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College students' methods for solving mathematical problems as a result of instruction based on problem… Santos Trigo, Luz Manuel 1990

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COLLEGE STUDENTS' METHODS FOR SOLVING M A T H E M A T I C A L PROBLEMS AS A RESULT OF INSTRUCTION BASED O N PROBLEM SOLVING by Luz Manuel Santos Trigo B.Sc. Instituto Polictecnico Nacional; Mex ico ; 1980 M.A. Universidad Nacional Autonoma de Mex ico ; 1984 M.Sc. Instituto Politecnico Nacional; Mex ico ; 1985 A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DECREE OF DOCTOR OF EDUCATION in THE FACULTY OF GRADUATE STUDIES MATHEMATICS EDUCATION W e accept this dissertation as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1990 ©Manuel Santos, 1 990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M A T H E M A T I C S A N D S C I E N C E E D U C A T I O N The University of British Columbia Vancouver, Canada Date O C T O B E R 1 1 , 1 9 9 0 DE-6' (2/88) ABSTRACT This study investigates the effects of implementing mathematical problem solving instruction in a regular calculus course taught at the college level. Principles associated with this research are: i) mathematics is developed as a response to finding solutions to mathematical problems, ii) attention to the processes involved in solving mathematical problems helps students understand and develop mathematics, and iii) mathematics is learned in an active environment which involves the use of guesses, conjectures, examples, counterexamples, and cognitive and metacognitive strategies. Classroom activities included use of nonroutine problems, small group discussions, and cognitive and metacognitive strategies during instruction. Prior to the main study, in an extensive pilot study the means for gathering data were developed, including a student questionnaire, several assignments, two written tests, student task-based interviews, an interview with the instructor, and class observations. The analysis in the study utilized ideas from Schoenfeld (1985) in which categories, such as mathematical resources, cognitive and metacognitive strategies, and belief systems, are considered useful in analyzing the students' processes for solving problems. A model proposed by Perkins and Simmons (1988) involving four frames of knowledge (content, problem solving, epistemic, and inquiry) is used to analyze students' difficulties in learning mathematics. Results show that the students recognized the importance of reflecting on the processes involved while solving mathematical problems. There are indications suggesting that the students showed a disposition to participate in discussions that involve nonroutine mathematical problems. The students' work in the assignments reflected increasing awareness of the use of problem solving strategies as the course developed. Analysis of the students' task-based interviews suggests that the students' first attempts to solve a problem involved identifying familiar terms in the problem and making some calculations often without having a clear understanding of the problem. The lack of success led the students to reexamine the statement of the problem more carefully and seek more organized approaches. The students often spent much time exploring only one strategy and experienced difficulties in using alternatives. However, hints from the interviewer (including i i metacognitive questions) helped the students to consider other possibilities. Although the students recognized that it was important to check the solution of a problem, they mainly focused on whether there was an error in their calculations rather than reflecting on the sense of the solution. These results lead to the conclusion that it takes time for students to conceptualize problem solving strategies and use them on their own when asked to solve mathematical problems. The instructor planned to implement various learning activities in which the content could be introduced via problem solving. These activities required the students to participate and to spend significant time working on problems. Some students were initially reluctant to spend extra time reflecting on the problems and were more interested in receiving rules that they could use in examinations. Furthermore, student expectations, evaluation policies, and curriculum rigidity limited the implementation. Therefore, it is necessary to overcome some of the students' conceptualizations of what learning mathematics entails and to propose alternatives for the evaluation of their work that are more consistent with problem solving instruction. It is recommended that problem solving instruction include the participation or coordinated involvement of all course instructors, as the selection of problems for class discussions and for assignments is a task requiring time and discussion with colleagues. Periodic discussions of course directions are necessary to make and evaluate decisions that best fit the development of the course. iii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES x LIST OF FIGURES xi ACKNOWLEDGEMENTS xii CHAPTER I: THE PROBLEM 1 Introduction 1 Background to the Study 3 Fundamentals in the Use of Problem Solving Instruction 5 Towards a Problem Solving Epistemology 14 Statement of the Problem 16 Rationale for the Problem 17 Research Questions 20 Limitations of the Study 22 Chapter Summary 23 CHAPTER II: REVIEW OF THE LITERATURE 25 Introduction 25 Interpretation of Mathematical Problem Solving 26 Problem Solving and Heuristics 30 Cognitive Science and Mathematical Problem Solving 33 Schoenfeld's Work on Mathematical Problem Solving 43 Chapter Summary 50 CHAPTER 111: DESIGN AND PROCEDURES 52 Introduction 52 Research Paradigm 52 The Pilot Study 55 First Phase of the Pilot Study 55 Background and Instruments 55 Discussion of the Questionnaire Results 57 Results from the Written Test 58 Summary of the Results of the First Phase of the Pilot Study 61 Second Phase of the Pilot Study 63 iv Rationale for the Second Phase of the Pilot Study 63 Background to the Second Phase of the Pilot Study 64 Analysis 65 Level of difficulty of the problem 66 Progress in obtaining the solution 66 The use of strategies 66 Recalling basic information 67 The use of graphs or diagrams 67 Looking back or checking the solution 68 Summary of the Second Phase of the Pilot Study 69 Reflections on the Second Phase of the Pilot Study 70 Third Phase of the Pilot Study 71 Introduction 71 Analysis 73 Features of Students' Approaches to the Interview Problems 77 Flexibility in Approaching the Problems 77 Consistency in Approaching the Problems 78 Effects of Class Instruction and Textbooks 79 Students' Conceptualization of Mathematics 80 Recalling Mathematical Knowledge from Previous Courses 81 Challenging Students' Difficulties 82 The Role of Context in Mathematics Learning 82 Mathematical Hierarchy and Its Influence in Learning 83 Methodological Elements of the Main Study 85 Subjects of the Study 85 Procedures 85 The Use of the Questionnaire 87 Problem-task Assignments 87 The Use of Interviews 87 The Thinking Aloud Technique 89 The Use of Written Tests 90 Class Observations 91 The Use of Nonroutine Problems 91 Implementation of the Written Tests 92 Limitations of the Written Tests 92 v Selection of the Interviewees 93 Analysis of the Information 94 Trustworthiness of the Study 96 Credibility 97 Transferability 98 Dependability 98 Confirmability 99 Chapter Summary 101 CHAPTER IV: ANALYSIS OF DATA REGARDING THE IMPLEMENTATION OF INSTRUCTIONAL ACTIVITIES ASSOCIATED WITH PROBLEM SOLVING 103 Introduction 103 Analysis of Instructional Activities Implemented in the Classroom 104 Features of the Educational System, the Mathematics Curriculum, and the Interaction between the Instructor and the Researcher 105 Components Associated with Problem Solving that Were Addressed by the Instructor and the Researcher 111 The nature of Mathematics and Problem Solving 111 The Role of the Instructor 112 Classroom Activities 115 The Use of Counterexamples in Problem Solving Instruction 116 The Students' Role Required in Problem Solving Instruction 118 Dynamics of the Classroom The Conceptualization of Mathematics and Learning Activities 119 The Instructor's Conceptualization of Mathematics 120 The Development of Class Interactions Between the Instructor and the Students 123 Evaluation of the Students' Work 127 The Use of Metacognitive Strategies 129 Chapter Summary 132 CHAPTER V: ANALYSIS OF DATA REGARDING THE STUDENTS' DISPOSITION TO PROBLEM SOLVING AND THEIR APPROACHES TO WRITTEN PROBLEMS 134 Introduction 134 vi Results from the Questionnaire 135 Results from the Assignments 142 The First Assignment Polarity Chart 142 Second Assignment The Concept of Function 144 Third and Fourth Assignments The Concepts of Limit and Derivative 146 Fifth Assignment Application of Derivative 148 Assignments on Metacognitive Strategies 149 Written Test 152 Introduction 152 Description of the Test 153 Analysis 154 Final Exam 158 Introduction 158 Analysis of the Students' Responses 159 Chapter Summary 162 CHAPTER VI: STUDENTS ' APPROACHES TO PROBLEM SOLVING 163 Introduction 163 Frame of Analysis 163 Questions Associated with the Frame of Analysis 167 Reading 167 Analysis 167 Exploration 167 Strategy Selection 168 Metacognitive Processes 168 Partial Evaluation and New Considerations 168 Planning and Implementation 169 Verification and Extension 169 Transition Period 169 Transfer Levels 169 Belief Systems 169 Categorization of the Students' Approaches to Problem Solving 170 vii Analysis of Patterns of Misunderstanding Exhibited by the Students 181 The content frame 182 The problem solving frame 182 The epistemic frame 182 The inquiry frame 182 Chapter Summary 187 CHAPTER VII: CONCLUSIONS, IMPLICATIONS, AND RECOMMENDATIONS 189 Feasibility of Mathematical Problem Solving Instruction 189 Consistency of Instructional Activities 189 The Students' Participation 190 The Extension of the Curriculum 191 Routine and Nonroutine Problems 192 The Use of Small Groups 193 The Evaluations of Students 194 The Use of Metacognitive and Cognitive Strategies 195 Instructional Directions 196 The Students' Expectations 198 Summary of Conclusions of the Main Study 200 Implications 203 Recommendations 208 Future Research 210 REFERENCES 213 Appendices for the Pilot Study Appendix PI Questionnaire 235 Appendix P2 Written Test 238 Appendix P3 Application of derivative 240 Appendix P4 Interview Problems for the Pilot Study 245 Appendixes for the Main Study 246 Appendix M l A Vee Diagram of the Study 246 Appendix M2 The Questionnaire 247 Appendix M3 Examples of Nonroutine Problems Included in the Assignments 251 Appendix M4 Metacognitive Assignments 257 Appendix M5 The Written Test 259 Appendix M6 Questions for the Instructor Interview 260 Appendix M7 First Interview Letter 262 Appendix M8 Second Interview Letter 263 Appendix M9 Introduction to the Interview 264 Appendix M10 Problems for the interview 265 Appendix M11 Organization of the Interview 266 Appendix M l 2 Second Part of the Interview 267 Appendix M l 3 Solutions to the Interview Problems 268 ix LIST OF TABLES Table 1. Overview of Activities Undertaken in the Main Study and Their Scheduling 86 Table 2. Selected Items of the Students' Responses to the Questionnaire 141 Table 3. Summary of the Results of the Assignments 149 Table 4. Results of the Final Exam 162 x LIST OF FIGURES Figure 1. The Abstraction Process: The Mathematical Idealization 9 Figure 2. Essential Contributors to Progress in Mathematics Instruction 45 Figure 3. Components of Data Analysis: Flow Model 100 Figure 4. Sources of Data and Considerations in Their Analysis 134 xi ACKNOWDLEGMENTS I would like to express my sincere appreciation and deep gratitude to those who directly provided me with invaluable guidance in the course of my graduate studies. Dr. Thomas Schroeder, my research supervisor, who was always available to discuss issues helped me immeasurably to develop and carry out the study. His expertise in mathematics and research methods was always evident in his advice. His swiftness and clarity to pin point the strengths and limitations of various aspects that emerged in the course of the study were crucial in focusing the direction of the study. His continual support and enthusiasm have made this period of my life more meaningful and always memorable. I would like to thank the rest of my committee, Dr. Walter Szetela and Dr. David Whittaker, for their insightful suggestions and constant encouragement. Their presence was deeply felt during my committee meetings. I am indebted to Dr. Z. A. Melzak for sharing his expertise in problem solving and advising me in the development of the study. His passionate commitment to revealing the history of mathematics and his elegance in presenting his mathematical knowledge remain sources of great inspiration to me. Of course, I am indebted to my fellow graduate students at the University of British Columbia with whom I discussed the ideas of the study. Their input and criticism helped me to clarify my ideas. I would like finally to thank Ms. Dianne Fouladi for her unconditional support, encouragement, and constant revisions of my dissertation. Luz Manuel Santos Trigo xii CHAPTER I THE PROBLEM Introduction In the process of learning mathematics, students deal with mathematical definitions, concepts, axioms, theorems, algorithms, and various strategies that are used in order to obtain the solutions of mathematical problems. Halmos (1980) maintained that solving problems is the heart of mathematics. Polya (1966) stated that "the solution of problems was the backbone of mathematical instruction since the time of the Rhind Papyrus" (p. 125). Kleiner (1986) pointed out that the development of mathematical concepts and theories arises from obtaining the solutions to concrete problems. Dieudonne stated that "the history of mathematics shows that a theory almost always originates in efforts to solve a specific problem" (cited in Kleiner, 1986, p. 31). Indeed, mathematical assignments, teachers' examples, and examinations all involve mathematical problems. If solving problems is important for students, it makes sense to investigate the effects of teaching aimed at developing strategies for solving mathematical problems. People who have recognized that problem solving is an important activity in the development of mathematics have focused their attention on designing and finding solutions to mathematical problems. Hilbert (1900) proposed 23 mathematical problems to the mathematical community. The process of seeking the solutions to these problems has notably influenced the development of mathematics. Descartes in the seventeenth century conjectured the existence of basic general rules for solving any type of problem in science. He presented general strategies in Rules for the Direction of the Mind and later in Discourse on the 1 Method which contained specific mathematical rules for solving problems. Although Descartes' goal was never completely realized, currently there is increasing interest in finding out what basic methodological strategies are used in solving mathematical problems. Melzak (1983, 1988) identified some strategies that are useful in approaching mathematical problems. He illustrated the application of one general strategy, "the bypass principle", with various examples from different areas including geometry, analysis, physics, technology, and telecommunications. The idea of the bypass principle is illustrated in the following example. Suppose that a partly damaged Latin text or inscription contains a statement to the effect that XIX times LXXVIII equals M?D??X?I?, where the question marks stand for illegible symbols. How would you restore it? In order to solve this problem, it is convenient to transform the operations into the decimal system in which the algorithms for the operations help solve the problem easily. Namely, there is a transfer of the problem into a different domain (bypass) and then a return to the original context. The present study investigates the effects of attempting to provide mathematics instruction based on problem solving. It is suggested that the study of the use of a problem solving approach will help students improve their methods of solving mathematical problems. The present study focuses on the strengths and limitations of this type of instruction. In this study, some episodes that explain main factors influencing the ways in which students solve problems are analyzed. The analysis has direct implications for mathematical instruction. 2 Background to the Study An analysis of exams, assignments, and classroom observations shows that students use a certain "methodology" when solving mathematical problems. Students' approaches in solving mathematical problems are influenced by the contexts in which the problems are presented. If the type of problem being solved has similar characteristics to a type of problem discussed by the teacher in the class, the students may apply methods similar to those presented by the teacher; however, if the context is changed and a structurally similar problem is stated in a different form, the students will often experience difficulties in solving that problem. A possible reason for this phenomenon is that students lack the basic strategies for solving mathematical problems. Schoenfeld (1985) found that students develop individual methods in accordance with their beliefs about mathematics and their experiences in solving mathematical problems. Hence, if instructors recognize and discuss the use of basic working strategies during instruction, students may develop better methods for solving mathematical problems. The importance of emphasizing the methodological aspect in solving problems is evident in the experiences of high-achieving students who were trained for the Mathematics Olympics in the United States. Students who did not follow a systematic method in solving the problems during the training were not able to solve problems in the actual competition even when some of the problems had been previously solved during the training (Schoenfeld, 1979, 1987c). This suggests that if students discuss the cognitive and metacognitive strategies used while working on the problems, then that discussion may help them to organize and recall the information needed for solving problems. 3 Polya (1945) identified basic stages in the process of arriving at the solutions of mathematical problems. The stages include understanding the problem, devising a plan for solving it, carrying out the plan, and looking back to check the solution. Polya suggested that if students are taught to use various heuristic methods in the framework of the basic stages, then they may become more successful in solving mathematical problems. Schoenfeld (1985) found that the heuristic methods suggested by Polya are general strategies that experts are able to recognize and apply when solving a mathematical problem. Schoenfeld also observed that for each strategy suggested, there are many more strategies involved that novices (students) fail to recognize and consequently do not use in a particular situation. For example, the strategy of "looking for a pattern by trying special cases" may be applied differently depending on the type of problem being studied. If the problem involves an integer parameter n, it is recommended that special cases be calculated, such as n = 1, 2, 3, or 4. If the problem deals with geometrical figures, then one should start by considering regular figures, such as triangles, squares, or semi-circles. However, if the problem asks for roots of algebraic equations, one may start by considering polynomials which are simple to factor. These examples illustrate the need for going beyond the presentation of general strategies. As a consequence, it is recommended that teachers know and discuss the richness and limitations of the use of the strategies. Schoenfeld (1988c) pointed out that discussing the role of the subject matter is important when considering the factors that are related to the learning process involved in the subject. For example, he suggested that it is important to discuss the difference between understanding the underlying mathematical ideas and becoming competent at performing the symbolic manipulation procedures in a mathematical domain. That discussion could 4 help clarify what type of learning is emphasized in the classroom and the type of result that could emerge from that instruction. Although there are some studies in which specific strategies have been investigated (Kilpatrick, 1967; Lucas, 1972; Schoenfeld, 1985), there is a need to implement both cognitive and metacognitive strategies in different contexts and to categorize them in accordance with the processes that students use in solving mathematical problems (Nickerson & Perkins, 1985; Perkins & Simmons, 1988; Schoenfeld, 1987b). The implementation of this type of instruction in regular classes could provide information about the potential of the use of problem solving instruction in learning mathematics. Fundamentals in the Use of Problem Solving Instruction Although almost every culture has exhibited some kind of mathematics knowledge and application, it is virtually impossible to come up with a satisfactory definition or complete characterization of mathematics. This difficulty is associated perhaps with the growth and extension of the discipline. Davis and Hersh (1981) pointed out that "the definition of mathematics changes. Each generation and each thoughtful mathematician within a generation formulates a definition according to his lights" (p. 8). Hersh (1979) recognized the importance of discussing aspects of mathematics that are related to its nature. He stated that "the issue then, is not, what is the best way to teach, but what is mathematics really all about ... controversies about high school teaching cannot be solved without confronting problems about the nature of mathematics" (cited in Lerman, 1990, p. 54). There is also a controversy regarding the foundation of mathematics. For example, the Platonist view assumes that mathematical entities are real and exist independently from the subject. These entities are not created and 5 will not change with time; any meaningful inquiry into them has a definite response whether the subject is able to explore this inquiry or not. "According to Platonists, a mathematician is an empirical scientist like a geologist; he cannot invent anything, because it is all there already. All he can do is discover" (Davis & Hersh, 1981, p. 318). Another view often identified as Formalism relates the development of mathematics to a set of axioms, definitions, and theorems. There are rules that are used to derive theorems and mathematical propositions or formulae. Although Formalists and Platonists hold opposite views about existence and reality, they do not disagree on what principles of reasoning are permissible in mathematical practice. There is another view, "constructivism", that affirms that mathematics can only be obtained by a finite construction. To summarize the impact of these different views of mathematics, Davis and Hersh (1981) stated: in the mid-twentieth century, formalism became the predominant philosophical attitude in textbooks and other official writings on mathematics. Constructivism remained a heresy with only a few adherents. Platonism was and is believed by (nearly) all mathematicians. But, like an underground religion, it is observed in private and rarely mentioned in public (p. 339). Mathematicians like Hardy (1877-1947) maintained that elegance and profundity are the main criteria for developing mathematics. In "A Mathematician's Apology", Hardy expressed, "...[/Judged by all practical standards, the value of my mathematical life is nil; and outside of mathematics it is trivial anyhow....And that I have created something is undeniable: the question is about its value" (cited in Davis & Hersh, 1981, p. 6 86). Hardy's view of mathematics is a view representative of those who endorse the "pure" nature of mathematics. There is another view of mathematics that emphasizes the application of mathematics. The basic principle underlying this approach is that mathematics is developed to meet a need for solving practical problems. This view of mathematics has influenced the design of mathematical curricula and the ways in which mathematics is presented to students. Ernest (1989) identified three different views of mathematics that are frequently observed in the teaching of this discipline: i) mathematics is not a finished product, but a dynamic discipline which is constantly expanding and readjusting to new problem solving situations (the problem solving view); ii) mathematics is a monolithic, static immutable product, which is discovered, not created (the Platonist view); and iii) mathematics is a useful discipline but an unrelated collection of facts, rules, and skills (the instrumentalist view). Ernest indicated that different views of mathematics produce different types of instruction. He stated: an active, problem-solving view of mathematical knowledge can lead to the acceptance of children's methods and approaches to tasks. In contrast, a static Platonist or instrumentalist view of mathematics can lead to the teacher's insistence on there being a single 'correct' method for solving each problem (p. 21). Robitaille and Dirks (1982) discussed three different orientations of the mathematical curriculum which are related to different views of mathematics: the French curriculum which emphasizes the formal type of mathematics, the British curriculum which emphasizes the application of mathematics, and the North American curriculum which emphasizes mathematical problem solving. 7 Although problem solving has been the direction of mathematical instruction in North America, its use is sometimes reduced to a unit added to the curriculum in which Polya's heuristic strategies are addressed while discussing mathematical problems. Lerman (1990) emphasized the importance of discussing the views of mathematics embedded in the actual teaching practice. He stated that "changes in mathematics education need to challenge fundamental assumptions about the nature of mathematics or else remain marginal in effect" (p. 54). The focus on problem solving activities in the classroom makes it important to incorporate a view of mathematics in which discussions of problems are considered essential in understanding or developing mathematical ideas. Cobb (1988) argued that an important goal of mathematical instruction is to provide learning conditions that help students develop more powerful structures than those they have at the beginning of instruction. These structures and the concepts could be discussed by presenting the mathematical content via problem solving (Schoenfeld, 1985, 1988b). Wimbish (1972) maintained that different views about the nature of mathematics should be part of the content discussed during instruction. He proposed to incorporate as part of instruction readings from mathematicians such as Halmos, Kline, and Whitehead in which views about the nature of mathematics and problem solving are addressed. If problem solving is a way to understand and develop mathematics, then it is important to discuss basic elements that are related to the process of developing mathematics. Davis and Hersh (1981) illustrated the relationship between the real and the ideal in the diagram shown in Figure 1. 8 REAL PHYSICAL IDEAL MATHEMATICAL Idealization Model Building Real World Verification A / \ / \ / \ / \ / IDEAL \ / OBJECT j. Mathematical Inference Implication to Real World A / | \ / I \ / I > 1/ ^ " It- " Figure 1. The Abstraction Process: The Mathematical Idealization (Davis & Hersh, 1981, p. 129) 9 In representing the "real world" there is an explicit or implicit conceptualization of "reality" and the idealized object. This is an issue that could generate controversy among the mathematical community. Davis and Hersh mentioned that "the so-called real world of experience, says Plato, is not real at all. We perceive the shadows of the external world and mistake the shadow for the true thing" (Republic VII 514-517; cited in Davis & Hersh, 1981, p. 129). Generalization is another aspect of mathematics that is worth mentioning. For example, the Pythagorean Theorem in two dimensions can be generalized in three dimensions. Davis and Hersh (1981) pointed out that "one benefit of generalization is a consolidation of information. Several closely related facts are wrapped up neatly and economically in a single package" (p. 1 35). Formalization is an aspect of mathematics that emphasizes the use of formal language for presenting mathematics. It is intended to make mathematical proof more rigorous. Lakatos (1976) discussed the difficulties that may appear while trying to formalize mathematics. He illustrated that the "proof" of the Euler-Descartes formula for a polyhedron V - E + F = 2 (with V as the number of vertices of a polyhedron, E, the number of its edges, and F, the number of its faces) which involves stretching the polyhedron on the plane led to the discussion of various counterexamples. These counterexamples resulted in the modification of the statement and consequently the proof was adjusted. Davis and Hersh (1981) discussed the notion of mathematical proof. They used the proof of the Pythagorean Theorem to illustrate some issues related to the purposes of proving a mathematical statement. For example, the process of proving includes constant criticism and judgment about the 10 strengths and limitations of the statement; it includes abstraction, formalization, axiomatization, and deduction. They stated: proof, in its best instances, increases understanding by revealing the heart of the matter. Proof suggests new mathematics. The novice who studies proofs gets closer to the creation of new mathematics. Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems (p. 151). However, this characterization of proof is often misunderstood in mathematical instruction. The concept of proof is normally presented as a polished product with no room for discussion from the students' side. The presentation of this type of proof in the classroom illustrates what Hanna (1990) calls "formal proofs". That is, proofs in which the validity of any statement depends on the axioms and their internal consistency. Schoenfeld (1988c) stated: for mathematicians, a "proof" is a coherent chain of argumentation in which one or more conclusions are deduced, in accord with certain well specified rules of deduction..., there is a great deal of flexibility in the way a proof argument can be written (p. 157). Hanna (1990) argued that the need for considering alternative ways that prove mathematical statements in the classroom may take different directions. Hanna considers it important to differentiate two types of proofs: proofs that prove and proofs that explain. Hanna stated that "a proof that proves may rely on mathematical induction or even on syntactic considerations alone. But a proof that explains must provide a rationale based 11 upon the mathematical ideas involved, the mathematical properties that cause the asserted theorem to be true" (p. 9). Lakatos (1976) suggested that mathematical ideas grow through speculation, discussion, and criticism. He stated that "informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculations and criticism, by the logic of proof and refutations" (p. 2). To Lakatos, proof in this context of informal mathematics does not mean a mechanical procedure which carries truth in an unbreakable chain from assumptions to conclusions. Rather, it means explanations, justifications, elaborations which make the conjecture more plausible, more convincing, while it is being made more detailed and accurate under the pressure of counterexamples (Davis & Hersh, 1981, p. 347). Kline (1980) also suggests that mathematical proof comes from intuitive advances based on a series of corrections of oversights and errors which eventually may lead to an acceptable proof for that time. However, no proof is final; he stated: [N]ew counterexamples undermine old proofs. The proofs are then revised and mistakenly considered proven for all time. But history tells us that this merely means that the time has not yet come for a critical examination of the proof. Such an examination is often willfully delayed (p. 313). Fawcett (1938) pointed out that the application of mathematical ideas in different contexts is unlikely if formal mathematics is studied purely on its 12 own terms. He carried out a study in which the concept of proof emerged from students' discussions. He stated: The concept of proof today is that all of our doctrines are built up on assumptions, definitions and undefined terms and because people's assumptions and definitions differ they disagree in their beliefs. In our concept [referring to the concept of proof endorsed in his study] there is no absolute truth but a doctrine is true within the limits of its assumptions (p. 142). In the same vein, Schoenfeld (1988b) stated that mathematics is a subject constantly changing; however, he added that the process of doing mathematics and thinking mathematically has remained invariant over many years. Kaput (1990) indicated that "the axiomatic method is no longer regarded among mathematicians as the primary method by which knowledge is generated, rather it is viewed more as a means by which knowledge constructed by other means can be efficiently organized" (p. 1, emphasis in the original). These means include the manipulation and elaboration of examples, the tuning of definitions, and the modeling of relations with non-mathematical phenomena. Lerman (1990) made reference to the importance of the values and consistency of mathematical results; he stated: Notions of proof, truth and rigour can be seen to be relative values, and there are conceivable alternative possibilities for the development of mathematics. There is no natural logical necessity to the state of mathematical knowledge at present. For as long as our results are useful, appropriate and consistent within our current understanding of the term, and given the existential position of our experience, we call them true (p. 55). 13 Wheeler (1990) argued that the complexity involved in the use of proofs makes this part of mathematics difficult to be studied in the classroom. Wheeler pointed out that "you will sometimes see that a proof is there because it explains something or because it establishes something unexpected, but then at other times when you prove something it's just a game because you already know what the result is" (p. 3). Wheeler went on to say "if you introduce kids to proofs with all these different purposes, and you also want them to understand indirect proofs of all types, mathematical induction, and so on that's a terribly sophisticated program" (p. 4). Towards a Problem Solving Epistemology Lakatos (1976, 1978) presented a view of mathematics in which conjectures or problems play a central role in the development of mathematical ideas. This view is essential for developing a problem solving epistemology in which the learner is actively engaged in understanding and developing mathematical content. Lerman (1990) referred to Lakatos' view of mathematics as the "quasi-empirical program" in which the growth of mathematical knowledge is seen as a process which involves conjectures, proofs, and refutations of mathematical ideas. In this view, there is room for uncertainty of mathematical knowledge as part of the nature of mathematics. Schoenfeld (1989) stated that "learning to think mathematically means coming to see mathematics as a vehicle for sense-making, and knowing how to use mathematics to make sense of things" (p. 81). Schoenfeld also suggested that the learning environment in which the students solve mathematical problems should become a culture of sense making. That is, solving mathematical problems is not simply getting answers but looking for 14 connections, approaching problems in different ways, generalizing solutions, and extending problems into other domains. From this perspective, mathematics is not a fixed knowledge of rules and procedures already developed and polished by experts, but an open area with room for negotiation and discussion of mathematical ideas or conjectures. Lave, Smith, and Butler (1988) suggested that the goal in learning mathematics should be to encourage students i) to generate dilemmas, ii) to create opportunities for invention and discovery, and iii) to understand patterns that help them to create mathematics, rather than to prescribe exercises on specific problem types and procedures. The basic assumption that supports a problem solving oriented course is that the growth of mathematics is based on finding solutions of mathematical problems. For example, the origin of calculus is associated with the need for solving specific problems, such as the drawing of tangents, the understanding of motion (instant velocity), and the notion of rate of change. These problems took specific form and were tackled from diverse perspectives. For instance, the problem of drawing tangents was recognized as the need to draw a tangent to any type of curve. As a consequence, the process of solving this problem was crucial for understanding several concepts such as function, limit, and derivative. Kieren (1979) argued that the relationship between society and mathematics should be an important component in the development of experiences in problem solving. The approach adopted for teaching calculus in the study focused on the discussion of problems as a means to developing understanding of calculus. The course also included a metacognitive component in which the students were asked to monitor their processes when solving problems. For example, the students would explain what they were doing, what they were expecting 15 to accomplish, and why they were using specific strategies. This type of activity seemed to be useful in monitoring the students' processes while solving the problems. Criteria for judging whether or not a mathematical statement is true are part of the epistemic component of mathematics and play an important role in the use of problem solving. For example, extending problems into other domains often requires analysis of critical points that need to be considered in the process of transforming the problems. As a consequence, discussion of these criteria should also be incorporated into mathematical problem solving instruction. Statement of the Problem The present study investigates the extent to which students who receive problem solving instruction are able to apply specific strategies for solving mathematical problems. A pilot study has already provided some information about the ways that students solve problems and the difficulties that students experience while working on some nonroutine problems. The present study explores the possible changes that may arise from instruction based on problem solving. It is assumed that instruction based on problem solving is not just a new method for teaching and learning mathematics but also a new way of thinking for instructors and students; therefore, the use of problem solving requires time and effort in order to become part of the students' and instructors' thinking. Nevertheless, it is possible to document the extent to which students change their ways of solving mathematics problems as a result of the instruction. The identification of the students' difficulties regarding the understanding of basic mathematical concepts when they use problem solving strategies is considered in the analysis of the study. The strategies considered in this study were first identified in the literature on 16 problem solving and were later suggested as possible alternatives for improving the students' ways of solving problems (Davis & Hersh, 1981; Polya, 1945; Schoenfeld, 1984; Silver, 1987). Rationale for the Problem When studying mathematics at the high school level, students are exposed to a wide range of mathematical content. They deal with properties of numbers, algebra, Euclidian and analytic geometry, and basic concepts of calculus. Throughout the process of learning mathematics, students use various methods to solve problems. When students begin their post-secondary education, there is an unspoken assumption that they should be able to understand basic mathematical concepts and use them in combination with various strategies to solve mathematical problems. Whether or not the mathematical requirements that students should have are high, in reality, the majority of the students struggle through their mathematics courses in order to complete them. For example, more than 20% of the students at the University of British Columbia fail their first year calculus courses (Lee, 1987). Although there are many factors that influence students' success in studying mathematics, the students' ability to recognize the use of certain strategies for solving mathematical problems may be an important component related to that success. Schoenfeld (1985) showed that instruction based on problem solving could improve first-year university students' performances in solving mathematics problems. Experts such as mathematicians approach problems differently from novices or ordinary students (Nickerson, Perkins & Smith,! 985; Schoenfeld, 1985; Silver, 1987). The methods most frequently used by experts and when and how they use them while solving mathematical problems could be 17 illustrated and possibly presented to the students in the classroom. Schoenfeld (1985) indicated that the monitoring process involved in the process of solving problems was an important strategy used by experts but was less evident in the novice students' processes. Schoenfeld (1988) stated: a common failing of novice problem solvers is that they go off on wild goose chases while working on a problem. Often, they pick one approach and persevere with it, despite clear signs that the effort is not yielding progress (p. 72). In explaining the way in which mathematics has developed, Hammond (1983) interviewed three working mathematicians specializing in different areas. They described their activity in mathematics as an ongoing process of understanding the nature, ideas and relationships among mathematical entities. This process involves the recognition of "simple" ideas that are examined and discussed from different perspectives. The original ideas are improved or modified based on the consideration of different examples and counterexamples. Even when a mathematical proof of a particular idea is the ultimate goal, mathematicians recognize that what happens during the process is essential to understanding that mathematical idea (Melzak, 1988). In describing the way a mathematician does mathematics, Schoenfeld (1983) pointed out that he or she first becomes familiar with a particular area; then he or she begins to suspect that something ought to be true. This provides some ideas with which to work. At this point, the mathematician starts to make sense of the suspected relationships and finally proves them. Schoenfeld also mentioned that during the whole process there may be any number of false starts, reverses, retrenchments, and modifications. Kline 18 (1985) went further when he stated that uncertainty and modifications are basic features of mathematics. Melzak (1988) pointed out that when a new proof is sought and various attempts end in failure, it is possible that very interesting serendipitous discoveries may be made during the search. The process of working with mathematical ideas often reaches a stage at which those ideas have been refined and are presented as a "final" product. This stage hides the struggles, problems, and improvements that were necessary for their development. Students very often are exposed to only this final stage; that is, they are asked to master the polished product. Moreover, the expertise of the instructor typically makes the presentation of mathematical content and problems smooth and straightforward, thereby hiding the methods and difficulties that someone had to overcome in order to make sense of and understand that mathematical content or problem. Students often get into trouble when dealing with mathematical ideas because they lack methodological tools for approaching mathematical problems. Students' understanding of mathematics is often conceived of as accepting and reproducing mechanically and passively what their teachers show them. Difficulties arise when students are asked to solve mathematical problems independently; they fail to accept that understanding mathematics means asking questions and clarifying meaning, making sense of their own ideas, and developing general strategies that can be applied in solving other mathematical problems. Although there is no unique way to approach a given mathematical problem, it is possible to identify basic methods that are characteristically used by experts in the process of obtaining the solution. For example, in dealing with mathematical ideas, experts employ trial and error; they speculate, organize, and transform the original ideas into different forms. 19 These activities which are carried out by experts may outline directions for describing essential strategies to be used when dealing with mathematical problems. The need for the implementation of mathematical problem solving instruction that incorporates activities used by mathematicians while working on mathematical problems provides the rationale for carrying out the present study. These activities are related to what Schoenfeld (1989b) called providing a microcosm of mathematical culture in the classroom. Research Questions In order to investigate the effects of the implementation of basic strategies for solving mathematical problems, it was decided to organize the analysis under four general questions. The information used in the analysis came from an instructor attempting to implement problem solving instruction in one of his calculus classes. 1. To what extent did the instructor provide appropriate conditions for problem solving instruction throughout the development of the study? What difficulties arose during instruction and how did they influence the development of the course? These questions focus on documenting the changes shown by the instructor while implementing learning activities related to problem solving. They also include an analysis of what the instructor agreed to implement and what he actually did during instruction. The sources of information used to analyze these questions include an interview with the instructor, notes from meetings with the instructor, and class observations. 20 2. To what extent did the students show mathematical disposition toward problem solving during the course? This question raises issues that include the students' confidence in using mathematics to solve problems and their views about problem solving. The analysis focuses on the contrasts between the students' responses to questionnaires given at the beginning and at the end of the course. Class observations, the students' work shown in assignments, and written exams are also included in the analysis. One of the written exams was the final exam designed by all the instructors who taught the course. This is taken as an indicator of how these students responded to what the group of instructors expected from the students. 3. How could the students' approaches to problem solving be usefully categorized in accordance with the categories employed by Schoenfeld (1985), namely: domain knowledge, cognitive and metacognitive strategies, and beliefs? How successfully did the students respond to instruction? The dimensions proposed by Schoenfeld have been adjusted and are used as a guide to analyze the students' approaches to mathematical problems. The sources of information used to answer these questions include the students" interviews and class observations. 4. What difficulties did the students encounter in the use of problem solving strategies when solving mathematical problems? Were those difficulties related to the class instruction? This question focuses on identifying patterns of misunderstanding exhibited by the students while solving mathematical problems and their relationships to instruction. It is complementary to question number three. The analysis points out the types of difficulty the students experienced during the process of solving mathematical problems and their relation to the instruction. The work of Perkins and Simmons (1988) is used to analyze the information. The sources of information for the analysis include the students' interviews and class observations. 21 Limitations of the Study The present study investigates to what extent students who received mathematics instruction based on problem solving were able to apply basic strategies for solving problems. The instruction included cognitive strategies that are found in courses up to and including the first year of university and special emphasis was given to the monitoring process involved in solving mathematical problems. These strategies were illustrated by discussing examples taken from calculus texts at the college level. The students were regular students with some interest in pursuing more mathematics courses in the future. The course is compulsory for those students planning to pursue science courses. It is important to mention that this was their first formal introduction to calculus that emphasized the basic concepts of the subject, that is, the concept of derivative, the application of derivative, and the fundamental theorem of calculus. Therefore, there were opportunities to introduce the students to some "new" calculus ideas that would encourage them to reflect upon ways of solving problems. The instructor who was in charge of the problem solving instruction had extensive experience in teaching mathematics at the college level. His formal education included graduate studies in mathematics and mathematics education. His general evaluation of the course was important in order to contrast the development of the problem solving course and the development of the courses that he had taught previously. This study was limited to the implementation of problem solving instruction to one group. The analysis of the data focuses on the extent to which the instructor provided conditions related to problem solving instruction and how the students responded to the use of strategies. 22 The nature of this study relies on qualitative methods and procedures. The means for gathering the information to be analyzed included an interview with the instructor, task-based interviews with six students, and class observations. Such means of data collection had proven to be useful in the pilot study. Therefore, the conditions of the study which involved the use of a small sample of students and the consideration of only one instructor limit the generalizability of the findings. The purpose of the present study is to investigate the students' approaches to mathematical problems as a result of problem solving instruction. The analysis of the data uses ideas from similar studies and relies on constant communication with other researchers and practitioners in the area. Although the frame of analysis will guide the researcher to analyze and interpret the information, it is also important to recognize that the researcher will be making adjustments in accordance with the development of the study. Schroeder (1983) stated that the ability of the researcher to discern and interpret the information is also an important component that influences the analysis and its interpretation. The study focuses on the processes used by the students and the instructor when interacting during specific mathematical tasks rather than making generalizations about large populations of students and instructors. However, as Marton (1988) indicated, this type of study provides useful detailed information about the participants' actions that cannot be achieved by other means. Chapter Summary This chapter deals with theoretical issues that explain the rationale for implementing a problem solving approach in the study of mathematics. It presents a historical background and epistemological components that are 23 associated with mathematical problem solving. It is suggested that a view of mathematics in which there is room for speculations, discussions, and criticism of mathematical ideas is compatible with a problem solving approach in which the students are encouraged to participate and to make sense of their ideas in order to solve mathematical problems. This chapter outlines the research questions that guide the development of the study. The research questions focus on 1) the implementation of learning activities associated with this instructional approach, 2) the exploration of the students' disposition toward problem solving, and 3) the extent to which the students use problem solving strategies when solving mathematical problems. Finally, this chapter describes the limitations of the present study. 24 CHAPTER II REVIEW OF THE LITERATURE Introduction Problem solving is an issue that has permeated the mathematical curriculum for several years and during the last 15 years has been the dominant model influencing the teaching of mathematics in North America. Kilpatrick (1988) pointed out that "problem solving has become a slogan encompassing different views of what education is, of what schooling is, of what mathematics is, and of why we should teach mathematics in general and problem solving in particular" (p. 1). Therefore, in order to discuss the literature related to this study it is necessary to present an overview of studies that have investigated the effects of implementing mathematical problem solving instruction and have influenced the design of the present study. The literature review will focus on tracing the evolution of the use of mathematical problem solving instruction. Many studies (Amit & Vinner, 1990; Marshall, 1989; Noddings, 1985; Pereira-Mendoza, 1975) have been carried out investigating the effects of incorporating problem solving strategies into classroom instruction. Although some results have not clearly indicated the benefits that students may receive from problem solving instruction, there are some recent studies that have shown that students may improve their methods of solving problems if they are explicitly exposed to problem solving strategies combined with instruction which incorporates metacognitive activity (Lester, Garofalo, & Lambdin, 1989; Schoenfeld, 1985; Silver, 1987). Therefore, the review of relevant literature will focus on the evolution of problem solving approaches, that is, the discussion of the aspects of problem solving that have proven to be 25 worthwhile to take into consideration. It will include studies that have investigated the use of problem solving strategies in the classroom and those which have used similar designs and procedures. The literature review is divided into three related parts: the first part is intended to explore various interpretations that are found in the implementation of mathematical problem solving. It also includes the seminal work of Polya in problem solving, as Polya's work is important in identifying the nature and use of several problem solving strategies. The second part will review some studies in cognitive science that are related to problem solving and that were important in deciding what activities to implement in the classroom. This section ends with a presentation of a model in which difficulties that students experience in the study of mathematics are explained. This model, proposed by Perkins and Simmons (1988), is used later in the analysis of the students' approaches to problem solving. The third part of the literature review focuses on the work of Schoenfeld, who has investigated the use of problem solving in different contexts. The work of Schoenfeld has shaped the design and the direction of the analysis of the present study; therefore, it is important to discuss the basic components of Schoenfeld's research that have influenced the present study. Interpretation of Mathematical Problem Solving "Problem solving" is a term used in various subjects and can encompass different meanings. In mathematics, Branca (1980) mentioned that dealing with problem solving may involve solving routine or nonroutine problems, applying mathematics to solve real world problems, solving puzzles, or testing conjectures that may lead to the development of new mathematical knowledge. In relation to the use of problem solving in mathematical 26 instruction, Branca identified three interpretations, that is, problem solving as a goal, as a process, and as a basic skill. The interpretation of problem solving as a goal is based on the recognition that problem solving is the main activity in the development and understanding of mathematics. Branca recognized that the main consideration here is that learning how to solve problems is the fundamental reason for studying mathematics. When problem solving is interpreted as a process, there is special attention to the methods, procedures and strategies used by the students while solving mathematical problems. The third direction, in which problem solving is interpreted as a basic skill, focuses on identifying specific types of problems, specific content areas, and specific techniques used to solve those problems. Schroeder and Lester (1989) in the same direction distinguished three possible interpretations that characterize and differentiate courses based on problem solving. Although there may be common characteristics among these approaches, the main focus of the course and the organization of the material are components that differentiate one approach from another. For example, Schroeder and Lester (1989) identified one approach as "teaching about problem solving". This approach emphasizes Polya's four stages identified during the process of solving mathematical problems. There is explicit discussion about these stages when solving the problem and discussion about basic heuristics for solving the problems. This approach shares some characteristics with the interpretation of problem solving as a process in Branca's description. The second approach, identified as "teaching for problem solving", focuses on the use or application of mathematical content. Therefore, the initial understanding of mathematical content is prerequisite to applying it in various contexts. As a consequence, problem solving 27 emphasizes the applications rather than the understanding of the mathematical content (this interpretation shares some features with what Branca called problem solving as a basic skill). The third approach to problem solving is that in which mathematical content emerges from a problem solving situation and in which this situation actively engages the students in the process of making sense of content. This approach is identified as "teaching via problem solving". It is suggested that this approach has similarities with the process of developing mathematics (problem solving as a goal). Schoenfeld (1989) suggested that it is possible to provide a class environment in which the students not only understand mathematics via problem solving but also develop mathematical content. Stanic and Kilpatrick (1988) identified three themes that have characterized the role of problem solving in the mathematical curriculum: i) problem solving as a context in which the value of problem solving is based on the idea that problems and solving problems provide conditions to achieve goals in the study of mathematics. The goals may include motivation, recreation, justification, or practice. ii) problem solving as skill in which there are a number of skills to be learned. There is a hierarchy of skills necessary to solve certain types of problems and emphasis normally is given to solving routine problems. iii) problem solving as art which represents the view of Polya. This view is discussed later in this chapter. Schoenfeld (1983c) suggested that mathematicians, cognitive psychologists, and mathematics educators may characterize the use of problem solving differently. For example, for a mathematician, a problem is a task in which there is an impasse that one does not know how to go about in order to solve it. Examples of these problems are found in books such as One 28 Hundred Problems in Elementary Mathematics by Steinhaus (1979). Therefore, the process of solving these problems is what mathematicians may call problem solving. In the discussion of problem solving from the cognitive side, Schoenfeld mentioned that the research questions of interest in this field are important, but that the treatment of mathematics needs to take into account other frameworks. The research questions include "how is it that people think about mathematics?" and "how does understanding of mathematics concepts develop?" One of the differences that Schoenfeld identified in the mathematical and cognitive-psychological perspectives is the concept of elementary mathematical structures. While the cognitive side focuses on multiple representations for base 10 that include discussions of iconic, enactive, and symbolic representations, for mathematicians the notion of structure of whole numbers involves terms such as "groups", "rings", and "integral domains". Schoenfeld (1983c) indicated that problem solving in the field of mathematics education has lacked coherent lines of inquiry and theoretical support that could lead to the exploration and implementation of various activities related to problem solving. Mathematics educators need to consider knowledge from other disciplines such as anthropology (cultural context) and psychology and from mathematics foundations in order to characterize mathematical problem solving. Schoenfeld (1983c) concluded that "problem solving face[s] a task of absolutely enormous proportions, calling for a synthesis of the best skill and knowledge from a collection of widely disparate disciplines" (p. 45). 29 Problem Solving and Heuristics Polya (1945) pointed out that the process of solving problems involves the use of heuristic methods. He illustrated the advantages of considering special cases, using diagrams, working backwards, and using contradiction methods. Schoenfeld (1987) stated that mathematical experts recognize and appreciate the use of the strategies identified by Polya, but novices may not recognize when and to what extent to use them. He suggested that this may be true because in each strategy there are many substrategies that may be applied differently depending on the nature of the problem. He stated that "Polya's characterizations were labels under which families of related strategies were subsumed" (p. 31). For example, the strategy that recommends the use of special cases may be applied to problems in which an inductive pattern is involved. Here, cases in which n = 1, 2, or 3 are good candidates to consider. The same strategy is applied differently if the problem deals with roots of polynomials. Here, it is recommended that one work with polynomials that are easily factored. However, if the problem is concerned with sequences involving recursion, it is useful to explore values such as 0 and 1. Kieren (1979) indicated that the use of heuristics, such as diagramming, making tables, and systematically listing information, could be seen as a mathematical model for a problem and as forms of representation. Heuristic strategies, such as drawing a diagram, breaking the problem into parts, and considering simpler cases, are central to representing a problem and designing a plan for the solution. Stressing heuristic strategies in problem solving instruction is related to the assumption that by imitating the ways that experts solve problems novices or students can become better problem solvers. However, some research studies have shown that this type of approach may be of benefit only to students who already have the cognitive 30 foundation upon which to build these types of strategies. There have been some courses in problem solving that have emphasized the use of heuristic strategies in different contexts. For example, Rubinstein has been in charge of a problem solving course at UCLA for more than 10 years. Rubinstein (1980) listed some of the objectives of the course including emphasis on general strategies for solving problems and mastering some specific techniques, discussion of problems in different contexts including real life problems, and reflection on the thinking processes involved in the process of solving problems. There is a book that accompanies the course in which there are different examples that illustrate some difficulties that may impede effective problem solving and some discussion about ways to overcome these difficulties. Unfortunately, there are no data regarding the extent to which the students are able to use the strategies learned in this course in other contexts. Real life problems are not easily represented as textbook problems and that may be an obstacle for their consideration. However, the activities suggested in the course may help students to consider more alternatives or approaches to the problems. A book on problem solving by Whimbey and Lochhead (1979) has been used in a problem solving course at Manhattan Community College (University of New York). The book describes various strategies for solving problems and provides several examples in which these strategies are applied. Although the types of problems found in the book are similar to those found in intelligence tests, the authors suggested that the book provides conditions for the students to acquire general skills that they can use in school courses or when taking tests. Several studies have focused on the use of more general strategies. An example of this type of study is the Productive Thinking Program of Covington, 31 Crutchfield, Davies, and Olton (1974) which emphasizes inventive, creative, or divergent thinking. Mansfield, Busse, and Krepelka (1978) reported results from the implementation of this study. They found that the Productive Thinking Program may be effective when applied in small groups by enthusiastic teachers. The Cognitive Research Trust (CoRT) program by De Bono (1983) which aims to teach thinking is organized into units which include organization, creativity, information and feeling, and action. The assumption in this program is that thinking is a skill that can be improved by exposure to the activities proposed in the program. This program has been implemented in various settings. For example, in Venezuela, the subjects showed gains after being involved in the program for some time (de Sanchez & Astorga, 1983). Griffith (1987) examined the heuristic processes used by teachers when solving mathematical problems. The results indicated that the differences between less effective and more effective problem solvers lay in the use of strategies that included modeling, planning, and the use of trial and error. For example, Griffith found that the more effective subjects used more modeling and planning processes, and were able to use trial and error more efficiently than less effective subjects. In addition, the more effective subjects spent less time analyzing the problems, asked fewer questions, and made fewer errors than the less effective subjects. Goldin and Landis (1985) employed interview techniques to elicit the heuristic processes used by a student while solving mathematical problems. They found that during the process of problem solving, the student was able to generate simpler problems when prompted to do so. The setting and procedures used during the interview helped the student pay attention to basic strategies of problem solving. For example, the student was able to 32 generate a sequence of related problems, to look for patterns, and to look back or check the solution. However, the student experienced difficulty in the phase of representation of the problems. Maher and Alston (1985) used small groups of students to investigate the use of the heuristic "think of a simpler problem". The students were asked to work on the problem "What is the remainder when 2 to the 50th power is divided by 2?" They found that the students were able to monitor part of the process while solving this problem. A complete discussion led by the teacher of the difficulties that the students experienced when working in small groups helped the students to overcome some of the misconceptions identified when solving the problem. In addition, the students had the opportunity to make comparisons and evaluations of their work. Jenkins (1988) investigated the effects of the teaching of three heuristic strategies (simpler problem, pattern recognition, and elimination) to high school students. The results showed that the processes of solving mathematical problems of the experimental group relied on identifying mathematical structures associated with a set of problems, whereas the control group depended on contextual details. Cognitive Science and Mathematical Problem Solving An important issue in cognitive science is the investigation of how knowledge is represented, stored, and used in different contexts. Researchers in these areas have explored ways in which problems can be represented, the role of specific and general strategies for solving problems, and the role of metacognition. All these aspects are central to the study of mathematical problem solving. Lester and Garofalo (1982) and Mayer (1982) pointed out that there is a 33 need for open communication among psychologists and mathematics educators allowing them to share and interpret research in order to develop a stable and useful body of knowledge about the nature, use, and limitations of problem solving. Schoenfeld (1985) suggested that mathematics educators cannot afford to ignore the psychological research in problem solving. Qualitative methodologies frequently used in cognitive psychology were only recently incorporated into mathematics education and they have proven to be useful in understanding the way that students approach problems. Indeed, the research reported by Schoenfeld is mainly based on the use of qualitative methods such as interviews with students. Mayer (1982) identified two stages in the process of problem solving: i) problem representation which depends upon linguistic, factual, and schema knowledge and ii) problem solution which is associated with algorithmic and strategic knowledge. Mayer suggested that students often experience difficulties at the representation stage even before getting engaged in solution procedures. Silver (1987) pointed out that problem representation is central to the problem solving process. He stated that an inaccurate or incomplete representation of the problem may make it difficult or impossible to solve. The quality of the solution is also related to the way that the problem is represented. Even though it may be argued that the process of solving a problem begins at the representation stage, Mayer suggested that making the differentiation is useful in order to identify what type of problem solving instruction should be emphasized. For example, he stated that "most instruction stresses the solution processes and subordinates knowledge of when to apply the procedures or how to represent problems" (p. 4). Davis (1986) indicated that "any mathematical concept, or technique, 34 or strategy - or anything else mathematical that involves either information or some means of processing information - if it is to be present in my mind at all, must be represented in some way" (p. 203). Davis went on to say: mental representation of a concept [depends] in a central way on: i) the ability to recall or to invent candidate examples, things that may be examples or may turn out to be counterexamples; ii) the capability of making judgements on exemplar candidates ('yes, that has property x' or 'no, that does not have property x') (p. 209). Lester (1982) suggested that cognitive psychology has a long history of studying human learning and instruction by focusing on mathematics. The development of theories of mathematical problem solving has been one of the areas of interest in psychology. Although there is no single methodology that characterizes the work done in cognitive science, it is possible to illustrate some research in this area that is directly related to mathematical problem solving. Schoenfeld (1982) stated that "the single most important reason to teach mathematics is that it is an ideal discipline for training students how to think" (p. 32, emphasis in the original). Learning to think mathematically means "coming to see mathematics as a vehicle for sense-making, and knowing how to use mathematics to make sense of things (including 'real world' and pure mathematical structure)" (Schoenfeld, 1989, p. 81). Mayer (1985) recognized the importance of integrating advances in problem solving from both psychology and mathematics education. He also suggested that methodological tools used by cognitive psychologists can be useful for the analysis of mathematical problem solving. He stated that "cognitive psychology can attempt to refine general theories of problem solving to learning and instruction for specific tasks; mathematics educators can attempt 35 to extend specific information about learning mathematical tasks to broader frameworks of problem solving" (p. 125). Mayer also discussed issues from the literature on cognition that may be relevant to mathematical problem solving. For example, students deal with knowledge representation in order to understand the problem and put the elements of the problem together into a coherent whole. Research in this area has shown that linguistic comprehension and knowledge of problem types are sources of difficulty in problem solving. Vergnaud (1984) suggested that representation is important in the study of mathematics because mathematics is essential in conceptualizing the world and because the use of homomorphisms is necessary in order to study mathematical structures. He identified three levels of the problem of representation and symbolization: the referent level, which deals with the real world as it is seen by the subject; the signified level, which refers to a theory of representation; and the signifier level, which refers to the use of different symbolic systems (graphs, diagrams, or algebras). Kaput (1984) recognized the importance of the role of representation in the study of mathematics. He mentioned the lack of a coherent and unifying theory that can explain students' difficulties in translating mathematical ideas and common experiences. He stated: having established the representation character of mathematics at an abstract level, our next step is to study the varieties of materially reliable forms, and the psychological aspects of those forms, by which the representations are achieved (p. 25). Silver (1987) discussed profiles of successful problem solvers. He focused on various characteristics, such as pattern recognition, 36 representation, understanding, memory schemas, and meta-processes. Good problem solvers recognize patterns within specific logic, that is, there is a link between the elements that form the pattern. For example, chess experts were able to reproduce the correct position of 20 or 25 pieces from an actual game, while ordinary players reproduced only 6 pieces. However, when the pieces were arranged randomly, there was no difference between the experts and the ordinary players in reproducing the correct positions (only about six pieces). Looking for patterns rather than recalling individual positions seems to be the main factor influencing pattern recognition. Silver (1987) suggested that the problem solver begins with an initial representation that evolves until he or she gets a representation that is adequate for solving the problem. He stated that "students who build similar representations for mathematically related problems are far more likely to notice their similarity and to use the relationship about one problem in solving the other" (p. 45). The ability of problem solvers to recall information needed to solve problems may depend on the ways that problem solvers organize their knowledge. The existence of memory schema, that is, "a cluster of knowledge that describes the typical properties of the concept it represents" (Silver, 1987, p. 45) influences the level of success in solving problems. Silver suggested that problem solvers tend to categorize problems as a means of retrieving important information from long term memory. He stated that "schemata are useful not only for retrieving clusters of related and useful information, but also for shaping the representation of problems" (p. 48). Research regarding the use of cognitive strategies has shown that extensive knowledge in specific domains appears essential to success in problem solving. This research has been extended to the study of managerial 37 or monitoring processes involved while solving problems, for example, the study of decisions regarding allocation of cognitive resources, the change of direction or methods, and the constant monitoring process involved in the entire process while solving the problem. These issues are referred to as metacognition, that is, the problem solver's knowledge about his or her own thought processes, control and self regulation when solving problems, and the system of beliefs and intuitions shaping the process. Kilpatrick (1987) emphasized the importance of formulating mathematics problems not only as a means but also as a goal of instruction. He pointed out that "the experience of discovering and creating one's own mathematics problems ought to be part of every student's education" (p. 123). This view supports the teaching of mathematics via problem solving. During the process of formulating problems, it is important to consider the aspects identified by students as problematic situations that can challenge their mathematical abilities. Kilpatrick discussed some preliminary ideas about the possible problem-formulating processes which included association, analogy, generalization, and contradiction. As an example which illustrates "association" in formulating problems, Kilpatrick referred to the concept maps developed by Novak and Gowin (1984). Kilpatrick stated: concept mapping is a process for making explicit one's associations with a concept, and accepting the given is a process for using those associations to pose new problems about the concept. Together they might provide mathematics teachers with a productive way of teaching problem posing as a school activity (p. 136-137). Indeed, during the design of the assignments that were used in the course of the present study, concept mapping was a useful representation for 38 organizing the main ideas associated with the concept of derivative. Kilpatrick mentioned that when students are asked to formulate the Pythagorean Theorem in three dimensions after it has been studied in the plane, they may use "analogy" in order to make that formulation. This theorem can also illustrate the "generalization" case when it is asked in n dimensions. The last idea "contradiction" is considered as a powerful problem-solving strategy. When used as "what-if-not", it may generate problems by contradicting one or more parts of the statement (Brown & Walter, 1983). For example, for the Pythagorean Theorem, "What if the theorem did not deal with right triangles?" or "What if the statement were not a theorem?" are some of the questions that can lead to other problems. Kilpatrick (1987) also suggested that the literature on problem representation may contribute to the process of generating problems. He stated, "how a problematic situation is represented will determine what problems can be derived from the situation as well as how they are solved once derived" (p. 139). Lave et al. (1988) suggested that problem solving should encourage more explicitly some kind of mathematical practice, that is, learning conditions in which the students might learn by becoming apprentice mathematicians to do what master mathematicians and scientists do in their everyday practice. This notion of mathematical practice is similar to what Schoenfeld called a microcosm of mathematical culture, that is, an environment in which students may study or perhaps develop mathematics in much the same way that mathematicians do (cited in Lave et al., p. 77). Lave et al. linked this type of practice with the work of Schoenfeld in doing problem solving with his students. Four dimensions are embedded in his problem solving approach: i) mathematical resources, ii) cognitive strategies, iii) 39 metacognitive strategies, and iv) belief systems about mathematics and problem solving. Lave et al. (1988) mentioned an ongoing project which is closely related to this type of mathematical practice at University of California, Irvine. The activities of the project involve asking the students (aged 5-12) to: a) find interesting patterns in numbers, shapes, and procedures, compare the patterns, extend them until they fail, and make them plausible; b) vary problems they are given; c) invent their own problems, and invent problems for others; d) turn their own mathematically promising noticing into investigations and pursue them across days or weeks; e) develop more than one solution to a problem, and more than one formulation of a solution; f) consider the character and strength of their understanding and how they came to that understanding; g) use the world to provoke and exemplify mathematical notions; and h) teach other students the way they are taught (Lave et al., 1988, p. 79). In order to solve a problem, the students interact with other students and make sense of the information to select ways of approaching the problem according to the students' resources. These resources include mathematical content, cognitive and metacognitive strategies, and systems of beliefs about mathematics and problem solving. Koplowitz (1978) found that undergraduate students who experience difficulty in solving problems do not have an appropriate sense of when they have solved a problem. For example, in his problem solving class, he asked his students to work on some mathematical problems. The students' solutions included different and 40 sometimes contradictory results. These results came from conceptually different ways of reasoning used by the students to interpret and approach the problem. Koplowitz observed that the students were confident with their solutions and were not surprised that other students arrived at different solutions. He stated "what intrigues me about these examples is not that the students devised incorrect answers for the questions, but that they were confident in those answers" (p. 306). Koplowitz suggested that more emphasis should be placed on encouraging students to reflect on what it is to solve a problem, that is, to compare one solution with another, to provide examples or counterexamples, and to develop a better sense of when a particular strategy can be used for a given problem. Ciffarelli (1988) investigated high school students' processes of constructing conceptual knowledge while solving algebra word problems. The main results indicated that students who were able to represent the problems and the activities involved in the solutions showed high levels of awareness of the structure of the problems. These students were also able to anticipate some of the solutions. Ciffarelli's study provides information about the importance of emphasizing ways in which the problems can be represented during the process of problem solving. Najee-Ullah (1989) studied mathematics beliefs of high school mathematics teachers while solving mathematical problems. Results of this study indicated that what teachers believed about mathematics influenced the ways they performed in problem solving. This study also showed that the teachers' beliefs about mathematics can be modified if they are exposed to a variety of experiences in which those beliefs can be challenged. Eyler (1989) investigated qualitative and quantitative differences of college freshman students in the use of metacognitive strategies while solving 41 mathematical problems. The results showed that successful problem solvers use more metacognitive strategies (both in number and quality) than unsuccessful problem solvers. In the exploration of difficulties that students experience in the study of mathematics, Perkins and Simmons (1988) introduced a model in which four frames of knowledge are discussed: i) content, ii) problem solving, iii) epistemic, and iv) inquiry. This model is used as a frame of analysis in the present study to discuss the main difficulties experienced by the students while working on two mathematical problems. The idea in the use of this model is to relate what types of mathematical experiences are associated with the ways in which the students solved or approached the problems. A description of the type of knowledge involved in each frame is presented in chapter six. What follows is an overview of some difficulties that students may experience in the process of learning mathematics and that are related to each frame of knowledge. Regarding the content frame, students may experience difficulty in the use of a specific concept. This difficulty is often associated with the students' ability to assess knowledge. The students may be familiar with a specific concept but lack the knowledge of the basic structure that is related to its use. It also may be related to the students' difficulty in dealing with confused knowledge or knowledge that has been recently discussed and not yet clearly conceptualized. Within the problem solving frame, Perkins and Simmons (1988) listed a set of counterproductive strategies that students may exhibit while studying mathematics. For example, the unsystematic use of trial and error methods, the use of a strategy for a long time even when no progress is made towards the solution, and the overuse of equation cracking strategies are some of the 42 common approaches associated with problem solving that students often exhibit. Some of the weaknesses associated with the epistemic frame include the students' lack of analysis to use examples and counterexamples to validate or disprove a particular mathematical statement. Finally, difficulties in the inquiry frame involve the students' resistance to explore more general cases in which the conditions of the problem are varied. Perkins and Simmons (1988) indicated that conventional instruction does not offer students the opportunity to participate in the formulation process of problems and this may limit them to make further analysis of the solutions. Schoenfeld's Work on Mathematical Problem Solving Schoenfeld (1987) recognized being initially inspired by Polya's book "How to Solve It". Polya in this book identified a four-stage model for discussing the process involved in solving problems, that is, understanding the problem, designing a plan, carrying out the plan, and looking back. The book contains examples in which the use of heuristic strategies is illustrated. Schoenfeld, who was trained as a mathematician, recognized that he himself had used several of those strategies; however, he stated, "I'd picked them up then pretty much by accident, by virtue of having solved thousands of problems during my mathematical career (that is, I'd been 'trained' by the discipline, picking up bits and pieces of mathematical thinking as I developed)" (Schoenfeld, 1987, p. 30). Although Schoenfeld recognized the potential of the strategies discussed by Polya, he found that people involved in students' training for mathematical competitions did not use Polya's ideas. The main principle used 43 by people responsible for training students in various mathematical competitions was that "one learns to solve problems successfully by solving a large number of problems" (Schoenfeld, 1979). In the analysis of the weaknesses of this approach, Schoenfeld indicated that some students may become successful with this method when solving similar problems but they may experience difficulty when the problems are set differently. Schoenfeld (1979) stated: a student may deal at length with a particular problem, and understand it completely at that time. Yet this does not guarantee that the "lesson" to be learned from it, or even merely the solution to it, will be retained in any way by the student. Lacking (or more precisely, being unable to access) the appropriate connections, the problem solver may later find himself staring at the same problem in total frustration, knowing that he has solved it before but is now unable to recall even the general method of approach (p. 39). The first task undertaken by Schoenfeld was to investigate why Polya's ideas were not working. Schoenfeld reviewed some studies carried out in cognitive science and artificial intelligence. He found that people in these areas had produced computer programs that were able to solve problems in areas, such as chess, symbolic logic, and calculus quite successfully. The strategies employed in these programs came from detailed observations of experts solving problems. These observations were arranged in a set of prescriptive procedures that the computers used to produce good results. Schoenfeld (1987b) mentioned that in order to understand the processes used by mathematical problem solvers and to provide direction for mathematical instruction, it is necessary to take into account the subject matter, the dynamics of the classroom, and the 44 learning and thinking processes. That is, it is necessary to incorporate knowledge from mathematicians, mathematics teachers, and mathematics educators, and cognitive scientists. Figure 2. Essential Contributors to Progress in Mathematics Instruction (Schoenfeld, 1987b, p. xix ) In consideration of these components Schoenfeld's central inquiry was then to investigate: "What level of detail is needed so that students can actually use the strategies one believes to be useful?" ( Schoenfeld 1987, p. 31). The results of this inquiry provided some information about why Polya's ideas were not working in the classroom. "The reason is that Polya's heuristic strategies weren't really coherent at all. Polya's characterizations were broad and descriptive, rather than prescriptive". Schoenfeld went on to say that, "in short, Polya's characterizations were labels under which families of related strategies were subsumed" (Schoenfeld, 1987, p. 31). He provided several examples in which a strategy characterized by Polya produced many similar 45 but basically different substrategies. The initial practical implication for teaching problem solving that came from recognizing the existence of many more strategies related to each strategy discussed by Polya was to design learning activities that i) identify the use of a particular strategy; ii) discuss the strategy in sufficient detail in a prescriptive manner; and iii) provide students with the appropriate degree of training. Although a prescriptive problem solving approach resulted in progress in the students' ways of solving problems, Schoenfeld (1985) recognized that this approach was not enough. For example, the students could know the strategy without knowing when to use it. As a result, Schoenfeld included a set of managerial strategies as a part of the problem solving instruction. Up to this point, the results obtained by Schoenfeld showed encouraging results in the use of problem solving. However, he noticed that his students were doing well because they were asked to use the strategies in a specific context. Even with complicated problems they knew the strategies, the content, and possible approaches to the problem. "I was getting good results partly because I had narrowed the context: students knew they were supposed to be using the strategies in class and my own tests" (Schoenfeld, 1987, p. 34). Similarly, Schoenfeld (1988c) described results from a well organized mathematical class in which the students did well on the Regents exam but showed lack of understanding of basic mathematical ideas. For example, the students in his study believe that the form or presentation of a mathematical solution is more important than the arguments; they also believe that all the problems can be solved in a few minutes, and that mathematics is learned passively. The next phase of Schoenfeld's research in problem solving began 46 when he examined videotapes showing his students solving problems in different contexts. He stated, "What I saw was nothing like what I expected, and nothing like I saw as a teacher. Generally speaking, we only see what students produce on tests; that's the product, but focusing on the product leaves the process by which it evolved largely invisible" (Schoenfeld, 1987, p. 33). Schoenfeld observed that students often do not use mathematical content that they are familiar with in order to solve a problem. They choose a direction immediately and often persist in using it no matter what. Schoenfeld suggested that in order to understand how students approach problems and consequently to propose learning activities that could help them improve their ways for solving problems, one has to extend the context and focus on basic factors that influence the ways in which that students solve problems. In several studies, Schoenfeld found that there are four dimensions that influence the ways that students solve problems: domain knowledge, cognitive strategies, metacognitive strategies, and belief systems. Domain knowledge includes definitions, facts, and procedures used in the mathematical domain. Cognitive strategies include heuristic strategies, such as decomposing the problem into simpler problems, working backwards, establishing subgoals, and drawing diagrams. Metacognitive strategies involve monitoring the selection and use of the strategies while solving the problem, that is, deciding on the types of changes that need to be made when a particular situation is deemed problematic. Belief systems include the ways in which students think of mathematics and problem solving. Schoenfeld (1987d, 1988b, 1989b) underlined the importance of relating the nature of developing mathematics to the process of solving mathematical problems. He mentioned that the main goal in learning 47 mathematics is to see the connections and to make sense of mathematical structures. In order to achieve this goal the students have to discuss their ideas, to negotiate, and to speculate about the possible examples and counterexamples that may confirm or disprove their ideas. Schoenfeld (1988) stated: for students to see mathematics as a sense-making activity, they have to internalize it as such. That is, they need to learn mathematics in classrooms which are microcosms of mathematical culture, classrooms in which the values of mathematics as sense making are reflected in every day practice (pp. 87-88). Schoenfeld (1989b) indicated that students should be aware of basic epistemological goals of problem solving instruction. For example, the students should understand that: i) finding the solution of a mathematical problem is not the end of the mathematical enterprise, but the initial point for finding other solutions, extensions and generalizations of that problem, and ii) learning mathematics is an active process which requires discussions of conjectures and proofs of mathematical ideas. This process may lead the students to the development of new mathematical content. Schoenfeld also indicated that it is necessary to consider classroom activities that are consistent with those epistemological goals. For instance, he mentioned that "helping the students to exploit what they do know, and to use their knowledge effectively, is a focus of my course" (Schoenfeld, 1989b, p. 80). Some of the class activities used by Schoenfeld in his problem solving classes involve: i) solving "new" problems (new to Schoenfeld) in the class in order to show the 48 students the decisions made during the process of solving the problems. These decisions are then discussed with the students. ii) showing videotapes of other students solving problems to the class in order to discuss the students' weaknesses and strengths shown while working on the problems. iii) acting as a moderator while the entire class discusses mathematical problems. Although the students are encouraged to select and try what they consider plausible, the moderator may provide some directions that are worthwhile to reflect on to the students. iv) dividing the class into small groups in which mathematical problems are discussed. The role of the instructor is to ask specific questions that help the students to keep track of their processes while solving the problems. These questions include those related to the subject matter and problem solving strategies. Schoenfeld recognized the importance of relating the learning activities at school with the activities that mathematicians or experts do when dealing with mathematical ideas. He stated: if you want people to emerge from instruction with the right sense of mathematics, then the mathematics classroom environment has to be a little cultural milieu where children take part in mathematics in a way that makes sense...in a way that will carry on outside the classroom (cited in Vobejda, 1987). Schoenfeld (1988) suggested that the main goal of instruction is to help students develop into autonomous learners. He also indicated that mathematical instruction should incorporate strategies for learning to read, conceptualize, and write mathematical arguments. Indeed, he identified a 49 fifth dimension, "learning activities" in which he discussed some strategies that could help students to read mathematical arguments. For example, students might frame their mathematical arguments in a three - phase sequence: convince yourself, convince a friend, then convince an enemy. In summary, Schoenfeld's work on problem solving incorporates a view of mathematics in which students are encouraged to make sense of mathematical ideas. The classroom activities suggested by Schoenfeld include open discussions of mathematical ideas between the students and the instructor. Schoenfeld also suggested that the students' analysis of mathematical problem solving should include their understanding of mathematical resources, the use of cognitive and metacognitive strategies, and their conceptualizations of mathematics and problem solving. Chapter Summary This chapter deals with the review of studies that helped to identify the types of activities related to problem solving that could help students improve their way to understanding mathematics. It starts with a summary of various interpretations of problem solving that are often found in the mathematics curriculum. The next part of this chapter dealt with studies that emphasized the use of heuristics in the classroom. This approach has probably been the most common approach to problem solving. It was discussed in order to identify the potential of the use of these types of strategies and also the limitations. Finally, the development of cognitive science and problem solving was discussed by focusing on methodological aspects and advances from cognitive science that could be useful for research in problem solving. A model proposed by Perkins and Simmons to identify patterns of misunderstandings experienced by the students while studying mathematics is 50 also discussed. The chapter ends with a discussion of Schoenfeld's work on problem solving which was important in the design of the study. 51 CHAPTER III DESIGN AND PROCEDURES Introduction This chapter deals with issues regarding the types of methods and procedures used in the study to collect and analyze the data. Results from the pilot study which were useful to decide the direction of the main study are also discussed in this chapter. It includes the rationale for selecting a specific paradigm of research and the strengths and limitations of using written tests, class observations, and task interviews as means for collecting the information to be analyzed. It also deals with the backgrounds of the subjects, the content involved, and the context of the study. Research Paradigm Krutetskii (1976) discussed various advantages that characterize the use of qualitative research as a necessary component for explaining and gaining insight into the phenomenon being studied. His research illustrated different examples in which students solved mathematical problems successfully by employing different means. While a quantitative analysis focused on the final product obtained by the students (solution) and organized the responses in the same group, a qualitative analysis investigated subtle differences found during the process of obtaining the solution. For example, in placing the digits 1, 2, 3,..., 9 in a "magic square", the approach which exploits subgoals, symmetry, and particular cases could be categorized as qualitatively different from the approach that uses only trial and error. However, from a quantitative perspective these two approaches might lead to 52 the right solution in the same amount of time and, therefore, be interpreted as the same. Miles and Huberman (1984) discussed advantages in the use of qualitative data. They expressed the opinion that: with qualitative data, one can preserve chronological flow, assess local causality, and derive fruitful explanations. Serendipitous findings and new theoretical integrations can appear. Finally, qualitative findings have a certain undeniability that is often far more convincing to a reader than pages of numbers (p. 22). Qualitative data can provide information about the difficulties that the students experience when learning mathematics and consequently this information could be used in planning mathematical instruction. Davis (1986) characterized several students' ideas about the concept of the limit of a sequence. For example, some students' responses were: i) "the limit is what the numbers get nearer to"; ii) "the limit is the smallest number that's not bigger"; and iii) "the limit is what you'd get if you went all the way out in the sequence". In interpreting these students' responses, Davis stated that "there is no presently-feasible way to translate this into quantitative terms. Although one could quote some numbers, their meaning would not be clear" (p. 215). Schoenfeld (1985) suggested that the use of small samples (groups of eleven students) could provide some clear implications for teaching. For example, in his experiment in which he used first-year university students, he concluded that explicit heuristic instruction does make a difference with regard to problem solving performance. 53 Davis (1986) used task-based interviews to investigate the origin of conceptual errors that students consistently make when dealing with basic mathematical concepts. He argued that the exclusive use of achievement tests to explain diverse phenomena in the classroom could give misleading results. "Achievement tests provide a certain kind of data, but they omit much more; they do not give a broad comprehensive picture of what is taking place" (p. 87). Hovdebo (1987), using the method of thinking aloud, was able to identify and categorize basic student characteristics for solving problems. Hovdebo's study was based on an analysis of how students responded to a single problem. Sherrill (1983) carried out a study to investigate the approaches students used to solve multi-step mathematical word problems. He interviewed 18 students and found that they did not read the entire problem carefully; instead, they tended to manipulate the facts involved in the statement of the problem. In addition, they did not check their solutions and failed to recognize the value of the use of heuristics. Sherrill's study illustrated findings that emerged from data gathered through interviews. The review of related studies suggested that a design which involves the use of task-based interviews is appropriate for meeting the objectives of the present study. A written test and a final examination provided additional information for the analysis of the students' methods of solving problems . Given that the sample for the present study involved only one class, it is important to note that the students' complete processes shown while working on the problems have been evaluated. In addition, the use of task-based interviews provided information regarding the types of students' difficulties and their understanding of mathematical concepts. Class observations and problem-task assignments contributed to the analysis of the interviews. 54 The Pilot Study An extensive pilot study was carried out prior to the development of the main study. The purpose of the pilot study was to explore the possible strengths and limitations of the means used to gather the information for the main study. The pilot was structured in three different phases. During the first phase a questionnaire and a written test were used to explore some characteristics of the students' approaches to problem solving. The second phase focused on the analysis of the students' use of particular heuristic methods while solving problems. Finally, in the third phase, there was interest in exploring the potential of using students task-based interviews to gather information regarding the students' processes for solving mathematical problems. First Phase of the Pilot Study Background and Instruments The exploration of the types of methods that students use when solving mathematical problems was the objective of the first phase of the pilot study. A class of 39 students taking a first-year college course in calculus was used for the pilot study. Two instruments (see appendixes P1 & P2) were designed to gather the information. The first was a questionnaire which included 38 questions arranged into five different groups. One group contained questions related to the students' views about mathematics and problem solving. The other four groups included questions regarding the types of strategies that students might consider during the process of solving mathematical problems. 55 The five groups of questions on the questionnaire were influenced by the work of Polya (1945) and ideas from Kilpatrick (1967) and Lucas (1972). Polya (1945) identified four stages that are basic in the process of solving mathematical problems. These stages include "understanding the problem, devising a plan, carrying out the plan, and looking back or checking the solution". According to Polya's framework, a set of questions was designed for obtaining information about each of the stages. It was not expected that the students would systematically use strategies described by Polya; however, the purpose was to collect some information about how the students were approaching mathematical problems. The idea was to identify major similarities and regularities in their ways of approaching and solving problems. The other instrument (the written test) involved a set of eight problems related to the concepts of function, limit, and tangent. The problems were selected in consultation with the instructor. He suggested that the orientation of the content was in accord with his course involving business-oriented calculus. The eight problems sampled the material found in the textbook used in the course, Applied Calculus by Dennis D. Berkey (1987). The fifteen-minute questionnaire was administered on January 24, 1989. The following day the students were asked to work on a set of problems for a period of 50 minutes. In the instructions which were clearly stated at the beginning of the study, students were told that the basic objective of this phase of the study was to explore the type of methodology that they were using in approaching the problems. They were asked to write down every idea that they had while working on a particular problem. 56 Discussion of the Questionnaire Results In the first phase of the pilot study, results from the questions on views about mathematics showed that 76% of the students were at least somewhat interested in studying mathematics. However, those students viewed mathematics as a set of rules which can be applied directly to solve problems. Ninety-four percent of the students believed that previous knowledge in mathematics determines the level of success in solving mathematical problems. This belief is consistent with the findings from a study carried out by Schoenfeld (1985). He identified certain beliefs that influence students' approaches to solving mathematical problems. As a consequence of these beliefs, students who feel that they do not have the necessary knowledge may give up on a problem quickly. Seventy-five percent of the students affirmed using the method followed by the teacher when solving a problem. In general, they tried to apply ideas used in similar problems discussed during the class. Sixty percent of the students said they ask for help when they have difficulties in solving mathematical problems. Regarding the use of a certain type of method in solving problems, it was found that fewer than 30% of the students were aware of having a method. More than 50% of the students believed that they need to improve their present knowledge of strategies for solving mathematical problems. The responses regarding the level of "understanding of the problem" showed that in general students are aware of the importance of understanding the problem before starting to solve it. However, few students, 20%, considered it important to restate the given problem in their own words. Sixty-two percent of the students maintained that they distinguish relevant from irrelevant information as a part of understanding the problem. The use 57 of graphical means, such as diagrams, was not identified as necessary to understanding the problem. Seventy percent of the students believed that changing the problem to a simpler form is an important step in getting the solution. It seems that students initially try to identify similarities with problems that they have solved before. In devising and implementing a plan for solving a problem, 78% of the students suggested that they explore ways which seem to lead to the solution quickly; however, 80% responded that they do not estimate the final answer to the problem before implementing a selected plan. This may suggest that the students' first attempts to solve problems are to implement quick ways without checking the plausibility of the solution. Polya (1945) suggested that the estimation of the answer helps to decide which way or plan is most suitable to pursue. Schoenfeld (1987) stated that most university students are not flexible in exploring ways to solve a problem. They normally pursue one method for a long time even when that method may not lead to the solution. Regarding whether or not students check the solution, it was found that few students go beyond solving the problem, even though they are normally aware of the importance of checking their work after they have solved the problem. The factor of time was mentioned by some students as an obstacle to checking their work. However, the students' ideas about checking the solution involve identifying whether or not there is a mistake during the process of solving an equation rather than making sense of the response or looking for other related problems. Results from the Written Test Problem one on the written test involved the use of the concept of function, the domain of functions, and the corresponding analytical 58 representation (equation). The graphical representations were given and the students were asked to write the corresponding equations and to distinguish which graphs were graphs of functions. In general, the students were able to write down the equation for each of the graphical representations. However, few of them identified which ones were functions. The domain, when it was identified, was described as real numbers even when some of the graphs clearly showed a different domain. It may be suggested that the students did not understand part of the question because most of them did not even mention the word "function" in their responses. The students probably believe that if they write the equation, then they automatically are dealing with a function; however, this issue needs further exploration. In designing problem one, it was expected that the students would use the geometrical test (that is, drawing lines parallel to the y-axis and checking the points of intersection) to decide whether or not that graph represented a function. However, no one showed that he or she knew how to use that technique. In general the students did not check the solution of this problem and they showed confusion in the use of the notation f(x). For example, a line parallel to the y-axis passing through (3, 0) was represented as f(x) = 3, and, in the same way, a line parallel to the x-axis passing through (0, 2) was represented as f(x) = 2. In general, they mistakenly identified the equation of a circle as a function. Problem two dealt with the absolute value of a function. The students were asked to sketch the graphs of two cases. The students did not have any difficulty in understanding this question. However, they approached the two cases separately without noticing the relationship between them. The students selected a few numbers and calculated the corresponding values. 59 They did not realize that they were dealing with a line with a slope of 45 degrees. When the function was changed, the students experienced difficulties. They explored only one of the two possible cases. It was evident that the students had enough information for solving this problem, but their lack of reflection or analysis regarding the use of that information shows that they failed to recognize any connection or similarity between the two cases. Problem three involved solving an exponential equation. In this problem, the students did not have any difficulty. However, few students explained why the first step was to equate the exponents. Problem four dealt with evaluating functions and introduced the concept of limit. The students found no difficulty in finding the value of f(0); however, they did not explain the meaning of the function when the values of x increase arbitrarily. The question of whether the employee's speed ever stopped increasing was answered without analyzing the equation; that is, there was no written evidence of any mathematical analysis. Problem five involved the use of the composition function. The students in general identified the composition; however, there were some problems in considering the dimensions of the problem. The relation between hours and minutes was not easily identified. That is, they did not equate units before solving the problem. This showed lack of attention to the key components of the problem. Problem six was about writing a function from given information. The students showed difficulties in writing the function. Their written work suggested that they did not follow a systematic order in representing the information. Moreover, they failed to distinguish the relevant information to be represented from the irrelevant information. 60 In problem seven, the students were asked to calculate the limit of two expressions. They solved this problem with a success rate of 90%. The solution of this problem is reduced to a simple calculation in which they have to factor an expression. Their success on this item showed that when dealing with problems in which a rule has to be applied or direct calculations have to be made students do reasonably well. Problem eight dealt with the concept of tangent. In general, the students solved part of the problem, but failed to write the desired equation. Few students represented the problem geometrically; they applied the procedure for finding the tangent without giving reasons for their approach. Although during instruction the geometrical meaning was emphasized, it was not clear whether the students really understood that geometrical meaning or only the mechanical part (algorithm). Summary of the Results of the First Phase of the Pilot Study The responses of the students to the questionnaire showed that they were aware of the importance of approaching mathematical problems systematically. The students also mentioned that the type of mathematics that they were interested in was the type in which rules could directly be applied. This was consistent with the rate of success in the written test in dealing with problems requiring direct calculation for obtaining solutions. Regarding the methods used by the students in solving the problems, it was found that few students approached the problems using a structured plan. It might be suggested that if the students perceived some similarities between the given problems and others seen before, they would use similar strategies to solve them. However, if the problems were different or involved different contexts, they would have difficulties in attacking the problems. Here the 61 absence of basic working strategies that could help students consider the problems from different perspectives was evident. A comparison of responses given on the questionnaire and on the written test suggested that the students recognized the utility of basic strategies in solving mathematical problems. However, they rarely used them in the actual process of solving problems. This phenomenon seemed to indicate that realizing the value of the strategies did not guarantee their use. The students' "methods" for solving problems seemed to involve considering each problem separately; they failed to recognize that problems that they have solved before are often particular or general cases that can be transferred and used in solving other problems. Some of the problems in the written test were similar to examples that had been used in the class to illustrate the concepts of limit and tangent. The students had been exposed to those examples just before the test was given. The students did considerably better on these problems compared with those on the rest of the test. This may suggest that when the content being studied is "fresh" and the problems are set in a familiar context, then the students seem to respond better than in cases where they have to recall and transfer knowledge from their other courses in order to approach the problem. It may be the case that the students' conceptualization of mathematics is reduced to sets of knowledge separated into different areas, such as arithmetic, algebra, and geometry. Each of these areas deals with seemingly different problems and the students do not notice the connections between these areas and their problems. 62 Second Phase of the Pilot Study Rationale for the Second Phase of the Pilot Study The preliminary analysis presented in the first phase of the pilot study provided some basic elements that characterize the students' methods in solving mathematical problems. It seems that the students' conceptualization of what mathematics is plays an important role in the methods that they apply when solving mathematical problems. Schoenfeld (1987) maintained that the way mathematics is presented to students during instruction influences the way that students perceive this subject. It may be recommended that the selection of learning activities in which the conceptual part (in addition to the mechanical part) of mathematics is shown would engage students in thinking of the importance of understanding the mathematical content beyond just considering it as a set of rules. The rationale for considering another phase of the pilot study rested on the need for exploring the students' approaches to problems when they are exposed to the use of problem solving strategies. Therefore, the next phase in this pilot study was to implement some basic heuristics (breaking the problem into parts, using diagrams, and working backwards) for problem solving that students were not using in the actual classroom. For example, it was decided to include specific heuristics when discussing the various stages involved in the process of solving mathematical problems. Research in problem solving has shown that students often get answers that do not make sense given the context of the problem. Students fail to explain or check whether or not what they have found is consistent with what was required in the statement of the problem (Davis, 1986; Polya, 1965; 63 Schoenfeld, 1985). Concerns of these types were also observed in the first part of the present study. To gather more information regarding how students conceptualize the use of diverse strategies, the students worked for 40 minutes on two problems (see appendix P3) involving the application of derivative. Afterwards the students were asked to hand in their work. In the same session they were asked to write down any relevant information that they could recall from the problems on which they had just been working. The aim of this part was to investigate the significance of retaining key components of the problems and success in solving the problems. Background to the Second Phase of the Pilot Study The first phase of the pilot study determined a need and, thus, provided a rationale for introducing specific strategies that could help students overcome their major difficulties when solving mathematical problems. The literature dealing with problem solving suggests that class instruction should include diverse examples that illustrate the importance of the use of different strategies. Schoenfeld (1985) recommended that the use of problem solving strategies should be presented in a descriptive way. This presentation should help students to identify the diverse applications and to conceptualize the limitations of such strategies. It is important for students to realize the extent to which a strategy should be pursued and to take into account the structure of the problem in order to make decisions regarding which strategy to use. These recommendations were taken into consideration when the materials for discussing the strategies during the class were prepared. The students were studying the applications of the derivative when the second phase of the pilot study was initiated. The instructor emphasized the 64 need to follow specific directions for solving problems related to several applications of the concept of derivative. The importance of identifying the variables of the problem, the basic conditions, and the importance of modeling the problems mathematically were issues considered important when solving the problems. In addition, the importance of discussing specific questions at different stages of the process when solving the problem was illustrated. According to the framework suggested by Polya, several examples were discussed during the class. The examples emphasized the need for selecting the key information from the problem, the importance of monitoring each stage involved when working on the solution of the problem, the use of graphs or diagrams, and the importance of contrasting the solution with the conditions of the problem. As a part of the evaluation of this phase, the students spent 35 minutes working on two problems related to the applications of the derivative (appendix P3). In the same session, after the students handed in their work, they responded in writing to eight questions. The questions were intended to explore basic information about the problems. The results are summarized in the following section. Analysis The data were analyzed by recognizing the students' major difficulties as manifested in their written work. It is important to note that the analysis was exploratory and based on what the students wrote during a limited time; moreover, the students did not write extensive explanations of their responses in the complementary questionnaire. This phenomenon limited the analysis. 65 Level of difficulty of the problem. The problems were similar to those used by the instructor when teaching the unit. Therefore, the level of difficulty was considered suitable for the class. The problems included some subtle parts that required basic awareness of real world conditions. For example, responding that the total number of employees should be 62.5 showed that the students failed to relate their mathematical response to the conditions of the problem. Indeed, students who were able to get that correct response often failed to make that connection. Progress in obtaining the solution. The students' responses indicated major difficulties when working on the problems. They did not display signs of approaching the problems systematically. Their written work was incomplete and often did not match the conditions of the problem. The students often started by defining the variable(s) involved in the problem in one way and then changing their meaning during the process of solving the problem. This may explain some of the difficulties that the students experienced in understanding what was asked in the original statement of the problem. The students experienced difficulties in translating the statement of the problem into the corresponding mathematical representation. It seemed that the students' conceptualization of some words involved in the statement of the problem, such as "average", "production", and "dimension", might be different from the contextual meaning embedded in the problem. This phenomenon was pointed out by some students when they were asked to indicate their major difficulties when dealing with application problems. The use of strategies. Even though several examples were illustrated to highlight the use of various means to organize the data, plan the solution, and 66 check the results, the students' written work did not show evidence that they followed a structured plan in approaching the problems. It may be suggested that the students are accustomed to working on problems in which a direct procedure leads to the solution. If a problem requires further analysis to understand the information, then the students may experience difficulties in exploring plausible ways to attack the problem. Recalling basic information. In general, the students were not able to remember basic information about the two problems. Fewer than 10 percent of the students stated the problem using the factual information of the original problem; nor did they use their own words to restate the problems. Instead, they tried to recall the same wording of the problem. It may be that the students were not expecting to write information about the problems and that they did not perceive the relevance of recalling the problems after they had worked on them. Whether or not the students solved the problems seemed to be unimportant to the students afterwards. Krutetskii (1976) found that capable students in mathematics were able to recall the information regarding the statement of the problem, the plan for obtaining the solution, and the solution. The use of graphs or diagrams. The use of graphs when working on problems that involve the applications of derivative may help students to understand the conceptual meaning of derivative. In general, the students did not use graphs to represent the problem. They failed to recognize that finding the maximum or the minimum means to identify a specific point that could have been represented geometrically. This representation could have helped them to 67 evaluate the solution, that is, to ascertain whether or not the solution was reasonable according to the conditions of the problem. In the second problem, the use of a diagram could have helped the students to point out the desired side of the rectangle. However, few students represented the problem graphically. The students did not distinguish and relate the conditions of the problem. Looking back or checking the solution. Polya (1945) recommended that the process of solving problems should include: i) checking the steps followed while solving the problem, ii) looking for different ways to solve the problem, iii) checking the plausibility of the solution, and iv) extending the problem to a more general domain. The students' written work does not show that they were aware of such strategies. They probably would have checked their work if they had had enough time to solve the problems. Nevertheless, the checking process was reduced to seeing whether or not there was a calculation or algebraic mistake when they were solving the problem. The students did not perceive the importance of solving the problem by using more than one method or the need to generalize or transfer the problem into a different domain. This phenomenon shows the students' reluctance to conceptualize other strategies for solving problems. The students seem accustomed to solving problems that normally do not require the use of different means for obtaining the solution, that is, problems which can be solved by applying a specific rule. The students may believe that all mathematical problems can be solved in the same way, and, as a result, they experience difficulties when they are asked to solve different types of problems. However, this issue needs further exploration. 68 Summary of the Second Phase of the Pilot Study The students overwhelmingly recognized that they had difficulties when dealing with problems related to the application of the derivative. The students believed that the only way to overcome these difficulties was by solving many problems of this type. However, research shows that practice without following a systematic approach does not necessarily help students to develop strategies for making sense of the problems and their solutions (Schoenfeld, 1979). The students' written work showed that they lacked basic strategies for organizing the pertinent data, did not reflect constantly throughout the process of solving the problem, and did not check or evaluate the solution. It is suggested that students should be encouraged to approach mathematical problems in such a way that an appreciation for systematically considering strategies could be developed. Moreover, it should be made clear that problem solving strategies are not a straightjacket and that flexibility in their use is required depending on the structure of the problem. A clear indication from the analysis of the data is that a problem solving approach in learning mathematics takes time to be assimilated and to be used by the students. Good examples must be presented during class along with different activities that reinforce and extend their use throughout the course. It is important to monitor the students' advances in the use of basic strategies. Students should conceptualize problem solving as a means to understanding mathematics and use this approach constantly. This is a crucial point when evaluating the strengths and limitations of this approach. The role of students' ideas about their understanding of mathematical concepts can provide important information during the implementation of instruction focused upon problem solving. 69 Reflections on the Second Phase of the Pilot Study There are various factors which influence the implementation of basic strategies for solving problems. Students' difficulties in adopting and using a problem solving approach are affected by their past experiences in solving problems. Problem solving is a way of thinking and a new conceptualization that requires reflection and constant evaluation of the problem being solved. When difficulties arise, it often means that the problem should be transferred into a different domain, or broken down into simpler problems in order to obtain the desired solution. The process of implementing problem solving strategies takes time and requires the use of diverse activities during instruction. For example, problems for the class discussions and assignments should be carefully selected. These problems should involve different levels of difficulty and show the power of considering various strategies for solving them. It is important to supervise students' work constantly. If possible, there should be personal interaction in which students should be asked to solve some specific problems. This will help give the instructor insight into the way his or her students are conceptualizing the problem solving approach. Instructors very often take for granted that their students will understand and apply specific strategies. They are often surprised to find out that what they expected from their students is never achieved. The students' written work is only part of their conceptualization; it is a limited indicator of the success or failure of the use of the problem solving approach. 70 Third Phase of the Pilot Study Introduction The limitations of the usefulness of the questionnaire and the students' written work shown in the test and complementary questions demonstrated a need for more information to support the concerns identified during the first two phases of the pilot study. Interviews in which three students were asked to solve three mathematics problems (see appendix P4) were carried out to explore the ways that students solve problems. It was deemed important to evaluate how the students would respond to the three problems and whether the interviews could provide more useful information for interpreting the previous results. Getting students to think aloud was a technique employed by the researcher. The findings were categorized by using the theoretical framework employed by Schoenfeld (1985). The categories that emerged from the data have direct implications for the teaching of mathematics. The aim of carrying out the interviews was to explore basic characteristics of strategies employed by students while solving mathematical problems. Three mathematical problems were used in the interviews (see appendix P4). One of the problems involved mathematics knowledge recently studied by the interviewees; the other two problems involved mathematical content studied in previous courses, such as algebra and trigonometry. The intent was to work on problems that involved content with which the students were familiar so as to focus on the use of different strategies for solving them. The interviews lasted approximately 75 minutes each. The students were asked to think aloud while solving the problems. A few questions were asked during the interviews to clarify what the students meant or were thinking about; however, the interventions were kept to a minimum. In the last part of the interview, some questions regarding mathematics and mathematics 71 teaching were asked. The purpose was to explore the students' conceptualization of mathematics and its teaching. Schoenfeld (1985) found basic categories which are related to the process of solving mathematical problems. These categories include: i) the domain knowledge that includes the basic definitions and procedures that the students have at their disposal. The definition of variables, polynomials, and rational numbers, procedures for graphing functions and for obtaining derivatives, and basic properties of inequalities are some examples identified in this category. ii) the heuristic strategies that students may use while solving problems. Polya (1945) discussed the use of various heuristics by presenting different examples. These strategies include exploring related problems, breaking the problem into parts, working backwards, and establishing subgoals. Although these strategies seemed to deal with general context, Schoenfeld pointed out that the means of implementation are very much domain specific. iii) the executive control that is involved in the metacognitive processes of solving problems. This category includes making decisions about what strategy to use, when to use it, and the extent to which it should be used. Schoenfeld (1988b) stated that "the general idea of executive control is that there is a feedback loop that consists of monitoring one's actions on line, assessing progress, deciding whether change needs to be made and taking action if the situation is deemed problematic" (p. 72). iv) the system of beliefs which involves the students' conceptualization of mathematics and problem solving. Schoenfeld stated that "one's notion of mathematics determines whether, and if so, how one uses mathematics in situations for which it is appropriate" (p. 73). 72 The categories identified by Schoenfeld were used as a guide to analyze the information gathered from the interviews. The analysis focused on the difficulties that the students experienced while solving the problems. Analysis Three students participated in the task-based interviews of the third phase of the pilot study. The interviews were audio recorded and transcribed word by word. The process of analyzing the information included identifying the strategies and the resources employed by the students while working on the problems. Since the researcher spent time observing the class, part of the analysis is also based on the development of the instruction. Each of the three students began reading the problems aloud, and then almost immediately, started drawing diagrams to represent the problems. For example, Maria's starting point was "First of all, I would draw a diagram and write the data...". After Maria had worked on the first problem for seven minutes trying to write the data in the diagram, she asked, "What type of problem is this? Do you want me to find...?" Robert similarly began by drawing an inaccurate diagram of the data. Drawing a diagram immediately after reading the problem tended to be the first step carried out by the students. The first attempts were inaccurate and showed lack of clarity in understanding the pertinent information of the problem. However, the inconsistency of the data written on their diagrams helped them to realize that their diagrams were not adequate or did not correspond to the given information. Making adjustments and clarifying relationships included in the data involved a process that required the students to go back to the original statement of the problem and to analyze more carefully the information provided in the data. 73 The students were able to represent the problem graphically in about 15 minutes. However, it was not clear to them what direction to follow in order to solve the problem. Maria mentioned, "I know that I have to use a mathematical relationship that involves the radius, height, and angle, but I do not remember which one." Another student stated, "I have represented the data graphically, but I'm not sure what formula to use." The three students approached the second problem by writing down the initial conditions given in the statement of the problem. They immediately got involved in finding the solution of the system of equations. They worked for about 20 minutes without success. Nevertheless, they persisted in solving the system. During this process, the students showed frustration when dealing with large expressions. Some of them even used their calculators to try to simplify the expressions. It was observed that the students were persistent but showed no progress in reaching the solution. At this point, it was suggested that they read the question again and write down all the information, including the unknown. When the students wrote the "sum of the inverses" algebraically, they immediately noted the linkage between the data and what was required in the problem. Robert showed major difficulties when working with algebraic operations. He was not able to relate the data to the question. He was confused with algebraic expressions such as (x + y) 2 * x 2 + y 2 . It was evident that this student was experiencing difficulties in applying basic algebraic concepts; however, it was surprising that this student was more fluent and accurate in providing the graphical representation of the bowl problem than the other students were. Comments such as "Are you sure?" or provision of some counterexamples helped him overcome some difficulties. That is, even 74 when Robert had difficulties in dealing with basic algebraic expressions, he showed progress when some of those difficulties were challenged. Maria, who noticed the connection between the data and the unknown after it was suggested that she reread the problem carefully, was asked to solve a similar problem. She paused for about five seconds, read the problem aloud, and then solved it immediately. The students had the mathematics background required to solve the problems. They were asked to talk about why they were not able to notice the relationship between the data and the question in problem number two. Maria responded, "I didn't think the last part of the question was related to the two pieces of information. I always think that one has to find the unknown first before substituting the unknown into the last question." This comment illustrates the mind set that students develop and apply when they are asked to solve problems that seem to match specific patterns. The third problem was similar to the types of problems that the students had studied in their last course (one month before). The students showed a systematic approach by drawing a diagram, identifying the mathematical concept related to the problem (surface area), and representing the problem symbolically. Part of the problem involved sketching a graph or a diagram of the problem. Here, the students experienced difficulties; they drew a straight line that did not correspond to the general form of the equation. The graph in this part was needed to answer the next part of the problem, that is, to estimate the lowest cost value. The students ignored the graph and found the lowest cost value by using standard procedures such as derivatives. When they were asked to interpret the result in the graphical representation that they had just sketched, they noticed the inconsistency and tried to fix the 75 graph. It seems that when students are challenged to make sense of their work, they may analyze the problem more carefully. Recent views of mathematics have recognized that the development of mathematics very often relies on the flexibility and openness of the mathematical ideas. In these views, there is room for false starts, speculation, guesses, and constant improvements of mathematical results (Davis, 1986; Hanna, 1983; Kitcher, 1984; Lakatos, 1976). Hanna (1989) pointed out that "... no proof is final, and indeed it is the essentially social process of negotiation of meaning, rather than the application of formal criteria from the outset, which leads to the improvement of a proof and its growing acceptance" (p. 20). This characteristic was only partly present in the initial work of the students. They guessed and speculated about the information. However, the students experienced difficulties in evaluating or changing the direction of the first attempt. Schoenfeld (1988b) suggested that what students believe about mathematics influences the process of solving the problem. "One's notion of mathematics determines whether, and if so, how one uses mathematics in situations for which it is appropriate" (p. 73). Schoenfeld (1985) argued that the students' classroom experiences influence the way they conceptualize mathematics. "If the bulk of students' experiences with particular mathematical ideas occurs in the classroom, then the students' mathematical words views -their abstraction of experiences with those ideas- will be based on those experiences (Schoenfeld, 1985, p. 185). In solving the problem, the students rushed to write down the data and then immediately worked on the solution. They often experienced difficulties because they were not using all the information provided in the problem. They pursued solving the problems 76 by exploring only one way. This may be related to the students' experience in solving problems. Examples of problems in which the first attempt at solution involves the use of a unique strategy that leads to the solution may influence the students' conceptualization for solving problems. For instance, they may believe that the initial strategy selected for attacking the problem always works in solving problems. In addition, students' conceptualizations may be further reinforced if the solution of assigned problems requires only the use of specific rules to solve them. As a result, the students may believe that the problems can be solved in that way. Features of Students' Approaches to the Interview Problems Flexibility in Approaching the Problems The interviews showed that the students initially selected a way of solving the problem and stuck to it without evaluating advances or progress made. For example, in the second problem, it was necessary to find the sum of the inverses of two numbers. All the students decided to solve the system of equations that represented part of the data of the problem. Although the students had difficulties solving the system of equations, they did not check other possibilities or use the information systematically. The students derived complex algebraic expressions during the process of solving the equations. However, this development only led them to check for possible algebraic mistakes rather than to explore other ways. It was clear that they were not confident with the type of expression with which they were dealing. This phenomenon seems to indicate that if their work involves exact numbers, such as integer numbers when obtaining square roots, students feel more confident with their work. 77 In exploring the difficulties found when solving problem number two, Maria expressed, "From what I have learned in the past, the teachers always teach us first to find the unknown. They didn't teach us to look at the question to see if there is any relation with the data before working the problem." In the same vein, Peter indicated: I recall when I ... when you are given two numbers with two equations, you try to solve for one variable in one equation, substituting it in the other one. Once I knew that the number was x or y, 3 or 8, 5 or whatever, I solve one over five, one over six, or whatever, and I solve it in that way. In explaining his lack of success, he mentioned, "I got a problem that I could not remember what the rule was here." Schoenfeld (1988) indicated that students often use the organizational form of their work as a criterion of validity. For example, some students did not accept a geometrical proof written in paragraph form (even when the argument was correct). The students believed that the proof had to be presented by using the "T" form. The "T" form was the way that their teacher had taught the class. Consistency in Approaching the Problems There is evidence that the students followed a specific plan in solving the problems. Nevertheless, they did not seem to evaluate the sense of what they were obtaining during the solution process. For example, the graph obtained in problem number three was contradictory to another part of the problem in which they were asked to estimate the lowest cost. The students seemed to approach the problem in parts without attempting to integrate the 78 parts. The plausibility of the responses seemed to be unimportant to the students. Polya (1945) recognized that constant evaluation and going beyond the solution are powerful steps in solving mathematical problems. Effects of Class Instruction and Textbooks Schoenfeld (1987) suggested that the way mathematics is presented to the students and the type of problems discussed during mathematical instruction influence the way that students approach problems. Instructors often select the examples to be discussed in class and work on them in advance. They know how to solve the examples and rarely speculate or begin with false steps to get to the solutions. In general, there is no time for discussion of the multiplicity of possible ways to solve the problems and the rationale for then selecting a particular strategy. The initial strategies invariably lead to the correct solution. The struggles or difficulties that the instructors had to overcome during the preparation period are never shown to the students. Illustration with examples is reduced to a presentation of a successful way to get the solution of a problem. Students may experience difficulties when working on mathematical problems because their methods do not run as smoothly as their instructors' methods. Szetela (1989) stated that "some students may in fact be good problem solvers but lack the confidence that success in problem solving should bring. Each new problem is viewed as much as an opportunity for failure as for success" (p. 1). A lack of confidence was apparent in the students' work. They constantly asked if the work was correct or not. Maria stated, "When I solve a problem, I follow the examples studied in class step by step. I also check the textbook examples. I usually don't have any problem working on the assignments. At the end of each 79 exercise, I check my solution with the solution given in the textbook. That indicates that I have solved the problem correctly...." Students' Conceptualization of Mathematics The last part of the interview included questions about mathematics and problem solving. The students consistently referred to the part of mathematics in which one can apply rules, make calculations with numbers, and use graphs to represent the problems. Since one of the major difficulties was to switch from one strategy to another when solving the problems, they were asked to elaborate on this issue. They mentioned the lack of discussion of problems that illustrate nonroutine cases. Maria pointed out that the understanding of the conceptual part of mathematics is not a necessary condition for doing well in the exams. For example, she mentioned, "The course that I am currently taking is much easier than my last course, because the instructor is interested only in manipulating numbers." Another student mentioned that what is needed in mathematics instruction is more emphasis on graphical representation. "A graph", he said, "helps you understand what is going on in the problem." Although the students may not have an elaborate view of mathematics compared with the views found in the mathematics literature, the findings suggest that they distinguish between performing mathematical operations and solving mathematical problems. Students are more interested in the application of algorithms or rules than in understanding the processes of doing mathematics. The students seemed more comfortable working with numbers (calculation) than finding mathematical relationships. Selden, Mason, and Selden (1989) pointed out that students may conceptualize the study of calculus as a set of rules or procedures for solving 80 routine problems. They stated that "students holding this view are likely to resist any emphasis on cognitively nontrivial problems, regarding them as distractions and irrelevant to the main business of a calculus course" (p. 50). Recalling Mathematical Knowledge from Previous Courses The process of solving the problems required the use of basic knowledge in algebra. The students experienced difficulties applying that knowledge. For example, Maria stated, "I can't apply my past knowledge; I have already forgotten even content from my last course." Peter said, "I don't remember the formula to solve the quadratic equation." When asked for possible reasons for that difficulty to recall past mathematical knowledge, Maria said: I don't remember the rule because it may be that I don't understand the concept quite clear. Maybe I just know the type of question that apply to the concept. I would like some explanation of the equation first, then apply the equation. What she pointed out is that the type of linkage between the understanding of the concept and its application rarely occurs in instruction. Schoenfeld (1985) described an example in which one of his students had been exposed to a specific problem before and could not solve it again after a period of time. This phenomenon occurred during the interview, and the students rarely made the effort to find the desired mathematics relationship on their own, even though the relationship could be easily discerned. 81 Challenging Students' Difficulties The three students experienced some difficulties when solving the problems at some points during the interview. However, with some direction they were able to continue and solve the problems. The degree of intervention by the researcher varied in accordance with the type of difficulty. For example, Maria, when working on problem number two, was able to identify the connection between the data and the unknown by simply being advised to read the problem and write the conditions again. Peter was not able to identify the connection immediately; nevertheless, he overcame some major difficulties when asked to make sense of his responses. It was clear that during the interviews the students were sensitized to the way in which they approached the problems. In the first problem, it was difficult to clarify difficulties that arose while working on the solution; the students wanted to make sure that what they were doing was correct. With the rest of the problems, they seemed to be more relaxed and more receptive to the researcher's comments. The Role of Context in Mathematics Learning The learning of mathematics depends in part on the presentation of various examples. The examples are problems embedded in a predetermined context. For example, the illustration of the application of derivative is often presented to the students by dealing with problems that characterize a certain type of structure. That is, the statement of the problem involves information to be modeled mathematically, and then treated by using the derivative techniques. Such examples require a great emphasis on the way that the problems are mathematically represented. If students succeed in representing the problem symbolically, then they will probably solve the problem quite 82 easily. The use of derivative in this type of problem becomes a routine exercise in which the application of an algorithm will eventually lead to the solution of the problem. This phenomenon may explain the students' lack of success in interpreting the sense of their solutions. They may see the application of derivatives as a secure way to solve that type of problem. Reflection on the importance of the use of derivatives may be encouraged by posing problems in a variety of contexts. For example, a task may start by providing some information from which students are asked to formulate the problem and to justify the need for using derivatives in their solutions. Mathematical Hierarchy and Its Influence in Learning Noddings (1988) mentioned that the understanding of mathematical concepts often requires the use of basic skills. For example, simplification of radicals, squares of numbers, and extraction of square roots are skills necessary for understanding the Pythagorean Theorem. As a consequence, if the students are fluent in these skills, they may focus their attention on concepts related to the Pythagorean Theorem rather than on struggling with small steps involved in the theorem. Noddings stated that "the human brain must give focal attention to the issue perceived as central and significant; subskills are executed in the periphery" (p. 248). The study of mathematics depends on the understanding of definitions, mathematical notation, facts, concepts, and diverse strategies that help make sense of mathematical relationships. Even though interrelationships among these aspects of mathematics exist, each aspect possesses a different level of difficulty to be understood and to be applied in different situations. Students often fail to recognize the level of importance of the parts of a determined 83 concept and consequently experience difficulties when applying that concept to a particular situation. The lack of hierarchy in the presentation of mathematical content is an aspect inherited from formal mathematics that still permeates mathematical instruction today. For instance, the study of the concept of derivative involves the understanding of several concepts, such as number, function, operation, limit, and continuity. These concepts are necessary to discuss the concept of derivative. Nevertheless, the interrelationships among these concepts and their level of importance are rarely part of mathematical instruction. A conceptual analysis showing their interdependence and use when discussing the concept of derivative may help students to recognize what parts of the concept need more emphasis or need to be reviewed. It may be suggested that the emphasis given during instruction to each concept should depend on the students' ability to understand these concepts. If the students fail to understand the concept of derivative and the instructor repeats the explanation many times without success, it may be because the students are experiencing difficulties in understanding another concept that is crucial to understanding the concept of derivative. 84 Methodological Elements of the Main Study The results of the pilot study indicated that it was important to try more problem solving activities in the classroom. It also provided an indication of the potential of the use of qualitative means for gathering the data and directions for the analysis. What follows is a description of the basic methodological elements of the main study and its trustworthiness. Subjects of the Study Thirty students from one class were the subjects of this study. The class was taught using a problem solving approach which emphasized the use of cognitive and metacognitive strategies for solving mathematical problems throughout the entire course. The students were involved in various activities which the recent literature in problem solving suggests could make a difference in the students' approaches to mathematical problems. For example, the students in this study were asked to evaluate their approaches to problems constantly, to reflect on the sense of their solutions, to extend the problems into different domains, and to criticize the statements of the problems. All these activities were modeled by the instructor (who also provided information for the analysis) during the class and were also monitored throughout the assignments. Procedures There were several instruments that were used to gather the information needed for this study. The pilot study showed that the initial data from the questionnaire would provide useful information for the analysis of the written tests and the design of the problem-task assignments. Interviews 85 provided insights about the concerns identified earlier. Therefore, the order of gathering the information for the main study was the same as that of the pilot study. In addition, class observations were carried out during the entire development of the study. Table 1. Overview of Activities Undertaken in the Main Study and Their Scheduling Activity Purpose Date Introduction to the Class Presentation of the study and the roles of the instructor and researcher to the class 05/09/89 Pre-questionnaire Elicitation of some students' ideas about mathematics and problem solving 06/09/89 Assignment on the use of polarity chart Monitoring the students' work 14/09/89 Metacognitive assignment Asking the students to reflect on their own work 14/09/89 Assignment on the concept of function Monitoring the students' work 21/09/89 Metacognitive questions Asking the students to reflect on their own work 21/09/89 Written test Exploring the students' approaches to five conceptual problems 18/10/89 Assignments on the concept of limit and derivative Monitoring the students' work 03/11/89 Metacognitive questions Asking the students to reflect on their own work 03/11/89 Assignment on applications of derivatives Monitoring the students' work 09/11/89 Students' interviews Exploration of the students' approaches to two mathematical problems 14-19/11/89 Post-questionnaire Elicitation of some students' ideas about mathematics and problem solving 06/12/89 Final exam Analysis of the students' responses to typical exam problems 07/12/89 86 The Use of the Questionnaire The results from the questionnaire used in the first phase of the pilot study provided initial information to characterize the students' ideas about problem solving and mathematics. This information was useful for designing the written test and the interviews. With some changes, the questionnaire was used again in the main study. These changes resulted from the pilot analysis and comments from other researchers. In addition, the questionnaire was given to the students at the beginning and at the end of the course. The information gathered via those questionnaires was contrasted in order to identify possible changes in the students' mathematical disposition. Problem-task Assignments In the pilot study, some specific problems were used to gather information about whether or not the students could recall information about the problems immediately after they had worked on them. This type of instrument proved to be useful in focusing on specific aspects of the process of solving problems. The same idea was used in the main study to gather information about the level of understanding of the problems, the use of strategies, the decision making process, the sense of the solution, and the possible extensions of the conceptualizations or understanding of problems into other domains. In addition, the assignments also included a set of metacognitive questions that the students were asked to reflect on while solving the problems. The Use of Interviews The interviews carried out in the third phase of the pilot study provided insights into categorizing the students' ways of solving problems. The interviews were important in probing the students' concerns identified in the two previous phases of the pilot study. Indeed, the findings were important in designing some of the activities to be used in the classroom. For example, the consideration of nonroutine problems and the need for more emphasis on different interpretations of the concept being studied were concerns identified in the interviews of several students. As a consequence, similar interviews were also part of the main study. Since the main study focused on the use of the problem solving approach, the interviews had an additional objective, that is, to evaluate whether or not the students had shown specific changes when solving the problems. In addition, the interviews enhanced the categories - identified in the pilot study - of ways in which the students approached the problems. Some students were interviewed (7 interviews were analyzed) towards the end of the course. The purpose of the interviews was to obtain information concerning the main ideas that the students use when solving problems and to probe the difficulties that they encounter when dealing with specific problems. Two problems which could be solved by using different means were used as a vehicle to elicit information about the students' understanding of a particular problem and their ideas about the way that they plan to solve the problem. The selection of strategies, the reasons for changing strategies while solving the problem, and the understanding of basic mathematical concepts involved in the problem were aspects to be explored during the interaction. The students were asked to think aloud when solving the problem; the interviewer asked clarification questions whenever required. The information collected from the interviews was contrasted with the information gathered from the assignments and class observations in order to 88 identify basic categories in the ways that the students approached the problems. The Thinking Aloud Technique The use of thinking aloud has become an important way to elicit the ideas that students use while solving mathematical problems. Krutetskii (1976) suggested that the interviewer should make clear to the interviewees that the interest is in the process used by them while working on the problems, and not in their final decisions. He mentioned that thinking aloud is different from explaining the solution aloud. Krutetskii stated that "sometimes the pupil (especially an older one) might think he is being asked to give an observation and description of his own mental process in solving a problem, an exhaustive account of how he is thinking" (p. 92). Therefore, it is necessary to tell the student that he or she should not try to explain anything to anyone else, he or she should pretend that there is no one in the interview but himself or herself. "Think aloud. You do this, don't you, when you are solving a problem alone at home?" (Krutetskii, 1976, p. 93). Ericsson and Simon (1987) suggested that the use of thinking aloud should include an initial warm-up. Here the subjects should become familiar with the equipment including microphones and tape recorders. In addition, the researcher should ask the subject to solve some problems and check the consistency of the data with the instructions. Ericsson and Simon also suggested that the researcher should monitor the subject's processes when thinking aloud. For example, after a pause of fifteen seconds or one minute, the researcher may intervene by saying "Keep talking" or asking "What are you thinking about?". 89 Perkins (1981) suggested that in order to guide the direction of the thinking aloud process, the students should be aware of basic principles. For example, the student should say whatever is on his or her mind, speak continuously, audibly, and telegraphically; also he or she should not overexplain or justify. Such recommendations were discussed with the students before the interviews and resulted in being useful for clarifying aspects that were expected during the development of the tasks. The interviewee were asked to relax and try some examples before they started working on the problems. It was decided that the interviewer would intervene only when there was something unclear or when the interviewer judged it necessary to gather more information at a specific moment during the interview. The Use of Written Tests Students often write tests as part of the evaluation of their achievement in a mathematics course. They often develop specific strategies in order to do well in such tests. They work under specific conditions, such as time restrictions, and most often emphasis is placed on obtaining the final answer. The pilot study showed that if the restrictions are relaxed, the students may provide information useful for exploring strengths and weaknesses in solving problems. The aim in using the written tests was to explore the ways that the students approached mathematical problems in a limited time. In addition, the tests were used as a system of reference in analyzing the possible changes after instruction. There were two written tests, one given to the students at the beginning of the course and a final exam given at the end of the course. 90 Class Observations The researcher observed some classes during the development of the study. The aim was to collect data that would help interpret the direction of the class. The parameters that were observed were: the type of discussion when solving problems, the use of strategies, the number of problems discussed, the time spent solving the problems, the organization of the class, and the students' participation during the class (type of questions or comments during class). An observation form was designed to gather this information. The activities used in the class using the problem solving approach were planned in advance; nevertheless, their implementation depended on the particular class environment. For example, a specific strategy might have required more examples or more discussion time than the planned time. All these events were recorded by the researcher and were used in the analysis. In addition to the class observations, the researcher met the instructor regularly to discuss the directions and development of the course. The Use of Nonroutine Problems Schoenfeld (1985) employed nonroutine problems to investigate the ways that students (with mathematics interest) solve problems. He found that the students improved their methods for solving nonroutine problems after they received instruction based on problem solving. How average students respond to nonroutine problems is an issue that needs to be investigated. The use of nonroutine problems in the present study helped to identify the extent to which students were able to transfer the content to various contexts, that is, to apply their knowledge from the course to solve the problems. In addition, since some nonroutine problems were incorporated in the instruction and were part of the assignments, it was important to consider 91 the reactions of the students. For example, it was important to document how the students approached the problems, the types of difficulties that they experienced, and how much time they spent working on the problems. Implementation of the Written Tests Two written tests were given to the students one month into the course and at the end of course. The first one contained six nonroutine problems in which the students had to consider basic transformations for getting the solutions. The idea was to evaluate the students' work under limited time. The problems involved the concept of function, limit, arithmetic sequence, and the use of exponents. Since getting the solution to each problem required the use of diverse strategies, the entire process while working on each problem was evaluated. The students had to write a final exam as a part of the evaluation of the course. It was decided that this final exam (designed by the course coordinator) could provide some information about the ways that the students in the problem solving course responded to problems that were different from the ones discussed during the class as well as those on the assignments. This information could also be used to contrast data gathered from the first written test. Among the parameters to be evaluated were: i) evidence of selecting the strategies, and ii) the extent to which the strategies were pursued (progress). Limitations of the Written Tests The first written test provided information regarding similarities and differences among the students. It also gave some indication about the types 92 of difficulties that they were experiencing in the process of solving the problems. Since all the students had previously taken similar mathematics courses, it was hoped that there would not be notable differences in the first written test. However, it was important to keep records of how each student did in the pretest. This helped to identify the types of difficulties that the students had and allowed the researcher to monitor their advances. The information gathered in the pretest was contrasted with the results of the final exam. However, the information collected from the written tests was not sufficient to explain the possible reasons for the students' difficulties. Another limitation of the written tests was the limited time that the students spent working on the problems: one hour for the first test and two hours for the final exam. This time was based on the time that the students normally spend on mid-term and final exams. The review of some studies also suggested that when students are asked to work for a longer time, the quality of their work decreases notably (Lucas, 1972; Schoenfeld, 1985). Selection of the Interviewees The students were invited to participate in the interviews through a letter (see appendix M6) which explained the objectives and general direction of the interview. It was stated that their voluntary participation would not affect their class standings and that the results of the interview would be kept confidential. Seventeen students responded as being willing to participate. Twelve students were actually interviewed, and seven interviews were selected for the analysis. The only criterion for the selection of the 12 students was their compatibility with the schedule proposed for the interviews. Two interviews were discarded, one because of the poor quality of the sound and the other because the interview was incomplete due to the 93 student coming late to the appointment. The researcher listened to the interviews several times and judged that in three interviews the subjects were more interested in explaining their processes than thinking aloud. These three interviews were not included in the analysis. In the end, seven interviews were selected for the analysis on the basis of showing clarity in the development of the interview. The 12 students who were interviewed received written feedback on their approaches to the problems (see appendix M i l ) . Analysis of the Information Six sources provided the information to be analyzed in this study: a questionnaire, two written tests, problem task assignments, class observations, and interviews. Each instrument contributed to the analysis in the following way: a) The questionnaire and the first written test provided information about the students' views about mathematics and problem solving. The questionnaire was given to the students at the beginning and at the end of the course. An item by item analysis for the questionnaire was carried out by using the "LERTAP" computer program. Directions for the use of this program were taken from Nelson (1974). The students' responses were contrasted and later related to the students' work shown in the first written test. This first part of the analysis characterized the students' initial approaches in solving mathematical problems and was important in analyzing the advances or changes shown by the students during the course. b) The problem-task assignments and class observations provided information about the development of the course. This information was analyzed by contrasting the students' work shown in the assignment with the activities 94 that were implemented in the classroom. The students' work shown in the assignments and written tests was evaluated by using an evaluation scheme developed by Schoenfeld (1985). He evaluated the problems by considering three categories : a) evidence of using an approach that might lead to the solution of the problem, b) evidence that the student had pursued that approach (pursuit), and c) the progress made by the student when solving the problem. Progress is divided into three levels: "some", "almost", and "solved" which were assigned in accordance with the progress made by the student when solving the problem. c) The final exam provided information useful for evaluating the students' work at the end of the course. It also helped in the analysis of the students' approaches to typical problems involved in the exams, [a), b), and c) contributed to the analysis presented in Chapter V.] d) The interview with the instructor provided information about his conceptualization of mathematics and his role in the implementation of the problem solving activities. e) The interviews with the students were transcribed word by word. This information was used to categorize the students' approaches to problems. The analysis was carried out using the theoretical framework employed by Schoenfeld (1985). A review of this framework is presented in the literature review (Chapter II). In the analysis, special emphasis was given to the types of strategies used by the students while solving the problems. This information was also contrasted with the initial categories identified in the pilot study. The students' interviews were also analyzed by considering the work of Perkins and Simmons (1988). This analysis focused on patterns of misunderstanding exhibited by the students while solving problems. The structure of the analysis considers four related dimensions: content, problem 95 solving, epistemic, and inquiry frames. All the names of the students cited in the analysis are pseudonyms. f) The researcher observed the class and recorded information regarding the type of examples discussed during the class, the students' and the instructor's interaction, and the type of difficulty exhibited by the instructor or the students during the class. This information was useful for the examination of the proposed research questions. Trustworthiness of the Study It is common to identify the strengths and weaknesses of research studies with a retrospective examination of the whole process involved in the research. For example, a statistical study relies on the randomness of the sample, external and internal validity, reliability, and the objectivity of the results. The nature of the study should define what criteria will help clarify the plausibility of the results. Lincoln and Guba (1985) indicated that the conventional criteria used to convince the audience (including oneself) of the value of qualitative studies differ from criteria used in quantitative studies. "Trustworthiness" is the term used by Lincoln and Guba to discuss the "credibility, transferability, dependability, and confirmability" of qualitative studies. These terms may be considered similar to the terms "internal validity, external validity, reliability, and objectivity" employed in statistical studies. Miles and Huberman (1984) pointed out that in a qualitative study, it is necessary to discuss the process involved in reducing the data, that is, the process involved in aggregating information, making partitions or reducing the information, analyzing, and interpreting the information. It is important to clarify the types of assumptions, criteria, decision rules, and operations 96 used for transforming the data in order to identify and support meaningful findings. The present study was redesigned as a result of the extensive pilot study carried out from September, 1988, to June, 1989. Some of the instruments for gathering information were used in the pilot study and subsequently adjusted for the main study. It is important to discuss the role of each criterion of trustworthiness within the context of the study. Credibility "Credibility" refers to the extent to which the results discussed in the study represent or reveal insights from the students. To meet this criterion, each category or claim identified in the analysis is supported and contextualized with various examples taken from the sources of data (interviews, class observations, problem-task assignments). An independent observer should agree that the findings are coming from the data and represent the students' ideas collected via different instruments. The internal consistency of the study offers grounds to believe that each claim or category is obtained by the analysis of the data from different angles. For example, the first phase of the pilot study provided some indication of the students' conceptualizations of mathematics and problem solving. The students' written work provided examples in which the students showed the use of those conceptualizations. Furthermore, the use of interviews allowed probing and gaining more insight into the nature of and possible influences on the processes of solving mathematical problems. 97 Transferability This criterion refers to the applicability of the study in other settings or contexts. The present study has already drawn some consideration from other instructors in the same college in which the pilot study was conducted. The findings discussed in the pilot study were presented in a departmental meeting in May, 1989. As a result, some instructors showed interest in carrying out similar studies or being part of the main study. In contrast, at the beginning of the study, only two instructors (out of twenty) showed interest in participating. In addition, it seemed that they wanted to see rapid improvements. It took time (two terms) to show that the research could be part of their instruction and of benefit to the students. Transferability is also enhanced by providing a comprehensive description of the study. This description will guide other researchers to carry out similar studies and make similar analyses in their contexts or areas of interest. Some instructors at the community college where the study is being carried out will apply part of this study in other areas, such as algebra and trigonometry. They intend to use the information provided in the preliminary reports of the pilot study which were presented to the Mathematics Department of the community college. Dependability The criterion of dependability refers to the nature of the data and procedures employed in the study. The methods and procedures to be used in this study were selected by taking into account ideas from different studies. In addition, the instruments were used in the pilot study and they appeared to be adequate for the type of analysis of this study. There are written records of the data. These will help any independent observer familiar with this type of 98 study to attest to the authenticity of the data and the validity of the procedures used to collect the data. Confirmability Any person in the field of mathematics education familiar with the research in problem solving should agree on the nature of the findings. If any major disagreement about a specific claim or category exists, the researcher should be able to clarify and provide sufficient grounds for that claim or category. Since the present study was conducted using actual mathematics instruction, the point of view of practitioners may help to clarify such disagreement. Although the intent of this study was to interpret and explain the students' approaches to mathematical problems that emerged from instruction based on problem solving, it is also expected that any other researcher familiar with the area using the same methods could arrive at similar conclusions. The systematic use of the instruments and the coding process used for the analysis of the information are important components that can help any researcher to trace the nature of the findings. Miles and Huberman (1984) considered three components associated with the analysis of the information that could help enhance the trustworthiness of the study, that is, data reduction, data display, and conclusion-drawing/verifying. 99 Figure 3. Components of Data Analysis: Flow Model (Miles & Huberman, 1984, p. 23) Data reduction involves explaining the activities used in the process of transforming the raw data, such as selecting part of the information, focusing on some specific strategies, and simplifying the information. For example, in the analysis of the students' approaches to problems, a set of questions was used to narrow and guide the analysis of the students' interviews. Miles and Huberman (1984) stated that "data reduction is a part of analysis that sharpens, sorts, focuses, discards, and organizes data in such a way that final conclusions can be drawn and verified" (p. 24). Data display includes the use of matrices, diagrams, or other means to focus on specific parts of the information. For example, a matrix which shows the strategies for solving problems used by the students helped organize the information for the analysis. Conclusion-drawing and verifying involve the recognition of regularities, patterns, or configurations that explain the findings of the study. 100 The process of analyzing the instructor's and students' interviews included the complete transcription of the tapes (verbatim). The transcripts . were used to identify trends of information that explain major events or categories. For example, in the interview with the instructor, it was important to identify information related to his conceptualization of mathematics and problem solving. For the students' interviews, it was important to relate information that was associated with the use of both cognitive and metacognitive strategies and the use of basic mathematical skills. At the beginning, the main ideas identified in the transcripts were coded and later discussed in accordance with certain categories. The initial analysis of the interviews was guided by a frame discussed by Schoenfeld (1983). However, the frame was later adjusted in order to incorporate information gathered from class observations into the analysis. The work of Perkins and Simmons (1988) was used to supplement the categories identified by Schoenfeld. The methodological aspects that were used to arrive at the results of the study were originally suggested by the review of related studies and later adjusted as a result of their use in the pilot study. Chapter Summary This chapter deals with methods and procedures used in the study to gather and analyze the information. It begins by presenting the rationale for the use of qualitative methods. It also discusses the results obtained in a pilot study carried out prior to the main study. These results were important in designing the main activities that were implemented during the study and also in identifying the strengths and limitations of the instruments that were used to gather the information. An outline of the main activities that were 101 undertaken in the study is also presented. The last part of this chapter deals with the trustworthiness of the study, that is, criteria that determine the credibility, transferability, dependability, and confirmability of the findings. 102 CHAPTER IV ANALYSIS OF DATA REGARDING THE IMPLEMENTATION OF INSTRUCTIONAL ACTIVITIES ASSOCIATED WITH PROBLEM SOLVING Introduction The focus of the study was on the implementation of problem solving instruction in a regular calculus class at the college level. The main purpose of this chapter is to describe and analyze events during the course that evidenced the relationship between the students' approaches to problem solving and instruction. Emphasis is given to the conditions provided during instruction that encouraged the students to learn mathematics via problem solving. The research questions of the study are used as a guide to discuss the results. Although the questions are analyzed separately, they are considered as a whole, and the discussion highlights interrelations among them. This chapter deals with the learning conditions provided by the instructor throughout the development of the study. Although there was extensive planning prior to the main study of the activities that could be included during instruction, it is important to analyze to what extent those activities were implemented. Thomson (1985) suggested that instructors' beliefs about mathematics and problem solving influence the way that they conceptualize and implement learning activities in the classroom. The need for documenting the directions of instruction was pointed out by Thomson (1988) when she stated: reports of instructional studies in problem solving have generally lacked good descriptions of what actually happened in the classroom (except 103 for those in which programmed instructional booklets were used) and have failed to assess the direct effectiveness of instruction (p. 232). Schoenfeld (1985c) stated that "learning involves the active process of constructing interpretations of what one sees, and what the student perceives may not be what the teacher intended" (p. 6). The sources of information that are used to analyze the direction of instruction include class observations, an interview with the instructor, and periodic meetings between the instructor and the researcher throughout the development of the study. Analysis of Instructional Activities Implemented in the Classroom The questions that will guide the analysis of this chapter are: i) To what extent did the instructor provide appropriate conditions for problem solving instruction throughout the development of the study? ii) What difficulties arose during instruction, and how did they influence the development of the course? These research questions are approached by first discussing the main premises that reveal the type of conceptualization of teaching that is embedded in problem solving instruction. This characterization is consequently contrasted with what the researcher observed occurring during instruction. The mathematical instruction proposed in the study emphasized the discussion of mathematical problems as a vehicle for understanding and developing mathematical knowledge. In this view, there is a conceptualization about the way in which mathematics is developed. This conceptualization is related to the way that students learn mathematics and 104 consequently to the way in which mathematics is taught. Therefore, to discuss the extent to which the instructor implemented or provided a problem solving environment in the classroom, it is necessary to discuss the epistemological bases that support the problem solving approach. Features of the Educational System, the Mathematics Curriculum, and the Interaction between the Instructor and the Researcher The instructor's education might be considered to be typical of mathematical instructors' education at the college level. He graduated with a major in mathematics, pursued graduate studies in pure mathematics, and has been teaching for more than eight years. His conceptualization of mathematics has been influenced by his experience as a student in this area. Grossman, Wilson, and Shulman (1989) recognized that the teacher's knowledge of the subject matter determines the direction of instruction. Kline (1975) examined the overall picture of the educational system in North America (particularly in mathematics) by focusing on the role of research and teaching at the university level; Kline suggested that students who complete studies in mathematics are not prepared for the teaching of this discipline. Although it may be thought that the most important role of the university is to fulfill research and teaching functions, it is interesting to find that research has become the catalyst of success and, consequently, the main activity in order to survive. "What the professor does in his research has little if any bearing on what he has to teach at the undergraduate and even beginning graduate level" ( Kline, 1975, p. 71). Because instructors who teach at the college level have to take mathematics courses throughout their formal education, it is important to discuss an overall picture that describes the type of experience that they may 105 get from their courses. Kline described several experiences that students often are exposed to in their mathematical courses. For example, at the elementary level, teachers may give "problems" in which students have to apply some operations without understanding why they have to perform certain rules; in such cases, "the students may not understand why the division of two fractions has to be performed by inverting the denominator and multiplying" (p. 7). In high school, teachers emphasize statements, such as "the sum of two integer numbers is another integer", "there is one and only one midpoint on every line segment", and "two triangles are congruent if the sides of one are equal respectively to the sides of the other". Although students may see these statements as obvious, it is not clear why teachers reiterate their validity. At the university level, students are lectured by professors who often transfer material from their notes to the blackboard. Students normally spend much time copying what is written on the blackboard. Kline went on to say that there is no interaction between the students and professors during class. Students' questions are often responded to by professors with statements, such as "it is trivial", "it is an immediate consequence of the former theorem", or "by using just contradiction". This general picture of some aspects of mathematical instruction may be related to the expectations that "successful" students who have passed through mathematical instruction may have when they become mathematics instructors at the college or university level. Kline also pointed out that professors at the university level are forced to publish several papers in order to survive in the system. As a consequence, mathematical research is the main factor that determines whether or not a professor will be promoted. Therefore, teaching becomes a secondary activity. Kaput (1990) indicated that the type of courses that mathematics 106 instructors take as a part of their studies are presented in axiomatic contexts. He stated: These courses normally leave little or no room for inquiry, and are often taught by faculty who present the material as a series of facts, definitions and theorems interrupted by text exercises, with a pedagogical attitude that implicitly regards these courses as a series of barriers to be overcome (p. 3). At the college level, instructors spend more time teaching than doing research; they normally have heavy instructional loads that include teaching three or four classes of 35 students each, marking assignments and exams, and often doing administrative activities that reduce the amount of time for the consideration of possible strategies that could improve the learning of their students. Although the instructor who was involved in this study might not have received any formal education in educational issues, he has been interested in improving his way of teaching. His interest in becoming familiar with the related literature to problem solving and its practical applications was evident in his actions. For example, before being formally approached for this research, he was interested in introducing class examples in which problems were set in different contexts as a means to relate the conceptual part of mathematics and its applications. In an interview carried out at the end of the problem solving course, he stated that: ...[it is important to make an] appropriate choice of problems to emphasize various aspects of mathematical pursuit, appropriate choice of problems to emphasize how concepts are employed, and that, prior to my work with you, was really the extent of my use of problem solving. 107 He was familiar with the work of Polya and he claimed to be using a problem solving approach in his classes; however, initial observations of his class indicated that his approach only emphasized his systematically solving various examples in front of the class without exploring what the students were doing. Thomson (1988) maintained that in order to characterize how teachers conceptualize the learning of mathematics, it is important to explore their views about the nature of mathematics and problem solving. She stated that: [l]n working with teachers, the main difficulties I have encountered have been related to the teachers' views of what constitutes a problem in mathematics, their views about the nature of mathematics in general and of problem solving in particular, their attitudes towards problem solving, and their beliefs about what it means to do mathematics (p. 234). Schoenfeld (1989) suggested that instructional activities that encourage students to participate, make conjectures, and defend their ideas are necessary in problem solving. During the class observation periods, there were indications that some of these activities were not present in his instruction. This suggested to the researcher that it was necessary to discuss issues with the instructor that could help clarify activities directly related to problem solving in order to share principles common to this approach. Initially, the researcher made comments about increasing the interaction with the students and spending more time discussing various ways to approach the problems given during the class. At the end of the second week of being observed in his class, the instructor realized that problem solving instruction involved more than the presentation of several problems to students. In the 108 periodic meetings (debriefing sessions) held with the researcher, he indicated that it was important to get information about how students were responding to his class. He also requested (during the pilot study) that the researcher illustrate the use of this approach in a one-hour class. This request might indicate that he did not conceptualize problem solving as a way of thinking that involves constant reflection on the part of the instructor as well as the students and as a daily activity rather than something that can be taught in one or two classes. Greeno (1989) described activities related to problem solving as a collaborative practice. He stated that "practice is an every day activity, carried out in a socially meaningful context in which activity depends on communication and collaboration with others and knowing how to use resources that are available in the situation" (p. 24). Although there is a risk involved in attempting to show the strengths of problem solving in a limited time, the researcher felt that it was important to respond to the instructor's request. A problem concerning the application of derivative was selected and discussed with the students throughout the entire hour. The activities during the class included exploring, discussing, and extending the problem using the students' ideas. At this point, the instructor was skeptical but still willing to explore more ideas about problem solving. The researcher judged that there was a need to discuss ideas with the instructor about problem solving in which the strategies and activities associated with the instructional approach were identified. The researcher decided to introduce the instructor to an article entitled "Confessions of an Accidental Theorist" by Alan Schoenfeld (1987). This article summarizes the evolution of Schoenfeld's ideas about problem solving, including the use of heuristics and metacognitive strategies. In this article, the strengths and limitations of Polya's work are discussed and used to introduce other 109 directions in problem solving. It also includes a rationale for considering students' beliefs about mathematics and problem solving in the analysis of problem solving. This paper was crucial in increasing the instructor's interest in problem solving, and, since reading it, he has spent time reviewing the research done by Schoenfeld and others in problem solving. At the end of the study, the instructor stated: ... because of interaction with you and the readings that you led me to and what not, I have made a point of distinguishing between performance and other activities. I have made a point of emphasizing how one can think about problems and of learning as a problem and then in an overview of the course of how one can think about the course as a problem. I have changed my expectations as I say to not focus so much on performance, but social aspects. Schoenfeld (1985c) stated that research studies dealing with teaching have neglected aspects that deal with the essence of the subject matter and its relationship with learning. He stated that there is a need to consider models for teaching in which the mathematical content should be confronted. Grossman et al. (1989) suggested that the process of teaching is influenced by the instructor's knowledge of the subject matter, the explanatory frameworks, the ways in which the subject matter is developed, and beliefs about the subject matter. Therefore, it is necessary to discuss the types of mathematics activities involved in problem solving and their relationships with the learning of mathematics. It is also important to discuss the extent to which the instructor implemented these activities in the classroom. The planning period prior to the main study included several meetings with the instructor. During these meetings, there was interest on the part of both the researcher and the instructor in developing a possible framework that 110 would characterize a problem solving approach. The results from the pilot study were used as a platform to discuss the main components that would be included in the learning activities for the main study. Several issues directly connected to problem solving were discussed and the main principles that would characterize a problem solving approach were agreed upon. The issues were discussed informally throughout the planning of the main study (June - August, 1989) and some notes were taken. Since it is important to evaluate the extent to which the instructor implemented the basic principles of problem solving in his class, it is necessary to examine what was discussed. Hence, what follows is a summary of the main foci of the discussions. Components Associated with Problem Solving that Were Addressed by the Instructor and the Researcher During the period July-August of 1989, the instructor and the researcher spent time discussing the findings of the pilot study. As a result of these discussions, they agreed that it was important to discuss practical and theoretical issues that were going to be part of the problem solving approach to be implemented from September to December of 1989. The main aspects included the following issues: The nature of Mathematics and Problem Solving There are various views about what mathematics is and how it should be learned. A main principle that emerged from the discussion was that mathematics is developed as a result of finding solutions to various problems. The researcher and instructor discussed the fact that, agricultural problems played an important role in the development of the Babylonians' and Egyptians' mathematics. The Greek problems, such as the quadrature of a 111 circle, the trisection of an angle, and the duplication of a cube, were sources of the development of branches of mathematics, such as modern algebra. Therefore, the identification and discussion of mathematical problems and the stages involved in arriving at their solutions are important in the process of learning mathematics. Hence, if the process for solving mathematical problems is a basic ingredient in developing mathematics, it makes sense to focus on mathematical instruction that emphasizes problem solving. Indeed, this was the main direction that would support the problem solving approach. During the adoption of a problem solving framework in teaching mathematics, it is necessary to underscore the basic principles that define the roles of the instructor and the students in the classroom. It is also important to discuss general conditions, such as the type of learning environment, the tasks or assignments, and the evaluation of instruction. The Role of the Instructor During the implementation of instruction based on problem solving, the instructor should be aware of the importance of considering key and strategic problems as a means of introducing and discussing the mathematical content and the problem solving strategies. A key problem can be approached by using different methods. These methods include ways to approach problems, such as algebraic, geometric, trial and error, simple case, or other approaches. A problem is cited in Davis (1986, p. 160) in which a student is given a picture of four squares, each divided into four smaller squares. The student is told that each square represents a chocolate bar and then is asked to share the chocolate bars equally among three people. This problem is an example of a key problem and could be used to explore the convergence of a geometric 112 series 1 + q + q +q +q + ... = y — with Ixl < 1. The concept of limit could also be introduced with a problem that involves the approximation of the square root of any positive integer. A magic square problem that consists of arranging the digits 1, 2,..., and 9 in a 3 x 3 matrix in such a way that all the rows, columns, and diagonals add up to the same number is an example of a strategic problem that also plays an important role in problem solving. For example, one way to solve this problem is by considering specific subgoals. The information given in the problem requires that each column must total the same number n. If there are three columns, then 3n = 1 + 2 + 3 + ... + 9 = 45. Therefore, n = 15. This gives direction for another subgoal; that is, if 15 is the sum, then what number should be located in the center? Why? and so on (Schoenfeld, 1988). There are various sources that provide key and strategic problems and instructors should be encouraged to examine such problems and adjust them to suit their classroom environments. Introducing key and strategic problems is an aspect of problem solving that should be accompanied by learning activities that engage students in what Schoenfeld (1989) called a microcosm of mathematical practice, that is, activities that motivate students to speculate, discuss, and defend their ideas and also that include examples and counterexamples that help the students to clarify and to understand the mathematical content involved in the process of problem solving. Polya (1945) suggested that the process of solving a problem should involve consideration of ways in which the problem can be extended to other domains. For example, the Pythagorean Theorem in two dimensions can naturally be extended to three, four, or more dimensions. Extending the problem also includes adding more variables, reducing or relaxing the original conditions, 113 or slightly modifying the statement of the problem. The instructor in the current study recognized the importance of incorporating mathematical problems that could show underlying mathematical principles during class instruction, but he also pointed out that these types of problems take much instructional time and that the proposed curriculum might not be covered if this instructional approach were pursued throughout the course. The instructor in an interview carried out at the end of the study stated : There is a lot of curriculum. There's a lot of stuff to cover and, yet in many cases, there is as much to cover because the students' experience is not as wide as would be optimal. And I don't know how to overcome that. Algebraic skills have to be addressed. They're not there, but many other skills aren't there. Arithmetic skills are few and far between The relationship between spending time in class discussions and covering the content is an issue that is related to what conceptualization of learning is being endorsed. DiSessa (1988) suggested that learning should not be viewed as mastering a properly organized collection of facts, but as conceptual change. He stated that"... . [t]here is plenty of opportunity to dig in somewhere, while leaving out some of the currently exorbitant pile of 'facts'. It may well be, and, indeed, I expect that more depth in learning may end up increasing coverage" (p. 49). He went on to relate ways that students could increase their understanding and cover more mathematical content independently. For example, students who have developed some true competence in an area may feel intrinsically motivated and eager to acquire more knowledge. Teachers may also find that discussing even one topic deeply during class instruction may guide students to learn other topics. As a 114 consequence, students may be more attentive and competent to learn outside of standard 'forced-learning' situations like school. He stated that "[i]n the light of these considerations, a refined slogan might be 'breadth will follow from depth' " (p. 49). Although the instructor was aware of the importance of giving more emphasis to the discussion of key problems than giving rules to the students, it was observed that he was concerned about whether or not the curriculum was going to be covered. As a consequence, the discussion of key problems often ended with an explanation by the instructor to ensure that the content for that day had been covered. Classroom Activities There are learning activities that have been shown to be helpful in engaging students in the process of problem solving. Schoenfeld (1985) recognized that the use of small groups is important in providing the conditions for motivating students to participate in the discussion of problems. The instructor agreed to incorporate the use of small groups during problem solving activities. He would monitor the students' interaction while working on the problems. Asking questions about the students' progress and making sure that all the students in each small group were participating were two of the strategies used to monitor the students' work. The use of small groups during the discussion of key problems in the classroom was an important ingredient of the problem solving instruction. It was observed that this activity encouraged the students to discuss their ideas regarding possible solutions of the problem. Research in problem solving has also suggested that in addition to discussing the cognitive strategies involved in the process of solving problems, students should pay attention to the metacognitive processes that they employ 115 while solving problems. Metacognitive strategies help students realize the need for shifting from the use of one strategy to another. For example, questions, such as "Why are you doing that?" or "Why are you using a specific strategy?", and "Where do you want to go by doing it?", have been found to be useful for the students while working on problems (Garofalo & Lester, 1985; Schoenfeld, 1987; Schoenfeld, 1988; Silver, 1987). Therefore, the instructor should reflect constantly on the extent to which the activities implemented in the class are suitable for encouraging the students to engage in the problem solving process. There should be a constant readjustment of what is planned and what is being implemented in the classroom. The participation of the students should provide information about their interests. This information should be used in the design and presentation of the problems. For example, the concept of function could be introduced by designing problems that involve information from various sources, such as the post office, supermarket, or field of sports. Designing the tasks around this information could increase and satisfy the curiosity and interest of the students. The discussion of the type of learning activities associated with problem solving provided the rationale for the consideration of small group discussions and the use of cognitive and metacognitive strategies in the instruction. The Use of Counterexamples in Problem Solving Instruction Problem solving is considered a way of thinking that provides conditions for students to learn, develop, and apply mathematical content. Schoenfeld (1989) suggested that problem solving enhances students' ability to think mathematically. This includes the students' ability to make 116 conjectures, to argue about mathematics using mathematical arguments, to formulate and solve problems, and to make sense of mathematical ideas. There are various strategies that are commonly used not only in solving mathematical problems but also in solving problems in other disciplines. These strategies include looking for patterns, breaking the problem into parts, and drawing a diagram. Perkins and Salomon (1989) indicated that the use of counterexamples is an important component that helps clarify ideas. "Philosophers, for example, seem to have a general cognitive skill: the strategy of looking for counterexamples to test claims" (p. 19). However, Perkins and Salomon suggested that the ability to find counterexamples may not be a general strategy but related to a specific content area. They stated that "a counterexample to a mathematical claim would have to be constructed appropriately from the premises of the mathematical system.... Different domains share many structures of argument, but bring with them somewhat different criteria for evidence" (p. 19). In mathematics, the use of counterexamples plays an important role in the development and understanding of mathematical content (Kline, 1980; Lakatos, 1976). Mathematical instruction should then incorporate the use of counterexamples as a means to understanding and applying mathematics. Perkins and Salomon (1989), in reference to the work of Polya, stated that many of the heuristics discussed by Polya are applicable to problems of all sorts. This may suggest that "problem solving could be viewed as a general ability and mathematical problem solving simply a special case" (p. 17). The use of particular tasks or assignments could help provide the conditions for the students to reflect on the application of several strategies in different contexts. Tasks should include routine and nonroutine problems, projects, and ways to monitor the students' progress when they were working on these 117 tasks. Silver and Smith (1990) recommended that it is important to discuss problems with multiple methods of solution during the class in order to engage the students in activities related to problem solving. They stated that "opportunities for students to create and discuss multiple solution methods for interesting mathematical problems constitute invitations to engage in high-level thinking in mathematics class, and they represent important opportunities for teachers to learn about and enhance their students' mathematical thinking and reasoning" (p. 36). During the interview carried out at the end of the study, the instructor expressed awareness of the importance of the use of examples and counterexamples during the class discussions. He also indicated that it was necessary to encourage the students to explore different methods of solutions and to consider other related cases as a part of the problem solving processes. The Students' Role Required in Problem Solving Instruction Provision of appropriate learning conditions in the classroom requires dynamic interaction between the students and the instructor. There are basic rules that could help implement activities related to problem solving. For example, students should be willing to learn mathematics and to participate actively in the discussion of problems while relying on mathematical arguments. The classroom atmosphere should be open and should encourage the students to initiate questions and suggest approaches to problems. They should make conjectures overtly and present solutions to the problems discussed in the classroom and should participate in all the activities during and after the class. The Professional Standards for Teaching Mathematics (National Council of Teachers of Mathematics, 1989) emphasized that 118 problem solving environments should also allow time for the students to puzzle, to be stuck, to try alternative approaches, and to dialogue with others and the instructor. Davis (1989) stated that the "basic job" of being a mathematics student includes: i) making sure that one understands the meaning of what is said or written; ii) checking whatever is said against one's own experience or one's own data; iii) asking questions and looking for examples and counterexamples; iv) searching for patterns, making connections with other problems or cases, and extending the results; v) using heuristic methods and monitoring strategies; and vi) learning about personal strengths and weaknesses in the way that one deals with mathematical situations. The instructor and the researcher spent time at the beginning of the course explaining to the students the purpose of the course and the main activities that were going to be implemented. A discussion of the roles of the students, instructor, and the researcher was also part of the introduction to the course. Dynamics of the Classroom: The Conceptualization of Mathematics and Learning Activities The historical development of mathematics suggests that it is important to emphasize the process of solving problems as a means to understanding mathematics. In order to characterize the use of problem solving in this study, a discussion of the type of conceptualization of mathematics endorsed by the instructor, the activities implemented during the classroom, the assignments used to monitor the students' progress, and the evaluation of the students' work undertaken in the course is presented next. The analysis is based on data gathered throughout the complete four month term (September-119 December). The sources of data were class observations, instructor and student interviews, periodic meetings with the instructor (twice a month), and student assignments. The Instructor's Conceptualization of Mathematics. Schoenfeld (1987b) pointed out that the way teachers conceptualize mathematics permeates the classroom activities that are implemented during instruction. He described his own experience as a student in which the instructor could not remember the binomial formula and showed the students how to figure it out. Learning activities that relate the sense of studying mathematical relationships are different from those in which the instructor gives only rules for solving problems. Schoenfeld (1987b) stated: the important thing in mathematics is seeing the connections, seeing what makes things tick and how they fit together. Doing mathematics is putting together the connections, making sense of the structure. Writing down the results - the formal statements that codify your understanding - is the end product, rather than the starting place (p. 28). Thomson (1988) pointed out that teaching is a human activity that involves experience, taste, and judgement. She stated that "in my view teaching is an activity that cannot be prescribed; it cannot be reduced to a sequence of predetermined steps to be learned as one learns, say, an algorithm" (p. 234). Therefore, there is room for the instructor to make instructional decisions that he or she considers suitable at a particular moment during the class. Hence, exploring the way the instructor thinks of mathematics and problem solving and analyzing the activities that he 120 implemented during the study could help to document the type of instruction that he provided during the study. There are indications suggesting that the instructor endorsed a view of mathematics that emphasizes the conceptual part of mathematics. For example, to the question "which aspects of mathematics would you mention in responding to the question 'what is mathematics?' " (in an interview carried out at the end of the course), he responded: ....I [would] probably start by turning the question around and say what is not mathematics, in an effort to try to immediately broaden the questioner's perspective, and in response to that, I would say it is not a matter of going through a few predetermined steps and ending up with an answer. I am not sure that... a clear description of what is mathematics is that easy, certainly not to those who would be asking it. However, I would include the uncertain nature of mathematical fact, call it. I would certainly include mention of how mathematical fact evolves, how it is changed ... with such examples as non-Euclidean geometry because it's relatively accessible, or perhaps more accessible, might be baseball mathematics, where our usual addition of fractions is thrown out the window with very good purpose. I would certainly include mention of attempts to understand or explain and try to distinguish that from absolute truth and discourage suggestion of absolute truth. One would have to include some mention of skills. There is no doubt about it, that one cannot do mathematics without a certain collection of skills and for the most part in undergraduate instruction that's the extent of [the] focus. However, I would want to go well beyond that and include notions of generalizations of patterns. I would like to emphasize the difference between various levels of mathematical activity: skill level, conceptualization level, validity level. In his response, the instructor differentiated the mechanical approach to mathematics instruction, that is, the identification of a determined 121 sequence of steps (rules) in order to understand content from the approach in which there is room for discussion, speculation, and criticism. His view of the nature of mathematics suggested that mathematics is a subject growing constantly and that there is not absolute truth. This view includes some points discussed by Kline (1980); however, his identification of various levels of mathematical activities (skill, conceptualization, and validity) may suggest that the instructor conceptualized three different approaches to teaching mathematics depending on the students' level of proficiency with the content being taught. The instructor showed awareness of the importance of going beyond the mechanical mode of instruction. The instructor is firm in his position on the teaching of mathematics at the undergraduate level. He considers the teaching of basic skills as important; he also believes that formal mathematics should not be the focus of undergraduate teaching. At the end of the study, he stated: I'm beginning to question more and more the utility in formal pursuits. It's only... formal mathematics is only useful if you have some reference points to evaluate its utility (my emphasis), and so only if you have some sort of understanding of perhaps some of the aberrations of, call it, intuitive mathematics; let's face it that it's really weird formulas just come in, and they formulate their theories after the fact. So no, I don't think that there is that much utility for it especially in first year calculus.... However, I do think that leaves lots and lots of room for many other valid mathematical activities and in particular conceptualization and applications of concepts in different settings; they're much much more universal than applications of algorithms. 122 The Development of Class Interactions Between the Instructor and the Students It is important to relate the activities that the instructor implemented during the problem solving instruction to his views of mathematics. For example, he relied on several examples to introduce each day's content; the students normally spent some time reflecting on the examples, but the instructor was always ready to answer any question from the students without exploring the students' difficulties. The instructor at one point formalized the definitions or theorems discussed during the class; often he demonstrated some of the theorems. This type of instruction occurred more often when the concept of derivative was used to present some of the formulae for obtaining derivatives. For example, all the rules for operating with derivatives (addition, multiplication, and division) were demonstrated by the instructor in one class. One student who probably did not follow the demonstration asked whether these types of proofs were going to be on the final exam; the instructor who might have known the purpose of this question responded, "No, but you have to know them".... Other students asked during the same class why the derivative of the product was not the product of the derivatives. The instructor responded with a formal demonstration of the expression that characterizes such a product. The researcher observed in this session that some students were experiencing some difficulties in understanding those demonstrations and that the instructor did not follow up some of the students' concerns. This type of intervention by the instructor sometimes happened in the course. In addition, when the students asked some questions regarding the development of the proof, the instructor only repeated the proof without exploring the students' difficulties. It seemed that even though the instructor was aware of the students' difficulties, he did not address these issues directly 123 during the instruction; perhaps, this was because of the limited time for covering the content. There are intricate steps in proving some of the operations with derivatives in which the students may experience some difficulties . For example, the necessity of adding "zero" when working on the product of derivatives may surprise some students. Some instructors believe that by presenting the proofs (in polished form) to the students in a transmissive way, the students will begin to appreciate that the use of rules is not arbitrary and then later on appreciate the nature of the proof. However, it is the researcher's opinion that more time should be spent discussing some of the students' difficulties during the class as well as more examples and counterexamples that show the use of the operations. The idea that students may be able to understand content through repetition of the same explanation has been challenged by many researchers. Blais (1988) indicated: Giving a maximum of explanation implies the creation of a listener-follower role for students. Such a role contributes to dependence, eliminates the need to think for oneself, and fosters the growth of learned helplessness. The habit of dependence and a belief in one's own helplessness ensures that the novice will remain just that (p. 628). Hence, if the students do not have a chance to reflect on why they experience difficulties when studying mathematics, the instructor's explanation may not be helpful to the students. Lester (1983) stated that "because teachers typically demonstrate only correct moves, students often come to view problem solving as the act of delving into a mysterious bag of tricks to which only a select few are privy" (p. 229). Blais (1988) in the same 124 vein wrote: conventional instruction fails, not because it is done poorly but because it includes the remedial processing option. Conventional instruction permits, allows, and sometimes blatantly encourages algorithmic activity that is separate and isolated from the perception of essence (p. 627). Although the instructor's view of mathematics included the idea that mathematics is learned through criticism and speculation, it was clear that the model of mathematics that he portrayed in his teaching presented mathematics as a well organized subject. For example, the instructor often introduced and presented the use of algorithms and rules in a sequential manner. For example, during the class several examples that involved the chain rule for determining the derivative of function were discussed. The strategy in attacking this problem was to apply the chain rule and the formulae for the derivative involved in the expressions. The students in the assignments in which they had to use the chain rule calculated derivatives without analyzing whether or not the functions were differentiable at the corresponding points. Thus, the message given to the students was that mathematics always works and that it is important to use the right approach to solve problems. The importance of engaging students in practicing basic skills is shown in an analogy which the instructor made between learning mathematics and playing golf. At the end of the course, he stated: .... I might give an analogy here. If you are learning to play golf, you first of all learn how to swing a golf club and have it hit the ball, and only when you get the ball flying you spend a lot of time on the practice 125 'till you're getting the ball flying and yet that's not the extent of what you do when you play golf, but that's an important part when you play golf. I agree that the emphasis we show on examinations is somewhat unfortunate and yet it's practice. Blais (1988) indicated that often students intend to follow their instructor's methods for approaching mathematics problems, and as a consequence, they normally expect to be told what to do; this is because the instructor has always provided them with the right answers or explanations. The instructor's ideas about the use of problem solving were linked to the way that he conceptualized mathematics. He maintains that doing mathematics is a social activity and that this aspect should be integrated into the problem solving instruction. He thinks that this aspect can be promoted by asking the students to work on the assignments together, asking the students to work in small groups during the class, and by discussing examples that show the application of mathematics in various contexts. However, in his actual class, there were few examples in which the students had the opportunity to defend their means of understanding or solving problems. Good, Grouws, and Mason (1990) suggested that many teachers generally believe that the use of whole-class teaching may be efficient for addressing the needs of students who vary in ability, and as a consequence, this approach remains the primary method for teaching. They stated that the use of small groups is often limited by the small amount of time and limited peer contact. Hence, Good, Grouws, and Mason indicated that: ... general advocacy of small groups may only result in more small-group practice of routine tasks rather than more time spent on exploration, interaction, or critical thinking. It seems important that process features that focus on the qualitative aspects of small-group instruction 126 receive more research attention (p. 14). Even when the instructor recognized the importance of discussing nonroutine problems in the classroom, he also recognized that the actual conditions of the college limit the use of problem solving. For example, he mentioned that the extent of the curriculum, the size of the class, and the testing practices are major concerns that impede trying new instructional approaches. He recognized that to discuss nonroutine problems on a daily basis during instruction takes time and that there is a risk of not covering the proposed curriculum. In addition, the final exams (designed by the department) normally include only routine exercises for which the students have to be prepared. Regarding the number of students, the instructor at the end of the course stated: Having class sizes of 38, 39 students ... it's difficult to do anything but present the traditional lecture format. And I think it's quite clear that traditional lecture formats are not very efficient learning tools, especially for today's student who doesn't engage in that way any more. They have to engage in other ways. Evaluation of the Students' Work Although the instructor recognized the utility of discussing nonroutine problems, he also suggested that the students' examinations should not include these types of problems. During the interview at the end of the course, he stated: I think that there is a good deal of... discussion of nonroutine problems in that it can emphasize how concepts can be applied, it can promote social interaction, it can promote the thinking aspect, the critical 127 analysis aspect of problem solving, and these are all things that are useful in order to learn. I agree that these things are not particularly easily tested and so this is not what comes up in examinations; however, I think that by pursuing them one can improve performance on examinations by knowing the stuff better by applying those things to it. He went on to say that he provided some coaching to the students on how to write exams in which they have to work quickly to solve 12 routine exercises in about two hours. Silver and Kilpatrick (1988) stated that: .... as long as test construction remains dominated by traditional views that put a premium on efficiency of measurement, including single scores for unidimensional measures having high internal consistency, problem solving will not be adequately assessed by tests (p. 182). Perkins (1987) argued that mathematical instruction should include problems in which the students have the opportunity to apply the content in different contexts. He stated that "education ideally aims not at the final exam but at the applications the students may make of what they have learned in further study and outside of school" (p. 50). For example, the problem solving strategies learned in mathematics should also be used in other domains, such as physics and chemistry classes, as well as beyond the school context. The students knew that even when they were asked to work on some nonroutine problems in the assignments and during class instructions, these problems could not be part of the final examinations. All the students were concerned about the final exam, and they constantly asked for the correct and most efficient procedures in order to do well in that exam. They often ignored 128 exploring the problem in more general domains or looking for other approaches. They knew that these types of activities are never included in the final examination. Although the instructor agreed to consider nonroutine problems in the assignments and class discussions, he rarely checked the students' progress in solving the assignments. The researcher, who was in charge of marking the assignments and giving written feedback to the students, periodically reported to the instructor and the students the students' strengths and difficulties. This report was always judged by the instructor to be satisfactory. The Use of Metacognitive Strategies Research studies in problem solving have indicated that it is important to consider metacognitive strategies as a part of the process of solving problems. Such types of strategies include being aware of one's repertoire of tactics or strategies and when, how, and why one should use them while dealing with mathematical problems (Garofalo & Lester, 1985; Schoenfeld, 1987). Garofalo (1987) stated: ... since metacognition has to do with awareness, its development requires one to observe what one does and to reflect on what one observes. Thus students must become watchers, analyzers, assessors, and evaluators of their own mathematical knowledge and behaviors (p. 22, emphasis in the original). Although the instructor acknowledged the importance of encouraging the students to monitor their problem solving processes, it was observed that in class discussions he emphasized the students' knowledge of concepts and procedures. The only explicit guide for asking the students to reflect on their 129 monitoring process was a set of questions provided by the researcher that accompanied the students' assignments. Garofalo (1987) stated that "if we want our students to become active learners and doers of mathematics rather than mere knowers of mathematical facts and procedures, we must design our instruction to develop their metacognition" (p. 22). Garofalo suggested that the teacher should ask the students questions that help them reflect on the type of error they make, the things they may do when difficulties arise, and ways to keep track of what they do. Although the instructor talked about the importance of thinking about mathematics and the importance of understanding the content, he also emphasized that it is important for the students to follow a particular sequence of steps while solving the problems. As a consequence, the students often were more concerned about whether or not they were following the expected progression of steps for solving the problems than about evaluating their progress with other approaches or means. Schoenfeld (1985c) stated that students often do not accept demonstrations of mathematical statements because they are not written in the format that they believe is the right one. He stated that "the result of such an emphasis on form is that students come to [believe] that it is the form of expression, as much as the substance of the mathematics, that is important" (p. 17). Schoenfeld (1985) recognized that it is important to show the students actual activities that experts carry on when dealing with mathematical problems. The instructor commented on an experience in which he intended to incorporate mathematical problems that were new to him in some of the class discussions. Thinking of the possible ways in which the problem could be solved and exploring some possibilities are activities that consume considerable time. Although the instructor outlined the structure of the 130 problem, discussed the potential of some strategies,and made significant progress towards the solution, there was not enough time to find the solution. Some of the students commented that the instructor had difficulties because he had not prepared for his class. Such comments made by the students who expected to be told immediately the right solutions decreased the instructor's interest in pursuing this type of activity, that is, trying to solve "new" problems for him during the class instruction. Henkin (1981) stated: .... one of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed....That's so unlike the true nature of mathematics (p. 89). The students* instructional expectations influence the way that they evaluate their instructor's practice. Hence, the students' opinions are important in deciding whether or not an instructor should be promoted. As a consequence, discussing new problems during the class may not be a common activity. The consideration of systems of evaluation that include the progress made by the students while working on nonroutine problems might be an important component to motivate students to work on these problems. There are several practical activities in which students should engage in order to be able to transfer the use of problem solving strategies into other contexts. Perkins (1987) suggested that the use of metacognitive strategies could help the students to use the strategies in other contexts; he also suggested that "it demands the conscious effort of the learner in seeking generalizations and applications beyond the obvious" (p. 51). In the same vein, Schoenfeld (1987) indicated that it is important for the students to select and pursue the right approaches, to recover from inappropriate choices, and, 131 in general, to monitor and oversee the entire problem-solving process. When activities which highlight and make these processes explicit become consistently part of the every day class instruction, then there will be an indication that the use of problem solving has become an important way to learn mathematics. Chapter Summary This chapter deals with the analysis of the main activities that were undertaken during the planning and implementation periods of the study. It starts with an overall picture of the educational system in mathematics, that is, the type of mathematical instruction that a student might receive in his or her formal education to become an instructor at the college level. This part was judged important by the researcher in order to analyze the instructor's actions during the course. The analysis in this session shows that the formal preparation received from mathematical courses does not address issues that provide theoretical and practical orientations to make instructional decisions. The next part deals with issues that were discussed between the researcher and the instructor. The pilot study indicated that it was important for both the researcher and the instructor to share basic principles that are related to the use of problem solving instruction. It was important to observe that the discussions of aspects related to the nature of mathematics helped both the instructor and the researcher clarify the links between problem solving and the development of mathematics. The discussion of the role of the participants in the study and the learning environment conditions helped to organize the instructional activities that were going to be implemented in the development of the study. Finally, this chapter deals with the extent to which several activities associated with problem solving were implemented in the actual 132 practice. The analysis shows that the instructor intended to incorporate several activities associated with problem solving into his class. The consistency in the implementation of this approach was often influenced by the time spent in class discussion, by the expectations of the students, by the extension of the curriculum, and by the evaluation of the students' work. 133 CHAPTER V ANALYSIS OF DATA REGARDING THE STUDENTS' DISPOSITION TO PROBLEM SOLVING AND THEIR APPROACHES TO WRITTEN PROBLEMS Introduction This chapter deals with the extent to which the students showed mathematical disposition toward mathematics and problem solving. It discusses the students' confidence in using mathematical ideas and problem solving strategies and their valuing of the study of mathematics. It focuses on the analysis of information gathered through the same questionnaire given at the beginning and at the end of the course, five assignments, two written tests, one of which was the final exam, and class observations. The following diagram indicates the sources of data used during the analysis. The shapes of the components do not have any specific meaning; they are used only as an aid to guide the reader. Data Sources Figure 4. Sources of Data and Considerations in Their Analysis 134 Results from the Questionnaire A questionnaire which included 45 items (see appendix M2) regarding the use of strategies for solving problems and views about mathematics and problem solving was given to the students at the beginning and at the end of the course. The purpose of the questionnaire was to elicit some students' ideas about mathematics and problem solving. In addition, the instructor indicated that the use of the questionnaire could be helpful for introducing basic terminology employed in problem solving and for encouraging the students to reflect on the use of some problem solving strategies. The questionnaire used in the pilot study proved to be useful for introducing basic ideas involved in the process of problem solving; however, some adjustments suggested by the instructor of the course were made. He also indicated the importance of giving the questionnaire in order to introduce the types of activities to be considered throughout the course. For example, some items, such as, "I restate mathematical problems using my own words", "I check the solution of each problem", and "I check whether or not I have used all the information given in the problem", were part of the regular instruction. An item-by-item pre- versus post-questionnaire analysis was carried out to identify trends in the students' responses. The students' responses were grouped in accordance with the use of strategies and ideas about mathematics and problem solving. The analysis focused on the comparison of the students' responses at the beginning and at the end of the course. Two issues were explored in the analysis of the students' responses: the use of problem solving strategies during the process of solving problems and the types of views of mathematics and problem solving. The information 135 regarding the students' use of problem solving strategies was arranged into four subgroups: i) understanding the problem, ii) design and exploration, iii) implementation, and iv) verification or looking back. The students' responses to the questionnaire were grouped into two nominal categories. One category included responses 0 and 1 and was interpreted as an indicator below the often option, while the second category included responses 2, 3, and 4 and was used as an indicator at or above the often level. The percentages given in the analysis to explain the students' responses represent the percentage of students that chose the category at or above the often level. For example, if 95% of the students responded that they use diagrams when solving problems, then that means that 24 out of 26 (total sample) students selected an option at or above the often level when responding to this item. There was indication that since taking the problem solving course the students have begun to question their views about mathematics and problem solving. For example, 79% of the students expressed interest in discussing mathematics problems with their classmates at the end of the course. Classroom interaction was an important activity throughout the course, and this might explain the increase in the students' response to this item (number 21) by about 21% compared with their responses given at the beginning of the course (58%). They also showed more interest in solving nonroutine problems (items 35 and 45) in which they had to use several concepts and spend time searching for the solution. This response changed from 79% at the beginning of the course to 95% at the end of the course. Although the results seem to indicate that the students are more aware of the importance of the conceptual part of mathematics and its applications, there is no assurance that the students have shifted to a new conceptualization of mathematics 136 completely. The students' responses in the questionnaire given at the beginning of the course showed that they characterized mathematics as a subject in which rules can be applied to solve any type of problem. In general, the students considered mathematics to be an objective discipline in which the content can be mastered. The students' responses given at the beginning of the course indicated that the students believed that the process of solving mathematical problems depends on the use of rules. Throughout the development of the course, there was special interest in showing the students that the majority of the problems require some transformation in order to be solved. The knowledge of mathematical algorithms, some rules such as the derivative formulae, and some heuristic strategies play an important role in solving problems. In addition, it was shown through the assignments, class examples, and the dynamics of the class that mathematics is an open discipline in which there is room for discussion, and speculation, and production of new results or problems, and that there may not necessarily be a direct way to solve a mathematical problem. These classroom activities could have influenced the students' responses at the end of the course when they expressed that the most important part of mathematics is to understand the concepts and apply them in different contexts. A more refined analysis of this issue is given in the next chapter. Regarding the use of a problem solving approach, the students became more confident about the existence of problem solving strategies and their use as a means of learning mathematics. For example, 63.7% of the students (item 32) responded that it is important to design a well structured plan before starting to work on the solution of a problem. The response at the beginning of the course was 44%. At the end of the course, 73.7% of the students felt that their present problem solving strategies were adequate(second category). 137 The initial response to this same item (number 13) was 63%. The students (95%) at the end of the course also responded that they were willing to persist in trying to solve the problem even when the problem seems to be difficult. The initial response to this item was 79%. This item seems to challenge one of the beliefs identified by Schoenfeld (1985) that the students normally give up if they cannot solve the problem in a few minutes. There was also agreement about the importance of reflecting on any type of difficulty experienced during the process of solving the problem. For example, 53% of the students in the post questionnaire responded that they always reflect on why a problem was difficult for them and discuss with the instructor or other students the difficulties found. The percentage of students' responses to this item (number 40) on the pre-questionnaire was 42%. The students showed differences in the use of strategies that lead to the understanding of the problem. For example, 42% of the students responded at the end of the course that they very often or always restate the statement of the problem using their own words; this was an increase of 21% compared to the students' responses at the beginning of the course (item number 1). Seventy-three percent of the students stated after the problem solving course that they very often or always try to find similarities with some familiar problems, while 63% of the students responded so at the beginning of the course (item number 8). Forty-seven percent of the students at the end of the course responded that they always read the statement of the problem several times before attempting to solve it, while only 35% of the students responded initially (item number 12). Regarding the design and exploration aspects, it was found that the students responded that they consider various heuristic strategies while designing a plan for approaching the problem. For example, at the end of the 138 course, 95% of the students responded that they use diagrams to represent the problem, while at the beginning of the course only 58% of the students acknowledged using diagrams (item number 5). Every student at the end of the course claimed to simplify the problem by finding simpler ways (breaking the problem down) and exploring ways which seem to lead to the solution quickly. The initial response to this item (number 6) was 73%. The students' responses show that they became more aware of identifying relevant information in the statement of the problem before attempting to solve it. The shift was from 75% of the students at the beginning of the course to 85% at the end of the course (item number 16). They also reported that they estimate more often the possible solution of the problem (item number 24), summarize the given data in a table or graph (item number 23), and frequently use patterns to approach the problem (item number 25). Polya (1945) pointed out that students should be encouraged to check whether the solution of the problem makes sense in the context of the statement and also to extend the problem into other domains. The students' responses regarding these issues showed that at the end of the course 42% of the students claimed to check the meaning of the solution of the problem, compared with an initial response of 32% (item 33). Sixty-eight percent of the students said that they check whether or not they have used all the given information in obtaining the solution. The students' response at the beginning of the course was 47% (item 27). Although during the class the instructor emphasized the importance of extending the problem into other domains or considering more generalized cases, few students (20%) recognized going beyond solving the problem. For this item only 5% of the students at the beginning of the course responded that they look back after having solved the problem. Schoenfeld (1988) pointed out that extending the 139 problem should be part of the process in solving the problem. However, students' instructional expectations which include grading policies and teaching practice in general in the school settings are often obstacles for considering and implementing other alternatives during instruction. For example, the instructor was concerned about the limited time for covering the entire curriculum and the time that the students normally spend taking the final exam. As a consequence, the extent to which nonroutine problems could be used in the classroom and in exams was limited by these concerns. Results from the questionnaire suggest that the students have begun to recognize that mathematics is not only a set of rules that can be applied to solve routine problems, but a discipline in which they can speculate, discuss and sometimes generate mathematics results. There is also an indication that the students recognize the importance of using problem solving strategies (cognitive and metacognitive) in the process of solving problems. Although the questionnaire results do not constitute proof of a major shift in the students' conceptualization of mathematics and problem solving, it may be suggested that the problem solving instruction influenced the students' views. The results could be interpreted as the initial recognition of a mathematical view in which problem solving plays a central role. The following chapters provide more information about the changes of the students' and instructor's views throughout the course. What follows is a summary of some selected items that are directly related to the use of problem solving strategies. They provided information about the responses that the students gave at the beginning and at the end of the course. 140 Table 2. Selected Items of the Students' Responses to the Questionnaire Students' Responses in percentages Problem Solving Activities At the beginning At the end Discussing problems with other classmates 58% 79% Solving nonroutine problems 79% 95% Following a well-structured plan 44% 63.7% Having adequate problem solving strategies 63% 73.7% Willing to persist with difficult problems 79% 95% Reflecting on difficulties 42% 53% Restating the problem using their own words 21% 42% Finding similarities with familiar problems 63% 73% Reading the problem several times 35% 47% Using diagrams 58% 95% Breaking the problem into parts 73% 100% Checking the meaning of solutions 32% 42% Checking whether or not the information was used 47% 68% Extending the problem into other domains 5% 20% 141 Results from the Assignments The underlying idea when designing the assignments was to monitor the extent to which the students were using the strategies discussed during instruction. It was important to evaluate the way that the students approached the problems independently, that is, without coaching from or interaction with the instructor. The assignments were given to the students every two weeks. The First Assignment: Polarity Chart The first assignment showed that the students experienced some difficulties when working on some of the problems that were set differently from the ones discussed during class, and it provided some feedback for the direction of instruction. The content involved in the first assignment involved a new concept, "the polarity chart", which is a geometric approach to determining the sign of some algebraic expressions. The students experienced difficulties in factoring various expressions, analyzing the domains for which square roots were defined, operating with radicals, and organizing the information on the chart in order to solve the problem. Although several examples were illustrated during the class and the students seemed to have grasped the use of this concept, it was evident that the students were not able to identify the strength of the use of this concept in different situations. For example, the first part of the assignment contained some problems in which the students were explicitly asked to find the polarity chart. The students' work showed clear understanding of the concept studied in class; however, they often failed to transfer the use of the concept into different contexts. For example, for 142 problems in which they were explicitly asked to find the polarity, the students worked systematically to solve them. However, for problems in which the term "polarity" was not explicitly stated but was necessary for the solution, the students experienced difficulties. In one problem, the students were asked to find the algebraic and graphic representations of a polynomial of degree 4 with roots 0, 2, and 3. The students were able to find the algebraic expression, but failed to recognize that the polarity chart could help them represent the problem graphically. During the class the instructor emphasized the construction of polarity charts given specific algebraic expressions. One of the problems on the assignment asked the students to determine the expression from a given polarity chart. The students identified this problem as the most difficult one, maintaining that it was different from those discussed during instruction. After a discussion of the results of the assignments with the instructor, he agreed to emphasize metacognitive strategies during instruction. For example, during small group discussions, the students were asked to verbalize what they were doing and why they were doing it, as well as to speculate on the possible results. It was decided that the students should be guided in how to consider problems in different contexts or forms, rather than left alone to make extensions from the problems. The researcher and the instructor met every two weeks to discuss the directions of the course. During one of the meetings, it was deemed important for the instructor to integrate the use of both cognitive and metacognitive strategies into the instruction. In addition, the process of evaluating the problems in the assignments would include recognition of the use of these strategies. The instructor encouraged the students to check the solution of the problems from different angles. For example, the need to estimate the type of 143 solution before solving the problems, the need to check the use of the data and for possible calculation errors, and the need to check whether or not the solution makes sense were illustrated. However, the students' work showed that they were not checking their work at all. For example, several students initially recognized the domain for which a radical involved in the problem was defined; however, these students did not utilize that information when writing down the final solution. That is, they included numbers for which the expression was undefined. Second Assignment: The Concept of Function The students received feedback from the instructor about the major difficulties identified in the first assignment. The instructor worked on some examples in which it was important to review each of the problems and clarify the basic aspects that were evaluated in them. The idea was to emphasize the importance of working through the problems of the assignment in a systematic way. The students were also encouraged to work on the problems in small groups. The second assignment included the concept of function and its basic operations. Some students experienced difficulty with the notation of functions. For example, defining the function in terms of s, (f(s)) and having to find f(2 + x) were difficult for some students. Their work showed the use of the two variables in representing the problem. This suggested that the students conceptualize the content in a specific context, including symbolic representation, the use of operations, and the type of problem associated with that content. If any one of these parameters is changed or represented differently, the students fail to recognize the important components of the problem. 144 One of the problems involved asking the students to sketch the graphs of some functions. The most popular approach was to assign some numbers and calculate the corresponding values of the function. Although during the class the students discussed several examples in which the material recently studied (polarity chart, graph of lines, parabolas, and circles) was used to graph several examples, they rarely used that information to sketch the graphs. Many students seemed to have used the calculator to obtain the graph without reflecting on the basic properties of those functions. The researcher discussed the students' work on these problems with the instructor whose initial position was not to let the students use their calculators. Indeed the use of calculators in calculus exams has polarized instructors' opinions even at the university level. In the present study, it was agreed that the context of the problems needed to be changed rather than imposing restrictions on the use of calculators. For example, ideas about emphasizing interpretation of graphs and about involving the students in discussions of patterns associated with the functions were aspects that emerged from the discussion and became part of the instruction. The concept of function involved discussion about correspondence between two non-empty sets. The students were able to recognize the main property of the concept of function. Nevertheless, the class and textbook examples that illustrated this concept emphasized the use of a formula as a means to representing some functions. This might have influenced the students' ideas about functions as only those which can be represented by formulae, or the students might have developed two kinds of conceptualizations, one which involves the notion of correspondence and the other regarding a formula representation. For example, the students experienced difficulties when they were asked to construct some examples of 145 functions. The students also experienced difficulty in one of the problems that involved estimation of the altitude of a rectangle inscribed in a triangle with the largest area. For this problem, the students needed to represent the problem graphically and to use their previous knowledge (similar triangles) to determine an expression that included the altitude. This might explain some of the initial difficulties experienced by the students; however, when the students considered some special cases and provided some values for the altitude, they were able to recognize the possible ways that could lead to the solution. This problem was especially interesting because the students were making different conjectures and discussing them with other students for some time before the assignment was due. Third and Fourth Assignments: The Concepts of Limit and Derivative Although the approach used to introduce the concept of limit was intuitive, the assignments included some problems in which the students were asked to discuss the existence of some limits. In general, the students focused their attention on points which seemed to be important for the problem. For Ix2 + ax - 2a 2l example, to discuss whether the limit of 3— exists when x goes to x - a Ix - al Ix ~r" 2al a, was approached by considering two cases, that is, limit of T—Z Ix - dl Ix H~ 2d! when x goes to a + and limit of zr—^ when x goes to a"; this analysis led the students to conclude that the limit exists only if I3al = - I3al, which is Ix2 + ax - 2a 2 l true when a = 0. Hence they concluded that limit of —— does not Ix2 + ax - 2a 2 l exist if a * 0 and limit of = 0 if a = 0. It is important to observe that these examples were particularly emphasized during the class discussions. 146 There was a problem in this assignment in which the students did not check the information and this led them to a wrong solution. The problem involved a function f(x) that was defined as x 3 - 3x 2 + 3 when x < 1, and as x 2 -2x when x > 1. The question was to discuss the limit of ^ ^ when x approaches 1. Although the students in general distinguished two cases, when x goes to 1 + and when x goes to 1", they calculated f(1) imprecisely; that is, when they were approaching 1", they used x 3 - 3x 2 + 3 to evaluate f(l); but when they were approaching 1 + , they used x 2 - 2x to evaluate f(1). It was evident that they did not realize that the function could not take the value of 1 when defined as x 2 - 2x. In addition, the students did not check the solution. A graphical representation would have helped them identify their mistakes. The assignment included some problems that focused on the interpretation of the derivative. For example, a problem that involves two vehicles travelling at different speeds was designed to motivate students to discuss one of the meanings of the concept of derivative. The students were asked to interpret the rate of change of the distance between the two vehicles at specific times (see appendix problem #11 part b). The students' explanations included: "The car is ahead of the truck and the distance between them is changing"; "The distance between the car and the truck is decreasing"; "The truck is catching up to the car at a rate of 9 m/sec at the instant that t = 1". In general, the students' responses showed that they tried to explain the phenomenon by interpreting the results obtained in the calculation of the derivative. 147 Fifth Assignment: Application of Derivative This assignment included problems regarding application of derivative, such as optimization problems and the drawing of curves. One problem reviewed the geometrical interpretation of basic concepts; that is, the students were asked to represent cases, such as a relative maximum where the tangent is not horizontal, a point which is not a relative extreme but where the tangent is horizontal, an inflection point where the slope is -3/2, etc. The students in general were able to represent these questions graphically and some of the students even added similar statements. In this assignment, there were indications that the students had developed an encouraging and supportive environment while working on the assignment in small groups. This last assignment included problems that were related to the application of derivatives, such as finding extrema of functions in practical problems and problems that involved rates of change in everyday life. Most of the assignments showed systematic approaches to solving the problems that included diagrams, symbolic representation of the corresponding relationships, identification of variables, discussion of critical points, and the checking process. The instructor who taught the problem solving course was also in charge of another calculus class. The main difference between the two classes rested on the input given by the researcher to the problem solving class, and the explicit and exclusive use of the metacognitive assignments in the problem solving class. Although the data gathered through the assignments and final exams showed that the problem solving class did better than the other group, it is necessary to gather more information in order to explore the origins of these differences. 148 Table 3. Summary of the Results of the Assignment Assignment # Maximum Average Grade Obtained Possible Prob-Solv Class Other Class 1 10 6.6 6.8 2 10 5.5 6.9 3 10 6.3 7.4 4 10 4.9 7.0 5 10 6.7 8.9 Assignments on Metacognitive Strategies Flavel and Ross (1976) have defined metacognition as "one's knowledge concerning one's own cognitive processes and products or anything related to them"; that is, "metacognition refers, among other things, to the active monitoring and consequential regulation and orchestration of these processes in relation to the cognitive objects or data on which they bear" (p. 232). Schoenfeld (1985) discussed the managerial activities (monitoring processes) used by the problem solver as one of the categories of problem solving. The problem solving instruction encouraged the students to monitor and constantly reflect on the use of diverse strategies during the process of solving the problems. In addition, a set of questions regarding the monitoring process was given to the students as part of each of the assignments (see appendix M4). The responses given by the students were organized in accordance with what they did in their assignments. The results showed that the students experienced difficulties reflecting on their own work. It may be suggested that the students work on the problems of the 149 assignments by selecting a strategy, and if the use of that strategy did not help them to reach the solution, they then explored other possibilities. They did not reflect on the direction that they were taking by constantly monitoring their work. The students were asked to reflect on the difficulties that they experienced while working on the problems. In general, they associated their success in solving the problems with the level of difficulty. If they could not solve a problem on the first attempt, then that problem was considered difficult. Few students indicated that difficulties found while solving the problems were associated with aspects of the problems, such as the context, the number of variables involved in the statement, or the representation of the problem. The problems in the assignments involved the use of cognitive strategies, such as breaking problems into parts, guessing and testing, and using diagrams. The students were asked to elaborate on the use of these cognitive strategies. Although they were able to recognize the strategies which they used in some of the problems, they were not explicit in discussing the strengths or limitations of the strategies in the process of solving the problems. They often mentioned the content that they used to solve the problem, for example, the use of quadratic equations or the concept of polarity or derivative. At the beginning of the course, the students did not recognize the process of monitoring their work as an important part of solving the problems. They thought that this was a mechanical activity that they normally did when they could not solve the problems. However, they later recognized that the self-reflection process is also important in making decisions about the choice of strategy to select and to what extent it should be used. Schoenfeld (1985) 150 has pointed out that experts constantly evaluate their work while solving problems which helps them design and explore some ways to solve them. The students often mentioned that they looked for several ways to solve the problems initially; however, they failed to discuss or test the potential of those possible ways. Perkins and Simmons (1988) pointed out that students sometimes make some conjectures about the problems, but they may do nothing to test their conjectures. They stated that "...[t]hey thus reason by analogy to generate a possibility - an intelligent move - but fail to deploy any kind of filter to check their possibility" (p. 310). The students recognized having difficulties in exploring extensions of the problems. However, they expressed the fact that they tried to look for similar problems that could be solved with the same approach. In addition, they worked on these assignments as a group and they recognized the potential of discussing the problems with other students. For example, some of the students' comments were that "Working as a team in solving these problems was much more enlightening and fun" and "We seemed to learn from each other as we solved the problems." Indeed, this is one of the goals of problem solving instruction, that is, to encourage the students to discuss and defend their ideas. These types of students' comments are consistent with findings obtained by Schoenfeld (1989b) in which the students expressed that when they have the opportunity to discuss and participate during instruction, they find the study of mathematics more interesting. 151 Written Test Introduction A conceptual analysis associated with the study of derivative showed that the concepts of function, limit, and continuity were directly related to the concept of derivative. This analysis suggested that the students' understanding of these concepts was important for studying the concept of derivative. Time is allocated for studying the concepts of function and limit before studying the concept of derivative in the calculus course for science students offered at the college level. Hence, there is an implicit assumption that students will grasp these concepts and become familiar with the basic terminology for the study of derivative after two or three weeks of instruction. The view of mathematics learning endorsed in the present study emphasized the process of making sense of mathematical ideas rather than simply mastering the mathematical content. Schoenfeld (1989) maintains that students are able to generate or develop mathematics content if they are engaged in what he calls a microcosm of mathematical practice. That is, students are engaged in the practice of mathematical sense-making by discussing problems in which there is room for speculation, examples, counterexamples, and discussion from the students' side. Here mathematics becomes an active subject that grows through criticism, open discussion, and negotiations. Kitcher (1983) pointed out that this view of mathematics is parallel to the historical development of mathematics. This view is contrary to that in which the polished content is given to the students to be mastered. The present study focuses on the extent to which students receiving instruction based on problem solving are able to use basic strategies (heuristic 152 and metacognitive) when solving mathematical problems. Among the instruments designed by the researcher for gathering the data for this study was a set of six problems (see appendix M4) related to the content studied during the first four weeks of classes. The content involved the concepts of function and limit, and their basic operations. This material was requisite for the study of derivative which was the next content to be studied in the course. The purpose of giving this test was to elicit some of the students' ideas about these concepts and about the ways that they approached these problems. That is, the type of strategy(ies), procedure(s), or evaluation that the students would use during the entire process of solving the problems. Description of the Test A common practice used to evaluate students' progress after mathematical instruction is to ask students to take periodic examinations. The instructor agreed to design a set of problems that were representative of the problem solving instruction; this test could be an indicator of the extent to which the students were able in a limited time to use the strategies discussed in class. The test included six problems related to the concept of function. Problems one and five were problems that involved the notion of evaluation of functions and limits. These problems had the same mathematical structure and required exactly the same type of analysis for their solutions. The difference lay in the type of phenomenon represented. The idea was to evaluate to what extent the students would distinguish the structure of the problems and would treat them similarly. Problem number two involved the recognition of when the given expression was not defined, that is, the importance of examining the denominator of the expression and using that 153 information to find the value of the parameter a. Problem number three was related to basic properties of exponents, factoring, and a system of equations. Although problem number three does not require much elaboration for its solution, it was important in order to evaluate the students' responses in a different context; that is, this problem did not fit directly with the content being studied. Problem number four involved the concept of limit and the concept of distance. Finally, problem number six involved the use of a step function and the relationship between its domain and range. Analysis Two structurally similar problems (one and five) required the identification of constants and variables in order to solve them. It was here that the students experienced difficulties when evaluating the expressions. For example, in problem number one, Vand m were the variables and were related through a specific expression. The first question required an evaluation of a change of mass from m 0 (initial mass) to 2m 0 while the second part required the use of the concept of limit when m changes. For the first part, the students' common responses included: i) substituting double mass as (2m 0/m); ii) substituting double mass as (2m0/2m) and concluding that the speed v is always the same; ii) substituting double mass as (m 0 /m) 2 , that is, as an exponent; iv) substituting double mass as m 0 /2m; and v) substituting double mass as m 0 /2m 0 . These responses suggested that the students did not identify which part of the expression was changing. There were at least two aspects related to this difficulty. One was the conceptualization of variable which required the 154 understanding of the phenomenon and its representation. The other source of difficulty was related to the formal notation used for functions. Notational problems were also identified in the problems of the assignments. For example, some students got confused when the same function needed to be evaluated with different variables such as x and s. In the second part of problem number one, the students were asked whether or not the object's speed could be arbitrarily large. The students' responses included three possibilities: the speed increases arbitrarily, has a limit, or is the same. The explanations given for each of the responses included: a) "no, because the highest speed is the speed of light"; b) "no, because the sign of the radical must be always positive", c) "no, because in the limit the expression goes to c", and d) "yes, because the momentum increases". In the second part of problem number five, the students were asked whether or not the teller's efficiency could become arbitrarily large. The students' responses included "yes" and "no". Their explanations included arguments such as: "yes because if one has more hours of experience, then one is more efficient", and "no, because no human being is able to increase his or her efficiency" arbitrarily large. The responses given in the second parts of problems one and five reflect different views about the problems. Some students based their explanation of V (speed) not becoming arbitrarily large on having accepted that the speed of light is the highest one. They did not take into account the specific expression given in the problem for their argument. It seemed that affirming the contrary would contradict what the students recognize as truth and that was sufficient evidence or argument for rejecting that possibility. 155 Another common explanation of the students was to use the sign of the square root. The students showed confidence about their belief that the radical must always be greater than or equal to zero; therefore, V could not be so large. Again, it seemed that the students based their explanations on what they knew from their experience about the domain of expression involving radicals rather than by exploring the actual relationship represented. In contrasting responses from problems one and five, it was observed that those students who approached problem number one correctly, in general, noticed the connection with problem number five. Nevertheless, there were some students who approached both problems differently. For the second problem, the most popular approach was to substitute directly x = 3 in the expression. This substitution made the denominator equal to zero, and the students concluded that a did not exist. It may be suggested that the students have dealt with limit problems in which only a direct substitution is required for solving the problem, and, in their experience, all the limits are obtained in that way. It was clear that some students were confident that there was no need to explore this problem from other angles after having obtained zero in the denominator. It is important to mention that during instruction the students worked on several examples in which the limit could be obtained only after transforming the expression. The idea was to show the students that there are limits that need more than simple substitution. However, the students' responses lead to conclude that even when the students are aware of these types of strategies it takes time for the students to assimilate these strategies and be able to use them whenever they are required. Perhaps, metacognitive strategies could help students recognize what strategy to use and when to use it. Problem number three (part a) was approached by focusing on the 156 system of equations represented by the exponents or by trying to solve the system directly, that is, without doing any transformation. The second part of this problem showed that some of the students were not familiar with the term "reciprocal" or failed to simplify the expression. The instructor suggested that the context made a difference in the students' response to this problem. That is, this problem is not directly related to the content in study; therefore, it might have presented some difficulty for the students who were expecting problems directly related to the course. In problem number four which involves two ants jumping in two different directions, each ant jumps one unit on its first jump, a 1/2 unit on its second, a 1/4 unit on its third jump, and so on. The students were asked to calculate how far apart the ants would be over an infinite amount of time. The students relied on their graphical representations. While a few students were able to solve the problem, other students did not recognize the limit and concluded that the ants were infinitely apart. It may be suggested that there are terms involved in this problem, such as, infinite and distance, that seem to be problematic for some students. For the last problem which involved a step function, the most common approach was to represent the problem algebraically. The students provided a single number as the response. However, the students did not realize that they were dealing with a step function and that the response involved an interval rather than only one number. In addition, this problem required some analysis to decide whether the given number ($1.75) was part of the range of the function. Only seven students made that linkage. An overwhelming concern manifested by the students was the time allotted for solving the problems. For example, for these six problems, the time limit was an hour. The students suggested that working on a problem 157 required some time for thinking of possible approaches, organizing some ideas, exploring, implementing and evaluating some strategies. They felt that it is difficult to approach the problems in that sense if the time limit is not flexible. The instructor's view is that the students have to be aware of this constraint and be able to work under these conditions. For example, he mentioned that his college allows two hours for the students to work on the final exam. As a consequence, the students should be ready for this task. In order to have a sense of the time available for the exams and the type of problems included in the final exams, some final exams taken by students from previous courses were checked. It was noticed that the problems are basically exercises that do not require any great elaboration to solve them. Schoenfeld (1985) stated that students generally believe that all mathematical problems can be solved in less than ten minutes. He also mentioned the way that mathematics is presented determines what students believe about mathematics. Final Exam Introduction The final exam was designed by the coordinator of the course. In theory, all the instructors teaching the course should provide ideas for the type of problems to be included in that exam. However, in practice, the coordinator alone designed the exam and circulated it to the other instructors two or three weeks before the exam was given to the students. If any instructor makes any comment or suggestion then the proposed exam might be redesigned. But if there is no response from the instructors, then the exam is considered suitable for the course. The exam included 12 questions that 158 involved applications of derivative, the use of rules to find derivatives, and graphing of functions. The time allotted for solving the problems on the exam was two hours. The analysis of the students' responses is presented in accordance with the type of problem and the corresponding progress made by the students. Analysis of the Students' Responses The students did reasonably well in all the problems that required the application of rules to get the derivative. For example, almost all the students solved or showed some significant progress in questions 2, 3, 4, and 5. These questions involved the use of the definition of derivative, the use of derivative formulae, or the use of limits to determine the continuity of a function. However, the students experienced difficulties in questions in which it was necessary to explore examples and counterexamples in order to determine the validity of each statement. For example, one of the problems was to determine whether or not the expression J(x3 + I)4 dx was a polynomial of degree 13. Only 10% of the students showed the corresponding transformations that led to the correct solution. It may be suggested that, in general, the students did not relate or think of the possible transformations that could have helped them to approach this problem. The problems that involved applications of derivatives were approached by the students systematically; however, they did not interpret the results or explore other related problems. This might have been due to the nature of the problems. For example, the statements included only direct questions that could be obtained by the appropriate selection of the type of interpretation of the derivative. That is, if the distance is described by the expression s(t) = 10t -1 3 , then to calculate the rate of change of the velocity at 159 t = 2 involves only the recognition of the derivative as a rate of change. Schoenfeld (1988b) pointed out that students often fail to relate or justify their responses with a mathematical argument. For example, in a mathematical contest that involved talented high school students, the students were asked: "[l]n which case does the plug cover a larger proportion of the hole, a square peg in a round hole or a round peg in a square hole?" More than 300 students took the exam and only 4 students provided explanations for their answers. However, the extent to which the students responded to this problem in accordance with what their mathematical instruction normally requires of them is an issue that could encompass a great deal of discussion. Schoenfeld (1988) stated that one's notion of mathematics determines whether or not one needs to justify responses with mathematical arguments. He went on to say that "students pick up their beliefs about mathematics from their classroom experiences with mathematics" (p. 75). This phenomenon was confirmed during the development of the present study. The instructor often gave examination problems that did not require justifications or other explorations. One of the problems involved finding the line tangent to the curve y 3 + xy = x 2 - 1 at the point where the curve crosses the positive x-axis. The students worked systematically by relating the slope of the line to the derivative, finding the point of contact, and using the corresponding equation of the line (slope, point). However, only 10 % of the students represented the problem graphically and checked their solutions. The instructor gave full marks even to those students who only obtained the equation of the line (29% of the students). It might be suggested that the students have not developed or incorporated a frame of thinking in which there is a need for examining or exploring the problems in different ways. The problems included in the final exam might be categorized as 160 routine problems that do not require transformations in order to be solved. The process of achieving their solutions involved only the recognition of an algebraic model (which sometimes is given) and the applications of derivative rules at different stages of the process. For example, the process could involve the calculation of the derivative, the critical points, or the use of criteria for determining maxima and minima. The instructor focused mainly on these aspects for the evaluation of the students' work. It is important to mention that the problem solving class did better as a group in every problem on the final exam compared with the other group that the same instructor taught during the same period. In the other group, there was less emphasis on activities related to approaches to problem solving. However, these results should not be considered as very significant in terms of the students' approaches to mathematical problems, as interviews indicated that the students frequently failed to use basic strategies that are important in the process of solving problems. It was observed that the problem solving class did better than the other class in problems that are typically included in the exams. In addition, the results showed that the students in the problem solving class became aware of the importance of considering activities that would help them to engage in discussion and reflection about mathematics and problem solving. This might be the initial point for improving their way of solving nonroutine problems. Table summarizes the results of the two classes on the final exam which was designed by the coordinador of the calculus course in consultation with all the instructors teaching this course. 161 Table 4. Results of the Final Exam Problem # Maximum Average Grade Obtained Possible Prob-Solv Class Other Class 1 6 3.4 3.2 2 10 6.3 6.3 3 10 7.7 7.6 4 10 8.5 7.7 5 8 5.8 5.7 6 8 5.5 4.7 7 8 5.5 4.7 8 8 3.2 2.7 9 8 3.5 2.3 10 9 5.1 3.9 11 7 3.6 2.7 12 8 2.5 1.5 Chapter Summary This chapter deals with the analysis of the data gathered through a questionnaire given at the beginning and at the end of the course, a written exam given in the second month of the study, five assignments, and a final exam given at the end of the course. The results indicate that the students became more aware of the importance of the use of several problem solving strategies. Although the analysis of results of the written test and the final exam show that the students often failed to explore possibilities that could have helped them solve the problems, it is important to mention that the quality of the students' work in the assignments showed significant progress. 162 CHAPTER VI STUDENTS 1 APPROACHES TO PROBLEM SOLVING Introduction Schoenfeld (1985) suggested that mathematics educators need to have more information about the ways students approach problems in order to consider learning strategies that would help students understand and create mathematics. This information could help instructors to make appropriate instructional decisions that best fit their classes. During the process of solving problems, students become engaged in different activities that reflect the ways that they see, understand, conceptualize, and think about mathematics and problem solving. The aim of this chapter is to categorize the ways that the students in this study approached mathematical problems. An underlying assumption in identifying or recognizing categories used by the students while solving problems is that mathematical problems are conceptualized in a limited number of qualitative ways and that it is possible to identify categories to characterize these conceptualizations (Marton, 1981, 1986). Frame of Analysis Maturana (1989) recognized that the process of observing a determined phenomenon involves the use of adequate language. Thus, for the discussion of the students' approaches to problems, a language for explaining and interpreting the ways that students interacted with the problems was adopted. This language is part of the theoretical frame used by Schoenfeld (1985) to discuss factors that influence the process for solving problems. Maturana (1989) distinguished two kinds of unities used to explain what is observed. Simple unities which take what is observed as a whole and composite unities 163 which involve their components and relationships. He stated that, when the unit is simple, the task is simple because one specifies properties and that is sufficient. However, a composite unity includes features such as organization and structure. He stated that "the organization of a composite unity refers to the relations between the components that make the unit what you claim it is. ... A unity is a composite unity of some kind only as long as its organization is an invariant" (p. 70). The structure of composite unities refers to the components and the relations that make a particular unit. "So the organization is invariant and is common to all the members of a particular class of composite unities, but the structure is always individual" (p. 71). The categories that emerged from the analysis of the information about how students approached mathematical problems illustrated the use of these types of unities. For example, in Maturana's terms, the identification of a specific type of strategy, such as drawing a table, simplifying the problem, or guessing and testing, would constitute a simple unit, while analyzing relationships associated with the use of the strategies would be a composite unit. Schoenfeld (1983) suggested that the process of solving problems involves the use of tactical and strategic decisions. Tactical decisions include the use of algorithms or heuristics, while strategic decisions are related to the direction and allocation of resources during the problem solving process. Schoenfeld found that experts' success in solving "new" problems is often related to the use of strategic decisions rather than tactical decisions. Strategic decisions are important components that characterize the way that experts approach problems; however, novices (students) seem to lack these types of strategies. Perkins (1987) pointed out that during the process of solving problems, individuals use thinking frames to organize and support their thought 164 processes. In explaining the origin of these frames, Perkins stated that "learners might become acquainted with a frame through direct instruction, they might invent it for themselves, or they might 'soak it up' from an enriched atmosphere, without the frame ever taking the form of an explicit representation" (p. 48). He distinguished three aspects of learning that influence the thinking process; first is acquisition in which the learner becomes initially familiar with the thinking frame and applies it to simple cases. The next aspect is to make this frame automatic, this stage comes after some practice. Finally, there is the transfer aspect in which the learner applies the frame across contexts which may be different from the original context of learning. These aspects were useful in recognizing some characteristics of the students' learning. The main sources of information that helped to categorize the students' approaches to mathematical problems were the task-based interviews that were carried out at the end of the study. The analysis of the task-based interviews was based on a coding system which led to the identification of episodes. These episodes represented a series of activities in which the student was engaged in specific actions. Schoenfeld (1983) focused the analysis of protocols on episodes that included: reading, analysis, planning, implementation, verification, and transition. These episodes were part of the framework analysis used in this study. The coding system was constructed by taking into consideration chunks of information that characterized each episode. This system differs from schemes in which a sequence of numbers or symbols is associated with a specific type of behavior identified in the process of solving problems. It is a more macroscopic type of system that integrates cognitive and metacognitive decisions used by the student while solving the problems. Schoenfeld (1983) stated that "in analyzing human problem-165 solving, exclusive attention to the microscopic level may cause one to miss the forest for the trees" (p. 348). Schoenfeld also argued that a framework for the analysis of protocols should include objective and subjective components. The identification of the loci of potential managerial decisions could lie in the objective part while the characterization of the origin of these managerial or executive decisions and identifying their strengths and their limitations could be part of the subjective part. Schoenfeld suggested that a set of questions associated with each episode could guide the analysis of the interview protocols. These questions were adjusted and used as a way to structure the information related to each episode. Indeed, it was useful to accompany each question with a checking box for identifying the episode and "evidence" and "comments" are headings for open-ended response. The idea of organizing the questions in this form was taken from a research project led by Schroeder (1989). All the interview transcripts were organized by taking into account phases identified during the process of solving the problems, phases that include: i) reading, ii) analysis, iii) exploration, iv) strategy selection, v) metacognitive processes, vi) partial evaluation, vii) planning and implementation, viii) verification and extension, xix) transfer level, and xx) belief systems. The last phase of the analysis is related to the students' ideas about mathematics and the problem solving course. To suplement the analysis the following form containing the questions associated with components of problem solving was used. 166 Questions Associated with the Frame of Analysis Reading 1. Did the student note all the conditions of the problem? [] immediately; [] after the second reading; [] experienced some difficulty; evidence and comments: 2. Did the student relate the statement of the problem to some specific situation? evidence and comments: 3. Did the student differentiate the relevant information of the problem? evidence, examples and comments: Analysis 1. Did the student focus his or her analysis on the relevant information of the problem? Q initially; [] focused on various aspects; []other; evidence and comments: 2. Did the analysis match the conditions of the problem? [] reflection on the data; [] more on the use of similar problems; evidence and comments: 3. Did the student integrate the entire information to decide what way to take? [] no integration; [] part; [] complete; evidence and comments: 4. Was there a coherent approach to the analysis? [] some inconsistency; [] little analysis; fj complete; evidence and comments: Exploration 1. Was there an explicit purpose in the exploration? Q directed by the data; [] by the possible solution; • other familiar problems; evidence and comments: 2. Was there a systematic exploration? fj with part of the information; [] all the information; [] other; evidence and comments: 3. Did the student consider several alternatives? [] based on the data; [] other; evidence and comments: 167 Strategy Selection 1. Did the student select an algebraic strategy? [] initially; [] sometimes; [j exclusively; evidence and comments: 2. Did the student select a geometric approach? [] initially; [] sometimes; [] exclusively; evidence and comments: 3. Did the student use a trial and error approach? •initially; [] sometimes; [] exclusively; evidence and comments: Metacognitive Processes 1. Did the student evaluate different alternatives to the solution? fj always; [j sometimes; evidence and comments: 2. Did the student monitor his or her progress? [] always; [] sometimes; evidence and comments: 3. Did the student shift from one strategy into another? Q sometimes; [] often; evidence and comments: 4. Did the changes in the use of the strategies help him or her solve the problem? • sometimes; [] often; [] always; evidence and comments; 5. What were the consequences of the presence or absence of the monitoring process? Partial Evaluation and New Considerations 1. Did the student match his or her resources with the information given in the problem? 2. Was there any new result that emerged from the exploration and analysis that the student used? 3. Did the student restructure the problem as a result of that new information and how did that influence the previous analysis and exploration? 168 Planning and Implementation 1. What aspects could indicate that the student followed a specific plan? 2. Did the plan seem suitable for that problem? Was the plan well structured? evidence and comments: 3. Did the student carry out the plan in a systematic way? evidence and comments: 4. Did the student monitor the implementation of the plan? evidence and comments: 5. Was there any type of error that impeded the implementation of the plan? evidence and comments: Verification and Extension 1. Did the student check the process involved in solving the problem? 2. Did the student check the plausibility of the solution? 3. Did the student show confidence with the solution? 4. Did the student relate the problem to other problems? [] simpler cases; [] more general problems; evidence and comments: 5. Did the student extend the problem into different domains? Transition Period 1. Was there any assessment during the solving process in which the student reconsidered the use of specific content or strategies? 2. Did that assessment change the direction of the process? Was it appropriate or necessary? Transfer Levels 1. Was there any evidence of the student transferring knowledge from previous courses in order to solve the problem? 2. Did the student relate the content used for solving the problem to other problems or situations? Belief Systems 1. Is there any indication suggesting what the student thinks about mathematics and problem solving in general? 2. Did the student elaborate on his or her experience(s) in previous courses and the calculus course? 169 Categorization of the Students' Approaches to Problem Solving The research question that is examined in the first part of this chapter is: How could the students' approaches to problem solving be usefully categorized in accordance with the categories employed by Schoenfeld (1985), namely, domain knowledge, cognitive and metacognitive strategies, and beliefs? The analysis is based on the students' approaches to two mathematical problems. Thinking aloud was the method used to gather information about the students' processes involved in solving the problems. The students were asked to work on the problems at the end of course. The problems were: 1. Find values of a and b so that the line 2x + 3y = a is tangent to the graph of f(x) = bx 2 at the point where x = 3 (line problem). 2. Find all rectangles with integer sides whose area and perimeter are numerically equal (rectangle problem). The students initially read each problem once and immediately started to recall similar problems that could help them solve the problems. For the first problem which involved finding two constants a and b that would determine a tangent line to a parabola, the direction that the students took was to make some calculations, such as working on the derivative and trying to relate it to the statement of the problem. For the second problem which did not involve current terms, the students spent more time trying to understand the statement before getting involved in getting the solution. It is suggested that the students' first approach is to search for what Perkins (1987) called "the low-road transfer", that is, looking for a context previously studied that resembles features of the problems. This approach is different from that which involves an intentional effort for understanding the problem and then exploring alternatives that may lead to the solutions. Perkins also suggested that self-170 monitoring helps the problem solver develop an awareness of what representations and transformations are required in order to solve the problem. For the first problem, the students calculated the derivative; this decision was made in response to the relationship between the tangent and derivative. During class instruction, the relationship between the derivative and tangent to a line was emphasized. It seems that the students learned this frame of thinking and wanted to use it mechanically. However, the calculation of the derivative did not suggest to the students how or where to use it. So, the students reread the problem and tried to understand the role of each of the parameters involved in the statement. The students initially approached the two problems without considering a specific plan. It seemed that the context in which the problem was set often provided them with the elements to design a plan. For example, Lydia expressed: "...it is easy if you have done a hundred in a book that are exactly the same kind; but when you get any of these without any reference it's harder to start." It was observed that self-monitoring processes which were sometimes initiated by the students themselves helped them to analyze the information provided in the problem more carefully. For example, reflecting on "What do I want to do?" led some students to transform the representation of the line into the "slope-point representation" and to analyze the relationship between the slope of the line and the derivative of the parabola. The students' analysis of the information was focused on information that they could easily transform. For example, in the rectangle problem, the students rushed to write down the formulae for the area and perimeter of the rectangle. For the line problem, the students calculated the derivative of the 171 parabola. It seemed that the familiar terms involved in the statement were used by the students as the initial point to calculate or add more information about the problem, even though the students often struggled using that information. For example, Alex after having calculated the derivative of the parabola asked whether "b" was a constant. This suggested that he got involved with several calculations but without being clear about the role of key information given in the statement of the problem. The students' exploration of ways for solving the problems relied on first using the familiar information either to represent the problem or to process the information through the use of specific concepts familiar to them. For example, for the rectangle problem, the students wrote down the expression of the area and perimeter and equated them. Up to this point, the students transformed the statement of the problem into the corresponding symbolic ab representation, that is, - y = 2(a + b). Although they isolated one of the variables included in the equations, they did not know how to interpret this result or finding. At this stage of the problem, they tried to reflect on or look for more information that could help them use this information. Some of the students even went back to the original equation and substituted the obtained value again. This may suggest that for the first exploration the students tried to fit the information given in the problem in some sort of frame that they were already familiar with instead of focusing on a possible new frame that could emerge from the information given in the problem. For example, they kept working on finding a solution for a and b in the rectangle problem from the equation, even though the transformations that they made to the equation did not show any progress towards the solution. It is suggested that the students did not use a systematic exploration to 172 consider plausible ways for solving the problems. They did not consider various alternatives before deciding which direction to take. It seemed that the students' only alternative in pursuing the solution was to relate the familiar given information to another familiar problem. Mike for example, found the derivative of f(x) and experienced difficulty in using it. He expressed: "I am not sure how to relate this to the equation of the line. I guess I need to substitute this value somewhere but I do not see where...." Although during class instruction the students spent time discussing the importance of representing a problem, the students showed no intention of representing the first problem graphically. For the problem that involved finding a and b , the students did not recognize that they were dealing with a family of lines and parabolas. It may be suggested that although the students may identify the general equations of the line and parabola they experience difficulty interpreting the meaning of the parameters involved in the equations. For the rectangle problem, the students drew a rectangle; however, it did not provide useful information to approach the problem. That is, the students did not use the representation to work on the solution. Polya (1945) recommended the use of guess and test to interpret the problem, to represent the problem, and to monitor the process while solving the problem. The students did not seem confident in using these strategies. They might use guess and test as the last attempt. For example, in the rectangle problem, they stuck to the algebraic approach for a long time trying to solve the equation. Although the students noticed that no progress was made when dealing with the algebraic expression, it was only when the interviewer asked them to try some particular cases that they realized that the testing method was useful in this problem. It was evident that the students used metacognitive strategies 173 throughout the entire process of solving the problems. Nevertheless, they often lacked basic mathematical resources, such as representing the problem geometrically or considering simpler cases for exploring plausible alternatives. Daniel while working on the rectangle problem stated, "...if I relate somehow a and b then probably I may not need to find their values. What about some boundaries? Am I wrong?" Daniel seemed to be willing to explore other alternatives; however, he experienced some difficulties in relating the algebraic expression to another context on his own. It seemed that the students used metacognitive questions as a response to some difficulties and not as means to explore the potential of their approaches. For example, they could have thought of a specific line and parabola for the problem that involved families of those curves (lines and parabolas) and a square for the problem that involved searching for all the rectangles in order to make sense of the information given in the problem (exploring particular cases). The students also had some expectations about the types of results that they could get at different stages when solving the problem. If the results did not match their expectations, they often changed the approach or spent quite a long time looking for an error that often did not exist. For example, Lydia, after she obtained the derivative and evaluated it, stated, "First thing, I think this looks like a wrong number; I must have done something wrong,....It does not look right. You know immediately. You think that is not like five, or four. Let me check if I have done something wrong...." Schoenfeld (1985) pointed out that the type of examples presented during mathematical instruction and the type of assignments that the students work on determine the students' ideas about problem solving. There is indication that the students normally monitored their processes as a response to a certain type of anomaly that they detected while working 174 on the problem. For example, if they experienced some difficulty in accepting what they were getting or if the approach used did not show progress, then they asked themselves some kinds of metacognitive questions. However, monitoring their processes not did always lead to a better approach to the problem. For example, realizing that they could not solve the relationship representing the equality of the area and perimeter of the rectangle, some of the students substituted the isolated value in the original equation which was exactly the same equation that they used to isolate one of the parameters. The students relied on matching their mathematical resources, mainly the ones recently studied, to the information given in the problem. They were also reluctant to consider a new frame for solving the problem. Although during class instruction the importance of considering several possibilities or ways for solving the problem before getting involved in a specific approach was emphasized, the students seemed to lack this kind of flexibility, and they showed consistency in considering only the approach that appeared to be "safe" in solving the problem. They failed to derive a possible set of strategies that could lead to the solution of the problems. There was no clear evaluation of what they were doing or where they were going while working on the problems; however, if their initial intent did not help them make any progress, then they often examined other alternatives. That is, total failure in solving the problem was normally the only outcome that motivated the students to search for other alternatives. New considerations for approaching the problems often came after having worked on the calculations that were related to some terms involved in the statement of the problem and not being able to use them to solve the problem. For example, "tangent to the graph" suggested to them to get the derivative of the function; "at the point 3" suggested to the students to 175 evaluate at 3; "the straight line tangent to" suggested the same slope. However, they failed to recognize the role of the parameters a and b and the relationship with the information that they obtained from the statement. It was evident that they knew the content involved in the problem but were unable to make the necessary connections for using it. It is suggested that the students made sense of the statement of the problem in parts without considering the problem as a whole. For the rectangle problem, the students also explored the familiar terms until they got an expression that related one side of the rectangle as a function of the other side. The students at this stage did not know how to deal with this expression and struggled to relate it to the statement of the problem. The students did not design a specific plan in order to solve the problems; they read the problems and engaged in specific transformations. They were not systematic in the implementation of strategies that could lead to the solution of the problem; rather, they selected a specific part of the problem which they thought could lead to the solution and persisted in using it until they did not know what to do next. For the tangent problem, they decided to work on some calculations (derivative, slope, and function evaluation) and spent quite a long time working on them but without having a complete picture about where and how to use that information. Maria after having represented the rectangle problem symbolically expressed, "I am not sure what it wants me to say, does it want me to say how long the length and the x's have to be? What would I do?" Even when the original plan for the interviews was to let the students work on the problems alone, there were a few interventions on the part of the interviewer. The interventions included asking some clarification questions, responding to some questions from the students, and providing some 176 metacognitive suggestions at various points in the development of the interviews. All the students eventually were able to solve the problems; however, they failed to check the sense of the solutions. Schoenfeld (1990) recognized that it takes time for the students and instructors to conceptualize other approaches to problem solving. For the tangent problem, the interviewer asked them to check the solution and elaborate on its meaning. The students' reaction was to review whether or not there was any algebraic or calculation mistake involved in the process that led them to get the solution. The interviewer then asked them to represent the problem graphically and it was not until then when they realized that the graphical representation could help them check the solution. The interviewer also asked them to change the values of the constants a and b and to explain what happens to the line and the parabola; the students then realized that they were dealing with a family of lines and parabolas. For the rectangle problem, the students initially hesitated to try specific examples in the expression in which one of the sides of the rectangle was isolated. It may be suggested that the students did not consider that the use of guess and test could help them solve the problem. They focused on searching for an algebraic approach that finally could provide the lengths of the sides of the rectangle. The lack of success in finding that algebraic approach led the students to explore some cases for one of the sides (a = ) and they were able to identify some numbers for the sides. It was then that they started to use this strategy to analyze the relationship. Finally, the students were able to limit the domain of the expression and consequently solve the problem. Although the students did not follow a systematic plan in order to solve 177 the problems, they showed several transition points at different stages of the process while solving the problems. For example, the initial interaction with the problems was to read the problem and explore or transform the familiar terms involved in the statement. The next stage was to become involved in calculations or representations of the problem in accordance with the terms that were familiar to the students. At this stage, the students often used metacognitive strategies as a response to some type of difficulty found when working on the calculation. As a result, they often went back to the original statement of the problem and checked for additional information or other alternatives that could help them to overcome the difficulties. It seemed that at this stage the students identified the important information of the problem and its relationships. The students actually understood the problem. The final stage involved getting the solution and checking the process involved in solving the problem. The students were asked to reflect on some of the difficulties that they experienced while solving the problems. They indicated that their first attempts at solving the problems were based on using their understanding of the problems that were discussed during the class. For the line problem, they mentioned that the term "tangent" triggered the use of derivative, whereas for the rectangle problem, the conditions of equating the area and perimeter suggested the use of an equation. Perkins (1987) identified two mechanisms for transfer: "low-road transfer" which may occur when the students solve a problem in one context and this triggers the students to solve another problem which resembles the previous one and "high-road transfer" which occurs when the students are able to apply the content in other contexts which may not be similar to the context studied in class. It is suggested that the students spent time exploring for "low road transfer" when they were asked to solve 178 problems. Schoenfeld (1985) pointed out that what students believe about mathematics and problem solving permeates the way they approach problems. The results obtained from the pilot study showed that the college students tended to think of mathematics as a subject in which the main goal is to apply algorithms or rules in order to solve problems. In addition, those students believed i) that the instructor should tell them how to solve problems during class, ii) that only good students in mathematics could understand what is going on in mathematics, and iii) that any mathematical problem could either be solved by using a previous example or not be solved at all. The instructor was aware of his students' ideas about mathematics and problem solving. It was important to consider some examples and learning activities that would encourage the students to question and reflect on such views of mathematics. To examine to what extent the students' views about mathematics changed after instruction, the students were asked at the end of the task-interview to reflect on the activities in which they had been engaged during instruction. The results showed that the students considered mathematics to be an important subject that could be applied in various areas. They found calculus a more interesting course compared to their previous courses in mathematics. They found the assignments challenging and interesting because there was much discussion about each of the problems. Lydia, at the end of the interview, stated: There is a lot of people in our class who are enthusiastic, people who argue their points; we argue the points when we think it's the right point. ... I think we are learning more because we understand what is going on, because we are thinking instead of just doing exercises from 179 the textbooks. It is suggested that the class interaction was of benefit for the students. For example, Don stated that there were many concepts that came out from the class discussions without his even realizing that they were there. For example, he stated: ... once in class, just before we were doing derivatives, where actually we were doing derivatives and nobody knew it . That was interesting because after when we started doing the actual...like we got the name for it, we already understood what was going on and then it was very fast after that... Another student Linda, in the same vein, when referring to the development of the class expressed: I would like to say that this class is different from the ones that I had before, and I like this one the most. This is the best class so far, and I guess it is because [there are] more assignments and more thinking about math than just doing it. In general, the students commented at the end of the task-interviews that as a result of the calculus course, they were motivated in discussing mathematical ideas with their classmates. They also indicated that it is important to understand the underlying concepts associated with the problems and not only to master the content. Although the students experienced some difficulties while working with the problems, it seems that their willingness and motivation showed during the course could be the initial point to construct a more integral way for problem solving. 180 Analysis of Patterns of Misunderstanding Exhibited by the Students The research questions that are discussed in this part are: i) What difficulties did the students encounter in the use of problem solving strategies when solving mathematical problems? ii) Were those difficulties related to the class instruction? In the first part of this chapter, the analysis of the students' processes used to solve the problems focused on relating these processes to episodes that identified main characteristics of problem solving. It is also important to explore in more detail the difficulties that the students experienced in solving the problems. For this purpose, a model presented by Perkins and Simmons (1988) was adopted for the analysis of the student's difficulties. A discussion of this model is presented in Chapter II. Perkins and Simmons introduced this model as an integrative model for explaining some difficulties that students normally experience in science, mathematics, and computer science. The model included four frames of knowledge (categories that distinguish kinds of knowledge), that is, i) content, ii) problem solving, iii) epistemic, and iv) inquiry. This model considers heuristics and metacognitive strategies within any of the four frames and what Schoenfeld identified as beliefs within the problem solving or epistemic frames. Perkins and Simmons suggested that heuristics, beliefs, and self-monitoring practices are orthogonal to each of the frames of knowledge. They stated: [l]n general, one set of contrasts addresses the form of the knowledge in question - strategic, background beliefs, autoregulative - whereas the four frames address what the knowledge in question concerns - matters of content, problem solving, epistemology, or inquiry (p. 314). 181 Although the models discussed by Schoenfeld (1985) and by Perkins and Simmons (1988) share the need to include different types of knowledge for analyzing students' understanding, there are some distinctions regarding the exclusiveness of the frames. It was important to analyze the types of students' misunderstanding identified during the first analysis of the interview in another context. This could provide some elements for contrasting the findings of the first part and generate recommendations for mathematical instruction. For example, the students' difficulties in using a systematic list to identify patterns could be related to the problem solving frame, while the students' difficulties in supporting the validity of a statement could be related to the epistemic frame. The main characteristics of the frames (content, problem solving, epistemic, and inquiry) are useful in order to follow the corresponding analysis. The content frame includes the terminology, definitions, and algorithms or rules related to the content. For example, algebra might include a variable, expression, equation, solution, and graph as important components of the content frame. It also includes the corresponding metacognitive strategy associated with the use of the content. Ways to recall information or to use the notation are associated with this frame. The problem solving frame includes specific and general problem strategies, managerial strategies, and beliefs about problem solving. This involves the solving of routine (textbook) and nonroutine problems. The epistemic frame includes a set of criteria used to validate the use or acceptance of a particular result, that is, the evidence or explanation that clarifies the use of a particular concept, rule, or procedure. The inquiry frame includes specific and general beliefs and strategies 182 that are used to extend or challenge the knowledge of specific content. Students may exhibit naive or limited notions regarding the four frames, they may show hardly any notions at all, or they may have no idea what the epistemic foundations of a particular subject matter are, or how to explore the domain completely. Therefore, it is necessary to analyze the students' understanding and ways of solving mathematical problems by focusing on these frames. Such focusing could help provide information about what types of activities should be included in mathematical instruction. The students showed difficulties in the use of symbolic language to represent concepts, such as functions, limits, and derivatives. Although they were able to recognize and use notation that was similar to the examples discussed during the class, that is, f(x) = 3x and g(x) = 3, they failed to apply the notation to cases, such as S(x2 - 3), and composition functions. The students discussed several examples in which some functions were graphed; however, they often failed to use the elements discussed during class that helped determine the graphs. They strongly relied on assigning some values to calculate some points of the graph. Perkins and Simmons (1988) stated that "as has been widely recognized, students do not approach subjects new to them with empty minds. They bring preconceptions that often rival and override those of the topic itself" (p. 308). The most common approach used by the students to calculate lim f(x) when x goes to a was to evaluate f(a) and check whether or not the result was a number. For example, when the students were asked to justify the existence 2x^ ™ 3sx I x 3 1 or non-existence of a so that lim ( — ) exists when x x -2x - 3 approaches 3, they relied on replacing the variable x by 3 in the given expression. Only 5% of the students recognized that the numerator and 183 denominator could have (x - 3) as a factor. That may indicate that the students' understanding of the content and ways to use it is limited to the types of experience that have been successful for them in a specific context; they failed to extend the problem or to examine other possibilities. It was observed that the students' use of problem solving strategies was related to their mathematical resources. The students recognized the importance of the use of strategies, such as trial and error, working backwards, breaking the problems into parts, and drawing diagrams; however, they often pursued the application of a strategy even when no progress was shown. For example, in finding all the rectangles with integer sides whose area and perimeter are numerically equal, the students spent much time trying to find a system of equations that could help them solve the problem. They used a guess and test approach after they had experienced difficulties in finding the solution by algebraic means. It was also observed that the students often mentioned several strategies that could have been used for solving the problems, but they failed to actually test their use. It was observed that the students adopted several tactics that they used to apply some concepts. For example, the concept of derivative was associated with the terms "tangent" or "maximum". As a consequence, in any problem that involved the term "tangent", they rushed to get the derivative of the expression and to try to use it in the problem. For example, one of the problems of the assignments involved approximating the rectangle with maximum area inscribed in a triangle; the students' response was to get the derivative of the algebraic representation that defined the area and to examine the critical points. This suggests that the students developed or adopted a response to the applications of some concepts and used it as a mechanism for solving the problems. 184 Students may become proficient in the use of rules or algorithms but fail to recognize the principles that support them. The epistemic frame is important because it may help students to judge and examine the mathematical ideas studied during mathematics instruction. For example, the students experienced difficulties discussing the validity of statements, such as, "if two functions have the same derivative, then they are equal"; "a rational function is continuous except when x = 0"; "if limit f(x) + limit g(x) exists when x approaches a, then lim f(x) and lim g(x) both exist when x approaches a"; and other similar statements. This may suggest that the students' examinations of these types of statements are based on "naive" intuitions and are not often based on the search for examples or counterexamples that could test those statements. It was also observed that the students intended to generalize results based on the exploration of a few cases. For example, since (f(x) + g(x))' = f'(x) + g(x)', then they thought that (f(x)g(x))' was (f'(x) g'(x)). That is, in general, the students lacked the strategies for criticizing the validity of mathematical statements. The students struggled when attempting to extend the problems into other more general contexts. It was observed that the students showed resistance when they were asked to explore other applications or linkages of the problems. Perkins and Simmons (1988) stated that "students, however, show little tendency to engage in problem finding and, indeed, conventional schooling offers few opportunities for such activity" (p. 314). Indeed, the students indicated that the ultimate and most important stage when solving problems was to get the solution. They did not show interest in pursuing the problems after they were able to solve them. They were aware that the type of practice that often is required during instruction to solve the problems, but not to go beyond getting the solutions. It is suggested that the students could 185 consider the exploration of other applications of the problems or content, if such activities become a regular part of the instruction and are components of the students' evaluation. For example, the problems should explicitly incorporate questions regarding extensions, applications, or transformations of the problems. The students often relied on the use of an "equation cranking approach" for solving the problems. They pursued some calculations or some data transformations in order to identify or relate an equation that they could fit to the problem. This approach often impeded the students' search for possible qualitative solutions to the problems. The students also showed awareness of strategies that have been useful for tackling problems from textbooks. Perkins and Simmons (1988) suggested that students evolve "naive" notions or straightforward and pragmatic generalizations for solving problems. For example, they often recognize a pattern of solutions embedded in the textbook problems. The textbook problems are organized in such a way that they are similar to the examples given for that section and one can recognize the material and the methods for solving them. In addition, the students' ways for checking the solutions of the problems were based on matching the solutions provided at the end of the textbook with their solutions. They did not use other strategies that could help them test their solutions. Therefore, the students' final target was to get the response given in the textbook. As a consequence, any intention to reflect on the sense of the solutions, to look for other applications of the problem, or to search for possible extensions was often reduced to finding the given solution. This phenomenon also influenced the confidence level of the students about the solution of the problems. For example, the students often asked for 186 the solutions of the problems on the assignments. When working on the assignments that did not include the solutions, they hesitated to discuss the likelihood of their solutions. Koplowitz (1978) stated that "if we are to help the students become better problem solvers we must first help them develop a better sense of when they have solved a problem and a better sense of when a particular method is appropriate to use in a given problem" (p. 307). Developing the students' confidence involves encouraging them to consider ways to evaluate their work and to approach the problems in several ways. Clement and Konold (1989) pointed out that at the beginning of the course the students might use the solutions of the problems given by the instructor as a means to verify their solutions. However, it is necessary to show the students that verifying the solution is part of the process of solving the problem and is, therefore, one of the responsibilities of the problem solver. Chapter Summary This chapter focuses on the analysis of the students' approaches to problem solving by considering information gathered mainly through task-based interviews. The analysis of the first part provides information about the use of mathematical resources, cognitive and metacognitive strategies, and the influence of the students' ideas about mathematics while solving the problems. The discussion leads to the conclusion that the learning activities implemented during the course have begun to influence the ways that the students solve problems. However, they have not developed a consistent and efficient approach in which they could select, explore, and monitor the activities used in their problem solving process. It is suggested that it takes time to overcome some of the students' practices for working on problems. For example, it was observed that the 187 students' first approach to the problems was to get involved in calculations but often without having a clear understanding of the problems; this strategy was inconsistent with the activities developed during class instruction. Although during the problem solving instruction the students were encouraged to explore various alternatives and to monitor their progress when using them, it was observed that the students mainly reflected on what they were doing when they reached an impasse while trying to solve the problems. They did not recognize the need to check their solutions and it was only when the interviewer suggested that it was important to verify and to understand their solutions that they did so. In the second part of this chapter, there was interest in contrasting the findings of the first part with an analysis that focused mainly on the students' difficulties in solving the problems. The results show that in order to help students to improve their ways of solving mathematical problems, it is necessary to pay attention to the mathematical content, cognitive and metacognitive strategies, ways of validating and using mathematical arguments, and ways of extending the problems. When all these ingredients consistently become part of mathematical instruction, then the students may develop their own frames for solving mathematical problems. These frames should be similar to what people in the field of mathematics use while working on mathematical problems. 188 CHAPTER VII CONCLUSIONS, IMPLICATIONS, AND RECOMMENDATIONS Several conclusions that focus on the feasibility of implementing problem solving instruction in regular college mathematics classes emerge from the present study. In addition, a set of conclusions related to the types of expectations that the students have about the course arises from the students' interviews and class observations. Both sets of conclusions are used as a basis for a discussion of the implications of the study and for the presentation of specific recommendations. Finally, a set of recommendations for future research is presented at the end of this chapter. Feasibility of Mathematical Problem Solving Instruction In order to present the conclusions that are related to the feasibility of implementing problem solving instruction, it is important to consider critical events that occurred in the classroom and the analysis of specific cases carried out in the previous chapters. The extent to which a problem solving course could be implemented in a regular college class depends primarily on the degree of consistency in which the main activities associated with this approach are used during the instruction. It is important to indicate that the implementation was also influenced by the particular conditions of the institution. Consistency of Instructional Activities The extent to which several instructional activities were implemented in the classroom is a component directly related to the feasibility of problem solving. The roles of the instructor and students were important in assessing the implementation. 189 The Students' Participation Originally, a main purpose of the course was to implement a problem solving approach in which the students would have the opportunity to participate in discussions involving the process of solving problems. The students could thereby speculate on possible solutions, test examples and counterexamples, and present their ideas about the content involved in the problems. It was observed that the instructor initially encouraged and motivated his students to participate during the class; however, the students' participation decreased notably as the course advanced. Two explanations are related to this phenomenon. One is the extent of the material that needed to be covered during the course, and the second is related to the difficulty of managing a new classroom situation in which there are more variables to consider in order to make classroom decisions. For example, to make the students' participation meaningful required spending time reflecting on the students' ideas and ways to link them to the content being studied. The instructor intended to direct the class discussions towards this goal initially, but, at the same time, he wanted to make sure that the students could identify sequential reasoning while discussing the content or problem. Byers (1984) stated that "there is usually the desire to keep ideas as simple and straightforward as possible within the framework of the course" (p. 35). Unfortunately, this direction was the most popular and desired among the students. It was also a challenge to motivate and show the students that it was worthwhile to spend some time reflecting on and discussing mathematical problems rather than just telling them the right solution. They initially were opposed to working on the problems on their own, but they increased their 190 interest in participating when they realized that they could advance and make more sense of their ideas as a group. The Extension of the Curriculum The instructor was concerned about the amount of material to be covered during the course. His plan included a well organized outline summarizing the material to be covered in each class. This expectation often changed the direction of the class discussion in the sense of focusing primarily on lesson content and ending the discussion by providing solutions to problems. The students also initially showed some reluctance to participate. It was observed that they expected to be told what to do rather than to try to understand and reflect on the problems themselves. This expectation on the part of the students was difficult to change. As a consequence, it is suggested that more work should be done on the selection of the content for class discussion and on the emphasis of the type of interaction between the students and instructor in the class. A conceptual analysis could be useful in identifying the key ideas and their relationships. In addition, more activities that encourage the students to participate in the class are needed. Research studies in problem solving have indicated that experts exhibit various phases while solving problems that include analysis of the information, representation, exploration of possible ways , design of a plan and implementation, and checking and extending the solution. In addition, experts often curtail the process of solving mathematical problems by discriminating and selecting the most likely way to solve the problem (Polya, 1945; Krutetskii, 1976; Schoenfeld, 1985). This phenomenon was observed during the class. The instructor often abbreviated the use of a strategy when discussing a problem with the students. As a result, many of the students 191 experienced difficulties in using it or they could not notice why the strategy did not help to solve the problem. Although the instructor fully intended to provide a problem solving environment in his class, he ended up giving more emphasis to finding the solutions of several problems and showing the students the content involved in the problems than in letting the students express their ideas and discuss the problems on their own. This suggests that it is difficult to overcome a way of teaching that has permeated the mathematics education environment for many years. Even when the instructor showed willingness to incorporate other activities into the classroom, he sometimes presented the solution of a problem or the proof of a relationship on the blackboard to the class without determining whether or not the students were following his presentation. Routine and Nonroutine Problems Schoenfeld (1985) suggested that the design of problems is a crucial component for engaging students in the discussion of problems. The problems should show the students the strengths of the use of the strategies and the importance of considering going beyond the solution. The problems in the assignments incorporated some of these ideas. It was noted that the students responded positively to the nonroutine problems included in the assignments, but they also realized that in order to get ready for the examination they had to practice routine problems. Since all the students wanted to get the best possible mark in the course, there were indications that the students intended to treat nonroutine problems as routine problems. That is, they were only concerned about the right and efficient approach to solving the problems without considering possible linkages or extensions. The students expressed the idea that ultimately the right solution is what counts in the examination 192 and they expected that the goal of instruction should be to prepare them for the exams. It is suggested that the students' concerns about the role of nonroutine problems in the classroom and assignments should be taken into consideration in order to encourage them to participate in their discussions. For example, more credits should be allocated to the process used by the students when working on these problems. The Use of Small Groups Although discussion of problems by small groupd of students during the class was not used throughout the entire instructional period as was expected, there were indications that it is a worthwhile strategy that should be part of the problem solving instructional activities. In addition, the progress shown by the students should be monitored by the group participants and the instructor should check the work on the assignments and provide corresponding feedback incorporating examples and counterexamples in order to challenge or support the students' ideas. In the present study, the instructor recognized the utility of using small groups during the class, and the students expressed the idea that by working in small groups they had the opportunity to discuss their ideas and discover different views about the problems. The role of the instructor was to coordinate activities and to ensure that all the students were participating. In addition, he provided some hints or direction for the discussions when the students experienced difficulties in understanding the problems. The results from the assignments were also discussed during the class and the most common approaches to the problems were discussed. The difficulties shown by the students in the assignments were also discussed. These types of activities encouraged the students to work on the assignments. 193 The Evaluations of Students There is also a need for designing or adopting an instrument to evaluate the progress shown by the students. Schoenfeld (1985) used an evaluation scheme that focused on the evidence and the extent to which an approach is used by the student. The scheme used for evaluating the students' work included aspects such as i) evidence in the use of the approach(es), ii) pursuit, and iii) progress (little, some, almost,and solved). This type of scheme was employed to grade the students' assignments. It is suggested that a scheme which focuses on the qualitative aspects involved in solving the problems could be used as a guide to adjust or construct an instrument that best fits the grading or marking of the problems. The work shown by the students in the assignments was an indicator of how the students were responding to the instruction. The students initially relied on the problems discussed during the class and used them as models to explore the approaches to the problems in the assignments. It was noted that the students' initial approach was to relate the problem to a previous one in which he or she knew how to get the solution. Polya (1945) recommended that the problem solver should relate the problem to simpler or similar problems. In the development of the study, it was observed that the students often became involved in calculations suggested by previous problems even though the structure of the problem was different and the only common components were some terms of the current problem. For example, during the interviews, the students became involved in various calculations that were familiar to them from the class discussions without having a clear picture of their use. It was suggested that the students expected that the problems on which they were asked to work could be solved in the same ways as the 194 examples had been solved. The exercises in the textbook are normally placed in a sequence in which the same content and methods could be used to solve them (Larson & Hosteler, 1987). It was also apparent from the students' work that they lacked confidence in their solutions; their certainty of their work depended on comparing their solutions with the "right" solution found at the end of the textbook. If their solution was the same as the one found in the textbook, then they believed that they had solved the problem correctly. This issue was addressed during instruction by encouraging the students to employ different approaches in order to check the solutions. As a consequence, some students often combined algebraic and geometric approaches in order to check the sense of their solutions, and they showed some increase in confidence in their work. For example, a graphical representation of one of the interview problems that involved the use of derivative could have helped students to check a solution obtained by algebraic means. Nevertheless, it was observed that students often neglected to explore other means in order to verify or check their solutions. The percentage of students who exhibited awareness of the importance of interpreting the sense of the solutions was only about 15%. The Use of Metacognitive and Cognitive Strategies The use of metacognitive strategies was emphasized throughout the development of the study. As a result, the students had the opportunity to discuss the problems, their approaches to them, and the difficulties that they encountered while solving them. This provided information about the extent to which the students were responding to the class activities. This confirmed the importance of encouraging students to discuss the 195 content, the strategies (both cognitive and metacognitive), the validity of the content, and possible extensions to other domains in order to learn mathematics. In addition, the problems should be designed in accordance with the students' interest. For example, the contexts for setting the problems might include information regarding postal rates, bus schedules, or growth rates. The idea is to motivate students by presenting problems in which they could relate to their own experiences, speculate about their solutions, and interpret or make sense of their solutions. Perkins and Simmons (1988) stated that "people learn much of what they have a direct opportunity and some motivation to learn, and little else" (p. 319; emphasis in the original). They also recommended that mathematical instruction should include explicit discussion of the content, problem solving, epistemic, and inquiry frames. A frame is defined as a guide to organizing and supporting thought processes. They stated that "instruction should involve explicit articulation by teachers and/or students of the substance of the frames and their relationships" (p. 321). Therefore, learning activities should include explicit discussion of strategies for studying the content, for problem solving, for validating or judging the content, and for extending the content and strategies into other domains. Instructional Directions The instructor was also aware of dealing with two instructional directions during the course; he recognized the importance of introducing the content via problem solving and consequently the need for spending time discussing nonroutine problems during the class. He also expressed concern about the students' performance on the exams, and he advised the students to solve many routine problems as a means to improving their speed while solving these types of problems in the exams. These two parts of the 196 instruction seemed to be contradictory. The instructor, however, also gave some weight in final grading to the nonroutine problems from the assignments, and the students showed some progress in approaching these problems. The students received feedback on their written work for each of the problems included in the assignments. In addition, there was a set of metacognitive questions to which the students were asked to respond for each of the assignments; the instructor also emphasized from the beginning of the course that the direction of the class was to emphasize problem solving strategies. All these activities influenced the ways in which the students approached problems at the end of the course. For example, there were indications that they considered various ways of solving the problems of the assignments; they often used diagrams to represent problems, and they also checked some of the problems. In general, the students receiving problem solving instruction showed evidence that they were aware of the importance of using different approaches for solving the problems. Although the final exam was not sufficient indication of the students' success in this course, the results showed that they had a higher average grade than the other calculus class and classes from previous years. The instructor also noted that the problem solving class was more motivated to participate and learn about the conceptual part of mathematics than his students had been in previous courses. An important conclusion that emerged from the study is that the problem solving instructor should be interested not only in the use of problem solving as means to teaching mathematics, but should also be convinced that it is possible to use problem solving throughout the entire instructional process of a course. In this study, for example, the instructor was interested in 197 "problem solving", but due to the extension of the curriculum and time limitations the classroom activities sometimes emphasized the transmissive approach to teach the students how to solve problems. As a consequence, it is important that the institution provide conditions in which instructors have flexibility to incorporate learning activities that encourage students to participate and understand mathematics rather than covering an extensive curriculum. Discussing new problems in class means that instructors need to spend extra time in the design and preparation of additional material; hence, the support of the institution is necessary to incorporate these ideas into mathematical instruction. The Students' Expectations The students who took the introductory calculus course had certain expectations about the teaching conditions and the type of evaluation for the course. For example, they were aware of the types of problems that the instructors normally introduced to their classes and of the fact that there was a final common exam that all of the students had to write. Therefore, they expected similar types of examination problems and evaluation. Although there is no explicit coordination among the instructors regarding the types of problems and activities to be used in the development of the course, there is a textbook which is used as a guide for presenting the content which is probably the best indicator for identifying the content to be expected from this course. In addition, the type of problem found in this book normally represents the type of problem included in the exams. Therefore, it is necessary to reconsider the selection of complementary materials, such as the textbook, assignments, and evaluations, in order to provide consistency in the problem solving activities inside and outside the classroom. 198 It is also important to spend some time analyzing the students' ways of solving problems; this information could help the instructor consider activities to help the students devise and construct their own strategies for solving problems and at the same time overcome some of the habits that often impede the consideration of other alternatives. Schoenfeld (1985) stated that students develop a set of beliefs about mathematics and problem solving. He also suggested that the way mathematics is taught plays an important role in shaping what students think about mathematics. DiSessa (1988) stated that "students come to courses with very well-developed prior conceptions that can interfere with the learning" (p. 48). Indeed, the expectations that the students had about the problem solving course was the first obstacle to overcome. For example, they expected to be told a set of rules for solving problems; they also expected the instructor to solve many examples during the class, and they expected to be asked problems on exams and assignments that were similar to those discussed during class. In fact, the students complained about the problems on the assignments that did not fit the "structure" of the class problems. It was observed that the students initially offered resistance to participating in the class discussion. During the first week, the participation of the students was centered on only five students. In addition, the type of interaction was limited mainly to responding to specific questions asked by the instructor, without any exploration of the reasons for those responses. This type of interaction might be considered superficial but could be beneficial if the students' ideas or explanations are also incorporated into the interaction. However, the class dynamics in the development of the study were often only superficial and more forms of class participation were needed during the instruction. In addition, there were some students (three or four) 199 who did not want to participate in the small group discussions. These students argued that it was a waste of time to carry out this type of activity. Thus, it is suggested that the small groups should be formed initially with students who are friends or get along and who do not feel threatened during the development of the discussion. It is also important to rotate the roles of the students while solving the problems; for example, the instructor should make sure that all the students in the small groups are part of the discussion and that all have an opportunity to express their ideas. The role of the instructor should be to monitor the progress shown by the students and to provide some guidance when the students experience difficulties. Summary of Conclusions of the Main Study Schoenfeld (1990) indicated that people initially resist the use of activities that are different from the ones that they normally carry out. The process of assimilating and implementing problem solving strategies in mathematics instruction should be seen as an ongoing process in which there is always room for improvement and adjustment. However, it is imperative to discuss the strengths and limitations of the present study in order to share experiences that could be useful for future research in problem solving. The discussion in this section will focus on basic principles embedded in problem solving instruction, the roles of the students and the instructor, and the development of the study. Emphasis will be placed on future directions for investigation of factors of mathematical instruction that could enhance a problem solving approach. An important challenge that appeared in the implementation of problem solving instruction was maintaining the congruity of the learning activities associated with this approach throughout the course. Kilpatrick 200 (1988) stated that problem solving encompasses views about education, mathematics, and instruction. These views could define different frames or approaches for the use of problem solving and consequently different types of learning activities associated with each approach. Although it might be the case that problem solving instruction is based on a combination of some or all of the above views, it is possible to recognize the direction of the approach by the focus or emphasis given during instruction. For example, initial interaction with the instructor in the present study gave some indication about the conceptualization of problem solving that he thought that he could include in the instruction of the class. This conceptualization included the presentation of several examples to the students in which the use of various solution strategies could be pointed out. It also included providing the students with examples which clearly illustrated the power of reflecting on other possible ways of solving the problems. However, the development of this type of approach showed variations throughout the course. It was observed that classroom activities were often teacher-centered and that there was little room for incorporating students' ideas about the problems into the instruction. This suggested that more work should be done in the planning process and that some collegial support and supervision should be undertaken. This involves discussions of activities and the extent to which they should be implemented. For example, it should be clear that the students' class interaction is an important component that could provide information about the direction of the approach and this information should be considered in the evaluation and adjustments of the learning activities. That is, each activity implemented during problem solving instruction should direct the students explicitly in the use of problem solving strategies and should be 201 monitored and evaluated in order to determine to what extent that strategy is becoming part of the students' frames for solving problems. This is different from the method in which several strategies associated with problem solving are identified and then tried in isolation during the class. This latter phenomenon was observed during the present study. For example, the instructor decided that for a determined period, strategies that involved small group interaction were going to be implemented; for another period, explicit discussion of the strategies involved in the process of solving problems was encouraged. This approach lacked the integration of the components embedded in problem solving. That is, a holistic picture of problem solving which is based on an ongoing implementation of activities that encourage the students to reflect and discuss mathematical ideas throughout the development of instruction was not present. An important component of instruction is open discussions between instructors and researchers about the directions and expectations involved in planning and implementing a problem solving course. Such discussions could include explanation of the rationale for providing a microcosm of mathematical practice in order to learn this discipline. Examples in which the evolution of mathematical ideas could be shown were sources of discussion, for example, Fawcett's work (1938), which dealt with the nature of mathematical proof as a means to engage students in mathematical discussions; Schoenfeld's work (1987), in which he discussed class instructional problems that included the magic square and Pythagorean Theorem; and interviews with experts that showed the development of mathematical ideas (Davis & Hersh, 1981; Hammond,1978; Silver & Metzger, 1989; Wieschenberg, 1984). The purpose of discussing these issues was to help the instructor coordinate classroom activities involving more student 202 interactions during instruction, more attention to the students' ideas, and the consideration of more problems that illustrated mathematical thinking. However, even when there was a substantial change throughout the development of course, there were several activities that were neglected in the instruction. For example, extending problems into different domains, checking solutions by using different means, and evaluating the students' monitoring processes while solving problems were issues that needed more attention. Schoenfeld (1990) recognized that people are hard to change, but the challenge is to continue working in this direction and to value the progress made and discuss what could be improved for the future. It was noted that in order to share a common frame for implementing problem solving activities with the instructor, it was necessary to spend considerable time discussing the basic principles of this approach. For example, the instructor's initial expectation was to implement a well-structured model for teaching mathematics that involved a specific sequence of activities. For example, in the pilot work done prior to the present study, he asked the researcher to teach problem solving in two or three days of classes. It seemed that he thought of problem solving as something additional to be taught to his students instead of as a way to teach mathematics. Hence, the necessity of selecting appropriate readings for discussing basic principles of problem solving is crucial for building a problem solving frame. Implications Results from the present study lead to the conclusion that it is possible to engage students in learning activities that help them reflect on the importance of considering strategies associated with the development of mathematics. However, in order to implement the learning activities 203 embedded in this approach, it is necessary to improve the communication between practitioners and researchers. It is important to identify the practical concerns that could limit the development of the problem solving approach. The use of problem solving involves the selection of problems to be discussed during the class and to be included in the assignments, the selection and implementation of learning activities, and the consideration of ways to evaluate the students' progress throughout the course. All these activities should be discussed among the instructors in order to construct a common framework that could be used as a guide for the introduction of problem solving. The role of the researcher(s) could be to provide some information about the results obtained in studies carried out in similar contexts. For example, Schoenfeld (1985) discussed some aspects that could be used to analyze students' approaches to problems. These aspects include: i) mathematical resources, that is, students' mathematical knowledge that might be used to attack the problem; ii) cognitive strategies, that is, the use of heuristic methods, such as drawing a diagram, breaking the problem into parts, or working backwards; iii) the monitoring process utilized by students while solving problems; and iv) belief systems, that is, the way students conceptualize mathematics and problem solving. These types of findings could be used by instructors as a reference in order to focus and evaluate the direction of their instruction. There were more instructors teaching the calculus course and only one was involved in the study. Even when one instructor could be the starting point in showing that it was worthwhile to try a problem solving approach to present mathematics, it was observed that it could also produce inconsistency in adopting such an approach throughout the course. It is suggested that in order to implement a consistent problem solving approach, it is necessary for 204 the course instructors to act as a group. That is, teaching problem solving should not be approached by an individual instructor; on the contrary, there should be a commitment among the instructors who teach the course to share the activities inherent to this approach. For example, the design of problems, the materials to be used in the classroom, and the learning activities to be undertaken are some components of problem solving instruction that should be discussed among the instructors. They should also periodically discuss the directions of their courses. In other words, teaching problem solving should be a shared task that includes designing learning activities compatible with this approach. For example, the examinations and procedures for evaluating the students' work should be consistent with the activities implemented during the class. As a consequence, the students could focus their attention on all of the activities involved in solving the problems. This is an important component, because students normally compare what they are expected to do with what other students from different groups do. The coordination of problem solving activities could be done among the course instructors. For example, as a result of the present study, the Chairman of the Mathematics Department and other instructors showed interest in the continuation of this type of research in a wider dimension, that is, involving more courses. The purpose would be to implement a problem solving approach in all the sections of the calculus courses. It is also important to encourage instructors to discuss epistemological issues related to problem solving. For example, the work of Hanna (1983) presented a view of mathematics that is related to problem solving. Davis and Hersh (1981) offered a collection of papers in which the history and development of mathematics and some implications for its teaching are discussed. Polya (1945) discussed a rationale for the use of the problem 205 solving approach and provided various examples that could be used in the classroom. The aim in discussing these sources is to build a common framework among the instructors that relies on a view of mathematics that is consistent with problem solving instruction. Some of the instructors at the institution in which the investigation was undertaken are already familiar with some of these ideas; however, they have not tried to implement a complete approach to problem solving. In addition, research studies indicate that a practical proposal for teaching problem solving could take different directions. Schoenfeld (1987) presented an overview of the evolution of ideas on problem solving. For example, he suggested that the strategies discussed by Polya are general and may not help students to improve their approaches to problems because they are broad categories that could generate many more substrategies. He provided several examples in which several strategies emerged from the list presented by Polya. Schoenfeld suggested that instructors should be aware of this generality and provide learning activities that address this issue. Hence, discussion of epistemological bases and possible practical instructional linkages for the use of problem solving could help instructors select the types of learning activities to be implemented in the classroom. Researchers should also participate in the discussion of these types of issues and become familiar with the educational environment at the specific school or institution. There should be a continual dialogue between practitioners and researchers in order to shape the problem solving approach in accordance with the real situation of that institution. As a consequence, the planning and implementation of the problem solving strategies could be approached in accordance with the limitations and opportunities emerging from that context. At the institution in which the study was carried out, there is interest at 206 all levels of the college in doing research that could help improve the students' learning of mathematics. There is a forum to propose and express alternatives in this matter; however, there is a need to focus on a feasible proposal and encourage the instructors to participate as a group. It is in this area that the role of researchers could become important. For example, they could engage the practitioners in a research project that includes specific goals to be achieved during actual practice. It is suggested from the present study that the implementation of a complete problem solving approach in the regular classroom could be done; however, this would require planning that involves the participation of all of the instructors and also the support of the administration. Another issue that arose during the development of the study was the evaluation of the students' work. A grading scheme was considered for evaluating the students' work on the assignments. Although at the beginning of the course the parameters to be evaluated were discussed with the students, they frequently did not show all the ideas used while solving the problems. They emphasized the final solution. For example, in the first assignment, the students failed to write down all the changes of sign while finding the domain of functions. As a consequence, they experienced difficulties constructing the graphical representation of functions. In addition, the students were aware of the percentage assigned to the assignment problems (15%) and this probably decreased their interest in spending time reflecting on and analyzing the problems or writing their ideas. The process of evaluating the students* assignments gave an indication of the types of difficulties that the students were experiencing. It also provided useful information for giving specific feedback on those difficulties. Although it was important to check the students' work carefully, this activity took much 207 time and the instructor probably could not devote so much time to the marking process (the researcher marked all the assignments). It is necessary to consider other alternatives in which the marking process could be shared with more instructors. The students also had expectations about the way that the exams would be designed. They knew that in order to do well in the exams they were required to use skills that could be efficient for solving the type of problem included in the exams. For example, a typical final exam included about ten "problems" to be solved in about two hours. Therefore, the students had to be ready to recall appropriate knowledge and use it quickly. There were no surprises involved in the process of solving the problems and the problems shared the same structure as the problems discussed in class. Consequently, the students directed their attention to the preparation for the exams and some of them refused to get involved in activities that seemed different from what the exams required. Unfortunately, these types of students' ideas were difficult to overcome. It might be the case that in order to engage the students in problem solving thinking throughout instruction it is necessary to consider a radical change in the design of exams and the ways of evaluating the students' work. Recommendations There are indications that the development of the present study has influenced the instructor's and students' ideas about mathematics and problem solving. Although there were several activities that were neglected or were not implemented in the expected way, it was observed that the instructor did incorporate more activities associated with problem solving that encouraged the students to reflect on problem solving processes. Therefore, it 208 is recommended that the instruction of mathematics be based on problem solving. It is also recommended that problem solving activities be planned and implemented in coordination with all the instructors teaching the course. For example, all the instructors teaching the same course could design the problems for the class examples and assignments. They could also meet regularly for discussing the progress of the implementation and any difficulties that may arise during the course development. In addition, the evaluation process should match the problem solving activities. For example, the students should be aware of the types of problems involved in the examinations (nonroutine problems), the type of work that is expected, and the parameters that are going to be evaluated. The structure of the class should incorporate classroom dynamics that include discussions of problems with small groups of students. All the students should participate and take different roles while discussing the problems. The instructor should coordinate the students' interactions and monitor their progress. It is also important to evaluate and share the students' ideas with the entire class. There was evidence that students need to discuss their ideas with other students and the instructor in order to clarify, defend, or use what may help them solve the problem. It was observed that the students often had ideas that were useful to explore the possible solution but they did not know how to use them. It is recommended that problem solving instruction should encourage students to discuss their ideas and work on problems with their classmates. The instructor should also challenge and discuss the students' ideas while solving problems. This type of activity could help students develop a frame for using and judging mathematical ideas (epistemic and 209 inquiry frames). Researchers in the area of problem solving should also work with the instructors on a regular basis. They should attend the meetings and probably suggest readings that could support and relate the ideas involved in a problem solving approach to the actual practice of doing mathematics. Future Research There are several routes that could be explored based on the results discussed in the present study. For example, it is important to involve all instructors teaching the same course in the process of designing and implementing a course based on problem solving. The basic components that should be discussed include the selection of problems for classroom discussions, the incorporation of cognitive and metacognitive strategies in the instruction, the consideration of learning activities in which the students have opportunity to discuss and present their ideas, and a frame for evaluation that provides a qualitative and quantitative analysis of the students' ways of solving problems. An important component in the students' processes of solving problems was the ways they expected to solve the problems. For example, when the solutions were not provided, they were not confident in their solutions. As a consequence, it is suggested that problems that involve strategies in which students have to give evidence to support the solutions or claims and to explain the rationale of using specific means in order to solve them should be incorporated in the instruction. It also should include strategies in which students are required to speculate or transform mathematics ideas in other domains or contexts; statements, such as "what if not", could guide the discussions. 210 More emphasis should be given to the use of metacognitive strategies. There is a need to explore the extent to which metacognitive strategies could help students identify their difficulties related to their mathematical resources or content knowledge. The instructor involved in the study often complained about the students' lack of mathematical resources. Hence, encouraging students to reflect on their strengths and weaknesses while solving problems could be an important component to overcome some of the difficulties. It is important to design mathematical problems that stimulate the students' interest. The problems should also include different levels of sophistication in order to encourage all the students to participate. For example, the students, in general, experienced confusion while working on the interview problems when they tried to apply a well-known solution procedure to the problem but did not show any progress. This type of experience might have blocked the students' interest in searching for other alternatives. Research carried out in problem solving has recognized the need of exploring with more detail the students' thinking while solving mathematical problems (Schoenfeld, 1987c). It is important to take into account the nature of mathematical thinking and its relationships with schooling. For example, the extent to which the nature of schooling provides the conditions for engaging students in problem solving environments that represent the nature of developing mathematics is an issue that needs to be investigated. There is also a need to investigate the transfer of students' mathematical knowledge into other domains. 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Foundation of cognitive theory and research for mathematical problem-solving instruction. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 33-60). Hillsdale, NJ: Lawrence Erlbaum. Simon, H. A. & Chase, W. G. (1977). Skill in chess. In I. L. Janis (Ed.), Current trends in psychology: Reading from American Scientist (pp. 194-211). Los Altos, CA: William Kaufmann. Smith, J. & Heshusius, L. (1986). Closing down the conversation: The end of the quantitative-qualitative debate among educational inquires. Educational Researcher, 75(1), 4-12. Spivak, M. (1980). Calculus. Berkeley, CA: Publish or Perish. Stanic, G. A. & Kilpatrick, J. (1988). Historical perspectives on problem solving in the mathematics curriculum. In R. I. Charles and E. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 82-92). Reston, VA: National Council of Teachers of Mathematics. Steen, L. A. (1983) Mathematics today. In L. A. 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Unpublished doctoral dissertation, Columbia University. Wimbish, J. G. (Ed.). (1972). Reading for mathematics: A humanistic approach. Belmont, CA: Wadsworth. 234 Appendices for the Pilot Study Appendix PI: Questionnaire Name male female Date. Course Instructions: For each of the following questions, please circle the numbered choice which most accurately reflects your response. The responses go from number 0 which corresponds to Never, 1 to Sometimes, 2 to Often, 3 to Very Often, and 4 to Always. Note: If you have any comment about a specific question, please write on the back of these sheets. 1. I like to study mathematics. 2. I am interested in studying mathematics. 3. I use a certain kind of method for solving mathematical problems. 4. I check the solution of each problem. 5. I use diagrams to represent the problem while solving it; 6. I simplify the problem by finding simpler ways of considering it. 7. I go beyond the solution asked in the problem. NEVER 0 0 0 0 0 0 0 ALWAYS 4 235 8. I try to find similarities in the statement of the problem with those of problems that I have solved before. 9. I first check ways which seem to lead to the solution quickly. 10. I explore different ways of approaching the problem before deciding on a specific strategy to follow. 11.1 like to work on mathematics problems to which rules can be directly applied to get the solution. 12. I read the statement of the problem several times before attempting to solve it. 13. I feel my present problem solving strategies are adequate. 14. I distinguish relevant from irrelevant information in the statement of the problem before I attempt to solve it. 15. I believe that previous knowledge in mathematics determines the level of success in solving a problem. 16. I prefer to solve problems for which mathematical proofs are required. 17. I study mathematics simply as a means to passing my mathematics exams. 18. I have difficulties in learning mathematics. 19. I prefer learning how to apply mathematical concepts rather than understanding why they work. 20. When solving mathematical problems, I try to follow the same approach my teacher uses in class. 21. I like to discuss mathematics problems with my classmates. 22. I ask some one for help when I have difficulty solving a mathematics problem. Appendix P2: Written Test Name male female Date Course Instructions: When working on the following problems, please show all the  work you do in order to get or approach the solutions. Write down your ideas, the methods you use and whatever means you employ in the process of getting or approaching the solution. To get the solution is not the only important aspect of the process; what you think, what you use, and how you use certain strategies are also important components. In approaching the problems, you may start working in any order but try to solve as many problems as you can. Problems 1. In Illustration 1, there are five figures drawn. For each figure find its equation, identify which figures are functions, and write down their domains by looking at their graphs. Illustration 1 238 2. Construct the graph of the functions defined by: i) y = Ixl and y = -Ixl; restrict the domain to the interval -3 < x < 3. ii) y = (1/2)(x + lxl). 3. Find the value of x in the equation 5 3 x + 4 = 5X~6. 4. A manager of the Canadian International Bank has found that after x amount of experience, a new employee at the counter can handle N(x) customers per hour, where N(x) = 32 - (24)Vx+T, x>0 i) How many customers can a new employee (no experience) handle per hour? ii) According to this model, does the employee's speed ever stop improving? 5. A manufacturing plant can produce one computer every 30 minutes. The total cost associated with the production of x computers is [C(x) = 30x + 0.02x2]. Write a function C giving the total variable costs resulting from t hours of computer production. 6. A producer of a certain software product sells the product according to a price structure that encourages volume purchases. For orders of up to five copies the price is $500 per copy. On orders of more than five copies the price is reduced by $10 for each copy in excess of five, except that all orders for more than 25 copies are filled at $300 per copy. i) Write a function P giving the price per copy in purchasing x copies of this software; ii) Graph the function. 7. Find: either i) lim (x - 4)/(x2-x-12); or ii) lim (x2 - 7)/(3 + 4x 2) x->4 x - > ° o 8. Find an equation for the line tangent to the graph of f(x) = 9 - x 2 at the point where x = 2. 239 Name: Course: Date: Male: Female: Part A Appendix P3: Application of derivative. Instructions: When working on the next two problems, try to identify the relevant information, write the variables, represent the problem mathematically (symbols), and carry out your plan step by step. Please write down all the means you use for solving the problems. You will have 35 minutes to work on these problems. 1. A computer company employs 50 people to work on an assembly line manufacturing computer components. The average daily production is 150 components per employee. If the number of employees increases, the average daily production (across all the employees) drops, because of crowding and sharing of equipment, by 2 components per new employee. What is the total number of employees the company should have for maximum production? 2. A gardener wishes to enclose a rectangular plot that has one side along a neighbour's property. The fencing costs $4 per metre. The gardener is to pay for the fence along three sides on his own ground and half of that along the property line with the neighbour. What dimensions would give him the least cost if the plot is to contain 1200 squares metres? 240 PartB Name: Course: Date: Male; Female: You have 20 minutes to answer the following questions, i) Regarding problem NUMBER ONE 1. What are the important data that helped you to solve the problem 2. What are the variables? 3. How can the problem be represented geometrically? 4. Restate the problem using your own words. 5. What was the most difficult part for you in solving this problem? 6. What aspects of the problem would you have liked to have been stated differently? 7. State a problem related to but more general than the one you just solved 8. What do you think you need in order to improve your strategies for solving this type of problems? 242 PartB Name: Course: Date: Male; Female: You have 20 minutes to answer the following questions, ii) Regarding problem NUMBER TWO 1. What are the important data that helped you to solve the problem? 2. What are the variables? 3. How can the problem be represented geometrically? 4. Restate the problem using your own words. 5. What was the most difficult part for you in solving this problem? 243 6. What aspects of the problem would you have liked to have been stated differently? 7. State a problem related to but more general than the one you just solved. 8. What do you think you need in. order to improve your strategies for solving this type problems? 244 Appendix P4: Interview Problems for the Pilot Study Problem #1 A hemispherical bowl with a radius of 20 cm contains water to a depth of 4 cm. Through what angle must the bowl be tilted before water spills? Problem # 2 a) The sum of two numbers is 28; the product of the numbers is 7. Find the sum of the reciprocal of the numbers b) Find the value(s) of (x + y) if x 2 + y 2 = 36 and xy = -10 Problem #3 The manager of a manufacturing company is interested in designing an oil can in the shape of a right cylinder. The cost of the material used in making the top and bottom of the can is 2c/cm 2 while the cost of the material used in making the sides is 1 c/cm . a) If r and h represent the radius and height of the can respectively, write an expression for the total cost of the metal used in manufacturing it. b) Assuming the can is supposed to hold 1000 cm 3 of oil, re-express the cost of the metal used in terms of the variable r alone c) Sketch a graph of your cost formula C(r) d) Estimate the dimensions of the can of least cost. 245 Appendices for the Main Study Appendix M l : A Vee Diagram of the Study Conceptual Focus Questions Methodological P h i l o s o p h y : i) Mathematical concepts and theories rise from obtaining the solutions of concrete problems. ii) Meaningful learning takes place in a problem solving environment. Pr inc ip les: i) Students are able to develop mathematics when instruction emphasizes open discussion and criticism of mathematical ideas. ii) Problem solving provides the conditions for the learning of mathematics. ii) Solving a mathematical problem is the beginning of finding alternate solutions of the problem that extend the scope of the problem1 iii) Learning to think mathematically means coming to see mathematics as a vehicle for sense-making and knowing how to use mathematics to make sense of things. Theory : The learning of mathematics takes place from the discussion of simple ideas and constant improvement through criticism and discussion of examples and counterexamples. The process of discussing ideas (solving problems) is influenced by cognitive resources, heuristic and metacognitive strategies, and belief systems. C o n c e p t s : mathematical problem, problem solving, mathematical epistemology, meaningful learning, heuristic and metacognitive strategies, qualitative differences. 1. What qualitative differences do students show in solving mathematical problems after receiving math-instruction via problem solving? 2. To what, extent does the instructor provide the conditions for problem solving instruction? Value claims: The study of mathematics is important for understanding social changes, such as, technological development and environmental and economic problems. As a consequence, the process of making decisions about the directions of society and their function might consider more alternatives Knowledge Claims: Mathematical instruction based on problem solving helps students develop and apply mathematical content. As a consequence, students may be more motivated and interested in studying mathematics. Transformations: Analysis of task-based interviews, questionnaire, assignments, pretest and posttest will identify to what extent students receiving problem solving instruction are able to use specific strategies for solving problems. Qualitative differences shown by the students when solving problems will be categorized and related to mathematics instruction. Records: class observations, pretest, posttest, students' task-based interviews, questionnaires, class assignments, and instructor's interview. E v e n t s : College students receiving calculus instruction based on problem solving are the subjects of the study. Mathematics problems are the vehicle to discuss mathematical content. Students will play an active role in solving mathematical problems. The use of cognitive (heuristic methods) and metacognitive strategies (control) will be part of the instruction. 246 Name. Date Appendix M2: The Questionnaire male female Course Instructions: For each of the following questions, please circle the numbered choice which most accurately reflects your response. The responses go from number 0 which corresponds to Never, 1 to Sometimes, 2 to Often, 3 to Very Often, and 4 to Always. Note: If you have any comment about a specific question, please write on the back of these sheets. 1. I restate mathematical problems using my own words. 2. I experience difficulty recalling information from my past courses that I need to solve mathematical problems. 3. I easily give up on the problem if I do not see any progress after five minutes. 4. I check the solution of each problem. 5. I use diagrams to represent the problem while solving it. 6. I simplify the problem by finding simpler ways of considering it. 7. I go beyond the solution asked for in the problem. 8. I try to find similarities in the statement of the problem with those of problems that I have solved before. 9. I first check ways which seem to lead to the solution quickly. 10. I explore different ways of approaching the problem before deciding on a specific strategy to follow. NEVER ALWAYS 247 NEVER ALWAYS 11. I like to work on mathematical problems to which rules can be directly applied to get the solution. 12. I read the statement of the problem several times before attempting to solve it. 13. I feel my present problem solving strategies are adequate. 14. I distinguish relevant from irrelevant information in the statement of the problem before I attempt to solve it. 15. I believe that previous knowledge in mathematics determines the level of success in solving a problem. 16. I prefer to solve problems for which mathematical proofs are required. 17. I study mathematics simply as a means to passing my mathematics exams. 18. I have difficulties in learning mathematics. 19. I prefer learning how to apply mathematical concepts rather than understanding why they work. 20. When solving mathematical problems, I try to follow the same approach my teacher uses in class. 21. I like to discuss mathematics problems with my classmates. 22. I ask someone for help when I have difficulty solving a mathematics problem. 248 NEVER ALWAYS 23. I summarize the data given in the problem by making a table, graph, or any other representation. 0 1 2 3 4 24. I estimate the final answer to the problem before I implement ways to approach the solution. 0 1 2 3 4 25. I search for patterns by considering familiar cases. 0 1 2 3 4 26. I use trial and error when looking for a way to find the solution of the problem. 0 1 2 3 4 27. I check whether or not I have used all the given information in obtaining the solution. 0 1 2 3 4 28. I check whether or not the solution seems reasonable to me. 0 1 2 3 4 29. I feel frustrated when I cannot use the mathematical knowledge I have to obtain the solution of a problem. 0 1 2 3 4 30. After I have found the solution of the problem, I think of other problems that can be solved similarly. 0 1 2 3 4 31. I feel uncomfortable when I am asked to solve mathematical problems. 0 1 2 3 4 32. I design a well-structured plan before I work on the solution of a problem. 0 1 2 3 4 33. I check step by step when carrying out the plan to reach the solution. 0 1 2 3 4 34. If I cannot solve the problem in about five to ten minutes, I easily realize that the initial strategy I selected may not be adequate and I immediately explore other alternatives for solving the problem. 0 1 2 3 4 249 NEVER ALWAYS 35. I like solving nonroutine problems in which I have to use several concepts and spend some time searching for the solution. 36. I consider a proper order of steps to carry out operations that lead to the solution of the problem. 37. I reflect on the method(s) I used to solve the problem after solving it. 38. I easily go back to the original form of the problem after having considered special cases in solving it. 39. When carrying out a plan to get the solution of a problem, I use knowledge I have gained from my current mathematics course. 40. If I require help to solve a problem, I reflect on why that problem is difficult for me. 41. I feel my instructor should discuss more problems in which students actively participate in the process of obtaining the solutions. 42. I feel students should give suggestions on how to write problems. 43. I find that the wording of problems makes the problems more difficult. 44. I go back to the statement of the problem when the original strategy I selected does not seem to be leading me to the solution of the problem. 45. I persist in trying to solve nonroutine problems even when they seem to be difficult for me. 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 250 Appendix M 3 : Examples of Nonroutine Problems Included in the Assignments 1. In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle? 2. Prove that if P is an interior point of a equilateral triangle ABC, and Pa, Pb, and Pc are perpendiculars to AB, BC, and AC, respectively, Then the altitude of AABC is the sum of the perpendiculars Pa, Pb, and Pc. 3. Find (x + y) if x 2 + y 2 = 36 and xy = -10 4. Find values of a and b so that the line 2x + 3y = a is tangent to the 2 graph of f(x) = bx at the point where x = 3. 5. The legs of a right triangle are initially 5 inches and 12 inches long. If the short leg is increasing uniformly at the rate of 1 inch per second, and the long leg is decreasing at the rate of 2 inches per second, how fast is the area changing when the triangle is isosceles? 6. Suppose p, q, r, and s are positive real numbers. Prove the inequality (p 2 + 1)(q2 + 1)(r2 + 1)(s2 + 1) pqrs 7. Is there any a so that 2x^ — 3sx | x ~ 3 ~ 1 lim ( 2^ 2X 3 ) exists when x approaches to three? 251 8. One leg of a right triangle begins to increase at the rate of 2 inches per minute while the other leg remains at 8 inches. How fast is the hypotenuse increasing when the first leg is 6 inches? 9. Prove that the rectangle of largest area that can be inscribed in any triangle has area equal to one-half the area of the triangle. 10. Assume that the ideal cigarrette contains 9 cm 3 of tobacco. What should be the dimensions of the cigarrete be in order to use the least amount of paper? 11. A truck is travelling north along Main St. at a rate of 13 m/s. Just before the truck comes to the light at 49th, the light turns green and a car waiting there accelerates at a rate of 14 m/s 2 also travelling nort. At the same instant the truck now accelerates at a rate of 10 m/s 2. Under these conditions the distance travelled by the truck and car respectively (measured in meters north of 49th) in t seconds after the truck goes through the intersection will be given by T(t) = 5 t 2 + 1 3t and C(t) = 7t2 +18. a) Determine the rate of change of the distance between the vehicles one second after the truck goes through the intersection and then three seconds after the truck goes throuhg the intersection. b) Describe the significance of the results from parts a) c) As in part a) but for times of four seconds and five seconds 252 d) Describe the significance of the results from parts c) e) Determine when rate of change of the distance between the two vehicles equals zero. What is the significance of this instant of time? 12. Discuss the differentiability of the function given below. x - x 2 if x<0 f(x) x if 0<x<1 I l2x-1l l if 1<x<2 13. The postage rate for airmail letters within Canada is given in the following table: Postal Rates weight: up to and including 30g 50g 100g 200g 300g 400g 500 g mail rate: $0.38 $0.59 $0.76 $1.14 $1.52 $1.90 $2.28 a) Write a function "AMP" (airmail postage) which determines the cost of sending a letter according to its weight. b) Sketch the graph of the "AMP" function. c) Find the domain and range of the "AMP" function. 14. A rectangle with altitude x is inscribed in a triangle ABC with the base b and altitude h. 253 a) Express the perimeter P and area S of the rectangle as a function of x. b) If the triangle ABC has 4 cm as its base and 3 cm as its altitude, "estimate" which of the inscribed rectangles in the triangle has the largest area. 15. Complete the last line of the following polarity chart and determine an algebraic expression which represents that chart. Is that the only algebraic expression for this chart? (Explain why or why not.) - 7 - 6 - 5 - 4 -3 - 2 - 1 0 1 2 3 4 5 6 7 • i | i | i i i i | | | i | • - ; + + + + + + + + + • + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +++++++++ J • + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + s • II • • • + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + • • U . J . -L. • + + +  i • I I I • • II • • • II • nd 6 nd 6 6 6 16. Write a polynomial which is of degree 3 and whose roots are 3, 4, and 0. Sketch its graphical representation. 1 7. A student must have an average of 80% to 90% on five tests in his calculus course to receive a mark of B. His grades on the first four tests were 98%, 76%, 86%, and 92%. What grade on the fifth test would give him a B in the course? 254 18. Let f be a real function. Represent graphically and explain the meaning of: i) f(x + 1) - f(x); ii) f(b) - f(a); iii) f(x + h) - f(x) j v ) f (cx) . v ) f c h ) . v i ) f (b ) - f (a ) I V ' f(x)' v ) f(x) ' V l ) (b-a) 19. Prove that the fuction x 2 + 1 at -1 < x < 0 f(x) = - x at 0 < x < 1 is discontinues at the point x = 0 and still has the maximum and minimum value on [-1,1]. 20. Discuss the following questions: i) If lim f(x) and lim g(x) do not exist when x approaches a, can lim [f(x) + g(x)] exist when x approaches a? ii) If lim f(x) exists and lim[f(x) + g(x)] exists when x approaches a, must lim g(x) exist when x approaches a? iii) If lim f(x) exists and lim f(x) g(x) exists when x approaches a, does it follow that lim g(x) exists when x approaches a. 21. Discuss the following questions: i) If f + g is differentiable at a, are f and g necessarily differenciable at a? (explain) ii) If f»g and f are differentiable at a, what conditions of f imply that g is differentiable at a? 255 22. Let f(x) = ax, if x<1 bx + x + 1, when x>1. Find a and b so that f is differentiable at 1 256 Appendix M4: Metacognitive Assignments Name Male female Date course Introduction: Throughout the development of this course you will be asked to reflect overtly on the way you solve mathematical problems. The course is oriented to emphasize problem solving activities. Several strategies will be discussed during instruction. It is expected that if you are aware of the basic stages involved in the processes of solving problems, that is, understanding the statement of the problem, devising a plan, implementing or carrying out the plan, and checking the solution, then you may develop a more systematic and effective approach to problem solving. As a consequence, you may recognize more easily the structure of a problem and apply better strategies for solving it. The purpose of the following questions is to gather information regarding your approaches to problem solving. It is important that you spend some time responding to these questions. You will receive some feedback about what may help you improve your ways for solving problems. 1. What problem (assignment*?) was the most difficult for you? What was the easiest one? Explain both answers. 2. What strategy(ies) did you use in solving each problem in the assignment? (an algorithm, breaking the problem into parts, using similar problems illustrated in class, trial and error, or others). Mention the specific problem and the strategy(ies) used. 3. Could you identify the key concepts that you used when solving the problems? Try to draw a diagram about how you think these concepts are related. 4. Did you check the plausibility of each solution, that is, whether the solution made sense to you? Was the solution something that you were expecting? (Explain). 5. Did you extend any of the problems into other more general cases or other contexts. Give examples if you did. 6. What type of activities do you think should be emphasized more during class that may help to improve your problem solving approaches? 7. Please write any comment(s) that you consider important in addition to these questions. 257 Second Form(This form replaced the first version) Additional Questions (Metacognitive Questions) 1. Which problem (assignment*?) was the most difficult for you? What was the easiest one? Explain both answers. 2. What strategy(ies) did you use in solving each problem in the assignment? (Applying an algorithm, breaking the problem into parts, using similar problems illustrated in class, trial and error, or others). Mention the specific problem and the strategy(ies) used. 3. As you might have noticed in solving mathematical problems, you often need to monitor and evaluate the entire process when working on the problems. Questions such as i) What am I doing now? ii) Is it getting anywhere?, and iii) What could I be doing instead? may help you to make decisions about what strategy to use or when you should change or use a different strategy. To what extent did you reflect on these questions when working on the assignment? (you may give some examples) 4. Did you check the plausibility of each solution, that is, whether the solution made sense to you? Did you check your calculations? Was the solution something that you were expecting? (Explain). 5. Did you extend any of the problems into other more general cases or other contexts. Give examples if you did. If not, please describe the difficulties you encounter in undertaking this activity. 6. What type of activities do you think should be emphasized more during class that may help to improve your problem solving approaches? Please be specific, for example, more examples from the instructor, more problems solved by the students during class, etc. 7. Please write any comment(s) that you consider important in addition to these questions. 258 Appendix M5: The Written Test 1. According to Einstein's Theory of Relativity, an object's speed V and its mass m, are related by the equation rest (i.e. rest mass). a) What is an object's speed when its mass has doubled? b) Can an object's speed be arbitrarily large? (why or why not?) 2. Is there an a so that (2 x 2 - 3 a x + x - a - l | x 2 - 2x -3 J x-* 3 exists? justify your answer. 3. a) Given 2 X = 8 y + 1 and 9 y = 3 X " 9 find the value of x + y. b) If the reciprocal of x + 1 is x - 1 , then x = ? 4. Two jumping ants start at the origin (i.e. the intersection of the two axes x and y). One travels along the positive half of the x-axis, while the other travels along the positive half of the y-axis. Each ant jumps one unit on its first jump, a 1/2 unit on its second jump, a 1/4 unit on its third jump, and so on; on any subsequent jump, the ant jumps only half as far as on the immediately preceding jump. How far apart will the ants be, assuming that their moves are frequent and take place over an infinite amount of time? 5. A bank manager has found that a new teller's efficiency N, measured in the number of customers per hour, and her experience x, measured in the number of hours are related by the equation Jx + l a) How many customers can a brand new teller handle in an hour? b) Can the teller's efficiency get arbitrarily large? (why or why not?) 6. When Jane mailed a letter, postal rates for a first-class letter were 20c for the first ounce or fraction thereof, and 17c for each additional ounce or fraction. If Jane spent $1.75 to mail a first-class letter, what was the weight of the letter? 259 Appendix M6: Questions for the Instructor Interview 1. Which aspects of mathematics would you mention in responding to the question: what is mathematics? 2. What classroom strategies should be emphasized in order to engage students in thinking of these aspects of mathematics? 3. What is the role of problem solving in learning mathematics? 4. How would you characterize or describe the use of problem solving in your class? 5. What type of problems did you select for your calculus class examples? 6. What expectation did you have of your students before you initiated your calculus class? 7. Which criteria did you use for designing your assignments? 8. In the interviews I carried out, some students expressed concerns about the types of problems that they should be solving during class and assignments. For example, some of the students mentioned that in the midterm and final exam they are asked to work on 10 problems in about two hours, that is an average of 12 minutes for each problem. Therefore, what is the purpose of the discussion of nonroutine problems during class or homework if at the end they are asked to spend no more than 15 minutes on each problem? 9. What is the role of the textbook in your class? Did you participate in deciding which one to choose? 10. To what extent does the length of the curriculum (material to be covered) restrict the activities that you plan to implement? 11. What would be an indication of students not understanding specific content when you are teaching your class? 12. Are there any constraints that impede the implementation of activities in your class in the way that you would like to carry them out? 260 13. Do you think that the exams and assignments accurately evaluate the students' understanding of mathematics? 14. How do you feel about the amount of time allocated to the class? How do you feel about the number of students in the class? 15. To what extent do you think the students' preconceptions (or ideas) of a specific concept influence their learning of that concept? 16. What do you think of the use of small groups in class when discussing mathematical problems? 17. In your calculus course, could you think of some concepts with which the students experience difficulty? What did you do to facilitate the learning of these concepts? 18. Could you summarize to what extent your actual practice as an instructor has changed in the last two years?. What are the major changes? What do you see for the future? 261 Appendix M7: First Interview Letter: Mathematical Problem Solving The purpose of the study is to investigate to what extent the students are able to use various strategies for solving mathematical problems. By participating in the study, the students will receive direct feedback about the types of methods they use, as well as their strengths and limitations when solving mathematical problems. As a consequence, the students may improve their ways of solving problems. The study includes several phases. One consists of asking individual students to reflect on the process of solving mathematical problems. If you agree to participate in this phase, you will be asked to provide some information about how you solve mathematical problems. You will be asked to solve three problems similar to the problems studied in the course M 153. The objective of this task will be to explore the use of strategies, the use of the mathematical content previously studied, flexibility in approaching the problems, and your ideas about mathematics and problem solving. The estimated duration of this task is about 1 hour. You will be asked to think aloud while solving the problems. Throughout the development of this task you may be asked clarification questions and/or questions probing your ways of thinking. This task will be recorded and will contribute to the data to be analyzed for the study. The result of this task will not affect your marks for the course. You can withdraw from this task at any time without any consequence to your class standing. There is no remuneration available for your efforts; however, addressing the issues questioned could very well be beneficial to your problem solving skills. The use of the data will be confidential, and at no time during the analysis or presentation of the results will the data be associated with the actual names of the individual participants. The data will be destroyed after the culmination of the study. If you are interested in volunteering for this part of the study, please write your name, telephone number, and the possible day(s) that you would like to participate. With that information, I will arrange the specific day and time. The room for this task interview will be A348b. Thank you in advance for your input into this study. Sincerely, Manuel Santos. NOTE: IF YOU DECIDE TO PARTICIPATE IN THIS TASK, YOU WILL BE INTERVIEWED ONLY ONCE FOR APPROXIMATELY 1 HOUR. HOWEVER, IT WOULD BE USEFUL FOR PROGRAMMING THE SCHEDULE IF YOU INDICATE SEVERAL POSSIBILITIES. THE INTERVIEWS ARE GOING TO BE HELD FROM xxxx TO xxxxx FROM 9:30 TO 4:30. NAME PHONE NUMBER DAYS AND TIME AVAILABLE 262 Appendix M8: Second Interview Letter November 16, 1989 Dear XXXXXX: Thank you for agreeing to participate in the interview. As was mentioned in the invitation letter, you will work on two problems similar to the ones studied in your current calculus course. When working on the problems, you will think aloud and I may ask you some clarifying or probing questions. In the last part of this interview, you will be asked to express your views about mathematics and problem solving. You do not need to study for this interview; what counts is the process of working on these problems. Moreover, this task will not affect your grades in the course. In doing this exercise, you will contribute to the data for a research study of which the purpose is to investigate how college students approach mathematical problems. Since other students will be interviewed in the next two weeks, it is very important that you do not talk to your classmates about the problems at least for this period (your collaboration is appreciated). I will give you a thorough analysis of the solution after the completion of the interviews. I will also give some feedback and advice on your approaches to problem solving. For this task you are scheduled on XXXXX, November XX. from XXX to  XXX in room A348b which is adjacent to the room where you take your calculus course. Thank you again for your participation. I am looking forward to seeing you on XXXXX. Sincerely, Manuel Santos. 263 Appendix M9: Introduction to the Interview This may be the first time that you are asked to think aloud when working on mathematical problems. The assumption in asking you to use this technique (thinking aloud) is that during the process of solving mathematical problems students often do not show diverse strategies or ideas that they think of and that probably are essential to the success in solving mathematical problems. Therefore, it is important to investigate your process of solving problems in order to design learning activities that best fit with your approach to problem solving. In using this technique, there are some points that you should consider when thinking aloud; they may help you to describe your ideas in an efficient way: 1. It is important that you describe your thinking in a natural way. That is, you should not explain to me what you are doing but just say what you are doing. 2. Try to speak continuously, do not elaborate your ideas or think whether they make sense to me, just keep talking even to say that you are drawing a blank. 3. I will be recording your ideas, then it is important that you speak loudly. 4. Do not worry about completing your sentence, what is important is to identify what your ideas are and what decisions you make while solving the problem. 5. This task should be a relaxing experience. Remember you will not be evaluated; what counts is the process when working on the problems. 264 Appendix M10: Problems for the interview Problem #1 Find values of a and b so that the line 2x + 3y = a is tangent to the graph of f(x) = bx 2 at the point where x = 3. Problem #2 Find all rectangles with integer sides whose area and perimeter are numerically equal. Appendix M i l : Organization of the Interview The identification of the key terms involved in the problem should be the initial point of the process of solving the problems. For problem #1, it is important to recognize the connections among the line, the slope of the line, the meaning of "tangent to the parabola in general and on a specific point", and the roles of the constants a and b. If a student shows serious difficulties in approaching this problem, I may suggest representing the problem geometrically by considering some values of a and b. Questions such as, "What is the slope of the line?", "What is a tangent of the parabola?", and "How are they related?" may be asked to guide the interview. The second phase may include the exploration of a possible plan by focusing on the application of the specific concepts identified in the initial stage. For example, questions such as, "What is the relationship among the slope, tangent, and derivative?", and "Is there any point in common between the parabola and the line?" may help the student to design a plan for solving the problem. The last phase when solving the problem will be to ask the student to reflect on the sense of the solution, to check the stages involved in the entire process, and to summarize what he or she considers to be the important events when solving the problem. For the second problem, the student may be asked to represent the function geometrically and identify the point of interest on the graph. Then the idea of limit should be explored; questions such as, "What is the limit of f(x) when x approaches the given point?" and "How is that limit related to the derivative?" may help the student approach the problem. For the last problem, it is important to clarify the type of rectangle of interest, that is, the one with integer sides and then to find a strategy for 266 comparing the area and perimeter of various rectangles. For example, the use of a table may be recommended for exploring some examples. In this problem, it is necessary to consider a systematic way for exploring the relationship between the area and perimeter of various examples. Appendix Ml2: Second Part of the Interview The students will be asked to elaborate on their conceptualizations of mathematics and problem solving. Some of the questions that may be considered in the second part of the interview are: 1. When I mention mathematics, what do you think of? Has the mathematics that you have studied been useful to you? 2. What type of problems do you like most to solve? What do you think about your mathematics tests? Do they accurately reflect your understanding of mathematics? 3. What do you think about your calculus class? Is it different from your previous mathematics courses? 4. What is your opinion about developing new mathematics? Can the students in the normal classroom create mathematics? or Is all mathematical content already developed and the students just have to learn it? 5. What is your idea about mathematical proofs? Are they important for studying mathematics? 6. When do you know that you understand something in mathematics? How important is memorizing in the process of learning mathematics? How confident do you feel about the mathematics that you have studied? 7. Could you describe your ideal mathematics teacher? 8. How do you feel about this interview? 267 Appendix Ml3: Solutions to the Interview Problems November 27, 1989 Dear , Thank you for having collaborated in the interviews. As I promised I have been reviewing them and have started to analyze them. Here are some initial comments on some of your approaches to the problems. For problem #1, I have observed that all of you read the problem quickly and recognized the term tangent as being associated to derivative. This recognition led you immediately to calculate the derivative of f(x). However, many of you did not know what to do with that derivative. It seems that in the initial phase some of you did not understand the statement of the problem completely. It is important to spend some time making sense of or analyzing the information given in the problem before you become engaged in calculations. It is recommended to explore particular cases in order to understand the problem. In this example, you could have given some values to a and b in order to have had an idea about which values you wanted to find. It is useful to work on the solution having an approximate picture of what you want to get. It was clear that all of you knew the content involved in the problem; however, some of you hesitated in approaching the problem, because the problem involved more variables than you may be used to finding in other problems. My advice here is that you should be more confident in your knowledge and not feel threatened by the statement of the problem. Guesses, approximations, and speculations are important activities that you should not hesitate to use in order to understand, criticize, or solve the problem. Some of you have received some additional questions for each of the assignments. The purpose is to make you reflect and consider the entire process while solving problems. Questions such as Where am I going?, Why am I doing this?, and What am I expecting from this? are useful for monitoring your process. I observed that it was difficult for you to shift from one strategy to another even when it seemed clear that you were going no where. Going back to check and to make sense of the solution of the problem should be an important aspect to consider for each problem. Checking does not mean only revising whether or not there is a mistake in your calculations but also determining the plausibility of your solution. For example, in problem #1 the graphic representation could give you an idea about whether or not 268 your solution makes sense; that is, go back to the statement of the problem and check the data, your solution and the sense in that specific context. The message here is that getting the solution does not imply that you have finished solving the problem. For problem #2, the major difficulty was to find a way for deciding what values of the sides of the rectangle made the area and perimeter numerically equal. I notice that you do not feel confident in using trial and error or particular cases to explore the relation that you got in solving this problem. You should be aware of the extensive use of these types of strategies by experts while solving or proving mathematical relationships. The key is that their use should be in a systematic way. For example, in problem #2, a table might have helped you decide what values you were looking for. Another approach could be transforming the expression you got into a simpler one. As a final comment you may consider working on your assignments and other problems together with your classmates. It is important to discuss, argue, or talk about the way you think of the problems. Research studies have shown that these types of activities improve students ways of solving problems and help them to recall mathematical knowledge studied previously. Thank you again for you time, and I wish you the best in your future studies. Sincerely, Manuel Santos. 269 Solutions to the Problems: Problem #1 Find values of a and b so that the line 2x + 3y = a is tangent to the 2 graph of f(x) = bx at the point where x = 3. Solution The equation 2x + 3y = a represents a family of lines having the same slope m = -2/3. The slope is obtained by transforming the equation into y = (-2/3)x + a/2; in which a/2 in the intersection of the line with the y-axis. The 2 expression f(x) = bx represents a family of parabolas with the vertex on the origin (0,0); the sign of b will determine the direction in which the parabola opens (up or down); the value of b determines the extent of the openness. Any tangent to f(x) at x will have 2bx as the slope (the interpretation of the derivative). Therefore, this slope should be the same as the one of the tangent line given, that is (-2/3). The point required is x = 3, so 2b(3) = -2/3 from which b = -1/9. At x = 3 and b = -119, we get y = f(3) = -1; since the tangent line intercepts f on (3, -1), then substituting the values of x = 3 and y = -1 in the line we obtain a = 3. Therefore, the values a = 3 and b = -1/9 give the line 2x + 3y = 1 and 2 f(x) = (-1/9)x . To check the sense of the solution we graph both equations. 270 Problem #2 Find all rectangles with integer sides whose area and perimeter are numerically equal. First approach If a and b represent the integer sides of the rectangles, then A = a x b & P = 2a + 2b. Since A must be equal to P, then axb = 2a + 2b, now isolating a we get a = (2b)/b-2. The problem now is to decide for which integer values of b we get a integer. 2b/b-2 can be written as 1 + (b + 2)/(b - 2). In order for this last expression to be an integer (b + 2)/(b - 2) must be an integer. We observe that the candidates must be greater than 2. By trying 3, 4, 5, 6, 7, 8, we observe that 3, 4, and 6 produce an integer for a, and after 7 it is impossible to get another integer for a because the difference between the numerator and denominator is a constant (4). Second Approach:  L W P A L W P A 1 1 4 1 3 4 14 12 1 2 6 2 3 5 16 15 1 3 8 3 3 6 18 18 1 4 10 4 4 4 16 16 2 2 8 4 5 5 20 25 2 3 10 6 5 6 22 30 3 3 12 9 5 7 24 35 Since the area is increasing more rapidly than the perimeter, we need go no further. 271 

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