seven piecesO Eight ‘a :? ‘(“( Id (‘L C C,O . 2 — ‘S f ) / — c... I + “> < Figure 5. Andrea’s written work for problem 1 (page 1). 60 t c 4 J - •)C L -ç < <Z €;)c: —Ia ‘5 — ( -t Figure 6. Andrea’s written work for problem 1 (page 2). 61 Becoming increasingly frustrated, Andrea declares, “I hate these questions, I never got them right in grade 10.” Subsequently, she states, “I can’t think of anything else except substitution.” After several more minutes, and unable to find a single solution to this problem, Andrea finally gives up. 4.2.2 Andrea’s Attempt at Problem 2 Andrea reads the problem and asks for clarification. “Do I just find all the combinations, or is it asking how many combinations?” When told that the problem is asking for the number of combinations, she replies, “These are hard problems! It’s easier to find the different types.” It is unclear what “types” Andrea is referring to. She proceeds to write the letters a to e to represent each of the digits. She thinks for a few seconds, then says, “I’m first going to try to find all the things that add up to four, because that seems easy to me.” She then lists the four combinations of three digits that add up to 4 (see Figure 7): 0+1+3 4+0+0 2+2+0 2+1+1 It appears that Andrea arrived at these four combinations in no systematic way. After listing these four, Andrea says, “There has to be more.” After some thought, Andrea finally concludes that she has found all the possible combinations for the first three digits. 62 Problem 2 l)avev Jones has lorgolten the number of his locker. The combination is a fluc—digit number. each digit being one o 11w numbers 0, 1. 2, 3, 4, 5 All he can remember about the correct nunibcris A • I he tjrst tinee dian s add up to 4; e’l he titth dtgt is create, than the fourth. Only knowing thts information, how many comb,naiiuns might be correct if repetitions of digits is allowed? ,. 2 ., a .1 ‘) ç.•’t AA C.; .1 Figure 7. Andrea’s written work for problem 2. 63 a Andrea then moves on to consider the fourth and fifth digits, and writes “e > d”. Suddenly, Andrea exclaims, “Oh, this is like probability, you can do the tree [diagram] thing.” Then, she says, “First, I want to find the possible combinations, then I’ll multiply it out, or whatever it is that you do.” However, Andrea moves away from the idea of using a tree diagram and does not return to it. Instead, she counts the number of permutations for the fourth and fifth digits if the fourth digit is 0, and states, “There are five possibilities for zero.” She then counts the number of permutations for the fourth and fifth digits if the fourth digit is 1, and concludes, “1 has four possibilities.” She continues to count the number of permutations if the fourth digit is 2, then 3, then 4. She does all this counting in her head, writing only “0 has 5 p’s”, “1 has 4 p’s”, “2 has 3 p’s”, and so on. Since she goes through this step for all the digits from 0 through 4, it seems that she does not see the evolving pattern. Andrea then adds, and correctly states that there are a total of 15 different permutations for the fourth and fifth digits. Andrea then returns to her four first-three-digit combinations and states that each has 15 possible permutations for the fourth and fifth digits. Andrea multiplies 4 by 15, to arrive at a final answer of 60 combinations. This answer is incorrect because Andrea considers only the four combinations of the first three digits, and not the possible number of permutations of those digits. 4.2.3 Andrea’s Attempt at Problem 3 Andrea reads the problem and exclaims, “I don’t know how to go about this!” She then says, “I think I’m just going to try this one out. I’m actually going to try to figure out . . all [the palindromic numbers] because I don’t know how to do it any other way.” She 64 quickly lists the palindromic numbers between 10 and 100 and notes that there are 10 of these. She then lists the palindromic numbers between 100 and 200 and notices 10 palindromic numbers again. “I see a pattern,” she states. “For every 100-number [interval], there will be 10 [palindromic numbers].” Continuing with this strategy, Andrea tries to count the number of 100-number intervals between 10 and 100 000, but loses count. “I’m going to divide 100 000 by 100.. .there’s a thousand [100-number intervals].” Thinking aloud, she reasons, “If there are 10 between 1 and 100, then there’s going to be 10 for every interval for 100.” As she does this, she begins to question her strategy, “oh, that doesn’t work for everything.” She decides that she will continue with this strategy up to 1000, because she believes that this strategy will work up to than point (see Figure 8). However, her uncertainty is clear as she states, “This is really confusing me.” Continuing, she notes that there are nine 100-number intervals between 100 and 1000. As she lists these nine intervals, she explains, “I have to see things more clearly.” Noting that there are ten palindromic numbers between 100 and 200, she calculates, “9 times 10 is 90.” She concludes that there are 90 palindromic numbers between 100 and 1000. Adding these to the nine palindromic numbers between 10 and 100, she correctly arrives at a total of 99 palindromic numbers between 10 and 1000. Working from 1000 onwards, Andrea lists the sequence, “1001, 1111, 1221, ..., 1991,” and counts ten numbers. Realizing that each first (and last) digit can be a digit from 1 to 9, she quickly calculates, “so, 9 times 10, so again, 90.” Thus, Andrea has correctly determined that there are 90 palindromic numbers between 1000 and 10 000. 65 Problem 3 A pniindrornw item her is one that reads the same from left to rinhi and from right to left. An example is 40764, Flow many palindromic munbers are there between 10 and 100,000> ,(1 1 ;:(f 2 l,f) i)) ‘‘it I2 Dl w, 2)1 / ‘ 1, (G 1j 11 = / _)c- S 2. ICa’ o’ N 7 c>C> 1 - 3 1 .1 -- i, L) .c- q 1 c: N •• // :2 cjL, 2L1 / 4 ) j1- \ 5 \ C> ‘1 2 c:;tc:) iii( ç) ‘—. 1 C’ Q( . ‘ ‘ 7 ‘ / \ 0 ( il,\kl\ II N -(v / Figure 8. Andrea’s written work for problem 3. 66 <>C2 Andrea moves on to larger palindromic numbers, and begins by brainstorming, “I’m thinking they’re all going to have ten now. .no, but then those are bigger numbers, those are . different. That’s up to 10 000.. .after that will be 10 001.” After some thought, she exclaims, “This is so irritating!” She struggles with it some more, and states, “This one is so confusing! I can’t write this one out!” However, seemingly with no other recourse, she starts listing five-digit palindromic numbers, starting with 10 001. After listing several, she exclaims again, “I can’t do this writing out!” She thinks a bit more, and decides to continue listing five digit palindromic numbers. She writes 12021, and says, “The middle [digit] has to go up to 9.” She goes back and says, “10 000 to 11 000.. .there will be 9.” She determines this by writing 10 101 and looking at the middle digit. However, in doing this, she has omitted 10 001. Andrea then writes, “11 [thousand] to 12 [thousand], 12 to 13.” Seeming to have come across a strategy, Andrea punches some buttons on her calculator. She determines that there are eighty-nine 1000-number intervals from 11 000 to 100 000 and there are ten palindromic numbers in each of the eighty-nine 1000-number intervals. She multiplies 89 by 10 to arrive at 890 palindromic numbers between 11 000 and 100 000. Finally, she adds the numbers she has arrived at: 99 (palindromic numbers from 10 to 1000), 90 (palindromic numbers from 1000 to 10 000), 9 (palindromic numbers from 10 000 to 11 000, this number is erroneous and should be 10), and 890 Q,alindromic numbers from 11 000 to 100 000). Due to her one error, Andrea’s final answer is 1088 palindromic numbers, one number off the correct total. 67 4.2.4 An Analysis of Andrea’s Approach Andrea’s difficulty with the first problem arises from the fact that she has mistakenly classified this problem as one amongst a group that she has solved before. In particular, Andrea’s previous mathematics course included a unit on solving systems of equations using various methods. Andrea mistakenly believes that this problem can be solved algebraically using a system of equations. Her belief here that specific types of strategies need to be used for specific types of problems is consistent with an observation made by Lerch (2004). It is interesting that Andrea does not consider any non-algebraic approaches to solving her system of equations, such as solving by graphing, even though her previous mathematics courses have very likely exposed her to both algebraic and non-algebraic methods to solving systems of equations. This seems reminiscent of Herman’s (2007) research, which found that, in solving mathematical problems where multiple representations are possible, students consider symbol manipulation to be “more mathematical” and consider the use of a calculator to produce graphs or tables as “cheating”. In addition, Shotsberger’s (1993) observation that some specific problem solving strategies were more natural to students than others seems to be in play here. I believe that Andrea’s “algebraic tunnel-vision” compounds her difficulty with this particular problem. As she struggles with the problem, Andrea is prompted twice to find “another way” or “a different method”. However, she seems to disregard both prompts and continues to struggle with algebraic equations. This seems to be clear evidence supporting Mamona-Downs’ (2002) claim that it is difficult for students to exit an algebraic mode of manipulation once they have entered it. It seems that Andrea’s approach here is based on recall of specific strategies for specific types of problems rather than on understanding the problem and developing a strategy to suit this specific problem. 68 Andrea initially reads the problem very quickly and immediately starts writing as she searches for a solution strategy. This is in contrast to Jacqueline’s approach, which generally seems to be to take a few moments to think about and process the problem before proceeding to an attempt at a solution. As Andrea manipulates her algebraic equations, it appears as though she has forgotten what the variables represent. In essence, she has extracted the mathematics and left the context of the problem behind. Despite not arriving at the correct solution, Andrea has much more success with problem 2. She begins by representing the five digits in the problem with five variables. However, she does not continue with an algebraic strategy. She uses these variables to help her visualize the problem and proceeds to an arithmetic guess and check strategy, similar to that used by Jacqueline on this problem. Andrea also separates the problem into three steps. She works out the number of permutations of the first three digits followed by the number of permutations of the last two digits, and combines these two results by multiplying to arrive at the final answer. Unlike Jacqueline, however, Andrea is less systematic in her solution process. Andrea seems to have found the four possible combinations of the first three digits through random guess and check (as opposed to systematic guess and check). Furthermore, after finding all four, Andrea intuitively believes that there are more combinations and attempts to find them through more guessing and checking. Andrea is more systematic in finding the permutations of the last two digits. However, her methods seem to rely on arithmetic and counting skills, and she does not seem to be looking for patterns. Despite this, Andrea’s reasoning in working this problem is essentially correct. She did not arrive at the correct solution because she neglected to consider all the different permutations of the first three digits. Instead, she simply used the number of combinations of those three digits. 69 Andrea indicates that she is uncertain as to how to begin the third problem, and decides to start listing palindromic numbers. It appears as though Andrea’s strategy here is to develop a list of palindromic numbers and to use this list to identify any emerging patterns or shortcuts. As she works through this strategy, Andrea seems to be assessing the successfulness and the effectiveness of her strategy. However, Andrea’s lack of success in identifying a pattern as her list grows longer is a source of frustration for her. She decides to break up the problem into smaller parts and uses arithmetic reasoning to calculate the number of palindromic numbers in shorter intervals. This works up to 9999, but when Andrea moves on to five-digit palindromic numbers, it quickly becomes clear to her that simply listing and calculating will be an onerous task, and that the identification of a pattern would be a more efficient way to go. She goes back, looks for a pattern to the five-digit palindromic numbers, and indeed comes across a successful strategy. Andrea uses this strategy to calculate the number of five-digit palindromic numbers and adds this to the numbers of palindromic numbers in the other, shorter intervals she had worked with. Again, her strategy is correct, but she arrives at an incorrect answer due to an error in reasoning. Hadar and Hadass (1981) describe this obstacle to arriving at the correct solution as not realizing the counting plan (i.e. making errors in following through with an established strategy). In this case, this error may have been compounded by Andrea’s relatively disorganized and unclear workings. Furthermore, it is interesting to note that, although Andrea’s final answer of 1088 is only one off the correct answer of 1089, Andrea speculates that this answer is incorrect. This is consistent with Eizenberg and Zaslavsky’ s (2004) observation that evaluating the reasonableness of an answer is difficult with combinatorial problems. 70 4.3 Tanja Of the three participants, Tanja seems to have the strongest mathematical background. Her solution methods are strategic and display some elegance. However, what appear to be relative simple errors cause her to arrive at the incorrect solution to two of the three problems. 4.3.1 Tanja’s Attempt at Problem 1 Tanja reads the problem and seems to understand it immediately. She takes some time to think about the problem and summarizes what the problem is asking for, stating that “I’m writing it out so that I can see it all in one spot, so I don’t have to go back to the problem every time.” After another short period of pondering, she decides, “I’m going to guess and check.” She starts with “9 x 8 = 72”, representing nine 8-slice pizzas, and determines that she needs to get down to a total of 60 slices by subtracting 12 (see Figure 9). She then states that replacing a pizza cut into 8 slices with one cut into 6 slices would decrease the total number of pizza slices by 2. However, Tanja drifts from this strategy and her subsequent calculations show that she seems to be randomly taking multiples of 8 and checking if she can somehow make up the difference to 60 by using multiples of 6 and 7. Interestingly, she finds a solution (cutting five pizzas into 6 slices each, two into 7 slices each, and two into 8 slices each), but excludes it as a plausible solution due to an arithmetic error. Tanj a reiterates her approach to this problem as “guess and check”. Continuing with her method, she manages to find another solution (cutting six pizzas into 6 slices each and three into 8 slices each) and recognizes it as a solution to the problem. 71 Problem I At Black fiot School, Friday is pizza day, and all of the 60 students look forward to tctting their own slice. Mrs. Richa.ls was part way through cutting each of the ten piz:zas when she (hseovered one of them had not been cooked. She and Mrs. Hendricks decided to cut some of the pizzas into seven pieces and cut sonic mm eigln pieces so that there would be enough pieces to go around. How man pizzas might have been cut into six pieces? Seven pcccs? bight pieces p ,>z et •-r .2 2 c, Figure 9. Tanja’s written work for problem 1 (page 1). 72 L’J CD C V - - -- 1 At this point, she asks, “Are there multiple solutions?” When told that there is indeed more than one solution, Tanja returns to a strategy that she had abandoned earlier. She begins with “9 x 7 = 63”, representing nine 7-slice pizzas, and describes her approach: “For every time I take away a pizza with 7 slices and add one with 8 slices, the [total] number of slices will increase by one and every time I take away one with 7 slices and add one with 6 slices, [the total number of slices] will decrease by one. So with this, I can find out.. .how many of the other pizzas would give me [a total of] 60.” Tanja then proceeds to employ this strategy. She first considers replacing one 7-slice pizza with an 8-slice pizza, bringing the total number of slices to 64. She states that she needs to decrease the total by 4 slices, so she replaces four 7-slice pizzas with four 6-slice pizzas. In this way, she arrives at her second solution: cutting four pizzas into 6 slices each, four into 7 slices each, and one into 8 slices. Using this same method, Tanja quickly produces the last two solutions to this problem. Tanja returns to her starting point of nine 7-slice pizzas, for a total of 63 slices. From this point, she considers that she “needs to take 3 off,” and produces her third solution: cutting six pizzas into 7 slices each and three into 6 slices each. Finally, going back to nine 7-slice pizzas for a total of 63 slices, Tanja replaces two of these pizzas with two 8-slice pizzas for a total of 65 slices. From here, she concludes that the total needs to decrease by 5, and proceeds to replace five of the 7-slice pizzas with five 6-slice pizzas. This brings her to the final solution: cutting five pizzas into 6 slices each, two into 7 slices each, and two into 8 slices each. However, Tanja does not realize that her task is complete, and continues to search for more solutions. Starting again at nine 7-slice pizzas, she replaces four of these pizzas with 74 four 8-slice pizzas for a total of 67 slices. She correctly concludes that this combination will not produce a solution because she will need to replace seven 7-slice pizzas with seven 6slice pizzas, but she only has five 7-slice pizzas to work with. Tanja then considers a new starting point: nine 8-slice pizzas for a total of 72 slices (see Figure 10). She considers that each replacement of an 8-slice pizza with a 7-slice pizza would decrease the total number of pizza slices by one, and each replacement of an 8-slice pizza with a 6-slice pizza would decrease the total number of pizza slices by two. As Tanja ponders this, she states, “I think there are six solutions, I’m not sure why.” She then systematically attempts to find two more solutions by looking at combinations of 6-, 7-, and 8-slice pizzas. Unable to find these last two “solutions”, Tanja correctly concludes that there are four solutions to this problem. 4.3.2 Tanja’s Attempt at Problem 2 Tanja reads the problem and says, “I’m going to write all this out here again.” She writes “a + b + c = 4” and “e > d” (see Figure 11). She thinks for a few seconds, then states, “Okay, I’m going to guess and check again.” She begins by focussing on the first three digits. “Of these [first three digits), what possible combinations can there be?” She considers three-digit combinations that add to 4, and begins her list with (see Figure 11): 0+1+3 1+2+1 2+0+2 She then continues this list with several permutations of these first three combinations. 75 ProNem 2 fiavey i fyy fbrgotteu the number of bi.s locker. The combination isa fi vedieit number, each dish beine one of the numbers 0, 1, 2, 3, 4, 5.Al] he can remember aboui the ccvnct nmn her is ‘The rsi three diiis idd up to 4 • 1 h,, hlth lit is tier lb in tfojth Only knowmns. this inlonnanon. how many combinations might be correct if repetitions of digits is allowed? lk e. C.) 3 4 .. 2 >d 2 2. , 3\ k ‘: ‘ Z 2 C, (1) 2 2. 2 3 (a2 t% 2i ti z - iS ) ) K 3 Figure 1]. Tanja’s written work for problem 2. 76 2 1 .- She identifies these first three combinations as possible arrangements that add to 4, and explains her reasoning as follows: “Out of the numbers 0, 1, 2, 3, 4, and 5, there’s only four numbers that are less than 4, and.. .that’s 0, 1, 2, and 3.. .and you can do ‘4 + 0 + 0’ and have that too, but I started with these ones first to get all the possible combinations, then I was going to do these [permutations of the first three combinations along with ‘4 + 0 + 0’].” Tanja has found the 4 three-digit combinations that add to 4, and proceeds to find all the permutations of these combinations in a very systematic manner. Starting with the combination “0 + 1 + 3”, Tanja writes 6 zeroes, arranged in two sets of three and written diagonally (Fig. 12). 0 0 0 0 0 0 Figure 12. Tanja’s two sets of diagonal zeroes. Tanja then fills in l’s and 3’s “around” these two diagonals to yield the six permutations of “0+ 1 +3”(Fig. 13). 013 031 103 301 130 310 Figure 13. Tanja’s six permutations of”0 77 + 1 + 3”. Tanja then quickly finds the three permutations of”1 + 2 + 1” using this same procedure. She explains that this combination has three permutations, and not six because “if I switch these two numbers [i.e. the two l’s in the first permutation], its still the same [permutation].” She repeats this procedure to find the permutations for “2 However, in finding the permutations for “2 + 0 + + 0 + 2” and for “4 + 0 + 0”. 2”, she mistakenly writes two diagonal sets of 2’s and fills in the empty spaces with 0 and 2 for one set and 2 and 0 for the other set, instead of using one diagonal set of 0’s and filling in the empty spaces with 2 and 2. This error yields six permutations instead of three, where some of the permutations have been repeated. Furthermore, Tanja does not catch this error. This results in an incorrect total of 18 permutations for the first three digits, compared to the correct total of 15 permutations. Tanja then moves on to the fourth and fifth digits. She systematically lists the 5 twodigit permutations where the first digit is 0 and the second digit is greater than 0 (i.e. 1, 2, 3, 4, or 5). She then lists the 4 two-digit permutations where the first digit is 1 and the second digit is greater than 1. Continuing in this manner, Tanja quickly comes up with the 15 possible permutations for the fourth and fifth digits. Finally, Tanja calculates the total number of permutations of the five digits. She reasons, “Any one of these [first-three-digit permutations] can have any one of these endings [referring to the fourth and fifth digit permutations].” She continues, “There’s 18 possible combinations for the first three digits.. .then there’s 15 possible combinations for the fourth and fifth digits.” She writes “15 x 18”, and multiplies without using her calculator to arrive at an answer of 270 combinations. This answer is incorrect because Tanja mistakenly calculated the number of permutations of the first three digits to be 18 instead of 15. Finally, she confirms her calculation on her calculator. 78 4.3.3 Tanja’s Attempt at Problem 3 After reading the problem, Tanja quickly lists all of the two-digit palindromic numbers (see Figure 14). She then continues her list with the first several three-digit palindromic numbers. She stops and counts nine two-digit numbers in her list. She then states, “In the hundreds, 10 palindromes.” She continues to work quickly as she lists several four-digit palindromic numbers, starting with 1001, and says, “In the thousands, 10 [palindromic numbersj again.” She then repeats the steps, and reasons that there are 100 five-digit palindromic numbers and 100 six-digit palindromic numbers. She adds her totals, and arrives at an answer of 229 palindromic numbers. She looks at this answer, and quickly realizes an error in her reasoning. She correctly states that she did not consider all of the palindromic numbers. Other than the two-digit palindromic numbers, she has actually only calculated the palindromic numbers that begin with 1. However, she does not realize that she has made a second error: she does not need to include six-digit palindromic numbers. Tanja builds on the work she has done, but starts anew in tabulating her totals. She begins by correctly noting that “in the tens,” there are nine palindromes. She proceeds to work on the hundreds. “10 for every nine.. .90,” she declares. She explains further, “There are 10 possibilities for hundreds,” and lists several: 101, 111, 121. These ten possibilities for each of the nine numbers in the hundreds...” As she explains, she lists, “100, 200, 300, 400, 500, 600, 700, 800, 900,” and states, “ten possibilities for each of these, so 10 times 9 is 90 possibilities.” 79 Problem 3 A puflndromw nu,nI,er is one that reads the same fron.i left to right and from right to left. An 1 10 and 00,000? example is 46764. flow many palindromic numbers are there between ZLZ U “ F (20 .4 tot Itt lit cr :o ii Ia :0 S.f . 1. :t ,&O trnS 4 uc lot / Zpy \O ‘. ?C( 14 cI Pt- %•,//(‘ c) (ti . t 22. v ) 4 J SI a 0 Figure 14. Tanja’s written work for problem 3. 80 ? i2 \5° Moving on to the thousands, Tanja says, “It would be the same thing as [the hundreds] because.. if you’re going to have 1001, these two [middle digits] can’t differ if . you’re going to have a palindrome, so you can only have 1001, 2002.. .all the way to 9999.. .only 90 possibilities.” Tanja seems to have a good grasp of this problem and is clearly confident in her abilities. For the ten thousands, Tanja quickly explains, “100 possibilities for 1 [i.e. 10 001, 10 101, 10 201, 10 301, ..., 19 991], there are nine of these [i.e. with a first and last digit of 1, 2, 3, 4, 5, 6, 7, 8, and 9], so that would be 900 here.” Recognizing a relationship between the five-digit and six-digit numbers like the relationship between the three-digit and four-digit numbers, Tanja correctly asserts that there would be 900 palindromic numbers in the hundred thousands. She explains this as follows: “The second and fifth digits would be the same number, and [the] third and fourth have to be the same number, so ten variation on [the second and fifth digits] and ten variation on [the third and fourth digits] would mean there’s 100 for every 1, and there’s nine possibilities [for the first and sixth digits].” Tanja then adds to arrive at a final total of 1989, and checks her arithmetic on a calculator. Although Tanja’s strategy is logical and she worked efficiently, she arrives at an incorrect answer because she includes numbers greater than 100 000. Due to Tanja’s proficiency with this problem, this incorrect solution may be attributed to her haste and carelessness in reading the problem and consequently her not realizing what the problem is asking for. 4.3.4 An Analysis of Tanja’s Approach Tanja approaches the first problem by opting for a strategy that she refers to as “guess and check”. Like Jacqueline’s guess and check strategy, Tanja’s guess and check is similarly 81 systematic and goes beyond arbitrary guess and check as described by Johanning (2007). In fact, Tanj a’s strategy involves the implementation of a procedure that draws on numerical patterns and incorporates very little guessing at all. Tanja’s strategy in this first problem is different from Jacqueline’s strategy. Whereas Jacqueline’s strategy in completing this problem may be described as effective, Tanja’s strategy possesses a degree of sophistication that Jacqueline’s was lacking. Tanj a seems to see relationships in the numbers used in this problem where the other participants did not, and uses these relationships in arriving at her solutions to the problem. An example of this occurs in this problem where she starts with nine 7-slice pizzas. She uses the idea that replacing a 7-slice pizza with an 8-slice pizza will increase the number of slices by one, while replacing a 7-slice pizza with a 6-slice pizza will decrease the number of slices by one to manipulate the total number of slices without changing the total number of pizzas. Tanja appears to be developing her solution strategy as she goes, guided by her confidence and her strong understanding of the problem at hand. Tanja goes as far as she can using the methods she has developed, arriving at the four solutions to this problem. At each step, Tanja checks her solutions to see that they satisfy the conditions laid out in the problem. Those solutions that do not satisfy the conditions in the problem are rejected. Looking at her solutions, Tanja intuitively deduces that there are more solutions to this problem. It is unclear why she believes this to be true. However, according to Eizenberg & Zaslavsky (2004), evaluating the reasonableness of an answer is a valid method for verifying solutions. Tanja goes back to her methods, but when they fail to produce further solutions to the problem, Tanja seems to attempt a more traditional guess at the solutions before giving up. Although Tanja has found all four solutions to this problem, she believes that there are more that she was unable to find. In problems like this one where 82 the number of solutions is not explicitly known by the solver, it is consequently difficult for the solver to know when they have finished solving the problem. In these instances, the solver needs to verify not only that their solutions are correct, but also that no other solutions exist. Therefore, the verification process is a critical step in these types of problems. Furthermore, I believe that identification of the correct and complete final solution as such is an important last step to problem solving. In Tanja’s case, she effectively rules out several invalid solutions through verification. However, her misconception that “there are six solutions”, and her belief that her solution is incomplete, diminishes the strength of the solution as a whole. Ideally, Tanja should be able to identify that the four solutions she has found form a complete solution to the problem through verification, without allowing her intuitive beliefs about the nature of the problem to influence her final solution. Tanja’s strong mathematical understanding is once again evident in her solution to the second problem. After reading and taking a few moments to understand the problem, Tanja attacks the problem and quickly develops an organized and elegant solution to the problem. Tanja’s solution is very systematic and it is clear that she has a good grasp of the problem. Like the other participants, Tanja splits the problem into three parts: permutations of the first three digits, permutations of the last two digits, and combining the results from these two parts to arrive at a final answer. In generating the permutations of the first three digits, Tanja presents a simple yet very creative procedure that should yield the desired results very quickly. However, in her haste, she repeats some of her permutations of the first three digits and consequently arrives at an incorrect answer to the problem. Based on her proficiency and clarity in working through the problems in this study, it is likely that this relatively simple error would not have occurred if Tanja took more care in the 83 implementation of her procedure for producing the permutations of the first three digits. Furthermore, this error would very likely have been caught if Tanja had spent more effort verifying her work critically. Tanj a’s strategy in producing the permutations of the last two digits is equally clear. She systematically lists all the permutations that start with 0, then 1, then 2, then 3, then 4, and she finds the total number of permutations. Finally, she multiplies the number of permutations of the first three digits with the number of permutations of the last two digits. It seems obvious to her that multiplying these two numbers will yield the solution to the problem. As was the case with the first problem, Tanja also refers to her solution method in this problem as “guess and check”. However, the only step that could possibly be described as “guessing” takes place at the beginning of the problem, where Tanja works within the constraints of the problem to generate the four combinations of the first three digits. In fact, it can easily be argued that this is not guessing at all, since she is using her mathematical knowledge to produce these combinations. Tanja’s solution to the third problem is equally organized. Furthermore, she does not appear to struggle with this problem as the other participants did. Interestingly, Tanj a does not refer to “guess and check” in the solution to this problem. After reading the problem, Tanj a starts listing palindromic numbers, starting with 11, seemingly to understand the problem and the nature of palindromic numbers better. She quickly realizes that there are nine two-digit palindromic numbers, ten three-digit palindromic numbers that begin with 1, ten four-digit palindromic numbers that begin with 1, one hundred five-digit palindromic numbers that begin with 1, and one hundred six-digit palindromic numbers that begin with 1. She initially believes that this is all that is required in this problem, but almost immediately realizes that she has neglected to consider three-digit, four-digit, five-digit, and six-digit 84 palindromic numbers that do not start with 1. She goes back and, once again in a very organized and logical manner, produces the correct number of each type of palindromic number. In doing this, she displays her strong number sense and her ability to see the numerical relationships. Unfortunately, for reasons unknown to the researcher, Tanja includes six-digit palindromic numbers when the problem was asking for two-digit to fivedigit palindromic numbers only. This error is possibly due to the wording of the problem. In reading that the problem is asking for palindromic numbers “between 10 and 100,000”, it is possible that Tanja mistook this to mean all palindromic numbers with two to six digits. If this is the case, then it is an example of Hadar and Hadass’ (1981) first obstacle to solving combinatorial problems: misinterpreting what the question is asking for. In addition, if Tanj a took more care and attention to understanding the problem (Polya’ s (1957) first step) before proceeding, it is likely that she would not have made this relatively simple error. Furthermore, it is also possible that this error would have been detected if she had returned to the word problem during the solution process to verify that she was on the right track. Regardless of this error, Tanja’s solution strategy is organized, efficient, and clear. 85 Chapter 5 DISCUSSION AND CONCLUSION The preceding chapters present an analysis of the mathematical strategies employed by three participants as they work through a series of combinatorial problems. This chapter will take an in-depth look at the strategies employed by the participants and the roles that these strategies play as each participant worked through the series of combinatorial problems. In addition, this chapter takes a step back, reiterates some of the maj or observations made in this study, and synthesizes them to produce a “big picture”. Finally, this chapter also discusses possible implications of this research for mathematics teachers and their teaching practices, as well as some unanswered questions that may lead to further research. 5.1 Problem Solving Strategies and Trends The three participants employed a variety of strategies in attempting the three problems in this study, leading to varying degrees of success. Some strategies were more organized and elegant than others. As the problem solving sessions were reviewed, several strategies and procedures stood out as attention grabbing or thought provoking. Some of these verified what other researchers discussed in the literature. Others were interesting, intriguing, or unexpected, as they relate to solving combinatorial problems. The strategies identified may not be specific combinatorial problem solving strategies. Instead, they are problem solving strategies that were observed in the context of solving combinatorial problems. These strategies are the use of algebraic notation and methods, guessing and 86 checking, looking for patterns, decomposing the problem into smaller parts, and verification behaviour. 5.1.1 Using Algebraic Representations and Methods In most cases, participants instinctively started each problem by representing it with an algebraic expression. In some cases, participants proceeded to attempt a solution through algebraic means. This was likely due to the strong emphasis on algebraic representation and manipulation throughout the secondary school curriculum in general and in secondary school classrooms in particular, and is consistent with Shotsberger’s (1993) observation that students found some problem solving strategies to be more natural than others. Although it was not taken into consideration when the problems for this study were selected, it turns out that none of the three combinatorial problems in this study were conducive to this strategy. In hindsight, this development was fortuitous because it forced the participants to think about and understand the problems outside of their mathematical comfort zone. As students approached each problem, they quickly realized that they were required to explore alternative avenues to the solution rather than simply applying a method in which they have had a great deal of practice. In fact, of the nine problem solutions attempted by the participants in this study, eight could be deemed successful to some extent. The only one that can clearly be classified as a failure is Andrea’s attempt at problem 1. In attempting this problem, Andrea relied exclusively on algebraic methods. This led to frustration and, ultimately, her giving up on the problem. Simply put, participants in this study needed to stray from the comfort of algebraic strategies in order to make any significant progress towards solutions to these problems. 87 Although participants generally seemed to realize very quickly in most instances that algebraic methods would be ineffective when tackling these problems, participants still used algebraic representations in some cases to clarify their understanding of the respective problem. This was especially evident in problem 2, where each of the three participants expressed the problem in terms of algebraic expressions. In doing so, participants seemed to be eliminating the “wordiness” of the problems and focusing on the mathematical relationships implied in each respective problem. Although this may be an effective method of understanding those mathematical relationships and developing a solution strategy, I believe that there is a danger of losing the context of the problem amongst the numbers and symbols. Once the mathematical relationships were clear and it became evident that to proceed with an algebraic strategy would only lead to a dead end, participants tended to refer back to the problem and to revise their approach with their new-found understanding of the problem in mind. Therefore, algebraic symbols were used in order to gain insight into the problem, but this representation was not carried through to a solution. 5.1.2 Guessing and Checking According to Johanning’s (2007) definition, guess and check is an iterative problem solving strategy where an initial guess at a solution is placed into the context of the problem and checked to see if it yields the desired solution. If it does not, then it is used as a guide to a better guess, which, when placed in the context of the problem, would ideally lead to something closer to the desired solution, if not the solution itself. This process continues until the solution to the problem is reached. This strategy is useful for solving relatively simple problems, but not effective for solving more complicated combinatorial problems. In 88 fact, problems 2 and 3 in this study were not conducive to this strategy at all because, even if a reasonable guess at a solution was found, there was no way to check the plausibility of that guess. Going strictly by Johanning’s definition, problem 1 is the only problem where guess and check may lead to a solution to the problem. The only participant to employ guess and check in a way that approached Johanning’s (2007) definition was Tanj a, during the initial stages of her solution to problem 1. In this case, it seems that Tanja successfully used guess and check to arrive at one of the four solutions to this problem, and possibly to better understand the nature of the problem. Although participants in this study frequently referred to their solution strategies as “guess and check”, few of then actually conformed to the strategy as described by Johanning. In general, participants in this study seemed to use the term “guess and check” to describe any solution method where there was no clear step-by-step process which led to the desired solution. When faced with this dilemma, participants tended to use the information presented in the problem to develop a starting point to the solution. In problem 1, this starting point generally took the form of a specific number of pizzas. However, this starting point was by no means an initial guess at a solution. Instead, it was a first step in a strategy that would, hopefully, lead to an initial solution. Since problems 2 and 3 asked for the number of permutations, and not the permutations themselves, an initial numerical guess was very difficult and its accuracy could not be gauged. Therefore, the starting point for each of these two problems was not a numerical value, but a “guess” at the first step of a solution strategy. In all three cases, participants tried to develop procedures that led to solutions. As these solutions emerged, they were “checked” with the information presented in the problem for consistency. Therefore, participants were not guessing at solutions and checking them with 89 the problem. Instead, they were guessing at solution processes and checking the solutions these processes produced with the problem. As procedures were manipulated and solutions emerged, participants simultaneously looked for more elegant solution strategies. These often took the form of patterns in the mathematical relationships that became apparent. In essence, what the participants referred to as guess and check was really an intermediary stage where the nature of the mathematics in the problem was explored in hopes of arriving at a more systematic, elegant, and efficient means of arriving at the solution. It was a method that would ideally lead to a “better method”. In general, it seems that the participants considered this “better method” to be one where a pattern in the mathematical relationships was identified and used to arrive at a solution efficiently. 5.1.3 Looking for Patterns It appears that, once participants realized that algebraic methods would not lead to the desired solutions in these combinatorial problems, they generally gravitated toward the identification of a pattern as their preferred solution method. However, patterns within the mathematical relationships of each respective problem were generally not obvious. Therefore, participants needed to explore the mathematical relationships within each problem in order to determine some practical patterns, which would ideally lead to the solutions they were looking for. This exploration stage was critical, and was referred to as “guess and check” by the participants. Furthermore, it often involved the development of a partial list of solutions that was used to gain an understanding of the problem and to identify patterns. Throughout this exploration stage, as indicated earlier, participants used information in the respective problems to developed procedures that would lead to a solution. More 90 importantly, however, they regularly examined their solutions and the procedures that led to them, with the hopes of identifying a mathematical pattern that would streamline the pathway to the solution. Participants who were successful in ultimately recognizing a pattern within a given problem would generally re-focus their attention to employing that pattern to arrive at a solution. This was evident as Jacqueline solved problem 2. When she recognized a pattern that would lead to a solution, she stated, “so there is a pattern”, and used the pattern to solve the problem. However, this explicit pattern recognition and subsequent employment of the pattern to reach a solution was not always present. As students explored each problem, it seems that participants were generally not taking steps or developing a procedure that would lead to the recognition of a pattern, but were hoping that a pattern would suddenly emerge and become evident. Participants who were not successful in identifying a pattern, as was the case with Jacqueline in problem 1 and Andrea in problem 2, resorted to following through with the existing solution strategy that had been developed. Although the participants were generally unsuccessful in using explicit pattern recognition as a primary strategy in arriving at a solution, the participants frequently recognized numerical patterns in the problems and used them in their strategies. For example, Tanja streamlined her solution to problem 3 by listing the first several palindromic numbers in the hundreds that begin with 1, then recognizing the pattern and subsequently determining the number of palindromic numbers in the sequence without having to complete the list. Furthermore, participants identified numerical patterns in the exploratory “guess and check” phase and used them to maintain some structure and organization in the procedure. These inherent uses of patterns within the strategies employed by the participants were common, and participants used them without explicitly stating that they were using patterns. 91 5.1.4 Breaking Up the Problem In conjunction with the search for recognizable mathematical patterns, participants also approached these problems by breaking them up into smaller, simpler cases. This is not surprising, since several research studies concluded that subdividing into cases was the most common problem solving strategy observed (Glass & Maher, 2004; Watson, 1996). In addition, De Hoyos et al. (2004) noted that this is a behaviour that results when problem solvers experienced uncertainty. They reasoned that looking at simpler cases reduced complexity, kept the situation manageable, and helped problem solvers to understand the problem. Furthermore, Sriraman (2003) observed that gifted students, whom are generally better problem solvers than other students, generally approached problem solving situations by starting with simpler cases of the problem at hand. The use of simpler cases was especially evident in problems 2 and 3. In problem 2, all three participants separated the 5-digit combination described in the problem into two parts: the first three digits and the last two digits. Participants found the number of permutations of each case while considering the constraints presented in the problem (often by using patterns), and combined these to arrive at a final solution. The use of cases in this problem was an obvious choice, given how the information in the problem was presented. However, it was interesting that participants knew to multiply the results from the two cases to arrive at their final solution even though these participants have not had formal instruction in combinatorics. Unfortunately, none of the three participants verbalized why they chose to multiply here. English (2005) found that the use of repeated addition when multiplication was a more efficient operation was a major obstacle for the young participants in her study as 92 they solved combinatorial problems. Evidently, the adolescent participants in this study have overcome this obstacle. Problem 3 was also conducive to the use of cases, even though this was less obvious from the wording of the problem. Participants generally separated this problem into four cases: two-digit, three-digit, four-digit, and five-digit palindromic numbers. Tanja also erroneously considered six-digit palindromic numbers. Within each of these cases, participants used numerical properties and patterns to determine the number of palindromic numbers. They then added the results from the four cases to arrive at their final answer. Again, it was interesting that the participants knew to add the results from each case here, given that they multiplied in problem 2. It seems that all three participants were tapping into their intuitive understanding of numerical relationships and used the context of the problems to make sense of those relationships. None of the subjects explained why they chose these operations as they were performing them, and it is unclear whether they would be able to articulate this reasoning if asked. In solving problem 1, the participants generally did not break up the problem into simpler cases in any significant way. This may be due to the nature of the problem. Whereas problems 2 and 3 were presented in a way such that cases are easily recognizable, problem I asked for several specific permutations under a set of given conditions. Furthermore, whereas problems 2 and 3 asked for a number of permutations, problem 1 asked for the permutations themselves. Interestingly, it appears none of the three participants even attempted to break up problem 1 into smaller parts. Participants did not express why they did not choose this path as an option. It is possible that intuition played a role in this decision as well. Finally, the fact that problem 1 was a partition problem according to Batanero et al.’s 93 (1997) implicit combinatorial model whereas problems 2 and 3 were selection problems may have had an impact as well. This study does not examine specific strategies that may be more conducive to each of Batanero et al.’s three classes of combinatorial problems. Further research into this aspect of strategies for solving combinatorial problems would be of interest. As they progressed through problems 2 and 3 in particular, it appeared that the participants generally broke up each problem and employed simpler cases wherever they could. However, it is interesting to note that none of the participants described solving a simpler problem as a solution strategy. It appears as though the use of simpler cases was an instinctive, unconscious step that occurred for these participants during problem solving. Furthermore, it seems that these participants had a good sense of where the employment of simpler cases would be possible and advantageous, and where it would not be. 5.1.5 Verifying Regardless of the strategy, an ongoing verification process was consistently observed throughout the problem solving sessions. It should be noted that the three problems in this study were conducive to verification. Thus, is may not be entirely surprising that the participants often looked back and critically examined their strategies to ensure that the processes in which they were engaged were valid and that they were progressing towards the desired solution. This process seems reminiscent of metacognitive processes described by Garofalo and Lester (1985) and by Montague and Applegate (1993). In addition, Lesh and Harel (2003) examined the iterative process of testing and revising during problem solving and Pape and Wang (2003) discussed the cognitive monitoring behaviour of their subjects. 94 There are clear parallels between the findings of these two latter studies and the observations made in this study. As the participants verified the validity of their reasoning, they also verified the accuracy of their mathematics. There seemed to be a positive correlation between the frequency of this verification and the degree of success experienced by the participant. Based on the creativity and elegance that Tanja displayed in solving the problems, it could be argued that Tanja demonstrated the strongest mathematical understanding of the three participants. However, she ultimately arrived at the wrong answer to two of the three problems. Tanja’s errors were relatively minor in nature and I do not believe that they reflected a lack of mathematical understanding. Instead, I believe that they could be better described as careless errors (often arithmetic in nature) that occurred possibly due to haste. Therefore, it is very likely that she would have caught these errors if she had spent more effort in verifying her work. On the other hand, Jacqueline, whose mathematical understanding does not seem to match that of Tanja’s, generally verified her work more often than Tanja. During this ongoing verification process as she progressed through each problem, Jacqueline was able to detect errors in her calculations and in her reasoning. This led to ongoing adjustments and corrections to her solution. As a result, Jacqueline successfully solved two of the three problems. It appears that verification occurred on two levels: a conceptual level, where participants were verifying their strategies and reasoning to ensure that they were “on the right track”, and a procedural level, where participants were verifying their mathematical operations in search of mathematical errors. It seemed that conceptual verification is used as guidance towards an effective solution strategy, whereas procedural verification was 95 important in arriving at the correct solution. Thus, Jacqueline, who engaged in verification more frequently throughout her solution processes, was generally more successful in arriving at the correct solution than Tanja, who appeared to have a stronger mathematical background, but did not verify her workings as frequently. Interestingly, Tanja completed the Principles of Mathematics 11 course with a higher grade that Jacqueline. If there is indeed a positive correlation between the frequency of verification during a problem solving task and the degree of success in problem solving, then these grades are anomalous with the findings in this study. It is possible that course grades alone do not accurately measure problem solving ability. Further research is required to examine the nature of this discrepancy. “Looking back” (or verification) was listed as the fourth step in Pólya’s (1957) model for problem solving. However, it should not necessarily be viewed as the fourth of four sequential steps. From this study, it was observed that all four steps in Pólya’s model seemed to happen concurrently. As each participant worked through each problem, they appear to jump back and forth between understanding the problem, devising a plan, carrying out the plan, and looking back. 5.2 The Big Picture Due to the size of the sample, this research should primarily be considered as an in depth case study of the specific participants involved. The intent was to develop a thorough understanding of the processes that three specific students engage in as they attempt to solve a series of combinatorial problems. Keeping in mind that these three students are by no means representative of any large population of learners, care must be taken not to generalize 96 beyond the limited population represented by this small group of participants. In particular, one must keep in mind that the three participants in this study were all completing their grade 11 year at a public school in British Columbia during the time of this study, had similar educational and socioeconomic backgrounds, and were all relatively successful as mathematics students. The goal of this research was to investigate the strategies that students turned to as they attempted to solve a series of combinatorial problems. From the observations made in this study, it was clear that students without formal training can be successful in solving combinatorial problems. It should be noted that the three problems used in this study were taken from resources that were intended for younger students. Given the difficulty that the participants in this study experienced in arriving at correct solutions, it could be said that combinatorial problem solving is not necessarily easier with more mathematical experience or understanding. Furthermore, it seemed that most combinatorial problems can be challenging to a wide range of ages and abilities. It would be interesting to observe how younger students solve the same three problems used in this study and to compare their strategies with the ones used by these participants. Although none of the three participants in this study had any formal training in solving this class of mathematical problems, they generally approached each problem in a similar fashion. Each participant instinctively turned to algebraic representations as a starting point. This seems to be a reflection of their mathematical training as well as a possible indication of their strengths and comfort levels. However, as it became clear that algebraic strategies would not work, participants turned to guess and check, use of numerical relationships and patterns, and decomposing the problem into smaller parts. It seems 97 participants believed that a systematic procedure may be effective in ultimately producing a solution, but the identification of a pattern would yield efficiencies in the solution process that a cumbersome solution strategy would not. Therefore, it appears that efficiency was a consideration in the minds of these participants as they solved combinatorial problems. In solving the problems, it should be noted that transferability of strategies from one problem to a subsequent problem was not observed. It seems that the participants treated each of the three problems independently from each other, and the strategies and methods used by each participant in solving one problem did not influence the strategies and methods used in solving subsequent problems. 5.3 Implications for Teaching Based on each participant’s initial approach to each problem in this study, it appears that algebraic methods and strategies were foremost in the minds of these participants. However, it was unclear whether this was due to an over-emphasis on algebraic manipulation in the curriculum, and therefore by classroom teachers, or whether students in general, and these three students in particular, tended to prefer algebraic strategies despite being instructed in a breadth of mathematical methods. It is likely that the reality lies somewhere between these two extremes. Taking a step back, it should be stated that there is nothing inherently wrong with algebraic manipulation. I believe that it is certainly a valuable tool in solving many mathematical problems. However, as has been demonstrated in this study, algebraic methods are not universally effective for all mathematical problems. Furthermore, it is but one branch on the vast tree of mathematics. Therefore, as classroom teachers, I believe it is important to ensure that we are presenting the prescribed curriculum in a balanced manner. 98 We should teach algebraic methods, but we need to give adequate instructional time and resources to other mathematical methods as well, particularly when it comes to problem solving. I believe it would be beneficial to instil in our students the notion that algebra is but one of a number of strategies that may be used in solving mathematical problems. In addition, it may not be the most effective or most efficient one. Despite the fact that the participants in this study were generally able to solve the problems without using algebraic methods, it would have been nice to see a more balanced approach, where the initial approach may not have been algebraic in nature. As is, it seems that the participants viewed other methods and strategies as second-class approaches that they could turn to when algebraic methods failed. In addition, classroom teachers should provide opportunities for students to experiment with a variety of problem solving strategies. It is too often the case that problem solving is embedded in lessons and units that emphasize one particular mathematical skill over others. This may be valuable in reinforcing a specific solution strategy over others, but alternative solution strategies should be discussed. believe that students need to be exposed to a variety of mathematical skills and strategies, and they should not be penalized for correctly solving a mathematical problem in a way that is inconsistent with the context of the curriculum content in which it is embedded. Furthermore, over time, efforts should be made to explore the pros and cons of different problem solving strategies in detail. Garofalo and Lester (1985) referred to Pólya’s (1957) four steps to problem solving as orientation, organization, execution, and verification. The importance of this last step was illustrated in this study. The data presented here seems to indicate that there is a positive correlation between the frequency of verification of one’s work and the likelihood of arriving 99 at a correct solution. Verification was effective in catching conceptual and procedural errors and in guiding students to the correct solution. Unfortunately, Pugalee (2004) observed that, when presented with a mathematical problem, students generally seem to overlook orientation, organization, or verification behaviours, in favour of execution behaviours. In this study, verification was somewhat lacking in some cases as participants solved problems 2 and 3 even though the nature of these two problems made them more conducive to the verification process than other mathematical problems. Therefore, in teaching general mathematical problem solving skills and strategies, classroom teachers should emphasize the important role of verification, along with orientation and organization, in addition to execution. I believe that it would be beneficial to students if they understand that all four of Pólya’s steps are equally important to successful problem solving. Furthermore, as explained by Passmore (2007), students should be made aware of and given opportunities to practice metacognitive processes such as self-regulation and monitoring that facilitate success in problem solving. 5.4 Future Directions This case study provides a detailed analysis of three specific participants as they manoeuvre their way through a series of combinatoric problems. The participants themselves were uniform in terms of educational background and academic success. Therefore, generalizations that come from this study are limited to students with similar educational backgrounds and academic successes. In order to develop a better understanding of problem solving strategies employed by a larger cross-section of students, further studies would be required. Based on the results of this study, several ideas for possible future study 100 come to the forefront. First, it has been noted that algebraic methods played a big part in the repertoire of problem solving strategies employed by the participants in this study. It would be interesting to examine how younger students without strong algebraic skills approach the same problems. Furthermore, it has been speculated in this study that the tendency towards algebraic methods by the participants was, in part, due to the emphasis on algebraic manipulation in most secondary school classrooms. The validity of this speculation should be examined in detail. In particular, it would be of interest to look at when, why, and how algebra “takes over” as the problem solving strategy of choice as students progress through school. Secondly, grade 11 students were selected for this study because they have had no formal training in the area of combinatorics. In British Columbia, this topic is covered in Principles of Math 12. It would be enlightening to examine the actual differences in problem solving strategies between students who have had formal instruction in combinatorics and those who have not. Specifically, it should be noted that none of the students in this study employed any combinatoric formulas when solving the problems. This was expected, since they have not been exposed to these formulas yet. Batanero et al. (1997) noted that students who have been exposed to formal instruction tended to rely solely on the use of formulas. It would be of interest to examine how this formal instruction affects students’ conceptual understanding of this class of mathematical problems. Finally, the participants in this study were described by their teacher as generally very good mathematics students, but not exceptional. Due to their relative uniformity, it was not surprising that their approaches to each problem were generally similar in nature. In comparison, one should also examine the strategies employed by mathematically weaker students and mathematically gifted students 101 with similar educational backgrounds in order to develop a better understanding of the mathematical trends of the larger population. In this study, it was observed that intuition seemed to play an important part in guiding the participants through each respective problem. For example, participants knew when to add and when to multiply in problem 2 in order to arrive at the correct solution. If questioned, it was unclear whether participants would be able to clearly articulate why they chose these operations. It would be interesting to examine this idea of mathematical intuition and sense making in students in greater depth. Specifically, the question of how this intuition develops, how it emerges during problem solving, and how it is sometimes abandoned for more formal, less understood, procedures needs to be explored. Despite the similarities between the three participants, each participant experienced a different level of success with the series of three problems. Jacqueline’s success was partly attributed to her frequent verification tasks as she progressed through each problem. In contrast, Tanja, who generally displayed more mathematical elegance and skill than Jacqueline, experienced less success than Jacqueline did because she verified her work less frequently. This study concluded that there is a positive correlation between the frequency of verification tasks and the likelihood of successful problem solving. This idea needs to be investigated in detail. Further study should look into the cognitive aspects of how verification tasks come about, the specific role of verification in problem solving, and, if it is indeed beneficial to successful problem solving in mathematics, how to instil this habit into mathematics students. In addition, the metacognitive differences between conceptual verification and procedural verification should be examined in detail. 102 In this study, it seemed that the nature of the problems themselves may have had an impact on the solution strategies. This is consistent with Garafalo and Lester’s (1985) research, which identified “task variables” (i.e. features of the problem-solving task) as one of three metacognitive components to problem solving. In this study, guess and check (as defined by Johanning, 2007) was more evident when participants were solving problem 1. In contrast, this strategy did not present itself as students worked through problems 2 and 3. In fact, it has been noted that guess and check would not be effective in solving problems 2 and 3, due to the nature of these problems. It should also be noted that problem 1 is classified as a partition problem by Batanero et al. (1997), whereas problems 2 and 3 are classified as selection problems. It is unclear whether this was coincidental or if there was some sort of correlation between these two factors. This point needs further examination. In addition, Batanero et al.’s third class of combinatoric problems, referred to as distribution problems, was not represented in this study. It would be interesting to look at each of Batanero et al.’s three classes of combinatoric problems from the perspective of the mathematical strategies that may be most conducive to each. Furthermore, it would be informative to investigate whether students approach each class of problems differently. If so, then we must also consider what those differences are. The area of combinatorics ideally allows for a variety of strategies in solving any one problem. This study showed that some strategies are favoured over others. In general, combinatoric problems provide an excellent source of material for research into mathematical problem solving strategies. As combinatorics is a relatively new branch of mathematics, the volume of research in this area is small in comparison to other, more 103 established branches of mathematics. Therefore, more research in this area is required in order to increase our understanding of our students’ problem solving strategies. 104 REFERENCES Batanero, C., Navarro-Pelayo, V, & Godino J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181-199. Biryukov, P. (2004). Metacognitive aspects of solving combinatorics problems. International Journalfor Mathematics Teaching and Learning. Retrieved May 27, 2006, from http ://www.cimt.plymouth.ac.ukljournal/default.htm De Hoyos, M., Gray, E., & Simpson, A. (2004). Uncertainty during the early stages of problem solving. In Proceedings ofthe 28111 Conference ofthe International Group for the Psychology ofMathematics Education, Bergen, Norway, 255-262. Eizenberg, M. M., & Zaslavsky, 0. (2004). Students’ verification strategies for combinatorial problems. Mathematical Thinking and Learning, 6(1), 15-36. English, L. D. (1991). Young children’s combinatoric strategies. Educational Studies in Mathematics, 22, 541-474. English, L. D. (2005). Combinatorics and the development of children’s combinatorial reasoning. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 121-141). New York: Springer Verlag. Fisher, L. (1982). Super problems. Palo Alto, CA: Dale Seymour Publications. Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B. Resnick (Ed.), The nature of intelligence (pp. 231-236). Hillside, NJ: Erlbaum. 105 Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24 (5), 645-657. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journalfor Research in Mathematics Education, 16(3), 163-176. Giblin, P., & Porteous, I. (1990). Challenging mathematics. Oxford: Oxford University Press. Glass, B. & Maher, C. A. (2004). Students problem solving and justification. In Proceedings of the th 28 Conference of the International Group for the Psychology ofMathematics Education, Bergen, Norway, 463-470. Radar, N., & Hadass, R. (1981). The road to solving a combinatorial problem is strewn with pitfalls. Educational Studies in Mathematics, 12, 43 5-443. Herman, M. (2007). What students choose to do and have to say about use of multiple representation in college algebra. Journal of Computers in Mathematics and Science Teaching, 26(1), 27-54. Johanning, D. I. (2007). Is there something to be gained from guessing? Middle school students’ use of systematic guess and check. School Science and Mathematics, 107(4), 123-133. Kloosterman, P., & Stage, F. K. (1992). Measuring beliefs about mathematical problem solving. School Science and Mathematics, 92 (3), 109-115. Lerch, C. M. (2004). Control decisions and personal beliefs: Their effect on solving mathematical problems. Journal ofMathematical Behaviour, 23, 21-36. 106 Lesh, R., & Hare!, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking and Learning, 5 (2&3), 157-189. Lester, F. K. (1985). Methodological considerations in research on mathematical problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving. Multiple research perspectives (pp. 4 1-70). Hillside, NJ: Erlbaum. Lovaszova, G., & Hvorecky, J. (2002). When there’s more than one way to get there... In Proceedings ofthe Seventh Asian Technology Conference in Mathematics, Melaka, Malaysia, 263-272. Mamona-Downs, J. (2002). Accessing knowledge for problem solving. In D. Hughes-Hallett, & C. Tzanekis (Eds.), Proceedings ofthe ,2 m’ International Conference on the Teaching ofMathematics at the Undergraduate Level), Hersonissos, Greece. Mason, L. (2003). High school students’ beliefs about maths, mathematical problem solving, and their achievement in maths: a cross-sectional study. Educational Psychology, 23 (1), 73-85. Meijer, J., & Riemersma, F. (2002). Teaching and testing mathematical problem solving by offering optional assistance. Instructional Science, 30, 187-220. Meyer, C., & Sallee, T. (1983). Making it simpler: A practical guide to problem solving in mathematics. Menlo Park, CA: Addison-Wesley. Montague, M., & Applegate, B. (1993). Middle-school students’ mathematical problem solving: An analysis of think-aloud protocols. Learning Disability Quarterly, 16 (1), 19-32. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. 107 Pape, S. J., & Wang, C. (2003). Middle school children’s strategic behaviour: Classification and relation to academic achievement and mathematical problem solving. Instructional Science, 31, 4 19-449. Passmore, T. (2007). Polya’s legacy: Fully forgotten or getting a new perspective in theory and practice? Australian Senior Mathematics Journal, 21(2), 44-5 3. Pine, S. E. B. (1996). Classroom video-recording: When, why, and how does it offer a valuable data source for qualitative research? Paper presented at the annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education, Panama City, FL. Polya, G. (1957). How to solve it (2’ ed.). Princeton, NJ: Princeton University Press. Pugalee, D. K. (2004). A comparison of verbal and written descriptions of students’ problem solving processes. Educational Studies in Mathematics, 55, 27-47. Resnick, L. B., & Glaser, R. (1976). Problem solving and intelligence. In L. B. Resnick (Ed.), The nature of intelligence (pp. 231-236). Hillside, NJ: Eribaum. Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 334-370). New York: Macmillan and National Council of Teachers of Mathematics. Schurter, W. A. (2002). Comprehensive monitoring: An aid to mathematical problem solving. Journal ofDevelopmental Education, 26 (2), 22-3 3. Serafino, K., & Cicchelli, T. (2003). Cognitive theories, prior knowledge, and anchored instruction on mathematical problem solving and transfer. Education and Urban Society, 33 (1), 79-93. 108 Shotsberger, P. G. (1993). An approach to analyzing the videotaped problem solving behaviours of individual students. Paper presented at the annual meeting of the American Educational Research Association, Atlanta, GA. Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students. Journal of Secondary Gifted Education, 16(3), 151-165. Van Dooren, W., Verschaffel, L, & Onghena, P. (2003). Pre-service teachers’ preferred strategies for solving arithmetic and algebraic word problems. Journal of Mathematics Teacher Education, 6, 27-52. Watson, R. (1996). Students’ combinatorial strategies. Teaching Mathematics and Its Applications, 15 (1), 27-32. Zimmerman, B. J., & Martinez-Pons, M. (1988). Construct validation of a strategy model of student self-regulated learning. Journal ofEducational Psychylogy, 80 (3), 284-290. 109 APPENDIX A Copy of UBC Research Ethics Board’s Certificate of Approval tU The University of British Columbia Office of Research Services and Administration BehavcuraI Research Ethics Board Certificate of Approval PRINCIPAl. INVE$TIOMOR DEPARTMENT Anderson, AG Cufficulum Studies INSTITUTIONtRI PIHERF RPSEARGI WILL St CARRIED NUER BO5-0340 our Vancouver School Board CO4NVESTIOATORE Yuan, Gary, Curriculum Studies AGENCIES — TITLE Secondary School Students’ Mathematical Problem Solving Strategies A7ROMAE DATE TERM WS 1 DOCuMENTS WCIUDSI) IN ISlE APPROVAL May 24, 2005, Consent forms / Contact letter The protocol describing the abovenamed project has been reviewed by the Committee and the experimental procedureswere found to be accept able on ethical grounds for research involving human subjects. Approval of the Behavioural Res4ach v one c, rjàilowing: Dr. JaretFrankish, Chair, Dr. Cay Hohmok, Associate Chair, Dr. Susan Rowley, Associate Chair This Certificate of Approval is valid for the above term provided there is no change in the experimental procedures 110 APPENDIX B Parent Consent Form THE UNIVERSITY OF BRiTISH COLUMBIA Fatatky of Education Departrnen of Curriculum Studica 2125. Main Mall Vancouvar, B.C. Canada V6T lZ4 Tel: (604) 822-5422 Fax:(604) 822-4714 UBC Project Title: Secondary School Students’ Mathematical Problem Solving Strategies. Dear Parent or Guardian: I, Gary Yuen, am a graduate student working on my graduate thesis in Mathematics Education under the supervision of Dr. Ann Anderson. I am inviting your child to prticipate as a subject in the research that. will form the foundation of this thesis. The purpose of this study is to learn about students’ mathematical problem solving strategies by observing and listening to students working individually on a mathematical problem. Your chIld is being invited to participate in this research because they fit the age profile of the students I wish to study. If you agree, to your child’s participation in this project. I will observe and interview your child during a series of three short sessions at School, during which he/she will be given one mathematics problem per session to solve. Each session will be approximately 15 to 30 minutes in length and will be videotaped. It is expected that the total amount of time required of your child may be as little as 45 minutes and will not exceed 90 minutes. The specific times and dates for the sessions will be arranged at your child’s convenience. The observations and intervIews will form the raw data that will become the basis of this research. Your child’s identity will be kept confidential through the use of pseudonyms throughout the process. All raw data, including videotaped data, will be locked in a secure location, and will only be accessible by myself and the members of my thesis supervisory committee. Upon completion, this thesis will be a public document. The information gathered in this study will in no way be used to harm or misrepresent your child. Should you consent to your child’s participation in this project, you have the’ right to refuse to be involved or to withdraw at any time. Such withdrawal or refusal to be involved will not jeopardize your child in any way. You do not waive, any of your legal rights ‘by signing the consent form. Since this study occurs outside of class activity on a voluntary basis, if you do not consent to your child’s participation, his/her involvement in normal classroom experiences remains unchanged. If you have any questions concerning any aspect of this project, the procedures to be used or the nature of your child’s involvement, please contact me at 778-896-3713. My thesis supervisor, Dr. Ann Anderson, can be contacted at 604-822-5298. If you have any concerns about your or your child’s treatment or rights as a research participant, please telephone the Office of Research Services at University of British Columbia, at 604-822-8598. Sincerely, -y Gary Yuen Version: April 15, 2005 page 1 of 3 111 ____ ____ ____ ________ ______ ___________________________ ____ ____ ____ ____ THE UNIVERSITY OF BRITISH COLIJMBIA LJjij1 Edueaflrn Department of Curriculum Studies 2125 Main Mall Vanertuver, B.C Canada V6T I Z4 i’d: 6O4 822-5422 Fax: 604) 822.47 4 j. PLEASE RETAIN THIS COPY FOR YOU R RECORDS Child’s name: Please check one box: U I CONSENT to my child’s participation in the study entitled: “Secondary Scho ol Students Mathematical Problem Solving Strategies” as described in the attached letter. U I DO NOT CONSENT to my child’s parti cipation in the study entitled: ‘Seconda ry School Students Mathematical Problem Solving Strategies” as described in the attached letter. Parent/Guardian Signature: Date: . ParentlGuardian’s Name: (Please print) I acknowledge that I received a copy of the letter and consent form for my own files . Parent/Guardian Signature: Version: AprS 15. 2005 page 2. of 3 112 ____ ____ ____ ____ _ ____ ____ ____ ________________________ ____ ____ ____ THE UNIVERSITY OF BRITISH COLUMBiA UBC Feuhy of £ducatian Department ofCurriculum Studies 2125 Main Malt Vntouver, B.C. Canada V6T 1Z4 Tel: (604) $22.5422 Fac: (604) 8224714 PLEASE RETURN THIS CONSENT FOR M TO THE RESEARCHER Child’s name: Please check one box: D I CONSENT to my child’s participation in the study entitled: “Secondary Scho ol Students Mathematical Problem Solving Strategies ” as described in the attached letter. C I DO NOT CONSENT to my child’s parti cipation in the study entitled: “An Analysis of Mathematical Problem Solving in Secondar y Students’ as described in the attached letter. Parent/Guardian Signature: Date: Parent/Guardian’s Name: (Please print ) I acknowledge that I received a copy of the letter and consent form for my own files. Parent/Guardian Signature: VersioO April 15, 2005 page 3 of 3 113 ____
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Problem solving strategies students use when solving combinatorial problems Yuen, Gary 2008
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Title | Problem solving strategies students use when solving combinatorial problems |
Creator |
Yuen, Gary |
Publisher | University of British Columbia |
Date Issued | 2008 |
Description | 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 |
Extent | 2086066 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-03-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0070808 |
URI | http://hdl.handle.net/2429/5535 |
Degree |
Master of Arts - MA |
Program |
Mathematics Education |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2008-11 |
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UBCV |
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