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Influences of metacognition-based teaching and teaching via problem solving on students’ beliefs about… Gooya, Zahra 1992

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INFLUENCES OF METACOGNITION-BASED TEACHING AND TEACHING VIAPROBLEM SOLVING ON STUDENTS' BELIEFS ABOUT MATHEMATICS ANDMATHEMATICAL PROBLEM SOLVINGbyZahra GooyaB.Sc. (Mathematics) University of Massachusetts in Boston, 1977M.A. (Mathematics Education) University of British Columbia, 1988A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESCURRICULUM AND INSTRUCTIONWe accept this dissertation as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIANovember 22, 1992Copyright Zahra Gooya, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of 412_114=r1:c.>_a_Liisc,LnceThe University of British ColumbiaVancouver, CanadaDate Dec e_ rr, )9er 2 '3 ; 1 (1'12 DE-6 (2/88)ABSTRACTThe aim of the present study was to investigate the effect of metacognition-basedteaching and teaching mathematics via problem solving on students' understanding ofmathematics, and the ways in which the students' beliefs about themselves as doers andlearners of mathematics and about mathematics and mathematical problem solving wereinfluenced by the instruction. The 60 hours of instruction occurred in the context of aday-to-day mathematics course for undergraduate non-science students, and that gave mea chance to teach mathematics via problem solving. Metacognitive strategies that wereincluded in the instruction contributed to the students' mathematical learning in variousways. The instruction used journal writing, small groups, and whole-class discussions asthree different but interrelated strategies that focused on metacognition.Data for the study were collected through four different sources, namely quizzesand assignments (including the final exam), interviews, the instructor's and the students'autobiographies and journals, and class observations (field notes, audio and video tapes).Journal writing served as a communication channel between the students:and the-instructor, and as a result facilitated the individualization of instruction. Journal writingprovided the opportunity for the students to clarify their thinking and become morereflective. Small groups proved to be an essential component of the instruction. Thestudents learned to assess and monitor their work and to make appropriate decisions byworking cooperatively and discussing the problems with each other. Whole-classdiscussions raised the students' awareness about their strengths and weaknesses. Thediscussions also helped students to a great extent become better decision makers.Three categories of students labeled traditionalists, incrementalists, and innovators,emerged from the study. Nine students, who rejected the new approach to teaching andlearning mathematics were categorized as traditionalists. The traditionalists liked to betold what to do by the teacher. However, they liked working in small groups and usingmanipulative materials. The twelve incrementalists were characterized as those whopropose to have balanced instruction in which journal writing was a worthwhile activity,group work was a requirement, and whole-class discussions were preferred for clarifyingconcepts and problems more than for generating and developing new ideas. The nineteenother students were categorized as innovators, those who welcomed the new approachand utilized it and preferred it. For them, journal writing played a major role in enhancingand communicating the ideas. Working in small groups seemed inevitable, and whole-class discussions were a necessity to help them with the meaning-making processes.The incrementalists and the innovators gradually changed their beliefs aboutmathematics from viewing it as objective, boring, lifeless, and unrelated to their real-lives,to seeing it as subjective, fun, meaningful, and connected to their day-to-day living. Thefindings of the study further indicated that most of the incrementalists and the innovatorschanged their views about mathematical problem solving from seeing it as the applicationof certain rules and formulas to viewing it as a meaning-making process of creation andconstruction of knowledge.iiTABLE OF CONTENTSABSTRACT^TABLE OF CONTENTS^ iiiACKNOWLEDGMENTINTRODUCTION.^Overview of the study^ 1Justification of the Study 3Description of the Instruction^Journal Writing ^ 4-Small Groups^ SWhole Class Discussion^ 5Research Questions^ 6Limitations of the•Study^Organization of the Chapters 8REVIEW OF RELATED LITERATURE^ 9Summary of Research aboutProblem Solving -^ 9What is a Problem?^ 9Classification of Problems^ 11Polya's Influence on Problem-solving Research andInstruction^ 12Heuristics 14Different Approaches to Problem-solving Instruction^ 15Metacognition^  17Studies of Expert and Novice Problem Solvers^ 19Metacognition and Beliefs^ 21iiiFrameworks For Metacognition^ 24The Influence of Metacognition-based Instruction onStudents' Belief System^ 26Using Writing in Metacognition-based Instruction^ 29Summary^ 30DESIGN AND PROCEDURES^ 31Rationale for the Research Paradigm: A Qualitative Approach^ 32Pilot Studies^ 33First Pilot Study^ 33Summary of results of the first pilot study ^ 34-Educational significance of the first pilot study.^ 38Second Pilot Study^ 39Summary of results of the second pilot study.^ 39Methodological Elements of the Main Study^ 41Subjects of the Study^ 41Description of MATH 335 42Sequence of the Instruction^ 42Nature of the Metacognition-based Instruction^ 43Journal writing.^ 43Small-group problem solving.^ 44Whole-class discussions 46Teacher as role model for metacognitive behavior^ 48Using videotapes.^ 49Teaching Via Problem Solving 50Social Norm of the Class^ 54Course Evaluation 57Summary^ 59ivData Collection Procedures^ 61Task-based Interviews 61Interview technique.^ 61Nature of the tasks. 62Use of Journal Writing^ 63Class Observation 64Quizzes, Assignments, and the Final Exam^ 66Data Analysis^ 68Data Reduction^ 70Trustworthiness 72-CLASSROOM PROCESSES^ 75Rationale for the Course 75Classroom Interactions^ 77Episode 1, May 8 78Episode 2, May 27^ 79Episode 3, May 30 80Episode 4, June 4^ 86Summary^ 90FINDINGS 91General Demographic Description of the Students^ 92Academic Status^ 92Age Range 92Mathematics Background^ 92Work Status^ 93Summary^ 93Description of Categories of Students^ 93First Category: Traditionalists 94Second Category: Incrementalists^ 96Third Category: Innovators 96Descriptive Statistics^ 98Findings Within Each Category^ 105Findings from the First Category, the Traditionalists^ 108The Instruction and its Components^  108The Students' Beliefs about Mathematics  111The Students' Beliefs about Mathematical Problem Solving^ 112Traditional Teaching"; the Traditionalists' Perspective^ 113Summary^ 114Findings from the Second Category, the Incrementalists ^ 115The Instruction and its Components^  116Time.^ 121 :Teacher's role  122:Journal writing.^  123--Small groups and whole class discussions.^ 125The Students' Beliefs about Mathematics^ 126The Students' Beliefs about Mathematical Problem Solving^ 128Mathematical Connections^  131Teaching Mathematics for Problem Solving: TheIncrementalists' Perspective^ 133Summary^ 134Findings from the Third Category, the Innovators^ 134The Instruction and its Components 135The teacher's role.^  143Journal writing.  143Small groups and whole class discussions.^ 146viClass environment.^ 150The Students' Beliefs About Mathematics and AboutThemselves as Doers of Mathematics^  151The Students' Beliefs About Mathematical Problem Solving^ 160Tests^ 167Teaching and Learning Mathematics via Problem Solving;The Innovators' Perspective^  169Summary^ 170Summary of the Findings^  171Postscript^  172CONCLUSIONS AND DISCUSSIONS^ 174Conclusions^  174Answering the Research Questions^ 178Educational Significance^ 180Implications for Future Practice 182Implications for Future Research^ 184Final Note^ 185REFERENCES 187APPENDICES^ 198APPENDIX A: Researcher's Letter to Students^ 198APPENDIX B: Student Consent Form^ 199APPENDIX C: Student Consent Form 200APPENDIX D: Course Outline-Summer 1991^ 201APPENDIX E: Course Materials^ 203First Dissection Puzzle 203Second Dissection Puzzle^ 204Graph of a =10%2^ 205vi iAPPENDIX F: Sample of Metacognitive Questions^ 206APPENDIX G: Math 335, First Assignment 207APPENDIX H: Math 335, First Quiz^ 208APPENDIX I: Math 335, Second Quiz 209APPENDIX J: Math 335, Second Assignment^ 210APPENDIX K: Math 335, Third Quiz^ 211APPENDIX L: Students' Selections for the Open Assignment as Percents ^ 212APPENDIX M: Final Exam--Spring 1991^ 213APPENDIX N: Students' Selections of Problems on the Final Exam asPercents^ 215APPENDIX 0: Interview Question A^ 216APPENDIX P: Interview Question B 217APPENDIX. Q: Sample of Teresa's Work^ 218APPENDIX R: Sample of Clam's Work 219APPENDIX S: Sample of Kent's work^ 226APPENDIX T: Thank you Letter to the Students at the End of the Course ^ 228viiiACKNOWLEDGMENTI first of all would like to thank my mother, Mrs. Batool Faghihi for raising me in such away that I always think I can do anything if I want to. Throughout my formative years,she was my role model and the source of inspiration. My mother lost my father while shewas in her late 30's, and she raised seven children all by herself. She was a high schoolprincipal, and at the same time she had energy to nourish us physically and emotionally.She was my foremost influential teacher and mentor. I would like to dedicate my study tomy mother since she gave me a chance to become a self-reliant innovator! She taught meto be tolerant of others' ideas and to respect people, including my students, for what theyare and not what I want them to be.I also would like to dedicate the study to the forty students who shared their ideas andthought with me. It was impossible to do such a study without the sincere involvement ofthese great individuals. They helped me identify the range of beliefs and the ways in whichthose beliefs were shaped by the instruction that they received. This recognition.adds toteachers' awareness of their students and that ultimately contributes to the teaching andlearning process.Dr. Hoechsmann deserves special thanks for his genuine help and thoughtfulness. Heprovided me with lots of insights and gave me a suitable and exciting context to conductmy study. I also wish to extend my appreciation to the Mathematics Department that gaveme financial assistance during my stay at the University of British Columbia, as well ascooperating with me to set up appropriate setting for the study.Every member of my committee helped me in unique ways. I sincerely appreciate theeffort that they made to help this dream of mine come true. First, I would like to thankDr. Owens who put up with me while I needed so much assistance and I always amgrateful to him for that. Last three months of his life and mine were crazy! We both hadhorrible deadlines to meet. I am glad that we both succeeded. Dr. Marion Crowhurstpatiently cooperated with me. I know that at times I was not reasonable and rushed intothings. However, she always had faith in me and kindly went through my work withouthesitations. She also gave me excellent advice specially in terms of data collection. Mystudy would lack insight and coherency without this advice. Thank you Marion for allyour help!Two people were extremely influential in my academic career at UBC. Dr. GaalenErickson gave me blessing to apply for the doctoral program. He appreciated thedifficulty that I was going through in terms of language and responsibility of raising myfamily. I thank you Gaalen for your understanding and your support!Dr. Thomas Schroeder had the most impact on my doctoral work. I got much neededconfidence to start my study. It was Tom who pointed out to me my strength andixcapabilities. He has scrutinized my work and has challenged me intellectually. Tom hasalso been a good friend and a great teacher. Thank you and God bless you Tom!I would like to extend my thanks to Dr. Frank Lester whose thorough analysis addedrichness to my study.I wish to thank Manuel Santos. I was motivated by his friendship and understanding. Weboth came to UBC sharing many difficulties such as differences in language and culture.However, we had similar world views concerning social and educational issues. Thiscommon understanding helped me to be more tolerant of conscious and unconsciousinjustices that I encountered frequently in this institution.I would like to thank May and Lynne Cannon for all their support and warmth. Theirkindness and friendship made my family's stay in Vancouver as smooth as it could be.The last three months of my life have been so intense, and all my wonderful friends havehelped me to bring the work to the end. I cannot name every one of them, only tell themhow much I appreciate their genuine friendship. However, I have to acknowledge anumber of them in particular. My special thanks to Mehrdad, Ashraf and Musa, Jahan andBahar, Hamid and Afshan, Mozzafar and Soheila, and Leah and Theresa for helping meenormously and in any ways that they could. Their friendship is beyond words.Last but certainly not the least is my own family. It was impossible for me to do my study,if I did not have as supportive, understanding, gentle, and loving husband as Bijan. He hasalways had faith in me and encouraged me to do what I thought I was capable of doing. Iwish to extend my appreciation to my exceptional children, Sahar and Ali who have put upwith their student parents for eight solid years! If they did not agree to let me come backto finish my study, I would not have done so! I also would like to thank my parents-in-law who have taken the best care of my children and my husband in last three months.Also my thanks go to my brother, my sisters, and their families who all have helped mychildren one way or another in my absence.To the memory of Hosein-ali Gooya, my father, who devoted his life to educatingstudents.xiCHAPTER 1INTRODUCTION.Mathematics, many researchers believe, is a natural mode of human thought. Asconcluded by the report of the National Research Council (NRC), the acts of patternmaking by children of all ages and making guesses after they observe things are naturalways of doing mathematics (NRC, 1989). People use and need to use mathematics fordifferent purposes. A wide range of fields of inquiry from social studies to technology andcomputer science are dependent on mathematics as a driving force. The NRC's report(1989), stated that "between now and the year 2000, for the first time in history, amajority of all new jobs will require post-secondary education" (p. 39) which meansalmost everyone, one way or another needs to learn mathematics. Therefore, problemsolving as a "lifeforce of mathematics instruction" (Driscoll, 1982) should be takenseriously into account. As emphasized by Stanic and Kilpatrick (1988) "problem solvingreally is for everyone" (p. 20), and the core of mathematical learning is the problemsolving process (e.g., Confrey, 1987; Thompson, 1985; von Glasersfeld, 1983; Nicholls,Cobb, Wood, Yackel, & Patashnick, 1988).Overview of the studyOver the past decade, problem solving has been recognized widely as a centralcomponent in the teaching and learning of mathematics. The National Council ofSupervisors of Mathematics' position paper (1977) stated, "Learning to solve problems isthe principal reason for studying mathematics" (p. 20). Similarly, in the National Councilof Teachers of Mathematics (NCTM) Agenda for Action in the 1980's it was emphasizedthat, "problem solving must be the focus of school mathematics in the 1980's" (p. 1).More recently, the Curriculum and Evaluation Standards for School Mathematics (1989)1focused on the urgent need for mathematical literacy among students. It stated, "(1) thatthey [students] learn to value mathematics, (2) that they become confident in their abilityto do mathematics, (3) that they become mathematical problem solvers, (4) that they learnto communicate mathematically, and (5) that they learn to reason mathematically" (p. 5).Furthermore the Curriculum and Evaluation Standards described mathematical literacy as"an individual's ability to explore, to conjecture, and to reason logically, as well as to use avariety of mathematical methods effectively to solve problems. By becoming literate, theirmathematical power should be developed" (p. 6).Despite all the emphasis on the importance of problem solving in teaching andlearning mathematics, the nature of problem solving and its place in school mathematicsrequire more detailed scrutiny. The NCTM's (1991) Professional Standards for TeachingMathematics suggests mathematical tasks, discourse, learning environment, and analysisas the "central dimensions of teaching mathematics" (p. 9) in which the goal is to developstudents' mathematical power. Teaching mathematics via problem solving helps studentsto gain such power (Schroeder & Lester, 1989). A number of studies have been done toidentify the ways that students solve problems. They make conjectures about the possibleexplanations for students' weaknesses in problem solving, and they also offer suggestionsto overcome some of those weaknesses.A review of the literature suggests a number of important factors regardingteaching and learning of mathematical problem solving. These factors are categorized byLester, Garofalo and Kroll (1989a; 1989b) as knowledge, control, affect, beliefs, andsocio-cultural factors. In order to make sense of what students do when they are engagedin problem solving activities, Schoenfeld (1987a; 1987b) suggests that it would be helpfulto look at the students' cognitive resources, their knowledge of problem-solving strategies,metacognitive or managerial skills, and belief systems. Traditional classrooms withteachers lecturing most of the time and students passively taking notes are not flexibleenough to allow an inquiry of that kind. Therefore, creating an appropriate learning2environment with a problem-solving approach seems to be essential for allowing suchdetailed investigation. This approach to the teaching of mathematics would help studentsto become engaged in a number of problem solving activities, as well as providingopportunities for teachers and investigators to look at the kind of factors that arementioned by Lester, Garofalo, and Kroll (1989b) and Schoenfeld (1987a).Justification of the StudyMathematics educators have been interested in problem-solving instruction formany years. Central to any question regarding the teaching of problem solving is theprimary question of understanding what people actually do when they solve problems.Within the past decade, investigation of the problem-solving process has led to a newfocus in some of the research literature. As explained by Silver (1982), "any seriouslearner and teacher of mathematics recognizes that ability to solve problems involves muchmore than acquiring a collection of skills and techniques.... The ability to monitorprogress during problem solving and awareness of one's own capabilities and limitationsare at least as important" (pp. 56-57). He characterizes these abilities as metacognitiveones.Metacognition, the knowledge and control of cognition, has gained a great deal ofattention from the research community since mathematics educators have realized thatnon-cognitive aspects of problem solving-performance are as important as cognitive ones.Therefore, there has been a growing interest in the study of the role of metacognition inmathematical problem solving. A number of studies have been conducted to investigatethe role of metacognition-based instruction to help students to become better problemsolvers. Lester, Garofalo, and Kroll's (1989a) study of seventh graders' mathematicalproblem solving was of this nature. They concluded that "instruction is most likely to beeffective when it occurs over a prolonged period of time and within the context of regular3day-to-day mathematics instruction (as opposed to being a special unit added to themathematics program)" (p. v).Considering this suggestion, in the Summer of 1991, I conducted a study in thecontext of a mathematics content course for undergraduate, non-science students, using ametacognition-based approach to instruction. Although the course was a content courseand not a problem-solving course per se, choosing appropriate problems as a means tointroduce and develop the mathematics concepts was an approach that made the mostsense for metacognition-based instruction to enhance students' cognitive andmetacognitive abilities.Description of the InstructionThe approach to instruction was teaching mathematics via problem solving inwhich the mathematical concepts were developed through discussing and solvingappropriate problems. The instruction used journal writing, small groups, and whole-classdiscussions as three different but interrelated strategies that focused on metacognition.These strategies provided the opportunity for the students to become more reflective ontheir own actions in class as well as to think about their thinking and talk about theirfeelings. They helped the students to increase their self-awareness as well as to gain inself-confidence, self-respect, and self-regulation that ultimately resulted in betterunderstanding of the mathematics concepts. A similar finding was observed by Lester andKroll (1990). "Control processes and awareness of cognitive processes developconcurrently with the development of an understanding of mathematics concepts" (p.156).Journal WritingThere was no structure for the students' writing. They were free to choose topicsof their own interest. They had to write at least two or three entries per week. However,the majority of them wrote about many issues situated in the course context. Journal4writing also served as a communication channel between the students and the instructorthat facilitated the individualization of instruction.In general, for many students, writing was an occasion of personalizing andinternalizing the learning processes. Journal writing provided the opportunity for thestudents to be more reflective and gave me a chance to study possible changes in thestudents' beliefs. I collected the journals every Friday and returned them the followingMonday giving students feedback and raising some issues with them. However, I need tosay that I spent at least thirty hours each weekend to keep my promise of delivering thejournals to the students.Small GroupsThe students learned to assess and monitor their work and to make appropriatedecisions by working cooperatively and discussing the problems with each other in smallgroups. I constantly asked the what, why, and how questions that Schoenfeld asked hisstudents (1985a, 1987a) while they were working in the groups. My role as an externalmonitor (Lester, Garofalo, & Kroll, 1989a; 1989b) was to coach them in a way similar tothat described by Schoenfeld (1987a), which was to help them to become aware of theirown resources, to appreciate them, and to use them proficiently. Small groups created theopportunity for me to point out to students a variety of problem solving strategies(heuristics).Whole Class DiscussionI asked the students to share their thoughts about given problems with the wholeclass after spending between ten and twenty minutes discussing them in their groups. I didnot try to lead the students to the solution. On the contrary, I constantly encouraged themto come up with as many different ideas about solving the problems as they could. Theywere expected to participate in the class discussions and be actively involved in theprocess of solving problems. My role at this stage was to coordinate the discussions, tohelp the students to utilize what they knew, and to help them become more reflective. The5students' responsibility was to try to make sense of the problems, to speculate, to makeconjectures, and to justify them. The class discussions helped them a great deal to becomemore self-regulated and better decision makers.I gave the whole class a problem and asked each group to work on the sameproblem simultaneously. This strategy was very useful when bringing the whole classtogether while discussing a problem. As a result of this activity, the students gained morecognitive and metacognitive knowledge of problem solving and a better sense ofresponsibility. They eventually realized that they would learn more by analyzing andsolving one problem thoroughly, than by doing many problems without reallyunderstanding them. It was pleasant and reassuring for them to see that although theirapproaches and solutions were different, they were acceptable and valid. Besides, theyhad to decide which approach made the most sense to them, since they had theopportunity to examine many possible alternatives. These events were used as a vehicle topromote self-regulation.Research QuestionsThe purpose of this research study was to address four interrelated questions:I.^What are students' usual beliefs about themselves as doers of mathematics, andabout mathematics and mathematical problem solving?2. How are these beliefs prior to instruction associated with students' approaches toproblem solving?3. In what ways do students' beliefs about mathematics and about themselves asdoers of mathematics change during metacognition-based instruction and teachingmathematics via problem solving?4.^In what ways do students' beliefs about mathematical problem solving and theirapproaches to problem solving change during metacognition-based instruction andteaching mathematics via problem solving?6Limitations of the StudyThe study was a case study to investigate the influences of metacognition-basedteaching and teaching mathematics via problem solving in the context of a day-to-daymathematics course. Generalizations from case studies are analytical and not numericalgeneralizations (Yin, 1989). Analytical generalization is based on a replication logic inwhich the researcher compares the theory he or she has used to frame the study with theempirical results from the study. Generalization therefore is governed by the nature of theresearch enterprise. This study has demonstrated the applicability of metacognition-basedteaching and teaching via problem solving. Those who share similar views about teachingand learning could generalize the findings to their own practice.The study was limited to the instruction of MATH 335. The syllabus wasspecifically tuned to interesting and engaging problems to make connections betweenmathematical ideas and develop them via those problems rather than putting muchemphasis on procedures. Students who had registered for the course introduction tomathematics (MATH 335) in summer of 1991 provided data for this study. The coursewas required for those who wished to apply for the Elementary Teacher EducationProgram at the University of British Columbia (UBC) and who had not taken amathematics course. However, it was open to all non-science students. The coursematerial and the textbook were chosen by the Mathematics Department. The case studymethodology enabled me to deal with a wide variety of evidence including students' work,interviews, and classroom observations.The study was not intended to determine beliefs about mathematics andmathematical problem solving held by students in general. However, it was designed tocharacterize the beliefs held by students who participated in this study before and after theinstruction. The aim was to investigate the interactions between the students and theinstructor and among themselves, the kinds of approaches that the students adopted in7solving problems, and the kinds of cognitive and metacognitive strategies that they used inorder to solve problems. The data analysis focused on the effect of the instructionalstrategies that were used in this course rather than generalizing about instruction andstudents. Nevertheless, the study provided detailed information regarding classroominteractions, classroom culture, students' actions while solving problems, and the nature ofinstruction that could potentially contribute to the research and teaching in this area.Organization of the ChaptersThe study is presented in six chapters. Chapter One is an introduction giving anoverview of the study. A review of related literature is presented in Chapter Two.Chapter Three gives the plan and procedures used in the study. The Fourth Chapterdescribes the nature of the interactions within the small groups and in the whole class.Findings of the study are discussed in Chapter Five. The final chapter presentsconclusions of the study.8CHAPTER 2REVIEW OF RELATED LITERATUREMathematics educators have conclusively expressed the importance of problemsolving in mathematics instruction. From the Agenda for Action (NCTM, 1980) to theCurriculum and Evaluation Standards for School Mathematics (NCTM, 1989), theemphasis has always been on problem solving. The latter stated that "problem solvingshould be the central focus of the mathematics curriculum" (p. 23). However, problemsolving in mathematics still needs a greater scrutiny. Lester, Garofalo, and Kroll (1989a)have discussed two possible reasons for difficulties with problem-solving competence.The first reason is the fact that problem solving is a complex cognitive activity which isinfluenced by non-cognitive factors. "A second reason why so many students have troublebecoming proficient problem solvers is the fact that they are not given appropriateopportunities to do so" (p. 3).The review of literature gives a summary of research in problem solving in general,and about metacognition in particular, to give a better understanding of the ways in whichthe present study has been designed and developed.Summary of Research about Problem SolvingThis section gives a summary of research regarding problem solving. The reviewwill help to examine some of the crucial aspects and views about problem solving. First anoverview of how different researchers define a "problem" will be given.What is a Problem?Although there is general agreement on the importance of problem solving amongmathematics educators, the same agreement does not exist on the nature of a problem.For example, in Halmos' view, "the major part of every meaningful life is the solution of9problems." He states that "it is the duty of all teachers, and teachers of mathematics inparticular, to expose their students to problems much more than to facts" (Halmos, 1980,p. 523). On the other hand, according to von Glasersfeld (1987) problems are definableonly by the subject (a student) and only arise when someone tries to overcome the barriersto achieve a goal. In this sense, most routine examples used by mathematics instructorscould hardly be called "problems." Halmos' position expresses a view that the definition ofa problem lies with the teacher whereas von Glasersfeld views the definition of a problemas resting with the problem solver.Many others have included the motivation and action of the problem solver as anessential part of their definition of problems. For example, in Polya's view, to have aproblem means "to search consciously for some action appropriate to attain a clearlyconceived, but not immediately attainable aim. To solve a problem means to find suchaction" (1962, p. 117). Similarly, in Henderson and Pingry's view (1953), the existence ofa question is necessary in order to have a problem, yet it is not considered as sufficient.As they say, "The additional conditions pertain to the individual who is considering thequestion." What may be a problem for one individual may not be a problem for another.A problem for a particular individual today may not be a problem for him tomorrow. Inline with Henderson and Pingry's view of "a problem for a particular individual," Lester(1980) states that "in order for a situation to be a problem for an individual, the personmust: (1) be aware of the situation, (2) be interested in resolving the situation, (3) beunable to proceed directly to a solution, and (4) make a deliberate attempt to find asolution. A mathematical problem, then, is simply a problem for which the solutioninvolves the use of mathematical skills, concepts, or processes" (p. 30, emphasis inoriginal). Schoenfeld (1981) points to the different individuals' perceptions of the problemas well, since he takes into account the views of mathematicians, mathematics educators,and cognitive psychologists. The view that the problem solver acts to define a problemalso is shared by a number of other researchers such as Radford and Burtons (1974),1 0Skinner (1966), and Newell and Simon (1972), all of whom are cited in the AlbertaEducation monograph (1987).As early as 1953, Henderson and Pingry's summary of the characteristics of aproblem and the role of a problem solver in defining the problem could encapsulate thecurrent writings about the nature of a problem. They state that:1. The individual has a clearly defined goal of which he is consciously aware andwhose attainment he desires.2. Blocking of the path toward the goal occurs, and the individual's fixed patterns ofbehavior or habitual responses are not sufficient for removing the block.3.^Deliberation takes place. The individual becomes aware of the problem, defines itmore or less clearly, identifies various possible hypotheses (solutions), and teststhese for feasibility. (p. 230)Classification of Problems During the process of learning mathematics, students are asked to solve a host ofproblem exercises. Mathematics textbooks have been and continue to be an importantsource of problems. If the aim of presenting problem exercises to students is to help themto learn mathematics and problem solving, repetitive mathematics problems are notsufficient. Beyond doing the repetitive mathematics problems, those that hold no"surprises," which Schoenfeld (1981) calls "exercises," students need to be confrontedwith more challenging and more interesting problems. Students often learn algorithms orprocedures for solving "textbook" problems but they cannot transfer this skill to a realproblem setting. Textbook problems are often good examples of "routine problems,"those that could be solved by substituting given data into a familiar problem that is solvedgenerally or by following step by step algorithms.Students' perceptions of a problem solution are associated with their conceptionsof problems. Seeing a problem solely as a textbook exercise gives legitimacy to students'assumption that there is only one solution to each problem. According to Schoenfeld(1987b), and the results of the Third National Assessment of Educational Progress1 1(Carpenter, Lindquist, Matthews, & Silver, 1983), the majority of junior high andsecondary school students believe that mathematics is mostly memorization, that there isusually one right way to solve every mathematics problem, and that mathematics problemsshould be solved, if at all, in a few minutes or less.Polya classifies mathematical problems into "problems to find" and "problems toprove" which are the extension of two distinguishable propositions of Euclid. "The aim ofthe first kind (the Latin name is problema) is to construct a figure; the aim of the secondkind (the Latin name is theorema) is to prove a theorem" (Polya, 1962, p. 119). Theprincipal parts of a problem to find are "unknown" and "data" that are linked together by"condition." An example of this type of problem is to find the rate of change of quantitiesthat are related. The principal parts of a problem to prove are hypothesis and conclusion."Prove that the bisectors of the base angles of an isosceles triangle are equal" is anexample of a problem to prove.Polya's Influence on Problem-solving Research and Instruction The problem-solving literature has been greatly influenced by the ideas of Polya.Although there are different interpretations of how to use his ideas, Polya's influence onthe last 30 years of study and research in problem solving is unquestionable. Thisinfluence is reflected in the vast number of researchers who refer to Polya as one of thesources from which their own studies were developed, such as Hoz (1979), Kulm (1979),Lester (1980), Silver (1982; 1985), Kilpatrick (1985), Noddings (1985), de Lange,(1987), and Schoenfeld (1985a).The intensity of the praise of Polya's work expressed by leading mathematicseducators reflects a reverence for the man and his work. According to Lester (1980),Polya's work is ". . . perhaps the most lucid thinking about mathematical problem solving"(p. 32). In 1945, Polya published How to Solve It, in which he introduced his four-stagemodel as a method of solving problems. Schoenfeld (1987b) stated that that book is12"a tour de force, a charming exposition of the problem solving introspections of one of thecentury's foremost mathematicians" (p. 30, emphasis in original).Instruction in general processes of problem-solving strategies was uncommon priorto Polya's fascinating book. But as Polya acknowledged, problem solving and the study ofheuristics have had a long past, and he hoped for a long future. Polya's work, in his ownview (1973), is in the spirit of Pappus', Descartes' and Leibniz' attempts to devise rules forthe direction of mind, a universal method to solve problems.In Polya's view, mathematics has two faces: (a) the systematic deductive science ofEuclid, and (b) an experimental inductive science. Polya notes that both of these aspectsare as old as the science of mathematics itself. However, mathematics in the process of"being invented," which means the experimental inductive science, has never beenintroduced to the students, the teachers, and the general public. Polya introduced problemsolving as a new way to understand mathematics. He and a few other mathematicians,among them Kline, questioned the validity of the formalist view in which mathematics ischaracterized as a closed deductive system (look to Dieudonne's address to O.E.E.C.,1961 as typical of this approach). Lakatos (1976) also objected to the formalists' view asthe only approach to mathematics and talked about informal mathematics as well. ForLakatos, "informal mathematics is a science in the sense of Popper; it grows by a processof successive criticism and refinement of theories and the advancement of new andcompeting theories" (cited in de Lange, 1988, p. 144). Mathematics education has beeninfluenced by informal mathematics ever since and has become more process, rather thanproduct, oriented in relation to problem solving and mathematical modeling.Polya's plan was based on his view of mathematics as an inductive, experimentalscience. He stressed that mathematical work consists largely of observations andexperiments; guessing at what might be true; feeling what ought to be true; testinghypotheses; looking for analogies; and building mental pictures. Polya described problem-solving as "a practical skill like, let's say, swimming. We acquire any practical skill by13imitation and practice . . . . Trying to solve problems, you have to observe and imitatewhat other people do when solving problems, and, finally, you learn to do problems bydoing them" (1973, pp. 4-5). Lester (1980) held the same view using baseball as anexample. 1Heuristics In order to help students become better problem solvers, Polya (1973) suggested alist of heuristics which helps students solve problems. Heuristics are plausible rather thanstrict, and as Polya said, heuristic is about the rules and methods of discovery andinvention. Thus, heuristics, which are tools available to the problem solver and whichexist independent of problem content, can be used efficiently as long as one distinguishesand maintains the distinction between these two kinds of reasoning, namely, heuristicreasoning and formal mathematical arguments.In general, problem solving research evidence supports the premise that teachingstudents heuristics enhances problem solving performance (Suydam, 1980), but the role ofheuristics in the teaching and learning about problem solving is viewed in several differentways. In particular, there are those who support the view that students' problem solvingperformance will be increased by teaching them via heuristics (Schoenfeld, 1979a;Goldberg, 1975; and Lucas, 1972, both cited in Schoenfeld, 1979b ). However, somecurriculum developers support the view that heuristics should be taught as individual skillsor techniques which are then applied to problem solving. In this regard Stanic andKilpatrick (1988) raise the following concerns about the nature and effect of someimplemented curricula:There are those today who on the surface affiliate themselves with thework of Polya, but who reduce the rule-of-thumb heuristics to proceduralskills, almost taking an algorithmic view of heuristic . . . A heuristicbecomes a skill, a technique, even, paradoxically, an algorithm. In a sense,1. It is interesting to see that after 28 years, even researchers' examples are from the samesporting contexts as Polya's.14problem solving as art gets reduced to problem solving as skill whenattempts are made to implement Polya's ideas by focusing on his steps andputting them into textbooks. (p. 17)Polya's intention in introducing heuristics was, as stated by Schoenfeld (1981), "touncover useful strategies for inquiry into (somewhat) novel situations" (p. 43). However,Schoenfeld (1987b) argues that Polya's heuristics are descriptive rather than prescriptive,and each heuristic could break down into many other strategies useful for solvingproblems. In Schoenfeld's view, Polya's descriptive heuristics are more useful for thosewho already know how to apply them. Yet these strategies are not detailed enough to beas useful for those who are not familiar with them. "Polya's characterizations were labelsunder which families of related strategies were subsumed" (Schoenfeld, 1987c, p. 42).Since there are many other components such as students' motivation and classroom culture(Schoenfeld, 1988a) which affect the students' performance on problem solving,Schoenfeld (1981), argues that "a teaching theory based solely on heuristics is doomed tofailure" (p. 43).Different Approaches to Problem-solving Instruction This section deals briefly with the different approaches to teaching problemsolving. Almost all mathematicians and mathematics educators believe that ability to solveproblems is an essential part of learning mathematics. However, their perceptions ofproblem solving influence their ways of interpreting the role of problem solving and itsapplicability in an actual classroom situation.One view is to consider problem solving as a goal to be reached independent ofspecific problems, their mathematical content and procedures or methods used to solvethem. Learning how to solve a problem, in this view, is the primary reason for studyingmathematics. The entire curriculum has been influenced by this view, and it has importantimplications for classroom practice. This view is what Stanic and Kilpatrick recognize as"problem solving as context," with five sub-themes namely: (a) problem solving asjustification; (b) problem solving as motivation; (c) problem solving as recreation; (d)15problem solving as vehicle; (e) and problem solving as practice in which all of these sub-themes seek to reach the valuable ends by the means of solving problems (for more detailsee Stanic & Kilpatrick, 1988). This approach resembles what Schroeder and Lester(1989) call "teaching for problem solving," in which the teacher usually introduces thebasic mathematical content before applying that content in finding the solution of severalproblems. The assumption in this view is that students should study the mathematicscontent first, then solve actual problems.A second view interprets problem solving as a skill in which the specific content ofa problem, kinds of problems and methods leading to solution should be considered. Aspointed out by Stanic and Kilpatrick (1988), "the problem-solving-as-skill theme hasbecome dominant for those who see problem solving as a valuable curriculum enddeserving special attention, rather than as simply a means to achieve other ends or aninevitable outcome of the study of mathematics" (p. 15). Stanic and Kilpatrick continue toargue that viewing problem solving as a skill does not provide opportunity for all studentsto get involved in solving non-routine problems, only capable students, since the skill ofsolving non-routine problems as a higher level skill is acquired only by those students whohave mastered skill at solving routine problems.This approach is similar to "teaching about problem solving" as Schroeder andLester (1989) call it. In this situation the teacher usually solves a problem and then he orshe talks about the process by identifying Polya's four stages. The key in thisinterpretation is talking about problem solving process and engaging students in solvingproblems. This view leads to adding a unit on problem solving to the school curriculum.British Columbia Curriculum Guide from K-8 (1987) is a good example for this approach.However, a third view sees problem solving as a dynamic, ongoing process. Forthose who hold this view, the final product is not as important as the methods, procedures,strategies and heuristics used by students to solve problems. This view resembles whatStanic and Kilpatrick call "problem-solving as art." The work of Polya gave rise to this16aspect of problem solving. Polya (1980) emphasized that problem solving is the "specificachievement of intelligence, and intelligence is the specific gift of man" (p. 1), and theprimary goal of education is to develop the intelligence and make teaching and learningmathematics "insightful." Polya believes that "to know mathematics is to be able to domathematics" (Polya, 1969, p. 574), like any other practical art that is learned through"imitation" and "practice." In Polya's view, teachers should illustrate the techniques andstrategies of problem solving and discuss them with the students. This view is almostidentical to what Schroeder and Lester (1989) call "teaching via problem solving," inwhich the mathematics content is introduced through discussions and extensions ofspecific problems.As a summary, "teaching via problem solving" can provide a situation in whichstudents do mathematics actively and creatively. This approach gives students theopportunity to use plausible reasoning by teaching them how. It is crucial to understandhow individuals learn to solve problems to be able to understand how such learning takesplace in the classroom. In the next section, I will discuss research on students' thinkingprocesses, their beliefs about mathematics, and how well they can control their actions insolving problems. These are the three main categories on which research onmetacognition has focused.MetacognitionCentral to any questions regarding the teaching of problem solving is the primaryquestion of understanding what people actually do when they solve problems. Within thepast decade, investigations of the problem solving process have led to a new focus in someof the research literature. As explained by Silver (1982), "any serious learner and teacherof mathematics recognizes that the ability to solve problems involves much more thanacquiring a collection of skills and techniques . . . . The ability to monitor progress duringproblem solving and an awareness of one's own capabilities and limitations are at least as17important" (pp. 56-57). He states that these abilities are metacognitive ones. Accordingto Flavell (1976), "'Metacognition' refers to one's knowledge concerning one's owncognitive processes and products or anything related to them. . . . Metacognition refers,among other things, to the active monitoring and consequent regulation and orchestrationof these processes . . .." (p. 232). Many researchers have reached conclusions similar tothose of Flavell and Silver and, as a result, metacognition and its implications formathematics education are now widely studied.Schoenfeld (1987a, p. 190), indicates three related but distinct categories ofintellectual behavior, on which the research on metacognition has focused:1. Your knowledge about your own thought processes. How accurate are you indescribing your own thinking?2. Control, or self-regulation. How well do you keep track of what you're doingwhen (for example) you're solving problems, and how well (if at all) do you use theinput from those observations to guide your problem solving actions?3.^Beliefs and intuitions. What ideas about mathematics do you bring to your work inmathematics, and how does that shape the way that you do mathematics?Schoenfeld's three categories of metacognition are reflected in the work ofGarofalo and Lester (1985). Based on Flavell and Wellman's (1977) categories, Lesterand Garofalo categorized knowledge about cognition according to how it influencesperformance in terms of the person, task, or strategy. In the person category,metacognitive knowledge consists of "What one believes about oneself and others ascognitive beings" (p. 164). Cognitive knowledge in the task and strategy categories refersto an awareness of one's knowledge about the nature of tasks and an awareness of one'sthinking strategies.The regulation of cognition concerns knowing how and when to use theknowledge described above. These metacognitive, or managerial skills, include planningstrategies, monitoring these strategies and being willing to abandon them when necessary,18and evaluating the outcomes for efficiency and their degree of effectiveness (Schoenfeld,1985a; 1985b).It is this latter aspect of metacognition that has the greatest impact on problemsolving in mathematics. There is not much purpose in knowing a variety of heuristics ifone does not know when to use them. Many researchers believe that it is an ability tomake these executive decisions that determines whether or not an individual will be asuccessful problem solver. By definition, metacognitive actions affect problem solving in aglobal sense and failure to adequately monitor and assess one's strategies can result in anunsuccessful attempt at reaching a reasonable conclusion. This provides one means foraccounting for the behavior of the person who knows the appropriate strategies neededfor solving a problem but yet is unable to solve it.Studies of Expert and Novice Problem Solvers The discussion about the types of problem and their structure has given rise to theextensive research proposals that aim to identify the characteristics of good and poorproblem solvers. Krutetskii's (1976) research on gifted students of mathematics indicatesthat a distinguishing factor between good and poor problem solvers is the students'perception of the important elements of problems. Differences between good and poorproblem solvers help to better understand metacognition and its implications for theteaching of problem solving. Many researchers have attempted to discover whatdifferentiates the novice problem solver from the expert. Krutetskii (1976), based on hisresearch of gifted students in mathematics, concluded that what distinguishes betweengood and poor problem solvers is their perception of the important elements of problems.Driscoll (1982) referring to Krutetskii (1976), identifies characteristics of good problemsolvers including1.^Good problem solvers sort problems according to their mathematical structure andnot by their content.192. Good problem solvers distinguish between relevant and irrelevant information anddo not focus on single variables within the problem.3. Good problem solvers are more active and use more problem solving strategiesand processes.Weak problem solvers in contrast will fail to exhibit the above behaviors and will also failto look back and assess their work. While he does not explicitly refer to metacognition,Driscoll does discuss elements of metacognitive actions. His survey provides a preliminaryindication that good problem solvers may use metacognitive skills that are superior tothose used by weaker students.The difficulty in doing research in metacognition, however, is that for many peoplemetacognition is a covert process. This situation is further complicated by the fact thatmost students have not been taught to use metacognitive skills such as managerial skillsand, therefore, it is difficult for them to articulate processes with which they are notfamiliar (Lester, 1986). Schoenfeld (1980) in an attempt to record and analyzemetacognitive activity during mathematical problem solving, found that expert problemsolvers possess efficient managerial skills while novice problem solvers do not. Theexperts, according to Campione, Brown, and Connell (1988) "find learning challengingand see themselves in control of their destiny. Weaker students [novices], in contrast,acquire fewer problem-solving strategies, are less aware of the utility of those strategies,and do not use them flexibly in the service of new learning. They are not convinced thatthey can control their performance and tend to be relatively passive in learning situations"(p. 94). Silver (1980) cites the identical situation when physics problem solvers wereobserved. He offers the possible explanations that expert problem solvers may have soincorporated metacognitive behaviors into their problem routines that they are difficult toobserve, or that these behaviors are only needed at certain points in the development ofexpertise.20Metacognition and Beliefs One issue directly related to metacognition which has been referred to previously,but not discussed in detail, is that of "beliefs." As mentioned earlier, Schoenfeld considersone's belief system to be an integral part of one's problem-solving structure, and the worksof Silver (1982) and Garofalo and Lester (1985) both include beliefs in their definitions ofmetacognition.According to a number of researchers, the link between metacognition and beliefsoccurs because one's world view affects the decisions that one makes. One's beliefsdetermine the context within which one selects from available "resources" and how onechooses to use those "resources" (Schoenfeld, 1983). An individual is unlikely to pursuecertain strategies if he or she does not believe that they will be successful. Therefore,although he or she may be aware of metacognitive behaviors, a person may make whatseems to be inappropriate use of them if they are inconsistent with his belief systems.In the context of mathematical problem solving, behaviors can be influenced by avariety of beliefs. A person's view of the nature of schooling and learning in general, thenature of mathematics and the learning of mathematics, the specific task itself, and one'sown mathematical abilities can all play a role in how one responds to a problem solvingsituation. For example, Silver (1982) suggests that "a person who believes that there is anunderlying structure to mathematics and that this structure is more important than thesurface details, will approach the study of mathematical material quite differently than astudent who does not hold this belief' (p. 21). Other beliefs which might positivelyinfluence how one approaches mathematical problem solving are the beliefs that there isusually more than one way to solve a problem, whether two methods are used to solve aproblem, they should result in the same solution, and that there exists a most concise wayto present a problem and its solution.It is perhaps equally important for educators to be aware of the mathematicalbeliefs that students hold which might hinder the problem solving process. A set of such21beliefs was reported by Lester and Garofalo (1982). Their findings indicated that manygrade three and five students believed that verbal problems are more difficult thancomputational ones, and that the size and quantity of numbers in the problems areimportant indicators of difficulty. Similarly, Driscoll (1982) reported that, in anassessment of thirteen and seventeen year old students, almost fifty percent agreed withthe statement that learning mathematics is mostly memorizing. Furthermore, almost ninetypercent agreed with the view that there is always a rule to follow in solving mathematicsproblems. Driscoll (1982) suggests that these students' conceptions of mathematics growout of a mathematical experience which usually emphasizes "memorization, regurgitation,and the conviction that the sole purpose for doing any mathematical problem is to get theright answer" (p. 63).Schoenfeld's (1987b) experience with college freshmen has led him to establish hisown set of commonly held beliefs. They are as follows: that formal mathematics has littleto do with real problem solving, that mathematics problems are always solved in tenminutes or less, and that only geniuses are capable of discovering or creating mathematics.Like Driscoll, Schoenfeld indicates that teachers must assume responsibility for theexistence of these kinds of perceptions and beliefs. As much of mathematics deals withabstract structures with which students have little real-world experience, the majority ofstudents' mathematical world views will be based on their experiences in the mathematicsclassroom. Many classrooms place a strong emphasis on writing mathematical argumentsin a prescribed form, which contributes only to the idea that "being mathematical" means"doing your steps." Likewise, most problem solving exercise sets are limited to thosewhich can be adequately discussed within a single class lesson (Schoenfeld, 1985b). Insummary, he states that teachers too often "focus on a narrow collection of well-definedtasks and train students to execute those tasks in a routine, if not algorithmic fashion.They test the students on tasks that are very close to the ones they have been taught ....22To allow them, and ourselves, to believe that they understand that mathematics isdeceptive and fraudulent" (Schoenfeld, 1982b; p. 29).Thompson's (1988) research indicates that the actions of teachers are a reflectionof their own views of mathematics. Therefore, not only must teachers be aware ofpotential student beliefs and how they are acquired, but they must also be aware of theirown. Teachers' beliefs about mathematics learning and teaching influence the nature ofthe mathematics environment in the classroom and the type of instruction that the studentreceives. As beliefs cannot be eliminated, Silver (1982) suggests some questions forpossible further study in this area: What beliefs are held by mathematics teachers? Whatare the beliefs of "excellent" teachers? and of course, What is the influence of beliefs onthe teaching of problem solving?The issues of metacognition and belief systems and their relationship to problemsolving in mathematics are currently the focus of interest for many researchers, yet theimplications for the mathematics teacher are still unclear. To date, most instruction inproblem solving has emphasized the development of algorithms and heuristics and hasalmost ignored the managerial skills involved in the process. As indicated above, teachersshould increase their awareness of their own beliefs and metacognitive actions.Schoenfeld (1985b) recommends that, at a minimum, teachers should demonstratethe decision making process when working on problems in class. Questions such as"What are the possible options here?" and "Is this strategy working out?" can serve asmodels to students and legitimize the process in their minds.Both Schoenfeld (1985b; 1987b) and Driscoll (1982) also encourage teachers tohelp students to verbalize their problem solving experiences. Schoenfeld indicates that thiscan be done with the class as a whole where the teacher acts as a manager. By carefullymonitoring the interactions, the teacher may be able to have the class engage in ametacognitive analysis of a proposed problem. This involves questioning all decisions,even when the development of the solution appears to be proceeding well. A postmortem23on the problem should also occur where a discussion of the representation of the problem,the use of control strategies and possible other solutions to the problem take place.Schoenfeld also advocates breaking the class into small groups to work onproblems while the teacher acts as consultant. The emphasis, however, is still on thestudents' awareness and control of their actions. Schoenfeld (1985a) has a poster in hisclassroom which contains the three executive questions: (a) What (exactly) are youdoing? (b) Why are you doing it? and (c) How does it help you? The purpose of thesmall groups is to bring out into the open what would ordinarily be covert processes andto encourage the proposal and discussion of alternate options. His suggestion fordetermining what the students' beliefs are is the repeated use of the question why. It is oneof the practical suggestions offered by mathematics researchers to date and is a reasonablyconcise method of explaining what the teacher must do. It must be noted, as Schoenfeldhimself points out, that his recommendations are based on anecdotal evidence and are notthe result of extensive research.Frameworks For MetacognitionThe major contributions of the study of metacognition to research on mathematicalproblem solving have been in the development of frameworks within which to betterunderstand and further that research. As previously stated, Schoenfeld (1984; 1987a) hasidentified three broad categories, namely: (a) "resources," or an individual's knowledge,such as facts, algorithms, and heuristics which have bearing on the problem; (b) "control,"or the conscious and unconscious metacognitive acts used; and (c) "belief systems," whichare the determinants of an individual's behavior. As he believes that decisions at themanagerial or control level may determine the success or failure of a problem solvingattempt, Schoenfeld (1983, 1985a) has focused his study of protocols on this type ofbehavior. By parsing protocols into chunks of consistent behavior, which he callsepisodes, he has developed a list of six managerial decision-points. These are reading,analysis, exploration, planning, implementation, and verification, and most of these24decision-points are self explanatory. According to Schoenfeld (1985a), "it is at thetransition points between these episodes (as well as other places) where metacognitivedecisions, especially managerial ones, can have powerful effects upon solution attempts"(p. 9). These six decision-points function as a framework within which the broad categoryof "control" could be investigated.Similarly, Garofalo and Lester (1985) have also designed a cognitive-metacognitive framework that considered it to be "directly relevant to performance on awide range of mathematical tasks, not only tasks classified as 'problems.' It is not a list ofall possible cognitive and metacognitive behaviors that might occur; rather, it specifies keypoints where metacognitive decisions are likely to influence cognitive actions" (p. 171).The framework contains four categories, namely: orientation, organization, execution, andverification. This framework is based on the work of Polya, Schoenfeld and others, and itis "intended as a tool for analyzing metacognitive aspects of mathematical performance"(p.172). The framework shows that "distinctive metacognitive behaviors are associated witheach category" (p. 171). Although Polya never explicitly mentioned metacognition, itseems there is a parallel between the above work and Polya's "four stage" plan. 2Perkins and Simmon (1989) talked about four kinds of knowledge, namely:content, problem solving, epistemic, and inquiry which, as suggested by Donn (1990),might be used as a framework to study students' misconceptions. The frameworkdescribes heuristics and metacognitive strategies inside all four kinds of knowledge. Thecontent knowledge consists of terminology, definitions, and relevant rules or algorithms toa content. Ability to recall and to use notations are of this kind. However, problemsolving knowledge includes general problem solving and managerial strategies, and beliefsabout problem solving. Furthermore, the epistemic knowledge verifies and justifies theuse of a particular concept or procedure. Finally, the knowledge of specific content is2 The "Four Stage" plan includes (1) Understanding the Problem; (2) Devising a Plan;(3) Carrying out the Plan; and (4) Looking Back.25extended or challenged by specific and general beliefs and strategies that are called theinquiry knowledge.The Influence of Metacognition-based Instruction on Students' Belief SystemMathematics educators have been interested in problem-solving instruction andperformance for many years. In particular, Lester, who has been involved in problem-solving instruction for twenty years, has recently studied young children's metacognitiveawareness as an important component of their mathematical problem solving.Metacognition, the knowledge and control of cognition, has gained a great deal ofattention from the research community since mathematics educators have realized thatnon-cognitive aspects of problem solving performance are as important as cognitive ones.Therefore, there has been a growing interest in the study of the role of metacognition inmathematical problem solving among these educators. A number of studies have beenconducted to investigate the role of metacognition-based instruction to help students tobecome better problem solvers. The following is the summary of some of the classroom-based research regarding metacognition in the context of day-to-day instruction.Lester, Garofalo, and Kroll's (1989a) study of seventh graders' mathematicalproblem solving was of this nature. This 12 week study involved students' written work,interviews of students individually and in pairs, classroom observations, andquestionnaires. The teacher played different roles at each phase of the instruction; amongthem was his role as monitor, facilitator, and a role model for metacognitive behavior.They found that students' beliefs are mainly shaped by classroom instruction andclassroom environment. They concluded that "instruction is most likely to be effectivewhen it occurs over a prolonged period of time and within the context of regular day-to-day mathematics instruction (as opposed to being a special unit added to the mathematicsprogram)" (p. v).Lester, Garofalo, and Kroll's (1989a) suggestion is in line with Schoenfeld's effortin his problem solving courses. Schoenfeld (1985a; 1987a) has done extensive work to26develop instructional techniques that promote different aspects of metacognition, namelyawareness of one's own thinking processes, control or self-regulation, and beliefs thatstudents have about mathematics and mathematical problem solving. He has dedicated afair amount of time investigating the third aspect of metacognition and exploring theorigins of beliefs that students bring into the classroom.Raymond, Santos, and Massingila, (1991) taught a course for prospectiveelementary teachers at Indiana University. The course was developed by a team underLester's direction (NSF Final Report # TEI-8751.478) and it was described by Raymond,Santos, and Massingila (1991) as: "teaching mathematics via problem solving bychallenging students to construct and/or reconstruct their [student teachers']understanding of mathematics" (p. 4). Their study was influenced by Lester, Garofalo,and Kroll's (1989a) of seventh graders. Raymond, et al. investigated the ways in whichstudents' belief systems might have been influenced by the instruction. They found that"the three basic components of T 104 [a mathematics content course] instructionalapproach (problem solving, cooperative learning, and written reflections) have thepotential to challenge students to question their mathematical belief systems regardingwhat it means to know, do, learn, and teach mathematics" (p. 1). The significance of thestudy might be its potential impact on students' future teaching actions. In conclusion,they suggested that more research is needed to study teachers' belief systems and theextend to which they will be positively influenced by mathematical instruction.Santos (1990) investigated the students' beliefs about mathematics, as well as theeffect of metacognitive instruction on students' problem solving performance at the collegelevel. In this study, he regularly observed the class while the instructor was teaching. Theinstructor became familiar with metacognition and problem solving through guidedreadings and discussions with Santos. He also developed many problems focusing onmetacognitive knowledge. Santos used both a modification of Schoenfeld's (1985)27framework and Perkins and Simmon's (1989) framework for analyzing his data and tostudy the students' misconceptions and difficulties.Santos' original aim for the course was to teach mathematics via problem solvingand to emphasize problem-solving strategies. He believed that this approach would givestudents a chance to discuss their ideas, to become more engaged in problem solvingprocesses, and to work more cooperatively in the class. Moreover, Santos developed anumber of nonroutine and interesting problems for the assignments which wereaccompanied by a series of metacognitive questions. Santos went through all the students'responses to the assignments and the metacognitive questions. Eventually, the students'approaches to mathematical problem solving were influenced by these activities. Thosestudents who received the instruction showed more awareness of the importance of usingdifferent approaches to solve mathematical problems. In addition, the study showed thatstudents who were exposed to metacognitive instruction scored higher on the final examwhich was the same across the college.In conclusion, Santos suggested that teaching mathematics via problem solvingshould become the approach to instruction rather than only a means to be usedoccasionally. He also concluded that more metacognitive strategies should be developedand be adopted throughout instruction. His recommendation is to design morechallenging course materials appropriate for teaching mathematics via problem solving.Campione, Brown, and Connell (1989) considered metacognition-based instructionas an alternative for instruction. They called it "reciprocal teaching," summarized byLester, Garofalo and Kroll (1989) as: "(a) instruction occurs in cooperative learninggroups; (b) instruction takes place in the context of learning specific content; (c) thestudents' attention is focused on solving a specific problem, not on monitoring, regulating,or evaluating actions per se; (d) students are not protected from error in their solutionefforts; (e) the teacher is allowed to be fallible; and (1) the teacher's role as a guide andmodel diminishes as students become more confident and competent" (p. 36).28In addition to the studies mentioned, several others have been conducted toinvestigate the different aspects of metacognition, concerning specific groups of people inspecific settings rather than regular day-to-day mathematics course. Notable studiesinclude: Hart and Schultz's (1985) study of three in-service intermediate teachers usingSchoenfeld's framework for analyzing students' metacognitive behavior, and Defranco's(1987) study of 16 mathematicians' problem solving and metacognitive behaviors.Using Writing in Metacognition-based InstructionThere is a growing sentiment for journal writing among researchers as a means tofacilitate learning of different subject matter. Vygotsky's (1962) view of the relationshipbetween language and thought lend support for the use of journal writing. As pointed outby Emig (1977) ". . . writing can engage all students actively in the deliberate structuringof meaning, it allows learners to go at their own pace, and it provides unique feedback,since writers can immediately read the product of their own thinking on paper" (quoted inBorasi and Rose; 1989, p. 348). Students could either have freedom to write any thoughtrelated to their mathematics or their writing could be fully structured and guided.The results of Crowhurst and Kooy's (1985) study show that journal writingduring the reading of novel indeed encourages students' thinking. "It encourages them toreflect upon and clarify their thoughts about and their responses to the literary work.Writing both requires and facilitates the clarification of thought" (p. 263). Langer andApplebee (1987) strongly recommended the use of writing in the school curriculum.Journal writing as a learning tool in the mathematics classroom has gainedpopularity among the mathematics and mathematics education research community.Borasi and Rose (1989) conducted a research study using writing in a regular collegemathematics course. In conclusion, they gave a taxonomy of potential benefits of journalwriting for both teacher and students and show how writing can be a means for moreindividualized teaching. Lester, Garofalo, and Kroll (1989a) used writing to investigatethe kinds of metacognitive decisions that the students made. In their study, Raymond,29Santos, and Massingila (1991) used reflective writing for the same purpose as well as todocument possible changes in students' beliefs about themselves as doers of mathematics.The number of mathematicians and mathematics educators who have extended theidea of journal writing throughout mathematics courses is growing. Burkam, Britton,Talman, Hartz, Buek, Rose, and White (all documented in Sterrett, 1990) are amongthem. They found that "writing to learn" has potential to contribute to mathematicsinstruction.The review of related research gives every reason to believe that writing givesstudents the opportunity to reflect upon their feelings, knowledge, and beliefs aboutmathematics. It will also be beneficial for teachers to read the students' journals becausereading the journals would help teachers to improve their teaching. Journal writing allowsfor individualized instruction, and it might help to create a friendly and supportiveclassroom environment.SummaryThis chapter gives a review of studies about the importance cognitive and non-cognitive factors pertaining teaching and learning of mathematical problem solving.Beginning with a summary of research about problem solving, it then gives an account ofmetacognition and its role in mathematical problem solving. It ends with a briefdescription of several classroom-based research studies using metacognitive strategies.Those studies helped to identify the types of metacognitive activities that would bebeneficial for enhancing students' understanding of mathematics, and the present study wasinfluenced by them.30CHAPTER 3DESIGN AND PROCEDURESThe review of the related literature indicates a number of important factorspertaining to teaching and learning of mathematical problem solving. Lester, Garofalo,and Kroll (1989b) categorize these factors as knowledge, control, affects, beliefs, andsocio-cultural factors. Silver (1985) expressed his concern for some of themethodological, conceptual, and reporting problems with research in this area. Amongthem were the lack of information on what the teachers actually did in the classroom whenteaching problem solving, the amount of control over the "teacher variable," theinsufficient assessment of the influence of direct instruction on students' problem-solvingbehaviors, and the absence of any instructional theory in guiding most of the researchstudies. A study of the role of metacognition in mathematical problem solving in twograde seven classes by Lester, Garofalo, and Kroll (1989a) addressed these concerns andprovided some insight into the research in this area. These researchers studied the effectof problem-solving instruction based on metacognitive strategies on students' problemsolving behaviors and concluded that the instruction would have been more effective if ithad happened in the context of regular day-to-day mathematics instruction rather than as aunit added to the mathematics program.Raymond, Santos, and Masingila (1991) taught a mathematics content courseknown as "mathematics for elementary teachers via problem solving," at IndianaUniversity and it was required for prospective elementary teachers. They adopted a non-traditional approach to instruction and used a number of instructional techniques designedto help students become more reflective. They documented some changes in students'beliefs by investigating the students' in-class and assigned written reflections.The purpose of the present study is to document students' beliefs aboutmathematics and mathematical problem solving as well as to investigate the possiblechanges in students' beliefs as a result of teaching through problem solving. This study is31similar to the Raymond, Santos, and Masingila's (1991) study. Yet there are a number ofsignificant differences in the variety of instructional techniques used, and the content of thecourse. Another aim of the present study is to address Silver's (1985) concern about thelack of information about what teachers' actually did in classrooms by documentingclassroom processes through audio and video-taping classroom sessions as well as takinginto account the researcher-instructor's daily journal in analyzing the data and reportingthe findings.The design of the study, the nature of the instruction, and the rationale forchoosing different instruments for gathering data are discussed in the following sections.Rationale for the Research Paradigm: A Qualitative ApproachAs has been indicated by many researchers, the processes of obtaining the solutionto a problem is more important than the final product. This indication gives rise to the useof qualitative approaches to research since it involves the close-up, detailed observationand documentation of the phenomenon under study (Yin, 1989). Krutetskii (1976) foundthat qualitative research was advantageous in studying students' individual differencesduring the process of solving problems. In line with Krutetskii (1976), Davis (1986)discusses the significance of qualitative data such as those obtained by task-basedinterviews for investigating the students' understanding of mathematical concepts. Thesedata cannot be replaced by the results of achievement tests. "Achievement tests provide acertain kind of data, but they omit much more; they do not give a broad comprehensivepicture of what is taking place" (p. 87). Qualitative methodology is an appropriateresearch paradigm with respect to the aims of the present study. It is crucial in studies ofthis nature to discuss the ways they are designed and redesigned during the natural courseof the inquiry. Two pilot studies were helpful in designing the study.32Pilot Studies Prior to the main study, two pilot studies were conducted to investigate thepossibilities and the limitations of the research setting, the data collection instruments, andthe interview techniques, and to revise the presentation of the course material. Thesummaries of the two pilot studies follow. All instances of emphasis in original materials,for example in students' responses, are indicated by underlining. Where emphasis has beenadded by the writer, italics are used. "I" stands for interviewer.First Pilot StudyThe first pilot study was conducted in the Fall of 1989. The pilot study had twomain purposes: first to test the thinking aloud technique for the task-based interview andsecond to evaluate the appropriateness of the non-routine tasks chosen. Two grade eightand two grade nine students participated individually in the interview. The students wereinterviewed in an informal setting outside their schools. Two of them were from the sameschool and the other two came from two different cities. The following two tasks werechosen for the interviewTask I:^What is the units digit for the following sum? 13 841 + 17508 + 24617Task II:^The last digit of the product 3 x 3 is 9. The last digit of the product3 x 3 x 3 is 7. What is the last digit of the product when thirty five 3's aremultiplied?The two cards with tasks written on them were Put on the table. Students had - achance to choose either one of them to start. The reason for not writing the two problemson the same card was my hypothesis that whichever task was given first, the studentswould presume that there had to be a reason for it. For example, the first problem mightbe easier. This idea was to prevent those potential feelings.All four interviews chose the second task first. For them, working with smallerquantities seemed to be easier. This belief is identical to what Lester and Garofalo (1982)identified in grade three and five students that problems involving larger numbers are more33difficult to solve. All the interviews were audio taped and transcribed for data analysisalong with the students' written work and my own notes. All four interviewees are givenpseudonyms for the sake of confidentiality.Summary of results of the first pilot study. The aim of this section is to share a few excerpts of the students' interviews to firstshow the appropriateness of similar tasks for the study that focuses on students' beliefsabout mathematics and mathematical problem solving, and to establish that the task-basedinterview was a suitable technique to be used in the main study.Students' beliefs are shaped in part by their teachers. Teachers and textbooks areoften responsible for formation of students' beliefs. Students rarely face a real "problem"in their everyday classes. As stated by Schoenfeld (1987b), "Even when students dealwith 'applied' problems, the mathematics that they learn is generally clean, stripped of thecomplexities of the real world. Such problems are usually cleaned up in advance—simplified and presented in such a way that the techniques the students have just studied inclass will provide 'solution' (p. 37).One of the common beliefs shared by all the interviewees came from whatSchoenfeld has described. Students have had little chance to "mess up" and engage in theprocess of solving problems. Teachers usually prescribe certain procedures to studentsand often their prescriptions affect students' curiosity and their eagerness for creativity anddiscovery. The interviewees were embarrassed by not "remembering" the procedure to dothe problem. Sara said, "You probably get someone that knows how to do it right at thebeginning."The students believed that school mathematics is mostly chasing after the "correctnumerical answers." They usually see the mathematics and problem solving through theirteachers' eyes. Teachers play a significant role in shaping the students' beliefs aboutmathematics. Mathematics could be fun or boring, depending on how it is presented tothem.34I^How do you like math?Sara^I liked it last year because I liked my teacher. . . this year math doesn'tinterest me. This year's teacher just goes by the book. It's really boring.Vicky, as well, said that this year she did not like mathematics because her teacher wasboring!Vicky^He just reads us the book that says something and then tells us how to doit . . . he is boring.I^What does he do that bores you?Vicky^You've got to go to his class! I don't know! He just gives us theworksheets, there is nothing in it, no excitement, anything . .All the students believed that mathematics is a set of rules and procedures that should bememorized. All of them first tried to remember how they solved the problem in class.Don even surprised me more when he said:Don^I remember that page in the math book in grade 7.I^What was in that page?Don^There was a way to do it [task II]. My teacher showed me how to do it .I remember vaguely in my math book that they did it [task II].In his view, problem solving could be reduced to memorization of series of rulesand methods offered by teachers and textbooks, and students' main responsibility was tofollow them step by step. This way of teaching would destroy students' confidence. It ishard to expect them to think about creativity and discovery when the teacher alwaysshows them what to do and asks them to spell out the same thing again in return. Most ofthe students believe that "teachers know the right way of solving the problems." It isunfortunate that they think there is only one solution to each problem and the best way todo it is the teacher's way because they certainly get the correct answer that way. Thestudents' beliefs about mathematics and problem solving could be summarized by thefollowing:35Belief 1:^Mathematics is a set of rules and procedures that should be memorized.Belief 2:^There is only one solution to each problem.Belief 3:^Mathematics is boring because it has no excitement.Belief 4:^The most important feature of a mathematical problem solving is findingthe correct answer.The task-based interviews also gave a chance to study the effect of students' beliefsabout mathematics and mathematical problem solving on their approaches to problemsolving. The students believed that following the teacher's way is the wisest thing to do inorder to get the correct answer. They are usually told what they have to do and they havehad few chances to say how they think. The transmissive mode of teaching leaves littleroom for students to think, to guess, to speculate, to conjecture, to reason, or to take anyrisks at all. These activities are not greatly valued in their classes. Students' beliefs aboutmathematics and mathematical problem solving directly affect their approaches to problemsolving. They usually try to remember the "right" formula for a given problem. This isone of the reasons that they do not see any point in looking for various ways to solve aproblem. In fact they believe that variations in methods of solving a problem indicate aweakness not a strength and there is always "the best way" for solving any problem, whichis "the teacher's way." This view was expressed very clearly by Vicky when she wasasked:IVickyIVickyIf you had this problem in class, what would be your teacher's way to doit?He would read the textbook. The book says how to do it and then we justdo it.Does everybody have different ways of doing the problems, or are they thesame?The first one [first problem] like everybody is gonna get the different one,like second one may be more and the third one will get better and then36like the tenth one . . . like everybody has the same thing as the teacher .. .it depends on how the teacher teaches.I^What do you like to do? Do you like to solve the problems in your way ordo you like to do what he says?Vicky^I just do what he says.I^Why?Vicky^Because that's what he says!I^Do you have all the trust in him?Vicky^Ya, the book is right.Ed said that it was safer for him to do what his teacher told him to do.Ed^I prefer going by teacher's method because if I go by myself and I do itmy . . . well, I just might do it wrong.The students believed that their success, which in their view was the correct answer,would be guaranteed if they followed teachers' way of solving problems. Fear of not beingsuccessful stops them from taking the risks which they cannot afford.Ed^The teacher is probably right. He knows better than me. Then if I go bymy way then I might get a faster way, but I might just do it wrong.Interestingly when an alternative way was suggested to him he responded differently.I^I was wondering whether you had a chance to do the problem in adifferent way with the teacher's presence . . . I mean if your teacher wassupervising you in a small group, to solve the problem together and discussit together, which way would you then prefer?Ed^I then prefer to do it my way because I remember it longer.His response was amazing. He naturally liked to do it his own way. He would"remember it longer" since he had been involved in the process of doing it rather thanusing the teacher's finished product.37The first pilot study informed me whether teachers' approaches to teaching andlearning mathematics inclined more toward negotiation than imposition (Cobb, 1988), thestudents would have a better chance to use their cognitive and metacognitive skills. Alsotheir views about mathematics and mathematical problem solving would be altered as aresult of the instruction.Educational significance of the first pilot study.It seems that if mathematics is presented to students via problem solving and ifthey are given a chance to think and to do, they learn more mathematics than by followingteachers' footsteps. In Sara's interview, she was stuck because she could not "remember"what to do. She said, "I know I was told how to do this!" yet she forgot. When I askedher; "Can you stop thinking about what you were told? Read the problem again and seewhat you can do," it did not take more than a minute or so before she said, "Hold on! .. .There is a pattern!" I asked her what happened, and she said, "Well . . . figuring out . Imean after awhile that you multiplied a number by 3 so many times . I probably wouldnot realize that there is a pattern unless I stopped and started over and looked back to seewhat I have done that makes it easier."The pilot study suggested that students would enjoy mathematics and learn morefrom it if mathematics teachers valued the processes of students' problem solving and if themain goal of school mathematics were not only the finished product of students' work, acorrect answer. Sara found different ways of doing the task only when she had anopportunity to try different methods. She commented, " If you say that this is the firstthing that came to my head, you try it and if it gets too hard you have to go back and .you have to force yourself to find an easy way to do it."It was a new thing for her to take a step back and "force" herself to find otherways of doing a problem. In fact, it would be a breakthrough for all students to gain moreconfidence and try to think, make decisions, and look for alternative ways when solvingproblems.38Second Pilot Study The second pilot study was carried out in the Winter term of 1991. I was ateaching assistant for Mathematics 335 (Introduction to Mathematics). The purpose ofthe second pilot study was to assess the potential research design, to revise the students'evaluation scheme, and possibly to alter some of the course material such as worksheets.The course met for three hours of lectures (Mondays, Wednesdays, and Fridays) and twohours of tutorials (Tuesdays and Thursdays) for 13 weeks. I was in charge of the tutorialsessions, although the instructor was always present. These sessions helped me assess theeffect of the instructional techniques and the instructor's role in the class and with regardto the students. Furthermore, the pilot study gave me the chance to revise my plans forstudent evaluation and to devise more suitable criteria for marking, since I was responsiblefor the marking of the students' monthly projects (three altogether) and the students'journals. The final exam was marked by the course instructor.Summary of results of the second pilot study. I suggested to the instructor collecting and marking the students' journals everysecond week. The instructor gave me the following criteria for marking the journals: (a)presentation, (b) integrity/originality, (c) completeness, (d) clarity, and (e) mathematicallevel. However, I realized that my perception of journal writing and its role for students'learning was different from that of the instructor. The marking scheme gave too muchweight to each criterion without much flexibility. I recognized that such criteria wouldlimit the students' creativity and structure their writing to a great extent. Also, givinginterim marks for the journals based on the rigid criteria did not seem to be a usefultechnique if the purpose was to help students to become more reflective. The pilot alsohelped me design a variety of quizzes and assignments to assist the students with theirlearning.I attended all the lectures and took field notes while sitting at the back of theroom. Those notes included my reflections on the course material and the students'39responses to the instruction in general. I regularly and constantly talked to the studentsand the instructor after each class. I gained insights from the students and always sharedthose with the instructor. Some of the course material for the main study underwent smallyet significant changes as a result of those reflections and the interactions. For example,one of the first worksheets that the students got was titled "Two easy questions aboutangles." The title disturbed many of students. They were concerned about why they hadso much difficulty and could not solve those problems if they were "easy." One studenttold me that she felt so stupid because the problem was supposed to be easy and still shecould not do it. These things might seem to have little or no significance on the wholeteaching and learning process. However, they sometimes cause great dismay and result inlower self-confidence for the students.I noticed that during the "lecture" hours, the students were not involved in theteaching process. Only students in the first few rows raised questions every once in awhile. The rest of the class were mainly taking notes and did not interact with others.However, in the tutorial sessions more students interacted with each other and with us(the instructor and I) while working in small groups. In those sessions, the problemshelped to clarify some of the mathematical concepts which were taught in the lectures, butwhich students had difficulty understanding.The pilot study gave me the confidence to teach mathematics via problem solvingrather than teaching mathematics for problem solving. The results showed that studentswould be more engaged and better motivated if the mathematical concepts were presentedto them through problems.40Methodological Elements of the Main Study The results of the two pilot studies gave considerable insight into redesigningdifferent parts of the proposed study. These studies supported a qualitative method assuitable for the purpose of the study. The following sections give a description of themethodological elements of the main study, an account of the trustworthiness, and anexplanation of the process of analysis.Subjects of the Study Students in the course introduction to mathematics (MATH 335) at the Universityof British Columbia (UBC) in the summer of 1991 were the subjects of this study. Forty-three students registered for the course; however, three students withdrew. One studentfound a good job that she could not refuse. Before the first quiz, a second one said, "I ama perfectionist, and I know that I can not do it perfectly; therefore I want to withdrawfrom the course." A third one withdrew after the second week of the course because itinterfered with her other courses due to the difficulty that she had with the registrar office.The remaining forty students filled out the consent forms (see Appendix B) givingpermission for their work to be used as the data for the study after they read my letterdescribing the study and informing them that pseudonyms will be used for the sake of theirconfidentiality (see Appendix A). Every student except one consented to be audio- andvideo-taped in the classroom. None of the students except one, had ever successfullyfinished any university mathematics course; yet a number of them had already earned theirbachelor's degrees in various fields, and the rest of them were in either the third or thefourth year of their program. Ten of the students were males, and thirty were females.The students' ages ranged from early twenties to mid-fifties. Most of the older studentshad enrolled in the course following a decision to change their careers and becomeelementary school teachers. None of the students except one, took the course as anelective; they all clearly said that they had to take the course because they needed it, not41because they liked it. At least half of the students worked long hours and were extremelytired when attending the class in the evenings.Description of MATH 335 Introduction to mathematics is a mathematics content course which is offered bythe Mathematics Department and designed for students who have a university educationthat does not include a mathematics course. MATH 335 is a required course for studentswho wish to apply for the University of British Columbia elementary teacher educationprogram and who have no other university mathematics course. However, the course isopen to all students with various background and educational purposes. The goal of thecourse has been to give the flavor of interesting and important mathematical issueswithout using calculus (see Appendix D for a copy of the course outline). The coursecontent is divided into three sections: geometry, numbers, and probability. The designerof the course has hoped to introduce the concepts in a more intuitive way to give studentsa chance to become more actively involved in doing mathematics. The emphasis of thecourse is not to "stuff' students with rules and facts but to engage them in the process ofdoing mathematics in a more concrete manner.Sequence of the Instruction The instruction took place Monday to Friday from 4:30 to 6:30 p.m. for six weeks.It started on May 6 and ended on June 14, 1991. This is in contrast to Winter term (thecourse is not offered in Fall term), when it is taught for 50 minutes per day, five days perweek, including three hours of lecture and two hours of tutorial for 13 weeks. Teachingassistants are normally in charge of tutorials and marking, except the final exam. In theSummer, I did not separate the lecture from the tutorial. The teaching assistants and Iworked with students all the time. The amount of lecturing was limited to 15 to 20minutes each day. The rest of the time was spent on working in small groups and whole-class discussions. There were also office hours for those students who needed extra helpoutside of class time. However, only a few students showed up at those hours, since most42of the students preferred to stay a little longer each night and ask their questions. I washappy to spare half an hour after the class time was over. The students were allowed totake a ten-minute break, but only a few of them stopped working and left the classroom.The first two weeks of the course were devoted to geometry, the next two weeksto numbers, and the next week and a half to probability. The course was reviewed in thesecond half of the last week, and the final exam was given on the last day of the class.Nature of the Metacognition-based Instruction The metacognition-based instruction used three different but interrelatedtechniques that focused on metacognition. These techniques are somewhat similar to, andyet distinct from the ones that have been used in the other studies mentioned in the secondchapter. These techniques were (a) journal writing, (b) small-groups problem solving, and(c) whole-class discussion. Along with these, two other less used but significanttechniques were (d) teacher as role model for metacognitive behavior, and (e) usingvideotapes. A short description of each of these techniques and the rationale for usingthem are given in the following sections. The social norm of the class is also discussed insome detail.Journal writing. Journal writing was one of the main activities in this course. The students wereasked to write at least two or three entries each week. There was no assigned structurefor writing the entries. Students were free to write about anything that they wishedregarding the course. At the beginning of the course, I explained journal writing as acommunication channel between the students and the instructor that facilitatesindividualized instruction. This potential benefit of journal writing was even moreimportant considering the diverse background of students.I had chosen a number of students' journal entries from the previous year to read inclass to give students a better sense of what I meant by journal writing regarding amathematics course. However, I changed my mind and let students interpret it by43themselves. I thought the chosen entries might become a norm for their writing.Moreover, the students asked me to give them some idea for writing in a mathematicscourse since they had never done writing in respect to mathematics. I therefore madesome suggestions of possible writing themes such as writing about (a) your feelingstoward mathematics, (b) reflection on the course and day-to-day activities in the class, (c)why you did what you did in order to solve a problem and how you did it, (d) what helpedyou to understand a concept or a problem or what hindered you from doing so, (e) yourfrustrations and difficulties regarding different issues in the course as well as your joyfuland happy moments of discovery and satisfaction.As a result of loosely structured writing assignments, I got a great variety ofstudents' responses which I would not have gotten from them otherwise. Some of thestudents wrote mainly about their difficulties with problems that they solved eitherindividually at home or cooperatively in the class, but others mostly reflected on their ownfeelings and beliefs about everyday activities in the class without quite putting them intothe course context. However, the majority of them wrote about many issues situated inthe course context. I collected the journals every Friday and returned them the followingMonday. It took me at least thirty hours each weekend to thoroughly read the journalsand respond to them accordingly. The evaluation of the journals is described in a separatesection under the heading "Course Evaluation."Small-group problem solving. Journal writing was designed to give students the chance to become morereflective on their own actions in class as well as the opportunity to think about theirthinking and talk about their feelings. For many students, writing provided an occasionfor personalizing and internalizing the learning processes. On the other hand, it wasexpected that students would learn to assess and monitor their work and make appropriatedecisions by discussing the problems among themselves in small groups. Therefore askingstudents to work in small groups seemed a reasonable way to achieve this goal and to help44students to become more reflective. I will give a brief description of why I chose smallgroups as one of the main instructional techniques.As I said earlier, the intention for the instruction was teaching via problem solving(I will explain it under the same heading). However, I believed that it was impossible toteach mathematics through problem solving without having students interact with eachother in the class. "Bucket- filling" teaching does not leave much room for discussion andcreativity. As Dewey once said, a child (and any student for that matter) is not a cup to befilled. Students are full of interesting ideas. One of the important roles of every teacher isto cultivate those ideas, to allow them to grow and develop. Literature is overflowingwith research findings regarding the role of small groups in students' learning (e.g.,Schoenfeld, 1989; Good, Grouws, and Mason, 1990; Maher, and Alston, 1985; Noddings,1985; Dees, 1985). There is great variation in the ways in which different researchers andteachers have used small groups in their studies and their classrooms.Vygotsky's (1976) ideas give a strong justification for the role of small groups instudents' development. He argued that working in collaboration with others helpsstudents to reach the "Zone of Proximal Development (ZPD)." Vygotsky (1962)explained that a child might be able to function up to a certain level, but workingcooperatively with others who are more capable might help him or her to function at ahigher level. The potential ability that a child (or person at any age) has and that could becultivated by some assistance but not in isolation, is the ZPD (also see Schoenfeld 1985a,1987a).In MATH 335 classes, students were asked to work cooperatively when solvingproblems. Some of the students reflected on this issue in their journals and expressed theirconcerns. However, not all the concerns were of the same nature. Some of the studentswere afraid of everyone else in the group finding out about their "stupidity," yet the otherstudents felt insulted by the idea. Melisa voiced that concern in her journal as, "I wasextremely taken aback by the suggestion that not only was co-operative learning okay, but45that it was required for this class." However, after about two weeks of debating andexperiencing, almost everyone felt comfortable about working in small groups, and theyeventually found a suitable one. The small groups were not structured by any means, andtheir sizes varied from three to seven or even eight members. The students liked theunstructured approach to formation of these groups. For example, Clara, in an interviewat the end of the course, said, "It was nice in the class the way that groups kind of formed.Everybody had their own little group that they worked well with." Many of the studentschanged their groups a number of times until they found a "right" one to work with. WhenI asked how they chose their groups and whether it just happened or they changed theirgroups at all, Clara said, "I did change it [my group]. I was working with Kent first, but Ifound that he was [pause]. . . he knew too much for me at first, so he kind of talkedbeyond me, but later on I could deal with him. I couldn't have at first because . . . I hadno idea what he was talking about." The interesting phenomenon is that after Clara feltconfident working with him, she did so, and they worked together from that time to theend of the course.Whole-class discussions. The pilot study showed that discussing what went on in small groups and sharingthe ideas was a necessary activity in the class. Whole-class discussions was one of themost helpful strategies to enhance the students' understanding of mathematics, and toinfluence their beliefs about mathematics and mathematical problem solving. However, ittook more effort to establish a social norm for whole-class discussions in which everystudent felt comfortable to get actively involved in the meaning-making processes, and toeventually become more self-regulated and a better problem solver.In the first three weeks of the course, I gave the students worksheets relevant tothe concepts that had been or were to be discussed by means of those problems. Later Iwill explain the ways in which they were prepared. The worksheets presented relativelyrich problems, ones which could be approached differently and which required some46planning and using of different problem solving strategies. However, different groupsworked at different paces, and this fact created some difficulties regarding whole classdiscussions. To be specific, the teaching assistants and I felt that the worksheets werebecoming the focus of the class discussions. No matter how many times we told them thatthose problems were not the end in themselves and that the important thing was theprocess of solving problems not just getting the correct answer, they were still worriedabout not finishing those worksheets during the class time. I therefore decided to give thewhole class a problem and ask every group to work on the same problem simultaneously.This technique was very useful in bringing the whole class together while discussing aproblem. It also relaxed the students since they were not worried about finishing theproblems as before or falling behind the others. Gradually they gained more cognitive andmetacognitive knowledge of problem solving, a better sense of responsibility, greater self-confidence, and ultimately the motivation to do some of the worksheet problems at homerather than holding the class responsible to do every single problem for them. The classobservation and the students' work provided evidence to support this claim. The studentseventually realized that they would learn more by analyzing and solving one problemthoroughly, than by solving many problems without really understanding them.After they all talked it over in their groups, one person from each group shared thesolution or approach of his or her group to each problem orally with the class while I waswriting it on the board (or they themselves wrote it on the board). Sometimes a problemwas solved or approached in four or even five different ways depending on its nature. Thepilot study and my previous teaching experience led me to believe that such activitieswould be extremely helpful in enhancing the students' metacognitive abilities. They had tomake decisions after they were confronted with the variety of approaches to the sameproblem. By examining different possibilities, they had a chance to see how one strategycould work better than the others and why. Everyone was encouraged to ask the personor group that solved a problem on the board for explanations. The debating and dialogue47helped the class to extend their understanding of the problem and consequently therelevant and related concepts, as well as to gain self-confidence and self-respect. It waspleasant and reassuring for them to see that although their approaches and solutions weredifferent, they were acceptable and valid. Besides, they had to decide which approachmade the most sense to them since they had the opportunity to examine many possiblealternatives. This process also helped self-regulation which is one of the crucial issuesregarding metacognition.However, sometimes nothing seemed to work, and no approach yielded a solution.This could lead to the students' frustration and disappointment. In such instances, I tookcontrol of the class discussions by summing up what went on in those sessions. Theintention was to help the students realize why they got nowhere, what possibly wentwrong, and what could be done. This activity helped them to see both the pitfalls and thelight through the awareness and the knowledge that they gained by examining the differentapproaches to the problem. Then they were better able to put everything in perspectiveand decide what to do to solve the problems. These events served as a vehicle to promoteself-regulation.Class discussions provided good opportunities to discuss the students' beliefsabout themselves, mathematics and mathematical problem solving. It was interesting tosee how those beliefs affected the ways in which the students approached the problems. Iwill discuss those in detail in the next two chapters.Teacher as role model for metacognitive behavior. Schoenfeld (1987a) once in a while used the "teacher as role model formetacognitive behavior" technique to model a good problem solving behavior (see alsoLester, 1988). His intention was to go through the solution processes in detail as if hewere confronted with the problem for the first time. He then discusses about theadvantages and shortcomings of this technique. One of the benefits for students is thatthey see appropriate problem solving behaviors that they might adapt. The main48disadvantage in his view is that "this kind of modeling approach is artificial and must beused sparingly" (p.200). Lester (1989a) taught the seventh graders as part of the"Metacognitive based instruction" project. He, too, occasionally adopted a similartechnique. At the end of the project, he expressed the same concern as Schoenfeld did,noting that there was hardly any new problem at that level (seventh grade) which he hadnot seen before. When he pretended to act like someone who was just confronted withthe problem and wanted to model good problem solving behavior, he admitted that hisaction was indeed artificial.I thought about the ways in which this technique might be utilized in my courseless artificially. I came to the conclusion that this strategy would be embedded in thewhole class discussion to some extent. In whole class discussions I did not limit thestudents to the assigned problems, and when the students or I were confronted with a newproblem directed to us from other students, we indeed had to demonstrate our problemsolving behavior. Sometimes these problems were extensions of the given problems, andoccasionally they were "what-if' questions which deserved genuine, rather than artificialproblem solving processes. Since the students were expected to discuss and speculate onthe varieties of possibilities regarding the problems, I eventually encountered unexpectedsituations that required "on-the-spot" thinking and decision making. As Schoenfeld(1987a) and Lester, Garofalo, and Kroll (1989b) mentioned, neither they nor I used thistechnique frequently. However, occasionally such incidents happened unplanned andproved to be beneficial.Using videotapes. Another technique that was adopted by Schoenfeld is the use of videotapes ofother students working on the problems. He describes that his students, after watchingthe video, empathized with students in the video and said "that could be me." Schoenfeld(1987a) notes, "That's precisely the point. It's a lot easier to analyze behavior when it'ssomeone else's, and then to see that the analysis applies to yourself' (p. 199). Such events49could help students to become more aware of their own thinking processes which is one ofthe main aspects of metacognition. This technique was used only once in the course, and Iwas quite surprised by its effectiveness.Before the course, I thought of using such a technique, but I was not sure where toget the appropriate videos suitable for the course. Fortunately something happened duringthe course that helped me to find one. In the second week of the course, Paul Cobb(1991) visited UBC and made two presentations. In the second one, which was mainly forteachers, he wanted to show a video clip of elementary students working on somemultiplication tasks. I decided to take all the students to that presentation (with theirconsent). Knowing Cobb through his work (Cobb, 1983; 1988; 1989), I believed that thetalk would help the students to become more reflective and would possibly stimulate theclass discussion.Amazingly, many students reflected on that video clip and identified themselveswith students in the video. They talked about that situation mainly in their journals andrarely in the class discussions. This activity helped them to become more aware ofmetacognitive issues. The point that I am trying to make is that this technique was indeedhelpful. However, finding and choosing relevant video clips is not an easy task.Teaching Via Problem SolvingThe instruction was not only based on metacognition, it also adopted the approachSchroeder and Lester (1989) call teaching via problem solving. The course coordinatorand I tried hard to find interesting and engaging problems and activities that woulddevelop mathematical ideas as students solve those problems. The problems were chosenin a manner to make connections between mathematical ideas and to lead one idea to theother. In teaching via problem solving, I gave no formula or procedures to students unlesswe as a whole group arrived at them.It is impossible to discuss every problem and activity that we did in class, but inthis section I would like to clarify what I mean by teaching mathematics via problem50solving using examples from three major parts of the course, geometry, numbers, andprobability, and then discussing one problem related to exponential growth in a greaterlength.The course started with two puzzles (Appendix E). In the dodecagon puzzle, wedeductively proved that the pieces of dodecagon could be rearranged to make a perfectsquare without losing the area. In the other example, the students saw that by rearrangingthe pieces into rectangle they lost or gained one unit. Some of them thought that therewas either an error sight or something wrong with their cutting. They tried everythingthey could. A number of them noticed that if they fixed the corners to make 90 angles,there was either an overlap or a gap in the middle. Kent suggested making sure if thediagonal was indeed a straight line by finding its slope at different parts. His insight ledthe class to see that the figure was not a perfect rectangle. Some of the students noticedthat by cutting the pieces of these squares and reassembling them, the area of so calledrectangles were always one unit more or one unit less than their corresponding squares.They finally realized that there was an interesting pattern between the side lengths and theareas of the squares and the side lengths and the areas of their corresponding imperfectrectangles. Those patterns involved the Fibonacci numbers. The students were fascinatedto see that many natural phenomena correspond to those numbers. These puzzles helpedus to have better understanding of mathematical reasoning. They were extremely helpfulin taking the class to the geometrical proof.The manipulative materials helped us to understand the concept of area andvolume. The whole class developed the ways in which to find the areas and the volumesof different polygons and polyhedrons. We eventually came up with working proceduresand formulas for those areas and volumes. Dissecting a cube into three irregular pyramidsshowed them that why the volume of a pyramid was 1/3 of the volume of the cube, and byextension, any prism. Using Cavalieri's Principle, the same factor 1/3 relates the cone tothe cylinder. Therefore, the 1/3 factor was not a mystery to them any more because they51actually saw it. We finally found the volume of hemisphere together, and all I did was torecord and organize their ideas. We then moved on to scaling and Cavalieri's Principle.Then finding the area of the unit circle took us to the section on numbers.Finding it by repeated doubling of the number of sides in a regular polygon was anexciting and engaging activity. The students were happy to calculate it as 3.141592 bythemselves. Many of them said it had always been a magic number for them prior to thecourse. Using Newton's method to find square roots of numbers was also engaging. Itwas presented as a way of changing a rectangle into a shape more and more like a square,by successively averaging its sides. As Peggy said, activities like that helped the studentsto have a greater appreciation for mathematics. Periodic and terminated decimalsapproached via Zeno's Paradox, and the ways in which to convert them into fractions gavestudents a chance to find many interesting patterns, and they eventually establishedformulas to facilitate their calculations. We found out where the numbers in logarithm andtrigonometric tables came from using graphs. For instance, we got the fractional powersof 10 by filling the gap between 10° =1 to 10' =10 using calculator. We then took onestep further and constructed the graph of a =1022. The graph helped enormously tounfold the mystery of logarithms (Appendix E contains a copy of the graph). The studentsgot a sense of empowerment knowing how the keys on calculators operate and wherethose magic numbers come from.I gave them simple problems to start a section on exponential growth and decay,including a problem involving doubling one's money everyday for a month, compoundinterest, growth of bacteria, and carbon dating. The students had special interest in theseproblems since their contexts were familiar to them. For example, there were greatdiscussions in the class regarding interest rates, inflation, budget deficits, cancer cells, thevalidity of carbon dating in identifying the age of artifacts, environmental issues, and soon. Problems of these kinds were extremely helpful in developing the mathematical52concepts and formulas. The following is an example of a problem regarding exponentialgrowth. (Appendix E contains two problems regarding exponential decay.)The cost of real estate in a certain city has gone up at a rate of 10% per year since 1975.If this trend continues, approximately what will a house be worth in 1995 if it waspurchased in 1985 for $120,000?Most of the students had little difficulty solving the problem because they werefamiliar with the context; they simply multiplied each year's total price by the factor of 1.1.I then suggested that they make a systematic list. Therefore, rather than multiplying theobtained price by 1.1 to get the new price, we multiplied $120,000 by 1.1, then the nextyear's price was increased by the same factor as ($120,000x1.1)x1.1, and so on. Theyeventually better understood the exponential growth, and they were able to easily arrive atthe formula to facilitate the calculations when large numbers were involved. The sectionincluded more and more complicated problems. I was surprised to see that theirunderstanding of the concepts allowed them to solve those problems without muchdifficulty.In teaching mathematics via problem solving I as the teacher had a special role toplay. I orchestrated the class discussions and monitored the students' work in smallgroups. The course material included many interesting problems and activities to developmathematical concepts. However, I was constantly looking for interesting and suitableproblems everywhere I could to accommodate the students' need. I tried to include arange of activities considering the students' special interests. Small groups wereparticularly appropriate to take those interests into account. For example, the relationbetween mathematics and painting, woodworking, baking, and games were among them.The students' journals provided me a unique opportunity to consider the students' needsconcerning different mathematical issues and to stress them during the instruction. I spentan enormous amount of time regularly to take every student's concern into account.However, these efforts made the study a "qualified success story."53All these activities required many hours of hard work and continuous reflection onwhat went on in the class to be able to improve the situation. However, this hard workpaid off enormously, and added much joy to my teaching.Many practical problems helped us developing the concepts and formulas inprobability. Without any lecturing or imposition in my part, those problems helped toestablish the concepts of combinations and permutations, and led us to develop formulasto facilitate the solution of problems involving larger numbers. Many students expressedtheir concerns about this section of the course in the beginning. However, many of themended up liking this last part more than other parts of the course. This section also madepossible the use of a number of problem solving strategies.Social Norm of the Class The most suitable environment--or "classroom culture", as Davis (1989) calls it--for teaching and learning mathematics, is one which is more natural and less artificial. Anumber of mathematics educators who have done classroom-based research (e.g.,Schoenfeld, 1985a, 1987a, 1987b; Lester, Garofalo, & Kroll, 1989a, 1989b; Raymond,Santos, & Masingila, 1991) have concluded that appropriate settings help students tobecome more aware of their thinking processes and to develop positive beliefs towardsmathematics. For example, it is easier for students to analyze another student's behaviorwhile solving problems, and be able to associate themselves with that when he or she issolving a problem for the whole class and when everyone else has chance to see what heor she is doing. Schoenfeld (1985a; 1987a) talks about the creation of a "microcosm" ofmathematics in a way similar to the way that mathematicians do mathematics. Lester,Garofalo and Kroll (1989a) found that students' beliefs are mainly shaped by classroominstruction and the classroom environment. These findings led to the creation of a "socialnorm" of the class which was an appropriate environment for metacognition-basedteaching and teaching via problem solving.54It was not easy for the students or me to set up such a social norm. It was difficultfor the students to change their views of the teacher's role. Many of them expected me totell them what to do all the time. In the beginning, a majority of them preferred "spoonfeeding" instruction since they were used to being taught that way more often. Theyreflected on the teaching method in their journals and expressed their dislikes or theirconcerns about this issue. Comments such as "Please give us a formula to solve theproblem . . . ," "tell us which is the way of solving the problem because multiple ways ofsolving it is confusing and a waste of time . . . ," "why are you so mysterious? Just tell uswhat to do," were not exceptional in the first two weeks. However, they diminished asthe course moved forward. My goal was to help students realize that their involvementand participation was a crucial contribution to the teaching and learning process, so Ireflected on their comments in reasonable detail and attempted to address those concernsboth through their journals and in the class.I was hoping that they would become independent and conscientious learnersthrough discourse and interaction with each other and that they would reflect on what theydid and why they did it. I wanted to show them that the teacher is not the authority in theclass who is always "right." On the contrary, my aim was to confront them with a teacheras a moderator, facilitator, monitor, friend, and role model who is also fallible, that he orshe indeed experiences the ecstasy of solving a problem after spending lots of time andeffort, as well as the agony of not knowing and being frustrated and confused.I tried hard to make sure that the students were actively involved in all the classactivities. I did everything I could to convince them that their participation in the classdiscussions was indeed beneficial to themselves and to other students. I talked to themindividually, wrote about it in their journals, and spoke to the whole class hoping that mostof the students would finally come along and participate in group discussions. Like mosttypical classes, only a few students were really engaged in the class discussion in thebeginning. Sometimes the active involvement of those students and their strength of55reasoning intimidated the other students. A number of them reflected on this issue in theirjournals. They argued that since those students were better able to solve the problems andexplain them to others, why should they "waste" the class time with their "wrong"solutions and "inadequate" explanations. My aim was to show them that there is notalways one correct way of solving a problem. Besides, everyone would learn a great dealfrom each others' ideas and even mistakes. I talked to them about the ways in which mostmathematicians did mathematics to came up with world shaking ideas. I challenged themyth that "some people have it and some people don't" concerning mathematics. The goalwas to let them to believe that everyone can do mathematics to some extent and that thebest way of doing it is to get involved and take responsibility for one's own learning.My role as a facilitator and monitor was crucial, especially in small-group work. Ihad to make sure that every single student was engaged in mathematical activities in thegroups. I believed that working in small groups would be effective, if and only if thestudents were all negotiating the meaning of what they were working on. The studentshad the responsibility to make reasonable decisions after discussing the problems in theirgroups. I knew that in group work, there was always a tendency for some people topassively follow the others in the group without being involved in the meaning-makingprocess. To avoid situations like that, I reserved the right for myself to intervene andmake sure that everyone knew his or her own responsibility.As the course moved forward, more and more students participated in the classdiscussions and felt less intimidated by others' ability. A major breakthrough wouldhappen when the students' fears dissolved, and they took responsibility for being part ofthe class discussions in a respectful and friendly manner. I did not expect all the studentsto come along and be part of the class discussions immediately since I believed that "oldhabits die hard." However, I was hoping that more students would become interested toat least try it. There were only a few individuals who never showed such an interest."Why don't you go to the board?" I asked one of them once. He laughed and said56"Because I am a chicken!" He was a private person and although he was in a small group,he never really engaged in the group's activity. My intention was to encourage thestudents as much as I could, but I did not want to force them to work cooperatively in thegroups. I would be glad if only a few of them changed their beliefs about the issue.However, I was certain that the class atmosphere had a major role to play in making thestudents feel comfortable expressing their ideas.One day the class was discussing a problem on probability. I wrote every small-group's responses on the board, and then I asked the students to discuss them and seewhich ones made more sense and why. Nina argued that her response made sense to herbecause "It sounds mathematical!" "I don't want to create any friction," said Jack, "but Ihave an argument against it." This was an example of the social norm that I hoped toestablish. Such a social norm would provide the students the opportunity to becomemetacognitively aware, to take control of their own learning, and finally, to help a greatdeal to achieve the goal of changing the students' beliefs toward mathematics andmathematical problem solving.In conclusion, establishing a suitable social norm for the class was indeed the mostdifficult task to do regarding the instruction. It took lots of effort and mutualunderstanding to create such a norm. My role as a teacher was crucial in the creation ofsuch a norm. I listened to the students carefully and appreciated the suggestions andcomments that I received from them. The students' suggestions and comments helped meto become more sensitive to their needs and to make decisions accordingly. It took atleast ten class sessions for everyone to realize that the desirable environment was one thatwas not only relaxed and friendly, but it was also engaging and thought provoking.Course Evaluation On the first day of the class I talked about the evaluation: 50% for final exam, thatwas required by the Mathematics Department at UBC; and the other 50% for assignments,quizzes, and journals. I did not have control over the percentage of the final exam.57However, I left it to the students to decide the weightings for other 50%. They agreed tohave 25% for quizzes and assignments ( six altogether), and 25% for the journals.None of the students was satisfied with the final exam at 50% of their mark. I didnot like it either since my intention was to provide an opportunity for the students to learnwithout being frightened by the final exam. I took the issue to the coordinator of thecourse. He and I decided to change the final exam's weighting from 50% to 40%, becausethe final was a two-hour exam in the Summer term as opposed to in the Winter term athree-hour exam (worth 50%). Therefore the weightings of journals and assignments(including three quizzes) changed from 25% each to 30% each.Except for part of the second and third quizzes, which were marked by one of theteaching assistants, quizzes and assignments were marked by me. The evaluation of thequizzes and assignments was not based on the "correct answer," but on the gist of thestudents' reasoning. The students were asked to explain why they did what they did andhow. The students were also asked to answer a series of metacognitive questions as partof their quizzes.The journal entries were collected every week to be read and commented on.However, no interim marks were given to the journals, since I wanted the students to feelfree and write the journal entries in any way that they wanted to. Otherwise, the markingscheme might dictate to them what to write rather than help them genuinely about theirfeelings, ideas, concerns, and criticism. Except for the second week's entries that wereread and commented on by both the teaching assistants and me, I read and commented onall the students' writing carefully and gave them my feedback, trying not to impose myideas on them.The criteria used for marking the journals were the maintenance of at least two orthree entries per week and volume of writing. By volume I mean the writing had to besubstantial enough to make a point. It was not enough for the students to write about58their "likes" and "dislikes" without any explanation. However, they were not denied anycredit for the mechanics or viewpoints expressed in their writings.One of the main purposes of journal writing was for the students to reflect on theexisting issues regarding the course, so it was crucial for the students to write regularlyevery week. Marks were taken off for late submission of the journals. For example, Tonynever wrote anything in his journal until the last day of the course when he handed thewhole thing in all at once. "I could not do it regularly because I am working," he said. Hejust wrote to fulfill one of the requirements of the course. This kind of writing could nothelp him to become more aware of the issues and be reflective, because it was only storytelling, full of fancy words without much meaning attached to them. Therefore, he got15% out of 30% for his journal.Towards the end of the term, I asked the students to evaluate their journalsthemselves and write about it in their journals. They had to justify the self-given markwith good reasons. After reading their journals, I was surprised with some of the studentswho gave themselves marks that were lower than what I thought they deserved, and I wasdisappointed with few others who gave themselves full marks (30 out of 30) and whoseonly justification was that "I worked hard and I deserve it" or "I really need to pass thecourse." Furthermore, a majority of the students had a kind of analytical approach andtheir self-given marks were well funded. However, experiencing this, I told them I saidthat I would reserve my right to adjust the marks if they were not appropriate referring tothose who asked for a mark without any justification.SummaryA metacognition-based instruction was designed to facilitate teaching and learningmathematics via problem solving. A review of related literature gave strong justificationfor the use of metacognitive strategies in a mathematics classroom to enhance students'mathematical understanding. Those strategies provided an opportunity for the students to59become actively involved in the meaning-making process through reflection andinteraction.In addition, as has been suggested by Cobb (1988), the idea of focusing instructionon negotiation rather than imposition was fostered. A wide variety of problems andactivities were prepared prior to the instruction, but as in the study by Cobb, Yackel, andWood (1989) of second graders' mathematical problem solving, those activities needed tobe developed and modified as the study progressed.60Data Collection Procedures Data for the study were collected through four different sources, namely quizzesand assignments (including the final exam), interviews, the instructor's and the students'autobiographies and journals, and class observations (field notes, audio and video-tapes).Task-based Interviews Six volunteer students in MATH 335 (introduction to mathematics) participated intwo task-based interviews, one in the first two weeks and the other in the last week of theinstruction. These six students were chosen after the collection of students' firstassignments and journals. These assignments and journals were used to judge the range ofthe students' mathematical abilities (low, medium, and high). The purpose of the task-based interviews prior to the instruction was to identify some of the students' initial beliefsabout mathematics and mathematical problem solving. The ways in which the studentssolved problems were also documented. In addition, students' knowledge of heuristicsand their ability to apply them in solving problems were investigated. The post-instructioninterviews were aimed at investigating whether the students' views about mathematics andproblem solving as well as their approaches to problem solving had changed as a result ofinstruction. Two other students were interviewed at the end of the instruction in additionto the six students who participated in both interviews. They were interviewed because ofthe substantial changes in their beliefs about themselves as doers of mathematics and aboutmathematics and mathematical problem solving as a result of the instruction.Interview technique. The thinking-aloud technique was introduced to volunteer students prior tointerviews. Perkin's (1981) method for thinking aloud was fostered. The students wereasked to say whatever was in their minds, whether it was a hunch, a guess, a wild idea, oran image. They were expected to speak continuously and audibly. Krutetskii (1976)warned that the thinking aloud is different from explaining the solution aloud. He alsorecommended that the students should know that in thinking aloud they are not explaining61anything to anyone, but talking to themselves. They could do this by pretending that therewas no one in the room except themselves.Appropriate hints such as "read the problem again," "try another approach," and"think about a smaller number" were provided if the students needed them. The aim wasto help the students go as far as they possibly could in revealing their understandings andbeliefs regarding mathematics and mathematical problem solving. The researcher did notpre-structure the direction of inquiry. The tasks provided the opportunity for students totalk about their beliefs and different approaches that they had toward problem solving indoing mathematics, as well as to demonstrate their managerial skills. A friendly remindersuch as "will you please keep talking" was helpful to encourage them to talk whileworking on a taskAll fourteen interviews were audio-taped and eleven of them were also video-taped. The first three interviews were not video-taped because they were carried out inmy office which did not have enough room to set up the camera. However, anarrangement was made to conduct the rest of the interviews in the classroom before orafter the regular class hours. All the students signed a consent form for being audio-taped,and all but one did so for video-taping. The person who did not agree to be video-tapedsaid that she was shy and might get nervous having the camera present. No other personwas present while carrying out the interviews. Pseudonyms were used for the sake of thestudents' confidentiality.Nature of the tasks. The tasks were two challenging and non-routine problems, namely "the hand-shake" and "the Fibonacci problem about the rabbits" or their extended problems (theproblems are presented in Appendices 0 and P). These tasks were chosen to provide thestudents with a chance to speculate, to pursue their hunches, and to conjecture. Therewas usually more than one reasonable approach to these tasks, and the students wereexpected to consider and use different strategies to do them. I thought the tasks could62help me to identify the mathematical processes and operations that the students applied inorder to solve them. The tasks required planning and reflection that would give me achance to observe the students' cognitive and metacognitive behaviors in action.In the pre-interviews, three students did the "hand-shake" problem and the otherthree did the "rabbits." In the post-interviews those who did "hand-shake" problem in thepre-interviews, were asked to do "rabbits" problem for the post-interviews and vice versa.The two students who only participated in the post-interviews, did both tasks in onesession.Use of Journal WritingThis study used journal writing as a means of communication between eachindividual and the instructor. I took this opportunity to write extensive comments(feedback) on students' journals hoping that the feedback would help students to reflect onwhat they did and to become more aware of their own thinking. It should be mentionedthat journal writing was part of the course evaluation.The students' journal entries provided the main part of the data for the study. Theyhelped me to monitor changes in the students' beliefs about themselves as doers ofmathematics, and about mathematics and mathematical problem solving, since the studentswrote them regularly and reflectively after classes.The data reduction regarding journal entries was an integrated part of the dataanalysis. Although I had read the journals regularly during the course of the instruction, Iread them again and again to make a better sense of the data. I summarized the students'writing in terms of the ways in which they utilized their journals, namely; if the studentsonly addressed cognitive issues, metacognitive issues, or both. The next step was todocument the changes (if any) in the students' beliefs about themselves as doers ofmathematics and about mathematics and mathematical problem solving, and the nature ofthose changes. Different themes emerged from the students' responses, and they arediscussed under the heading "data plan" later in this chapter.63In addition to the students' journal entries, my own reflective journal provided datafor the study as well. I wrote regularly during the course of the instruction. My journalwas particularly useful in analysis of the classroom processes. The journal entries helpedme to make more sense of the audio and video-tapes of the classroom processes.Moreover, the students' autobiographies that they wrote at the end of the first classbecame an important part of the data for the study. They gave valuable informationregarding the students' demographic and mathematical background. The autobiographiesprovided a unique opportunity for me to monitor changes in the students' beliefs aboutthemselves as doers and learners of mathematics, and about mathematics and mathematicalproblem solving.Class Observation Two doctoral students in the Department of Mathematics and Science Educationwere hired by the Mathematics Department at UBC as teaching assistants for the course.They both had extensive teaching experience in Canada and abroad. Hazel (pseudonym)assisted me in the first four weeks of the course, and Roy (pseudonym) in the last fourweeks of the six-week long course. They both observed the class and Hazel took fieldnotes occasionally. In addition, Roy more frequently contributed to the class discussionswhile Hazel's contribution was mainly in small groups (as it happens without a specificplan). In particular, Roy gave a lecture to introduce logarithms. Many students reflectedon his teaching method in their journals. Samples of these reflections will be discussed inChapter 5.The aim of the class observation was to document the class interactions whensolving problems as a whole-group or in small groups. I also kept my own journal andwrote immediately after each class about my observations and understandings of theclassroom processes. In addition, the assistants and I talked after each class to reflect onwhat went on in the classroom and to discuss the relevant issues regarding the instructionand the students. The discussions with the assistants and my reflections on classroom64processes provided valuable data for the study. The purpose of documenting theclassroom processes was to investigate the ways in which the students solved problems,the extent to which they used problem solving strategies, the amount of control that theyhad in choosing or abandoning the strategies, the time allocation for each problem, and thenature of their participation.The last four weeks of the class sessions were audio-and video-taped using twoportable tape-recorders and one camera which was set up in the back of the class andoperated by the teaching assistants and myself. The procedure was to get the whole classin the range of the camera as much possible, and to focus it on the different groups whilethe group members were discussing a problem. I moved the tape-recorders from group togroup as the students were working together. Towards the end of the course, I askedeach group to say the name of its members aloud, to be taped while writing them on apiece of paper. This helped me to match the students' voices with their written work.The class sessions were video-taped for two reasons. The first was a technical oneto ensure the "fidelity" of the data, which according to Lincoln and Guba (1985) is "theability of the investigator later to produce exactly the data as they become evident . . . thegreatest fidelity can be obtained using audio or video recordings" (p. 240). The secondreason was an educational one in the sense that Davis (1990) discussed. Videos couldshow to teachers and student teachers the possibilities for a wide range of classroombehaviors . They would acquire a "powerful kind of knowledge" by viewing the tapes.He went on to say that "giving this knowledge to students (or to other researchers) is amajor part of the job of mathematics education" (p. 21). It is worth mentioning that all thestudents except one signed a consent form letting their audio and video-tapes to be usedfor the educational purposes (see Appendix C). The one who did not consent to be video-taped, always sat in one corner, and I made sure that she was not in the range of camera.65Quizzes, Assignments, and the Final ExamAll the students did three "take-home" assignments and took three "in-class"quizzes which were worth 30% of their final mark. The aim was not to test the studentsfor what they did not know. On the contrary, the goal was to engage them in the processof solving a number of challenging and non-routine problems, ones that did not haveimmediate solutions. These non-routine problems helped me to investigate possibleimprovement in the students' methods and approaches while they received themetacognition-based instruction. Two of the assignments and the three quizzes eachconsisted of only one non-routine problem or task. The non-routine problems could offerthe opportunity to identify the students' ability to transfer their content knowledge to othercontexts. The quizzes and assignments along with the metacognitive questions thataccompanied them, provided data regarding the students' beliefs, awareness, andmanagerial skills. (See Appendix F for a copy of the metacognitive questions.) Thequizzes in particular, offered valuable data for the study, since the students' work wasevaluated under time constraint, which is more often the case in normal classroomsettings. The following is the short description of the ways in which the quizzes andassignments were selected.The first assignment which asked for a proof of the Pythagorean Theorem (seeAppendix G), was given on the first Friday of the course and was collected on thefollowing Monday. This assignment helped me to investigate the students' conception ofproof, as well as their ability to do a proof. I talked to those who did not understand theproof at all and asked them to do it again after we discussed it until they were convincedthat they understood it. I changed the marks of those who did a better job in the secondtime to encourage them to seek understanding and to try to make sense of themathematical concepts. However, I made it clear that the same procedure would notcontinue and my intention was to modify it for the other quizzes and assignments.66The first quiz, the one about "length and volume" (see Appendix H) was given onthe second Friday of the class and returned on the following Monday. The first half hourof the class was planned for this quiz, but it took about one hour. The quiz consisted ofone rich problem with five assumptions. The problem was taken from the resource book(Jacobs, 1982) required for the course and was one of the problems in the third"worksheet." While the students were working on this problem in their small groups, Inoticed a wealth of different approaches to this problem and decided to use this problemfor the quiz to collect all those interesting solutions. It also would help the students to berelaxed since they had already seen the problem. The students were allowed to discussthe problem in their small groups. However, they had to write it individually. Workingtogether gave them confidence to overcome the fear of an exam, which most of them hadcommented on in their journals.The problem for the second quiz, about "volume and proportionality" (seeAppendix I), was taken from Schoenfeld (1987b), and was given to the students in thefirst half hour of the third Friday of the course. Like the first quiz, the students worked intheir small groups and then wrote the solution individually. This quiz was also markedover the weekend and returned to the students on the following Monday.The second assignment (see Appendix J) was given on the fourth Monday and wasreturned to the students on the following Friday. The task was to prove that the squareroot of ten was irrational using indirect proof. Some of the students who got low marksfor this assignment, asked permission to do it again. I told them that I did not mind at allgoing through their assignments for the second time. However, I would average their newmarks with the previous ones to obtain the mark for their second assignment.The third quiz was a problem with two parts on the content that had been coveredin the probability section (see Appendix K). It was given on the fifth Friday of the class.A week before the test, the students were told that they had to solve and write the lastquiz individually. Many of them complained and expressed their concerns. However, they67appreciated the idea after I explained that the purpose was to help them get ready for thefinal exam which had to be written individually.The third assignment was a special one since the students were free to choose anyproblem or topic that seemed interesting to them. I suggested the following: (a) apply amathematical concept to a real life situation, (b) formulate your own problem and try toanswer it, (c) choose any problem or topic related to the course that you are particularlyaffected by or disappointed by and explain why, (d) evaluate the course, (e) critique theteaching method, and others. However, it was totally up to them to decide whatever theywanted. This variation of the students' responses added richness to the data. Appendix Lcontains a table showing students' selections for the third assignments.The final exam was given on the last day of class, and the students had to write itindividually. The exam consisted of essay questions and multi-step problems, and itprovided data for the study (see Appendix M). The intent was to evaluate the students'selections of strategies, their approaches to problems, and the coherency of their writtenresponses. These aims were achieved, for example, by selecting open-ended problems thatask for students' explanations. Appendix N contains a table showing students' selectionsof the final exam.All the assignments, quizzes, and the final exam provided data for the study. Theyadded to my insight about the students and helped me to better understand the students'problem solving behaviors under different constraints and conditions.Data Analysis Data for the study were provided through four different sources, namely quizzesand assignments (including the final exam), interviews, the instructor's and the students'autobiographies and journals, and class observations (field notes, audio and video-tapes).Several cognitive-metacognitive frameworks that research has shown to be appropriate foranalyzing the data emerging from a study of this type, such as Garofalo and Lester's68(1985), and Schoenfeld's (1985, 1987a), were considered. However, those frameworksonly served as helpful and general guidelines and were by no means exact or limiting.In the early stages of analysis, I tried to make sense of the data that emerged fromthe students' written tests and assignments, their journals, the interviews, and theclassroom processes separately, and then put them together to draw final conclusions.When this proved unfruitful, I took the advice of Strauss and Corbin (1990) to look at thedata from a different angle. I got the idea of looking for common themes that mightemerge by looking at student by student data.I made a new file for each student consisting of the student's autobiography, thejournal entries sorted by dates, the student's assignments and quizzes, the final exam, noteson classroom interactions taken from video-tapes, and the excerpts of his or her interview,if applicable.This new scheme gave me a chance to look at each individual's changes from thebeginning of the course to the end. Despite individual differences, the students appearedto be falling into three distinct categories. The first category is representative of thosestudents who not only showed little or no interest in different ways of teaching andlearning mathematics, but were also resistant to them. The second category consists ofstudents who were inconsistent. They sometimes showed enthusiasm for new ways oflooking at mathematics and mathematical problems. However, they were concerned thatit might be too late for them to look at things from different perspectives. Finally, thethird category represents a group of students whose beliefs about themselves as doers ofmathematics, as well as their beliefs about mathematics and mathematical problem solving,became more positive during the course of instruction.The description of each category, the range of the students they represent, thenature of changes in the students' beliefs, and the ways in which the changes happened aredescribed in Chapter 5.69Many common themes emerged from the students' responses considering varioussources of the data. The analysis focused on the following themes: (a) instruction and itscomponents, (b) students' beliefs about mathematics and about themselves as doers andlearners of mathematics, (c) students' beliefs about mathematical problem solving, (d)mathematical connections, (e) and tests or other forms of evaluation. The studentsreflected on different components of the metacognition-based instruction, namely: (a)journal writing, (b) small groups and whole-class discussions, (c) time, (d) the teacher'srole, (e) and classroom environment.The findings of the study are organized based on these themes. However, eachtheme is not necessarily discussed within each category. The analysis consists of theeffects of the metacognition-based instruction and the instructional techniques as well.Data Reduction My approach to the data reduction was first to study each student's file separately.Every student was like a puzzle, and different sources of the data were the pieces of thosepuzzles. The more I studied the sources of the data, the better I was able to put the piecestogether.The journal entries were the most important set of data, since they were mainly thereflections of the students on everything else that was going on in the course. In addition,the students' autobiographies proved to be a valuable source of the data considering thepurpose of the research that included the study of change in each individual frombeginning of the course to the end. My first attempt after reading the journals was just tocluster them, which involved rewriting most of them and using different colored markersto write the themes that kept emerging from the students' writing. The next step was tomake sense of what the students wrote and look for commonalities and differences amongthem.The assignments, the quizzes, the final exam, and the metacognitive questions thatwere part of the last two quizzes were helpful in investigating the ways in which the70students solved mathematical problems, the problem solving strategies that they used, andthe effect of the exams on their self-confidence. Different solution approaches were alsoclustered.The following example shows how the data that were provided by the first quizwere reduced for the purpose of analysis. The students used a variety of differentapproaches in finding the area of hexagon, octagon, and dodecagon. The studentsdiscussed the problem in groups and wrote it individually. I first collected all the differentapproaches to finding those areas, first in terms of each student, and then across thegroups. I was interested to see how students inspired each other in groups and negotiatedthe meaning and at the same time asted as decision makers. Working in small groups inexam situations showed how well students understood the essence of group work andtheir responsibility and their active involvement in that regard.The students' assignments that they did individually at home gave me a chance tolook at their work in a totally different situation. The first assignment asked students toprove the Pythagorean theorem mainly using cut outs and manipulatives, and the secondone involved proof by contradiction. I clustered the students' approaches to these twodifferent kinds of proofs based on the nature of their argument, and their conceptions ofmathematical proof.The data from the third assignment, which was an open one, were organized interms of the students' selection of problems and issues. I was interested to find out howstudents might possibly pose problems on their own and discuss the solution processes. Ireduced the data in terms of the students' selection of problems.Responses to each final question were collected and then reduced according to theproblem solving strategies that students used, their understanding of mathematical proof,the generalizability of the proof, and so on. The managerial skills that studentsdemonstrated in their final exam were also clustered.71Procedures for analyzing the students' interviews followed the same format asthose for the video-tapes of the classroom processes. I used some parts of the interviewsto clarify and enrich the findings from the other sources. For example, I transcribed theways in which the students solved the tasks, the kind of problem solving strategies thatthey used, and the evidence of managerial skills. I also talked to them about their beliefsabout themselves as doers of mathematics, and about mathematics and mathematicalproblem solving. I compared and contrasted the data from both pre- and post-interviewsto help me in analyzing the possible changes in students' beliefs about mathematics andmathematical problem solving.Chapter 4 provides a report of the data that were obtained through the audio- andvideo-tapes of classroom processes and my reflective journal that I wrote each night afterclass. I watched the video-tapes of the classroom processes several times until I feltcomfortable with my understanding of the processes. The next step was to transcribe theparts that most clearly showed the nature of the whole-class discussions and captured theessence of the cooperation within small groups. I regularly contrasted the video-tapeswith my reflective journal. This exercise proved to me the authenticity of these two setsof the data.TrustworthinessLincoln and Guba (1985) argue that different research paradigms require differentcriteria for trustworthiness. They explain the basic issue in relation to trustworthiness as:"How can an inquirer persuade his or her audiences (including self) that the findings of aninquiry are worth paying attention to, worth taking account of? What arguments can bemounted, what criteria invoked, what questions asked, that would be persuasive on thisissue?" (p. 290). In general, they used the term "trustworthiness" to discuss credibility,transferability, dependability, and confirmability of qualitative studies. These terms72correspond roughly to internal validity, external validity, reliability and objectivity inquantitative studies.The results of the present study are based on the students' ideas and responses andthat gives "credibility" to the study. All the claims and categories in the analysis aresupported by the students' input coming from various sources of data, namely: (a)interviews, (b) journals, (c) classroom interactions, and (d) assignments, quizzes, andtests. This triangulation enhances the internal consistency of the study and shows that thefindings are based on the analysis of the data which were gathered through differentinstruments.Before the study was even finalized, some of the results and the findings wereapplied to the same course or other mathematics courses through informal discussionswith other mathematics instructors. This applicability is known in qualitative studies as"transferability" (Lincoln & Guba, 1985). The purpose of this chapter has been to give ascomplete a description of the study as possible to help other interested researchers toconduct similar studies and possibly to shed more light on the research in this area. Thisdescription would increase the transferability of the study.Studying and scrutinizing a number of relevant studies (e.g., Schoenfeld, 1987a,1988b; Lester, Garofalo, and Kroll, 1989a; Raymond, Santos, and Masingila, 1991) led tothe selection and development of the methodology and the data collection procedures forthe present study. Furthermore, the appropriateness of the techniques for the analysis wasattested to by the pilot studies. The record of the data gives the opportunity to anyindependent observer who is familiar with the nature of the study to examine theirauthencity and the validity of the collection procedures. This accessibility of the data andthe procedures accounts for the "dependability" of the study."Confirmability" refers to the agreement with the findings on the part ofmathematics education researchers familiar with similar studies. Furthermore, the views ofmathematics instructors about the findings are crucial, since the study was carried on in a73context of mathematics instruction. I believe that I am able to give adequate grounds torefute possible disagreement about any part of the findings.74CHAPTER 4CLASSROOM PROCESSESIn Grade One the teacher split us into three groups; the Beavers, theRobins, and the Ducks. I was a Duck and I have never forgotten thatfeeling. I also realize this gave me a math block or phobia. Today I canunderstand the theories behind physics and chemistry, yet still carry withme a fear of math. I know I will overcome this phobia of math; hopefullythis class will do this.^ (Lilian's autobiography)The main emphasis of this study was to investigate the effect ofmetacognition-based instruction on the students' beliefs about themselves as doers ofmathematics and about mathematics and mathematical problem solving. Themetacognitive techniques (or strategies) that were used to implement the instruction weredescribed in Chapter 3. This chapter focuses mainly on the classroom processes and thestudents' interactions with each other and with the instructor, both in small groups and inwhole-class discussions. These findings are based on the data that were provided byvideotapes of classroom processes and my reflective journal that I kept every day duringthe course of the instruction. The chapter is divided into three sections, namely: rationaleof the course, the nature of the class interactions, and a summary.Rationale for the CourseOn different occasions, the coordinator of MATH 335 and I had severalconversations regarding teaching the course. We talked extensively before I startedteaching in the summer of 1991. It was interesting to hear his points of view aboutteaching and learning mathematics in general, and about MATH 335 in particular. "Whatdo we really want these students to accomplish" was the main theme of our discussions. Iwas relieved to learn that we had somewhat similar aims for MATH 335. The course was75originally designed for prospective elementary teachers, but it was open to all non-scienceuniversity students.The students taking the course would soon become elementary teachers and teachyoungsters. As teachers, they will be confronted with children who do not separate thesubjects. They are in "school" to do things and learn things. These energetic and creativeelementary students like to work with puzzles, and have fun. Mathematics is best learnedwhen it is "fun" and appealing to students. Therefore, the course should expose theprospective teachers to various positive and enjoyable experiences regarding mathematics.Most of them were mature learners, either in their third or fourth year of an undergraduateprogram, or they already had obtained their baccalaureate degrees. Moreover, they havelimited yet diverse experiences with mathematics. The diversity has to be acknowledgedand the course designed in a manner to accommodate the students' needs.The rationale justified the role of manipulative materials, classroom activities, smallgroups, and whole-class discussions as integrated parts of teaching and learningmathematics. These activities and discussions were helpful to achieve one of the goals ofthe instruction that was to encourage students to "do" mathematics. Hands-on activitiesand class discussions helped the students to encounter contradictory situations, whichmeans enjoying the success and feeling the difficulty, and learn that they both werepleasant parts of every experience. In addition, the instruction was to bring aboutmathematical connections to make the mathematical teaching and learning moremeaningful. Furthermore, the classroom environment allowed them to become risk-takersand see that everybody was capable of understanding some sorts of mathematics. The aimof the course was to show them that not only mathematicians "do" and "enjoy"mathematics, but also that non-mathematicians can "do" mathematics and deserve to have"fun" with it. Therefore, the course included many little fun activities and used them toexplore and develop a number of subtle mathematical concepts. It gave the students achance to see the attractive and creative side of mathematics.76Small groups and whole-class discussions helped the students to see the dialecticalnature of mathematics. It helped them experience mathematics as a social activity that islearned through social interaction. The students were encouraged to create and to domathematics by themselves without relying on outside authority, the teacher, to tell themwhat to do all the time, and to value the process of doing mathematics rather than being somuch concerned about the finished product, the "correct answer." In conclusion, therationale was to help the students see that they would not become musicians just bystudying the theory of music, only by actually getting involved in the process of makingmusic, and so it is with mathematics.Classroom InteractionsInstruction was based on the students' interactions with each other and theinstructor either in a whole-class setting or in the small groups. It took a long time and alot of effort to make the students believe that it would be worth their while if they activelytook part in the discussions. Some of the students were annoyed and frustrated with thenoise level. Lynn and few others asked for less interruption, which meant no or lessquestioning, and more direct instruction and guidance on my part. I believed if they sawany benefits, they would like the idea of interacting with each other in the class.Sometimes students ended up working by themselves. In such cases, I had to ask themrepeatedly to work in groups. Eventually, working in small groups and participating inwhole-class discussions became a norm and the students felt comfortable participating inthem. However, a few students resisted working in small groups and participating inwhole-class discussions.Bringing about some changes in terms of classroom processes was not easy andrequired a tremendous effort both on the students' part and mine. To avoid redundancy, Ihave chosen a few episodes based on video-tapes of the class and my reflective journal, to77portray the interactive nature of the course. In these episodes, "I" represents me, theinstructor.Episode 1, May 8 The following happened on May 8 [third day of class]. Nina came to the board todo a problem. She solved the problem correctly. However, as soon as Jim asked her whyshe did what she did, she wiped it out.I^Why did you wipe it out?Nina^Because I was wrong.I^Why?Nina^I don't know.I^You must have had some reason for that . . .Nina^I did it because I had to do something with these numbers. Okay, I canmultiply 1.2 by 50 rather than dividing 1.2 by 50.I^How did you get that 50?Nina^I don't know!Nina was not the only one who would say that she did not know and could notunderstand anything regarding mathematics because she had "math phobia." In thebeginning many students would sympathize with her and thought that she was speaking ontheir behalf. In fact it was more acceptable to say that "math is dreadful," and wash theirhands of it, without taking any responsibility for their own learning or even trying toimprove the situation. The instructor's goal was to challenge this view. I aimed to showthem that mathematics was not "dreadful" as they thought it was and also to help them togain more self-confidence in doing mathematics, to be able to defend their ideas ratherthan feeling vulnerable. Nina's responses to Jim's question was a manifestation of herinsecurity and low self-confidence regarding mathematics which was typical of themajority in the class.78Episode 2, May 27What Kent did on May 27, was only one example showing how the studentsconfidently expressed their ideas in class and were able to defend them. On May 27, wewere discussing the irrational numbers and how to calculate it 3 . A number of studentswere arguing about whether a "true circle" exists and what was the concept of limit. Kentshared his understanding with the class:Kent^If you keep slicing and slicing the sides of polygons, you get smaller andsmaller slivers and you won't be able to see it after a while. I can see thatyou can do that forever. So as it could go forever.The students asked for more explanations, and Kent continued:Kent^If you take this part of the number line [he pointed at the interval between3 and 4] and blow it up by a microscope, you can go this way [within theinterval] forever. 3.14 is somewhere here but it can go forever.Melisa^Is it the matter of complete and incomplete numbers? Because irrationalnumbers never end like It. . . .The calculation of it gave the class a chance to have a more in depth discussion ofrational versus irrational numbers and decimal expansions. Jack's question about TE was agood transition to "proof by contradiction." He said he was told in high school that it wasequal to 22/7, so he was confused. Jack's question was discussed in class at length. Itgave us a chance to more specifically discuss the idea of indirect proof. Therefore, thequestion was the contradiction between IL as an irrational number and as being equal to22/7 which was a repeating decimal, a rational number. In discussions like this one,students were engaged in a meaning-making process, a number of mathematical ideaswere connected, and the students' mathematical understanding was developed.3 There was a banner on the wall that had TL to many decimal places. When we wanted tocalculate 1C, Nina looked at the wall and said: "Here is Tt. We don't need to do it anymore."79In general, there was a big difference between what Kent did on May 27, and whatNina did on May 8. Nina immediately wiped out what she wrote on the board when hersolution was questioned either by the students or the instructor. Nina (and many othersfor that matter) thought she was wrong, since she was not confident to defend what shedid. As the course progressed, the students gained more confident, since discussing theirideas and defending them was the day-to-day practice in class. Class environment, orsocial norm played an important role in helping the students to become more independentand self-reliant, and consequently more confident. Therefore, what Kent did became anorm rather than an exception.The development of the social norm of the classroom was an interestingphenomenon. After the first half of the course, the students found it easier to worktogether in the small groups and on different problems. Therefore, it was difficult toattract all the students' attention to the same problem when it was discussed in class as awhole. After the first half of the course, I gave the whole class the same problem to workon for 10 or 15 minutes. We then discussed the problem in class. In this way, I was ableto pull the whole class together and more students interacted with each other. Thefollowing few episodes describe the nature of those interactions.Episode 3, May 30 I put on the following problem on the board and asked the students to spend 15 to20 minutes on it in small groups.If someone offers you a job that pays one penny on the first day and doubles themoney every day after that for 30 days, would you accept the offer?After all the groups had a chance to discuss the problem among themselves, I then askedevery group to put their solution attempts on the board to be discussed by the whole class.I constantly told them that the purpose was to discuss ideas and not necessarily to judge80final products. The following dialogue is a snapshot of the interaction regarding the aboveproblem.I^Now I would like each group to share their ideas with the whole class.Clara^I will just tell you what we've done in our group. But we haven't got theanswer.I^That's fine. We don't want a finished product. We just want to hear fromeach other.Clara^People worked it out on calculator as 2x2x2x .... Second way that we didwas . . . and then we took like for the first day it was only one penny andthen the second day it doubled. It was 30 days. The second day was 2,and that was for 29 days. So we put 2 29 , and it worked out on calculator,and both answers were the same.(What Clara's group did was to multiply two 29 times and then add one. The otherway was to use xY function on the calculator and add one at the end).I^So did you get the sum of the money the person got at the end of the30th day?Clara^Yes.Melisa^[Raising her hand] What we did [another group] was make a tablehaving three columns. The first column was a penny and then secondcolumn was doubling that money and the third column was accumulatedpay. What we noticed was for example, on 11th day, you earned $10.24 aday and the sum of the first 10 days which was for previous days, so... like the 30th day, you'll be actually earning 2 3° but minus one cent.That would be the sum of what you've earned from the first day to the 30thday.I^Thanks . . . so what about you [pointing at another group]?Shirley^I didn't quite understand what the 10th day has anything to do with it.81Melisa^Oh, just to give an example.Patrick^[From Melisa's group] Well on the 10th day you earn $5.12 and theaccumulative pay on the 10th day was $10.24 and on the 11th day youmake $10.24. You make one penny more than you've already made.Melisa^^That was accumulative pay from the 1st day to the 10th day which is equalto the pay on next day. But you actually make one more penny.Patrick^So, on the 10th day, you make 2 9 . That's how you earn $5.12.The total pay was $10.23 which was 2' ° —1 . . . So, on the 16th day, I used2' s . The accumulated pay was $655.36 which is 2 16 - 1. So the formula is2 x- 1 penny.Jack^[from another group] We did it as: on day one you make 1 penny, on day 2you make 2 pennies, day 3 you make 4, so we noticed the pattern on theright hand column. The 1st day is 2 ° , 2nd day is 2`, 3rd day is 2 2 , . . . , soafter 30 days, that's equal to so many cents . . . is equal to 2 29 . When youpunch that into calculator, that was how many cents you would earn.Nina^[Same group as Patrick] So more like what you guys did [pointing to theClara's group].Patrick^That's how much money you made on the 30th day.Jack^The problem we had and I've still seen is that the power is one less at theday, and originally we had 2 3° . But because you start at 0, you don't startat one, that's where the difference comes in.Nina^Could I say something? If we said 2 29 , is that the same as if we just saidthat it's 30 days, so it's 1 3° ? Would that give us the same answer? Itwouldn't!? Would it?!!Lana and few others said "NO! NO!" and explained why it was not correct to dosuch a thing.82Nina^So we have to make sure that you got the first accumulative pay beforeyou add the other powers. So if you make 5 cents on the first day, the 2ndday you make 10 cents, it will be 10 29 , what you would make. .. right?The interesting point was that she wanted to extend the pattern and check to seewhether it would work for different problems with the same structure. However, the classdid not discuss Nina's proposed problem at length, since they first wanted to finish theproblem at hand. Furthermore, we discussed what she said within her own group. Shirleythen continued the discussion.Shirley^If it had been 31 days which you've gone to 2 3° . So you just automaticallylost the top one.Lana^[Same group explaining to Shirley] . . . on 1st day you get one penny,the next day you gonna get an increase, so on the 2nd day you get 2pennies. So to get down to 29 . . . that's your pay for 29th day, but youhave to add it to everything that you've accumulated . . . so that comes upto 2 3° - 1 That's all we did.Patrick^I don't argue with that. 2 29 is how much you get on the 30th day.Melisa^But your question was much you get on the 30th day if you keepthe job ... well if I was making that much money, I'd better take the job.Patrick^^Did I ask Zahra if you want the total amount of money that you'll be paidfor a month is work?Carmel^So the answer does it mean the last day of everything or we have to add upevery day on the top of that?I^It depends on how you interpret the problem. Are you looking for the lastday's pay or for the accumulated pay?Carmel^That was the question. What did you want? Last payment?I^That's one way of interpreting it. You could either do that or find the sumfor the salary.83Roy [TA]^In the real life situation, what do we mean by salary or wage?Jim^Normally we only get paid after each day of work.Roy^So in that case you gone interpret asPatrick^So in that way it would be 2 29 .The discussion continued for a while. The students shared their ideas about theways in which to carry out a plan for either interpretation of the problem. The classdiscussed different ideas that were expressed by small groups. I then talked a few minutesabout the problem to conclude the discussion.I^So overall, is there any question about this problem?Nina^What is the correct answer?I^It depends on how you look at the problem.Nina^But there must be a right way of looking at the problem ... I don't know.I wasn't given any answer.Leah^Why?Nina^[Laughing] I don't know! I was looking for a correct answer.I^I mean there are two answers to this depending on how you wouldinterpret the problem.Nina^So you either think of it as . . . [interrupted by Patrick],Patrick^. . . amount of money that you get paid on the last day or all the money that[you] get for 30 days.Nina^Okay! I got it.I^Everything is relative. I mean [the] correct answer depends on how youconceive the problem.Nina^What if we get the question on the final exam? How do we really interpretit?84(Unfortunately, the fear of the exam sometimes interfered with the meaning-making process. However, this chapter is not the suitable place to discuss the purpose of"evaluation.")This problem was an interesting activity to introduce the idea of exponentialpowers and logarithms. Many students expressed their concerns about the topicimmediately after they received the outline of the course. Some of them talked aboutlogarithm as being "dreadful." However, the problems of this kind provided anopportunity for the students to better understand the topic and even enjoying it withoutdreading it. For the students, a solution to this problem was conceivable. If nothingworked out, they could always punch 2 and hit the multiplication button 30 times on theircalculators to get the answer. However, the problem stimulated many thoughts regardingexponential powers and logarithms, and the students did not feel bored.In the second pilot study, the instructor posed the same problem. Most of thestudents thought that the problem was difficult and beyond their grasps, because it wasrelated to exponential powers and logarithm. Moreover, the instructor solved the problemby himself, without any interactions with students except students' amusement at startingwith only a penny and ending up with such a large number. He spend a great amount oftime calculating 23° by breaking it down into numbers with smaller exponents. For manystudents, calculation became the main focus of the problem. They therefore suggestedusing a calculator to save the class time, as the aim was only to calculate 2 3° . However, ifstudents in that class had had a chance to discuss the problem, and struggle with it, theywould probably have gotten much more out of it. Nevertheless, they silently took notes tomake sure that they could produce the solution (and answer for that matter) on the exam.This problem stimulated good class discussions, as well as providing anopportunity for them to use a variety of problem solving strategies (heuristics) to solve it.For example, a group of students thought of a similar problem with smaller numbers tomake a better sense of the problem, and another group made a systematic list, while third85group was looking for a pattern. In the meanwhile, they had a chance to see how thingswere growing exponentially. Furthermore, the students learned how to use logarithm andxY functions on their calculator rather than punching 2 and hitting the multiplicationbutton 30 times.Episode 4, June 4 On this day, I started the class with a problem to introduce the "fundamentalcounting principle" and then continued with a True or False question to talk specificallyabout permutations and combinations.In how many ways can you answer a T or F exam that has three questions?I posed the problem and asked the students to spend 10 to 15 minutes discussing itin their groups. After group-discussions, every group presented its solution attempts tothe class. I wrote all those suggesting ideas while students were presenting them. Then,the class as a whole discussed each solution. The discussion helped the class to reject thesolutions that did not make sense to them and keep those which did. The followingexcerpt is taken from the class interaction on June 4:Sandra^26, because there are two ways of answering [each question].I^What do you think about it [asking the whole class]?Nina^It sounds mathematical [every one laughed].Jim^1 don't want to create friction here. I only have an argument against it. Ithink that's the number of possible answers that you can have when youconsider [pause] like in each case you are only allowed to have threeanswers out of the six possibilities. That's [referring to 2 6 ] suggesting thatyou are able to have six answers out of six possibilities. But you have toeliminate three because of the fact that you can only have three answers insuch an exam. You can't have for example, . . . the questions can not be86True and False. They either have to be True or False. Does that make anysense? I don't know if that's clear.This episode shows how we were able to establish a social norm for the class, inthat every one had a voice and we all respected each other's opinion. We, as a whole,gradually developed a mutual understanding among ourselves. Jim's response to Sandrawas interesting: "I don't want to create friction here. I only have an argument against it."After a good discussion about why 2 6 could not be an answer to the problemposed, I said:I^Now, is it O.K. to say that this one [2 6 ] is not correct?[interrupted by Jim]Jim^Sandra! The way of looking at it would be to say that if you want to lookat the number of possibilities, then that would be appropriate, . . . becausethen you would have six T, F, T, F, T, F. But when you look at thepossible answers, then you have to do it with 3.I^How different are they? You said the number of possibilities and numberof answers.Jim^What? You mean the distinction that I made?I^Yes.Jim^Well, the fact is that there are six; there are 6 possible responses:T,F,T,F,T,F [pause] but you can have one answer for each one. So that's.. . ya!Nina^So 2 possibilities for each question.Jim^Ya! 2 possibilities for each question.Nina^And 3 questions, so there are only 8 possible . . . [interrupted by Jim]Jim^I'm only trying to make distinction between possibilities and answers. Youcan only have three answers, but there are 6 possibilities.Jim could relate to Sandra and understand why she thought of 2 6 . It would bepossible simply to tell her that she was wrong and that would be the end of the discussion.87However, the whole-class discussion gave Sandra a chance to see her confusion, ratherthan letting those ideas go underground (Cobb, 1991). Jim had no difficultyunderstanding the problem, instead he talked about a hypothetical situation to makeSandra realize where she went wrong. Then Carmel talked about the reality of True orFalse questions and they could only be answered either "True" or "False" and not both.Kent extended the problem to an exam with 3 choices (as opposed to only True orFalse), and said:Kent^In other words, [if] you can say True, False, or True/False, you could!Then it logically tells us that there are 3 possible choices and threequestions, then it would be 3 3 . Am I right?Class^Ya!Kent^Okay, then it seems we've got some principle here . . .. You could havemultiple choice questions with 5 possible answers and 3 questions, then wehave 5 x 5 x 5. Three questions and 5 possible choices.Lana^Are you saying that there are 3 choices?Kent^No! That would be 3 x 3 x 3.Kent continued to explain a problem with 5 questions and 5 different answers veryclearly. After a good discussion about this problem Kent said:Kent^Well, I'm just trying to find the underlying structure of the logic of this andseeing if I plugged in other numbers from different questions and still itcould be true.I^So 2 6 doesn't work for the True or False exam with 3 questions?Kent^If it were 6 questions, it would.Patrick^And with 2 possible responses for each questions.It was interesting to see how the class developed an understanding of the countingprinciple. Kent and Patrick discussed a situation in which 2 6 was an appropriate response.Furthermore, I turned to the problem of having an exam with 3 questions and 2 responses88each, to address Patrick's concern about 2 3 , one of the solutions to this problem. Patrickargued that in a problem involving 5 shirts and 3 pants, the number of choices thatsomeone could make was 5x3. However, in this problem we were talking about the sameprinciple but the solution seemed different. "What does the 2 in 2 3 stands for? Whatabout 3?", I asked the class.Many students got involved in the discussion to find the reason for the differencebetween those two responses. They knew there were 2 choices to make for each one ofthose 3 questions, therefore the total number of choices was 2 x 2 x 2, the same argumentfor the number of choices relating to 5 shirts and 3 pants was 5 x 3. However, thepresentation of a solution for the first problem (that was also suggested by the students)confused some of the others including Patrick. Patrick expected to have the samepresentation for the problem involving shorts and pants since they both had the sameunderlying structure. I told them 2 3 was only a short cut to present the number ofchoices, otherwise we did exactly the same thing in both problems, that was multiplyingthe number of choices for each question, or for pants and shirt, to find the total number ofchoices to be made: 2 3 = 2x2x2 ways of answering 3 questions with 2 choices for each,and 5x3 ways of combine 5 shirts and 3 pants.Solving problems like this might not take more than 5 minutes in a regular class.However, if we did not discuss the problem, I would not realize the difficulties thatstudents were facing in understanding a problem that might look simple and easy. Itwould be easier to introduce the concept (in this case counting principle), solve a fewexamples showing the class how to apply the certain rules and formulas, and then askstudents to solve a list of similar problems. However, in such situations Sandra's answerwould simply be "wrong" and the "correct" answer would be put on the board for display.Of course, students would copy it down in their notebooks. Patrick had to make sure towrite the total number of choices in an exponential form if each case had the same number89of choices (similar True or False question), and to multiply them if they were different(similar to shirts and pants).Nevertheless, the metacognition-based instruction provided an opportunity for thestudents to freely express their ideas and discuss them both in small groups and in class.We only solved a few problems in each class, we developed many mathematical conceptsthrough those few problems.SummaryThe class interactions were stimulating and there were many interesting episodesto talk about. However, it is not possible to address all the problems that were discussedin the class. The purpose of giving these few snapshots was to portray the nature ofinteractions, and to show the nature of the class environment— its social norm. Such anenvironment allowed the students to participate actively in the meaning-making process.Many mathematical concepts were developed via problem solving. In addition, a numberof problem solving strategies were discussed within the contexts of interesting problems,whenever it was appropriate. For example, in a problem related to the counting principle,many students used one or more of the following problem solving strategies: Looking fora similar problem, using smaller numbers, acting out the problem, guessing and checking,making a table or systematic list, looking for patterns, and others.In general, the students left the class feeling more confidence in their mathematicalunderstanding. They realized that everyone was capable of doing mathematics to someextent, and that no one deserved to be a "duck." Self-confidence and self-reliance playedimportant roles in the students' learning. The students learned to be critical regardingmathematical problems, and as Ben said at the end of the final exam: "I learned to neveraccept anything without asking 'why'." He continued: "That's the best thing I've learnedfrom this course. I learned to learn. I got my confidence back, and that's just great!"90CHAPTER 5FINDINGSThe findings of the study are based on 40 students who enrolled in MATH 335(introduction to mathematics) in the Summer of 1991 at the University of BritishColumbia. The main purpose of the study was to monitor changes in the students' beliefsabout themselves as doers of mathematics, and their beliefs about mathematics andmathematical problem solving as a result of metacognition-based instruction. The chapteris organized into six sections, namely; (a) general demographic description of the students,(b) description of three categories of students, (c) descriptive statistics concerning all thestudents, (d) findings within each category and comparison of the findings acrosscategories, and (e) summary of the chapter.Direct quotes are taken from the students' work, including their journal entries, tofacilitate the interpretation of the data. The students' quotes are underlined whenever the-student underlined, or added emphasis in some other way. Italics indicate that emphasishas been added by the writer.Direct quotes from the students' journals are shown by date (e.g., Jim's May 13journal entry is shown as "Jim, May 13" except for the last journal entries that are shownas "Jim's last journal entry." Some of the students dated their entries in terms of weeksthat they were in the course. In that case, I have used the date-range of the week to avoidambiguity. For example, the first week entries are dated May 6-10. In a few instances thedates of the entries were not clear, and in these instances I have used a question markinstead of the date.91General Demographic Description of the StudentsThis section presents academic status, age range, mathematical background, andwork status of the students. The section ends with a summary.Academic Status The students who took MATH 335 were all non-science students. Of the 40students, 26 provided information about their specific academic status. Twenty one outof 26 students had baccalaureate degrees in either Arts (B.A.), Physical Education(B.P.E.), or Education (B.Ed.). One student had already finished her first year in themagisterial program in Educational Psychology. The other five students were in their thirdor fourth year of an undergraduate program.Age Range The students' ages ranged from early 20's to early 50's. Those who were 40 andabove were updating their teaching credentials. The dates of the last mathematics coursesthat they had taken ranged from 1966 to 1988, except for one student who took Algebra11 at night school in a local college in 1989.Mathematics Background Some of the last mathematics courses these students took before this course wereas follows: Grade 10 Mathematics, Algebra 11, Algebra 12, Grade 12 Mathematics,Grade 13 Mathematics, High School Mathematics (as they called it), Calculus, and FiniteMathematics. The mathematics background of some of the students is not known, datafor 36 of the 40 students were available for this purpose. Three of these 36 students failedtheir last high school mathematics courses once, and one student failed it three times. Onestudent took "Mathematics for Elementary Teachers," at a university in England in 1966,to fulfill the requirements for a general degree in education (B. Ed.). One student tookCalculus in 1966 and needed MATH 335 to update her teaching credentials. Only onestudent successfully finished first year calculus at a local college. He did not need MATH92335 to enroll in the Elementary Teacher Education Program. However, he took thecourse to enrich his mathematical background. Two students took Finite Mathematics atthe University of British Columbia and they both failed the course.Work Status I could not quantify the data for this section due to the nature of the information. Icollected the information in a narrative manner. Personal information about the studentswas obtained through their journals and after-class discussions. Most of the studentseither had full-time jobs, or other courses to attend, or both. Some of them were singleparents. Their jobs varied from working full-time at a cemetery to performing on stageevery night after class. There were five performing artists and three substitute teachers inthe class.Summary The students were distinctly different in many ways, including their academicstatus and mathematical background. However, they shared a common goal of wanting towork with children. Some of the students chose the teaching profession because theyconsidered it a rewarding mission. Those students wanted to help the youngsters since thefuture of the nation is in their hands. However, some other students wanted to enter theElementary Teacher Education Program because of the job market and economicsituation.Description of Categories of StudentsThree categories of students emerged from the data collection and data reductionprocess. Each category consists of the students who shared similar beliefs aboutthemselves as doers of mathematics, and about mathematics and mathematical problemsolving. Also the members of each category had common beliefs about the ways in whichthe metacognition-based instruction either altered their approaches to problem solving or93failed to make significant changes. Three categories of students ranged from resistingchange to accepting change to embracing change. The three names, "traditionalists,""incrementalists," and "innovators" have been chosen based on the commoncharacteristics, as labels for the categories. The following is a description of the threecategories and a discussion of the idiosyncratic cases within each one of them.First Category: Traditionalists Nine students belonged to this category which is characterized by those who eitherrejected the new approach to instruction and explained why it did not work for them orthose who were never influenced by it for reasons not related to the instruction. Adescription of the former group is given here.Dan was a visual artist. He related all his journal entries to his art: "I also have aparticular interest in the geometry section as it relates to painting." He mainly participatedin the activities that he enjoyed the most. For example, he did well in geometry becausehe was fascinated by the subject, but he hardly participated in activities regarding rationaland irrational numbers. He said, "Ordinarily I would be uncomfortable with dealing withmoney and numbers. . . . Numbers drive me crazy!" I could not judge the influence of theinstruction on him. However, the instruction gave him a chance to utilize mathematics inthe context of his art.Joan had a full-time job: "I wasn't here at all this week as I had to work." Shemissed almost half of the classes due to her job situation, and she came to classes late.She left early because of another class after the mathematics course. She alwayscomplained and never tried to do anything to improve her school situation. Several timesshe made appointments to meet me in my office, but she failed to keep them. Joan handedin most of her assignments and journal entries late. She kept saying she just wanted to passthe course. She did not have time to "think" about and reflect on what we did in class.Her responses to the metacognitive questions, which were part of the second and thirdquizzes, showed her lack of interest in mathematical issues. When asked, "Would you94restate the problem using your own words?" Joan wrote; "The way the problem is statedis good."Like Joan, Tony was often absent or late, and in addition, did not have time tokeep his journal entries. He produced the journal in one day to meet the requirements ofthe course. Therefore, I gave him only one third of the journal mark without anyfeedback, except one page of comments explaining the purpose of journal writing. Hisjournal had nothing to do with "reflection," it was a story written in a rush.Sarah was the only student who did not agree to be video taped during the classsessions. She was a private person. She wrote very little in her journal entries, and herwork was not reflective.Lynn had a full-time job and other courses that made her extremely tired and latemost of the time. However, unlike Joan, she hardly missed any classes.Nina was unique in many ways. She talked all the time and she always insisted onsaying that she did not know anything and she could not understand anything related tomathematics. However, she was quite capable of doing and understanding mathematicswhen she changed her attitude.Pamela looked at things in a mechanical way. Journal writing was only arequirement that she had to fulfill. In her last entry she wrote, "I think I'm all Ijournalled'out." Fifteen minutes into the final exam she wanted to leave. I asked her to read theproblems carefully and not to give up so soon. It worked and she managed to respond tomany questions. I told her that she could keep the copy of the exam if she wanted. Sherefused and said, "I don't want to look at it ever again."In general, the traditionalists were resistant to change. They preferred to havemore structured instruction even though some aspects of the instruction, such as workingin small groups and using manipulative materials, helped the majority of them to gain moreself-confidence.95Second Category: Incrementalists The twelve students in this category expressed their appreciation for some of thecomponents of the instruction. However they questioned its lack of "direction" and"guidance." These students considered the instruction as a series of teaching strategiesthat should or should not be adapted depending on the situation. Beth's last journal entrycaptured the views of the people in this category: ". . I believe this with any teachingmethod; it must be used in moderation."Kayla, Lilian, and Terri expressed their "fear" of mathematics. However, Teresahad never had "any serious difficulty with the subject." Shirley "liked" it, and Jim, whosehobby was designing buildings (his post-interview), "enjoyed" mathematics although he"never got good grades in high school."Some of the incrementalists were optimistic about doing well in the course. Beth,Leah, Carmel, Lana, Keith, Diana, and Lilian hoped to become "more confident,""understand mathematical processes and thoughts," and "gain more confidence in mathcomputation" by taking the course.Lilian, the only student to fail the course, tried very hard but was working longhours and had another course to attend after the course. She gained more self-confidenceas she said, but needed more time and more responsibility towards her own learning.Overall, the incrementalists were more homogenous than the students in the othertwo categories. In fact, there was not an idiosyncratic case in this category to bediscussed separately. The students in this category asked for a more "balanced"instruction that adapted some aspects of the metacognition-based instruction and some"structured" instruction, as they called it.Third Category: Innovators All nineteen members of this category were definitely influenced by the instruction.With the exception of Patrick who said, "Math always came easy to me. The onlyproblem I had in the class was with identities. I know why, however; it was because Isimply did not memorize those ugly things (identities)" (Patrick's autobiography), theeighteen others expressed their concerns about mathematics at the end of the first class.Clara's response was typical: "I have to say I approach this course with my stomach in abit of turmoil" (Clara's autobiography), but she then continued, "I hope I will do it! . . . Ialso hope that I will enjoy it, not just tolerate it to get a grade." The most importantfactor helping the innovators to benefit from the instruction was their willingness to tryand overcome their fears. "I am nervous about this course," said Barbara, "but I amprepared to work hard. I even hope to overcome some of the blocks to understanding"(Barbara's autobiography).The innovators were different in many ways. The variation among them is worthmentioning.The youngest and the oldest students in the class were among the innovators. Theinterviewees were chosen on the basis of their performance on their early work in thiscourse, and on their beliefs about mathematics. Two of the interviewees who wereinitially among high achievers fell into this category. Furthermore, an interviewee fromthe medium range ended up in this category. In addition, I interviewed two more studentswho showed noticeable changes from the beginning to the end of the course. Therefore,five interviewees were eventually characterized as innovators. There were two performingartists (musician and stage actor) and one painter in this category. Their approaches toproblem solving were articulate and interesting. Their interest in art helped them toefficiently utilize the mathematics they learned. They wanted to learn by "doing" not justmemorizing the facts and formulas.Patrick was unique in his own way. He consistently showed positive attitudestoward mathematics. At times, Patrick was frustrated and confused, but he was neverdisappointed. Melisa wrote frequently in her journal about how had she wanted toconstruct her own meaning since she was in elementary school, but the system did not lether. "I used to ask [my grade 4 teacher] 'but why?' The teacher said, 'It doesn't matter97why Melisa. Just do it as I did" (Melisa's autobiography). Patrick and Melisa were moreenthusiastic about new ideas, studying the mathematical connections, and striving forfurther extensions of problems and concepts. They shared some of those ideas throughtheir written work, and their journals became a channel for individualized instructionthrough reflection and interaction. Their eagerness was well described by Melisa, "Ibecame quite excited because of the whole new can of worms that opens up in my mind."Clara's journal was superb. She solved every problem using multiple approaches.The managerial skills were well manifested in her solution processes. While she was theonly student in the whole class who got 100% on her journal, assignments, and quizzes,she froze on the final exam and could not perform. She got 57% on her final exam andher overall mark was 83%.In general, the innovators were all willing to try, and that was crucial. Sometimesthey were upset and worried and sometimes they were happy and satisfied.Metacognition-based instruction provided them an opportunity to gain more self-confidence and to realize that they were capable of doing mathematics. They also changedtheir beliefs about problem solving since they experienced multiple approaches to solve aproblem. These students adapted the metacognition-based instruction as a new teachingapproach. In fact, they shifted their paradigm and saw the teaching and the learning ofmathematics from a new perspective. That was the reason for referring to the members ofthis category as "innovators", those who were open to change and were actively involvedin the meaning-making processes.Descriptive StatisticsThe present study was qualitative in nature. Its main purpose was to monitorchanges in the students' beliefs about themselves as doers of mathematics and aboutmathematics and mathematical problem solving. Nevertheless, descriptive statistics will98help the readers to have a better picture of the individuals in the class. The informationfrom each category is presented in table form. The following three tables give adescriptions of the background of the students in the three categories. In addition,attendance patterns, final exam scores, and final course grades are shown.99Table 5.1Descriptive Statistics Regarding the TraditionalistsName Age Math Date Degree In Ed Exam Final AttendPam 20's G12* - 46 66 +Nina 20's H.S - B.A. 2Yr 55 75 +Lynn 20's - - 34 63 UALora 20's G12 2Yr 59 71 +Sandra 20's - - - 60 71 +Dan 30's H.S 1976 B.A. UPD 63 67 +Joan 30's Apply 39 58 UAMarion 20's Gil - B.A. Apply 68 75 +Tony 20's G12 B.PE. Apply 54 52 A/LNO = 9 Average Exam = 53% Average Total = 66%Notes.- = No data are available* = Failure in the courseMath = mathematics backgoundHS = High School MathematicsDate = date of last mathematics courseIn Ed = in Elementary Teacher Education program2Yr = Already in the Elementary Teacher Education ProgramUPD = Updating teaching credentialApply = applying for the Elementary Teacher Education programExam = Final exam scoreFinal = Overall grade/course final markAttend = Students' attendanceL/A = Mostly late, a few absencesA/L = Mostly absences+ = Full attendance without tardiness100Table 5.2Descriptive Statistics Regarding the IncrementalistsName Age Math Date Degree In Ed Exam Total AttendKeith 20's Calculus - 3rdYr Apply 66 76 +Lana 30's A1.11 1980 B.A. Apply 93 89 +Teresa 20's A1.12 3rdYr 86 84 +Kayla 20's B.A. 2Yr 55 68 VALeah 30's H.S 1978 B.A. 61 74 +Terri 20's A1.12 1986 B.A. 2Yr 63 78 +Lilian 20's H.S 1979 - 22 47 A/LBeth 20's A1.11 1984 2Yr 65 75 +Jim 30's - B.A. - 80 85 +Shirly 40's G13 60's B.? 2Yr 39 64 +Carmen 20's Finite* - 69 71 +Diana 20's A1.12 B.A. Apply 81 79 LNo = 12 Average final = 67% Average total = 74%Note. Codes for this table are on the next page.101Notes for Table 5.2 - = No data are availableA = Repeat high school mathematics at night school* = Failure in the courseMath = mathematics backgoundAl = AlgebraHS = High School MathematicsDate = date of last mathematics courseIn Ed = in Elementary Teacher Education programApply = applying for the Elementary Teacher Education program2Yr = Already in the Elementary Teacher Education Program3rdYr = Third year undergraduate programExam = Final exam scoreTotal = Overall grade/course final markAttend = Students' attendanceL/A = Usually late, a few absencesA/L = More absences, Usually lateL = Late+ = Full attendance without tardiness102Table 5.3Descriptive Statistics Regarding the InnovatorsName Age Math Date Degree In Ed Exam Total AttendPenny 20's G12 1988 3rdYr 60 77 +Linda 20's G12* - B.A. 2Yr 44 70 +Dave 20's Gil 1984 2Yr 49 63 LMelisa 20's G12 - B.A. - 80 83 +Barbara 40's H.S 1966 B.?. 2Yr 81 83 +Kent 30's G12 B.A. 94 90 +Rita 40's Calculus 1966 B.Sc. UPD 70 79 +Carmen 20's A1.11 1981 - 47 64 +Patrick 30's G12 1979 B.A. 93 93 +Rose 40's 1968 B.Ed. UPD 85 85 +Peggy 50's College 1966 B.Ed. UPD 69 84 +Sophia 20's H.S 1981 91 91 +Bob 40's G12 1968 B.Ed. UPD 98 90 +Sally 20's H.S 1985 B.A. 68 78 +Clara 40's A1.11 70's 2Yr 57 83 +Sarah 20's Finite* 2Yr 58 78 +Jack 20's A1.12* 1986 4thYr 84 85 +Ben 20's - - Apply 72 87 +Sheila 30's G12* Apply 78 81 +No. = 19 Average Exam = 73% Average = Final = 81%Note. Codes for this table are on the next page.103Notes for Table 5.3.- = No data are availableA = Repeat high school mathematics at night school* = Failure in the courseMath = mathematics backgoundAl = AlgebraHS = High School MathematicsFinite = Finite Mathematics at university levelDate = date of last mathematics courseIn Ed = in Elementary Teacher Education programApply = applying for the Elementary Teacher Education program2Yr = Already in the Elementary Teacher Education ProgramUPD = Updating teaching credential3-4Yr = Third or fourth year undergraduate programExam = Final exam scoreTotal = Overall grade/course final markAttend = Students' attendanceL/A = Usually late, a few absencesA/L = More absences, Usually lateL = Late+ = Full attendance without tardy104Findings Within Each CategoryA number of different themes emerged from the students' work. Those themeshelped to organize the findings and give partial answers to the research questions. Theexcerpts from the students' written and oral responses are generally representative exceptfor several idiosyncratic responses that are discussed separately.The main themes that emerged from the students' responses were their reaction tothe instruction and its components including (a) small-group work, (b) whole-classdiscussions, (c) journal writing, and role of the teacher and the classroom environment.Almost everyone talked about the role of small groups and whole-class discussions in theirlearning. Many students did not write about the role of journal writing per se,nevertheless, journal writing played an important role since it gave them an opportunity todiscuss their point of views.Students journal entries were categorized as dealing mainly with cognitive issues,or metacognitive issues, or both. Seven students focused primarily on cognitive aspects ofthe course and went through the solutions of the problems discussed in class to make surethey understood those problems. Eleven students reflected mostly on metacognitive issuessuch as their beliefs about themselves and about mathematics. However, twenty studentskept a balance between cognitive and metacognitive issues in their journals. Thesestudents wrote about their awareness and beliefs, and used the problems to discussmetacognitive issues in context. This way of utilizing the journals helped them to improvetheir managerial skills to a great extent. Appendix R contains a sample of Clara's journalentry regarding an "exponential decay" problem. Clara wrote questions that she askedherself including "What do I know," "[This] is what we know, this is what we want toknow," and so on. The way in which she solved the problem is a good example of thosewho used their journals to address both cognitive and metacognitive issues.The following three tables show the number of traditionalists, incrementalists, andinnovators who used their journals in cognitive, metacognitive, or both [balanced] ways.105Table 5.4Ways of Utilizing Journals, the TraditionalistsCognitive^Metacognitive^Balanced^Total3^ 1^3^7*Note.*Two journals did not fit into this classification. One was written at the end of the courseand it was more as a report and not as a reflection on what went on in the class, and theother journal was specifically written about mathematics and painting in the thirteencentury.Table 5.5Ways of Utilizing Journals, the IncrementalistsCognitive Metacognitive^Balanced^Total 1^ 5^6^12Table 5.6Ways of Utilizing Journals, the InnovatorsCognitive^Metacognitive^Balanced^Total3^5^11^19106The roles of manipulative materials and the teacher will not necessarily bediscussed at length in all the three categories.The other themes that became apparent and that the students kept referring towere their beliefs about mathematics and mathematical problem solving and themselves asdoers and learners of mathematics, mathematical connections, and tests in a mathematicscourse. The following gives an overview of the different themes and sub-themes that arespecifically discussed within each category.The traditionalists (first category) discussed three themes in particular, namely (a)the instruction and its components, (b) students' beliefs about mathematics, (c) andstudents' beliefs about mathematical problem solving. The incrementalists and theinnovators also discussed these three themes at great length.The incrementalists (second category) reflected on some of the aspects of theinstruction in particular. Those aspects of the instruction that are discussed in the secondcategory are as follows: (a) time, (b) the teacher's role, (c) journal writing, and (d) smallgroups and whole class discussions. "Mathematical connections" is another theme that isdiscussed under a separate heading only in this category.The innovators (third category) talked about class environment as a maincomponent of the instruction. Furthermore, the innovators reflected on themselves asdoers and learners of mathematics which is discussed under the heading with "students'beliefs about mathematics." "Tests" as a separate theme is discussed only in this category.Since most of the responses were the result of the students' reflection on whathappened in the course, I do not discuss "reflection" as a separate theme but as one that isembedded throughout the whole discussion.107Findings from the First Category. the Traditionalists An interesting phenomenon that emerged from the traditionalists' responses was anembedded definition of "traditional teaching." The students' responses will be discussed,and a "student-based" definition of particular kinds of instruction will be provided.The Instruction and its Components I liked being told what to do instead of discovering it for myself by trialand error. Again this must be a legacy which I have inherited from myown school days. (Nina, May 30)Journal writing served many purposes including better communication between thestudents and the instructor. "I had written honestly and tried to express my understandingand concerns. I appreciate the opportunity to do this," wrote Lynn in her last journalentry. On a number of occasions, students asked for help with their journals regarding thedifficulties they had with specific problems. However, some of them did not considerwriting as a reflective process to enhance their learning. These students did not utilizetheir journals fully. For three of them, journal writing was just an added requirement to befulfilled. Pamela, in her last journal entry, said "I think I am all journalled [sic] out!" Bythat Pamela meant she had a certain number of entries to reach the criterion (I did not askthem to write for the sake of writing). Joan's entries were not reflective. They were shortand only reported, without discussion, the topics that were covered in some of the classes.In her entry of June 5 she wrote, "I wasn't here all three days (whole of probability), but Idid get the sheets to be done." The interesting point was that she knew how much effortshe made for those entries. "I really didn't have time to complete in the way I should have.Many times I had to write it 3-4 days late and thus it was not as accurate or up to date asit could have been. Also I didn't write very much each time." This was her last journalentry and yet her understanding of journal writing was the same as when she started thecourse: to do the sheets (hand-outs) and do them accurately. This shows her mechanicalapproach to the idea of writing as opposed to using writing as a reflection on what she108did, why she did it, and how those things happened. She did not even commit herself tothis mechanical approach due to her other engagements irrelevant to the course.The traditionalists did not feel the necessity to elaborate and discuss mathematicalideas. Joan's responses to the metacognitive questions regarding the second and thirdquizzes were typical, short and inarticulate. For example, the third question asked:"Would you restate the problem using your own words?" She wrote, "The way theproblem is stated is good," as if the question asked for a correction. She did not thinkabout the question as a metacognitive activity to promote awareness.The traditionalists were generally impatient with whole-class discussions, with theexception of Nina who kept talking unless other students asked her to be quiet so thatother students could speak. Lynn's response on May 29, sheds more light on this claim."Roy took control of the learning environment for over an hour. He lectured so that theusual incessant questioning by the students, was unnecessary and generally unacceptable."Lynn did not see the need for any kind of discussion. She even devalued the others'eagerness. "This portion of the lesson was only confusing when the class becameobsessed with debating their viewpoints. I find this a waste of time and frustrating to 90%of the other class members who would like to move forward. There is a little bit of a 'star'syndrome happening" (Lynn, June 4). Lynn did not respect the others' viewpoints asgenuine interactions of ideas, but rather she reduced all their effort and enthusiasm to a"star syndrome" and a sign of "obsession." Another point is that she did not have muchinteraction with other students to make such a generalization about 90% of the students'who "liked to move forward." (She usually left immediately after each class due to herfull-time job about which she frequently wrote in her journal). Finally, I could not figureout why she wanted to "move forward" since she knew the students were responsible onlyfor the material covered. She sometimes lost patience with the other students and became"angry" with them. "It makes me angry that people choose to make the class difficult bycreating extra variables. . . . Am I unreasonable? . . . This was a very straightforward109question that was needlessly dragged out," she continued. Her journal entry shows thatshe did not like any speculation and prediction regarding mathematical problem solving,and secondly, that she allowed herself to identify with other students' needs withoutinteracting with them. Nevertheless, her journal entry of June 6 revealed her concern forwhole-class discussions. "All the arguing that goes on often leaves me more confusedabout things I was very sure of in my life previous to this class!!!!" She was moreconfused because she deliberately did not want to take part in the debating and thediscussion process. "This 'matching' question was awful. I immediately set up a matchingquestion on my sheet. I probably would have demonstrated this to the class. However, Igot so frustrated with those who do waste class time with their 'self-important' .. .explanations that I could be hypocritical!!". So for her and others like her, whole classdiscussions were nothing but confusing and frustrating.Although many other students in this category were impatient with whole classdiscussion, they acknowledged the benefit of working in small groups. In her May 7journal entry, Lora expressed her gratitude for working in small groups. "I liked the ideaof working in groups. I find by explaining my method/solution to others, I betterunderstand it myself. It also helps me to lessen my anxiety." Marian, wrote in her lastentry, "One thing I have learned from the course is the importance of working together."She repeated the same statement in her post-interview.Many students expressed difficulties that they had with geometry prior to thecourse. However, working with manipulative materials helped the majority of the studentsto gain a better understanding of geometry. "I enjoyed participating in today's class. Ifind it very helpful to use cut-outs/manipulative when doing mathematics. I find it easierto comprehend the purpose and steps of the activity," wrote Lora in her May 7 journalentry. Sandra addressed this issue after the class went through the proof of thePythagorean theorem: "It is much easier to imagine relationships using a model (3-dimensional) rather than using only your mind to imagine a shape." In general "hands-on"110activities, as the students called it, gave them a chance to see mathematics as a "fun"activity. "I feel that I actually learned something, and that math is not as hard as it seems.. . . Thanks for a great mathematical experience. It was a lot of work but a lot of fun too,"wrote Lora in her last journal entry.The Students' Beliefs about Mathematics The traditionalists shared a common belief about mathematics. Lynn said, "Math isnot interesting enough to persevere." For Nina, mathematics was a subject that she didnot like much. Marian was a bit "apprehensive" about it, and Lora had "anxiety/phobiawhen it comes to math." All of them expressed these feelings in their autobiographies atthe end of the first class. These thoughts showed the traditionalists' belief aboutmathematics prior to the instruction. Pamela explained how she developed such belief:I guess you could say I have a phobia  about math. In elementary school Ienjoyed it very much and did very well, but once I got to high school Iseemed to struggle with it from the very beginning. Once I reached Grade12 I hated it and I had to repeat Math 12 three times before I finally passed(just barely). I still panic whenever I am in a situation where I have to doany calculating of any sort. If there is any way to get out of a situationwhere I have to do math I will find it! I want to be comfortable in this areaso I hope this class will help me with it.(Pamela's Autobiography)Pamela's description of the development of her "hatred" of mathematics wasamazing. It made me wonder how students' beliefs were influenced by the ways in whichmathematics was presented to them. She was honestly voicing many other students'beliefs. However, she was hoping that the course would help her and others by "beingtold" as opposed to being actively involved in the sense-making and meaning-makingprocess. Nina's response on the second day of the course is representative: "It seems thatI need somebody to constantly explain things to me. So far I find it very hard tounderstand things on my own."111The Students' Beliefs about Mathematical Problem SolvingThere are different definitions for "procedural learning" and "traditional teaching"in the literature. Nevertheless, the traditionalists defined these terms clearly andindependently. A descriptive checklist for "traditional teaching" based on the students'responses will be given at the end of this section.The emphasis of the metacognition-based instruction was to develop themathematical concepts through problem solving. The trend was to start the class with aproblem. Then by discussing it within the small groups and collectively in the class, thestudents would develop an understanding of the concepts involved in solving thatproblem. This teaching method required active involvement of the students. They werealways encouraged to ask questions and to see the dialectical nature of mathematics. Myrole as a teacher was more as a monitor and a facilitator and not as the class authority.The students in this category rationally rejected this approach to teaching mathematics andasked for more "direction," "lectures," "structure," and "guidance." Marian voiced thisclearly in her May 9 journal entry as well as in her post-interview: "I am having a bit ofdifficulty in an unstructured class. I suppose I have to get used to a lab-oriented classrather than a lectured class setting." Lora felt she was on her own "with little guidance. Idid not even know where to begin on most questions." On the second day of the course,Lynn wrote;I'm afraid at this phase of my education I need fewer instructions and moredirect knowledgeable guidance. It is not easy for students to beoveraggressive in their beliefs and unfortunately can confuse ingenuity orinnovation with correctness. Am I obsessed with rules? I really feel I needmore concreteness! Sorry. . . . For those who are trying to hold their'shaking' interest in math in check, side points such as discoveringFibonacci in a very specific case are distracting.^(Lynn, May 7)The important thing is that she questioned her dependency on the rules. "Am Iobsessed with rules?" but she did not want to find a way out of the situation. Nina firstwrote how she "liked to be told," and then talked about the roots of her dependency and112reliance on rules. She held the same belief all through the course as did the othertraditionalists:I must admit one thing. I found Roy's instruction exceptional andextremely helpful. I liked being told what to do instead of discovering itfor myself by trial and error. Again this must be a legacy which I haveinherited from my own school days. In high school our math instructorstood in front of the class and told us mathematical facts, much like Roydid today, and that is how we consequently learnt. I know that there is abetter way to teach math by allowing students to have a period ofexploration and discovery. However, there is only six weeks to teach amath class with so many topics, I think direct instruction is a better way togo. I may be wrong but for myself this way would work better.(Nina, May 30)The point is that although I categorized these students as traditionalists, theyrejected the approach mainly for pragmatic reasons. They were afraid of "wasting time"as Lynn said, and of not having enough time to finish the course material. They were notrisk-takers, and sometimes felt that it was too late to change. "I need a lecture reviewingthe steps" said Lynn, who continued, "Too much pride and stubbornness and frustrationresults from adults being responsible for their own learning! I really want to understandthis. Please provide more guidance!". It is true that old habits die hard. She held thesame belief at the end of the course, ". . . There was a method to your madness Zahra! Istill wish I was taking this class as a 12 year old, I might have been a math major!!" wroteLynn in her last journal entry.Traditional Teaching": the Traditionalists' Perspective The traditionalists defined "traditional teaching" in mathematics based on theirbeliefs about mathematics and mathematical problem solving. They wanted me to firstteach the concept, then solve a few problems on the board step by step, and then givethem more exercises to practice the method of solving the other problems. "I really hopedto see more examples worked out step by step on the board. I feel I'm wasting too muchtime doing the 'wrong' thing" wrote Lynn on the third day of the class. She later113expressed her belief more bluntly and even asked for a panacea. "I want to know theformula that will help me solve a task. I'm not saying that there is only one method ofsolving every type of question, but I would like to be able to confidently employ amethod" (May 24).The main characteristics of "traditional teaching" regarding mathematics, based onthe Traditionalists' views, were as follows:-Teacher to be in "control of class" and to give "guidance" and "direction" tostudents-There is no need for "incessant questioning"-Students "like to be told" by teachers what to do-"Frustration" is a sign of weakness and forces students to give up-Time should be used "wisely"-Problems should be solved by teachers on the board "step by step"-The "correct answer" is the main emphasis of problem solvingSummary Although the traditionalists did not change their beliefs as a result of themetacognition-based instruction, they definitely were influenced by it. Even their oldbeliefs were challenged by it at times. Journal writing was a communication channelbetween most of them and myself, writing gave them a chance to become more reflective.For example, Pamela solved a problem in class and reflected on it in her journal."Somehow that doesn't make sense. If the pyramid is only 2.5 cm tall and 5 cm width,how can it hold something that is 20.82? I am confused" (Pamela, May 22). Marian,Lora, and Lynn reflected on different issues in their journals frequently. I tried tostimulate discussion as much as I could by relating to them through their journals. Thebenefits of working in small groups and with manipulative materials were acknowledgedby most of them.114Generally, traditionalists liked a "direct instructional approach" as it was called bythe students, or some of them referred to it as "procedural knowledge." Marian wroteabout this approach in her last assignment, and talked about it in her post-interview: "It isdifficult to get from A to C if one does not know B. What I am saying is that teachersneed to refresh students' memories more, particularly on areas they have forgotten.Sometimes going back a few steps sheds light on the present situation." In this approach,the teacher is the sole authority and has the final word on directing students step by step.This approach considers mathematics to be cut and dried, with concepts presented inisolated fashions with more emphasis on rules and procedures. In this view, mathematicshas little or no connection with day-to-day life and mathematical concepts are treated asisolated facts. What Nina wrote in her May 9 journal entry illustrates this view. "I foundit very difficult to work backwards because I am so conditioned to applying the rule or thetheorem to the exact situation. . . . I always thought that when the problem was finished itwas over and done with and we didn't have to depend on it any longer.... I alwaysthought that when we are given a rule, unusual things could not happen." For her, ruleshave an absolute role which do not leave room for generalization and extension inmathematics.Findings from the Second Category, the Incrementalists The students in this category, the incrementalists, were influenced by themetacognition-based instruction to some degree. They liked some aspects of theinstruction which worked well for them, and were opposed to the other aspects thatseemed unrealistic to them. They regularly compared traditional and the metacognition-based instruction regarding teaching and learning of mathematics. The incrementalistsconcluded that a "balanced" instruction using these two approaches would be morepractical for teaching and learning mathematics. This section illustrates some aspects ofthis duality.115The Instruction and its Components Most of us were taught mathematics by a direct instruction approachwhere the emphasis was on getting the right answer. If you know thecorrect formula your answer would match with the answer in the back ofthe textbook. Math was never something that could be explored orspeculated about, which produced a great deal of anxiety.(Kayla, May 8)Kayla beautifully portrayed her beliefs about mathematics and mathematicalproblem solving, and the ways in which they were influenced by the kind of instructionthat she described. She and many others tried to look at the teaching and learning ofmathematics from a different perspective. At times they thought that the interactiveinstruction that allows discussion and values innovation was the way to go. "I guess thebiggest gift a teacher could give to a student is the ability to look at, think about, andsolve any problem on his or her own. . . . My understanding is that the method discussedwould be steering children towards thinking for themselves and working things out, andnothing could be more valuable" (Leah, May 21, reflecting on a video presentation by PaulCobb). However, there were times when they became frustrated and disappointed. OnMay 23, Carmel reflected on the same presentation and wrote;I can see that this method of instruction is being used in our mathematicsclass. To be honest, at times I find it frustrating. I'm not sure what Ishould be working on, I also feel like I'm far behind everyone and notunderstanding the lessons fully. Maybe I just need to get used to it, butafter 3 weeks you'd think I would already be adjusted. I guess changingjust isn't easy.^ (Cannel, May 23)It is absolutely true that "changing just isn't easy" for anyone, but it is conceivable.Besides, change was less likely to happen if people were satisfied with the currentsituation, or they did not question it. "Let's talk about the possible ways that would helpyou to catch up and not feel frustrated or left behind (as you said). Please come and seeme," I wrote in her journal. We sat down and talked about these issues at length. I116believe interaction and communication helped to reduce frustration, and promoteawareness and understanding.Carmel voiced many other students' opinion when she said "changing just isn'teasy." Lana expressed this difficulty in her May 24 journal entry, "I have 'moments' ofunderstanding the logic of the reasoning behind the example we went through. But thenafter a while it all starts sounding like some philosophical game of words and semantics. Iguess I see it this way because I find it just beyond my grasp and can't appreciate theprocess. I find this learning method frustrating because it is so foreign to me."Nevertheless, she tried to make sense of the "learning method" and become more familiarwith it. "When we do a problem in class, it makes sense, but this understanding is always'a posteriori.' It is like sitting in front of a work bench full of tools but not knowing whichones to use for a given project. If someone said, 'this is what you want to do . . . ,' then Imight have a better idea of what tool to pick up in mathematics . . .." Concepts, solutionmethods, and formulas were among those mathematical tools that Carmel wanted to learnmore about. Familiarity with those mathematical tools was a necessity; however, Carmelwas waiting to be told what to do with those tools. Furthermore, the next step would beto learn how to use those resources efficiently, that is to know when to use them, why,and for what reason, skills that are called "managerial."The incrementalists sometimes tasted "moments of understanding", as Lana said,and at other times became tired and gave up: "I'm not sure why we weren't just told theprinciple in the beginning. Why are we made to guess the theory? . . .. Personally I like tobe told the right thing the first time, then practice using it in equations," wrote Carmel inher June 5 journal entry. Furthermore, she explained why: "Children may like the'guessing style,' but we're adults." Nonetheless, the "guessing style" helped Beth (May15) to have a "concrete understanding" of the "formula or method."I find that in this class we are working backwards compared to the way Iwas taught in high school math class and this may be part of my confusion117and misunderstanding at times. In high school we were just given aparticular formula then asked to apply it in various situations. However, inthis class I feel we are going through a guess and check process in order tocreate a formula and in the end we have a more concrete understanding ofthe formula or method rather than just memorizing it.^(Beth, May 15)Incrementalists were in a constant conflict between their previously held beliefsabout teaching and learning mathematics, and the new approach that they were confrontedwith. Tern's journal entry of May 13 illustrates this conflict. "I still feel that I couldbenefit from more direct instruction, as I find the exercises a bit difficult to undertakebased on my notes taken during class." Furthermore, she compared the class with her highschool Algebra class where any talking was "no! no!" and said, "My mind got a muchneeded break from memorizing facts once again." Tern continued the same discussion inher May 22 journal entry: "We need to get away from assigning students page after pageof boring questions to answer. . . . We need to show them math can be fun." Themetacognition-based instruction created a quandary situation for the students. On onehand, traditional instruction was straightforward and easier to deal with--predictable but"boring." On the other hand, the new approach gave them "a much needed break" tomove away from facts and formulas and see that "math can be fun." Furthermore, theytalked about "balanced" instruction and how to incorporate certain aspects of the twoapproaches into instruction.The incrementalists were influenced by the metacognition-based instruction butthey were not ready to make a transition. The ways in which the incrementalistssynthesized the metacognition-based instruction was to consider it as a set of teachingstrategies, and not necessarily a different way of looking at teaching and learningprocesses. Shirley explains; "It would have been helpful if we were told to hand ourhomework in the beginning. I found it difficult to see concepts through the manycalculations we had to do at times" (Shirley, June 14). She wanted me to first teach theconcepts, then give them rules and formulas, and the students would apply those rules andformulas in the homework problems. Developing those concepts through problem solving118was difficult for her and other incrementalists. The point was that their understanding ofthe instruction was "teaching mathematics for problem solving," which meantunderstanding the concepts, learning the rules and the formulas, and then "applying" themto a problem or a question. Kayla's comment in her June 5 journal entry shows herexpectations of the instruction. "The worksheet questions are much more difficult thanthey need to be. If the goal is for us to apply what we know to a problem or a question, itshould be made very clear."Most of the other incrementalists were clear in expressing their views. They knewwhat the goal of the instruction was. However, they tried to compromise and solve theconflict by proposing a "balanced" and "modified" instruction. By that they meant toadapt some aspects of traditional instruction and some aspects of the metacognition-basedinstruction. "This class gets frustrating because although I can see the benefits of groupdiscussion and letting the students share their ideas and discover math for themselves, itseems as if there's a lack of guidance and clear instruction. I think that there needs to bea balance between instruction and 'self-discovery' (for the lack of a better term) . . .."That was Teresa's solution (May 30) for a more "balanced" instruction to accommodatetheir needs. Beth's advice was more elaborate:I think the teaching method of helping the students realize their thoughtscan sometimes be over used. I believe this with any method, it must beused in moderation. In this situation in particular, as a student, it can befrustrating to be asked 'what do you think?' in response to a question, it islike answering a question with a question. In a hypothetical situation astudent may be completely confused and not have a clear idea at all. . . . Ithink that when a child obviously has no clear conception of something,asking them 'what do you think?' may only increase the frustration whichmay affect their attitude towards mathematics. I do see this method usefulthough, once the child/individual has developed an understanding of thebasics involved in a particular math problem. My response to this opinionis taken strictly from personal experience. If I know when I am going toask a question and the response is going to be 'what do you think?' (or aform of this question) I am very hesitant to ask and this is when I feel Ineed the most help.^ (Beth, June 6)119There are a number of interesting issues in this passage. She was in a quandarysituation. Her entries clearly show the conflict. She was struggling with different beliefsand that was the main distinction between the incrementalists and the traditionalists. Thetraditionalists were characterized by those who rejected the new approach and did notshow much enthusiasm for looking at mathematics from a different perspective. However,the incrementalists were moved by the new approach. On one hand it made them feelfulfilled and satisfied, while on the other hand they felt that it was not practical. Thisconflict was a sign of change that was taking place, but "changing just isn't easy." (Cannel,May 23). The instruction helped Beth to some extent but it did not cause a drastic changein her beliefs about mathematics and mathematical problem solving. The instruction (ormethod as she called it) was to help the students understand better. It was not somethingto apply after they had developed an understanding because the "method" was not an endin itself. Constructing meaning by individuals through reflection, interaction, anddiscourse was a great activity to acquire knowledge and to develop further understanding.There are more examples to show the characteristics of incrementalists. Forexample, Keith in his May 9 journal entry wrote, "I personally like this method becauseyou are able to discuss each others' ideas and come up with a single answer. I wish moreclasses were set up this way to give students the opportunity to express themselves."However, he started his June 10 journal entry writing that "According to Zahra . . . " whenhe wanted to discuss a statistics problem and then continued to solve that problem himself.The emphasis of the instruction was on the sense-making process, that no one shouldaccept anything because someone else said it is correct but because it seems reasonableand makes sense. Nevertheless, by referring to me, he wanted to legitimize what he wasdoing.The instruction influenced their beliefs about mathematics and mathematicalproblem solving and gave them a chance to learn mathematics and be more "critical" aboutmathematical problems. Carmel explained in her last journal entry:120I guess after all I got a lot out of this course. More than I can recalllearning in any other mathematics class. At first I didn't like this teachingstyle but I now feel that my views have changed. What I have learned inthis class is solid in my mind. I can be critical and really think out themathematics problems. . . . Although I must admit some aspects of thecourse slipped by.^ (Carmel's last journal entry)Shirley did not say anything about change, but she got a chance to see a community ofpeople who showed some positive views about mathematics. "It was an enormousamount of material for that length of time . . . I would not wish this on a friend although Imet some wonderful people in the class and enjoyed their enthusiasm for mathematics"(Shirley, June 14). Teresa, in her last journal entry, commented on the interactive natureof the instruction, an opportunity she had not experienced in her previous mathematicsclasses. "It was nothing at all like what I expected. It was definitely not your traditionalmath course! I liked it in that the atmosphere was relaxed and unlike the usual lectureformat that so many . . . university courses have, there is a lot of opportunity to interactwith others in class." Moreover, Lilian, who knew she would fail the course, expressedsimilar feelings about it in her June 3-10 journal entry: "Believe it or not, taking mathagain has had a rather therapeutic effect on my life."Time. Some people were concerned about "time." Jim wrote "a constructive critique ofmathematics 335" as his last assignment. In that critique he talked about the positive andthe negative aspects of the instruction. "The teaching method of this course gave studentsan opportunity to get to know each other, to realize that not everybody goes about solvinga problem in the same way, and to challenge one another." He then continued, "Thedrawback in implementing such a method is that it is often time consuming. In certainclasses we only managed at times to cover two problems. Therefore, one has to watchone's timing. Another drawback is that in many classes we failed to actually go over manyof the problems that were given to us. A solution to this would be to make certain that we121go over all of the problems related to a particular topic before moving on to anothertopic." I responded to this concern in his journal, "It is time consuming, but time is notwasted. I think if you understood the two problems well, you'll be able to do the rest ofthem on your own. But if you didn't thoroughly understand them, it doesn't help you ifeven 10 of them are solved in the class, because still the 11th one gives you trouble."What Beth wrote was a good reply to Jim's concerns about "time."Some people may feel that much time may be spent on one problem, butthey have to understand that a fully developed concept can be extended andfurther developed more than a superficially learned concept. So in reality,the same amount is learned to a deeper understanding if promoted by themethod in which children discuss their problems and share their ideas.(Beth, May 21)This journal entry reflected on a video clip from Paul Cobb's presentation at the Universityof British Columbia. Interestingly enough, the visual aspect of the presentation helped herand many others (who all commented on the presentation) to relate to the childreninvolved in the video presentation and enable them to see the benefit of similar approachesto teaching and learning mathematics. In fact she believed that the investment in timehelped them gain a better understanding of the mathematical concepts.Teacher's role. The "teacher's role" was another issue that directly related to the instruction.Some of the incrementalists expressed their gratitude for group work and discussion inclass, but when it came to specifics, they preferred to be told. Teresa, in her June 4journal entry explained. "I know that these kinds of questions [referring to the secondquiz] are meant to get one thinking, solving, and explaining, but I am just not sure aboutwhat it is I'm supposed to think and explain!". "What do you mean by that" I wrote, and Icontinued, ". . . I can't tell you what to think, but I may ask you to give an explanation ofwhat you did and, why you did it, and how you arrived at and these questions force you tothink!" My purpose was to challenge her dependency on external authority to tell her122what she should do and when. Holding such a view interferes with the process ofbecoming an independent and active learner.Beth had a different view about the teacher's role in class. In response to apassage from the "Republic" by Plato, she wrote in her May 8 journal entry, "The dialoguewas interesting, but I felt Socrates, the teacher, did too much talking. Perhaps he couldhave encouraged the boy to come up with the facts he had the boy acknowledge." Beth'sexpression showed her reflection on the teacher's role; what it is and what it should it be.She was moving away from teacher-centered instruction towards more student-centeredinstruction, that teacher was more a facilitator, monitor, and most importantly a friendrather than a distant authority in the class.Lilian's reflection on her second quiz on May 27 was interesting; it showed amutual trust between two friends: student and teacher. "The last quiz [second] I walkedaway with a feeling of guilt because at our table three out of the five seated knew how tosolve the problem. I mainly copied for the first little bit (round in square). Soon however,the problem made sense." I could speculate that good communication between thestudents and myself was partly responsible for this honesty and openness. It showed myrelative success in establishing a friendly relationship between the students and myself.Journal writing. The idea of journal writing in a mathematics class was new to many students.However, some of the incrementalists like Lana wanted to give it a chance.This idea of keeping a journal may be a good one. I've never thought ofexploring how I feel about math in my regular diary. Guess I assume Ialways have to write about 'bigger' thoughts. However, exploring thesesmaller, 'micro' thoughts and behaviours may be a more productive andsuccessful way to deal with the larger problem of confidence in myacademic abilities . . . we shall see.^(Lana, May 7)Diana misunderstood the whole purpose of journal writing. She said some peoplewrite more extensively because they don't understand the discussed problem as opposed to123those who do understand, therefore: "they do not have very much to comment about . . .It is more honest and it is a better reflection of how a student really feels if they simplystate that they understand and don't have much to comment on" (Diana, June 6). Herdescription of journal writing suggests the only purpose of writing is to give either areport of what they do not understand or to ask for clarification of several minor points. Iasked Diana to talk about "better reflection" and discuss it in more detail, since Iwondered how a person can judge the "betterness" of a reflection. However, Diana didnot respond to my feedback, in either writing or verbally, and since she usually came toclass late and left early, I did not have a chance to discuss this matter with her.With the exception of Diana, all the other incrementalists utilized journal writingmore than the traditionalists. A majority of the incrementalists used journal writing asmeans of individual instruction. I would not have been able to help them the way I did, ifthose incrementalists like Keith did not address the difficulties that they had regardingspecific topics. "The entire process was very confusing for me because I wasn't too sureas to why we started at bar graph #8 and not for example bar graph #2?" (Keith, June 10-11). He then discussed another topic in detail in the same journal entry and wrote, "Pleasecorrect me if my definition is incorrect! . . . " (Keith, June 10-11). I responded to hisrequest by going through the problem in his journal in detail, and wrote, "This is yourresponsibility to make sure that your problems are solved in the class. Please don't let itgo unless you are sure!" The journal writing helped him and many other incrementalists toreflect on what we did in class, and whether the solution processes made sense to them ornot. Lilian even wrote in her June 12 journal entry that the course and journal writing hada "therapeutic" effect on her life. Terri's views about the role of journal writing wasrepresentative:I asked you my questions in my journal entries. In this respect journalwriting has been very effective, there are several types of journals and notonly is this journal a personal journal, it is also a learning journal. Journalwriting has been a very effective means of learning and helping the teacher124teaches better in this class. Therefore, I am definitely going to encouragemy students to write in their journals after math classes. . . . Teachers morelike to give each and every student individual attention but, with today'sovercrowded classrooms this isn't possible. So if the students use theirjournals as a means to reflect on how they are feeling as well as to askquestions which they didn't get a chance to do during the class, the teachercan give each student individual help. This type of journal also enables theteacher to monitor the class progress rather than always testing thestudents through quizzes or exams. . . . At first we all moaned and groanedabout having to write journal entries 2-3 times a week, but I'm glad that wehad to, because I learned a lot from the experience. It was also nice to getyour feedback and I'd like to thank you for your time and effort.(Terri, June 12)Small groups and whole class discussions. A number of incrementalists appreciated the role of whole class discussions morethan others, but some were confused by it. "I was confused when everyone was arguingabout all the different ways people can sit in chairs [referring to a problem concerningpermutations that we acted out in the class] . . . I think it is a great idea to back up one'sideas when discussing the equations" wrote Shirley in her June 5 journal entry. She gotconfused but she appreciated the role of discussion. However, Teresa did not have thesame appreciation for the whole class discussions.I really don't think that this group method works for everyone. Somepeople just work better on their own (because working in groups frustratesthem or confuses them.) But it's difficult to give everyone a choice in thismatter I suppose. I guess it's just that sometimes I feel as if I almost knowwhat I'm doing but when hearing other people speak, it only serves toconfuse rather than clarify.^ (Teresa, May 30)In general, the incrementalists wrote briefly about the role of whole-class discussions andwere less reflective in this regard.Except Beth, Teresa, and Lilian, the incrementalists rarely commented on thespecific role of small groups. For Beth, working in small groups gave her assurance. "Itwas interesting to see how different people attacked the problem (different thinkingpatterns have always interested me!). I found that when I was doing the problem I wasnot entirely sure that I was correct, but when I explained it to someone else, I would either125notice my errors or realize that I was correct in solving the question" (Beth, May 15).However, Teresa was afraid that she had become dependent on the others. "I think thatworking math problems in groups, etc. has been a good idea because we can learn fromeach other; but one thing that it is missing is that I found I was starting to rely on someoneelse to show me how to do something instead of sitting down and trying to think of a wayto do it myself' (May 26). Lilian raised another important issue in this regard: "I havenoticed that in group situations I feel awkward and stupid if I can't contribute to the teameffort or offer anything of value. When I felt this way I became quiet and sometimes, I'venoticed, I try to avoid the task at hand" (May 23-30). I commented on Lilian's concernand told her that every "why" that she asked would contribute to the "meaning-making"process of the group.All the incrementalists acknowledged the role of manipulative material in theirbetter understanding of the mathematical concepts especially geometry. Kayla's journalentry on May 8 is representative; "I found that exploring shapes by actually cutting themout and manipulating them a very interesting activity."The Students' Beliefs about Mathematics The first and the third research questions concern the students' beliefs aboutmathematics and possible changes in their beliefs as a result of the instruction. Thepurpose of this section is to study the effect of the instruction on the incrementalists'beliefs and to discuss the nature of the change.All twelve students in this category, wrote in their autobiographies at the end ofthe first class, about their views and their experiences with mathematics. Terri, Shirley,and Jim all said they enjoyed mathematics although they never got good marks in theirmathematics courses in high school. However, Kayla "never enjoyed it", and Beth said"Math has never been one of my strengths ... ," as did Keith. Leah did not see any pointto studying mathematics at all. "I also know that 'you' don't need math in the real world,or at least the one I was in. Not with calculators and computerized cash registers," but126she continued to say "I would like to understand mathematical processes and thoughts"(Leah's autobiography). Lana got good marks for her mathematics courses but: "despitethese successes I am convinced I cannot understand the principles behind algebra. So far,I feel I have gotten by strict rote memorization of rules and formulas." Carmel took FiniteMathematics and failed the course; however, she said: "I hope to achieve more confidencewith my mathematical ability. I'm looking forward to the next six weeks." Diana washoping as well to overcome her "fear" of mathematics. For Lilian, the mission was moredifficult since she had "math phobia," but she was "determined to overcome this phobia."Teresa never had "any serious problems in the subject and enjoyed some aspects of it butwas never interested enough to continue studying it." Her belief about mathematics wasfirm and clear, she believed that mathematics was "more objective" than other subjectmatters. Therefore, it was easier to teach mathematics than, let us say, literature.Teresa's beliefs about mathematics like many others, altered as the coursedeveloped. She wrote about the process in her May 15 journal entry.Before this class, I always felt that math was a purely objective subjectwhere one could only be right or wrong. In a way, this is true. But I thinkthe problem is that many people think math to be like this, which in turn,leads them to believe that not only must there only be one correct answer,but there must only be one correct method in which to reach that answer.We're puzzled when our first try at a problem does not work, and when wefind a solution that seems too 'simplistic,' we lack confidence in our answer.But it's difficult to opt out of that frame of mind.^(Teresa, May 15)Shirley expressed her views and wrote, "I am becoming a little more intrigued thantortured . . . well I am getting more interested in math but still finding it difficult. . . Iseem to need to go over things many times. .. . I enjoy the inspirational messages" (June10). Everything is relative in this world, and change is a matter of degree.After the first few classes when we were all working with concrete materials, Terriwrote in her May 9 journal entry. "If you have positive experience with math throughmanipulative materials, then you experience the same degree of success." Kayla who127"never enjoyed math" before, felt that she was making some progress in overcoming hermathematics anxiety. "I feel I am making better progress this week. I think I am probablyovercoming some of the math anxiety I was experiencing" (Kayla, May 17). Kayla wrotein her last journal entry, "It is important for me to leave behind my negative feelings sothat students don't pick up on my negativity." She directly addressed the role of a teacherin the students' beliefs about mathematics. "I can honestly say my attitude about math hasdramatically improved since the beginning of the course!" This alone can partially answerthe third research question that the students' beliefs about mathematics were influenced bythe metacognition-based instruction, for as Teresa wrote in her last entry "math can beenjoyable."The Students' Beliefs about Mathematical Problem SolvingI have to feel a certain amount of success in problem solving to want to continue.(Kayla, May 9)Some of the incrementalists showed a considerable change in their beliefs aboutmathematical problem solving. The aim of this section is to portray that change.The incrementalists, like the traditionalists, started the course asking for themethod of solving problems. "I wish we could have one direct approach or formula touse. That the problem solving could be more economical and fast. I understand we havemany ways of approaching problem solving but I am concerned about not being able torecognize what to do when given area or height to find," Lilian wrote in her May 13-17journal entries along with other examples4• Lilian's view was as extreme as Lynn's view, atraditionalist who was also looking for a panacea to solve all her problems. They bothwere so concerned about time that they could not relax to enjoy the activities in the class.4 Lilian, the only one who failed the course, missed many classes due to her full-time joband other courses that she was taking. She took the same course in the following year(1992) and earned an "A." Lilian said she started the course the second year with differentbeliefs and that helped her to get through easily. She even "enjoyed" the mathematicscourse.128However, Teresa's belief was less extreme and more representative of theincrementalists. "All that is needed is the knowledge of basic rules ... and an applicationof these rules, plus the information given about the problem pieces together to know thecompletion or answer to what one is trying to solve" (Teresa, May 7). Theincrementalists, as Teresa described, were struggling with the idea of mathematics as beingobjective, or mathematics as a creative endeavor. Sometimes they felt that they couldsave time by seeking the formula or method to solve problems since they thought it was"more economical and fast" as Lilian described.There was a constant battle between these two different views about mathematicalproblem solving among the incrementalists. They often made a comparison betweentraditional and the metacognition-based instruction and discussed different aspects of bothapproaches. They showed some interest in being taught in the latter way, but were moreconservative regarding "new" ideas. Lilian and a few others were exceptions among theincrementalists who were persistent in their views about mathematical problem solving.They were afraid they would be more confused if dealing with a different approach.More confusion followed me when I would have to see the many ways ofproblem solving on the board. I would tend to find this frustrating becauseultimately what I would want is one direct way of problem solving.However, even when we did get formulas I would have difficulty inmemorizing the procedure or know when to apply what and where.(Lilian, June 10-14)Lilian was in such conflict that it could be a sign of change! On one hand, "one directway" looked an easy way out, but on the other hand, she felt the need for certain skills thatwere metacognitive in nature, such as the ability to use her resources efficiently.On the contrary, for the majority of the incrementalists the traditional approach tomathematical problem solving did not make much sense. Keith, in his May 8-9 journalentries explained, "I think math is not just only applying a law with a bunch of numbers,but rather, one must understand the reasoning behind the law in order to fully answer the129question properly." Furthermore, the metacognition-based instruction offered them a newperspective, but it was not enough for them to make a transition.The incrementalists were happy to understand the reasons for what they did whilesolving problems. Kayla wrote in her May 20 journal entry, "I found it important to keepasking 'why do you think it is this way?' or 'what is another way to do this?' . I realizehow important it is to understand why." The new approach had something to offer toKayla and other incrementalists and helped them alter their beliefs about mathematicalproblem solving, Kayla's case was an illustrative example. On the first day of the courseshe said she "never enjoyed" mathematical problems. Later on, she wrote on her thirdquiz: "Once I got started and was able to explain what I was doing and why, I began toenjoy solving the problem." This was a big achievement for her, and an answer to thefourth research question that, indeed "the students' beliefs about mathematical problemsolving were changed as a result of the metacognition-based instruction."Students' approaches to mathematical problem solving was shaped by their beliefsabout mathematics. For those who saw mathematics as "objective" as Teresa described itin the beginning of the course, mathematical problem solving was as straightforward asapplying certain formulas or following a method to arrive at some answers. This approachdid not leave any space for reasoning and creativity. In this view, there was a one-to-onecorrespondence between mathematics and numbers, while everything else seemed"unmathematical." Teresa's journal entry of May 13 was illustrative:When trying to find the area of the dodecagon, we figured a way of findingan answer, but we were doubtful because it seemed to be a slightly toounmathematical, way to solve it, however, in comparing our answers withother groups who did the problem another way, the answers were thesame. I've also realized that there are many different ways of approachingmath problems.In fact their way of finding the area of the dodecagon was innovative and interesting.They constructed the dodecagon using pattern blocks (one hexagon, six squares, and six130triangles,) and found the area of each one of them, and then added them up to get the areaof the dodecagon. However, her group members had a hard time seeing that their solutioncould be mathematically sophisticated and elegant. (Appendix Q contains a copy ofTeresa's work.) Shirley's last entry shed more light on this view. Shirley said that shecould do many things but she was worried about "the actual math, not the ideas," by thatshe meant calculations, using formulas, and plugging in numbers. It is sad that "the ideas"of solving a mathematical problem in her view did not count as "actual math."The incrementalists gradually saw the benefit of having different approaches toproblem solving. They experienced the importance of problem solving processes asopposed to emphasis on the "correct answer." "I know that getting the answer correct isnot the most important factor when figuring out problems, . ." wrote Shirley in her June5 journal entry, "the process is more important." They started to see that the multi-approach to problem solving gave them "strength" and increased their mathematicalpower. "I think my strengths are that I try and look at the problem from severalperspectives ... I also can see things spatially" (Kayla, May 14). In addition, she was alsoaware of her "weakness." "My weakness is that if I experience some difficulty I have ahard time trying to overcome this." That awareness was the reason that she was able toovercome that "weakness." The emphasis that traditional teaching and evaluation puts onthe finished product, namely the "correct answer" forces students to be prudent and avoidtaking risks.Mathematical Connections "Mathematical connections" was a theme that the incrementalists talked aboutfrequently. They said that they had difficulty relating the mathematics that they hadlearned in their schooling, namely a set of rules and procedures, to anything outside themathematics class. However, most of the incrementalists were excited to see the "real-life" application of mathematics that they were doing in class. In the beginning of thecourse, what Leah wrote in her autobiography was typical. "I also know that 'you' don't131need math in the real world, or at least the one I was in, not with calculators andcomputerized cash registers." The instruction provided an atmosphere for the students toallow them to see the applicability of mathematics in their daily lives, and to seemathematical connections. Keith tried to link the mathematics that he learned in class toeveryday life around him. In his May 13 journal entry Keith explained how he measuredhis bedroom's height applying a method of measuring height that only needed a sheet ofpaper and a pencil. "I tried the formula at home to determine the height of my bedroomand found it to be very accurate. Thus, math isn't something you only apply in theclassroom, but it's also used in your everyday life:Terri was excited to see that she could use the mathematics that she learned inclass to do a real problem concerning ordinary people. "It's neat little tricks like this[finding the probability of having one boy out of seven children] which keep me interestedin and liking math" (Tern, June 6). Mathematical connections increased their knowledgeand their appreciation of mathematics. "Today we learned about sines and cosines. It'sgreat to finally know what those buttons on my calculator are actually for ... " (Lana,May 27), in her last assignment Kayla explained what gave her a greater appreciation ofmathematics. "Math has seemed to me unrelated to most things, when I can relate aconcept or a theory to an actual person and period in history it holds greater meaning. Igain a greater appreciation and interest in solving the problem when I have something torelate it to" (Kayla's last assignment).For some incrementalists, mathematics made sense only if it was practical andrelevant to their lives. Diana wrote about the "practicality" of mathematics in response toone of the metacognitive questions that was part of the second quiz, "how did you likesolving the problem?" Her response to the question was that "I enjoyed it because it waschallenging for me and practical in that it is a problem that could be applied in everydaylife to be more efficient." However, her response to the same question regarding the thirdquiz was that "I really didn't like it because it is confusing and mind-boggling, not only to132figure out just what the question is asking but also to come up with such large numbers."For Kayla, Diana, and others in their camp, "large numbers" were not practical.Therefore, they disliked them and saw no point of dealing with such problems:I still don't fully understand the question on page 387 [Jacobs, 1982]. Amajor reason why I think I have difficulty is because I don't see theimportance in finding out the answer. I am asking myself why do we wantto know how many pages to produce a 100,000,000,000,000 . . . poems.To be honest I don't care! I think because the nos. are so huge. . . .I justdon't find this problem interesting.^(Kayla, June 6)It was important to see the connection and the relevance between mathematics and reallife. However, one had to be careful not to reduce the necessity of the sense-makingprocess of looking for connection and relevance, to a strictly utilitarian view ofmathematics. It would be difficult to enjoy anything if you always looked for theimmediate benefit. This attitude hindered further innovation and creativity.Teaching Mathematics for Problem Solving: The Incrementalists' Perspective The incrementalists defined "teaching mathematics for problem solving" in theirown words which were based on their beliefs about mathematics and mathematicalproblem solving. They wanted me to teach the concept first, solve a few problems on theboard step-by-step, and then give them more exercises to practice the method of solvingsimilar problems. However, this approach is more flexible than "traditional teaching" inthat students are involved in the meaning-making process but to a limited degree. Themain characteristics of the "teaching mathematics for problem solving" based on theincrementalists' views were as follows:-A "balance" between traditional instruction and metacognition-based instruction-"Discovery-learning"-Active participation in small groups-"Guided inquiry"-"Time" is wisely spent133-All problems should be solved in the class-Mathematical connection is an important part of mathematical learning-The teacher's role is more like a mentor than the sole authority in the classSummary The incrementalists were in a quandary situation. The majority of them like Jimand Beth solved the conflict by proposing "balanced instruction, "meaning to adopt someaspects of traditional and some aspects of the metacognition-based instruction.Nevertheless, the other incrementalists were not able to resolve the conflict easily. Theinstruction did not change their world views, although it influenced their views or at leastmade them re-think teaching and learning mathematics.The incrementalists utilized their journals to discuss both cognitive andmetacognitive issues. They conclusively benefited from working in small groups one wayor the other. However, they barely commented on the whole class discussion. Theytalked about mathematical connections at great length. Although most of them were nothappy about writing the exams, only Keith, Lilian, and Shirley reflected on this anddiscussed it in their journals. The incrementalists did a great job in elaborating on theexams and answering the metacognitive questions which were part of the second and thirdquizzes. They were reflective in their problem solving, and their journals and examsprovided an opportunity for us to communicate with each other in a reflective manner.The reflective communication helped them to become more aware and acquire bettermanagerial skills.Findings from the Third Category. the Innovators I think math has been taught like 200 years ago, militant, deductive, no discourse. All weare trying to do is to find the right answer. It doesn't really matter anything else, how toget there. That's a very Victorian attitude. (Marris's pre-interview)134The nineteen students in this category, the innovators, were all influenced by themetacognition-based instruction to different degrees. For some of them, the instructionwas a way of teaching and learning mathematics that they had always asked for but nevergot. For the others, they did not mind a different way of approaching mathematics thatcould help them understand the mathematical concepts better. Although the innovatorsdid not reject the new approach, there was a considerable uncertainty and frustrationamong several of them. It took longer for some of them to see the effect of the instructionon their learning. The majority of innovators wrote about how the instruction helped themto gain more self-confidence and self-worth. They conclusively reflected on the role of thejournal writing in their mathematical learning. Many of them wrote about the effect of thewritten and oral feedback on enhancing their understanding, self-confidence, and self-worth. The classroom environment was a major theme that emerged from this category.Unlike the traditionalists and incrementalists, the innovators frequently wrote about theeffect of the classroom environment on their learning. The establishment of the socialnorm of the class was partly due to the constructive suggestions that the students madethrough their journal entries.This section gives details concerning the different themes that emerged from thestudents' work in this category. A student-based definition of "teaching mathematics viaproblem solving" will be given later.The Instruction and its Components For a number of students, the instruction was the way of teaching and learningmathematics that they had longed to have. They always wanted to be involved in themeaning-making processes in their mathematics classes. The innovators' desire was toconstruct meaning by themselves and enrich their understanding by seeking answers totheir "whys." This had not necessarily been encouraged in their past: "I used to ask [mygrade 4 teacher] 'but why?' The teacher said: 'It doesn't matter why Melisa. Just do it asI did' (Melisa's autobiography).135The instruction for most of them was "all new." Melisa wrote in her May 21journal entry, "I realized a while ago that part of the thrust of this class isn't just uslearning about math. This class is also a model of a teaching method. From myperspective this is all new" (Melisa, May 21). Nevertheless, experiencing a "new" waywas what Patrick was afraid of: "The concern I have with this course stems from anexperience my mother had at UBC.. . . She nearly failed . The course approached musicin a way she was not used to. I hope I do not have similar problems" (Patrick'sautobiography). However, Barbara was curious: "I find the approach so different. I amvery interested whether it will help to overcome the difficulties I have in math. I amtypical of the poor math student" (Barbara, May 10). Furthermore, her curiosity wasgradually replaced with endorsement. "Alternative methods that I had to try to bringabout my understanding were suddenly acceptable as ways of doing problems. There wasno longer just one correct way. Talking to one's peers was also acceptable" (Barbara,May 13-17).The innovators were able to make a comparison between the "new" instruction andwhat they had already experienced in their schooling. "I didn't find Roy's teaching methodall that wonderful. I felt like I was back in high school. I much prefer to discover stuff formyself. It takes longer but I think I understand better in the long run" (Melisa, May 30).The innovators were therefore more influenced by the instruction than the traditionalistsand the incrementalists, since they somehow got what they were looking for: "It's too sadthat schools don't teach kids to think about why they are doing what they are doing inmath . . . it makes it more accessible and understandable to more students. . . . I knowwhen I went to school . . . the right answer was the only important thing . . . and I neverdid well at it" (Clara, May 22). Barbara mentioned the role of interaction, "Thus it was adelight to discover the new approach to teaching math in a concrete and moreindividualized way . . . not only through direct teaching but also through interaction with136materials and their peers" (Barbara, ?). Clara discussed the possibility of having suchinstruction in the elementary schools:This new way encourages personal thought and I like that. I can really seethe benefit of this approach to math in the classroom ... from the verybeginning. Children don't need to memorize facts and procedures . . .rather they need to understand why .... We are having to explain the whyof things . . . The why is really important . . . this really makes it differentfrom a high school course . . .^(Clara, May 22-26)Being encouraged to ask "why" and find answers provided an opportunity for the studentsto overcome their "fears" of mathematics: "It's wonderful the way the [my] fears are beingput aside via being taught the 'whys' of math. . . . That's what I really like about this class,nothing is accepted because a book says that it is true" (Melisa's last journal entry).Some of the students were concerned but hopeful that the instruction would workfor them. "The first class, was different from what I expected. I assumed it was going tobe the same as we took in high school. Instead it was a fact finding mission. I'm not sureI understand the significance of it. It is interesting and different and not unpleasant, justdifferent. I just hope I get it all" (Sheila, May 7). Sheila's positive feelings helped her tobe patient with this "different method." "I still find the method unusual and different thanwhat I'm used to. Fm used to structure. I realize that this class makes you think on yourown. Great concept! but again, I have to learn to be patient" (Sheila, May 8th).However, her patience was fragile, "I didn't understand what was expected or what I hadto do. The more I got frustrated, the more I gave up. I don't like feeling this way. I'mnot sure how to deal with it. Maybe I'm just not too smart when it comes to math"(Sheila, May 9). The main purpose of the instruction was to alter students' old beliefs--notto reinforce them. I wrote to her: "But your feeling has nothing to do with yoursmartness. You got your feelings from your experience. As you said, you are not 'usedto' this way of learning." Her frustration was a legacy from many years of "bucket filling"teaching, "tell me what to do, I will do it." The question was whether the instruction137could help her to "deal" with it or not: "I'm still a little concerned about what is wanted"(Sheila, May 14). She was still struggling with her previously held beliefs that werechallenged by a different "method." It took a while for Sheila and some others includingPeggy, to feel confident with the "new method." Peggy wrote on her May 14 journalentry, "I am finally beginning to feel a little more confident in the class . . . though thereare still times when I feel I am 'sinking.' The innovators tried hard and gained a lot. Theywere active in class and their contributions to the small groups and whole-class discussionswere highly appreciated. The innovators wished they had learned mathematics throughproblem solving from the beginning: "I wish I had learned math this way the first time"(Sheila's last journal entry). Perhaps it would have helped them develop a strongermathematical background. Peggy explained her difficulty in the absence of a strongmathematical background: "My problem, I can see, is my great lack of math background,so I have nothing to draw on. I can now work out what I need to do, but I don't have themath knowledge to do this, which I find frustrating" (Peggy, May 22). Peggy found ananswer to what she needed and why, but she did not have enough resources to know whatfrustrated her. However, she was getting better at experiencing success in the course,since her awareness and her confidence were increasing. She was able to identify heremotions towards mathematics, her knowledge about it, and most importantly what shewould need to deal with these issues. One of the contributions of the instruction was tohelp the students gain more confidence and feel more comfortable as the courseprogressed. Reflecting on Paul Cobb's video presentation, Peggy explained how thathappened.The lecture [Cobb's presentation] yesterday was very interesting and I cannow see why you have been "instructing" us in a similar manner. The ideathat we "discover" for ourselves . . . rather than being told a way, to do itthis way also involves more discussion between people in groups. Alsopeople solve problems in different ways. I certainly feel more comfortablein this type of situation and not afraid of making mistakes but tryingdifferent ways, though at times I still get frustrated! (Peggy, May 22)138Sophia pointed out a fundamental difference between this "new method" and the"traditional math class" as she called it. "It is certainly not like a traditional math class as Iremember them from high school, when the instructor would work out long equations onthe black board and all the students would follow along and copy and accept things to betrue, as given. It seems that in this class we are given a lot of time to discover thingsourselves and draw conclusions. I'm finding that it's a more creative and understandingway of learning," wrote Sophia in her May 8 journal entry.Traditionally, students would do what Sophia described: passively take notes whilethe instructor was "lecturing" without much interaction. Most of the questions raisedwere for clarification of a few minor things that students were unable to read on the boardor did not hear properly. Ben was not at all sure about the way the instruction wasdeveloping away from the "old way." "I know it is good to ask questions for clarificationbut I really believe a lot of people would benefit from taking a couple of minutes to lookat the example, think about it, and reflect on what was done and why" (Ben, May 13).They said that dialogue and debate were not usually a part of traditional teaching andstudents did not play a major role in the meaning-making process. However, as the courseprogressed, he saw the benefit of interactive instruction and said, "I liked Roy as a teacher,very interesting. But I felt like I was back in grade 13. The teacher doing the talking, thestudents just copying the work down" (Ben, May 29th). In such a setting, he would listen,"copying the work down", and only ask questions "for clarification," not for furtherdevelopment and understanding.On June 3, Sally did what Ben experienced in his grade 13: took notes withoutbeing really involved in the class activity. However, she did not like it and reflected onthat in her journal. "Today, I felt like I was writing down notes frantically withoutunderstanding what I was writing. So now, when I reread my notes it is very difficult totranslate them into something comprehendable" (Sally, June 3). What Sally did wasimportant. She tested the efficacy of what she called a "new method" against what139seemed to be a trend in most of the mathematics classes. Sally realized that she couldcomprehend more with the "new method" and was glad to find out that the "new way ofteaching was being carried out in elementary schools." By reflecting on Paul Cobb's videopresentation, she explained why she thought the "new way" was more productive.I had no idea that this new way of teaching was being carried out inelementary schools. And, I am very glad that it is because the way I wastaught math was not a success (I've always been afraid of math, and notgood at it). Therefore, since a lot of people seem to feel this way, it isgood that the teachers are opening up to new ways of teaching . . .. There ... will be people who are not looking far enough into the future to see thebenefits of this method. These people might get angry at the fact that theteacher does not simply tell the children to memorize. But, I feel that it isimportant for children to be able to think for themselves. I know that in mymath history I could never do this, and never really had to. Therefore, Ifind it hard now (in this class) to think out how to solve a problem on myown.^ (Sally, May 21)The way Sophia dealt with the instruction was interesting. She was able to clearlydiscriminate between two different approaches to instruction. She then internalized theinstructional processes and wrote more specifically about how that worked for her. Thefollowing excerpt shows how she and others in her camp empirically concluded that theinstruction would work for them. "I was very pleased by the way that this classprogressed for me today. . . . The problems assigned to us ... gave our whole group agood sense of accomplishment. I think I'm now getting used to the methods of learningthat we'll be using. At first I was feeling like it took us an awfully long time to grasp justone idea, but as long as we're progressing at the rate expected, I guess that's fine" (Sophia,May 9).The innovators were more critical and more observant about issues regarding theinstruction. They talked about a variety of issues concerning the course and outside of it.For example, almost all of them reflected on Paul Cobb's video presentation and comparedit with what was going on in the course:140The video and discussion on math teaching methods was very interesting. Iam glad I have experienced at least two weeks of this method as it iscompletely new to me . . . I can see there are many things being learnedincluding social skills such as listening, sharing, communication, working ingroups without intimidation, etc. The method seems to be time consumingand I feel a teacher would have to be very careful to maintain the goal ofthe lesson as well as dealing with unexpected points of value that come up. . . I feel it is important for children to work on their own as well ... weneed to learn independence and self reliance. This method is so much morefun than the old way . . . So briefly, I think a combination of methods mayhelp. The final analysis is that an effective teacher has to be able to usevarious methods and to know her students well.^(Rita, May 21)Rita v.ns the only innovator who talked about variation in methods. However,what she said was slightly different from what the incrementalists said. Her main pointwas flexibilit - and spontaneity in teaching. I told her that I truly believed teaching was anart and there 'as no "method" written in stone to be followed by all teachers, and teachershave to be fle able enough to switch from one way to another depending on situations andstudents' neec,s. However, it does not mean that teachers can switch their world views.The majority of the incrementalists asked for more "structure" and expressed theirconcerns about "time." However, except Sophia and few others who sometimes thoughtthat discussing problems took longer than they expected, Ben was the only innovator whowas worried about it. "I think there should be more structure so we can go through themquickly and get on with the day's new work" (Ben, May 15). I pointed out to him thatthere was no reason for him to be worried about time. He knew the final exam would onlyinclude the material covered. I wrote that Ben should stop to ask himself what would bethe importance of "the day's new work" if he did not develop much understanding.In the beginning of the course, Penny said she preferred to be told by the teacherwhat to do. "It wasn't clear to me, what the teacher expected from us . . . I did not knowwhat she [teacher] wanted us to do" (Penny, May 8). However, her last journal entryshowed that she changed her views about the instruction as the course progressed. "Thepart I enjoyed the most was working with actual figures [manipulative/models] instead of141the pen and paper method where the teacher is the only one that taught. I liked the waythat we taught each other things" (Penny's last journal entry).The students were pleased to learn how certain formulas were derived. "Thiscourse is so neat the way it has gone behind the formulas we used in high school withoutquestioning, and now we are learning how they actually came about" (Bob, May 23journal entry referring to calculation of ic). He talked about mathematical connections andthe purpose of teaching mathematical concepts to young children and adults. Rose wroteabout the emphasis on applying a formula to a certain situation and finding "the correctanswer" as the main activity in her previous mathematics classes: "I remember some ofthese [mathematical concepts] were taught when I was in high school. . .. We just appliedthe formula and found out the answer. We would never know or understand why theformulas are applied and worked out that way. But now we do not concentrate on merelygetting the correct answers. We need to understand what these formulas mean anddiscover how they are set" (Rose, May 24)."Models help me to do better. When I have got them in my hands, I can do waybetter," said Penny in her post-interview. The role of manipulative materials wassignificant in making sense of the formulas being developed and used. "I must admit thatthis afternoon was fun. To have the 'hands on' experience was excellent and much moremeaningful than trying to understand the theory by writing down information. I hopemany of the lectures will focus on the 'practical' aspect of math" (Peggy, May 7). Roslin'scomment on the role of manipulative materials was representative: ". . . Anotherimplication is that in solving this kind of complicated problem, it is vital to have the objectin hand. Then you can have a clearer idea of what it looks like. It is even better to draw itor make one ourselves. Then we can feel it, reset it, or rearrange it to understand better.. . . In other words, we have to look at the object perspectively [sic] and imaginately [sic]"(Rose, May 24).142The teacher's role. The traditionalists wanted me to direct them all the time and tell them what to do.As a matter of fact, they considered limited "lecturing" and my non-authoritarian role inthe class as a weakness rather than a strength of the instruction. However, the innovatorshad different views about the teacher's role: "Even the instructor, from time to time, isfound emerging [immersingl herself among the groups, giving the appropriate limitedguidance hints. She does the least talking and instructing. In fact, there is no lecturing"(Rose, May 10). A number of innovators elaborated on this issue. They liked to "do it"by themselves. "This was a neat lesson because the teacher could have merely told us thisfact but instead she let us 'do it'" (Linda, May 7). Rose's observation about the teacher'srole is representative. She wrote more specifically about traditional and metacognition-based instruction in general and the instructor's role in both settings in particular:It is not in the traditional and conservative way as we learned before, usingand memorizing the multiplication table, or being told by the teacher whatwas right or wrong for the answer. . . . The teacher is not just going to sitback and relax. In fact, she has an important role to play in helping andadvising the children as required. She is like a guide or consultant, goinground the pairs and listening in. She observes their difficulties andmistakes, both individual and general problems. In the light of this, she willbe able to shape both in the class discussion later and activities in furtherlessons.. . . I, personally, advocate this method and I think it is aworthwhile attempt. After all, as E. Stevick says: "Students are not apower supply, in such a way that the power used by one diminishes thevoltage available to the rest."^ (Rose, May 24)Journal writing. All the students in this category considered the journal writing to be significantpart of their learning. Many of them received individualized instruction through theirjournals. However, in the beginning of the course, there was confusion, uncertainty, andeven resistance towards keeping journals in the mathematics class. "I disagreed out andout with the use of journal for this course. I felt that such a tool would be useless in oursituation. What good would writing our feelings down about math, for anyone except143make them feel worse?" wrote Melisa in her "critique of the course." She then explainedhow the uncertainty and resistance phased out and she was soon able to see the benefits ofkeeping a journal. "I began with reluctance, and found it difficult to do. But then Irealized that I could write about the things I did not understand and in doing so I furtherrealized that things became more clear if I did this. I utilized my journal to solveproblems" (Melisa's last journal entry).Each student benefited from journal writing in a unique way. Clara wrote abouther own writing experience, "I guess that writing the journal will be helpful because inthinking about what I was going to write . . . I suddenly realized that a few things hadbecome clear" (Clara, ?). Journal writing for Ben was a means of self-assessment. "Thejournal was my way of assessing how well I understood a new concept. It also providedme the opportunity to express my views on different events/occurrences in the classroom"(Ben's last journal entry). For some people, journal writing served as a review andorganizer. "It [journal writing] has helped me to review what we have been doing, andsort things out in my own mind, this I have found very helpful" (Peggy, June 13).Sandra was the only student in the entire class who asked to have more than 30%of the final mark for journal writing, but no one agreed with her. Her journal wasextremely lengthy and detailed, it even had a "table of contents"! She wrote, "Personally,this journal has been a guide for me which has aided me in focusing on my strengths andbuilding my confidence in math" (Sandra, June13). In response to my request for the self-evaluation of their journals at the end of the course she wrote, "As my journal . . . showsmy progress and development and the effort I have put into it, I feel I have earned a markof 28/30 = 93.33333...%! which is a rational number, sorry no time to give you a proof!"(Sandra, June 13) I promised her that "I'll be rational too!"The innovators were the only ones who wrote extensively about my writtencomments in their journals and how that feedback helped them to develop a betterunderstanding of mathematics.144Feedback is likely to enhance students' motivation because it allows themto evaluate their progress, to understand the level of their competence, tomaintain effort toward realistic goals, to correct their errors and to receiveencouragement from their teachers. Sometimes it even develops a betterunderstanding between the teacher and students because they cancommunicate to each other more deeply if the situation is not allowed inclass due to lack of time and opportunity in the classroom. Thus it createsfriendship and fosters a better relationship between them.(Rose's last journal entry)Jack talked about an important aspect of the written communication between thestudents and the instructor. ". . . The feedback is important because the instructor needsto know the areas of concern of the student, and the student can be helped through writteninformation or comments by the instructor" (Jack, June 10).Journal writing has become a bandwagon. Many students shared a negativeexperience regarding journal writing in some courses. They were asked to keep theirjournals in many classes, but instructors did not really go through them in detail and whatthey usually got was a "check mark" on their journals. The fact that I responded to theirjournal entries pleased them enormously. Rose explained:Being a teacher for many years, I know it is much easier to give grades andmarks than giving some written feedback on a piece of work. Besides, it isvery time-consuming and painstaking. To me, the journals we write is justlike the channel for communication and the feedback we receive is morethan an extrinsic kind of motivation, like marks, grades, school reports, testresult and teacher approval. There is understanding, assurance, comfort,and console as well. Success at it helps build up a prestige in my own eyesand in the eyes of others.^(Rose's last journal entry)I wrote to her, "I truly believe that students don't learn from their grades, rather they learnby reflecting on their doing and understanding. I hoped that the feedback would servesuch a purpose, to open up a dialogue between the students and the teacher and to createa more friendly environment that learning is fun and natural rather than being boring andartificial."145For the majority of the innovators, writing was a personal endeavor that theyenjoyed doing. They did not take it as "another requirement" to be done, as thetraditionalists did:I really liked writing in my journal. It helped me sort out my own problemsand thoughts, and provided a form of communication between you and me.I appreciate that you took the time to read our journals and respond tothem all! You seem to care about how your students are doing in yourclass (whereas practically all of my previous profs don't seem to care).(Sally's last journal entry)The students' comments helped me to know the students better and become moreaware of their needs. They also made constructive suggestions in terms of instructionaldecisions. "So it seems I would like it a bit more if you (Zahra) would sum up the ideas orproblems that we've been working on at the end of the class" (Sophia, May 9). Suchcomments made me more conscious of the class needs in general. I carefully consideredthem in order to improve the learning environment. As Peggy mentioned, this was a "two-way exercise" in which both the students and the instructor were actively involved in themeaning-making process. Peggy's observation was representative:I had always thought that the comments you wrote on the journals weregood, relevant, and showed that you obviously read them and cared. Alsothat you were concerned for us as people not just bodies in the class. Forme the comments were encouraging and helpful. I am very grateful for thetime and effort that you put into your comments. If there were little or nocomments that would make me feel that you weren't interested or maybedidn't even look at them. So therefore why should I put time and effortinto the project. I think that it is a two-way exercise, if you show interestthen I will put the effort into doing something.^(Peggy, June 11)Small groups and whole class discussions. "Another significance of this course is that activities dominate the entire two-hourlesson. Learning is not a passive process. If students are to learn, they must be activelyengaged in doing, not just listening but comprehending" (Rose, May 10). However, a fewstudents like Sally initially had reservations about the group work. "I am quite nervous146about taking the class, especially since we have to do our work in groups so everyone cansee how much I don't know" (Sally's autobiography). Soon, as the course progressed,they realized that they should not have been worried about "not knowing" since no onewas a "real whiz." "I like working in groups, that's been very helpful and it makes us allfeel like we're in the same boat and that none of us are real whizzes" (Sophia, May 9).Besides, they all had something to share. "Talking about it [mathematical concepts andproblems] in groups helps enormously. We all have something to contribute even if it isonly our ignorance" (Barbara, May 10).The innovators actively participated in the small groups and the whole classdiscussions. For them, the group activities were crucial components of the instruction andnot just the classroom routines. "It [the course] taught me how to work in groups,sharing ideas and suggestions. While I have done group work in many different classes,the work done here was much more intense and definitely thought provoking.Concentration and patience skills were also worked on in class," wrote Ben in his lastjournal entry. The group work helped the innovators to develop a deeper understandingof mathematics. "It was not hard to figure out the mechanics of it [a problem], but I didn'treally know why I'd done it, but as we talked about it in class . . . it became clearer"(Clara, June 10).The group work provided an opportunity for the students to hear different ideasand discuss every possibility. "I like the way you are going through each possibility andmaking it clear why one explanation works and the other doesn't" (Barbara, May 31). Thediscussion helped them realize their conceptual difficulties as well. "I would not have seenmy mistake if there had been no class discussion (Patrick, June 12).Peggy explained how the cooperative nature of the group work helped her groupmembers with their quizzes. "It was good to work as a group because we generated theidea together on how to solve it then worked on our own to find the answers .. ." (Peggy,May 28). Sophia wrote how a diversity of ideas enriched their discussions and increased147their understanding. "Our group is working well because it seems that different people aregood at different things and we've all been able to help each other out" (Sophia, May 23).The small groups had the potential to make some of the students dependent on theothers. However, the best way to overcome that possible difficulty was to resolve thiswithin the groups. The innovators gradually became more aware of their responsibilitiesand took the matter in their hands. Rita was an enthusiastic and active participant in allthe class discussions. Her group worked together very well. She and her group membersreflected in the second quiz on what that the students did in their groups and addressedthis issue. Rita's reflection was coherent and typical:We received the quiz question and very shortly I suggested that we find thedifference in area left unused . . . The others [in group] agreed that thisseemed logical so we proceeded to do that. I didn't even know I wasthinking about it until I woke up in the night as if something had bit me andthe thought in my mind was oh no! we didn't go far enough . . . so weshould have gone one step farther to show the ratio of square/circle tocircle/square. . . . I felt really stupid and embarrassed that although I had agood suggestion, I was too careless to carry it far enough. But then I alsofelt a bit defensive in that one of the others could have cued in too. Groupwork can be stressful if you mislead someone.^(Rita, May 27)Rita raised a number of interesting issues in her journal entry. The main point was the wayin which she reflected on what she did in class. Reflection on what went on in the classgave her a chance to have more insights about the problem and be able to see where shewent wrong and why. Another point was the role of the small groups. She felt "stupidand embarrassed" because she did not solve the problem properly. However, everymember had the right and duty to argue with others in the group if something did notmake sense. The group work and the regular reflections helped them to realize that groupwork meant sharing ideas and not following a leader. As Rita wrote in her May 21 journalentry, "We need to learn independence and self-reliance."Whole-class discussions helped the innovators to become better decision makerssince they were constantly confronted with different ways of doing things. They had to148decide which ways would make more sense to them and why. "On Tuesday we startedlearning about factorials and the class went crazy! It was all rather incredible to see aroomful of people who were perceiving a situation in two quite separate ways, and tofurther see how difficult it was for each side to explain its view to the other" (Melisa, June5). The discussions helped the innovators to see that there was not only one "correctway" of doing a variety of problems but several ways. The discussions gave them achance to examine a diversity of approaches to same problems. They saw a host ofreliable and "correct" ways of solving mathematical problems that were all different but"correct." Barbara's response to my comments in her May 13-17 journal entries wasinteresting: "Perhaps I wasn't stupid after all, that problems I have experienced in tryingto learn math may have been as a result of thinking that there was only one correct way tosolve a problem, and that everyone (or most people) somehow had access to that methodwhile I didn't."The group work, as Barbara mentioned, helped the students to alter their viewsabout themselves as doers of mathematics and to overcome their fear of mathematics. Inaddition, Linda re-emphasized Barbara's views when she said, "I was dreading this class ... but it turned out to be quite fun. It wasn't as difficult as I thought it would be, but I thinkthe sole reason for this is the class from the beginning developed a cooperative nature. Bythis I mean everyone felt insecure with math but we all shared our knowledge together tocome up with the answers" (Linda, June 13).Melisa's "in-depth evaluation of the course" summarized the different feelingsexpressed by different students in this category about the role of group work in thestudents' learning. Therefore, I would like to end this section with an excerpt from herwork:I was extremely taken aback by the suggestion that not only was co-operative learning ok, but that it was required for this class. I felt insulted,because it's seemed to me that co-operative learning was for people whodid not understand, or who were not clever. The first week I stewed when149someone asked me to explain something until when I realized that I didn'tthink any less of them, and that they understood my explanation. I turnedimmediately and asked someone to explain something I already understood.This was illuminating because I got to understand the problem in a wholenew manner, which enriched my understanding. I would advocate that co-operative learning was essential to a student's success in this class. Youlearn so much faster this way, and I think you learn more. It is quiteexciting intellectually to bounce far fetched ideas off each other only to findout that your idea isn't so far out and that other people have similar ideas.. . . Talking out our ideas facilitates our understanding.(Melisa's last assignment)Class environment. One of the main characteristics of this category was the students' views about theinstruction. The innovators did not consider it as a set of teaching strategies to be adoptedseparately depending on the circumstances. They thought about it as a new way ofapproaching the teaching and learning of mathematics. The innovators realized that alongwith the instructor, they played an important role in the meaning-making process. Theyalso were aware of the importance of the classroom environment in metacognition-basedinstruction. The "relaxed" atmosphere let them feel comfortable. ". . A very 'friendly andwarm' class; one in which I felt very comfortable and at ease. I was afraid of being 'theonly one who didn't understand' but this wasn't the case in this situation" (Peggy, May 7).The "friendly" class gave them a chance to become more confident: "The classroomatmosphere was very relaxed; you are eager to help us understand, and it was fun to learnin groups. Also, I really like the way you don't pressure us in any way (i.e. you don't askpeople to come up to do their answers on the board, instead the people volunteer) . .This relaxed atmosphere allows me to admit to not understanding something (withoutfeeling like an idiot) so that something can be done so that I do understand" (Sally, June13). This was an interesting expression by Sally since she was "nervous" about working ingroups in the beginning of the course. This was certainly a change in her beliefs aboutherself, and she attributed the change to the relaxed atmosphere of the classroom.150A number of innovators, among them Clara, apologized to me for the lack of orderin the class: "However, I think in a sense, with these hands-on experiments andopportunities to discuss problems, we [the students] sometimes forget our manners as webecome excited . . . So I would like to apologize for our apparent rudeness.... It really isexcitement in a sense" (Clara, June 1). It was very considerate of her to feel like that, andI told her in any interactive situation this was inevitable. "I believe that this is a non-traditional way of teaching, therefore it requires a non-traditional class environment.Besides, it is not "rudeness" and you shouldn't apologize for that! I take it more as"excitement" as you said. I think if we accept to do group work and whole classdiscussions, we shall gradually establish the social norm which is suitable for this kind oflearning environment . .." (my comment on her June 1 journal entry). The reality was thatall of us together helped to establish such a "social norm."The Students' Beliefs About Mathematics and About Themselves as Doers of Mathematics I suppose one of my biggest gains from this course has been an increasedappreciation for what mathematics is all about.(Bob's last journal entry)All the innovators had experienced some difficulty concerning mathematics withthe exception of Patrick to whom "math always came easy." The innovators' beliefstowards mathematics were intertwined with their beliefs about themselves as doers ofmathematics. Some of them did not do well since they believed they were not capable.Barbara's story was representative. "I am a mature student. . . . I have always thought Iwas deficient in my thinking about math ... my thinking processes are good in other areas. my academic experience with math is abnormally low" (Barbara's autobiography). "Ithink that I thought of my brain as split into thinking about words and thinking aboutnumbers and that somehow the latter part was damaged" (Barbara, May 6-10). Her effortin doing well in mathematics failed as she developed such a belief about herself, "Aftertaking math for a few years I stopped believing that I could do it and stopped trying"(Barbara, May 28). Later in the course she explained why: "I was afraid that I couldn't151solve any of the problems. . . . Eventually I approached all problems with the secret beliefthat I couldn't do them, that I had to fail because I was stupid in math" (Barbara's lastjournal entry).Furthermore, the instruction helped Barbara to change her beliefs about herself asa doer and learner of mathematics.I think I have always equated slowness in assimulating ideas with lesserintelligence. I think this belief is generally held in this culture as well.. ..But if we all end up in the same place but at varying times, if our learningstyles are met, then ultimately what is the difference.. . . I have sincediscovered that I am capable even if I am slower than most.(Barbara's last journal entry)The major gain was that she no longer considered her performance (marks) as the soleindicator of her capability regarding mathematics:I would like to say that regardless of how well I do in this course, I thinkthat I have learned a lot, the most important thing being that I no longerbelieve that I am stupid or deficient in mathematics understanding. Ihonestly never thought I could understand as much as I have ... eventhough I originally hated the thought of taking this course, I have found itto be of great value.^ (Barbara's last journal entry)A number of innovators did "poorly" in mathematics because they did not like it."Math has never been my favorite subject, in fact I've always done poorly especially inhigh school. I am a classic example of the girl who, once she becomes a teenager, hasgreat difficulty with math. I failed Grade 11 and Grade 12 Math. Therefore I took it insummer school" (Linda's autobiography). She did not explain why she considered herselfa "classic example," but the point was that she had difficulty with mathematics and it wasnot her "favorite subject." Later on Linda explained that mathematics was difficult for herbecause she could not relate to it. "I think math can be fun (in the sense of challenging) ifand only if I know what I am doing. Math is a difficult subject for me because sometimesI cannot see things visually, henceforth I get frustrated and give up." She then gave anexample to clarify her point: "When I first looked at the figure . I was intimidated. But152once I looked at the figure I realized that I could use my previous knowledge of thePythagorean theorem to solve the question" (Linda, May 14). Being involved in themeaning-making process helped her to change her beliefs about mathematics. "This surewas a switch for me explaining math instead of sitting there getting confused andfrustrated. I liked it, and I hope most of the lessons will work like that. Then maybe I willnot dread math and begin to like it" (Linda, May 9). As the course progressed she wrote,"I enjoyed working on this [problem related to volumes]" (Linda, May 16); and in her May23 journal entry, "I'm not totally in the dark, and I am becoming more confident withmath." Her reflective writing showed that she gained more self-confidence when she wasable to "do" mathematics, and that played an important role in altering her beliefs towardsmathematics."I swore that I would never look at math/algebra again but I guess I was wrong,"wrote Penny in her autobiography. Some of the innovators like Penny were afraid ofmathematics and did not want to face it again because their experiences with mathematicswere generally unpleasant. Clara and Penny described how the lack of success inmathematics left them fearful. "I guess I have math phobia. I am worried about takingthis course and wondering if I will be able to understand it. My memory of math is that itwas the first and only subject I 'failed,' wrote Clara in her May 8 journal entry. It wasdifficult to start a course with these bitter memories. However, more mathematicalunderstanding helped her to leave those memories behind. "It is with great fear and dreadthat I began it [the course], and while math is still not easy for me, a lot more isunderstandable that I ever thought!!" (Clara's last journal entry). More importantly, shechanged her belief about herself as a doer and learner of mathematics. Clara realized thatgood marks were not the indicators of her mathematical ability. "Well, no matter whathappens in this course, one thing that has been good about it is that it has made me look atwhy I thought I couldn't do math. I have discovered, in fact, that I can do a lot moremathematically than I ever thought possible" (Clara's last journal entry).153Dave realized that his involvement in mathematical activities would lead him to abetter understanding of mathematics. "Coming into this class, I had real concerns aboutmy ability to do math. . . . I still feel very 'shaky' about it. It was a relief to hear that thiscourse is a 'hands on' approach which allows me to see and to do math, not just listen"(Dave's autobiography). Previously mathematics had been presented to many students asa purely objective and rule-oriented subject. Rose was surprised to see that "mathematicsis not dull and boring as my first impression of it" (Rose, May 9). Dave was glad to hearthat he could "see" and "do" mathematics as opposed to memorizing the facts andformulas.Coming to this class on May 6, I was very worried that this would beanother math failure for me. It was a pleasant surprise to have many ofthese fears alleviated. All through elementary school I was terrified ofmath and would literally sweat when the teacher said to open ourmathematics books. On May 7 we began to 'play' with a couple of puzzlesand do a kind of math I have not really experienced before. It was actuallyenjoyable and calming to cut and tape in math class! . . . We spend most ofthe time doing the math. . . . This may be an incorrect thought but I get thefeeling that we are free to experiment and come up with different answers.If they are wrong (the answers) then there seems to be a lot of time givenfor each of us to discover the answer. I like that!^(Dave, May 7)Doing mathematics as Dave described, was enjoyable for innovators, yet many ofthem had not done it for years. Clara wrote, "Which part of the brain is involved inmathematical thinking? I can feel the wheels creaking into action ... long unused"(Clara, May 9). However, they did better when they were actively involved in themeaning-making processes and were able to "accomplish" something. "I was notfrustrated because I felt I actually accomplished something (even though I was only ableto complete one question)" (Ben, May 16). Furthermore the sense of "accomplishment"gave them more self-satisfaction and they gained more self-confidence. ". . . [I was]feeling quite pleased with myself and obviously having a great sense of achievement andsatisfaction" (Peggy, May 16). As Bob wrote in his journal, "This course is so much like154running long distance, it's incredible;" but he also said, "No gain without pain!!" (Bob,May 30).Jack felt "uneasy" in the beginning. "During the first class of Math 335 I feltuneasy and somewhat nervous. . . . I feel my mathematical background is weak" (Jack'sautobiography). Furthermore, the class activities helped Jack to become "easy" withmathematics and even "enjoy" it. "I am enjoying the work much more now because I amable to understand why specific questions give certain results" (Jack, May 23).In general, the innovators held beliefs about mathematics similar to the otherstudents in the class. Their beliefs were shaped by the ways in which mathematics hadbeen presented to them--abstract with no practical aspect. Besides, some of them believedthat there were people who could learn mathematics, there were others who could not,and that was the end of the story. "The instructor did well at making us feel that . . .anybody can learn math. It is not something you are born with as the previously acceptednotion that settled into my brain from high school. I actually felt a little excited aboutmath for the very first time" (Jack, May 7). However, some of the innovators wereoptimistic. As Ben expressed in his autobiography, "I am hoping that I will become morecomfortable with the whole math process." The others were willing to try their best to dowell in the course. "I passed the course (Algebra 11) with a great deal of stress andnightmares. However, that was ten years ago, and I hope that I have reached a higherlevel of maturity which will enable me to apply myself fully" (Carmen's autobiography). Anumber of them were even more persuasive. "This is so confusing, but I'm not about togive up yet. I've got this far so I'm sure if I persevere, I will make it through to a betterunderstanding" (Ben, June 5). Sandra's "determination" was exceptional. "I am a visuallearner. I know that writing about my difficulties and successes with math will bebeneficial. I need to learn strategies and how to approach math. I am willing to learn asmuch as I can . . . I am determined to apply what I can to learn more about math"(Sandra, May 7).155The students' work showed that their beliefs about mathematics were shaped bythe ways mathematics was presented to them. When Bob wrote about the influence of theteacher and instruction in his success or failure in mathematics was typical:In my background, math has gotten me very depressed at times as well asvery uplifted . . . A combination of poor attitude (adolescent) andquestionable teacher, resulted in my failure of the course [Grade 11 Math]and the need to repeat it in the following year. . . . I was quite scared totake Grade 12 Math to complete my university entrance. Well I had such agood teacher for math in Grade 12 that I ended up getting an honors markat the end. After feeling quite discouraged about [mathematics], I endedup feeling very good and positive towards math. (Bob's autobiography)Later in the course, Bob raised an important question about the nature ofmathematics. "We are now into it and how it was formed, developed, discovered; what'sthe right word?" (Bob, May 23). I wanted him to respond to his question. "What do youthink?... ," I asked. These questions were crucial, since different approaches to teachingand learning of mathematics were influenced by people's beliefs about the nature ofmathematics. The students were actively involved in the formation, development anddiscovering of mathematical ideas.The metacognition-based instruction provided an opportunity for the social andindividual construction of mathematical knowledge. The students learned mathematicalconcepts by "doing" not "memorizing." The majority of innovators were apprehensiveabout geometry and probability in particular. "Geometry presents the greatest fear. I havealways just memorized the necessary information and then copied it out without everdeveloping a true understanding" (Ben's autobiography). Furthermore, being able to "do"things in class and using manipulative materials helped the students to see the "fun" side ofmathematics. "I was very apprehensive about taking this course, as I haven't donesecondary math for over 20 years. I must admit this afternoon was fun" (Peggy, May 7).It also helped them to gain a better understanding of geometry and probability.156If . Anyways, probability was quite a dark cloud over my head a week ago and now it'sgray with a few sunny periods" (Bob, June 11, referring to whole class discussion and thedemonstration regarding probability problems). Sophia said, "I really enjoyed the lessonon Pascal's triangle. It's really fascinating to discover the relationship between thenumbers and how the numbers can be used to help solving problems" (Sophia, June 6). Itwas a pleasure to see the "fear" of mathematics gradually replaced by fascination as thecourse progressed.Kent and Patrick, musician and stage player respectively, reflected particularly ontheir mathematical background. These two artists both expressed their appreciation forgeometry. Kent wrote: "Geometry . . . I enjoyed very much and algebra . . . I did notenjoy. Perhaps the visual aspect of geometry and its seemingly more tangiblecharacteristics was more appealing. Algebra seemed more abstract and 'on paper' and alsoa less creative endeavor" (Kent's autobiography). Patrick also found that mathematicsalways came easy to him. The similarity of their mathematical interests and their reasoningwas interesting. Kent and Patrick were well aware of their likes and dislikes regardingmathematics. They were eager to learn and gain more mathematical power. "I think thatmy mathematical tower is increasing more in width than it is in height. It was very narrowbefore I started this class," wrote Patrick in his May 24 journal entry.Melisa reflected on her beliefs towards mathematics prior to and after the course."This was without a doubt, one of the more difficult courses in my university life. Thiswas primarily because the class was math (and I had a hang-up about the subject) and thatit was being taught very differently from any class I've been taught before. I came to theclass hating math, dreading math, and now I rather like it" (Melisa, June 10).Melisa's enthusiasm was phenomenal. At her work, she utilized the mathematicsshe was learning in class and she was excited. She wrote, "I found myself today at work,in the midst of doing a display, trying to figure out how much paper I needed to do adisplay, and at what angle to cut it. Well!! Pythagoras came right in and I knew exactly157what/where to cut!" (Melisa, May 6-10). She then continued, "It isn't awful or scary, Ilike it! I am having some difficulty with working and doing this class, but I find myselfthinking about what we are discussing in the class during work." Melisa regularly usedher journal to extend the problems discussed in the class.Patrick and Melisa were similar in certain ways. They were eager to extend theproblems and concepts that were discussed in class. I was happy to help them developnew ideas through their journals. "I had a fantastic moment of realization over theweekend," wrote Patrick in his June 10 journal entry when he found out why in binomialsituations the Mean and the Standard Deviations were n/2 and .J/2 respectively. Iwrote to him, "Isn't that great to discover things like this? I really admire yourperseverance, enthusiasm, and endeavor in learning mathematics." Patrick was alsoenthusiastic after he developed the formula for the area of an n-sided polygon by himself.He did an excellent job of extending what he knew and came up with a subtle equation:"Before this class began I did not know what a mathematician meant by 'this is a beautifulequation!' I now know what is meant by such a statement. It has to do with the area ofpolygons and circles An = (sin (13 360/0 I/2n" (Patrick's last assignment). Melisa told methat she was so happy to discuss the mathematical ideas in a family gathering that wasdominated by many "male scientists!" (Melisa's post-interview).One of the characteristics of the innovators category was that its members wererisk-takers who welcomed challenges. They were persuasive and eager. As Ben said theywere not "about to give up." Patrick's example was not unique: "Boy was I everconfused today--the matching test question. It's a scary feeling but I'm glad it happened. Inow know the feeling of total confusion and I think that will help me with my teaching. Inow appreciate what others go through. Peggy, Melisa, Ben, Rita, Barbara, Kent, andClara all reported that they woke up in the middle of the night experiencing similarincidents.158The course offered the students a new perspective. The innovators' beliefs aboutmathematics were influenced by the metacognition-based instruction. "This has been themost unique course that I have taken. I have never thought of math as being creativeendeavor, and yet I am convinced that what we did for last six weeks was definitelycreative" (Carmen's last journal entry). "I suppose one of my biggest gains from thiscourse has been an increased appreciation for what math is all about and how I need tobring this enthusiasm into the classroom," wrote Bob in his last journal entry. It wasrewarding for Bob to extend his scope and to increase his "appreciation" for mathematics.Bob explained how class discussions and interactions helped him to "see the light."". .. I've been amazed when I think back and just the way the communication took placewas quite something. Math seems like that. You're in the dark not knowing what to doand then one word is said and the light comes on and it's like the darkness has completelydisappeared" (Bob, May 13).Carmen summarized her beliefs about mathematics prior to and after themetacognition-based instruction. Her summary was fairly representative of innovators'views of mathematics and themselves as learners of mathematics. I will end the sectionwith it.What I am saying is that math should be meaningful. It is meaningful, andit should be taught in a meaningful way. Up until this course, math hadbeen a terrifying ordeal of numbers . . . I couldn't translate words intonumbers. That is what math is all about. It is the ability to see theuniverse speak the language of math. If you want to learn the secret of it,you must first learn to speak the language. This math class has served as atrip to another country. At first everything is foreign, and you ask yourselfwhy am I here? Why could I have not gone to some place where theyspeak English and isn't so foreign, and feels more comfortable. But onceyou have stayed in the country for a while, the language becomes a littleeasier to understand. You begin to understand a bit of the language andbecome one step closer to understanding the whole culture of the country.That's what this course has been. It has been a chance to experienceanother foreign part of this world that makes up our journey. It has beennice traveling with you Zahra. Thank you; you have served as an English to159math dictionary. Without you I would have been really lost in space.(Carmen's last journal entry)The Students' Beliefs About Mathematical Problem SolvingThe difference between a true understanding and knowledge of procedureis the difference between following directions on a map without any ideaof where you are going or what the map really represents, andunderstanding what the map is and why the symbols are used in aparticular way and also having at least some idea of where you aresupposed to end up .... I think you can apply this analogy tomathematics problems.^ (Barbara, May 13-17)Generally, the innovators came with the same belief about mathematical problemsolving as the traditionalists and the incrementalists, that usually there is only "one correctway" to solve problems. Moreover, they believed that the main goal of problem solvingwas the "correct answer" and that it could be obtained by applying certain rules andprocedures and plugging in formulas. This belief was shaped by their experiencesregarding teaching and learning mathematics: "When I did math . . . ten years ago, wewere given formulas and told to apply them in a certain way to arrive at some abstractconclusion" (Carmen's last journal entry). However, the innovators were distinctlydifferent from the traditionalists. They were willing to try another approach tomathematical problem solving since they felt the "traditional approach" was not adequate.However willing they were to try a different approach, it was not easy for some ofthem to leave behind what they grew up with. After all, it was easier for them to solveproblems using certain formulas. "I am good at plugging in formulas and solving theproblem, but when it comes to written problem solving questions, I have difficulty"(Penny, June 3).The static view of mathematical problem solving and the dependency on formulasmade Linda wonder whether there was a "solution" to solve all the problems. "Well,160today was interesting. I was honestly hoping we would not have to construct geometricalshapes from other shapes but unfortunately I was wrong. I am absolutely hopeless . . .because I just cannot decipher them. Is it possible for the brain not to see where thesolution lies?" (Linda, May 7). I was curious to know why she was thinking that way."What is a solution?" I wrote to her and continued, "Why do you think that the solutionlies in the brain? Don't you think that you are solving problems by going throughprocesses rather than just calling them from your brain or your memory?"The students' beliefs about themselves as doers of mathematics and towardmathematics and mathematical problem solving were all tied together. Change in one'sbeliefs about mathematics would result in a change in the beliefs about himself or herselfas a doer of mathematics and towards mathematical problem solving. For example, theywould not feel "stupid" anymore if they were not able to comprehend "one" way ofsolving a problem because they were aware that there are other possibilities. "Perhaps Iwasn't stupid after all, that problems I have experienced in trying to learn math may havebeen as a result of thinking that there was only one correct way to solve a problem, andthat everyone (or most people) somehow had access to that method while I didn't"(Barbara's response to my comments on her journal entries for May 13-17).Most of the innovators liked the practical approach to mathematical problemsolving. They became interested in doing things. "I feel better about the math coursetoday. . . . I could actually do some questions. Of course one of the big discoveries is thatthere are different ways to prove things . . . I like the hands-on approach!" (Clara, May 9).The innovators' involvement in class activities facilitated their understanding of theprocesses of doing problems rather than memorizing facts and formulas. "I feel thatwithout understanding the ideas and concepts of math, the calculations won't make sense.There is no point in memorizing a formula or concept if it is not initially understood"(Sandra, June 11). Barbara wrote about how it would feel applying formulas withoutreally understanding the processes. "If I were doing a problem by rote with a formula, I161would do the first few steps on a journey to a destination. But I would always reach apoint where I couldn't take the next step because the way was dark. I just couldn't make aconnection between the previous steps and the next one" (Barbara, May 21-24).The innovators, unlike the traditionalists, became eager to find out "why" they didwhat they did. It was no longer sufficient for them to learn "how" just by following whatthe instructor told them to do. Barbara reflected on her previous experience withmathematical problem solving. "I understood how to do things but not why I did them.And I often forgot how to solve problems or mixed various methods up, because I'd forgetthe procedures and formulas or get them mixed up" (Barbara's responses to my commentson her May 28 journal entry). Therefore, they benefited more, as Peggy explained, whenthey were actively involved in the sense-making process. "I think this is much morebeneficial to students than 'learn by being told,' as you remember much more when youactually do it yourself' (Peggy, May 7). Learning about different approaches to solveproblems helped them to gain more confidence. Sandra explained, "Since I feel moreconfident and sure of myself when I approach a math question, I don't get fazed as easilyas when I started the course. As I don't get as easily frustrated as compared to at thebeginning of the course, I find it interesting to learn how other people approach figuringout the areas" (Sandra, May 17). Their confidence helped them to become better problemsolvers: "This is the first math course in which I have been starting to feel confident. Ican do mathematical problems that would have been inconceivable a few years ago"(Dave, May 13).Most of the innovators were amazed to see different approaches to solving theproblems. As Jack said, "It is amazing to me that there are so many different ways ofsolving problems" (Jack, May 23). Yet, Carmen observed, "Everyone tackled theproblems in different ways" (Carmen's last journal entry). Learning about different waysto solve mathematical problems broadened their perspectives. "I know now that there aremany ways to get the same answer and it is not just a matter of using a formula. One has162to know why they are doing something and not just plug in formula after formula" (Linda,June 13). The more they actively participated in the meaning-making processes andexperienced various approaches to solving problems, the less they became "stuck." AsBarbara explained, "For me, I don't seem to get 'stuck' in the way that I did earlier in thecourse. . . . I think that is because II. . . read the problem carefully, write all the importantinformation down, relate the question to what we have been learning in class, try tosimplify the problem" (Barbara, June 6). The innovators acquired managerial skills andbecame better decision-makers when they were confronted with the many approaches tosolving problems, and they gradually became better problem solvers. Peggy's work wastypical. "It took quite a while to work out what the problem was, what I already knew,and how I could solve it!" (Peggy, May 14).The innovators enjoyed learning mathematics through games and puzzles: "It'sbeen really great to learn by doing puzzles. . . . Is this really math??" (Sophia, May 16).Mathematics through games and puzzles helped them to get away from a view consideringmathematics to be everything but fun. The games and puzzles also helped them tounderstand the mathematical problems better. "Calculating puzzles is a lot more fun thanthe traditional pen and paper method. Cutting and pasting the different shapes challengedmy intellect to a certain degree, and at the same time made me learn more about geometry.By being more practical allowed for more learning" (Penny, May 8). Rose reflected on thecreative aspects of mathematics. "As soon as I entered into the classroom, I found it waslike a handicraft lesson. . . . After all, I have found the lesson is full of fun and creative andit can be introduced through games, puzzles or paradoxes. But most important of all, theyare not used only for entertainment but to draw the learners to the acquisition of thefundamental ideas of math" (Rose, May 9).As the course progressed, the innovators became more aware of their own thinkingand learning styles. Their managerial skills improved tremendously, and their beliefs aboutmathematics and mathematical problem solving changed considerably. The students' work163showed that the metacognition-based instruction helped them to gain metacognitiveknowledge and become better decision-makers. "My problem is that I have troubledeciding which way to attack the problem, which method to use, but after workingthrough some of the questions on the sheet you gave me . . . it is beginning to sort itselfout in my mind" (Peggy, June 6)A number of innovators solved their final exam problems by reflecting on whatthey did, why they did it, and how. This was a big achievement considering theirdisappointment at having to write a "stressful" final exam. Kent's and Jack's exampleswere illustrative. There was one problem (B5) on the final exam that was new. Thatmeant that the class had not gone through a similar problem. Kent chose this problem.The ways in which he solved it were interesting. He wrote questions that he askedhimself, "Will this help me? Can I use this?" He tried to reason it out and then decide.(Appendix S contains a copy of his work.)Jack (referring to the same problem) first got the shaded area using thePythagorean theorem involving long calculations three times. He then crossed all threeout and gave an elegant solution and wrote, "Mark this." Jack was very pleased about itand after the final exam he told me, "This is good for your study," and laughed. Thesetwo solutions showed his awareness and his managerial skills to help him come up with anew and better solution.The instruction provided the students with an opportunity to learn differentproblem solving strategies and to use a variety of approaches to solve problems. "Thequestion that you gave us to work on about the book of poems was very interesting and itseemed very difficult, until I made it into an easier question by saying that if each page hadthree lines and there were five pages in the book, each line would have 5 combinations,5x5x5" (Peggy, June 7). She then solved the problem beautifully. Sophia wrote about thestrategies that she used to solve some of the problems. "The most helpful way of solvingproblems for me is to determine whether the problem is similar to one that we've done in164class or to reduce the numbers that are easier to work with or to visualize" (Sophia, June6). The interesting point is that she realized what was helpful for her. She did not say shehad to do it because it worked for others, but because it worked for her.In teaching mathematics via problem solving the mathematical concepts weredeveloped through the mathematical problems. Finding volumes of different solids, usingthe concept of scaling, and developing concepts in probability are only a few examples.Teaching via problem solving also created an opportunity for the students and me to use anumber of problem solving strategies and use them efficiently. Innovators acknowledgedand used those strategies more than other students. The ways Peggy and Patrick solvedthe problem of "one hundred thousand billion poems" are two examples of how theinnovators used different problem solving strategies including, (1) using smaller numbers,(2) breaking problems into different parts, and (3) using a systematic list. In teachingmathematics via problem solving, they learned those strategies in a natural way, ratherthan out of context in a hypothetical situation which would have been more artificial andless appealing.Experiencing a variety of approaches to solve problems increased their confidenceregarding problem solving. "[When] I get confused by irrelevant information, ... I think Ishould just start [solving the problem], and if I don't figure it out, try another approach"(Sandra, May 10). This experience increased the innovators' ability to explain things toeach other more productively and to feel challenged rather than hopeless or stuck. "If oneexplanation doesn't work, I'm challenged to think a new way to explain the problem"(Patrick, May 24). He was getting away from the traditional teaching that repeated thesame old thing, in the same old way, until it sank in.The students were constantly encouraged to ask themselves what they were doing,why they were doing it, and how it could help them. This helped them to becomereflective at all times and give reasons for what they did. It also increased their sense ofresponsibility towards their own learning. Melisa's reflection on her second quiz was a165good example. "Actually it was really interesting, and in retrospect, even though I got theright answer, I think I won't do well because I didn't raise my concerns about the question,and we were encouraged to do this" (Melisa, May 27) In her journal, she went throughthe quiz again and responded to the metacognitive questions that were part of the quiz.Most of the innovators expressed their dislike for exams of any sort. However,they were reflective even under exam pressure. Peggy wrote about her feelings towardsfinal exams and how she tended to panic in such situations. However, she found a way tohelp herself, and she wrote, ". . . I must remember, what is the problem! What do Ialready know? How can I solve it? and not go and do things that are not required in thequestion and make it more difficult for myself' (Peggy, June 7).Penny's example was interesting. She felt a responsibility for explaining everythingthat she did, even the formulas that she used to solve the problems on her final exam. Sheused the formula for finding the volume of the cone (Al on the final exam) and she wrote,"The reason for this is that in class we observed Zahra put water into a cone and we couldsee that the water going into the cone was exactly 1/3 of the cylinder." She constantlysaid that she liked to see things to be able to make sense of them. It was rewarding thaton the final exam she could relate to what she "saw" in class. Besides, she felt that shecould not even use a formula without explaining "why."The innovators practiced the same activity even when they were taking notes. "Ifound that one thing I have to do with my notes is to write out the details of what I amdoing and why" (Peggy, May 17).Melisa's reflection on her beliefs about mathematical problem solving after theinstruction summarizes what the innovators have said in this regard. "This class has beenlike an explosion of ideas about what I know and feel, not just about math but about allthings. When we break down the barriers of how we know, and acknowledge that thereare other ways of knowing, we learn quicker, and I think, adapt more easily to newlearning strategies" (Melisa, May 26).166Tests In the end all the enjoyment of discover) , is buried because I'm so worried aboutwhether I'll do the [final] exam okay.^ (Clara, June 7)The innovators raised a number of issues concerning "tests" in general. Amongthem were: the purpose of evaluation, fear of exams, and the students' perception ofexams. "I have never been able to overcome in a test situation, and it is one reason why Ithink I did so badly in math in elementary school. So much of the education was based ontest scores that whatever real understanding I might have had was not seen" (Barbara,May 28). They complained that the nature of metacognition-based instruction and theexam did not match. "Throughout this course we have been encouraged to talk overeverything, work together, don't worry about the right answer, understand the process,etc. Then we are given a final exam, and in two hours without the benefit of our notes orinteracting with others, we are expected to pass the course" (Clara, June 6).The students knew they could pass the course if they did a reasonable job duringthe course, and that the final exam would not be any different from what they had done inthe class. However, the exam situation was stressful. The difficulty was that a few ofthem felt desperate that they could not "remember" on the exam. "I have to try andremember for the exam. I get very nervous and anxious" (Peggy, May 30). Clara's casewas extreme and she was ready to leave the final exam after 15 minutes. I encouraged herto stay and do as much as she could. She was the most articulate and hard workingstudent in the class. Her contribution to the teaching and learning processes wasenormous. Clara knew she could not fail since she had earned full marks for her journal,assignments, and quizzes before the final, that was 60% of her total mark. Nevertheless,she was so overwhelmed on the final exam that she could hardly utilize what she knew.However, Peggy and many other innovators tried to put their fears toward examsbehind them. "I must get myself back into a more positive frame of mind. At the momentI am feeling really overwhelmed by this test paper. I can't sleep at night worrying over the167. . . final exam. I know that this is not good and will not help me, but at the moment this ishow I feel" (Peggy, June 4). Melisa had better success in this regard. "The final is still thebig scary test that it's always been! And I get queasy every time I think of it. But, youknow, I really feel that I've done the best I could in this course. . . . I'm not sure why, butI'm not worried about the exam any more" (Melisa's last journal entry).Many innovators considered exams to be something different and necessarilydifficult--an insurmountable task. Sandra explained, "The [first] math quiz today was veryfair. I was expecting a question I had never seen before" (Sandra, May 17). Kentreflected on the same quiz and wrote, "My math quiz went extremely well. In fact, I donot recall any other time after a test feeling so good" (Kent, May 17).The majority of the innovators also believed that exams focused mainly on finishedproducts, which meant getting "correct answers," more than on problem solvingprocesses. Some of them were annoyed to see the process problems on the exam. "Todaywe had a quiz in math. I found the quiz to be quite confusing because one could perceivethe question in many different ways. I think that the questions should have been clearer. Idid not know if Zahra wanted me to do calculations in figuring out the answer or justwriting statements to describe what I thought was the right answer" (Penny, June 8). Thestudents were used to having everything "set" and "fixed" in mathematics, so when thequestion was flexible, they got confused. They were surprised to learn that their solutionprocesses were worth more than the product, the "correct answer." "I'm glad we getcredit for our ideas of how and why we calculate and solve the questions" (Sandra, June8).Carmen's honest reflection on her third quiz questioned the purpose and thevalidity of evaluation in general. "I had frightened myself so much that the negativereinforcement manifested itself in fear. . . . I managed to do well on the quiz but I'm not100% convinced that I understood what I was doing? Is that possible?" (Carmen, June10).168There are a number of interesting issues in this short passage that requirediscussion. The first issue is the old and ever existing question of evaluation: why do weevaluate students, what do we evaluate them for, and how? (A partial answer to thisquestion would requires another doctoral dissertation!). The second is the fact that somepeople are good exam takers, "I managed to do well," as Carmen said, and the marks theyget do not necessarily reflect their conceptual understanding of mathematics. Last but notleast is Carmen's honesty. I was impressed by the way she talked about her mark. I couldconclude, based on her writing, that the "relaxed environment" and a friendly relationshipbetween the students and the instructor allowed her to express herself openly.Teaching and Learning Mathematics via Problem Solving: The Innovators' PerspectiveA working definition for "teaching and learning mathematics via problem solving"emerged from the innovators' reflection on the instruction. The innovators welcomedmany aspects of the instruction. They were tired of experiencing their failure regardingmathematics. The innovators were not satisfied with the ways in which mathematics hadbeen presented to them. The course offered the innovators a new perspective that allowedthem to gradually see the more innovative and practical side of mathematics. Theyfinished the course believing that mathematics could be a shared knowledge and that it hasmostly been developed out of practical needs. The innovators were happy to have anopportunity to develop their own ways of solving problems, and to see the importance ofprocesses in problem solving rather than mere emphasis on finished products, "correctanswers." They realized the necessity of having dialogue and discourse in meaning-making processes regarding mathematics through group work and whole-classdiscussions. A better communication with the instructor was another vital aspect ofmathematical learning, and journal writing made that possible. They felt comfortabledeveloping the mathematical concepts through problem solving. The main characteristicsof the "teaching mathematics via problem solving" based on the innovators' views were asfollows:169-A "new approach" to teaching and learning mathematics-Dialogue and discourse are integrated components of the instruction-Students are involved in the "meaning-making" process-Whole-class discussions are not "structured" but "monitored" by teachers-Mathematical problems can be approached in various ways-Problems can be solved in different ways-Problem-solving processes are more important than finished products-Mathematics would be more "fun" and "meaningful" if it is presented throughgames and puzzles, and be connected to day-to-day life-The teacher is a facilitator, monitor, and friend and not a sole authority in class-The teacher is fallibleSummaryThe innovators' beliefs about mathematics and mathematical problem solving priorto the instruction were not very positive. Many of them considered themselves "deficient"in mathematics and thought they were just not capable of doing mathematics. Theirbeliefs were influenced by the ways in which mathematics had been presented to them,without connections to their day-to-day life.However, the innovators expressed their dissatisfaction of the ways in whichmathematics had been taught in the past. They were willing to try a "new approach" thatmight help them to become more comfortable regarding mathematics. The innovatorsconsidered the metacognition-based instruction a whole new perspective to teaching andlearning mathematics. They reflected on different components of the instructionextensively and consciously. The innovators' reflection on the instruction showed thattheir beliefs about themselves as doers of mathematics, and about mathematics andmathematical problem solving were changed to a great extent as a result of a.-- metacognition-based instruction.170Summary of the FindingsThe traditionalists rejected the "new approach" to teaching and learningmathematics. They either did not care for a new experience and felt more comfortablewith the "traditional" approach, or they said they were too old to try a differentinstructional approach. However, their rejection was not the sign of an appreciation forthe "traditional" instruction. Rather, they were used to an old habit that would die hard.The traditionalists were not risk-takers, they preferred to stick to the tradition as it wasmore predictable and less innovative.The incrementalists were in a conflicting situation. The majority of them solvedthe conflict by proposing "balanced instruction," the adaptation of some aspects of"traditional" and some of the "metacognition-based" instruction. The instruction did notchange their world views drastically. However, their beliefs about mathematics andmathematical problem solving were influenced.The innovators' beliefs about themselves as doers of mathematics, and towardsmathematics and mathematical problem solving were changed as a result of themetacognition-based instruction. It was a great achievement for them to becomecomfortable in a mathematics class and gain enough confidence to discuss themathematical ideas and make appropriate decisions. However, the process of change wasnot easy. In many instances, there was a great deal of frustration and confusion.Overall, although the students enjoyed working in small groups, whole-classdiscussions did not appeal to all of them. The traditionalists did not see the necessity forclass discussions. However, those discussions helped the incrementalists to develop theirunderstanding of mathematics, while the innovators considered discussions as anintegrated part of the teaching and learning process.The incrementalists and the innovators utilized their journals to facilitate theirlearning, but the traditionalists' view of journal writing was more mechanical. They merely171considered the journal writing another requirement to be met.Each student, without exception, expressed his or her appreciation for usingmanipulative materials to enhance mathematical learning one way or another.PostscriptThe students fell into three distinct categories based on the data that were gatheredthrough different sources. Out of 40 students, 9 of them belonged to the first category;the traditionalists, 12 of them fell into the second category; the incrementalists, and theother 19 students; the innovators, were in the third category. The following event at theend of the course confirmed the selection of categories to a great extent.The 36 students responded anonymously to a questionnaire about evaluation ofcourse and instructor on June 13, 1991, one day before the final exam. Surprisingly, thenumber of students falling into the same three categories based on the nature of theirresponses matched closely with the number of students that belonged to those categoriesthat emerged from the analysis of the data. In fact, 19 students' responses had the samecharacteristics as the innovators, 11 as the incrementalists, and 6 as the traditionalists.This last source of data was valuable to attest to the validity of the categories. Thefollowing anonymous excerpts crystallize the range of the students' responses.A majority of the students talked about the positive aspects of the course, and howit gave them a chance to "try to do math without being pressured. The course allowed formany possible ways to get answers."However, there were two sharply distinct opinions on the ways in which thecourse/instruction affected different groups of students. For the majority of the students,it was a "good course to cure math phobia,"[because] . . . it made [them] face [their] mathanxiety and to some extent overcome it. It made [them] see that [they were] able to solvemathematical problems." The instruction gave the students an opportunity to "see" and to172"do" mathematics rather than memorizing facts and formulas, using them for a period oftime, and then forgetting them: "It is a visual course. You can see how things are appliedand work out as opposed to just memorizing formulas." They expressed their gratitudefor having a chance to discover things by themselves. "I really appreciated being able todiscover things in class as opposed to being told what something was [for example] ... itand deriving 7C." Many of the students wrote that the instruction (the course) offered thema new perspective regarding teaching and learning mathematics. However, the course andthe instruction was so foreign for a number of them, (the traditionalists), to the degree thatit was ". . . frustrating and an insult to [their] intelligence."173CHAPTER 6CONCLUSIONS AND DISCUSSIONSMathematics educators have been interested in problem-solving instruction formany years. Central to any question regarding the teaching of problem solving is theprimary question of understanding what people actually do when they solve problems.Within the past decade, investigations of problem solving processes have led to a newfocus in some of the research literature. Metacognition, the knowledge and control ofcognition, has gained a great deal of attention from the research community sincemathematics educators have realized that non-cognitive aspects of problem-solvingperformance are as important as cognitive ones. Therefore, there has been a growinginterest in the study of the role of metacognition in mathematical problem solving.This chapter gives a conclusion of the study, including answers to the researchquestions. Its educational significance and implications for future practice and research.are also discussed.ConclusionsThe aim of the present study was to investigate the effect of metacognition-basedinstruction and teaching via problem solving on students' understanding of mathematics,and the ways in which the students' beliefs about themselves as doers and learners ofmathematics and about mathematics and mathematical problem solving were influenced bythe instruction. Metacognitive strategies that were included in the instruction contributedto the students' mathematical learning in various ways. The instruction used journalwriting, small groups, and whole-class discussions as three different but interrelatedstrategies that focused on metacognition.Lester, Garofalo, and Kroll (1989a), expressed their concerns about the lack ofdata regarding students' backgrounds, and about the credibility of students' self-reports, in174their study. In the present study, an effort was made to overcome these concerns. Thestudents' autobiographies provided much valuable data regarding the students'backgrounds and the knowledge and beliefs that they brought to the class. In addition,this data source was extremely helpful in analyzing the possible changes in the students'beliefs about themselves as doers of mathematics, and about mathematics andmathematical problem solving. Furthermore, the students' journal entries provided the richand authentic data used in analysis. Moreover, the second and third quizzes wereaccompanied by a set of metacognitive questions to partially address the difficultyregarding students' self-reports mentioned by Lester, Garofalo, and Kroll (1989a).Journal writing served as a communication channel between the students and theinstructor, and as a result facilitated the individualization of instruction. In general, formany students, writing was an occasion of personalizing and internalizing the learningprocesses. Many of them appreciated the opportunity to write about related cognitive andmetacognitive issues as the course progressed. Journal writing provided the opportunityfor the students to clarify their thinking and become more reflective. Journals were alsoan excellent data source for studying possible changes in the students' beliefs aboutthemselves and about mathematics and mathematical problem solving.Small groups proved to be an essential component of the instruction. The studentslearned to assess and monitor their work and to make appropriate decisions by workingcooperatively and discussing the problems with each other. In addition, small-groupscreated an opportunity for me to offer the students a variety of problem solving strategies(heuristics) and to help them to become aware of their own resources, to appreciate them,and to use them proficiently. Most importantly, as one of the students wrote, "working ingroups really helped to take the frustration out of the course because it is math."Whole-class discussions raised the students' awareness about their strengths andweaknesses. The discussions also helped students to a great extent to become betterdecision makers. These discussions provided an opportunity for the students to become175more reflective of their own actions in class as well as to think about their thinking andtalk about their feelings. Whole-class discussions helped the students to increase theirself-awareness as well as to gain more self-confidence and self-respect, and that ultimatelyresulted in a better understanding of mathematical concepts and increased theirmathematical power.In general, the metacognition-based instruction gave me a chance to teachmathematics via problem solving. Starting the class with a problem that all groups couldwork on simultaneously was a worthwhile activity for developing mathematical concepts.This strategy was useful for bringing the whole class together while discussing a problem.As a result of this activity, the students gained more cognitive and metacognitiveknowledge of problem solving, and a better sense of responsibility for their own learning.The students eventually realized that they would learn more by analyzing and solving oneproblem thoroughly, than by solving many problems using only a set of rules andprocedures, and by plugging in formulas, without really developing a deep understandingof problems and concepts. It was pleasant and reassuring for the students to see thatalthough their approaches and solutions were different, they were acceptable and valid.Besides, they had to decide which approach made the most sense to them since they hadthe opportunity to examine many possible alternatives. These activities were used as avehicle to promote self-regulation.Three categories of students labeled traditionalists, incrementalists, andinnovators, emerged from the study. Nine students, who rejected the new approach toteaching and learning mathematics were categorized as traditionalists.. The traditionalistsliked to be told what to do by the teacher. They did not see any point in spending timeand energy developing mathematical ideas. Instead, the traditionalists preferred learningcertain rules and procedures and becoming more fluent in applying formulas in specificsituations. However, they liked working in small groups. The twelve incrementalists werecharacterized as those who propose to have balanced instruction in which journal writing176was a worthwhile activity, group work was a requirement, and whole-class discussionswere preferred for clarifying concepts and problems more than for generating anddeveloping new ideas. The nineteen other students were categorized as innovators, thosewho welcomed the new approach and utilized it and preferred it. For them, journalwriting played a major role in enhancing and communicating the ideas. Working in smallgroups seemed inevitable, and whole-class discussions were a necessity to help them withthe meaning-making processes. The students in each category shared common beliefsabout mathematics and mathematical problem solving.The incrementalists and the innovators gradually changed their beliefs aboutmathematics from viewing it as objective, boring, lifeless, and unrelated to their real-lives,to seeing it as subjective, fun, meaningful, and connected to their day-to-day living. Sincethey were actively involved in the meaning-making processes, the innovators sawmathematics as shared knowledge that could be constructed by themselves to a greatextent. They realized that mathematics was not as straightforward as they thought, thatthey could guess, speculate, and make conjectures regarding mathematics. As the courseprogressed, the incrementalists and the innovators did not feel hopeless dealing withmathematical issues. On the contrary, most of them believed that everyone couldunderstand and do mathematics to some extent.The findings of the study further indicated that most of the incrementalists and theinnovators changed their views about mathematical problem solving from seeing it as theapplication of certain rules and formulas to viewing it as a meaning-making process ofcreation and construction of knowledge. They learned that there were many differentapproaches to most of the mathematical problems and that they could all be sound andpromising. The students developed certain managerial skills to decide which approachmade the most sense to them and why. However, the traditionalists did not like the waysin which the course progressed. They rejected the teaching approaches throughout the177course except for working in small-groups and using manipulative materials, that allstudents liked to do.Answering the Research Questions The findings provided partial answers to the four interrelated research questions.The following summarizes the answers to these questions.1. What are students' usual beliefs about themselves as doers of mathematics, and aboutmathematics and mathematical problem solving?Generally speaking, prior to the instruction, the students believed that mathematicswas objective, difficult, often unreachable and incomprehensible, a set of isolated facts andformulas that had no connections with real-life situations. That some people were good atit and some others were deficient when it came to mathematics. Many students eventhought that some people could do mathematics and others could not. Except for a fewindividuals, mathematics had always been scary and boring.2. How are these beliefs prior to instruction associated with students' approaches toproblem solving?With the exception of a few, the students considered mathematical problem solvingas difficult and dry. They mostly believed there was only one correct solution and onecorrect answer for every problem. Some of the students even wished to find a formula tosolve all the problems. They thought of problem solving as nothing but knowledge ofprocedures. They did not see much flexibility in solving mathematical problems. Theyconsidered problem solving to be imposed on students rather than being negotiated. Manyof the students did not attempt to solve any mathematical problem on their own, since theybelieved that they were deficient in mathematics and solving mathematical problems wasan unfeasible task.1783. In what ways do students' beliefs about mathematics and about themselves as doers ofmathematics change during metacognition-based instruction and teaching mathematicsvia problem solving?The beliefs of the incrementalists and the innovators about mathematics werechanged to a great extent, as a result of metacognition-based instruction that aimed toteach mathematics via problem solving. Teaching mathematics via problem solvingprovided an opportunity for them to see mathematics as shared knowledge and a "humanendeavor." They eventually believed that mathematics could be fun, practical, engaging,and an interesting activity that everyone can do and understand to some extent.Moreover, the innovators' appreciation for mathematics increased as they got moreinvolved in the meaning-making processes with mathematical ideas. However, thetraditionalists did not like to get involved in many activities towards construction ofmathematical knowledge. They continued preferring to be told how to use the formulasand plug in the numbers to get the correct answers.4. In what ways do students' beliefs about mathematical problem solving and theirapproaches to problem solving change during metacognition-based instruction andteaching mathematics via problem solving?Teaching mathematics via problem solving provided an opportunity for theincrementalists and the innovators to "see" and to "do" mathematics instead of onlylearning certain procedures and formulas. They became more critical and more sensitiveto mathematical problem-solving. Many of them, especially the innovators, tried to justifytheir reasoning by answering what they did, why they did it, and how they did it whilesolving mathematical problems. Teaching mathematics via problem solving provided theopportunity to present mathematical concepts in the context of engaging problems.Furthermore, the solution processes helped them to develop those concepts in ameaningful way. In addition, their knowledge of mathematics was extended and theirappreciation for mathematics was increased after they had a chance to see a host of179different approaches to solving the same problems. Most of the incrementalists and theinnovators eventually learned that those approaches could all be equally valid and correctas long as it made sense to them and they could justify those approaches with solidreasoning.Educational SignificanceThe present study benefited from the recommendations of Lester, Garofalo,and Kroll's (1989) research on the role of metacognition in mathematical problemsolving of seventh graders. They wrote about two fundamental premises of theirstudy that, "metacognitive processes develop concurrently with the development ofan understanding of mathematical concepts (assumption 1) and that metacognitioninstruction is more likely to be effective if it takes place in the context of learningmathematics (assumption 3)" (p. 118). Taking these premises into account, theyreflected on their study and addressed the following concern for further research inthis area of research. "More particularly, the instruction was largely isolated fromthe regular mathematics curriculum . . . For the most part, the problem-solvingsessions had little or no direct relation to regular mathematics instruction and manystudents did not view them as a being central part of their mathematics class" (p.118).I sincerely thought about these two assumptions and realized that it was indeed myintention and my goal to carry out research in "regular mathematics instruction."However, I realized that it would be more work and I would probably face more difficultyand less success, considering all the constraints that exist in a regular day-to-daymathematics class. Nevertheless, I was interested to try and see whether it was at allpossible to do such a study under those circumstances.180The present study was about teaching an undergraduate mathematics course. Thestudy was unique in that it had many constraints that classroom teachers face in theirregular day-to-day teaching. I chose neither the course syllabus nor the textbook. Theclass size of 40 students was beyond my control. The course had a mandatory final examthat was worth 40% of the students' final marks. However, the study showed thatmetacognition-based instruction was applicable considering all those constraints. Therewere other constraints or limitations that some of students were concerned about. Forexample, no one was happy to come to class for two hours every weekday in hot summerevenings, for six weeks. Many students expressed their dismay about the intensity of thecourse, especially since most of them had full-time jobs, and some of them had othercourses to finish in that summer. The study was carried out while facing all thoseconstraints. In fact, I welcomed those constraints hoping that the findings of the studymight contribute to teaching practice more broadly. Classroom teachers also do not havemuch control over the conditions similar to those I had in respect to teaching thismathematics course.I have had many conversations with different teachers about bridging a gapbetween research and practice. On a number of occasions, they have complained that thefindings of various studies have not reflected the reality of their classrooms. Theseteachers were argued that the day-to-day teaching has limited their flexibility in terms oftrying different approaches to instruction and doing more qualitative evaluation ofstudents' work. Their point was that the research settings were usually well chosen andresearchers were not accountable for students' achievement on exams. These teacherswere also doubtful about the feasibility of group discussions with too many students intheir classes. This study was conducted in a regular mathematics classroom in which theteacher/researcher was limited by the same constraints as these teachers were. Therefore,the findings of the study might give practicing teachers some idea about the ways in whichmetacognitive strategies could be adopted in their own classrooms.181Implications for Future PracticeThe present study employed several metacognitive strategies to teach mathematicsvia problem solving. Those strategies proved to be useful in enhancing students'understanding of mathematical concepts and promoting their self-regulation and self-confidence.The study showed that journal writing was an effective means to enhance students'awareness. It also helps teachers to become more aware of their students' beliefs aboutthemselves, mathematics, and mathematical problem solving. Students' writing provides aunique opportunity for teachers to acknowledge individual differences among students,and to help them in more effective ways. Journal writing could be utilized in variousways. For example, students can correspond with each other and reflect on each others'writing. It could be on-the-spot reflection on certain issues, whether it is structured,unstructured, or semi-structured. However, the main point is that teachers should readstudents' journal entries regularly and respond to them accordingly. The mostdiscouraging action of a teacher is to ask students to write without giving studentsfeedback. Students in the study appreciated the fact that their writing was valued, andtheir writing was as important to the teacher as it was to them. It is obvious that goingthrough students' writing is extremely difficult and requires lots of time and effort.However, as was expressed by Rishel (1990), "although the workload may increase by alinear factor with this kind of teaching, the rewards are sure to go up also--by anexponential factor" (p. 33). Therefore, on the basis of the present study, I stronglyrecommend the use of journal writing in mathematics classes. I also advocate working insmall groups and having whole-class discussions for facilitating students' mathematicallearning, by providing detailed information about the role of these two activities in thestudy. Furthermore, the classroom culture (social norm) which teachers establish plays animportant role in facilitating and monitoring studetns' learning and has an extremelyimportant effect on students' learning.182The following gives an example of how I have implemented the findings of thestudy in my own teaching practice: I had a chance to teach MATH 335 again in thesummer of 1992. This teaching assignment provided an opportunity for me to reflect onmy study and to use metacognitive strategies in a more systematic way.I started the course by giving the whole class the same problem to work on, and Iconsistently did the same thing throughout the course. The study showed that, despite therichness of the selected problems, worksheets had the potential to become the focus of thecourse instead of being a means to enhance students' mathematical learning. To overcomethe difficulty in the second summer, I gave the students one problem at a time, rather thangiving them worksheets with many problems. In this way, I was able to pull the wholeclass together to work on the same problem. I also was more experienced in applyingcertain metacognitive strategies namely, journal writing, small-group work, and whole-class discussions.However, the most important contribution of the present study to my teaching inthe following year was a greater insight that I got about students' beliefs about themselvesas doers of mathematics, and about mathematics and mathematical problem solving. I alsolearned how those beliefs would influence students' mathematical understanding. Althoughthe second class was different from the first one in terms of students' characteristics andpersonalities, one thing seemed to be unchanged: there was the same range of beliefs as Iobserved in the study. The study taught me to be more appreciative of students'difficulties and to be more sensitive to the variety of conceptions (some would saymisconceptions) that students had about some of the mathematical concepts.The study showed that students were frustrated when they did not see muchconnection between mathematics and their real world. This finding encouraged me tocollect more mathematical problems dealing with day-to-day life to connect the students'mathematical world to their daily experiences.183To conclude this section, I would like to share my observation regarding theteacher's role in conducting classroom-based research. Teachers should have a chance tosee more emphasis on metacognition in their training program. Metacognitive activitiesare teachable and learnable. Teachers and researchers should work together to developcognitive-metacognitive frameworks considering social and cultural differences of theirwork places. Crosswhite (1987) suggested that researchers can provide practicingteachers with a "ready-to-wear" framework for promoting the growth of metacognitionamong students. However, there is no panacea to give to teachers, and any researchwithout active involvement of teachers is doomed to failure in the long run. The complexarea of thinking and reflecting cannot be reduced to a set of algorithms. There have beena number of promising research studies that focus on real classroom settings and haveoffered different practical models. These models can be adopted and used depending ondifferent socio-cultural background of students and the knowledge that they bring into theclassrooms.Implications for Future ResearchThe three metacognitive strategies and many other strategies could be adopted,developed, and extended to enhance students' cognitive and metacognitive skills. Thefollowing includes a number of suggestions for further classroom-based studies in thisresearch area considering students' socio-cultural backgrounds.A number of studies have been conducted to investigate the use of journal writingin matheamtics courses. The ways in which journal writing are utilized might varyconsidering students' age and socio-cultural backgrounds. Therefore, more research isrequired to study the possibilities and limitations of journal writing across age and culturalgroups. In addition, further research is needed for finding suitable criteria for evaluatingjournals since journal evaluation is a difficult task and research in this area is in itsinfancy.184As was suggested by Lester, Garofalo, and Kroll (1989a), the best setting in whichto conduct such research is the context of a regular day-to-day mathematics course. Smallgroups and whole-class discussions proved to be essential components of the instruction.One area that requires great attention is the establishment of a suitable social norm for theclass in which students have the chance to work cooperatively in small groups and todiscuss the ideas with the whole class. The establishment of such a social norm, orclassroom culture, was among the hardest tasks to achieve in the study. However, itsestablishment paved the way for adopting a number of metacognitive strategies fruitfullyand productively. Further research is needed to refine the processes of conducting classdiscussions and establishing classroom culture (or social norm) in which students developbetter managerial skills and to gain more self-confidence. In addition, theteacher's/researcher's role in conducting whole-class discussions and establishingclassroom culture is crucial and requires more detailed scrutiny in regular classroomsettings.Moreover, further research is needed to investigate the ways in which students'beliefs about themselves as doers of mathematics, and about mathematics andmathematical problem solving might be influenced by video presentations using a varietyof approaches to teaching and learning of mathematics. The study showed that videopresentations provided a unique opportunity for students to analyze other students'behaviors in the presentations while they are involved in the problem-solving processes,and relate to them. These video presentations give them a chance to become morereflective on their own problem solving behaviors.Final NoteThe study of covert behavior of human beings is difficult and complex, but it isfeasible and exciting. Researchers need to focus on the problem of bridging betweentheoretical and practical aspects of the role of metacognition. If they work more closelytogether, researchers and practicing teachers might unveil these covert behaviors to agreat extent. Students' beliefs are mainly shaped by instruction and the classroomenvironment. Teachers' beliefs about mathematics and mathematical problem solving havea major effect on students' metacognitive awareness. I therefore, wish to see moreresearch being done in this research to shed more light onto this complexity of humanlearning.Teaching mathematics through problem solving and using metacognitive strategiesis time consuming, but it is enjoyable and rewarding. The teacher's role in this approach asfacilitator, monitor, coach, and role model is significant. In fact, teachers are key elementsin such an approach. 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(1983). Problem solving in the mathematics curriculum: A report,recommendation, and an annotated bibliography. Washington, DC: MathematicalAssociation of America.Schoenfeld, A H. (1984). Beyond the purely cognitive: Belief systems, social cognitions,and metacognition as driving forces in intellectual performance. Cognitive Science, 7(4),329-363.Schoenfeld, A H. (1985a). Mathematical problem solving. Orlando, FL: Harcourt BraceJovanovich.Schoenfeld, A H. (1985b). Metacognitive and epistemological issues in mathematicalunderstanding. In E. Silver (Ed.), Teaching and learning mathematical problem solving:Multiple research perspectives (pp. 247-266). Hillsdale, NJ: LEA.Schoenfeld, A H. (1987a). What's all the fuss about metacognition? In A H. Schoenfeld,.Cognitive Science and Mathematics Education (pp. 190-215). Hillsdale, NJ: LEA.Schoenfeld, A. H. (1987b). Confessions of an accidental theorist. For the Learning ofMathematics, 7(1), 30-38. Montreal.Schoenfeld, A. H. (1987c). A brief and biased history of problem solving. In F. R. Curcio(Ed.), Teaching and learning: A problem-solving focus (pp. 27-46). Reston, VA:NCTM.Schoenfeld, A. H. (1988a). Problem solving in context(s). In R. I. Charles and E. A. Silver(Eds.). The Teaching and assessing of mathematical problem solving (pp. 82-91).Reston, VA: LEA/NCTM.Schoenfeld, A. H (1988b). When good teaching leads to bad results: The disasters of "well-taught" mathematics courses. Educational Psychologist, 23(2), 145-166. LEA.194Schoenfeld, A. H. (1989a). Teaching mathematical thinking and problem solving. In L. B.Resnick & L. E. Klopfer (Eds.), Toward the thinking curriculum: Current cognitiveresearch. 1989 Yearbook of the Association for Supervision and CurriculumDevelopment (pp. 145-165).Schoenfeld, A. H. (1989b). Ideas in the air: Speculations on small group learning,environment, and cultural influences on cognition, and epistemology. InternationalJournal of Educational Research, 13(1), 71-78Schroeder, T. L. & Lester, F. K. (1989). Developing understanding in mathematics viaproblem solving. In P. R. Trafton & A. P. Shulte (Eds.), New directions for elementaryschool mathematics: 1989 Yearbook (pp. 31-42). Reston, VA: NCTM.Silver, E A.; Branca, N A.; & Adams, V. M. (1980). Metacognition: The missing link inproblem solving? In R. Karplus (Ed.), Proceedings of the Fourth InternationalConference for Psychology of Mathematics Education (pp. 213-230). Berkeley, CA.University of California.Silver, E A. (1982). Knowledge organization and mathematical problem solving. In F. K.Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 15-25). Philadelphia, PA: Franking Institute Press.Silver, E. A. (1985). Research on teaching mathematical problem solving: Someunderrepresented themes and needed direction. In E. A. Silver (Ed.), Teaching andlearning mathematical problem solving: Multiple research perspectives (pp. 247-266).Hillsdale, NJ: LEA.Silver, E A. (1987). Foundations of cognitive theory and research for mathematics problem-solving instruction. Cognitive Science and Mathematics Education, 33-60.Silver, E. A. (1990). Contributions of research to practice: Applying findings, methods, andperspectives. In T. J. Cooney & C. R. Hisch (Eds.), Teaching and learning mathematicsin 1990's: 1990 Yearbook (pp. 1-11). Reston, VA: NCTM.195Stanic, M. A. & Kilpatrick, J. (1988). Historical perspective on problem solving inmathematics curriculum. In R. I. Charles and E. A. Silver (Eds.), The teaching andassessing of mathematical problem solving (pp. 1-22). Reston, VA: NCTM.Sterrett, A. (1990). Using Writing to teach mathematics. Washington, DC: TheMathematical Association of America.Strauss, A. & Corbin, J. (1990). Basics of qualitative research. Grounded theoryprocedures and techniques. California: SAGE Publications, Inc.Sydam, M. (1980). Untangling clues from research on problem solving. In S. Krulik and R.E. Reys (Eds.), Problem solving in school mathematics: 1980 Yearbook (pp. 34-50).Reston, VA: NCTM.Talman, L. A. (1990). Weekly journal entries-an effective tool for teaching mathematics. InA. Sterrett (Ed.), Using writing to teach mathematics (pp. 107-112). Washington, DC:The Mathematical Association of America.Thompson, A. G. (1985). Teachers' conceptions of mathematics and the teaching ofproblem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problemsolving: Multiple research perspectives (pp. 281-294). London: LEA.Thompson, A G. (1988). Learning to teach mathematical problem solving: changes inteachers' conceptions and beliefs. In R. I. Charles & E. A. Silver (Eds.), The teaching andassessing of mathematical problem solving, vol. 3 (pp. 232-243). Reston, VA:LEA/NCTM.von Glaserseld, E. (1983). Learning as a constructive activity. In J. C. Bergeron &Herscovics (Eds.), Proceedings of the Seventh Annual Meeting of North AmericanChapter of the International Group for the Psychology of Mathematics Education (pp.41-69). Montreal.von Glasersfeld, E. (1987). Radical constructivism as a view of knowledge. Paperpresented at AAAS Annual Meeting. Chicago, USA.196Vygotsky, L. S. (1962). (E. Hanfmann & G. Vakar, Eds. & Trans.). Thought andlanguage. Cambridge, MA: MIT Press and Wiley.Vygotsky, L. S. (1978). (M. Cole, V. John-Steier, S. Scribner, & E. Souberman,Trans.).Minds in society. Cambridge, MA: Harvard University Press.White, A. T. (1990). A writing-intensive mathematics course at Western MichiganUniversity. In A. Sterrett (Ed.), Using writing to teach mathematics (pp. 34-38).Washington, DC: The Mathematical Association of America.Yin, R. K. (1989). Case study research: Design and methods. Beverly Hills, CA: Sage.197APPENDICESAPPENDIX AResearcher's Letter to StudentsDear student;With the permission of the Mathematics Department, I am doing a study on "The influenceof selected instructional strategies on students' understanding of mathematics and problemsolving ability." The study is being conducted as part of my dissertation for the doctoraldegree in mathematics education.The study includes two parts. One consists of asking volunteers to participate inindividual task-based interviews before and after the course of the instruction. Those ofyou who agree to participate will be asked, during the interview, to discuss with me howyou solve mathematical problems. Each interview will not take more than one hour andwill be audio and video taped. The second part of the study consists of documentation ofinstruction and class interactions by taking field notes and video-taping the class sessions.Also your written work and exams will contribute to the data to be analyzed for the study.Participation or non-participation in the study is voluntary, and will not affect yourmarks for the course. You can withdraw your data from the study at any timewithout any consequence to your class standing. The data obtained in this study willbe kept strictly confidential. Pseudonyms will be used in any report of the study. The datawill be destroyed after the culmination of the study.In working with students last year in this course, they reported that this experience wasmost beneficial, to their understanding of mathematics and their problem solving skills.Please indicate your consent or refusal, to participate in the study in the attached form. Itwould be helpful if you return the form as soon as possible. If you have any questions,you may contact my research supervisor Dr. Owens at 822-5318 or myself (Zahra Gooya)at 822- 2351. I appreciate your consideration of this request and I thank you in advancefor your input into the study.Sincerely,Zahra GooyaGraduate studentDepartment of Mathematics and Science EducationUniversity of British Columbia198APPENDIX BStudent Consent FormI received a copy of Ms. Gooya's letter describing the project.I understand that my agreement or rejection will not in any way affect my status inclassroom interaction or in academic assessment.I do agree to participate in the study of the influence of instructional methods on students'understanding of mathematics and problem solving by offering myself as informant in thetask-based interviews.SignatureI will give permission to Zahra Gooya to use my written work as part of the data for herstudy.SignatureI agree to be in the range of the video camera while the class sessions are being video-taped.SignatureDate199APPENDIX CStudent Consent Form(Regarding Video tapes)I received a copy of Ms. Gooya's letter describing the project.I understand that my agreement or rejection will not in any way affect my status inclassroom interaction or in academic assessment.I will give permission to Zahra Gooya to use the video tapes of class interactions with mein it for professional development, but not commercial purposes.SignatureDate200APPENDIX DCourse Outline-Summer 1991MATH 335 ( Introduction to Mathematics)1. From Meno to Pythagoras. The first math lesson on record: Socrates shows how todouble a square. Gradually generalizing the method, we finally arrive at the theorem ofPythagoras. No numbers yet: Lengths and areas need not always be quantified.2. Diagonals and Heights. If you can find square roots, Pythagoras facilitates indirectmeasurement. Concentrating on lengths related to regular polygons, but also trying somethree dimensional objects. Making use of scales and proportion.3. Area and volumes. Stacking strips or slabs, you can use heights to determine areasand volumes. Prime example: triangles and pyramids. Thence (regular) polygons andpolyhedra.4. Cavalieri's Principle. Reinterpreting slices of pyramids, we find the area under aparabola. Comparing a hemisphere (slice-by-slice) with a cratered cylinder, we find thevolume of a sphere.5. Rational vs. Irrational Numbers. We compute square roots by repeatedly averagingthe sides of rectangles (Newton), and approximate rc by repeatedly doubling the number ofvertices of a polygon (Archimedes). We show that periodic decimals represent rationalnumbers (cf. Achilles and tortoise), and prove that many square roots (e.g. -,./f) areirrational.6. The Lore of Large Numbers. We find spatio-temporal representations for several verylarge quantities, like the national debt. Scientific notation and orders of magnitude. Thetechnique of "counting factors" gives easy estimates for products and quotients.7. Fractional Powers: Logarithms. Using square roots, we make a table of lOr , , wherer = k/32 with 0<k<32. The resulting graph gives a way of writing any positive number as10a , with obvious benefits for multiplicative calculation.8. Growth and Decay. Many quantities evolve according to a scheme of the formQ(t) = Qoa tat, where a>1 (growth) or a < I (decay). We look at a typical crop of suchproblems.2019. Multiplicative Counting. Multiplying the number of choices available at each stage ofa composite task. Counting with or without replacement. Permutations. Probabilities asrelative counts.10. Binomial Coefficients. If k elements become interchangeable, the total count isdivided by k!. Reshuffling "words" with repeated letters. Binary words. Moreprobabilities.11. From Pascal to Gauss. Histograms derived from different lines of Pascal's trianglehave a strong family resemblance. Extrapolating the limiting case. Probabilitiesinterpretation.12. Testing and Sampling . The normal curve shows how the values of many (butcertainly not all) random variables tend to cluster. Of course, it is also useful in computingbinomial probabilities, especially when the number of "trials" is large. Standard Deviationversus sample size.202APPENDIX ECourse MaterialsFirst Dissection Puzzle203Second Dissection Puzzle204Graph of a = 1 0%2n a^ n an010203041.07461.15481.24091.3335171819203.39833.65173.92434.2170OS 1.4330 21 4.531606 1.5399 22 4.869707 1.6548 23 5.233008 1.7783 24 5.6234-09 1.9110 25 6.043010 2.0535 26 6.4938 .11 2.2067 27 6.97E312 2.3714 28 7.498913 2.1483 29 8:05E4•-•^•^•^14 2.7384 30 8.6596 '`15 2.9427- 31 9.3057-*^0.-^4^18. 8.16= 322 10.000 '205APPENDIX FSample of Metacognitive QuestionsNameDate^ANSWER THE FOLLOWING QUESTIONS REGARDING THE SECOND QUIZ5 (it takes about tenminutes):1. What information helped you to solve the problem?2. What difficulties did you have in solving the problem?3. Would you restate the problem using your own words?4. What do you think you needed to know in order to solve the problem?5. Would you state a problem which is related but more general than this problem?6. How did you like solving the problem?5 These questions were included in both second and third quiz.206APPENDIX GMath 335, First Assignment(May 10, 1991)Proof of the Pythagorean Theorem1) State the Pythagorean Theorem;2) prove the Pythagorean Theorem (As you know, there is not only one way or the wayprove this theorem. It is up to you to prove it in a way that makes the most sense to you.)3) What do you think a mathematical proof is? What do you think about thegeneralizability of a proof?207APPENDIX HMath 335, First Quiz(May 17, 1991)Find the area of each of the following polygons. Assume that all sides have unit lengthand work out the answers in the corresponding square units. You have to show all yourwork and explain that WHY you did WHAT you did and HOW would it help you. Youcan work in groups and use your notes. However, you have to write the solution upindividually.208APPENDIX IMath 335, Second Quiz(May 27, 1991)Which fits better, a square peg in a round hole or a round peg in a square hole?You should explain that WHAT are you doing? WHY are you doing it? And HOW is ithelping you?209APPENDIX JMath 335, Second Assignment(May 31, 1991)(a) Show that the equation x 2 =10y2 can not be satisfied by any two whole number xand y.(b) What does this say about -ii-a?210APPENDIX KMath 335, Third Quiz(June 7, 1991)How many different license plates are possible if each contains three letters followed bythree digits? How many of these license plates contain no repeated letters and no repeateddigits? You have to show all your work and explain that why you did what you did andhow would it help you.211Appendix LStudents' Selections for the Open Assignment as PercentsGeometry Numbers Counting Critique Pose*Traditionalists (9) 22 45 11 11 11*Incrementalists (12) 18 9 64 9 8*Innovators (19) 16 32 42 11 5Class (40) 18 28 42 10 7Note.Pose = Students' self-posed problems*These categories of students are defined in Chapter 5.212APPENDIX MFinal Exam--Spring 1991Mathematics 335Instructor: Zahra Gooya^ Time: 2 hoursPart ASelect TWO of the following four topics. Each counts for 20% of your mark.Al. Explain Cavalieri's Principle. How could it be used to find volume of a hemispherewith a circular base of radius R.A2. State the Pythagorean theorem. Show why it is true and how it is useful by themeans of an example.A3. How can Pascal's triangle be used to determine binomial probability? What is therelation between binomial probability and the bell curve?A4. Choose a topic in the course which struck you as particularly interesting or useful (itmust be a topic that you have not or will not be using in any other parts of thisexamination.)(a) Set a question or a problem on this topic.(b) Solve the question or the problem, giving clear explanations and showing all the stepsused to arrive at the answer.Part BBl. A toy in the shape of a hemisphere on top of a cone. The cone has sloping edge of13 cm. Its base matches the top of the hemisphere, which has radius 10 cm.(a) Make a sketch of the toy.(b) Find the height and the volume of the toy.B2. The number 1/7 and 0.3333... are both rational numbers. Is the number0.012345679012345679... a rational number? If you agree that it is a rational number,write it as a fraction of two integers. If you disagree that it is a rational number, provethat it is an irrational number.B3. A committee of 5 is to be chosen from a group of 7 females and 6 males. In howmany ways can the committee be chosen such that:(a) Any 5 are chosen, irrespective of sex.(b) All 5 are females.(c) All 5 are males.(d) 3 are females and 2 are males.213(e) There is at least one female in the committee.B4. A couple just had a new child. How much should they invest now at 12%(a) Compounded monthly in order to have $60,000 for the child's education 16 yearsfrom now?(b) Compounded annually in order to have $60,000 for the child's education 16 yearsfrom now?(c) Which one is a wiser investment?B5. Find the area of the shaded region if the vertex angle of a square with sidesmeasuring 4 units is trisected.214Appendix NStudents' Selections of Problems on the Final Exam as PercentsAl A2 A3 A4 B1 B2 B3 B4 B5*Traditionalists (9) 22 33 44 33 67 44 100 54 33*Incrementalists (12) 50 83 42 17 75 25 92 33 75*Innovators (19) 42 100 32 26 74 26 84 47 68Class (40) 40 80 38 25 73 30 90 44 62Note.*These categories of students are defined in Chapter 5.215APPENDIX 0Interview Question ATen delegates from ten countries meet at a conference on world peace. If each delegateshakes hands with the delegate from each of the other countries, how many handshakeswould take place?216APPENDIX PInterview Question BA pair of rabbits one month old are too young to produce more rabbits, but suppose thatin their second month and every month thereafter they produce a new pair. If each newpair of rabbits does the same, and none of the rabbits die, how many pairs of rabbits willthere be at the beginning of each month? What happens in the first six months?217APPENDIX QSample of Teresa's Work(Finding Area of the Dodecagon)APPENDIX RSample of Clara's WorkClara solved the following problem on her June 5 journal entry.5. The carbon of a fossil bone contains only 20% of the normal amount of C...Approximate the age of the bone.219221Clara solved the following problem on her June 7 journal entry.8. To anesthetize a dog, 30 mg of sodium pentobarbital is required for each kilogram ofbody weight. If the drug is eliminated exponentially from the blood stream, with a half-life of 4 hours. Approximate the dose which will keep a 20 kg Alsatian anesthetized for45 minutes.223225APPENDIX SSample of Kent's work(Regarding Pmhlern RS nn the Finn] Pvnm\226227APPENDIX TThank You Letter to the Students at the End of the CourseDearI wanted to take this opportunity to thank you for participating in my study. Thestudy could not become a reality without your involvement. I know that there weremoments of frustration and dissatisfaction while we, as a whole class were trying tonegotiate the meanings for better understanding. Yet there were moments of enjoymentand discoveries that could hardly happen if I simply imposed my ideas on you. I greatlyappreciate your patience while working with me for six hard working weeks. I can onlytalk about myself to say that I truly enjoyed these exhausting weeks! I will dedicate mystudy to you as a potential teacher and I am sure that you will make a wonderful one. Ihope that your future students will have many opportunities to express themselves and willnot accept anything until they find answers to their "why's"!I will miss you!Zahra Gooya228


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