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Conceptions of probability held by preservice teachers of secondary school mathematics Koirala, Hari Prasad 1994

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CONCEPTIONS OF PROBABILITY HELD BY PRESERVICETEACHERS OF SECONDARY SCHOOL MATHEMATICSbyHART PRASAD KOIRALAB.Ed., Tribhuvan University, Nepal, 1976M.Ed., Tribhuvan University, Nepal, 1980M.Ed., University of Hull, England, 1986A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Curriculum Studies)We accept this dissertation as conformingto the required standard.THE UNIVERSITY OF BRITISH COLUMBIAAugust, 1995© Han Prasad Koirala, 1995In 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 cu;cu Studi’esThe University of British ColumbiaVancouver, CanadaDate Ac-&jw.it 18, i.9SSDE-6 (2)88)AbstractThe purpose of this study was to explore the qualitatively different conceptionsof probability held by two groups of preservice teachers of secondary schoolmathematics at the University of British Columbia. The study also explored theconsistency of participants conceptions of probability and their views on theutility of formal probability in solving everyday problems.A set of written tasks, pair-problem-solving tasks, and interview tasks relatedto probability were given to the participants. A total of 40 preservice teachersparticipated in the written component, 16 of whom also participated in the pair-problem-solving and the individual interview components.It was found that the preservice teachers held qualitatively differentconceptions of probability. Their conceptions of probability were grouped intoformal and non-formal. Formal conceptions of probability included the use ofprobability concepts such as independence and randomness, and the use ofprobability formulas, rules, and applications in solving problems. Non-formalconceptions of probability included participants’ use of everyday experiences andheuristics as well as the use of science knowledge in solving probability problems.The participants’ conceptions of probability varied widely among tasks dependingon whether the tasks appeared to be taken from probability textbooks or from aneveryday context. Many participants stated that knowledge of formal probabilitywas not useful in solving everyday problems.Two main condusions were drawn from the results of this study. First,preservice teachers hold qualitatively different conceptions of probability thatlargely depend on contexts determined by tasks and settings. Second, students’understanding of probability may be influenced by their non-formal conceptions,and these should be used in teaching formal probability concepts.-U-Table of ContentsAbstract.iiTable of Contents iiiList of Tables viiAcknowledgments viiiCHAPTER 1 1Introduction 1Conceptions of probability: What do they mean2 3Why study preservice teachers’ conceptions of probability? 4The purpose of the study and the research questions 6Theoretical framework of the study 7Data sources and pilot study 8Organization of the chapters 9CHAPTER 2 10Review of Related Literature 10Research on students’ understanding of probability 10Developmental research on probability 12Piagetian Studies 12Fischbein 14Green 18Heuristic research on probability 20Representativeness heuristic 21Availability heuristic 24Critiques of heuristic studies 25Further development of heuristic studies 26Research on preservice teachers’ conceptions of probability 28Diverse conceptions of probability: A historical account 29Formal and non-formal conceptions of probability: Another dimensionin looking at students’ conceptions of probability 32Overview of Chapter 2 34— 111 —cHAPThR3 .36Research Methodology and Procedures of the Study 36Theoretical framework for the methodology of the study 36Generalizability of the study 37Semi-structured interviews 37Pair-problem solving 39Principles guiding research interviewing and pair-problemsolving 40Methods of data analysis 41Research Method 43Development of instruments 43Construction of tasks 43Field tests 44Criteria for selecting tasks for pair-problem solving andinterview 45Description of tasks 45Data collection procedures 49Conducting written tasks 49Selection of preservice teachers for pair-problem solving andinterview 50Procedures for pair-problem solving and interview 52Data transcripts 54Data analysis 54Data review and validation 57Discussion of transcripts with preservice teachers 57Discussion of researcher’s interpretation with preserviceteachers 58CHAPTER 4 59Preservice Teachers’ Conceptions of Probability 59Qualitatively different conceptions of probability 59Non-formal conceptions of probability 61Using everyday experiences in solving probability problems 61Using a representativeness heuristic in solving probabilityproblems 71Using science knowledge in solving probability problems 75- iv -Formal conceptions of probability.79Using independence and randomness in solving probabilityproblems 79Using formulas in solving probability problems 83Overview of Chapter 4 87CHAPTER 5 89Consistency Analysis of Preservice Teachers’ Conceptions of Probability 89Consistent formal:The case of Paul 92Consistent non-formal:The case of Alan 97Inconsistent:The case of John 103Overview of Chapter 5 109CHAPTER 6 111Preservice Teachers’ Perceptions about the role of Probability in theirEveryday Problems 111Participants’ views about the role of probability in solving everydayproblems 111Probability in university and probability in everyday life: Twodifferent phenomena 112Knowledge of university probability enhances non-formalprobability 116Preservice teachers’ methods of solving everyday problems 122Overview of Chapter 6 124CHAPTER 7 125Conclusions, Discussion, and Implications 125Conclusions 125Question one: What are some qualitatively different ways in whichpreservice teachers conceptualize probability2 126Question two: In what ways do preservice teachers’ conceptions ofprobability vary across tasks and settings? 130Question three: What are preservice teachers’ views about the role ofprobability learned at the university level in solving everydayproblems? 132Discussion of critical issues arising from the study 133Role of contexts in solving problems 134Understanding probability and using it in everyday life 135Implications for teacher education 137Implications for curriculum and instruction 139Recommendations for further research 141References 142Appendix A - Written component 160Appendix B - Pair-problem solving 164Appendix C - Interview protocol 166Appendix D - Extract from John’s interview transcripts 169-vi-List of TablesTable 1. Written and pair-problem-solving tasks 46Table 2. Mathematical concepts associated with the tasks 48Table 3. Background information of preservice teachers selectedfor the pair-problem solving and the interview 51Table 4. Formation of pairs 53Table 5. Preservice teachers’ formal and non-formal conceptionsof probability 60Table 6. Examples of preservice teachers’ consistent andinconsistent conceptions across tasks and settings 91-vi’-AcknowledgmentsFirst and foremost, I would like to express my sincere thanks to myresearch supervisor, Dr. David F. Robitaille, without whose help I would not be atUBC and, therefore without whom I would never have accomplished this thesis.His constant interest and encouragement throughout my doctoral studies at UBCwere always much appreciated.I wish to thank all of my committee members Dr. P. James Gaskell, Dr. AnnAnderson, and Dr. Tony Clarke for their encouragement and guidance throughoutthis thesis. Particularly, Ann’s and Tony’s page by page comments on earlierdrafts of my thesis were very helpful in organizing this thesis in the present form.My sincere thanks to the 40 students at the University of British Columbiawho voluntarily participated in the study. Your generous sharing of time andideas has helped me and I hope, by that, will also help others.My sincere thanks are also due to Dr. Brent Davis for helping me to collectdata, and also for making useful comments in one of the earlier drafts of my thesis.My thanks are also due to Mr. Jim Mennie who helped me to modify some of thetasks used in the study and also to collect data for the study. I also like to thankDr. David Pimm for helping me to shape the study by providing valuablecomments on my proposal when the study was only crawling.My special thanks to Edward, Tami, Christopher, and Logan Robeck whowere really our family friends. My family and I thoroughly enjoyed all we didtogether. You became an important part of our lives and we will remember youforever.My sincere appreciation is also due to Katy Ellsworth and Tami Robeck fortheir help in editing the language of this thesis. I would also like to extend mythanks to all the graduate students of the Department of Curriculum Studies. Thehelp from Edward Robeck, Sandra Crespo, Cynthia Nicol, George Frempong, andJoel Zapata were important for the present work. George’s and Joel’s help torevise tasks and Ed’s help to design some tasks are always appreciated.Finally and most importantly, I would like to give my greatest thanks to mywife, Sita Koirala, and children, Shrijana and Pratistha, for their concern andunderstanding of my work. Thank you for understanding why I was not able togive as much time as I would have liked.Thank you very much for all your help!— viii —CHAPTER 1IntroductionMathematicians, statisticians, and mathematics educators considerprobability and statistics to be valuable for people in understanding theireveryday world (Borel, 1962; Dörlfer & McLone, 1986; Good, 1983; Jacobsen,1989; McGervey, 1986). Dewdney (1993) and Paulos (1989, 1994) have arguedthat people who lack a basic understanding of probability and statistics are oftenfunctionally innumerate. Probability and statistics are also considered importantfor advanced post-secondary studies; for example in the natural and socialsciences. Dörlfer and McLone (1986) claim that probability and statistics are notonly important in many occupations but also relevant “for a general criticalunderstanding of many facts in economics, politics, ecology and other fields” (p.70). Falk and Konold (1992) argue that “probability will be as important in the21st century as mastering elementary arithmetic is in the present century” (p.151)1.There is a growing movement to introduce elements of probability at bothsecondary and elementary school levels in North America (MathematicalSciences Education Board, 1990; National Council of Teachers of Mathematics,1980, 1989; Shulte, 1981)2. For example, the Curriculum and Evaluation Standardsfor School Mathematics published by the National Council of Teachers ofMathematics (NCTM, 1989) recommends that probability be taught at all gradelevels in the United States. Despite this recommendation, probability does notseem to be widely taught in schools in North America (Romberg, 1992;1t is necessary to mention, however, that they have not recommended replacing arithmetic withprobability. The statement emphasizes the role of probability in students’ everyday lives.2 North America, probability is underrepresented in school mathematics curriculum andinstruction compared to other statistical topics such as graphs, averages, dispersion, andregression. This study focuses only on issues related to probability, although when researchershave discussed them together it is difficult to separate probability and statistics.—1—Shaughnessy, 1992). At present, it appears that very little instruction inprobability is occurring at the classroom level (Garfield & Ahlgren, 1988;Shaughnessy, 1992).One suggestion as to why teachers do not teach probability is that many ofthem do not have adequate preparation for its teaching (Fabbris, 1988; Jacobsen,1989; RAde, 1985; Steinbring, 1989; Zuliani & Sanna, 1988). According toSteinbring (1989), most mathematics teachers think that mathematics is anobjective and hierarchical discipline and so are not equipped to “handle theinexactitude and uncertainty” (p. 203) inherent in the study of probability.Researchers contend that the teaching of probability should be a part of teachereducation programs in mathematics so that teachers will be prepared to teach it(Juraschek & Angle, 1981; Râde, 1985; Steinbring, 1989).Based upon a literature survey, there is a lack of research on the teachingof probability in teacher education programs. There are no studies focusing onexploring teachers’ knowledge about how to teach probability, nor are therestudies that have explored inservice or preservice teachers’ understanding ofprobability.Bramald (1994) has provided some evidence that secondary schoolmathematics preservice teachers find it difficult to understand basic concepts ofprobability. However, his report is based solely on one snapshot of his classroomteaching and does not constitute systematic inquiry into preservice teachers’knowledge about probability. Preservice teachers’ knowledge andunderstanding of probability have implications for the teaching of probability inschools in the future; it is therefore important to gain insights into preserviceteachers’ understanding of probability.-2-Conceptions of probability: What do they mean?In studies related to students’ understanding of particular phenomena,various terms such as concepts, conceptions, and conceptualizations have been used(Linder, 1989; Robertson, 1994a, b). According to Robertson, concepts are publiclyheld sets of rules or norms about the use of terms. Conceptions are anindividual’s interpretation of concepts, that is, the ways that individuals makesense of concepts within their individual and social frameworks. Conceptionsinclude the understanding of concepts that may or may not be consistent withthat of experts, but may be shared by some other individuals in a certaincommunity. Conceptualizations, however, are “idiosyncratic personalunderstanding” (Robertson, 1994, p. 26) which are not necessarily shared with theexpert community or another social community.Conceptions include elements of both concepts and conceptualizations.They involve both mathematically correct and incorrect understandings of atopic held by a person. In mathematics education research, Thompson (1992) hasidentified several elements of conceptions such as “conscious or subconsciousbeliefs, concepts, meanings, rules, mental images, and preferences” (p. 132) heldby students. Since these elements are used in mathematics education, as well asbeing closely related to the definition used by Robertson, Thompson’s definitionof conceptions is used in the present study.In the light of Thompson’s definition, conceptions of probability includestudents’ formal knowledge of probabilistic facts, concepts, rules, and theoremsas well as students’ non-formal beliefs, meanings, intuitions, experiences,understandings, and images related to probability. Conceptions of probability,for example, include how students make sense of probability concepts (such aschance, randomness, and independence) that are usually accepted by themathematical community. Conceptions of probability may also include students’-3-idiosyncratic views of everyday chance phenomena such as games, lotteries,accidents, weather, and so on. This study documents both the formal and non-formal conceptions of probability held by preservice teachers.Why study preservice teachers’ conceptions of probability?Shulman (1986) argues that teachers require subject matter contentknowledge, pedagogical content knowledge, and curricular knowledge. Subjectmatter content knowledge is the knowledge gained by understanding facts,concepts, procedures, and structures of the discipline. This knowledge is gainedprimarily by studying mathematics content courses in mathematics departments.Pedagogical content knowledge includes “the most regularly taught topics inone’s subject area, the most useful forms of representation of those ideas, themost powerful analogies, illustrations, examples, explanations, anddemonstrations” (p. 9). The latter also includes “an understanding of what makesthe learning of specific topics easy or difficult” (p. 9) for school students and theconceptions of the topic that the students hold. Pedagogical content knowledgeis usually gained in mathematics methods courses.In terms of Shulman’s categories, preservice teachers’ conceptions ofprobability are part of subject matter content knowledge and their understandingof students’ conceptions of probability are part of pedagogical contentknowledge. Both kinds of knowledge interact with each other in the teachingdomain. Preservice teachers’ conceptions of probability, their understanding ofstudents conceptions of probability, and the interaction between these twoinfluence how preservice teachers view the teaching of probability. Gaininginsights into their conceptions of probability can, therefore, be valuable foreducators involved in the preparation of teachers.Investigating students’ conceptions is an important step for educators whobelieve in a constructivist view of learning. According to this view, students-4-subsequent mathematics learning and understanding are guided, filtered, andshaped by their existing conceptions of mathematics (Ausubel, Novak, &Hanesian, 1978; Confrey, 1991; Fischbein, 1987). Viewed from this perspective,preservice teachers’ beliefs, meanings, rules, mental images, and preferences playa substantial role in their learning of subject matter content knowledge orpedagogical content knowledge about probability. Each preservice teacherconstructs and reconstructs a wide range of mathematical understandings as hisor her existing conceptions interact with the mathematics taught by a professor.Therefore, mathematics professors and mathematics education professors requireinsight into how preservice teachers conceptualize probability so that probabilitylearning and the teaching of probability can be made more meaningful.The importance of studying preservice teachers’ conceptions of probabilityis also supported by researchers at the University of Wisconsin, such asCarpenter, Fennema, and their colleagues (Carpenter & Fennema, 1991;Fennema, Carpenter, Franke, & Carey, 1993; Fennema & Franke, 1992). Theirinnovative teacher training program is based on “cognitively guided instruction”which is founded on the argument that “instructional decisions should be basedon careful analyses of students’ knowledge and the goals of instruction”(Carpenter & Fennema, 1991, p. 11). To extrapolate this viewpoint, then, anunderstanding of how preservice teachers conceptualize probability is useful formathematics professors in their teaching in the same way that the understandingof school students’ conceptions is useful for school teachers. These researchersnot only emphasize the importance of gaining insights into students’ conceptionsbut also argue that teachers’ conceptions affect how they view teacher educationprograms and how they teach. Thompson (1984, 1992) and Battista (1994) alsosupport this viewpoint. Hence, preservice teachers’ conceptions of probabilitycan provide useful information for mathematics professors as they plan and-5-design strategies for teaching probability, and for mathematics educationprofessors as they plan and design method courses to help teachers learn to teachprobability in schools.The purpose of the study and the research questionsShaughnessy U992) argues that the success of a reform movement inprobability education relies on how well teachers are educated to teach this topic.If teachers do not have sufficient background knowledge and understanding ofprobability, they will not be able to teach it successfully. He therefore calls forresearch on teachers’ conceptions of probability as an important step ininvestigating the teaching of probability in schools. He suggests that researchers“gather information from teachers at both the pre-service and in-service levels”(p. 489). Despite Shaughnessy’s call to gather information about inservice andpreservice teachers’ understanding of probability, researchers have not givensignificant attention to this area. The purpose of the present study is to addressShaughnessy’s concern and provide a picture of preservice teachers’understanding of probability. Specifically, this study seeks answers to thefollowing questions:1. What are some qualitatively different ways in which preservice teachersconceptualize probability?The aim of this question is to portray the varieties of reasoning used bypreservice teachers to solve probability problems. Their reasoningsportray how preservice teachers think about or understand probability.2. In what ways do preservice teachers’ conceptions of probability varyacross tasks and settings?This question has two components. First, in what ways are preserviceteachers’ conceptions of probability consistent or inconsistent across-6-various tasks? Second, in what ways do their conceptions vary inindividual and pair settings?3. What are preservice teachers? views about the role of probability learned atthe university level in solving everyday problems?This question addresses the issue of whether or not preservice teachersthink that the probability courses they took at the university level werehelpful in solving everyday problems. The responses obtained from thisquestion will provide a picture of their conceptions of formal probabilityin coping with everyday problems.Theoretical framework of the studyThis study draws upon a constructive perspective that argues individualsconstruct knowledge based on their personal and social experiences (Cobb, 1989,1994; Confrey, 1991). Cobb (1994) argues that “mathematical learning should beviewed as both a process of active individual construction and a process ofacculturation into the mathematical practices of wider society”(p. 13). Davis,Maher, and Noddings (1990) provided a similar opinion stating that “eachlearner has a tool kit of conceptions and skills with which he or she mustconstruct knowledge to solve problems presented by the environment” (p. 3).Consistent with the perspectives presented by these researchers, constructivismin this study is viewed in the following ways:1. An individual constructs knowledge by attempting to make sense of his orher environment and by actively reorganizing his or her experiences(Cobb, Yackel, & Wood, 1991; Corifrey, 1991; Resnick, 1983; Simon, 1994;von Glasersfeld, 1984, 1989).-7-2. Mental action by an individual is situated in his or her historical, cultural,and institutional settings (Bishop, 1988; Carraher, 1991; Lave, 1988; Lave &Wenger, 1991; Saxe, 1991; Walkerdine, 1988, 1990, 1994; Wertsch, 1991).3. Students bring prior experience to instruction, and these experiencesshould be utilized in teaching (Ausubel, Novak, & Hanesian, 1978;Carpenter & Fennema, 1991; Confrey, 1990a, b, 1991).Having adopted a constructivist view of learning, this researcher wasinterested in exploring each individual’s conceptions of probability duringdifferent phases of data collection and the analysis of the study. As a result, thisstudy has elicited preservice teachers’ qualitatively different conceptions ofprobability.Data sources and pilot studyData for the study were gathered through written tasks, pair-problemsolving, and individual interviews with two separate groups of preservicesecondary school mathematics teachers during the 1994 summer and fallmathematics teacher education programs at the University of British Columbia,Canada. The summer group was used as a pilot group for the study. Ninepreservice teachers from the pilot group responded to a written task related toprobability. Four of the nine participants, exhibiting a range of opinions andexperiences, were then selected for the pair-problem solving and the individualinterviews. Participants with different opinions and experiences were selected inorder to increase interaction between them in the pair-problem solving activitiesand to explore qualitatively different conceptions of probability. All the pairproblem solving and interview sessions were audiotaped. The tapes were thentranscribed and analyzed in conjurction with the written responses. The analysisof the pilot data demonstrated that the instruments and the data collection-8-procedures were robust and successful in depicting a variety of ways in whichpreservice teachers conceptualize probability.The same procedures used for the pilot study were used for the mainstudy. A total of 31 preservice teachers of secondary school mathematicsparticipated in the written component in the fall of 1994. Twelve of theparticipants participated in the pair-problem solving and the individualinterview tasks. As in the pilot study, all the pair-problem solving and interviewsessions were audiotaped, and then transcribed and analyzed.Analysis of participants’ responses to written, pair-problem solving, andinterview tasks indicated that they held different conceptions of probability.However, some participants in the pilot group held interesting conceptions ofprobability which differed from those of the main group. For example, aparticipant in the pilot group provided responses to tasks and settings that weremarkedly less consistent than those provided by the participants from the maingroup. Because of this, the pilot group’s conceptions of probability were reanalyzed and included with the results of the main group. Since the dataobtained from the pilot group were included with the data obtained from themain group, this study represents conceptions of probability held by 16preservice teachers. A more detailed description of methodology and datasources can be found in Chapter 3.Organization of the chaptersThere are seven chapters in this dissertation. The first three chapterspresent the theoretical foundations and methodology for the study. The nextthree chapters present the results related to the three research questions. Thefinal chapter draws the results together in relation to the literature and discussesthe study’s contribution to probability education both in terms of research andteaching.-9-CHAPTER 2Review of Related LiteratureThe literature on probability can be organized around four themes: theneed for teaching this topic in schools (Jacobsen, 1989; Pereira-Mendoza & Swift,1981), the current status of probability in school mathematics curriculum(Ahigren & Garfield, 1991; Kerkhofs, 1989; Scheaffer & Burrill, 1988), suggestionsfor teaching probability (Biehier, 1991; Gnanadesikan, Scheaffer, & Swift, 1987;Higgo, 1993; Holmes, 1988; Kerkhofs, 1989; Newman, Obremski, & Scheaffer,1987; Schupp, 1989), and the difficulties that students face in understandingprobability concepts (Garfield & Ahigren, 1988; Green, 1983; Hope & Kelly, 1983;Piaget & Inhelder, 1951/1975; Shaughnessy, 1977, 1981, 1992). The present studyis related to the fourth theme. Hence, in this chapter, the literature related tostudents’ understanding of probability is reviewed. In the absence of research onpreservice teachers’ conceptions of probability, most of the review in this chapteris focused on school and college students’ conceptions of probability.Research on students’ understanding of probabilityFischbein (1975) and Scholz (1991) provide a historical overview ofresearch on students’ learning of probability. Research paradigms discussed byScholz can be summarized under two headings: behaviorist and cognitive.Early behaviorist research on probability relied on stimulus-responsetheory where individuals’ actions in uncertain situations were believed to bebased on the reinforcement they received. This paradigm equated learningprocesses with a black box model where inputs and outputs observed in theexperimental situation were used to assess people’s understanding of probability.Individuals were “perceived as being governed by random processes” (Scholz,1991, p. 215) rather than by their mental processes. The behavior of a research-10-subject was analyzed “according to the standard interpretation of the situation orproblem” (Scholz, 1.991., p. 241). Individuals? cognitive acts were ignored becausethey were not observable. People’s responses were assumed to be context andculture free. Researchers? judgments were either eliminated or minimized as faras possible. The behaviorist paradigm tried to control both subject andexperimenter subjectivities. This paradigm perceived “subject-task relationship”as “ontological objectivism,” which assumed that an individual’s consciousnesscan be separated from outside reality (Scholz, 1991, p. 237).In the cognitive research paradigm, the notion of ontological objectivismand the black box model of learning are rejected. The cognitive researchparadigm assumes that students have their own ways of thinking and that theirthinking can have a significant impact on their learning of mathematics. Inprobability, research conducted by Piaget and Inhelder (1951/1975), Fischbeinand colleagues (1975, 1.984, 1987, 1991), Green (1983, 1988, 1.989), Kahneman andTversky (1972), Tversky and Kahneman (1971, 1973, 1982a, b), Konold andcolleagues (1989, 1991, 1993), Scholz (1991), and Shaughnessy (1977, 1981, 1992)can all be categorized as cognitive research although there is a large variationamong the approaches taken. Green’s studies are basically research surveysconducted with relatively large samples of students. Kahneman and Tversky’sresearch relied on surveys but was conducted with smaller samples of students.Fischbein and his colleagues used both surveys and interviews. Konold and hiscolleagues conducted interview studies with relatively small samples of students.All these researchers reported students’ understanding of probability in variousways.There are basically two types of research on students’ understanding ofprobability concepts: developmental research on precollege students such as thatinspired by Piagetian studies and heuristic research on students, conducted with-11 -both school and college students. Each of these is discussed in greater detailbelow.Developmental research on probabilityPiaget, Fischbein, and Green each conducted studies of children’sunderstanding of probability from a developmental perspective. The researchconducted by each of these individuals relating to children’s understanding ofprobability concepts is discussed below.Piagetian StudiesPiaget had a tremendous influence on research about children’s thinking.He was one of the earliest to study children’s conceptions of a wide range ofmathematical topics such as number, space, time, motion, and chance. Hisresearch approach was a major breakthrough from the behaviorist researchparadigm because of its attention to cognitive processes. Piaget posited thatchildren’s cognitive development occurs in stages, starting from the sensorymotor stage then moving to the pre-operational stage, the concrete operationalstage, and finally into the formal operational stage. Children who solvedmathematical problems requiring logical reasoning (typically those at least 11years of age) were considered by Piaget to be at a formal operational level.This view, espoused by Piaget, is evident in many of his books includingone co-authored with Barbel Inhelder, The Origin of the Idea of Chance in Children(Piaget & Inhelder, 1951/1975). This was the first systematic, qualitativedescription of aspects of children’s probabilistic thinking. The book describestasks (such as throwing counters, drawing marbles, mixing beads, tossing coins,rolling dice, spinning spinners, and so on) that were used to explore students’conceptions of chance and randomness.-12-In Piagetian studies, researchers ask children for their ideas about naturalevents and listen carefully to what the children say (Piaget, 1929). Piaget wasinterested in exploring the children’s notions of reality and causality (Solomon,1994) with little or no influence from the researcher. The results of “clinicalinterviews” using these tasks with children of various ages indicated that preoperational and concrete-operational children could not grasp formal probabilityconcepts; and only by the age of 13 or 14 did children acquire basic probabilityconcepts: for example probability as a ratio (Inhelder, 1976; Piaget & Inhelder,1951/1975). Piaget and Inhelder found that children predicted certain outcomeson the basis of immediately preceding events and not on the basis of deductiveanalysis. Children believed that random events could be controlled by “smart”people and they could not distinguish between random and necessary outcomes.In Piaget and Inhelder’s (1951/1975) view, the quantification of probability, orchance, and combinatorial thinking occurred only at the formal operational stage.Because of this, Lovell (1971) recommended that probability not be taught inschool prior to grade six.Some researchers have criticized Piaget’s stage theory (e.g. Donaldson,1978; Fischbein, 1975) and his structuralist view of learning (Ernest, 1991;O’Loughlin, 1992; Walkerdine, 1988). However, his theoretical premises thatindividuals construct knowledge by acting on their environment (Ellerton &Clements, 1992; Vergnaud, 1990) and that different individuals think in differentways (Confrey, 1990b; Vergnaud, 1990), as well as his use of clinical interviewing(Ginsburg, 1981, Ginsburg, Jacobs, & Lopez, 1993; Konold & Johnson, 1991) as amethod for investigating children’s thinking have become basic elements formany researchers interested in children’s conceptions of mathematics andscience. In addition, Piagetian studies of children’s conceptions of different-13-topics have provided not only information on how children think but thefoundations for constructivist views of learning.FischbeinLike Piaget, Fischbein was also interested in the development ofprobabilistic reasoning in children. His main area of interest was how intuitionand logical structures can influence children’s probabilistic thinking. Hecriticized Piaget and Inhelder for dealing only with logical and formal aspects ofchance and probability and for ignoring “interpretive, explanatory, and problem-solving predispositions generated by the intuitive base” (Fischbein, 1975, p. 6).Fischbein (1987) defined intuition as “immediate, apparently self-evidentcognitions” (p. 204). According to him, “Intuitions are always the product of personalexperience, of the personal involvement of the individual in a certain practical ortheoretical activity” (Fischbein, 1987, p. 213, emphasis in original). In his research,Fischbein concentrated on the relationship between intuitive and formal thinkingin children.Fischbein and his colleagues worked in the area of probability becausemany probability concepts are either intuitive or counter-intuitive for manystudents. Fischbein (1975) distinguished between the concepts of chance andintuition of chance. Re-analyzing Piaget and Inhelder’s data relating to differentexperiments on children’s conceptualizations of probability, Fischbein claimedthat even pre-operational children before the ages of 6 or 7 did have someintuition about chance. As early as 1967, Fischbein, Pampu, and Minzat (1975)conducted experiments with children as young as 6 through 14 years of age toverify this claim. A total of 188 children were interviewed. Five inclined boardswith different systems of progressively branching channels were constructed.Out of these five boards the first three were equiprobable and the last two werenon-equiprobable. The problem posed to the children was, “I am going to drop a-14-marble in here (pointing to the top of the main channel)—where will it come outat the bottom?” (Fischbein et al., 1975, P. 162). After the subject’s first response heor she was asked, “Will it always come out there, or can it come out in otherplaces?” (p. 162). Another question followed, “If I drop the marble a lot of times,one after the other, will it come out at each place the same number of times orwill it come out of some places more often than others?” (p. 162). Children wereasked for their reasons for each answer.Fischbein and his colleagues were surprised to learn that it was theyoungest children who gave the greatest number of correct responses for thelayouts with equiprobable routes. For non-equiprobable routes, correctresponses increased with age. According to Fischbein (1975), for older children“chance was nothing but ambiguity and uncertainty” (p. 73). For youngerchildren ambiguity and uncertainty did not affect the outcomes. Fischbein andhis colleagues indicated that school experiences based on a deterministic view ofmathematics were not helpful for developing appropriate formal concepts ofprobability. Their findings questioned the legitimacy of Piagetian stage theorythat children’s probabilistic conceptions develop only at the formal operationalstage.3In another study, however, Fischbein and Gazit (1984) reported thatchildren’s probabilistic thinking improved with age and instruction. When askedthe probability that the sum of the digits on the roll of two dice is even, 44% of 5518 1grade seven children gave a correct answer of or Only 3% of the 70 grade5 children and 16% of the 160 grade 6 children gave a correct answer to this task.All these children were tested after receiving instruction in probability. The3Piaget’s stage theory can also be criticized on the basis of other research indicating that evenadults believe that chance phenomenon can be controlled (e.g., Langer, 1982).-15-results indicate that children’s performance substantially increased with age,although the problem was difficult for the majority of students at all grade levels.In response to another question—whether a consecutive sequence ofnumbers such as :1, 2, 3, 4, 5, 6 or a random sequence of numbers is more likely towin a lottery—teaching seemed to have a significant influence at any gradelevels. But the effect of age was not as substantial as in the first question. Thequestion was found difficult by a majority of the students at all grade levels.More than 40% of the children thought that a random sequence of numberswould have a higher probability of winning the lottery. Although children gavemixed responses to different items, Fischbein and Gazit concluded that theteaching program in probability had a positive influence on children’sperformance.In a more recent study, Fischbein, Nello, and Marino (1991) administeredtwo sets of questionnaires with problems related to certain, likely, possible, andimpossible events to 618 elementary (9— to 11—year old) and junior high school(11— to 14—year old) children in Pisa, Italy. The results of this study were notsubstantially different from the previous studies.Although students’ facility generally increased with age and instruction,there were certain items in which the facility rate decreased with age. Forexample, in the rolling of two dice children were asked, “Are you more likely toobtain 5 with one die and 6 with the other, or 6 on both dice? Or is theprobability the same in both cases?” (p. 532). Of the elementary children, 73%responded that the probability of obtaining (5, 6) or (6, 6) would be the same.This error was made by approximately 90% of the junior high children. Theresponses were similar to those given in a coin tossing problem in which childrenwere asked, “When tossing two coins which result is more likely: to get heads-16-with one coin and tails with the other, or to get heads with each of the two coins?Or is the probability the same for both results?” (p. 532).The above tasks of recognizing equally likely cases were difficult, not onlyfor Fischbein’s subjects, but also for scholars throughout the history ofprobability. For example, even eminent mathematicians such as d’Alembertbelieved that the occurrence of 0, 1, or 2 heads in a throw of 2 coins would beequally likely (Borovcnik et al., 1991). D’Alembert failed to recognize theimportance of order or permutation in the determination of equally likely cases.Similarly, another well known mathematician, Leibniz, reasoned that producinga sum of 11 and 12 on a roll of two dice was equally likely (Glickman, 1990). Liked’Alembert, Leibniz did not recognize the importance of the order of the events.Although the recognition of equally likely cases in compound events(when two or more coins or dice are tossed together) is difficult, students’ lack offacility in the studies conducted by Fischbein and colleagues may have resultedfrom the language used in the questions. In the present researcher’s view, thelanguage of the questions in their studies was not simple, and the children didnot have an opportunity to ask questions. Similarly, students’ responses werecollected through questionnaires and the researchers did not have anopportunity to ask questions of, or discuss difficulties with, the children.The studies conducted by Fischbein and his colleagues did not producethe same results as those of Piagetian studies. In Piagetian studies, developmentof the children’s cognitive thinking was almost assured with an increase in age.In Fischbein studies, cognitive growth was not necessarily a product of age. Itdepended on the nature of the problem and the language used. Nevertheless, intheir attempts to understand children’s conceptions of mathematical structures,both Fischbein and Piaget emphasized the importance of understandingchildren’s ways of thinking.-17-GreenGreen’s studies of elementary students aged 7—11 (1988) and of secondarystudents aged 11—16 years (1983) in England were large-scale surveys devotedspecially to children’s understanding of probability. Green’s tasks for elementarychildren included a Piagetian raindrop problem, and a total of 1600 childrenparticipated in this task. In the Piagetian raindrop problem, about how 16raindrops would land on the flat roof of a garden shed divided into 16 squaresections of equal size, children’s facility in answering ranged from 31% for 7— and8—year olds to 37% for 10— and 11—year olds (Green, 1989). Although a significantproportion of young children demonstrated some understanding of randomness,their understanding did not increase substantially with age.In an earlier study, Green (1983) administered a survey to 3000 secondarystudents that consisted of items on randomness, tree diagrams, a snowflakeproblem, spinners, and Piagetian marble tasks. The results for the snowflakeproblem, which is equivalent to the raindrop problem, were particularly striking.Green found that the percent of students choosing a random distribution ofsnowflakes declined with age. Students’ facility for this item was approximately25% for 11—year olds and 20% for 14—year olds. Although Green (1989) notedthat students’ responses on randomness varied on the snowflake, raindrop, andcounter problems, he concluded that randomness was not well understood bystudents. Similarly, children’s understanding of probability as a ratio was alsoweak. For example, on a task that asked students to decide whether or not therewould be the same likelihood of picking a black marble from two separate boxesof marbles when one box contained three black marbles and one white marbleand the other contained six black and two white marbles, nearly 50% respondedsummary of results about children’s understanding of randomness from both studies isreported in Green (1989).-18-that there would be a greater chance of picking a black marble from the box withsix black and two white marbles.Like Piaget, Green concluded that children did not understand probabilityas a ratio. In addition, like Fischbein, Green found that children had difficulty inunderstanding probabilistic language such as at least, certain, or impossible.Green’s studies complemented those of Piaget and Fischbein by providing resultsfrom large samples of students.In the United States, data from the National Assessment of EducationalProgress provide some information about children’s knowledge of probability.Carpenter, Corbitt, Kepner, Lindquist, and Reys (1981) and Brown, Carpenter,Kouver, and Swafford (1988) reported the results of probability items from twoNational Assessments. Both reports indicated that students seem to have someprobabilistic intuition at an early age. However, the majority of the students didnot show a grasp of simple probability concepts in terms of ratio. For example,only about half of grade 7 students and two-thirds of grade 11 students gave acorrect answer in response to the question that asked the probability of picking ared object from a jar containing 2 red and 3 blue objects (Brown et al., 1988). Inan item involving compound events that asked the chance of getting at least onetail when two fair coins are flipped, only 5% of grade 11 students selected thecorrect response of 3 in 4 (Brown et al., 1988). Brown et al. (1988) reported that70% of the students who responded to the items selected “1 in 2”. It appears thateven grade 11 students did not go beyond an intuitive grasp of probability.Probability was found to be difficult for students in British Columbia, aswell. According to a large-scale study conducted by the Ministry of Education,Ionly 45% of grade 10 students selected the correct alternative of when asked tocalculate the probability of obtaining two heads when a person flips two quarters(Robitaille, 1990). As in the American case, 38% of students selected an incorrect-19-response of for this item. The probability of getting a total of 5 on the roll oftwo dice, was found to be even more difficult for grade 10 students. Only 15%Iwere able to select the correct response, A quarter of students stated that theydid not know the answer, and another quarter gave a response of (Robitaile,1990).Nevertheless, the results from the National Assessments and the B.C.Mathematics Assessment show only how students perform on specific items, anddo not provide enough information to make substantial claims about students’understanding of probability in general, for at least two reasons. First, therewere only a few items devoted to probability. Second, the items were multiple-choice, and students’ thought processes were not probed further.Green’s studies are more comprehensive than the U.S. NationalAssessments and the B.C. Mathematics Assessment because of the use of a widevariety of items related to students’ understanding of probability. Nevertheless,the same criticisms apply to Green’s studies. Most of the test items in his studieswere multiple choice and children’s ideas were not elicited.The studies conducted by Piaget, Fischbein, and Green indicate that thedevelopment of probabilistic knowledge in children is not linear. Rather, it iscomplex and involves children’s personal experiences as well as socio-linguisticfactors.Heuristic research on probabilityKahneman and Tversky have proposed a research paradigm called aheuristic approach. Heuristic research is designed to understand people’s intuitivebiases and cognitive representations of a particular task. Like the developmentalresearch conducted by Piaget, Fischbein, and Green, Kahneman and Tversky’sresearch emphasizes the need to understand people’s ways of thinking (Scholz,- 20 -1.991). On the basis of various studies, usually conducted with college students,Kahneman and Tversky argue that people estimate the likelihood of certainevents by using different heuristics, which are often different from thoseproposed in accepted probability theory. They refer to these as judgmentheuristics, which basically fall into two categories: representativeness heuristic(Kahneman & Tversky, 1972) and availability heuristic (Tversky & Kahneman,1973). Subsequent studies conducted by Tversky and Kahneman, and others,have confirmed the use of judgment heuristics by children and adults of differentages and abilities (Agnoli, 1987; Kelly, 1986; Shaughnessy, 1977, 1981, 1992;Tversky & Kahneman, 1982a, b). Since these heuristics have the potential forinforming our understanding of the ways in which people conceptualizeprobability, it would be useful to describe them further.Representativeness heuristicPeople applying a representativeness heuristic, base their argumentsabout how well an outcome represents the characteristics of its parent populationand whether the outcome was generated by a random process. An event A isjudged more likely than an event B if A appears more representative than B(Kahneman & Tversky, 1972). For example, in one study Kahneman and Tversky(1972) asked the following question to grade 10, 11, and 12 students in Israel:All families of six children in a city were surveyed. In 72 families the exactorder of births of boys and girls was GBGBBG5.What is your estimate ofthe number of families surveyed in which the exact order of births wasBGBBBB? (p. 432)If the judgment was made based on a formal view of mathematics, bothsequences would be rated equally likely. But 75 of 92 subjects judged that the5G=girls, B=boys-21-number of families with the exact order of birth BGBBBB was much less than 72,the median estimate being 30. Similarly, in another item, the same subjectsjudged that the sequence BBBGGG was significantly less likely than GBBGBG. Thesequence GBGBBG was regarded more likely than BBBGGG or BGBBBB for tworeasons. First, unlike the sequence BGBBBB, it has an equal number of B’s and G’s.Second, B’s and G’s appear to be randomly distributed in GBGBBG, but not inBBBGGG and BGBBBB.Similarly, in a study conducted by Shaughnessy (1981) in the UnitedStates, 50 out of 80 college undergraduate students prior to instruction inprobability said that the sequence BGGBGB would be more likely than thesequence BBBBGB. Only 2 students selected the latter sequence and 18 studentssaid that all the sequences had the same chance of occurring. Shaughnessy alsoasked students to provide written reasoning for their choices. They reasonedthat the first sequence “fits more closely with the 50:50 expected ratio of boys togirls” (Shaughnessy, 1981, p. 91).According to Kahneman and Tversky (1972), a representativenessheuristic is manifested in the so-called gambler’s fallacy or the negative-recencyeffect. As early as 1951, Jarvik identified the gambler’s fallacy which states that,in the case of two equally likely events A and B, subjects tend to predict outcomeA after a long run of outcome B. The thinking behind this fallacy is that theoccurrence of A will result in a more representative sequence than anotheroccurrence of B. Hence, chance is viewed “as a self correcting process in which adeviation in one direction induces a deviation in the opposite direction to restorethe equilibrium” (Tversky & Kahnemann, 1982a, p. 7).A belief in representativeness also leads to the “conjunction fallacy”(Tversky & Kahneman, 1982b). Tversky and Kahneman asked the following- 22 -question to undergraduate and graduate students in Israel, Canada, and theUnited States:Linda is 31 years old, single, outspoken, and very bright. She majored inphilosophy. As a student, she was deeply concerned with issues ofdiscrimination and social justice, and also participated in anti-nucleardemonstrations.Please rank the following statement by their probability, using 1 for themost probable and 8 for the least probable.6(5.2) Linda is a teacher in elementary school.(3.3) Linda works in a bookstore and takes Yoga classes.(2,1) Linda is active in the feminist movement.(3.1) Linda is a psychiatric social worker.(5.4) Linda is a member of the League of Women Voters.(6.2) Linda is a bank teller.(6.4) Linda is an insurance salesperson.(4.1) Linda is a bank teller and is active in the feminist movement.(Tversky & Kahneman, 1982b, p. 92)Since the description of Linda was constructed to be representative of anactive feminist and not representative of a bank teller, Tversky and Kahnemanexpected that the mean rank of the compound last statement would fall betweenthe mean ranks of its components. Interestingly, the result was as expected byTversky and Kahneman. What is also interesting is that many subjects who hadtaken statistics and probability courses rated it as more likely that “Linda is abank teller and is active in the feminist movement” than Linda is either a bankteller or an active feminist. The subjects, therefore, violated the conjunction ruleof probability, namely P(A and B) P(B). Some researchers, however, argue thatthe conjunction fallacy can be easily corrected. For example, Agnoli (1987)6The numbers in parentheses are the mean ranks assigned by the subjects.- 23 -claimed that even children of 11 to 13 years of age can be trained to avoid theconjunction fallacy by using Venn diagrams.Availability heuristicPeople applying an availability heuristic base their decisions mainly ontheir previous experiences of a particular instance or “by the ease with whichinstances or occurrences can be brought to mindt’(Tversky & Kahneman, 1982a,p. 11). For example, if a person has been involved in a car accident, he or she ismore likely to estimate a higher rate of car accidents than a person who has neverbeen involved in an accident (Shaughnessy, 1981, 1992). Similarly, many peoplewill say that it is more likely for an English word to begin with the letter k, forexample “key”, than to have k in the third position, for instance “like” (Kahneman& Tversky, 1972).Similarly, Tversky and Kahneman’s subjects estimated that morecommittees of 2 members can be made than committees of 8 members from agroup of 10 people. In their research, the median estimate for committees with 2members was 70 while the median estimate for committees with 8 members was20. In fact, the number of committees that can be formed out of 1.0 people, with 2or 8 members is equal. According to Tversky and Kahneman (1982a) morecommittees of few members comes more easily to mind than committees of manymembers, from the same group of people.The vast majority of Shaughnessy’s (1981, 1992) undergraduate studentsalso said that more committees of 2 members could be made than committees of8 members from a group of 10 people. Neither Kahneman and Tversky norShaughnessy reported whether their subjects had studied combinatorial logicprior to participating in their research. It would certainly be interesting to see ifresearch subjects would still continue to say that more committees of 2 members- 24-can be made than committees of 8 members from a group of 10 people afterlearning combinatorial principles.Critiques of heuristic studiesKahneman and Tversky’s insights into how people think in makingdecisions have been very useful for mathematics educators. However, theirstudies lack the clarity that would have been provided by describing the researchsetting and subjects’ characteristics in detail. Furthermore, they have beencriticized for methodological and conceptual shortcomings. There is insufficientevidence, for example, to determine whether the research subjects comprehendedthe text, how they perceived the situation, and what meanings they attached tothe concepts and tools used in the research (Scholz, 1991).Borovcnik and Bentz (1991) reviewed students’ responses to items used byresearchers such as Green and Shaughnessy. Borovcnik and Bentz (1991)explained various forms of reasoning that students might have used inresponding to items. Even when a student’s answer was correct, it was difficultto know whether the student used a correct strategy or a naive heuristic to comeup with the answer. They assert that students’ incorrect answers might be due tolinguistic and communication problems rather than lack of understanding ofprobabilistic concepts.In much the same way, Nisbett, Krantz, Jepson, and Fong (1982) arguethat, unlike Kahneman and Tversky’s studies, many probability models orproblems used in “social-psychological” studies do not have a single correctanswer. Even if an error is determined, it is difficult to know without actuallyprobing students’ reasoning processes, whether the error is because of faultyreasoning, incorrect models, or incorrect prior beliefs, even when standardstatistical problems are presented. Shaughnessy (1993) criticizes psychologicalresearch in general where “tasks are posed in a multiple-choice, forced-answer- 25 -format, where subjects’ understanding of the task and their actual thinkingprocesses are masked” (p. 72).Scholz (1991) criticized the heuristic approach to research from atheoretical perspective for not considering any connections or mutualinterdependencies among different heuristics. He proposed a “structure andprocess model of thinking’ in which subjects’ thought processes and theircomprehension of texts and tools could be determined by tracing connectionswith what the subjects had said before. In addition, connections betweendifferent heuristics could be made by inferring the “complementarity of intuitiveand analytic thinking” (p. 229). Scholz’s proposal is based on an information-processing theory which metaphorically relates the human mind to a computer.However, Shaughnessy criticized Scholz’s proposal by stating that it was solelycognitive. Scholz’s model ignores students’ subjective beliefs and the contextualsituatedness of peoples’ probabilistic thinking.Further development of heuristic studiesThe studies conducted by Konold and his colleagues considered thatpeoples’ conceptions of probability were contextual and situated in theframework and the problem setting on the one hand, and in their personalhistory and belief systems on the other. Unlike Kahneman and Tversky’sresearch questions about how people arrive at probabilistic judgments, Konoldwas interested in how people interpret probability problems.Konold et al. (1993) claimed that when given probability tasks, high schooland college students did not reason normatively; that is to say, the respondentsdid not reason according to accepted principles of probability. They conducted anumber of studies of high school and college students, and found that students’conceptions of probability were inconsistent across various tasks. Many studentswho responded correctly to multiple-choice test items reasoned incorrectly in-26 -written justifications and interview settings. Moreover, their studies indicatedthat students were not consistent even on the same multiple-choice test whenitems were phrased differently, for example changing “most likely” to “leastlikely.”According to Konold and his colleagues (1989, 1991, 1993), therespondents in their studies reasoned from an “outcome approach”. That is, therespondents believed that their task was to determine what the next outcomewould be in a single trial of an experiment rather than to estimate what waslikely to occur if the experiments were repeated several times. Participants whoreasoned from an outcome approach in Konold’s studies were using a non-statistical conception of probability (Shaughnessy, 1992).Based on a review of school and college students’ probabilistic reasoning,Shaughnessy (1992) identified four types of conceptions of probability: non-statistical, naive-statistical, emergent-statistical, and pragmatic-statistical.1. Non-statistical. Indicators: responses based on beliefs, deterministicmodels, causality, or single outcome expectations; no attention to orawareness of chance or random events.2. Naive-statistical. Indicators: use of judgmental heuristics, such asrepresentativeness, availability, anchoring, balancing; mostlyexperientially based and nonnormative responses; some understanding ofchance and random events.3. Emergent-statistical. Indicators: ability to apply normative models tosimple problems; recognition that there is a difference between intuitivebeliefs and a mathematized model; perhaps some training in probabilityand statistics; beginning to understand that there are multiplemathematical representations of chance, such as classical and frequentist.- 27 -4. Pragmatic-statistical. Indicators: an in-depth understanding ofmathematical models of chance (i.e. frequentist, classical, Bayesian); abilityto compare and contrast various models of chance; ability to select andapply a normative model when confronted with choices underuncertainty; considerable training in stochastics; recognition of thelimitations of and assumptions of various models. (p. 485, emphasis inoriginal).Participants in developmental research such as those conducted by Piaget,Green, and Fischbein and subjects in the heuristic research reported byKahneman and Tversky held either non-statistical, naive statistical, or emergentstatistical conceptions of probability. None of these researchers reportedresearch participants having pragmatic statistical conceptions of probability; thiswould indicate that the students lacked understanding of formal probability.Research on preservice teachers’ conceptions of probabilityMost of the existing research in probability has focused on elementary,secondary, and first year college students’ conceptions. Little attention has beengiven to preservice teachers’ conceptions of probability. The only study ofsecondary school mathematics preservice teachers’ understanding of probabilitywas reported by Bramald (1994). He selected the following item from Green(1982) and assigned it to his class of preservice teachers.Which of the following results is more likely?(1) Getting 7 or more boys out of the first 10 babies born in a new hospital(2) Getting 70 or more boys out of the first 100 babies born in a newhospital(A) They are equally likely(B) 7 out of 10 is more likely-28 -(C) 70 out of 100 is more likely(D) No one can say.(Green, 1982 cited in Bramald, 1994, P. 86)Bramald (1994) was quite surprised to learn that the majority of hisslitdents, from a total of 31 in his class, thought that both events, 7 out of 10, and70 out of 100 were equally likely. In further discussion, it was evident toBramald that his students, like the school students, lacked a basic understandingof formal probability. Only three students chose the correct answer that 7 out of10 is more likely than 70 out of 100. However, Bramald did not state norspeculate why students might have thought the way they did. The participantsmight not have thought about probability concepts, and might have simply used7 70the concept of equivalent fraction. Since the fractions and are equivalentthey may have decided that the statements were equally likely.Without further probing students’ thinking processes, Bramald (1994)argued that preservice teachers are “pushed too quickly into the traditional routeof manipulation of probabilities and then churning out ‘right’ answers, perhapsin the hope that understanding will follow later” (p. 86). His study providessome insights into how preservice teachers of secondary school mathematicswith reasonably strong backgrounds in mathematics think about probabilityproblems. However, his report is based on one snapshot of his classroom anddoes not provide any detail about students’ conceptions of probability.Diverse conceptions of probability: A historical accountPhilosophers and mathematicians have held different conceptions ofprobability throughout history (Daston, 1988). At present, there are four distinctconceptions of probability that are derived from its historical tradition: dassical,-29 -frequentist, subjectivist, and structural (Borovcnik et al., 1991; Hawkins &Kapadia, 1984; Kapadia, 1988; McNeill & Freiberger, 1994; Steinbring, 1991b).Classical probability was first defined by Pierri Laplace (1749—1827) as “theratio of favourable cases to the total number of equally possible cases” (Hacking,1975, p. 122). Classical probability is obtained by making an assumption ofequally likely cases and is sometimes referred to as a priori, Laplacian, ortheoretical probability. Although the credit for defining classical probability isgiven to Laplace, his definition was criticized because of the use of the word“possible,” which has a dubious meaning. Contemporary mathematicians haverefined this definition. For example, Kline (1967) expresses this as “if, of n equallylikely outcomes, rn are favorable to the happening of a certain event, the probability of theevent happening is rn/n....” (p. 524, emphasis in original).Frequentist probability was “connected with the tendency, displayed bysome chance devices, to produce stable relative frequencies” (Hacking, 1975, p.1). Some people refer to this as a posteriori, experimental, or empiricalprobability. It is defined as the ratio of the observed frequency to the totalnumber of trials in a random experiment over the long run. Mathematically, itinvolves the theory of limits and convergence.Subjectivist probability has been common since medieval times although itreceived wide recognition only in the twentieth century. According to it“probabilities are evaluations of situations which are inherent in the subject’smind” (Borovcnik et al., 1991, p. 41). Daston (1988) defines it “as the intensity ofbeliefs” (p. 188). Also known as personal probability, it assumes that humanbeings are capable of estimating the probability of certain events and makingadjustments when additional data are obtained. Mathematicians and statisticianscall it Bayesian probability named after Thomas Bayes (1701—1761).Mathematically, it is associated with Bayes’ theorem.-30 -Structural probability, like other systems of formalist approaches tomathematics, is based on a system of axioms, definitions, and theorems(Borovcnik et al., 1991). The basic axioms of probability were first developed byAndrei Kolmogorov in 1933 (Daston, 1988). According to Kolmogorov, the basicaxioms of probability are P(E)O and P(S)=1 where P and S represent an eventand a sample space respectively (Kolmogorov, 1956). Based on these axioms andother terms, the addition rule, the multiplication rule, and other rules andtheorems were developed and used to solve probability problems.Mathematicians, logicians, and philosophers have debated which of theconceptions of probability is superior. Yet, these conceptions were closelyassociated with epistemological issues concerning how probability is learned.For example, philosophers or mathematicians who believed that knowledge canbe derived deductively in a formal fashion, argued for classical and structuralconceptions of probability. Those who believed in inductive reasoningsupported the frequentist approach and those who believed in personaljudgment supported the subjectivist approach to probability.The different conceptions of probability that emerged have had significantinfluence on the teaching and learning of probability (Borovcnik et al., 1991;Hawkins & Kapadia, 1984; Kapadia, 1988; McNeill & Freiberger, 1994;Steinbring, 1991b). The subjectivist conception of probability seems to becompatible with a constructivist view of learning because of its emphasis onpeople’s updated experience and knowledge in estimating probability. Thesubjectivists claim that they can only measure the degree of a person’s belief, notan absolute truth. As in the constructivist view of learning, the subjectiveconception allows people to continually adjust their beliefs as they come to knowmore about the world. However, it is important to note that a constructivist viewof learning does not support the argument that people’s views and beliefs can be-31-updated or modified with a set of formulas as it is done in subjective probabilityby using Bayes’ theorem.Formal and non-formal conceptions of probability: Another dimensionin looking at students’ conceptions of probabilityThe historical debate about different kinds of probabilities (classical,frequentist, subjective, and structural) has influenced the research and teachingof probability. The conceptions that emerged throughout history can be used tocategorize student’ conceptions of probability. However, with the exception ofsubjective probability all these conceptions focus mostly on formal aspects ofprobability. Furthermore, a subjective approach to probability ultimately relieson Bayes’ theorem (Moore, 1990), which is perplexing even for professionalmathematicians and researchers (Falk, 1992; Shaughnessy, 1992).By focusing merely on historical views to categorize students’ conceptionsof probability, there is a danger of losing important elements related to students’conceptions of probability, such as the representativeness heuristic (Kahneman &Tversky, 1972; Tversky & Kahneman, 1982 a, b), the availability heuristic(Tversky & Kahneman, 1973, 1982a), the outcome approach (Konold, 1989, 1991),and non-statistical, naive-statistical, emergent-statistical, and pragmatic-statistical conceptions of probability (Shaughnessy, 1992).Shaughnessy’s (1992) categories attempt to make a connection amongvarious conceptions of probability. He categorized Konold’s outcome approachas non-statistical conception and Kahneman and Tversky’s different kinds ofheuristics as naive-statistical. He used the label emergent-statistical to indicatesome understanding of formal probability and pragmatic-statistical to indicate anin-depth understanding of formal probability. The difficulty with Shaughnessy’scategories is the amount of overlap among the conceptions. It is difficult todetermine whether a particular conception is non-statistical, naive-statistical,-32 -emergent statistical, or pragmatic statistical. In addition, like Confrey (1991), thepresent researcher is not comfortable labeling a student’s model as amisconception or naive conception from an expert point of view, because it failsto credit the student’s perspective. Instead of labeling students’ conceptions asmisconceptions, naive conceptions, or pragmatic statistical conceptions, analternative model of grouping students’ conceptions is proposed for the purposeof this study.This alternative model is to group students’ conceptions of probability intoformal and non-formal ones. Formal conceptions of probability will be definedas those in which students make explicit reference to and use of mathematicalconcepts and techniques. All other conceptions of probability documented in thestudy will be defined as non-formal conceptions.Formal conceptions of probability include all the elements of pragmatic-statistical conceptions of probability discussed by Shaughnessy (1992). Ifsomeone uses probability concepts and techniques directly from their school oruniversity courses they will be classified as formal conceptions. However, theuse of a term such as chance, independence, or randomness is not enough toclassify a student’s conceptions as formal ones. It depends on how students haveused those terms in solving problems. Steinbring (1991a, 1993), for example,provides examples of how meanings of probabilistic concepts differ even if thesame words are used. Steinbring draws on a constructivist perspective andargues that the meanings of chance constituted by social interactions are differentfrom the formal meaning of chance. Non-formal meanings of chance stem fromeveryday personal and social experiences, which are related to “luck, bad luck,fortune, etc.” (Steinbring, 1993, p. 20). A student expressing a non-formalconception might regard a chance event as very rare, or nearly improbable, butone that can occur. In contrast, a formal conception of a chance event would be- 33 -associated with randomness and independence and does not necessarily imply asmall probability of events (Steinbring, 1993).Non-formal conceptions are developed by individuals from theirexperiences outside formal setting of university and schools. For the purpose ofeveryday problem solving they may be even more viable than formalconceptions of probability that are generally acquired through school anduniversity education. For the purpose of this study, the outcome approach, therepresentativeness heuristic, and the availability heuristic will all be grouped asnon-formal conceptions of probability because none of these are based on schoolor university knowledge of probability. Other subjective conceptions ofprobability, such as chance, luck, and weather forecasting will also be grouped asnon-formal conceptions of probability.For the purpose of this dissertation, formal and non-formal conceptionsare not viewed as mutually exclusive. They are used for analytical purposes onlyand there are qualitatively different conceptions of probability within them.Overview of Chapter 2One of the major constraints in teaching probability in schools is students’difficulties in understanding probabilistic concepts as evident in various studies,such as those conducted by Piaget, Fischbein, Green, Kahneman and Tversky,Shaughnessy, and Konold. It is not only school and college students who facedifficulties in understanding formal probability; school teachers may also beexpected to face such difficulties. However, teachers’ understanding of formalprobability both at preservice and inservice levels has not been documented inthe literature.This researcher argues that preservice teacher& conceptions of probability,both formal and non-formal, are important for professors who have to plan and- 34-teach probability. This study provides some of the qualitatively differentconceptions of probability held by preservice teachers.-35 -CHAPTER 3Research Methodology and Procedures of the StudyThe purpose of this chapter is twofold. The first is to provide a theoreticalframework for the research methods used in the study. The second is to providedetails of the research design: specifically, development of the instruments, datacollection, data analysis, and data review procedures.Theoretical framework for the methodology of the studyThe methodology for this study derives from the constructivistperspective proposed by Guba and Lincoln (1989, 1994) and Schwandt (1994).Schwandt (1994) argues that constructivist research focuses on how meaningsare created, negotiated, sustained, and modified within a specific context ofhuman actions’ (p. 120). The importance of contexts and interactions asproposed by Schwandt is also emphasized by Guba and Lincoln who state that,‘individual constructions can be elicited and refined only through interactionbetween and among investigator and respondents.” (Guba & Lincoln, 1994, p. 111,emphasis in original).In order to strengthen continuous interaction between investigator andrespondents through context-specific tasks, constructivist-oriented researchdemands a qualitative and interpretive approach. In order to achieve this in thepresent study, the researcher used three sources: written tasks, pair-problem-solving tasks, and semi-structured performance task interviews. Consistent withthe constructivist and interpretivist approach to research, this study does notclaim to have discovered a universal truth about preservice teachers’ conceptionsof probability. Rather, it provides an analysis of their thinking about probabilitywithin the context7of the study.7Context in this study is viewed as a product of task and setting.-36 -Generalizabiity of the studyThis study uses “naturalistic generalization” or “readers’ generalization”(Lincoln & Guba, 1985; Stake, 1978, 1994) and analytic generalization (Firestone,1993; Lather, 1991; Yin, 1989). Generalizability in the naturalistic sense is “thedegree to which findings derived from one context, or under one set ofconditions may be assumed to apply in other settings or under other conditions??(Shulman, 1981, p. 8). Riecken (1989) expressed the same view by saying that thetransferability of knowledge claims made by one study “to other situations willdepend on the degree of ‘fit’ between contexts” (p. 13). The results from a studymay serve as a working hypothesis for other settings that are sufficiently congruent(Erickson, 1992; Lincoln & Guba, 1985). In this kind of generalizability, thereader compares his or her own experience with what is claimed in the research.It is the reader, rather than the writer, of the report who determines whether theknowledge claims pertaining to the study will be transferable to his or hercontext (Erickson, 1992; Shulman, 1981).In analytic generalizations, researchers attempt to generalize their findingsto a broader theory. The results from a study contribute to the theory becausethe researcher can find the weak points in the theoretical position in order torevise and extend the theory (Lather, 1991). The present study draws uponanalytical generalization and argues for the role played by tasks and settings inthe exploration of students’ conceptions of probability.Semi-structured interviewsIn semi-structured interviews the initial tasks and questions are selectedby the researcher but the phrasing of questions and their order may be altered tofit the characteristics of each participant and setting. The major characteristic ofsemi-structured interviewing is that it provides the interviewer with a substantialamount of freedom and autonomy for probing students’ thinking (Denzin, 1989;-37 -Fontana & Frey, 1994; Posner & Gertzog, 1982). Although an interviewergenerally selects the topic, it is the interviewee’s responses which determine thedirection of the interview. The initial questions of the interview are quite generaland the interviewer changes subsequent questions based on students’ responsesto the initial questions. Interviews are “guided by the prepared questions, butnot bound by them” (Rowell, 1978, cited in Posner & Gertzog, 1982, p. 199). Theinterviewer listens carefully to what the student says, and observes what thestudent does. Based on what has been heard and observed, he or she “makes andtests hypotheses about the student’s thinking” (Ginsburg et al., 1.993, p. 242).Curricular theorists such as Schwab (1989) and mathematics educatorssuch as Davis (1971) have suggested that researchers use an in-depth, semi-structured interview method if the intention of the research is to explore deeperlevels of students’ understanding. According to Schwab (1989) even if tests areused to assess students’ understanding of subject matter knowledge, they shouldbe supplemented with interviews so that the students’ modes of thought whileacting on tests can be investigated.Semi-structured interviews have been shown to be particularly effective inrevealing students’ understanding of mathematics and science content (Confrey& Lipton, 1985; Davis, 1971; Ginsburg, 1981; Ginsburg, Jacob, & Lopez, 1993;Linder, 1989; Lythcott & Duschl, 1990; Piaget, 1929; Posner & Gertzog, 1982).Since a constructivist assumes diversity of experiences among participants, he orshe examines carefully “the student’s use of examples, images, language, definitions,analogies etc. to create a mode which may well transform the interviewer’s ownunderstanding of the mathematical content in fundamental ways” (Confrey, 1991, p.115, emphasis in original).It is important that interview tasks be thought-provoking and interestingso that students will engage in the tasks (Confrey, 1991; Confrey & Lipton, 1985;-38 -Denzin, 1989; Ginsburg, 1981). According to Confrey, the tasks in the interviewsshould require more than recalling facts, rules, and procedures. She emphasizesthat a problem itself is not an important factor in probing students’ conceptions;it is only a means of creating interactions between the interviewer and thestudent.Pair-problem solvingIn this study, pair-problem solving can be considered as a special type ofgroup interviewing. In the pair-problem solving, participants work on problemswith each other rather than being asked questions by the interviewer. Theinterviewer asks questions only if the conversation between the participantsstops without any conclusion or if the interviewer is interested in some specificissues that arise in the course of the problem solving. The inclusion of pair-problem solving in the study can add all the major benefits of groupinterviewing. In particular, the construction of meanings by constant interactionand negotiation among individuals can be enhanced if the study includes bothpair-problem solving and individual interviews.The action of the interviewer or the respondent is very much shaped bythe immediate action of the other in a specific sociocultural context (Clandinin &Connelly, 1994; Eisenhart, 1988; Erickson & Shultz, 1982; Gee, Michaels, &O’Conner, 1992; Hedges, 1985; Jones, 1985a, Mishler, 1986; Woods, 1992).Because of differing sociocultural contexts, results from the individualinterviewing and the pair-problem solving can be very different. Hence, pairproblem solving contributes to the study by providing a different context forexploring the consistency of students’ subject matter knowledge.Pair-problem solving can be helpful in understanding an individual’sthinking process because talking with other people “often helps people toanalyze their own attitudes, ideas, beliefs and behavior more penetratingly and- 39 -more vividly than they could easily do if just alone with the interviewer”(Hedges, 1985, P. 73). Hence, pair-problem solving allows individuals to analyzetheir own thinking in terms of the context set by the pair.Pair-problem solving also has the advantage of making the workingsituation less strange for interviewees and thus less stressful. The fact that theinterviewer does not always need to ask questions can be considered as anotheradvantage of pair-problem solving. The respondents ask each other differentkinds of questions than those of the researcher and therefore could reveal newtypes of understandings (Denzin, 1989).Principles guiding research interviewing and pair-problem solvingThe following principles were applied in the study during the pair-problem solving and the individual interviews:1. The researcher listened more than talked (adapted from Denzin, 1989;Erickson & Shultz, 1982; Wolcott, 1990) because as Denzin (1989) states,“understanding draws upon shared experiences. Persons can’t shareexperiences if they don’t listen to one another” (p. 109).2. The researcher checked the meaning of words, phrases, or sentences ofwhich he was unsure (adapted from Jones, 1985a). The researcher wasalways sensitive to contradictory statements and asked questions to gaininsight into what the preservice teachers actually believed (adapted fromLythcott & Duschl, 1990; Posner & Gertzog, 1982).3. The researcher was not satisfied with only the right answer. Instead hetested the strength and consistency of the preservice teachers’ views byusing repetition and countersuggestion (adapted from Ginsburg et al.,1993; Piaget, 1929; Posner & Gertzog, 1982). Follow-up questions were not- 40 -intended to elicit answers, but to explore, clarify, and extend preserviceteachers’ tacit knowledge, doubts, and assumptions.4. The researcher attempted to develop “rapport, trust and friendship,sociability, inclusion” (Woods, 1992, p. 375) and showed interest inpreservice teachers’ concerns, feelings, and cognitive orientations.An illustration of these principles can be found in an extract from aninterview transcripts provided in Appendix D.Methods of data analysisThe data analysis was based upon the constant comparative method(Guba & Lincoln, 1989; Lincoln & Guba, 1985) and the interactive model(Huberman & Miles, 1994). In addition, it drew upon the probability researchconducted by Konold and his colleagues (1989, 1991, 1993) to analyticallyexamine students’ conceptions of probability.The purpose of analysis is to make sense of data by identifying patterns,categories, or themes that are meaningful for the interviewees, researchers, andpotentially useful to readers (Jones, 1985b; Eisenhart & Howe, 1992; Goetz &LeCompte, 1984; Merriam, 1985; Pitman & Maxwell, 1992). Jones (1985b)suggested that data can be organized by predetermined categories butresearchers should still remain “sensitive to the unanticipated categories whichderive from the concepts of the research participants” (p. 58). She contends that“categories do not just ‘emerge’ out of data as if they were objectively ‘there’waiting to be discovered” (p. 58). Different people, with different perspectives,construct different categories to make sense of the data (Denzin, 1994; Holstein &Gubrium, 1994; Jones, 1985b; Lincoln & Guba, 1990; Novak & Gowin, 1984;Rosenau, 1992).-41 -Jones (1985b) also suggests making connections to the concepts andtheories one is working with to “confirm, elaborate, modify or reject theme (p.59). Categories, constructs, and themes are constructed and reconstructed as newdata are collected (Eisenhart, 1988; Spradley, 1979). Since the responses from theinterviewees are “produced in the context of a developing sequence ofinteraction” (Hammersley & Atkinson, 1983, p. 193), it is important to interpretdata in light of the interview contexts and settings.After the collection of data and the construction of categories, theresearcher should go to the participants to check whether they agree with themeanings constructed by the researcher (Guba & Lincoln, 1989, 1994; Lincoln &Guba, 1985). Guba and Lincoln (1989) call this process “member checking,”whereas Hammersley and Atkinson (1983) call it “respondent validation” (p. 195).Lather (1991) supports this process of validation by arguing that the researchershould go back to respondents and discuss how the data are viewed by theresearcher “both to return something to research participants and to checkdescriptive and interpretive/analytical validity” (p. 57).Member checking allows the researcher to present his or her conjecturesand hypotheses to respondents for comments, elaboration, corrections, andrevisions (Guba & Lincoln, 1989). It provides an opportunity for the researcherto verify his or her constructions about the responses and also provides a chancefor respondents to correct errors or provide additional information andclarification about their data. Member checking also provides additionalinformation about whether respondents are consistent in their meanings aftersome period of time.Member checking can be used to clarify ambiguities and confusion thatmay have occurred during data collection. This can take a form of stimulatedrecall interviews that provides an opportunity for both researcher and-42 -participants to reflect on “uncomfortable moments” which occurred during thedata collection period (Erickson and Shultz, 1982). In addition, it helps theresearcher to understand what parts of the interviews or the pair-problemsolving were thought salient and important by the research participants.Research MethodData in this study were collected through three types of instruments: awritten task, a pair-problem solving task, and a semi-structured interview.Details of their development are provided.Development of instrumentsEach instrument’s design was based on a review of literature and theresearcher’s previous experience in teaching probability. The tasks werefinalized during field testing.Construction of tasksSeveral items used by previous researchers to examine students’understanding of probability (for example Kahneman & Tversky, 1972; Konold etal., 1993; Fischbein, 1975; Fischbein & Gazit, 1984) were reviewed, and theirapplicability for exploring preservice teachers’ understanding of probability werediscussed with colleagues and committee members. In addition to these items,several tasks designed by the researcher went through the same sort of testingand review.A total of five tasks8were selected for the written and the pair-problem-solving components and a total of nine tasks, which included two from thesecomponents, were selected for use in the interview. The written componentincluded both multiple-choice and open-ended tasks. The written tasks requiredpreservice teachers to justify their reasoning about their solutions. The interview8Tasks included written items as well as mampulatives, such as boards, marbles, dice, andthumbtacks. For the purpose of consistency everything will be called a task.- 43 -included tasks representing a variety of contexts such as coin tossing, caraccidents, lottery winning, spinning a spinner, rolling thumbtacks, and dicethrowing. These tasks were selected because they were deemed to be useful bythe researcher in exploring preservice teachers’ conceptions of probability andthe results could be compared with the results obtained by previous researchers.Field testsA formal field test of the tasks assigned to the written, the pair-problemsolving, and the interview components was carried out with two mathematicseducation graduate students at the University of British Columbia in May of994. Both graduate students had majored in mathematics and physics for theirbachelor’s degrees. Both had taken a probability course either at high school oruniversity. The graduate students were chosen for the field test because theywere readily available and had backgrounds in mathematics and probabilitysimilar to the preservice teachers who would participate in the study.The field test of each component was conducted with these graduatestudents. The written, pair-problem solving, and individual interview sessionstook approximately 35, 1.5, and 90 minutes respectively. Each pair-problem-solving session and interview was audiotaped, transcribed, and analyzed.Both students found the tasks in the written component to be fairly clearand gave similar responses to each other. Thus the written component did notundergo major revision. In the pair discussion, however, both students agreedwith each other and insufficient interaction took place. The pair-problem solvingwas changed to encourage more interaction. Some tasks in the interview werefound to be lengthy and confusing for both students and these tasks were eitherrevised or eliminated from the interview.-44-Criteria for selecting tasks for pair-problem solving and interviewTwo criteria were used in selecting tasks for the pair-problem solving.First, the written and the pair-problem-solving components were not to beidentical although some tasks could be repeated to explore students responses indifferent social dynamics. Second, the tasks had to be interesting andchallenging to the graduate students.The tasks for the interview were developed based upon four criteria. First,the interview tasks had to be interesting enough to engage the interviewees in asustained conversation to reflect on their understanding of probability. Second,the contexts of the interview had to vary from verbal mathematical problems tomanipulatives and games. Third, the tasks could not be too hard or too easysince this might hinder interviewees’ thinking processes. Fourth, the interviewsession had to last for a reasonable duration from both the interviewee’s andinterviewer’s perspectives: about 45—60 minutes.Description of tasksThe written component and the pair-problem-solving component eachconsisted of five tasks. Description of tasks is shown in Table 1.The first task in the written component was called the birth sequence task,and asked participants to select the most likely sequence of births in a family.This task was similar to the head-tail sequence task, in the pair-problem-solvingcomponent, wherein participants were asked to determine the most likelysequence of alternatives that might result from ffipping a fair coin six times. Thesecond task in the written component, the rectangular spinner task, askedpreservice teachers to decide the most likely section where the spinner wouldstop. The third task in the written component, the five-head task, requiredpreservice teachers to decide the sixth outcome if the previous five outcomes-45 -Table 1. Written and pair-problem-solving tasks.Written tasks Pair-problem solving tasksBirth sequence task Head-tail sequence taskRectangular spinner taskFive-head task Five-boy taskLottery and car accident Lottery and car accidenttask taskCircular spinner task Circular spinner taskBank teller taskwere all heads when a fair coin was tossed. The five-boy task in the pair-problem-solving component required the same reasoning. The fourth task in thewritten component or the third task in the pair-problem solving, the lottery andcar accident task, required participants to decide whether they were more likelyto win a jackpot in the B.C. lottery 6/49 or be killed in a car accident. The finaltask in the written component or the fourth task in the pair-problem solving, thecircular spinner task, required preservice teachers to select two numbers out ofeight so that they could have the best possible strategy to win the game. Thefinal task in the pair-problem solving, the bank teller task, required participantsto determine the number of tellers needed if one to six customers randomlyarrive at the bank (see Appendix A and B for further details).There were five tasks in the interview component, the head-tail sequencetask, the lottery task, the board and marble task, the dice task, and the thumbtacktask. The first task, the head-tail sequence task, was repeated from the pairproblem solving to explore changes in participants’ conceptions from the pair tothe individual setting.- 46 -The remaining four were new. In the second task, the lottery task, thepreservice teachers were asked to resolve a conflict between two students whohad opposing views about selecting numbers for a lottery 6/49. In the third task,the board and marble task, the participants were shown a wooden board with asystem of channels progressively branched toward the base of the board andwere asked where a marble dropped from the main channel, would exit at thebottom. In the fourth task, the dice task, the participants had to choose whetherthey wanted to be a player or a dealer in a dice game (sometimes called a chucka-luck”) in which three dice are rolled to determine whether the player or thedealer wins the bet. In the fifth task, the thumbtack task, the preservice teacherswere asked whether a thumbtack will point up or point down when it is droppedor rolled. Finally, preservice teachers were asked to provide their opinions as towhether they found university or school knowledge of probability useful insolving everyday problems (See Appendix C for details of the interviewprotocol).It is clear from the above description that some tasks that required thesame sort of mathematical reasoning were included within and across settings inorder to determine how preservice teachers’ thinking would be similar ordifferent across tasks and settings. The tasks and their relation to each other inwritten, pair-problem solving, and individual interview are shown in Table 2.Preservice teachers required knowledge of randomness and independenceto come to a mathematical solution in the birth sequence task and the five-headtask in the written component. The same reasoning was required in the head-tailsequence task and the five-boy task in the pair-problem solving. Knowledge ofrandomness and independence were also required for the head-tail task, thelottery task, the board and marble task, and the dice task in the interview setting.- 47-Table 2. Mathematical concepts associated with the tasks.Tasks Written Pair-problem InterviewsolvingBirth sequence task R, I, 0Rectangular spinner task R, I, SFive-head task R, ILottery and car accident C, QtaskCircular spinner task E, AHead-tail sequence task R, I, 0Five-boy task R, ILottery and car accident C, QtaskCircular spinner task E, ABank teller task R, I, SHead-tail sequence task R, I, 0Lottery task R, I, CBoard and marble task R, I, PDice task R, I, 0, MThumbtack task R, I, SLegend:A = Addition rule C CombinatoricE = Recognition of equally likely cases I = IndependenceM = Multiplication rule 0 = Recognition of orderP= Physical symmetry Q = Probability as a ratioR = Randomness S= Angles in sectors-48-Some tasks required an understanding of the addition rule and the multiplicationrule of probability. The repeated tasks within and across different settingshelped the investigator to probe further how each preservice teacher thoughtabout probability in a different context.Data collection proceduresAfter the development of the instruments, data were collected from twogroups of preservice teachers of secondary school mathematics in the summerand fall of 1994 at the University of British Columbia. At the beginning of thesedasses, the researcher told preservice teachers the purpose of the research anddistributed a student consent form. The data collection began as soon as thepreservice teachers consented to participate in the study.Conducting written tasksAll preservice teachers, nine from the summer group and 31 from the fallgroup, who wished to volunteer were asked to complete the written componentof the study. The purpose of the written tasks was to identify preservice teacherswho had diverse ways of thinking about probability. The participants wereasked to provide some background information such as the number of courses inmathematics, statistics, and probability they had taken and what their majorsubject was. They were requested to provide accounts of their reasoning for thesolutions of open-ended tasks. The written responses were analyzed for a rangeof opinions regarding participants’ beliefs, knowledge, intuitions,understandings, and experiences related to probability.The responses provided by the preservice teachers in the writtencomponent were mostly consistent with the mathematics of probability. Out of40 who participated in the written component, 32 said that all four options(BGBBBB, BBBGGG, GBGBBG, and BGBGBG) were equally likely in the birth sequence-49 -task. Similarly, 34 out of 40 chose the mathematically correct response that thespinner would stop at section 3 because it had the greatest central angle. In thefive-head task, most preservice teachers (34 out of 40) responded that theoutcome heads or tails was equally likely to come up in the sixth trial even ifthere were all heads in the previous trials. The same proportion (34 out of 40)stated that it was more likely that they would be killed in a car accident ratherthan winning a jackpot in the lottery 6/49. Among all the written tasks, thecircular spinner task evoked the widest variation. Out of 40 preservice teachers,22 stated that they would choose any two numbers because all of them had thesame probability of winning; eleven preservice teachers argued that they wouldchoose numbers that were side by side. Five stated that they would choosenumbers that were opposite each other.Although participants’ responses for all tasks were mainly formal in thewritten component, their justifications of the answers varied. These variations,both in answers and in justifications, were considered in the selection ofparticipants for the pair-problem solving and interview components.Selection of preservice teachers for pair-problem solving and interviewThree criteria were used to select preservice teachers for the pair-problem solvingand individual interviews: that the preservice teachers had taken at least 10mathematics courses in their undergraduate degrees, that they had taken someuniversity or secondary school probability courses or had some backgroundknowledge in probability, and that the responses they provided to the tasks inthe written component differed from one another’s.A total of 16 preservice teachers, 4 from the summer group and 12 fromthe fall group, were selected for the pair-problem solving and the individualinterviews (see Table 3 for their mathematics and probability backgrounds).- 50 -Table 3. Background information of preservice teachers selectedfor the pair-problem solving and the interview.No.of No.ofuniversity university Year themathematics probability probabilityAcademic courses courses courses werePseudonyms qualifications taken taken takenAlan BSc Elect. Eng. 10 1 2nd, Grade 12MAAnne Math/German 10 2 2ndBABill Math/Psychology 20 3rdFeng BASc Mech. Eng. 10 1 3rdGary BSc Math 18 1 3rdGita BSc Math/Biology 12 1 3rd, Grade 12Ivan BA Math/English 18 1 3rdBSc 3rd (Part of aJane Chemistry/Math 12 1 chemistrycourse)BSc Math/Comp.John science 16 3 2nd, 3rdLisa BSc Stat 16 1 2ndMary BA French/Math 14 3 2nd, 3rdMate BSc Math 14 3 2nd, 3rdPaul BSc Math 17 2 3rdReid BSc Physics/Math 10 2 3rd, Grade 12Ruby MSc Math 25 2 2nd, 6thRuth BA Math 20 2 3rd, Grade 12-51-Eight pairs of preservice teachers, two from the summer group and sixfrom the fall group, were grouped by the investigator. Each pair consisted ofpreservice teachers with diverse backgrounds and opinions so as to produceinteractions that would help broaden the exploration of students’conceptualization of probability. For example, Gita and John were pairedbecause Gita used mathematical reasoning in the written component whereasJohn used elements of the representativeness heuristic. In another pair, Ruby’sreasoning was based on mathematical concepts while Ruth’s reasoning movedbetween mathematical and everyday reasoning.This kind of pairing was not always possible because there were morestudents who demonstrated mathematical reasoning than everyday reasoning.In such cases, pairs were made using other criteria, for example the number ofmathematics and probability courses they had taken at university and their majorsubjects. The pairs are shown in Table 4.Procedures for pair-problem solving and interviewThe main purpose of the pair-problem-solving component was to gaininsights into preservice teachers’ probabilistic thinking in a different socialsetting from that of the individual setting of the written and the interviewcomponents. The main purpose of the individual interview was to probe ingreater depth participants’ beliefs as revealed in the written tasks and pair-problem solving, and to explore further their conceptions of probability, whileremaining open to the possibility that new conceptions might emerge.In the pair-problem solving, participants were asked to discuss possiblesolutions to tasks. That is, pairs discussed and attempted to solve tasks together.The investigator observed the pairs’ work and made notes. Only in certaincircumstances, for example if pairs stopped talking or reached an agreement veryquickly, did the investigator provide prompts. The discussions were audiotaped.-52 -Table 4. Formation of pairs.Pair No. Participants and relevant selection information1 Gita (mathematical)John (elements of representativeness heuristic)2 Feng (mathematical, mechanical engineering graduate)Reid (mathematical, physics graduate)3 Alan (Elements of gambler’s fallacy)Ivan (mathematical and everyday reasonings)4 Anne (conflict between mathematical and everydayreasonings)Gary (mathematical)5 Bill (mathematical and everyday reasonings)Mate (mathematical)6 Jane (elements of representativeness heuristic)Paul (mathematical)7 Lisa (elements of representativeness heuristic)Mary (conflict between mathematical and everydayreasonings)8 Ruby (mathematical)Ruth (conflict between mathematical and everydayreasonings)Each tape of the pair-problem solving was analyzed to make sense ofpreservice teachers’ thinking about probability problems in a pair setting. Noteswere made by the researcher during this analysis. Each interview was guided bythese notes and the interview protocol (see Appendix C) prepared by theinvestigator. In the interview, respondents were asked to read tasks and providetheir opinions. Subsequent questions were based on respondents’ initialreactions to the tasks,-53 -During the interview, each preservice teacher was allowed to askquestions related to the tasks. They were also allowed to perform experimentswith marbles for the board task, roll dice to answer questions on the dice task, orroll thumbeacks to answer questions on the thumbtack task. They were alsorequested to think aloud so that their ideas could be audiotaped. Theinvestigator also made observation notes immediately after each interview wasconcluded, These notes were added into the data transcripts.Data transcriptsBefore the tapes were transcribed, the researcher listened to them to get asense of the pair-problem solving and interview sessions. It was noted that therewere repetitions of words and sometimes sentences in the tapes. In some casesthere were personal conversations with students unrelated to probability. Exceptfor those repetitions and personal conversations, all other portions of each tapewere transcribed. However, words such as “ya”, “huhuh”, “right”, “you know”,“well”, “um” etc. were not transcribed unless they were meaningful in thecontexts of the interview and problem solving. The researcher also added hisnotes on participants’ interviews or the pair-problem solving to thetranscriptions. In some cases notes were added to the transcripts directly fromstudents’ written work during the interview or the pair-problem solving.Data analysisAudiotapes from the pair-problem solving and the individual interviewswere transcribed together with the researcher’s summary of the importantconcepts and patterns that appeared (Hedges, 1985). A summary of eachpreservice teacher’s responses to the pair-problem-solving tasks and theinterview tasks was presented in words and sentences in large matrices. Asimilar summary of participants’ responses to the written tasks was prepared.-54-These summaries were then used to categorize participants’ responses intoqualitatively different conception of probability within formal and non-formalconceptions.Conceptions were called formal if there was a clear indication of explicitreference to and use of mathematical concepts. Conceptions were called non-formal if there was no explicit reference to and use of mathematical concepts.Examples of preservice teachers’ formal conceptions of probability includethe following:1. Use of a mathematical formula, for example, the probability of winning a49,jackpot m the lottery 6/49 is(49 — 6)! x 6!2. Use of a mathematical technique, for example, the probability of gettingany given sequence in the head-tail sequence task is because theprobability of getting each event is and there are 6 events that areindependent of each other.3. Use of a mathematical idea, for example, the spinner in the rectangularspinner task is most likely to stop in number 3 because the section withnumber 3 has the largest central angle and the spinner is spun a largenumber of times.4. Use of an experiment, for example, to determine whether the thumbtack isgoing to land point down or point up one should throw or roll thumbtackat least 30 times and calculate the relative frequencies. That is the ratios ofpoints up and points down with the total number of trials.5. Use of a mathematical reasoning, for example, “Any combination of sixnumbers is exactly as likely as any other combinations of six number inthe 6/49.”-55 -The conceptions which could not be determined as formal werecategorized as non-formal. For example:1. The participants’ references to their experiences of playing or observinggames, observing customers at bank, predicting outcomes based on luck,and so on.2. The use of representativeness heuristic, that the sequence THTHHT, in thehead-tail sequence task, were most likely because the number of headsand tails were equal, and the events in the sequence were random.3. The use of knowledge from disciplines other than mathematics.In certain cases, however, when it was unclear as to whether theirresponses fell into either the formal or non-formal category, responses werecategorized as jointly formal and non-formal. For example, in the head-tailsequence task, if preservice teachers recognized that the previous five outcomesdid not influence the next outcome because the ffip of a coin was independentand random, then they were categorized as a combination of formal and non-formal conceptions. This response might be from their learning of independenceand randomness in school, university, or everyday experiences with coins.In some cases, the preservice teachers responses to certain tasks werecategorized as a conflict between formal and non-formal conceptions because theparticipants indicated that they were not sure as to which reasoning they shouldapply for solving the task. For example, in solving the lottery task, a responsewas categorized as a conflict between formal and non-formal, if a participantstated that intuitively a random sequence of numbers was more likely than aconsecutive sequence of numbers to win the jackpot in the lottery 6/49, butmathematically any six numbers was equally likely to win it.-56 -The categories were then interpreted in terms of the following questions:What are the similar or different ways in which the preservice teachersconceptualize probability in the written, the pair-problem solving, and theinterview settings? How do preservice teachers’ responses vary from one task toanother in the individual and pair settings? How does one participant modify orelaborate his or her conceptions of probability with respect to anotherparticipant’s conceptions? What are participants’ views about the role ofacademic probability in their everyday lives? Based on these questions theresearcher wrote his interpretations of each preservice teacher’s probabilisticthinking.Data review and validationTwo methods were used for reviewing and validating the data in thestudy. First, an opportunity was provided for each preservice teacher tocomment on his or her transcript. Second, an opportunity was provided for eachparticipant to discuss the researcher’s interpretation of his or her conceptions ofprobability.Discussion of transcripts with preservice teachersTranscripts for both the pair-problem solving and the individualinterviews were given to the respective preservice teachers for their commentsand elaboration. They were asked to clarify words, phrases, or statements thatwere not understood by the researcher. The participants were allowed to modifycertain statements.The preservice teachers returned the transcripts together with theircomments within a period of one to two weeks. Some preservice teachers didnot make any comments and agreed fully with what was written in thetranscripts. Some others only made editorial comments. A few made substantial-57-comments and clarified their statements during the pair-problem solving and theindividual interviews. The modifications made by the participants helped theresearcher to determine their views of probability further.Discussion of researcher’s interpretation with preservice teachersThe researcher’s interpretation of each participant’s probabilistic thinkingwas distributed to that person for comments and criticisms. Everyone provideda response to the researcher’s interpretations.Some preservice teachers were happy with the researcher’sinterpretations, although they commented that they may not have come to thesame conclusion as the researcher, Some others wanted some clarification aboutthe researcher’s interpretations of their thinking process. Only two preserviceteachers disagreed with some of the researcher’s interpretations of theirprobabilistic thinking. The reasoning they provided helped the researcher toprobe their conceptions further. The final version of the analysis, interpretations,and conclusions of this study fully incorporates the preservice teachers’comments on the researcher’s interpretations.-58-CHAPTER 4Preservice Teachers’ Conceptions of ProbabilityThis chapter presents the analysis and interpretation of data for the firstresearch question:What are some qualitatively different ways in which preservice teachersconceptualize probability?In order to answer this question, the qualitatively different conceptions ofprobability that the preservice teachers demonstrated during various tasks andsettings of the study are discussed. The conceptions described in the study arederived from 16 preservice teachers responses on the written tasks, the pair-problem-solving tasks, and the interview tasks.Qualitatively different conceptions of probabilityBased on the analysis of data obtained from the various tasks and settings,several themes are developed regarding preservice teachers conceptions ofprobability. These themes are grouped into formal and non-formal conceptionsof probability. The themes and the associated conceptions are presented asdistinct for analytical purposes only.Preservice teachers’ formal and non-formal reasoning about all tasks in thewritten, pair-problem-solving, and interview components are shown in Table 5.Table 5 shows that the preservice teachers’ responses varied based on tasks. Allthe participants, for example, used formal reasoning in the rectangular spinnertask. Similarly, the majority of the participants reasoned formally in the birthsequence task and the lottery and car accident task in the written component, andthe five-boy task and the circular spinner task in the pair-problem-solvingcomponent. In the dice task, however, the majority of them either reasoned nonformally or provided conflicting reasoning between formal and non-formal-59 -Table5.Preserviceteachers’formalandnon-formalconceptionsofprobability.BST=Birth-sequencetaskFHT=Five-headtaskCST=CircularspinnertaskFBT=Five-boytaskLT=LotterytaskDT=DicetaskRST=RectangularspinnertaskLCT=LotteryandcaraccidenttaskHTI’=Head-tailsequencetaskBfl=BanktellertaskBMT=BoardandmarbletaskTT=ThumbtacktaskF=FormalthinkingN=Non-formalthinkingFNFormalandnon-formalthinkingwithoutconflictF<->NFormalandnon-formalthinkingwithconflictWrittenPair-problemsolvingInterviewBSTRSTFHTLCTCSTHTLaFBTCSTBflHflLTBMTDTflAlanFFNFNNFNNFFNFF*->NFNNFNAnneF4-*NFFeNFF<-*NFFNNFFFNFeNF÷-*NFNNFNBillFFFFNFNFFNFFFNFFFNF<-*NFNFengFFFFFFFNFFFNFFFNFNGaryFFFNFFFFNNFFFNFFFF<->NFNGitaFFFNFFFFNFFFNFFFFFNIvanFFFNNFNFFNFFFNFFFNFeNFNJaneNFNNFFeNFNF<-,NFFNF<-,NFeNFNNFNJohnFeNFFFFF<->NFNFFNFNF<-*NFeNFNF<->NFNLisaNFFNFF<->NFNFFFNFeNFNNFNMaryFeNFFNFFF<-*NFNFFFFF<->NFNF<->NFNMateFFFFFFFNFFFNFeNFFNFeNFNPaulFFFFFFFFFFFFFFFNReidFFFFFFFFFFNFFFFFNRubyFFFFFFNFNNFFFNFFFNFeNFNRuthFFFNFFeNFNFNFeNFeNFNFeNFeNFNFeNFNLegend:—60-conceptions. Their responses were mixed with formal and non-formal reasoningin the five-head task during the written component, the lottery and car accidenttask and the bank teller task during the pair-problem-solving component, andthe board and marble task, and the thumbtack task during the interviewcomponent. Unlike the dice task, the participants did not demonstrate a conflictbetween formal and non-formal thinking during these tasks.It appears that the preservice teachers provided different types ofresponses to different tasks. If the task could be solved directly by mathematicalrules or formulas, they applied formal reasoning. If the task appeared to be froman everyday context, and no mathematical rules or formulas came to their mind,they used non-formal reasoning. For example, they demonstrated a formalconception of independence and randomness on the rectangular and circularspinner tasks which were similar to textbook problems. But many of the sameparticipants did not demonstrate a formal conception of independence andrandomness on the dice task which was associated with an everyday context.Qualitatively different conceptions of probability demonstrated by theparticipants within their non-formal and formal thinking are discussed below.Non-formal conceptions of probabilityAlmost all the preservice teachers at some point interpreted probabilitybased on their observations of how things worked in their everyday lives. Threeconceptions—using everyday experiences, using the representativeness heuristic,and using science knowledge in solving probability problems—are discussed inthis section.Using everyday experiences in solving probability problemsExperiences gained by people in different facets of their everyday lives,such as playing or observing games in casinos, playing lotteries, observing the-61-flow of people in banks, and making everyday decisions are considered aseveryday experiences for the purpose of this analysis. The experience thepreservice teachers gained in university mathematics or probability courses arenot considered everyday experiences. The preservice teachers’ use of theireveryday experiences was particularly evident in the dice task during theinterview component and the bank teller task during the pair-problem solving.Preservice teachers also used their everyday experiences in the lottery and caraccident task, which was asked during both the written and the pair-problem-solving components.According to the mathematics of probability, the dice task can be solved inthree ways: (1) by calculating the expected value for a player, which comes toabout a loss of 8 cents for every dollar; (2) by calculating the probability that aplayer will win, which is about 42%; and (3) by calculating the probability that adealer will win, which is about 58%. If someone correctly uses his or her schoolor university knowledge of probability, he or she would come up with thesolution that the dealer has the advantage.Although the majority of the preservice teachers attempted to solve thedice task formally in the beginning, most of them abandoned their attempts andused everyday experiences to decide whether they would want to be the dealeror the player. All the respondents knew the probability of simple events, for1example that the probability of getting a number m the roll of a die isHowever, most of them had difficulty in finding out the probability ofcompound events, such as the probability of getting the same number in two orthree dice simultaneously. As a result, only three of 16 participants usedappropriate mathematical strategies in solving the dice task; all the rest eitherattempted to solve it by using their everyday experiences or by usinginappropriate mathematical strategies.-62-While attempting to solve the dice task, some preservice teachers usedtheir personal experiences of playing or observing dice games and similar gamesin the real-world. For example, Anne stated that she would want to be the dealerbecause “most gambling casinos seem to win” and John stated that it is thedealers who eventually win money. Lisa said the same thing by stating “whenyou play cards with your friends the chances are a lot of times for the dealer towin.” Ruby stated the following from her experience of observing casinos:Well, I’ve not gambled but I’ve been to casinos just to watch people. Ithink it’s fun to watch them. They get really excited with games and theykeep betting higher and higher amounts of money. Most of them couldn’tpay any more because they lost it. Of course I could be unlucky but mychances are better if I’m the dealer.Based on her experience of observing games in casinos, Ruby was prettysure that the dealer was going to win. At her first sight of the task, she was notconcerned with calculating the mathematical probability of the dealer or theplayer of winning.Ruby was also drawing on the issue of luck when she stated “I could beunlucky but my chances are better if I’m the dealer.” Her reasoning implied thatluck plays a very important role in gambling. She thought that probability alonecannot be used to predict what is going to happen. The issue of luck, she wasusing in solving this problem, was drawn from everyday experiences and notfrom formal experiences received during university mathematics and probabilitycourses.Similarly, Mary and Ruth argued that winning $100 depends very muchon luck. Mary thought that if she happened to bet a large amount, say $10, at thetime when her number turns up then she would win. Moreover, if she were-63 -lucky, the same number could turn up in two dice or even in three dice. It isimportant to note that Mary was not confusing chance with luck. She wasdifferentiating between chance and luck by stating that “the chances are that oneout of 6 times that I roll this dice it is going to be a 6. Luck is choosing the timeit’s going to be.” If she were lucky, Mary thought, she could still win the moneyfrom the dealer even if the dealer had a higher chance of winning. Ruth was alsoconcerned about luck. She stated that “if you win on the time when it’s yoursand you win like $20. And you lose on the other 5 times it’s only $1; then youlose $5.” Then she thought how lucky she would be to win this large an amount.Her solution to the problem ultimately relied on luck. Hence, Mary’s and Ruth’sfinal decisions to be the player were not based on the calculation of probabilitybut on a feeling they developed from their everyday experiences.Anne’s and Lisa’s view was that probability should be balanced with riskand control. Anne, for example responded to the dice task as follows:The dealer has to risk 3 times in ... every roll of the three dice. He’s riskingthree times the amount of money that the player bets. [Thinking] Yeah,the dealer takes greater risk. But ... it favors the dealer in terms of theother five numbers coming. So it’s just a matter of evaluating whether thatends up putting a balance between player and dealer or whether or not itstill favors the dealer. For me, in a situation like this, like I said, comesdown to also other factors that don’t have to do with necessarilyprobability, but just where I can see factors that I can control [laughs].Anne’s decision to be the player in the game was not because shecalculated probability or the expected value mathematically but because shecould control the amount of the bet as a player. Anne did not worry too muchabout the probability calculation because as a player she had the power of-64-controlling the bet. Like Anne, Lisa also talked about control. However, unlikeAnne, Lisa stated that she wanted to be the dealer because “the possibility iscontrolled by the dealer.” According to Lisa, the landing of the dice depends onhow you throw the dice. It is notable that both Anne and Lisa showed theirawareness that what they said was contradictory to the mathematics ofprobability. Nevertheless, they were quite happy in solving the problem withthe feelings they had developed in their everyday experiences.In the bank teller task, however, all pairs of preservice teachers attemptedto use their mathematical knowledge as far as possible. Usually, they made atable or a diagram to look at the flow of customers arriving at the bank. Manypreservice teachers looked at a worst possible scenario of six customers arrivingper minute and proposed four tellers or five tellers to serve customers. However,at this point the reliance upon pure mathematical knowledge was suspendedbecause they immediately questioned their solution in terms of cost efficiency. Abank manager in a real life situation would try to reduce the cost of banking asfar as possible. Employing five tellers at the bank would not be a cost-effectivesolution.When the preservice teachers attempted to think of a new strategy to solvethe task they were distressed by the fact that 1 to 6 customers could arrive at thebank. There was no way to know how many customers would arrive at aparticular minute. Although they stated that they could use the concept ofrandomness from their probability courses, they decided that the number ofcustomers arriving at the bank is not random in a real life situation. Hence, theyfelt that using a simulation did not provide a useful solution to the problem. Forexample, Ivan and Alan had the following discussion about the task:Ivan: Right, you don’t know. But again if you look at other things ... tosolve a math equation or to make this a math question the average-65 -rate is three and a half. Overall of time it’s three and a half.... For aparticular second or for a particular hour you might have lots ofpeople. I mean if you look at when banks are busy—in theafternoon and in the evening. And so if you really want to be moretechnical you want less tellers in the morning, less tellers twoo’clock in the afternoon, more on lunch, and more on evening.More at three o’clock. Basically three and a half people coming in aminute is what you’re working with.Alan: These are interesting questions because it says the numbers ofcustomers arriving varies between 1 and 6 customers per minutebut it doesn’t say what the distribution of those is over time.... Wemade a simplifying assumption which is that the numbers 1 and 6are evenly distributed over time. There could be periods whereyou get six customers a minute or an hour. That information is notgiven to us.Ivan and Alan used their everyday experience of observing customers atdifferent times and decided that the distribution of customers arriving at thebank is not uniform throughout the day. They stated clearly that they decided towork with the average number of customers to come to a mathematical solutionwhile knowing that the solution might not be useful in real life situations. Ivanand Alan gave a clear message that they make many simplifying assumptions inmathematics that may not be helpful in dealing with the complexities of real lifesituations. Anne and Gary had a very similar opinion on the problem.Most pairs were not satisfied with their mathematical solution to the taskespecially because their everyday experiences differed from the mathematicalsolutions. Some participants argued that they would not go for a mathematicalsolution to the problem because it would not provide an appropriate-66 -representation of a real life situation. However, preservice teachers werecomfortable with building a mathematical model through observation of a reallife situation in order to arrive at a more useful solution to the problem. Forexample,In a real life situation, if I was the manager I’d ... observe it for a couple ofdays, record the observation of waiting time and decide according to that.I won’t bother to make this so complicated. (Gita with John)In the real world, it’s going to be more complicated. The only way toassign the number of tellers is by observing what actually happens fromday to day. Even if you decide once, you have to make changes from timeto time. (Reid with Feng)The above pairs argued that the solution to the problem becomes moremeaningful when the decision is made by actually observing the number ofarrivals at the bank for some days. They argued that the number of customersarriving at the bank at a particular minute is not random. Assuming the arrivalto be random is risky for making good practical decisions. The preserviceteachers were arguing that everyday experiences are more important than themathematics of randomness in solving this task.Similarly, the majority of the participants used their everyday experiencesin the lottery and car accident task asked in the written and the pair-problemsolving components of the study. For example, Lisa stated that there is a lot ofnews on various media about car accidents and deaths. But there is hardly anynews about the people who win the jackpot in the 6/49 lottery. Hence, for Lisa,the chance of being killed in a car accident was much higher than winning ajackpot in the 6/49 lottery.-67-However for some of the preservice teachers, like Feng, mathematicslearned at the university helped to come to a solution to the problem. In thewritten component Feng stated that “the probability of winning a [jackpot] in6/49 is about 1/14000000. I am sure the rate of a fatal car accident is muchhigher than that,” She knew that the probability of winning the jackpot in theB.C. lottery 6/49 is approximately 1 in 14 million. Nevertheless, she used hereveryday experience to decide that the ratio of being killed in a car accident inB.C. was much higher than 1:14000000.Some of the preservice teachers; like Anne, Bill, Ivan, Jane, and Mary,disregarded probability calculations for the lottery and car accident task. Theystated that it was more likely that they would be killed in a car accident becausethey did not buy lottery tickets but they did drive cars. For example, Jane statedthe following when she was working with Paul:Paul: First of all we have to find out the probability of being killed in acar accident.Jane: I didn’t have much problem with this question.... I don’t playlottery 6/49. So the probability of me ever winning a jackpotwithout buying a ticket is not going to happen.Jane was quite satisfied with her answer. But Paul argued that theyshould use mathematics to come to a decision rather than simply avoiding it.Paul stated that they could make assumptions to solve the problem, such asbuying one lottery ticket per week. However, Jane indicated several problemssuch as:Jane: You’ve to drive a car once a week too or walking.... To me theopporti.rnity should be equal in both cases in order to be able tocompare ... Again that would depend on how many people buy the-68 -tickets and ... how many people are driving cars and how manystreets you’ve to cross..., Actually you have to know moreinformation.Paul: Yeah. There’re several interpretations for this question.... We haveto make some assumptions such as the probability of one personbeing [killed] in a car accident. You have to assume that ... eachindividual in British Columbia assumes equal risk. Which is nottrue, so it’s a wrong assumption. However, we have to startsomewhere. The second you’re born you have a continualprobability of “whatever,” throughout till the time you die.Clearly, the task was not solvable by a simple mathematical model. Paulbelieved in making assumptions if the tasks were not well formulated. Heattempted to convince Jane that since they were mathematics graduates theyshould be providing formal solutions even if the solutions were unrealistic. Gitaand John went through the same sort of discussion. Gita stated that having a caraccident depends on “whether you drive or not? What time of the day youdrive? Probability of being killed if you are in an accident,” and so on. Johnsupported her and argued that they could not provide a formal solution to theproblem that would be meaningful in the everyday world. Even after a solutionwas offered, the preservice teachers suspected that the solution would notactually reflect what would happen in the everyday world.In some other tasks, for example, the circular spinner task, the board andmarble task, and the thumbtack task, some of the preservice teachers used theireveryday experiences and not the mathematics of randomness. Ruth, forexample, had the view that probability could be controlled by people. In thecircular spinner task in the written component of the study, Ruth stated that the-69 -landing of the spinner depended on the amount of force applied to the spinner.She stated,I would pick 4 and 5 because if the spinner is not spun hard enough, itmight not go around to 6, 7, and 8. It is very likely that it would get past 1,2,3 even with a very light spin. (Ruth in the written component)Ruth, however, made it clear that “if it was going to be spun hard, thenany two pairs of numbers would have an equal chance of winning.” Ruth’sthinking was similar in the thumbtack task. She thought that the way in whichthe thumbtack lands depends on the height of the drop. She stated that the tackwould be more likely to land point up if it was dropped from a larger heightbecause the flat side of the tack was heavier.Like Ruth, Ivan thought that results of experiments could be controlled bypeople. In the board and marble task he sent the marble in any number 1,2,3, or4 he wanted, most of the times. He controlled the movement of his finger andstated that he was studying the bounces made by the marble. He argued that hewould be more comfortable if the experiment was replaced with a computersimulation. Ivan argued that when experiments are manually conducted byhuman beings, the outcomes cannot be purely random. Gary also stated that themovement of the marble could be changed by changing the place where it wasdropped from and the initial force applied to it. In all these examples, thepreservice teachers were stating that there are many human factors involved indetermining probability, and so simple mathematical solutions could not beuseful for everyday purpose.As discussed by the preservice teachers, the lottery and car accident task,the dice task, and the bank teller task could be solved mathematically. Many ofthe preservice teachers, however, chose to make a decision based on their-70 -everyday experiences. Although these preservice teachers had strongbackgrounds in mathematics and probability, they were quite satisfied with theanswers they developed from their experiences. They were actually aware thatthe mathematics of probability differs from the non-formal reasoning that theyhad developed in their everyday experiences. Nevertheless, except in sometasks, their first instinct was drawn from everyday experiences rather than fromformal knowledge of mathematics, They tried to use formal knowledge onlywhen they were not satisfied with their non-formal solutions or when theresearcher requested that they think formally.The preservice teachers’ everyday experiences were very important in theway they approached each task. In many cases, the preservice teachers’ everydayexperiences and their mathematical knowledge conflicted with each other. Someof them hypothesized that the researcher would value their formal rather thantheir non-formal knowledge of probability, so they attempted to develop aformal answer as far as was possible. Others stayed with their non-formalanswers based on everyday experiences. In some tasks, the preservice teachers’everyday experiences were so strong that they looked at the tasks from theireveryday perspective rather than from a formal mathematical perspective.Using a representativeness heuristic in solving probability problemsAccording to Kahneman and Tversky (1972), the representativenessheuristic has two components: (1) the outcome in the sample should look like itsparent population and (2) events in an outcome should appear random. Lisa’sreasoning involved both characteristics. In the head-tail sequence task, in theinterview setting, Lisa stated as follows:Okay, first of all I thought that all four outcomes are possible and then Irealize that ... it also deals with order. When you say for example (b) is-71 -i-I1-Ii-nTr... the first one of course is half/half, either half head or tail. Thenif you want to produce a second head I think the probability ofproducing a head and tail would be different because since the first time isnot a tail so I think increases the probability of having a tail instead of ahead. That’s why I’m thinking that the ... four outcomes will not beequally likely. You have to look at the order to decide which one is mostlikely.... Here the first one is head and then tail then following four heads.I’m trying to think about ... a possible answer and also ... I try to use mycommon sense to decide. I think the first two are not most likely. (c) and(d) are probably likely in the sense they’re more even like three tails andthree heads but the order is different. It looks that (c) is more randomthan this one [dl. So ... I feel somehow that the random is more likely. Ifyou want to produce an order that is most likely I would say probably (c).Lisa argued that the most likely outcome should have approximately anequal number of heads and tails that were randomly distributed. Her argumentfollowed the logic of the representativeness heuristic. Jane, another preserviceteacher, used the first component of the representativeness heuristic in the head-tail sequence task in her interview setting as follows:Jane: Basically my reasoning is—(a) and (b) ... seem to be a little bit lesslikely of occurring than (c) and (d). I don’t see much differencebetween (c) and (d) because they’re just sort of flipping back andforth between. My general understanding of flipping a fair coinwill be that if you have a large enough sample, you’re going to getbasically about fifty percent of each type showing up. So in thatsense ... there are two things that I’m thinking of; partly the samplesbecause there is only a string of 6—kind of too small to be able to-72 -tell. But given that I still think I know what the right answershould be ... which is (e).Han: Which of the following orderly sequences is least likely to resultfrom flipping a fair coin six times? [shows the same sequencesHTHHHH, HFIHTrT, THTHHT, HTHTHT]Jane: In my feeling ... (a) would be less likely.... The length of the string ofthe same thing occurring is probably less likely.Jane stated that the right answer should be (e) wherein all sequences areequally likely especially because of the relatively small sample size. If the samplesize was large enough, she would think that a sequence with nearly an equalnumber of heads and tails would be most likely. When asked which sequencewas least likely Jane stated that a sequence with a long same-side run, such asHTHHHH, would be least likely. Jane stated that since the coin is fair a most likelysequence should have approximately an equal number of heads and tails. Shewas concerned with whether the outcome looked like its parent population, notwith whether the event head or tail appears as randomly distributed. Hence,Jane was using the first component of the representativeness heuristic to come toher solution to the task.Jane and Lisa also used the gambler’s fallacy in the above task. Thegambler’s fallacy is discussed in this section because it satisfies the firstcharacteristic of the representativeness heuristic. The gambler’s fallacy statesthat, in the case of two equally likely events, A and B, subjects tend to predictoutcome A after a long run of outcome B. The gambler’s fallacy is closely relatedto the representativeness heuristic because people using this fallacy expect anequal number of heads and tails in an outcome. Jane said that since there werealready five heads a tail was more likely to come because a fair coin was used.Otherwise, she suspected the coin might not have been a fair one. Lisa was also73 -using the gambler’s fallacy when she said that the probability of getting a tail inthe second trial increased because a head occurred in the first trial.The most consistent user of the gambler’s fallacy in the study was Alan.He used the gambler’s fallacy in the dice task as follows:Han: Now it’s time to decide. Now, you have the opportunity ofbecoming a dealer or a player?Alan: I’ll be the player.Han: Why is that?Alan: Because I can look at the patterns and I can get some ideas of whatmight be happening.... The longer we play the game the morelikely I think I am to win. [thinking] I think I’d still be the playerbut the reason for being the player is that I can look for cases wherea certain number has not come up for a long time. Those cases arelikely to come up. I’ll start betting on them. That’s the strategy thatI would use.Alan reasoned that he expected the probability of the occurrence of anevent to be higher when it had not occurred for a long time. Alan not only saidthis but also applied the fallacy in choosing the numbers when the game wasplayed. He was a player and the researcher was a dealer. Alan chose numberswhich had not appeared on the previous rolls of the dice and argued that sincethose numbers had not occurred in the previous throw they had a slightly betterchance of occurring next time. Alan used the gambler’s fallacy in other tasks,such as the circular spinner task and the lottery task, as well.Feng’s response resembled the gambler’s fallacy when she argued that shewould keep doubling the bet and eventually win the money from the dealer. Buther response was different from Alan’s. She was concerned that the total amount-74 -of money she had was only $100, which would not be sufficient to keep doublingthe bet. Her reasoning was somewhat similar to Gary’s who stated,You’re stuck with a hundred dollars. I’m not really sure. I’ve noreasoning as to why I say this, I think I rather be the player.... I’mthinking more in terms of go on forever because there is only a hundreddollars. I think if it went on forever it would be to my advantage.But Gary knew that $100 would not be enough, even to double 10 times.So Gary was not satisfied with the answer he provided as to why he wanted to bethe player. Finally, he remembered that he should have calculated the expectedvalues for the player to win. However, Gary stated that he forgot how tocalculate expected values and maintained his non-formal answer.Using science knowledge in solving probability problemsMany preservice teachers studied physics in their undergraduate degrees.As a result, their interpretation of some probability problems relied on theirphysics knowledge. Their use of physics knowledge was particularly evident inthe case of the thumbtack task. To answer the question of whether it is morelikely that the thumbtack would land with point up or point down the preserviceteachers used terms such as stability, gravity, momentum, and so on. Forexample, Alan’s view is depicted in the following transcript:Alan: I’d say point up is more likely.Han: Why is that?Alan: Just because the weight of the head is going to be more likely tocause it a fall like that.... This [point up] looks more stable than that[point down]. So if we try [drops the thumbtack] it’s more likely tofall on the stable state.... I think the tack will more naturally go inthat position [point up].- 75 -Han: Okay. But when you drop this it can also bounce. Right?Alan: Right. [drops the tack]. But as it bounces it is going to naturallyrotate the heavy end down. So I don’t think it’s random.Bill’s reasoning was very similar to that of Alan.Bill: With the point up.Han: Why?Bill: Because it’s heavier at this end. It’s going to fall that way. There ismore weight at the bottom of the pin. So gravity is going to pullthat towards bottom. Well, it might go down but the odds are.That’s what my reasoning is. It’s going to wind up like that [showshow it winds up].Han: So you don’t consider the fact that it can bounce?Bill: Even if it bounces ... it’s just how much momentum it has when itgoes to make a flip or something along the table. That’s what Ithink. And you can work it out.Similarly Lisa used her knowledge of the center of gravity to solve thistask,Lisa: Okay. I’ll say this [point up] is more likely.Han: Why?Lisa: The reason is—it’s about physics. It looks that the center of gravityis based down here [center of the tack]. You can even feel it. Sowhenever you throw it I think it’s more likely that it’s going to turnup [point up] this way rather than this way [point down]. This isnot balanced [stable].Han: Have you considered the bouncing factors when you drop it fromhere?-76 -Lisa: No. But it may be. I didn’t think about that. But when you say it Irealized when it bounces—I think because the center of gravity is atthe bottom. So even when it bounces, I think, provides theopporftmity to land upside down. So, I think, this [point up] ismore likely.Similar to Alan, Bill, and Lisa, several of the other participants such asFeng, Ivan, and John, argued that the tack would land point up because the flatside of the tack is heavier and more stable than the pointed part. However, Reidmentioned that the way the tack lands depends on the curvature of the tack andthat the bounce factors indicate that the tack should land point down. The abovetranscripts indicate that the participants did not consider the thumbtack task as apurely probability problem driven by a random process.Likewise, some of the preservice teachers used their knowledge ofgenetics to solve the five-boy task in the pair-problem-solving component. Theyargued that the five-boy task may have had to do with a genetics factor ratherthan with a random phenomena. Anne, while working with Gary, stated thatsome couples may have only boys or girls because of hereditary factors. Reidalso provided a similar statement when he was working with Feng. He firststated that the probability of having a boy or a girl for John and Tracy in the sixthtrial should be equally likely. But a little later he stated,But it’s also a matter of genetics. If they have already five boys thenactually the likelihood of having another boy is more because it isdetermined by male sperms. If the trend is already there for boys there isa tendency towards male children. (Reid, pair-problem solving)Working with Ruth, Ruby also thought that genetics may have somethingto do with birth patterns in John and Tracy’s family. She stated that because of- 77-genetics factors some couples have girls after girls or boys after boys. Shethought that if it was a genetics factor that was influencing the birth of babies inJohn and Tracy’s family, then it was more likely that they would have a boy inthe sixth trial as well. However, she argued that she needed more data aboutJohn and Tracy’s family trees to come to this conclusion. Otherwise, she wouldsimply assume that the chance of having a boy or a girl at a particular birth was50% and so John and Tracy would have an equal probability of having a boy or agirl in the sixth trial. The discussion between Jane and Paul was similar to Ruthand Ruby. Like Ruby, Jane argued that the chance of having a boy or a girl mayhave actually been affected by male sperms and female fertility rather thanaffected by random processes.The preservice teachers were using their knowledge of physics andgenetics to solve some of the tasks posed in the study. When the task was relatedto throwing an object, they used their knowledge of physics, and when the taskwas related to the births of babies, they used their knowledge of genetics. Intalking with the preservice teachers, it was at times difficult to determinewhether they developed this knowledge from their science classes or from theireveryday experiences. It appeared that such knowledge resulted from bothscience knowledge and everyday experiences. What was clear from theresponses was that, for some preservice teachers, the use of probabilityknowledge, such as independence and randomness, was not applicable to thiskind of problem.Although many preservice teachers used non-formal conceptions ofprobability in some tasks, they also used formal conceptions of probability inother tasks. Table 5 shows that participants used formal mathematical reasoningin many tasks of the study.-78 -Formal conceptions of probabilityAll the preservice teachers used their formal mathematics knowledge tosolve particular probability tasks, such as the rectangular spinner task. Thepreservice teachers who used formal knowledge used such concepts asindependence, randomness, and the addition and multiplication rules ofprobability. Two themes of conceptions: probability as independence andrandomness and probability as formulas, are discussed.Using independence and randomness in solving probability problemsThe formal solutions to all tasks in all components of the study requiredan understanding of independence and randomness. Most of the preserviceteachers demonstrated an understanding of independence and randomness invarious tasks. Gita, for example, in the head-tail sequence task in the interviewsetting reasoned thatIt’s equally likely because the probability of having a head doesn’t dependon ... a previous throw. The probability of getting a tail is not going to begreater if you had a head in the previous throw. If you had a head in thefirst time, the probability of getting a head in the second time is still onehalf. Doesn’t matter if you’ve 10 heads in a row, the probability of gettingthe eleventh head is still one-half and the probability of getting theeleventh tail is still one-half.... What happened in the past doesn’tinfluence the present throw.Paul’s reasoning was similar to Gita’s in the lottery task. He had to advisestudents on whether a consecutive sequence or a random sequence of sixnumbers was more likely to win a jackpot in lottery 6/49.Paul: I’ll advise them that there is absolutely positively no difference inprobability between 1,2,3,4,5, 6 or 1,2,3,4,5,49 or 3,5, 10,36,2479 -and whatever. Makes absolutely no difference. Any combinationof six numbers is exactly as likely as any other combination of sixnumbers in the 6/49.Han: Do you think that the students will be convinced?Paul: Convinced of that?Han: Yes. How would you convince the students?Paul: One can think of this as there’re 49 chips in a row. And the object isto pick six of those. They’re arbitrarily numbered 1 to 49. Thosenumbers are strictly arbitrary. So they don’t affect the likeithood ofone chip being picked over another. Therefore these numbers arejust as arbitrary. The probability of picking a 1 is exactly the sameas picking a 47 or 48. Therefore picking 1,2,3,4,5, 6 is no more orless likely than picking any other six numbers.Paul demonstrated a good understanding of independence andrandomness in the above task. Later in the discussion Paul argued that no onecan say for sure whether a sequence of numbers is random or not. Becausewhether or not a number is random depends on how it was selected. If theselection of numbers was random, Paul argued, the sequence of numbers israndom even if they appeared consecutive. If the selection was not random, thesequence of numbers is not really random even if it appears random. Paul had acomprehensive understanding of random numbers.But in the dice task several of the preservice teachers who attempted tosolve the problem inappropriately used formal concepts of independence andrandomness. Because the concept of independence was not properly utilized,Feng, Gary, and Mate thought the player had only 3 chances of winning out of 18possible chances. The following transcript between the researcher and Mateillustrates this issue.-80 -Han: I think you have the feel for the game. Now it’s time to decidewhether you want to become the dealer or the player.Mate: [Thinks for a while]. I’ll probably choose to become the dealer....When a person is putting down money on one number to occurout of 18 possible different outcomes here you have 3.... I’dprobably think better off being the dealer in this case because theodds are in your favor. More or less, you’ve 15 out of 18 chancesthat a number other than ... one of the l’s, or whatever, will roll uphere on dice.Han: Is it 15 out of 18 chances?Mate: This is 1 in 6 chances, that is 1 in 6, that is 1 in 6. ... You have a 1 in 6chances that a 1 will be up here on this dice, 1 in 6 chances that a 1will be up there on that one, and 1 in 6 chances that a 1 will be upon that dice. So there is a 5 in 6 chance that a number other than 1will appear on this dice. Same on this one. Same on that one. Sothe odds of losing are better than the odds of winning on any givendice.However, Gary, Feng, and Mate were all concerned that their method ofreasoning was perhaps mathematically incorrect. Feng thought that theprobability for the player would not be that low. Gary thought that since thedice were independent of each other he had to look at each die individually.Later in the discussion Mate also thought that the dice were independent of eachother and so he could not add probabilities. However, their knowledge ofindependence and randomness did not lead them to apply the multiplication ruleof probability.-81 -In the same way Bill, Reid, and Ivan got incorrect solutions because theyused the addition rule of probability, when they should have used themultiplication rule.My chance of getting 1 in each die is one in six. And there are three dice,so my chance of getting one in three of them is 3 in 6. That’s fifty/fifty.Then my chance of getting 2 ones is, I guess, 1 in 36. Oh! hold on it ismore than that [long pause, performs some mental calculations]. I don’tremember how to do that. Let’s see [Long thinking]. I don’t know.Anyway, my odds are already fifty/fifty just by getting 1 to 1. Two l’swhatever the probability is on top of that one 1. And the probability ofgetting three l’s, even though that is very low (1 in 6) that is also addedto the total. So my odd of winning is better than fifty/fifty. (Reid)There is a 1 in 6 chance you’ll get your number here and 1 in 6 chanceyou’ll get your number here and 1 in 6 chance you’ll get your number here[shows different dice]. So the chance of me getting 1 is 3 in 6? Yes....Actually I’m supposed to be the player. Because if I said I, my chance ofgetting a 1 is 1 in 6, 1 in 6, 1 in 6. Right? So I should win half the times.But the chance of me getting two l’s should also be added to that. Thechance of my getting two l’s is lower than my chance of getting one 1,obviously, or three l’s is lower than chance of getting one 1. But the sumof all those chances is higher than 50%.... It doesn’t matter which number Ipick. But my chances as a player winning are higher than that of thedealer winning. (Ivan)Bill, Reid, and Ivan were all confused in their attempt to solve the task.All reasoned that the probability of obtaining a number on a die is . Since,there are 3 dice, the probability of obtaining a number on one of those 3 dice is-82 -or 50%. But the correct probability calculation places it at about 35%E(i 5 5 751x X=All these preservice teachers were aware that each die wasindependent of the other, and that the landing of a number on one die does notinfluence the landing of a number on the other dice. They also stated that theyknew the multiplication rule of probability. However, when they were engagedin solving the problem the association between independence and themultiplication rule did not seem to occur to them.Further, they did not appear to realize that the addition rule of probabilityholds in the case of mutually exclusive events, not in the case of independentevents. The six faces of a die are mutually exclusive. However, one die is notmutually exclusive of the other, Rather, each one is independent of the other.Some participants like Gita and Paul demonstrated an understanding andappropriate use of independence and randomness throughout the tasks andsettings. Some others like, Bill, Feng, Gary, Ivan, Mate, and Reid thought aboutindependence and randomness but could not correctly apply them in someprobability calculations. These participants could appropriately useindependence and randomness in some tasks, like the rectangular spinner task,but not in other tasks, like the dice task.Using formulas in solving probability problemsSome preservice teachers attempted to solve all tasks in the written, thepair-problem solving, and interview components by using some mathematicalformula. Reid, for example, simply applied a mathematical formula ofcalculating the probability to the birth sequence task. Since there were six birthsReid stated that the probability of each sequence is When the number oftrials was increased to 20 from 6, Reid was very quick to say-.-. Anotherparticipant, Paul, argued in the same way.-83 -Reid and Paul not only applied mathematical rules and formulas as far aspossible but also enjoyed making rules and formulas themselves. In the lotteryand car accident task, for example, many of the preservice teachers used theireveryday experiences and intuitions to come to a solution. But Reid and Paulcame up with the following formulas to calculate the probability (Pr) of beingkilled in a car accident. According to ReidNumber of people killed in a year x Average human life spanPr= Total populationAccording to PaulDeaths in one yearPr = . x Expected years yet to liveTotal populationAfter writing the formulas, each of them stated that they could obtain datasuch as how many people were killed in a car accident in a particular year andwhat was the total population of B.C. in that year. Similarly, data about averagehuman life span could be found.The formulas provided by Reid and Paul are similar to each other exceptin some differences in language. Unlike some of the other preservice teacherswho simply took the ratio of deaths to population without considering humanlife span, Reid and Paul thought that the ratio did not give a good estimate ofbeing killed over a lifetime. When this matter was brought up in the interviews,they argued that they could be killed in a car accident during their lifetime soitwas essential to find out their expected years to live and multiply this figure bythe ratio.Reid and Paul’s methods are mathematically more complex because theyalso considered the role of human life expectancy in order to estimate theprobability of being killed in a car accident. However, for them the probability ofbeing killed in a car accident was simply plugging different numbers into the-84-formulas they provided. Because of their reliance on a mathematical procedure,they did not think of other complex issues inherent in the likelihood of beingkilled in a car accident (such as whether one drives and in which part of thecountry one lives) that other preservice teachers thought about. Their formulascould not provide justifications for issues such as: Is the probability of beingkilled in a car accident equal for all people who live in B.C.? According to Paul,not all people who live in B.C. have an equal probability of being killed in a caraccident. He stated, however, that a mathematical solution to the task cannot beobtained without making the assumption that every one who lives in B.C. has anequal probability of being killed. In the pair setting when Paul was working withJane, he stated that he had to make wrong assumptions if the task was not wellformulated. He stated that although the probability of being killed in a caraccident was not equal for everyone, he had to assume that the probability wasequal for everyone in order to solve the problem.Although many preservice teachers used their everyday experiences andintuitions during the dice task, three preservice teachers used universityknowledge of probability to reach a solution. Gita was one of the threepreservice teachers who used correct mathematical procedures to solve thisproblem. She argued that she wanted to be the dealer because the dealer had agreater probability of winning. On a piece of paper she wrote that X X =—x—x—=—; —x—x—=—. The total is —. There are three differentways of6 6 6 216 6 6 6 216 216obtaining these probabilities. So the probability for the player is 3 x216She concluded, “I think the probability of getting 1, in one way or the other, is 93out of 216.... 43% of the time you’re going to win and the dealer is going to winthe rest of the time. So I think I’d rather be the dealer.” Gita’s logic was based onformal mathematics. She was correct according to the formal mathematicalprinciples of probability except for her failure to account for the fact that there is-85 -only one way of getting the same number on all the dice. Gita’s error is notsurprising because the recognition of order in such a task has been difficult forworld famous mathematicians such as d’Alembert, Leibniz and so on (Borovcnik,Bentz, & Kapadia, 1991; Glickman, 1990). Nevertheless, she noticed her errorwhen she was given her transcript to comment on, and provided the91mathematically correct solution ofJohn’s strategy was similar to that of Gita. At the beginning of theinterview John wanted to be the dealer because usually it is the dealers who winthe money. Later when the researcher asked him if he could come up with amathematical solution he calculated the probabilities as follows:Let’s see. It’s like 3 independent trials, right? So the probability of gettinga number in one die is for the three dice will be times timesThe probability of getting 3 wins is considerably lower, that would beThe probability of 2 wins and 1 loss would be —i- times ... The probabilityof I win plus the probability of 2 wins plus the probability of 3 winsequals plus plus 4- equals --. [laughs]John felt very happy about his solution because his non-formal reasoningof becoming a dealer actually matched with his formal calculation. However, he155did not look at the different ways that one can get X X and so on. Hissolution was therefore incomplete. But the strategy he used was mathematicallysound.Another preservice teacher, Ruby who was totally satisfied with her nonformal reasoning in the first interview, decided to work on the problem further.When the researcher returned the interview transcripts for further comments andelaboration, Ruby looked at a probability text and realized that she should have-86-calculated the expected value if she wanted to come to a mathematically correctanswer. When she returned the transcript she provided a mathematical solutionas follows:E(Player)= _53 + 52 + 2 x 5 + 3 =6 6Since the expected value for the player was negative, Ruby commentedthat she should be the dealer. She said in the follow-up interview that she wasglad that her non-formal reasoning in the first interview was not incorrectmathematically. Ruby’s strategy was correct. However, as did John, she did notrecognize that there are three different ways of getting a number once and twice.As stated above the inability to recognize the different ways one can getcertain outcomes has been a problem throughout the history of probability. Thesame thing was true in the case of these preservice teachers who have training inmathematics and probability.Overview of Chapter 4In this chapter preservice teachers’ qualitatively different conceptions ofprobability were discussed. These conceptions were grouped under formal andnon-formal. It was argued that the preservice teachers used non-formalconceptions such as using everyday experiences, the representativeness heuristic,and knowledge from other disciplines in solving probability problems. Theirformal conceptions of probability included knowledge of independence andrandomness, and knowledge of mathematical formulas and procedures.Examples of the participants’ use of everyday experiences were prevalentin the lottery and car accident task in the pair-problem-solving component and inthe dice task in the interview component. A few of the preservice teachersdemonstrated other kinds of non-formal conceptions such as therepresentativeness heuristics and the gambler’s fallacy.-87-Formal conceptions of probability were used by the majority of theparticipants in many tasks, such as the rectangular spinner task, the lottery andcar accident task, and the circular spinner task in the written component, and thefive-boy task and the circular spinner task in the pair-problem-solvingcomponent. For example, in the rectangular spinner task, all of the preserviceteachers took a formal approach by looking at the greatest angle in differentsections of the spinner. Similarly, they used a combinatoric formula to find outthe probability of winning a jackpot in the B.C. 6/49 lottery.Only a few preservice teachers demonstrated complex formalmathematical thinking infused with their knowledge of independence,randomness, formulas, axioms, and proofs in probability. Although thepreservice teachers could explain what they meant by independence andrandomness, they were confused when attempting to integrate these conceptswith the addition or multiplication rule of probability in some tasks, such as thedice task.It is interesting to note that the majority of the participants expressed theirdesire to use mathematics as far as possible to solve probability problems in theuniversity setting. In the dice task, for example, they attempted to solve theproblem mathematically. However, almost all the preservice teachers could notprovide a mathematically correct solution to the problem because of theirinability to relate independence and randomness to the addition rule and themultiplication rule of probability. On the other hand, the majority of thepreservice teachers solved the bank teller task mathematically, but wereunsatisfied with the applicability of the mathematical answer to a real lifesituation.-88-CHAPTER 5Consistency Analysis of Preservice Teachers’ Conceptions of ProbabilityThis chapter presents outcomes of analysis of the data for the secondresearch question:In what ways do preservice teachers’ conceptions of probability varyacross tasks and settings?The data for this question were generated from the preservice teachers’responses to the tasks presented in different settings9of the study: the written,the pair-problem solving, and the interview. The outcomes of the analysispresented in this chapter are based mainly on the tasks that required anunderstanding of independence and randomness to solve them correctly.The birth sequence task in the written component was mathematicallyidentical with the head-tail sequence task presented in the pair-problem solvingand the interview settings. Similarly, the five-head task in the writtencomponent was mathematically equivalent to the five-boy task in the pair-problem-solving component. The lottery task and the dice task in the interviewwere closely related to the head-tail sequence task and the five-head task in termsof independence and randomness.The analysis focuses on how the preservice teachers’ responses variedacross similar tasks in the same setting and how the responses varied acrosssettings on the same or similar tasks. For the purpose of the analysis, theirresponses were classified as consistent or inconsistent. If participants usedformal reasoning correctly or incorrectly to solve at least half of the tasks in allthe settings, they were classified as consistent thinkers. They were also deemedto be consistent thinkers if they used non-formal reasoning in a cluster of tasks9settings and components are used interchangeably.-89 -throughout all the settings. For example, if they used the gambler’s fallacy in atleast half of the tasks in which this fallacy was possible then they were labeled asconsistent thinkers.If they switched from non-formal reasoning to formal reasoning or viceversa and stated that they were confused as to which reasoning they should havebeen using to solve the task, they are referred to as inconsistent. The preserviceteachers’ inconsistent reasoning was often due to a conflict between their formaland non-formal reasoning.Table 5 in Chapter 4 shows the use of formal and non-formal reasoning tothe tasks posed in this study. As can be seen in Table 5, the participants’responses varied widely between formal and non-formal conceptions on thetasks posed within and across settings. In some tasks, almost all the preserviceteachers used formal conceptions, and in other tasks they used their non-formalconceptions.Although the preservice teachers’ reasoning differed in different tasks,some of them, like Paul and Reid, were consistent in using their formalmathematical reasoning to solve all the tasks. Alan was consistent in using non-formal reasoning, particularly the gambler’s fallacy, in solving a cluster of tasksin the study. However, other preservice teachers, like Anne, Jane, John, Mary,and Ruth demonstrated inconsistency in solving the tasks. Three preserviceteachers’ conceptions of probability are provided below to illustrate consistentand inconsistent cases of solving the tasks posed in this study. Paul is chosen toillustrate a case consistently using formal mathematical reasoning to solve thetasks. Alan is chosen to illustrate a case consistently using the gambler’s fallacywhile attempting to solve the tasks. John’s case provides an illustration of howpreservice teachers can change their conception depending on the tasks andsettings. The cases of Paul, Alan, and John are summarized in Table 6.-90 -Table 6. Examples of preservice teachers’ consistent andinconsistent conceptions across tasks and settings.Components Tasks Paul Alan JohnWritten BST F F F NRST F F FFHT F G FLCT F F FCST F C FPair-problemsolving HTI F G F-*NLCT F FN FNFBT F G FCST F F FNBTT F FN FNInterview HiT F F F -* NLT F F-G F->NBMT F F-*G F NDT F CiT FN FN FNConsistent Consistent InconsistentFormal Non-formalLegend: BST = Birth-sequence taskFHT = Five-head taskCST = Circular spinner taskFBT Five-boy taskLT = Lottery taskDT = Dice taskRST = Rectangular spinner taskLCT = Lottery and car accident taskHTT = Head-tail sequence taskBTT = Bank teller taskBMT = Board and marble taskrr = Thumbtack taskF = Formal thinking N = Non-formal thinkingF N Formal and non-formal thinking without conflictF ÷÷ N Formal and non-formal thinking with conflictG = Non-formal thinking (Gamblertsfallacy)-91 -Consistent formal: The case of PaulPaul’s reasoning consistently incorporated the mathematics of probabilityacross tasks. In the head-tail sequence task in the written component, hereasoned that all four sequences were equally likely because each event, head ortail, is equally likely. Similarly in the five-head task, he chose that the outcome(head or tail) was equally likely because “the previous five outcomes have nobearing on sixth outcome.”In the pair-problem solving, Paul’s reasoning was consistent in the head-tail sequence task and the five-boy task. For instance, Paul read the head-tail taskaloud and answered as follows during the pair-problem solving:These are all independent of one another and so there is no one morelikely than the other one. It’s more likely that you’re going to have ... threeheads and three tails than you’re going to have say 6 tails or 6 heads.However it’s because it’s orderly, they’re all equally likely. Because eachone is independent of the other one. So on the first one it’s 50/50 heads ortails, second one is 50/50, third one is 50/50, fourth one is 50/50. Sothey’re all the same. Do you agree Jane?Jane was largely a non-formal thinker. At this point she did not readilyaccept Paul’s formal reasoning. Jane was thinking that the sequence THTHHT orHTHTHT was more likely to occur than the sequence HTHHHH because thenumber of heads and tails in those sequences were equal. However, Paul arguedthat all sequences were equally likely in the following way:Okay.... But (e) [all sequences are equally likely] is correct, right? I thinkwhat you’re saying is that expectation of large sample you’re going tohave relatively close to 50/50 each way and that’s true. That’s a fact.However when you’re given an orderly sequence then that eliminates this-92 -because they each become independent of one another. Therefore any oneis just as likely as any other one.Finally, Jane agreed with Paul. In a discussion with Jane of theresearcher’s interpretation of her conceptions of probability, she stated that Paul’sreasoning was highly mathematical. She thought that mathematical ideas werevalued in university education and research and so she decided to agree withwhat Paul had said. Jane also stated that Paul was a very good mathematicalthinker and she could trust him regarding his mathematical capability.In the five-boy task, Paul argued that having five boys previously “hasabsolutely no bearing on the sixth”. He argued that the present birth would beindependent of the previous births “much like flipping a coin and getting headsand tails.” He stated that the chance of getting five boys in a row is only 1 in 32.But it is the same for other outcomes as well.In the interview setting, Paul recognized the head-tail sequence taskimmediately and argued that all sequences were equally likely. Later in thecourse of the interview, the researcher asked him about his written response tothe question “which of the following orderly sequences is most likely to resultfrom flipping a fair coin 100 times?” Four alternatives HTHHHH...( 99 heads and Itail), HHE-I...rfT (50 heads and 50 tails), THTHI-ITTHTHHT... (50 heads and 50 tails),and HTHTHT... (50 heads and 50 tails) were given in this problem. Paul arguedthat all sequences were equally likely because they were independent events. Inthe interview setting, he stated,As long as it’s orderly there is no difference at all. If it’s not ordered thenthere is a difference.... If you flip a coin a hundred times ... your mostlikely solution would be 50 and 50. That would be your expected value.It’s unlikely that you would get exactly 50/50. However, that’s the most-93 -likely possibility—50 heads and 50 tails. However, if you order them,then it eliminates that. Because on any given flip they’re all independent.So on any given ifip, it can be either heads or tails. It’s a fifty/fifty chance.Therefore any order of heads and tails is ... equally likely. Doesn’t matterhow many flips you have, Doesn’t matter at all. It’s all equally likely.When asked to clarify his statement that the “most likely solution wouldbe 50 and 50,” Paul talked about the expected value. He argued that whileflipping a fair coin 100 times, the expected value would be approximately 50heads and 50 tails. In general, Paul’s reasoning was consistent with mathematicaltheory of probability. He did not change his conception even when the numberof trials in the sequence was changed from 6 to 100. On top of that, Paulcalculated the probability of each sequence to be - when there were six eventsin the sequence. Paul had no difficulty extending this to—-if there were 100events in the sequence in a particular order.As in the head-tail sequence task, Paul reasoned mathematically inselecting a sequence of numbers for the lottery 6/49. He argued that “there isabsolutely, positively, no difference in probability between 1, 2, 3, 4, 5, 6 or 1, 2, 3,4,5,49 or 3,5, 10,36,24 and whatever.” For Paul “any combination of sixnumbers is exactly likely as any other combination of six numbers in the 6/49”because each number in the draw is independent of the previous one. He arguedthat the sequence of numbers 1, 2, 3, 4, 5, 6 is arbitrary just as is any othersequence of numbers. Paul’s argument did not change even when the researcherprovided a counter suggestion saying that no single consecutive sequence ofnumbers has ever won a jackpot in the lottery 6/49. Instead Paul arguedconsistently that “it makes no difference what six numbers the person picks as-94 -long as they are all between 1 and 49 and there are no repeated numbers.”Hence, Paul’s reasoning was consistent throughout the tasks in all the settings.Although Paul correctly applied the concept of independence andrandomness in almost all tasks, he had some difficulty solving the dice task. Likemany of the other preservice teachers, his strategy did not consider the differencebetween mutually exclusive and independent events while solving the dice task.However, his solution was quite sophisticated compared to the other preserviceteachers. First, he thought that it was a tie game. Later, he reasoned that theplayer had the advantage in the game. The following transcript helps to give asense of his mathematical thinking:If I assume that each number that shows up on the die is different, ... thenit’s clearly a game of fifty/fifty. I’ll get it half the time. You’ll get it halfthe time. There are three dice and six opportunities. I’ll pick one of thethree that wins. I win.... If I pick one of those three that doesn’t show up,then I lose. So it’s fifty/fifty. That’s assuming that each number thatshows up on a die is different. For example, ... if I pick a numberrandomly from I to 6 that’s fifty/fifty chance of winning, fifty/fifty chanceof losing because the numbers that are up in the dice are, let’s call them 4,5, and 6.... So I have got fifty/fifty chance. But that’s making a falseassumption that they’re all different.... If it’s a repeated number ... like 5, 6,and 6 to keep it kind of easy ... then my chance of winning are 1 in 3. Andmy expected winnings are going to be $2+$1 divided by 3 which gives theIexpectation of a dollar.... If they’re all the same, there is a chance thatI’m going to win and I’ll win $3 + 0 over 6. So it’s 50 cents again.... I’mwrong here I think. Let’s assume I’m right. Okay, now. The chance that-95 -1 5 4they’re all different is T times times The chance that two are thesame and one is different is [long pause] .. times I times 3.6 6Later Paul calculated that if all the numbers were different then the5420.probability would be I x x = if two numbers were the same and one51 15number was different then the probability would be — x — x 3= —; and if all three6 6 36numbers were the same then the probability would be x 1=1. And finally the6 6 36expected value was x (0.50) + .!.. x (1.0) + (0.50) =. Since this was the36 36 36 72probability for him to win as a player, he decided to be the player in the game.Paul’s thinking was formal because he applied the knowledge of expectedvalue from his university probability courses. However, he did not come to amathematically accepted solution to the problem because he did not treat eachdie individually. He reasoned that the probability of his number turning up inone of the three dice was 50%. In fact he was using the addition rule ofprobability. He had strong background knowledge of expected value and othertheorems in probability, but he could not arrive at the correct mathematicalsolution because of an incorrect assumption.With the exception of the dice task, Paul applied the concept ofindependence and randomness throughout the tasks in all the settings: written,pair-problem solving, and interview. He was also consistent in the discussionwith the researcher which took place during the transcript checking and theresearcher’s interpretation of his conception. When the researcher questionedPaul concerning his solution to the dice task, Paul recognized that he had notlooked at each die independently. He stated that he would obtain a differentanswer if he had to solve the problem again because he would look at the diceindependently.-96 -Paul valued a formal rather than a non-formal approach to solvingproblems. The only time he used non-formal reasoning was in the thumbtacktask. However, the non-formal reasoning he was using in the thumbtack taskwas based on physics knowledge rather than everyday experience. He enjoyedworking on the probability problems if they could be solved mathematically. Hewas actually frustrated by some of the tasks in which the assumptions were notclear. Paul stated that he enjoyed looking at mathematical problems in atheoretical way but he did not “enjoy haggling over whether a tack is going to fallone way or the other way.” He clarified that he preferred “to be given someassumptions even if they’re false” and let his “mind explore the possibilitiesbeyond that.”As has been discussed, Paul consistently used formal reasoning to solveproblems. Unlike many of the other preservice teachers, conflict between formaland non-formal reasoning was not evident in Paul’s thinking.Consistent non-formal: The case of AlanAlan was one of the preservice teachers who was consistent in using non-formal reasoning, particularly the gambler’s fallacy. In the five-head task in thewritten component, Alan stated that “over a larger and larger number of flips,there will be a tendency for the number of heads and tails to be equal. Thereforeit is more probable that a tail will occur as the number of flips resulting in headsincreases.” Alan reasoned that since the occurrence of a head or a tail wasequally likely and a head had occurred five times already it was more likely thata tail would occur in the sixth trial. Alan was using the gambler’s fallacy in thistask.Alan also used the gambler’s fallacy in the circular spinner task in thewritten component. He showed this fallacy in the following way:-97-It doesn’t matter which two are picked initially. However, if a patterndevelops that shows a certain number coming up often, I would avoid itnext time. I would also look for numbers which never come up, and betthose on successive roands.Alan consistently used the gambler’s fallacy in many tasks in the pair-problem solving and the interview settings. Alan’s reasoning in the head-tailsequence task in the pair-problem-solving component was based on thegambler’s fallacy. When he was working with Ivan in the pair-problem solvingof the head-tail sequence task he reasoned as follows:Ivan: It doesn’t matter which order that it goes in. The probability is50:50 for each trial.Alan: One thing I was thinking when reading this question for the firsttime, I recognized this question before: ... is it not a lowerprobability, for example you will roll/flip all heads or all tails?[pause] That’s the thing that’s got me about this one becausethere’re certain patterns in here [shows the sequence HTI-1i-IHH]which seem to me to be less likely.Ivan: But are they? Is it not just chance, chance, chance?Alan’s argument indicates that the sequence HTHI-{1-{E-I was less likelybecause the number of heads and tails in this sequence were not evenlydistributed. But Ivan did not agree with him, so Alan continued to argue asfollows:Another way to think about it is: ... If you think about how this sequenceoccurred—if you bet in the middle here ... you see here, we have i-rr, I-IT,I-IT, what would you bet the next two are going to be heads? You say wellit is not very likely. So that’s what worries me about this. Again, I’m not-98-thinking clearly about probability here.... Something intuitive about thissays the possibility of having a sequence that has only one tail seems lesslikely than some even distribution.But Ivan did not agree again. Ivan argued that “in terms of a grandscheme there should be an even distribution.” Furthermore, Ivan argued that ifthe run of a string, of say three hundred tosses or more, are made then everysingle sequence HTHHHH, HHHTrr, THTHHT, or HTHTHT would be there. WhenIvan increased the number of tosses to 300, Alan was more comfortable inagreeing that all sequences were equally likely. However, in the five-boy task,Alan repeated his gambler’s fallacy as follows:Alan: My thinking is that the nature tends to be in equilibrium. So if theyafready have five boys, I think it’s more likely to have that the sixthchild will be a girl.Ivan: From the scientific point of view, it’s completely random. It’sabsolutely completely random whether or not their sixth child willbe a boy or a girl. So there is no more likelihood to be a girl or aboy. The actual chance is either XX or XY. I mean the first one isgiven by the whatever, mother, and determined when it’s given bythe father. Whether or not it carries Y chromosomes or Xchromosomes and it’s 50/50. It’s completely random.But Alan was not convinced by Ivan’s argument. Alan went on to arguethus:Alan: It seems to me that the more children they have the more likely it isthat their population would resemble the total population.Ivan: Yes. But that doesn’t mean that the next one has to be a girl.Alan: True, doesn’t have to be, but it’s more likely.-99 -Ivan: No. It can be a boy and there are 300 more kids, then maybeeventually it will. It doesn’t have to be though. Again, it could be astring of B’s: boy, boy, boy, boy.... That’s where they happened tobe a sequence in a grand scheme thing. Somewhere else, there issomebody having six girls.Once again Alan stated that he would agree with Ivan if the number ofbirths was large enough, say 300. If there were 300 births, then Alan would thinkthat the ratio of boy to girl would be 50:50.In the interview setting, Alan recognized the head-tail sequence task andmaintained the same solution he negotiated with Ivan in the pair setting. Alanreasoned that even with 100 trials in the head-tail sequence task, all thesequences were equally likely. Alan was thus changing his conception based onthe interaction he had had with his peer. However, when the task was changed,Alan did not maintain his new conception.In selecting a sequence of numbers in the lottery 6/49, Alan stated thatwhether the sequence was consecutive or random it had the same chance ofwinning a jackpot “because the number coming up is random and there isnothing about the previous number coming up that has anything to do with thenext number coming up.” Later the researcher asked what Alan would say if thestudent was thinking that no string of consecutive numbers had ever won thejackpot and that it was therefore unlikely to win. Alan argued the opposite andshowed his gambler’s fallacy. He stated that “because consecutive numbers havenever shown up they are slightly more likely to occur than [the random sequenceof numbers].” He elaborated this argument as follows:Alan: Let’s say you just pulled out lotto 6/49 numbers to infinity. You’lleventually get all the outcomes over a long period of time. As you-100-continue to increase the number of times you had an outcome, youwould eventually see an even number of all outcomes. ... In otherwords, let’s say you’ve thrown ... the total number minus 1, then I’dsay the chances are that you get the missing outcome in the lastthrow are fairly good. And the more chance you have to choose thenumbers, the more likelihood is you’re going to have an evendistribution.Han: You said that here, too [shows his response in the written task].Alan: Yes.Han: What is your conclusion? What would you tell to them?Alan: I would be inclined to say that if the six consecutive numbers havenever occurred then they might choose the six consecutivenumbers. That means slightly more likely. But it’s a very slightamount because the chances of occurring them at a given throw ispretty small.... You might be only slightly improving your chances.Alan also demonstrated the gambler’s fallacy in the board and marbletask. He observed that the marble did not come to No. I when he dropped itseveral times. He stated that he was positive that the marble would start to cometo No. 1 more frequently because it had not for a long time. He stated, “if I wasnow to bet, I would start betting on 1”. After dropping the marble for 24 times heconcluded that the board was reasonably fair even if the marble went to No. 1only twice.In the dice task, Alan decided to be the player in the game. When askedfor his reasons, Alan provided an argument which also resembles the gambler’sfallacy:- 101 -Han: Now it’s time to decide. Now, you’ve the opporftrnity of becoming adealer or a player?Alan: I’ll be the player.Han: ‘‘Vhy is that?Alan: Because I can look at the patterns and I can get some ideas of whatmight be happening. ... The longer we play the game, the morelikely I think I am to win. [thinking] I think I would still be theplayer but the reason for being the player is that I can look for caseswhere a certain number has not come up for a long time. Thosecases are likely to come up. I’ll start betting on them. That’s thestrategy that I would use.Alan reasoned that the probability of the occurrence of an event would behigher when it had not occurred for a long time. Alan also applied the gambler’sfallacy in choosing numbers during the playing of the game, when he was playerand the researcher was dealer. He chose numbers which had not appeared onthe previous rolls of the dice.Although Alan was inconsistent in solving some tasks, like the lottery taskin the interview setting, he was consistent in his use of the gambler’s fallacyacross many of the tasks in all three settings—the written, the pair-problemsolving, and the interview. He temporarily withdrew this fallacy in some of thetasks, for example when he was working with Ivan in the circular spinner task,but he continued to use it in other tasks and settings. His fallacy was temporarilychanged because of peer interaction with Ivan. After the pair-problem-solvingcomponent, Alan mentioned that he had a hard time convincing Ivan. Alanthought that Ivan’s thinking might have been mathematically more sound thanhis, but he was not sure.- 102 -In a follow-up interview, Alan mentioned that he had developed thisbelief (gambler’s fallacy) from his experience in playing black jack in his real life.He stated that he had used the same strategy in a gambling situation and hadactually won money several times. Alan was concerned that his way ofreasoning might not be consistent with the mathematical theory of probability.However, he also stated that during his school and university courses, he hadnever had an opportunity to use probability to analyze everyday problems;because of this he felt that he lacked the necessary skills for applying probabilityto everyday problems. Alan also confided that he had difficulty conceptualizingmathematical probability because he could not make mental models to solveprobability problems, as he could when solving problems in number theory orgeometry.Inconsistent: The case of JohnJohn’s reasoning was inconsistent within some and across many tasks inthis study. His response to the birth sequence task in the written component wasinfluenced by the representativeness heuristic, even though he said that “havinga boy is as equally likely as having a girl” and also “each birth is independent ofany previous births.” Despite his awareness of independence and randomness inthe birth sequence task, John thought that the sequences BBBGGG, GBGBBG, andBGBGBG would be the most likely. This implies that the sequence BGBBBB wouldbe the least likely. John chose these sequences as most likely because theycontained equal numbers of boys and girls. He was, therefore, not correctlyutilizing his formal knowledge of independence and randomness in the task.Unlike his response to the birth sequence task, John’s response to the fivehead task was driven by his formal knowledge of independence andrandomness. He stated that both outcomes (heads and tails) were equally likely.He provided the following reasons for his selection:-103-The sixth outcome is equally likely to be a head or a tail because the coin isfair. Also all trials (flipping of a coin) are independent from other trials.Just because you happen to flip 5 heads does not mean that your sixth flipwill be swayed either to heads or to tails.Hence, John was inconsistent in his treatment of the birth sequence taskand the five-head task during the written component of the study. When he wasworking with Gita in the head-tail sequence task during the pair problemsolving, she reminded him of the importance of order in the task. He wasquickly convinced by her suggestion and agreed that all the sequences werelikely. The following transcript depicts the pair-problem-solving situation:Gita: I would say that they are all equally likely.... What do you think?John: I don’t think so because in sequence HTHFII-IH there is only one tail.The probability of getting a head or a tail is about 50% and so this[HTFIHHHI would be more unlikely than the other sequences, forexample, which have three heads and three tails.Gita: Yes, but if it is the order (first you get the head and second you getthe tail) then I don’t think that the number of heads or tails in thetotal matters. I’m not sure if the question is asking that.John: Oh! Actually, you know what? That changes my mind.He agreed with Gita that all the sequences in the task were equally likely.John carried this change of conception to the individual interview setting wherethe head-tail sequence task was presented again. As he had concluded in thepair-problem solving with Gita, John stated that all the sequences were equallylikely. Upon further probing, however, he was not highly confident about theequal likelihood of all these outcomes.-1O4Han: Do you mean that even with 5 heads and 1 tail in one sequence and3 heads and 3 tails in the other sequence does not make anydifference?John: Urn-. I still have some doubt, I guess, obviously about choosingthat they are all equally likely. I still have to say that they are allequally likely, but I’m not quite sure if my answer is right. Thereare different ways you can look at it. I’m not sure which is the rightway. The mathematical way is heads and tails are equally likelyand each event is independent and you come to the conclusion thatthey are all equally likely. Or you could take a count of how manyheads and tails you get. For (b), (c), and (d) [sequences HHHT’fl’,THTHHT, and HTHTHT] you got 3 heads and 3 tails. So those willhave the same probability. And ... if you think that in 6 throws of acoin you should get 3 heads and 3 tails is a fallacy. You could getanything really.The above excerpt shows that John was not sure whether all sequences areequally likely. Although John showed a change of conception in the pair-problem solving, he was not confident as to whether the question was concernedonly with the independence of the flips or also with the total number of the flips.When the researcher asked whether the outcomes would be equally likely if thetotal number of trials was 20 and there were 19 heads and 1 tail in one sequenceand 12 heads and 8 tails in the other sequence, John replied that he would changehis answer. John thought that when the number of trials was increased up to 20,then the sequences with roughly half heads and half tails would be more likelythan the sequences with an unbalanced number of heads and tails, for example,nineteen heads and one tail. He elaborated this view as follows:-105-Han: Do you think that all the sequences would be equally likely, if thetotal number of trials was 20 and in (a) I had 19 heads and 1 tail andsomewhere, in (c) for example, I had 12 heads and 8 tails or 10heads and 10 tails? [shows the examples of sequences HTHHHH...,HTHTHI-ll-IThTHI-ll-I...]John: Urn. No, I think it changes as the number of trials increases. But thething is it’s a fallacy to assume that you get 10 heads and 10 tails in20 trials.Han: I understand that. But what about comparing those 19 heads and 1tail and 12 heads and 8 tails? Do you think that they’re equallylikely?John: No. Intuitively I’d not think so. I would expect that 12 heads and 8tails would be more likely. I’m trying to think how you cancalculate the probability of these sequences mathematically, but Ikind of forgot. [shows the head-tail sequence task] Intuitivelyspeaking, I’d still think that these three sequences (b), (c), and (d)[sequences HHHTrT, THTHHT, and HTHTHTI are more likely than (a)[HTHE-IHH], that (a) is less likely. Urn-, but that’s intuitive. I’mthinking that any of these sequences are equally likely because thenumber of trials you’ve got is small, you only have six trials; andthis whole thing that the trials being independent and theprobability of heads and tails being equal, I just have to say they’reequally likely.Han: Even in 20 trials?John: If you extend this to 20 trials, I might change my answer. If youincrease the number of trials, I would go more with my intuitionand say those sequences have roughly half heads and half tails.- 106 -What I would try to do is somehow reflect that in the mathematics.But right now I can’t quite figure out how to reflect.As can be seen in the above transcripts, John stated that the probability of1getting each sequence was when the researcher requested a mathematicalsolution. However, he did not extend this to the case when the number of trialswas 20.In the above two transcripts, John expressed his concerns about twoissues: First, you cannot expect the same number of heads and the same numberof tails in a sequence with a small number of trials. If that happens, one has todoubt its validity. His concern was clear when he stated that it was a fallacy toassume that you would get 3 heads and 3 tails in 6 trials or 10 heads and 10 tailsin 20 trials. Although he thought this was a fallacy, he still asserted that theoutcomes with an equal number of heads and tails would be more likely than theoutcomes with an unequal number of heads and tails. The second issue is relatedto the number of trials. If the number of trials was small, John thought that thesequences, whether they appeared balanced or not would have the sameprobability of happening. However he thought that if the number of trials waslarge enough, the sequences where heads and tails were balanced were morelikely to occur than the unbalanced sequences. This indicates that although Johnwas aware of independence and randomness, he did not look at each eventindependently, especially if the number of trials was larger. His faith in the lawof large numbers overshadowed his knowledge of independence andrandomness.John’s struggle in the above transcript was to find a resolution between hisnon-formal and formal thinking. He indicated that he sometimes used nonformal reasoning and sometimes formal reasoning. John found it easy to make a- 107 -decision if his non-formal and formal thinking did not conflict with each other,but when they confficted he lacked confidence in his decision.John continued to show both non-formal and formal thinking in thelottery task. For this task, which involved choosing six numbers for a lottery6/49, John gave the following reasoning:I would probably talk to them about different ways of looking at them.You could look at it mathematically and say that the probability of gettinga certain number is equally likely as getting any other number. So in asense it doesn’t matter which number you pick. You know you are stillgoing to have the same chance of winning as picking any other numbers.Then again, you can look at the past history of the numbers drawn from6/49. You can keep track of what the winning numbers are and then lookfor trends or patterns of the numbers that were chosen a lot. You cancome up with certain numbers that seem to be chosen quite frequently.Later, the researcher provided him with information from Luck magazinestating that no consecutive number sequence has ever won a jackpot. John statedthat a consecutive sequence had a smaller chance of winning the jackpot than anyother random sequence of numbers. The conversation was as follows:Han: Not a single consecutive number has won the jackpot so far. Whatdo you think now?John: That’s actually crossed my mind. You know whenever I see thelottery winnings, fve never seen the consecutive numbers. Um-,I’m getting into ethics here. I would have to agree with the otherstudent who thinks that the chance of getting a sequence of sixconsecutive numbers in the lottery is smaller than getting a randomsequence of any six numbers. Then again, if I’m advising these-108-students I would only present the information and let them come toa decision.Han: What’s your personal thinking?John: A random sequence is more likely than choosing six consecutivenumbers. That’s my opinion.Han: What’s the reasoning for that?John: I’m just basing my decision on the fact that intuitively sixconsecutive numbers seem unlikely to me and by the informationyou’ve just provided it has never happened. My reasoning is purelybased on intuition. Mathematically speaking, any six numbers isequally likely, right? There’re mathematical ways of looking atthings and there is your intuition. Sometimes you have got toseparate the two or make compromises or whatever I don’t know.Once again, John was struggling with whether to use non-formal orformal reasoning. He made it clear that mathematically, any six numbers chosenfor the lottery had an equal chance of winning. But his non-formal reasoning didnot support that. However, his choice of non-formal or formal reasoningdepended on the setting and the person whom he was talking.Overview of Chapter 5This chapter described both consistent and inconsistent reasoning that thepreservice teachers used in solving the tasks posed in the study. In varyingdegrees, most tasks evoked conceptual dispersion that included non-formal andformal conceptions of probability. For example, the rectangular and circularspinner tasks evoked a much smaller conceptual dispersion than other tasks suchas the dice task. This means that preservice teachers’ responses varied on tasks-109-within and across settings. However, variation among responses to differenttasks was not the same for all participants.Some participants, like Paul, used consistent mathematical reasoning inmost tasks. Others, like John, used inconsistent reasoning that was formal, non-formal, or a mixture of both. The participants were either confident of theirformal mathematics and were able to solve problems using it, or they hadforgotten elements of mathematics that they were taught and had to rely on theirnon-formal reasoning.When the preservice teachers recognized inconsistencies in theirexplanations they admitted to them arguing that human beings are inconsistentin their thinking. The majority of them commented that their thinking differeddramatically when they solved problems in their everyday lives and when theysolved problems in their university examinations. Some of the preserviceteachers who changed their conceptions from non-formal to formal, stated thatthey tried to do so because the research was conducted in a university settingwhere formal mathematics is valued more than non-formal.In the pair-problem solving, formal thinkers always dominated the non-formal thinkers. Paul dominated the discussion with Jane. Jane agreed withPaul because she thought that he was a highly formal thinker. Alan temporarilychanged his gambler’s fallacy to formal reasoning in the head-tail sequence taskand the five-boy task. Similarly, John changed his conception based on adiscussion with Gita who applied formal reasoning to solve the problems.Although, Jane, Alan, and John all temporarily suspended their non-formalreasoning when challenged by their peers, they continued to use non-formalreasoning in the interview setting, which followed the pair-problem solving.- 110 -CHAPTER 6Preservice TeacherstPerceptions about the role ofProbability in their Everyday ProblemsThe third research question was presented in Chapter 1 as:What are preservice teachers’ views about the role of probability learnedat the university level in solving everyday problems?The data for this question were drawn from two types of questions askedof the preservice teachers, First, the preservice teachers were asked to providetheir opinions as to whether they found university knowledge of probabilityuseful in solving their everyday problems. Second, the preservice teachers weregiven a variety of tasks to solve as described in Table 2 on page 49. Thepreservice teachers’ responses to different tasks and settings are thoroughlydescribed in Chapters 4 and 5. It is not intended to repeat those in this chapter;the gist of the responses will be provided as well as a detailed description of theirviews of the role of probability in solving everyday problems. The participants’responses to the tasks summarized in this chapter provide some insight into howthey use formal probability in solving everyday problems, in addition to howthey view the role of formal probability.Participants’ views about the role of probability insolving everyday problemsThe statements of participants in this study suggest that their knowledgeof formal probability is not useful in solving everyday problems. However, twokinds of responses were obtained from the preservice teachers. About half ofthem thought that probability in university settings and probability in everydaylife are two different phenomena. The rest argued that knowledge of university-111-probability strengthens reasoning skills, which in turn helps in solving everydayproblems.Probability in university and probability in everyday life: Two differentphenomenaSome preservice teachers in the study commented that they nevercalculated probabilities while solving everyday problems. They responded thatthe university probability courses were not useful because they were tootheoretical and hence had little or nothing to do with everyday problems. Gita,for example, expressed her feelings in the following way:You probably do [use probability in everyday problem-solving] but I don’tthink about using it.... You don’t think, “I’m using probability,” you thinkof it as using common sense. I don’t play the lottery because the chance ofwinning the lottery is so low. You might not work out the probability butjust intuitively you know that the probability is low. There are 49numbers and I pick only six numbers. The chance of my winning numberis very low. You know that. You don’t need to sit down and figure outthe probability.Gita’s comments indicate that she considered university probability to bepurely formal—not closely related to her everyday life. Gita thought that hernon-formal thinking was enough to decide that the probability of winning alottery was very small, and so it would not be worthwhile to spend money onlotteries. The above transcript indicates that Gita’s non-formal thinking is aproduct of her common sense and some formal knowledge of probability.However, in a further discussion Gita elaborated that if she did use formalknowledge of probability to solve problems then it would be what she learned inGrade 12, not what she learned in university probability courses.-112-Like Gita, many of the other preservice teachers such as Alan, Anne, Jane,John, Mate, and Ruth stated that the probability they learned in universitycourses was not helpful in solving everyday problems. They argued that theycould not use formal probability in their everyday world because the probabilitycourses at the university did not emphasize solving everyday problems. Theythought that if their probability courses were intended to help them in solvingeveryday problems, they might have used it in their everyday lives. Accordingto them, the purpose of the probability courses was to prepare them for othercourses in mathematics, statistics, physics, chemistry, and engineering.The preservice teachers also argued that they might have been able to useprobability in their everyday problems if their understanding of probability wasgood. Unfortunately most of the preservice teachers admitted that they did nothave a strong conceptual grasp of probability. According to these preserviceteachers, the courses in probability were taught in a rote fashion with noemphasis on conceptual understanding. They rarely had time to clarify conceptsthrough discussion. For example, Gita stated the following:It was taught in a very formulaic way. You look at the problem. Youidentify it. You match it with a formula and solve it that way. I neverunderstood the formula and I never understood how the formulas werederived. I could never remember them. So I don’t think I’ve used themproperly. I used them more maybe in terms of how to read problems. Tolook where the events are independent or dependent or things like that.More than trying to fit them into a formula. No, I don’t think that coursewas very helpful.Although Gita was highly critical of the way probability was taught at theuniversity level, she was one of the few participants who used formal-113-mathematical reasoning in most of the tasks posed in this study. When this issuewas raised with Gita in a follow-up interview, she mentioned that her Grade 12probability unit was much more helpful in solving the tasks because she hadseveral opportunities to experiment with coins, dice, and spinners in that unit.However, another preservice teacher, Alan, forgot what he had done in Grade 12.He did remember the probability course he took at university and stated that itdid not provide any help in solving everyday problems. He mentioned that hisprofessor did not communicate well and spent most of the time writing “lots andlots of equations that didn’t make sense” to him. He spent his time in the courseroutinely following the formulas and applying them to problems in a rotemanner without actually understanding why. Hence, for Alan, using probabilityformulas in everyday life was removed from his way of thinking. It is interestingto note that he did not use formal probability in solving everyday-oriented tasksposed in the study. Instead, he used non-formal reasoning to solve them.Ruth did not like university probability because she actually found itcounter-intuitive to her thinking. According to her many things in probabilitydid not fit with her “feelings.” Ruth’s “feeling” is the same as non-formalreasoning. Ruth stated that in the everyday world human feelings play animportant role in the solution of a problem, while in university courses feelingsdo not have much place. Ruth found that her feelings drawn from everydayexperiences were incompatible with the probability theories she had studied atuniversity. Because of this, she followed the formulas and knew theirapplications, but without much understanding. In a follow-up interview, Ruthmentioned that in everyday life she always relied on her feelings, while inuniversity and similar settings, she tried to apply her formal knowledge ofprobability. Ruth further explained that formal probability at a university does-114-not have much place for peopl&s everyday feelings. In much the same way, hereveryday world did not make use of formal probability.Anne thought that probability was an interesting subject. However, shedid not believe that teaching probability without conceptual understandingwould be helpful in everyday life. Anne described her probability class asfollows:Even though I found it a very fascinating subject ... He presented in such atheoretical [way].... It seems to tie with a lot of things, but he didn’t makeas many connections..., It was very much torture. Just learn the formulaand, you know, we weren’t necessarily understanding why.It was evident that all the preservice teachers mentioned above sawapplications of university probability as different from chance phenomena thatpeople encounter in everyday lives. John, for example, saw probability as a“purely academic” matter that is unrelated to his “day-to-day life.” Although hetook risks in his day-to-day life and knew that those were related to probability,he actually found it difficult to explain exactly how school and universityknowledge of probability helped him to make judgments in his everyday life.John mentioned that he would rather use intuition and personal experience tosolve everyday problems involving probability. According to him, solutions toeveryday probability problems do not require the use of the mathematics that helearned at university. Moreover, he found that formal probability was tootheoretical to be applicable to everyday problems.Many of the preservice teachers stated that they would try to useuniversity knowledge for questions asked in a university setting or in othersimilar settings. However, in solving problems encountered in everyday life theywould feel secure relying on everyday experiences and intuitions. Anne argued-115-that in everyday life she used “human probability” instead of universityprobability. Human probability is taken to mean the same as non-formalprobability—that is thinking about chance phenomena in the everyday worldwithout explicitly referring to formal probability. For example, how likely is itthat her sister will call her before going to the grocery store? How likely is it thatthe bus is going to come within the next 10 minutes? For Anne, non-formalprobability was different from university probability because the former is notbased on formulas, procedures, axioms, or theorems. Instead, it is based onpeople’s everyday experiences and intuitions.Knowledge of university probability enhances non-formal probabilitySome of the preservice teachers were more positive about their learning ofprobability at the university. They reasoned that although probability learned atuniversity is not directly applicable to their everyday lives, it enhances thethinking skills and problem-solving abilities which can be applied to theireveryday problems. Therefore, they argued that they use probability learned atthe university in an indirect manner. For example, Lisa stated the following:I guess in a sense you use it but you don’t ... realize. You can think whatcould be the percentage of something happening, how likely it’ll happen.I think everyday you think about something, you figure out how to solveproblems, how to make decisions. I think that part, probability part, is apart of thinking process. You just don’t realize it.... I will not look at [thefact that]: there are so many combinations, which one would be mostlikely? I don’t look that way because I’m not very comfortable inpredicting outcomes.Lisa thought that the probability she had learned at university washelping her to make decisions in an indirect manner. However, upon further-116-probing, she was not sure whether the assistance in making decisions had comefrom university probability courses or from her everyday experiences. Lisa’sstatement implies that her non-formal thinking was influenced by the probabilitycourse she had taken at university. That may be the reason why Lisa’s responsesto the probability tasks in this study were characterized by a conflict between hernon-formal and formal thinking.Reid had views similar to Lisa’s regarding the use of probability ineveryday life. The following transcript illustrates the ways in which Reidthought that probability was useful in solving everyday problems:Reid: Even though I don’t remember the exact formulas and so on, I stillgot through the thinking process in looking at complex problems.You’re not just bringing your intuition, but you’re bringingknowledge and experience.Han: Do you use your university or school knowledge of probability tosolve your everyday problems?Reid: Yeah. For instance, you drive into school. And you say, well I’monly going to be here for 2 minutes to pick up some books. So whydon’t I just park in the meter parking and not put any money into it.Because what’s the probability that someone will come along andgive me a ticket in 2 minutes? I can make some in-depth thinking.Let’s say this guy comes every hour. The likelihood that he’s goingto come by in those 2 minutes is very small. Even if you aren’tmaking the actual number calculations all the time, you’re makingsome relative calculations in your mind. And you do this sort ofthing all the time in real life.Han: Is that because of your university mathematics or you do thatanyway?-117-Reid: Good question. That’s difficult to say. But it certainly comes beforeuniversity education. It’s a sort of common sense.Reid was one of the few consistent mathematical thinkers in the study. Hestated that he would use intuition and formal probability to solve problems ineveryday life. Moreover, his intuition was influenced by formal probability.Being a formal thinker, Reid thought that he would use probability in hiseveryday life. However, his example of parking a car at a meter did not actuallyrequire a university level understanding of probability. People could solveReid’s problem without knowing probability that is taught at universities. It isnotable that even Reid, who was an advocate of university probability, wasactually providing an example of the non-formal probability that Anne spoke of.Mary and Bill also provided examples of non-formal probability. Mary’sexample of non-formal probability can be seen in the following transcript:Han: How often do you use probability in everyday life?Mary: Actually I think I use probability every day.... Like the chances areif I go to the university to work on my project, nobody will botherme. Whereas if I stay at home and try to work on my project, mysister will be there. The chances are she’s going to come in andbother me a lot and I’m not going to get much work done. So Ibetter go to the university.Han: Do you need university probability to decide those sorts of things?Mary: [pause] To decide those kinds of things, no I don’t. But ... to meprobability gives a kind of way of looking at things which is sort ofdifferent than [everyday intuitions].... In the courses I took, therewas a lot of material done in a very short period of time. And myretention from these courses about probability is actually very low-118-[laughs]. I think if I had just taken a course in probability ... I thinkcourses like that provide students the better opportunity to learnthe probability to practice. I didn’t get much practice in myuniversity learning probability to effectively use it in everyday life.Although it gave me a better intuition. Like I’ve a feeling for itwhich I didn’t have before,Mary thought that her intuition may have been influenced by heruniversity experience in probability. She expected that her non-formalconception of probability was a function of her everyday and universityexperiences. However, Mary’s example of her sister interrupting her when shewas studying is a good example of non-formal probability. Mary’s reasoningstemmed completely from her everyday experiences and not from the kind ofprobability learned at university. Mary admitted that she did not use universityprobability to decide whether or not she should go to the university to work.Mary was concerned that her probability course did not use everydayproblems such as coin tossing and dice rolling. She had never had anopportunity to solve probability problems that may be encountered in everydaylife, She thought that if she had had an opportunity to practice solving everydayproblems in her university courses she would now be better at solving everydayproblems. Bill, however, argued the opposite. He thought that the probabilitycourse he had taken at university was useful because it helped to eliminate hisintuition. This is clear in the following transcript:Han: You’ve done some probability at your university. How often doyou use probability in your everyday life?Bill: Pretty often.Han: Pretty often? Can you give me an example?-119-Bill: [pause] No. [laughs] Okay, go to the bus this morning and it saysthat it’ll come at 7:21 or 8:21. But based on previous experience Iknow that it doesn’t usually come right at 8:21. It usually comes acouple of minutes later. So, even at 8:21 I’ll still run to the bus stopand probably catch a bus. You know the odds are better that Iwould [catch the bus].Han: Interesting. Do you need university probability to make thatdecision?Bill: No. [laughs] [pause] When do I use that? If I’m feeling reallykeen. If someone asked me a question and I really wanted to knowthe answer, I would probably look at my textbooks and use aformula. But my probability course did teach me not to rely onyour intuition kind of things. There is a lot of other factors thatinfluence, like the bus schedule doesn’t mean that the bus is goingto show up. Because there is traffic. Maybe it broke down. Maybethey have more buses on schedules because it’s a busy time.... Thereare lots of other factors. Like we did a birthday problem beginningof stat class.... He [the instructor] made a bet with the class that atleast two people in the class will have the same birthday. And thefact that you’ve a class of larger than 20 people ... the odds arebetter that two people in the class are going to have the samebirthday.... Your intuition might tell you that there is only a 1 in365 chance that you have the same birthday as [someone else]....You can’t always rely on your intuition. This thing I got out of mystat class.Bill argued that he no longer relied on his intuition after taking probabilitycourses at the university. The above transcript indicates that Bill’s probability-120-course was designed to show that students’ intuitions were faulty. For example,the birthday problem was used by the instructor to argue that students’ intuitionsare often inferior to mathematical probability. Bill was convinced from hiscourse that he should use mathematical reasoning in solving probabilityproblems. It is interesting, however, that the example (waiting for the bus) heprovided was an example of non-formal probability not an example of universityprobability.There were a few other students who argued that university probabilitywas useful in solving everyday problems. However, like Reid, Mary and Billsaid they had a hard time providing a suitable example of a problem andexploring how they would use it.It is notable that all the students mentioned above, except Bill and Reid,did not enjoy the learning of probability in university courses. Two otherparticipants, Gary and Paul, enjoyed probability at the university level. Theyexpressed themselves as follows:Oh! It was fantastic. We had an incredible teacher. Very, very difficult. Itwent like: binomial theorem, multinomial theorem.... He made us actuallyprove this.... Really, really difficult course but I really learned a lot.... Iwish I could have taken a little bit more after. There was so much so fast.A lot of it just went in and out. At least I know that there is a way tofiguring out, like you are not remembering how to do expected values. Iknow there is a way to do that. I can always go and look at it.... I reallyenjoyed it. Really interesting! (Gary)I enjoy taking up things in my own mind.... Taking a theoretical, and howdoes it apply. I enjoy that. But I don’t enjoy haggling over whether a tackis going to fall one way or the other way. I mean I’ve got to be given some-121-assumptions even if they’re false. Then let my mind explore thepossibilities beyond that. (Paul)Both Gary and Paul enjoyed probability, not because probability had a lotto do with their everyday lives, but because they could make hypotheses andcompare them with theory. In general, Gary and Paul both enjoyed theirprobability courses at the university because the courses were mathematicallysound. Their arguments showed that they were both deductive thinkers anddrew conclusions based on axioms, assumptions, and hypotheses. Althoughthey found university probability sound and useful from a formal perspective,they argued that the mathematics of probability was not useful in everyday life.According to Paul, in real life “you go on with your intuition.” Thus almost allthe participants believed that the probability they learned at the university wasnot useful for them in solving problems that they encounter in their everydaylives.Preservice teacher& methods of solving everyday problemsSome tasks, such as the lottery and car accident task in the written and thepair-problem-solving components, the bank teller task in the pair-problem-solving component, and the dice task in the interview component, were relatedto the preservice teachers’ everyday world. All the preservice teachers eitherdrove cars or were familiar with car accidents. All had been to a bank atdifferent times of the day and had observed a line up. All but one knew how toplay the lottery 6/49. Some played card games, such as black jack, and someobserved people playing card and dice games in a casino. The preserviceteachers could have used their university knowledge of probability in solvingthese tasks. However, as has been discussed in Chapters 4 and 5 many of the-122-preservice teachers did not use their university knowledge of probabffity todetermine solutions for these tasks.As discussed earlier many of the preservice teachers, Alan, Anne, John,and Ruth for example, mostly used non-formal reasoning in solving theprobability tasks presented in the study. They basically developed non-formalreasoning from their past experiences in their everyday lives. However, to someextent their non-formal reasoning was framed by their school and universityknowledge also.In a task like the dice task, the majority of the preservice teachersattempted to solve the problem mathematically. However, they found it difficultto solve. They stated that the probability of getting a number in the roll of one1 5die is . So the dealer had chance of wrnnmg the bet. However, they wereconfused as to how they would calculate the probability when there were threedice. They could not figure out whether they had to apply the addition rule orthe multiplication rule of probability. They were highly confused with theprovision that the player could win double or triple the amount of money he orshe bets if the same number occurs in two or three dice. Because of suchconfusion and conflict they could not solve the problem formally and they stayedwith their non-formal reasoning.Many of the preservice teachers could not do the probability calculationformally because their probability learning was not conceptual enough to applyto everyday problems. Moreover, they forgot the mathematical rules andtheorems from their probability courses. Mary and Ruth, for example, believedin luck only when they failed to solve the problem formally. In the secondinterview both of them mentioned that they wanted to solve the academicproblem formally because both of them felt that formal knowledge was morevalued in university teaching and research. However, in their everyday lives- 123 -they believed in luck and feelings. Hence, for these preservice teachers,probability at university and probability in everyday life were two differentphenomena with only a small overlap.Overview of Chapter 6Most of the preservice teachers in this study argued that the knowledge ofprobability they gained at university was not useful in solving problems thatthey might encounter in their everyday lives. Of those who thought that theirprobability knowledge would be useful in solving everyday problems, mostcould not give a suitable example of how university probability was useful. Onthe contrary, the examples they provided were more suitable to illustrating thatthe probability they learned at the university contributed very little to problemsthey had to solve in their everyday lives. Mary’s example of her sisterinterrupting Mary in study, Reid’s example of parking a car at a meter, and Bill’sexample of waiting for a bus are all related to chance factors. However, thesolutions to these problems do not require university knowledge of probability.Even the preservice teachers who argued for the benefits of universityknowledge of probability realized that the benefits were not direct orimmediately applicable. They argued that their knowledge of universityprobability helped to enhance their thinking skills which in turn made thembetter problem solvers. However, many participants did not come up with amathematically acceptable solution to different tasks, such as the dice task, thatwere contextualized in a real-world situation. Many of them either went withtheir non-formal thinking or applied inappropriate mathematical procedures andstrategies. When this issue was raised, they stated that probability learned atuniversity was not readily applicable to simple everyday problems.-124-CHAPTER 7Conclusions, Discussion, and ImplicationsThe purpose of the study was to investigate preservice teachersconceptions of probability (first question), the consistency of their conceptions(second question), and their views about the role of university probabilitycourses in solving everyday problems (third question). A variety of tasksinvolving probability were given to 16 preservice teachers in three differentsettings: written, pair-problem-solving, and interview situations. The outcomesof the analyses and interpretation of the data with respect to each researchquestion has led to a number of conclusions which are discussed in this chapter.ConclusionsIn relation to the first research question—What are some qualitatively differentways in which preservice teachers conceptualize probability?—non-formal andformal conceptions of probability with qualitatively different conceptions withineach category were identified. In answer to the second research question—Inwhat ways do preservice teachers’ conceptions of probability vary across tasksand settings?—it was found that their conceptions varied depending on the typesof tasks and settings as well as their personal experiences. Inquiry into the thirdresearch question—What are preservice teachers’ views about the role ofprobability learned at the university level in solving everyday problems?—led tothe conclusion that the preservice teachers viewed university probability as beingnot closely related to their day-to-day lives. The conclusions to each of theresearch questions are discussed in greater detail below.-125-Question one: What are some qualitatively different ways in which preserviceteachers conceptualize probability?The preservice teachers held qualitatively different conceptions ofprobability. These qualitatively different conceptions were grouped under non-formal and formal conceptions of probability. Non-formal conceptions ofprobability included the preservice teachers’ use of everyday experiences andheuristics in solving problems. Formal conceptions included the preserviceteachers’ knowledge of independence, randomness, probability formulas, rules,and applications.Both non-formal and formal conceptions identified in this study areunique compared to earlier studies reported in the field of probability research.Some of the non-formal conceptions such as the outcome approach discussed byKonold (1989, 1991) were not found to be used by the participants in this study.Rather than attempting to answer a particular problem based on one trial, all theparticipants in this study appeared to believe in the law of large numbers,although their conceptions of large numbers were different. The number of trialsfor a large number varied from 30 to a million. For participants who werestrongly influenced by formal probability, a large number required at least ahundred trials. For others, it was a matter of context depending on tasks andexperiments.Non-formal conceptions of probability such as the representativenessheuristic and the gambler’s fallacy were found to be used by some of theparticipants in this study. However, the conceptions as discussed by theseparticipants seemed to differ from the ones reported in previous research in atleast three respects.First, not all the participants who used a representativeness heuristicconsistently thought that the outcome in the sample should look like its parent-126-population, or that the events in an outcome should appear random. Instead, theparticipants who used this heuristic focused on the first characteristic that asequence with nearly equal number of heads and tails is most likely to occur. Inthe head-tail sequence task for example, the sequences THTHHT and HTHTHTwere thought to be equally likely because both of them had the same number ofheads and tails. If the preservice teachers had used the representativenessheuristic in the sense that Kahneman and Tversky (1972) have described, theparticipants would have stated that the sequence THTHHTwould be more likelythan the sequence HTHTHT because the former sequence appears to be morerandom than the latter which has a systematic (alternate heads and tails)occurrence of heads and tails.Unlike Kahneman and Tversky’s studies, some of the participants whostated that the sequence with nearly equal number of heads and tails would bemost likely to occur did not hold the same view when the number of trials wasincreased from 6 to 20 or to 100. Participants in this study thought that with alarger number of trials all the sequences would be equally likely. Hence, therepresentativeness heuristic was used only in the case of small samples. Thesestatements provided by the participants indicate that the representativenessheuristic is far more complex than has been suggested by previous researchers.Third, the participants used these heuristics because of everydayexperiences of success in solving real life problems. For example, one participantwho consistently used the gambler?s fallacy got this idea from his experience ofplaying blackjack and other games in a casino. He often won money in real lifeby using this strategy. He made it clear that his reasoning might not beconsistent with the mathematical theory of probability, but was fine for him inhis everyday life.-127-One of the uses of formal probability discussed in the literature is that ithelps people in making reasonable decisions in their everyday lives (Borel, 1962;Dörlfer & McLone, 1986; Good, 1983; Jacobsen, 1989; McGervey, 1986). But thisparticipant thought that his decision, while not making use of formal probability,was reasonable because he frequently won money. Thus for him the heuristicswere not fallacies; instead they were successful strategies for solving everydayproblems. Nevertheless, he stated that he would not use the gambler’s fallacy ina university examination because it is not valued in that context. The reasoningprovided by the participants indicated that they viewed the mathematics ofprobability or the heuristics in terms of contexts and situations.The participants in this study also demonstrated conceptions similar to theavailability heuristic. Most of the discussion provided in Chapter 4 regarding theuse of everyday experiences in solving probability problems is related to thisheuristic. Researchers argue that people use previous experiences withoutevaluating how appropriate such experiences are in solving problems (Tversky &Kahneman, 1973, 1982a; Shaughnessy, 1992). According to these researchers, thelack of evaluation may mislead people’s attempts to solve probability problems.However, in this study, the participants’ argued that they had to use theireveryday experiences in solving some of the problems because formal probabilitywas not always useful. The participants provided various examples of why theythought that mathematical solutions to the problems were not applicable to reallife situations. For example, their suspension of a mathematical solution to thebank teller task because of its lack of applicability to the real life situationindicates that the participants in this study were thinking beyond a simplemathematical model of probability.By the same token, the preservice teachers’ argument that a person wasmore likely to be killed in a car accident than win a jackpot was based on a non-128-formal conception that is not incompatible with the mathematics of probability.The same is true for preservice teachers’ decision to be the dealer in the dicegame because, “It is the dealer who usually wins money.” Their solutions to theproblems as well as the reasoning processes they used to come to the solutionswere sensible when viewed in terms of everyday life situations. Hence, theavailability heuristic is connected with a person’s everyday life and so may bemore useful than formal mathematics in thinking about problems related to theirpersonal life.In addition to reporting the use of these heuristics in different ways thandescribed in earlier research, this study has added some new and qualitativelydifferent conceptions of probability. One such conception is the participants’ useof knowledge from other disciplines in solving probability problems. Some ofthe preservice teachers used their knowledge of physics and genetics to solvesome of the problems. It appeared that the participants’ thinking aboutprobability was interwoven with knowledge from different disciplines.In some of the tasks, the participants’ use of formal probability concepts,such as independence and randomness, and probability formulas and rules, wasreasonably easy to see. The participants’ responses discussed in Chapters 4 and 5indicate that they had a more complex understanding of formal probability thanstudents in the earlier studies, such as those reported by Piaget and Inhelder(1951/1975), Fischbein and colleagues (1975, 1984, 1987, 1991), Green (1983, 1987,1988), Kahneman and Tversky (1972), Tversky and Kahneman (1973, 1982a, b),Shaughnessy (1977, 1981, 1992), and Konold and colleagues (1989, 1991, 1993).The participants’ mathematically correct responses were not surprising becausethey had better backgrounds in mathematics and probability than the schoolchildren and first year college students in earlier studies. However in this study,many of the participants who correctly used probability concepts, such as- 129 -independence, randomness, and the addition and the multiplication rules, couldnot use these concepts correctly on some tasks, such as the dice task. Theirdifferent responses to similar tasks in different settings indicate that theirconceptions were situated in contexts generated by tasks, settings, and differentsoda! dynamics.Question two: In what ways do preservice teachers’ conceptions of probabilityvary across tasks and settings?Preservice teachers’ conceptions of probability varied across tasks. Theyused formal conceptions if a task appeared similar to a textbook-type problemand could be solved by a relatively straightforward application of a mathematicalformula or a rule. They used non-formal conceptions if the tasks appearedsimilar to everyday experiences. On some tasks, for example in the lottery taskand the dice task, their conceptions of probability were characterized by aconflict between the non-formal thinking they developed from their everydayexperiences and the formal probability they learned at university.Not only did the participants’ responses vary across tasks, but theirresponses also varied across settings. In the pair-problem solving, the preserviceteachers who used non-formal conceptions agreed with those who used formalconceptions. However, many of the participants who seemingly changed theirconceptions toward the formal ones in the pair-problem solving went back totheir original conceptions when probed individually in the interview setting.It is also interesting to note that the participants’ responses were usuallyformal in the written component, and more non-formal or a mixture of both inthe pair-problem solving and the interview components. The participants’formal conceptions were challenged when different social dynamics were createdin the pair-problem solving and the interview components. When their formalconceptions were challenged by the researcher in the interview, they began to- 130 -shift their conceptions to non-formal ones. In much the same way, they began toshift their non-formal conceptions to formal ones when challenged.The role of the tasks was also an important determinant of how theparticipants viewed probability. On a textbook-like task formal thinkers alwaysdominated the non-formal thinkers. When formal thinkers were able to apply amathematics formula to solve a task, non-formal thinkers usually supportedthem because in their view formal mathematics would be valued in such tasks.Once the participants changed their conceptions from non-formal to formal intextbook-like tasks, they used this formal conception in solving other tasks, too.But that was not true for everyday-oriented tasks. On tasks set in an everydaycontext, participants’ non-formal conceptions were difficult to change. Theyfound that non-formal conceptions were more helpful than formal ones if thesolutions had to be applicable to real life situations.These findings, derived from the preservice teachers’ responses generatedfrom different tasks in various settings, provide a richer understanding ofstudents’ conceptions of probability. Research conducted by Piaget and Inhelder(1951/1975), Fischbein and colleagues (1975, 1984, 1987, 1991), Green (1983, 1987,1988), Kahneman and Tversky (1972), Tversky and Kahneman (1973, 1982a, b),Shaughnessy (1981, 1992) all concentrated on students’ understanding ofprobability by giving different tasks but not providing different settings. Thosestudies provide some insights into how students’ reasoning varied in differenttasks, but fail to show how their reasoning changes across different settings.This study has reported not only qualitatively different conceptions ofprobability but has also provided examples of how and when participants’probabilistic reasoning varies. When the participants moved from individualwritten tasks to the pair-problem solving and then into the interview setting,-131-their responses varied based on social interactions between the respondents andalso between the respondents and the researcher.Question three: What are preservice teachers’ views about the role ofprobability learned at the university level in solving everyday problems?The participants generally viewed the teaching of university probability asa highly theoretical area which did not emphasize conceptual understanding.They viewed the lack of emphasis on conceptual understanding of formalprobability as contributing to their inability to use it in solving everydayproblems. That was one of the reasons why the preservice teachers relied onnon-formal probability in solving problems that they encountered in their day-today lives. Despite an ever increasing appeal for a need to learn and teachmathematics with understanding (Hiebert & Carpenter, 1992; Skemp, 1976),conceptual understanding of mathematics does not seem to have been achievedfor these participants at the university level. This seems to be a concern whichneeds to be addressed by mathematicians and mathematics educators who areconcerned about students’ conceptual understanding.Some of the participants in the study thought that they were unable to useformal probability in solving everyday problems because their probability classesnever emphasized its use in that context, They thought that if they had hadopportunities to solve everyday problems in their probability classes they wouldhave been able to use it in their everyday lives. According to them the use ofeveryday problems in probability classrooms would have provided them withbetter understanding of probability, which in turn would help them in solvingeveryday problems. Although reform documents such as Everybody Counts(National Research Council, 1989), Curriculum and Evaluation Standards for SchoolMathematics (NcTM, 1989), and Professional Standards for Teaching Mathematics(NCTM, 1991) emphasize the role of everyday contexts in the teaching and- 132 -learning of mathematics, the importance of such contexts appear not to havebeen valued in the teaching of probability at the university level.Some of the participants in the study thought that university probabilityenhanced their reasoning skills which in turn helped them solve everydayproblems. However, the examples of problems that they gave did not requireuniversity knowledge of probability. Instead, the examples they providedillustrated how university probability was not required to solve everydayproblems. This indicates that the participants’ immediate thinking abouteveryday problems had very little to do with university probability.By probing the participants’ views about the role played by universityprobability in everyday problem-solving, this study has provided additionalinsights into students’ conceptions of probability. Their examples of probabilityproblems from their everyday contexts were useful for the researcher in probingtheir conceptions further.Discussion of critical issues arising from the studyIn Chapter 2, it was noted that the major thrust of research into students’conceptions of probability was concentrated at elementary and junior high schoollevels. A relatively small number of research projects were also carried out withfirst year college students. This study was unique in that it focused on preserviceteachers who had strong backgrounds in mathematics. This study contributes tothe literature by raising two different issues. First, it highlights the role ofcontexts in students’ methods of solving problems and second, it providesinsights into the relationship between an understanding of formal probabilityand its use in solving everyday problems.-133-Role of contexts in solving problemsThis study has provided information regarding preservice teachers’understanding of probability by portraying how they think differently aboutproblems in different tasks and settings. Their solutions to the problemsdepended on tasks and settings and also depended on their personal andinstitutional experiences.Many of the participants used non-formal conceptions in solving everydayproblems, but they began to shift toward formal ones when they were askedwhat answer they would give on a mathematics examination. They indicatedthat, for examination purposes, they would suppress their non-formalconceptions because only formal conceptions are valued in that context. Thissuggests that to be successful in an academic environment, they believed theyshould be able to suspend their non-formal conceptions.Although the participants indicated their tendency to suspend their non-formal conceptions for the purpose of university examinations, some non-formalconceptions of probability do not conflict with formal conceptions, and theformer may help in the learning of the latter. For everyday purposes like waitingfor a bus and parking a car the calculation of formal probability may appearridiculous, although a calculation can be done. In order to solve such problems anon-formal conception of probability appears to be enough. In more abstractproblems, formal probability may be essential. Instead of devaluing students’non-formal conceptions, educators may serve their students better by makingthem aware that competing conceptions of probability and their meaningful usemay depend on contexts.- 134 -Understanding probability and using it in everyday lifeThe participants in this study stated that their probability courses did notvalue their experiences and intuitions, and thus they were not confident insolving everyday probability problems from a formal perspective. This viewexpressed by the preservice teachers, that their probability learning was notmeaningful because they never had an opportunity to reflect on theirexperiences, is consistent with the arguments presented by many constructivisteducators (Ausubel, Novak, & Hanesian, 1978; Bishop, 1988; Confrey, 1990a, b;Fischbein, 1975, 1987). These educators argue that students come to class withtheir own prior experiences and they cannot develop a conceptual understandingof formal knowledge if their prior knowledge is not elicited and extended.The basic premise of a constructivist view of learning is that students’prior experiences should be recognized, respected, and used in teaching.Otherwise, students will continue to show incoherence, inconsistency, andfragmented knowledge of subject matter as was evident in this study.Probability courses for preservice teachers therefore should aim at a strongconceptual emphasis by providing opportunities for them to connect formalknowledge to their everyday experiences in the context of social interaction inthe classroom. As indicated earlier, the participants in this study argued thatformal probability would have been more meaningful to them if the teaching hadattempted to establish a link between everyday chance phenomena and formalprobability. This view is consistent with the views argued by educators such asGallimore and Tharp (1990) and Hennessy (1993). Stigler and Baranes (1988)have argued that students can be comfortable in solving school problems andapplying them in the real-world only if school activities are similar to thepractical activities performed in day-to-day life.-135-The arguments provided by the participants in this study and by othereducators imply that if a link can be established between students’ non-formaland formal conceptions, there is some hope that students can understand formalconcepts and in turn be able to apply that knowledge to problems that theyencounter in their everyday lives. However, as Walkerdine (1990, 1994) argues,educators should be aware that the everyday practice of mathematics isdiscursively different from school practice and the link between these two is notreadily obvious. For example, the participants’ examples of everyday chancephenomena, such as waiting for a bus, parking a car, and the rise and fall of stockprices do not follow a mathematical principle of randomness, so it is difficult toconnect this to formal probability learned in school and university. Whether theuse of preservice teachers’ everyday experiences in the teaching of probabilityhelps them solve their everyday problems is a question that was not directlyaddressed in this study and would need to be pursued.It is important that mathematics programs for prospective teachersprovide an opportunity for them to make sense of what it means to learnmathematics. This is different from the practice of simply learning definitions,rules, formulas, and routine applications that seems to be prevalent in theteaching of mathematics from school to university levels (Battista, 1994). Aprogram of mathematics should emphasize problem-solving “in an environmentthat encourages students to explore, formulate and test conjectures, provegeneralizations, and discuss and apply the results of their investigations” (NCrM,1989, p. 128). Students’ intuitions and everyday experiences do not alwayshinder such mathematics learning but they could be helpful in making suchprograms meaningful to students (NCTM, 1989).The quotation from the NCTM Standards emphasizes a new vision ofmathematics learning that enhances conceptual understanding. Many of the-136-participants argued that they could not use formal probability in everydayproblem-solving because they did not understand it conceptually. There is notenough evidence from this study to suggest that an emphasis on conceptualunderstanding helps students in effectively solving everyday problems.However, it is certainly an interesting area for further study because manymathematics educators have argued that the learning of mathematics shouldemphasize conceptual understanding (Hiebert & Carpenter, 1992; Skemp, 1976)while others have stressed that one of the purposes of doing mathematics is to beable to solve everyday problems by using it (Dörlfer & McLone, 1986; Jacobsen,1989; Paulos, 1989).Implications for teacher educationEducators who argue for the teaching of probability in schools anduniversities are interested in understanding how prospective and practicingteachers conceptualize probability (Shaughnessy, 1992). There is very little in theliterature to inform educators about how preservice teachers with strongbackgrounds in mathematics think about probability. This study provides aportrayal of two groupings of qualitatively different conceptions of probabilitythat a group of preservice teachers held. The various conceptions demonstratedby the participants in this study can inform both mathematics professors andmathematics education professors about how people with reasonably strongbackgrounds in mathematics solve probability problems.The study provides evidence that university students, even after takingmany mathematics courses, including a probability course, continue to use nonformal reasoning to make sense of problems related to probability. Insightsgained into how probability is conceptualized by preservice teachers canfunction as points of discussion, both during the teaching of subject matterknowledge of probability in university mathematics or statistics departments and-137-the teaching of pedagogical content knowledge of probability in mathematicseducation departments. Some instructors may find that their own students mayhold conceptions of probability similar to those identified in this study. Since theconceptions in this study are identified from the preservice teachers’ reflectionupon their everyday and mathematical experiences they are likely to bemeaningful and relevant to other preservice teachers who have similarexperiences. All this information will help instructors plan and teach courses inprobability and probability education.By knowing some of the different ways university students conceptualizeprobability, mathematics professors may respond to their students’ difficulties inunderstanding formal probability by providing alternative explanations of theprobability concepts to be taught. Mathematics education professors may be ableto help preservice teachers by addressing the differing conceptions of probabilityheld by other preservice teachers and also held by school students. Students andprofessor may wish to explore such questions as: To what extent do school anduniversity students’ conceptions differ and what do these differences pose forteaching? Furthermore, mathematics professors may be willing to discuss whattypes of probability conceptions their own group of students hold and whetheror not these conceptions are mathematically correct. Since students’ conceptionsserve as a powerful force in their understanding of mathematics (Borasi, 1990),discussion of these conceptions may prove to be beneficial for preserviceteachers’ understanding of probability. The diversity of the preservice teachers’conceptions elicited in the study provides a rich context for examining theteaching of probability. For example, students in a probability classroom couldbe asked to evaluate the appropriateness of using everyday experiences orscience knowledge in solving probability problems such as the lottery and caraccident task and the thumbtack task posed in this study.138-Furthermore, mathematics and mathematics education professors canobtain insights into their own students’ conceptions by using varieties of tasks indifferent settings such as those presented in this study. The eliciting ofconceptions held by their students may help professors to extend students’experience-based thinking to more formal mathematics.Implications for curriculum and instructionThis study has implications for the field of curriculum and instruction. Aquestion that arises from this study is the meaning of literacy in probability:What does it mean to be literate in probabffity? Are university mathematicsgraduates literate in probability? Can a person be considered literate if he or sheknows how to solve textbook probability problems but cannot apply them insolving everyday problems? Can someone be called literate if he or she can solveeveryday problems without using formal probability? The works of Dewdney(1993) and Paulos (1989, 1994) imply that being literate in probability meansbeing able to apply formal probability to solve problems related to chancephenomena that people encounter in their everyday lives. If being literate inprobability is being able to apply it in solving everyday chance phenomena byusing formal probability, then schools and universities do not seem to helpstudents to be literate in this regard, as expressed by the participants in thisstudy.However, if being literate means to be able to solve everyday problemswithout using formal probability, then many people without any knowledge offormal probability can still be considered literate in probability. There may bepeople without training in formal probability who are better at predictingphenomena in real-world situations, such as weather forecasting, stock prices,interest rates, etc, than the ones who have a good knowledge of formalprobability.-139-This finding has an important implication regarding the nature ofprobability and its teaching at school and university levels. If people with strongbackgrounds in mathematics and probability do not find their knowledge ofmathematics useful in the solution of everyday problems, why would ordinarypeople and students with little mathematics background do so? Perhaps morefundamentally, is a formal conception of probability useful in an everydaycontext? If the purpose of teaching probability is to help students solveformulated textbook-like problems, then an emphasis on formal conceptions ofprobability is certainly necessary. However, if the purpose of teachingprobability is to help students in solving problems that can be encountered inday-to-day life, then more than an emphasis on formal conceptions seemsneeded,Based on the findings of this study, a question can also be raised about aconceptual change view of learning. In classroom teaching, students andteachers bring differing cultural background, and personal experiences (Bishop,1988) which significantly affect classroom discourse. Because of this socialinteraction, students may demonstrate a change of conception in classrooms andexaminations. But in everyday thinking, they may continue to adhere to theirnon-formal conceptions. The results of this study provide evidence that studentsnever entirely dismiss their original conceptions. Even after studying severalcourses in university mathematics and a course in university probability, theparticipants in this study continued to utilize their everyday non-formalconceptions while solving problems related to probability.-140-Recommendations for further researchIt could be expected that the preservice teachers might demonstratedifferent conceptions of probability if their probability courses had providedthem with an opportunity to compare their non-formal and formal conceptionsof probability. However, how students would develop their understanding offormal probability if their non-formal conceptions were explored and usedduring teaching has not been researched. A useful project for further studywould involve eliciting students’ conceptions of probability prior to instructionand exploring how those conceptions would change over time if their non-formalconceptions were used in teaching.The participants’ responses in this study were a function of tasks andsettings. A second project could be to conduct a study of students’ views aboutprobability tasks and settings. It would be interesting to investigate whystudents’ conceptions are non-formal during certain types of tasks and settingsand formal in others. Exploring students’ conceptions of tasks and settings canbe beneficial for teachers and professors who teach probability because theresults from such studies may help them in selecting interesting, thoughtful, andchallenging tasks for their teaching to be used in various settings.A third implication of this study could be to introduce an undergraduatemathematics course for future teachers that emphasizes eliciting theirconceptions through interviews and journal writing, and using the informationthus obtained in teaching. Important research questions can be: What dostudents think about mathematics learning and teaching prior to instruction andhow do their views about mathematics evolve and change during and afterinstruction? 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An exploratory survey of teachers of mathematicsin the state upper secondary schools in Italy: Some results. In R. Davidson &J. Swift (Eds.), Proceedings of the Second international Conference on TeachingStatistics (pp. 120-126), Victoria, BC.: University of Victoria.-159-Appendix A - Written componentBackground InformationCircle either Yes or No in the following questions. If the questions are notfollowed by Yes or No, please write your responses.Student’s ID: Sex: M/FAre you a preservice teacher? Yes / NoAre you an inservice teacher? Yes / NoHow many university graduate or undergraduate mathematics courses have youtaken?Your highest academic qualifications:Have you taken any statistics course? Yes / NoHow many courses?At what levels?Have you taken any probability course? Yes / NoHow many courses?At what levels?Have you had any teaching experience? Yes / NoHow many years of teaching experience?Have you ever taught probability? Yes / NoIf Yes, how many years?At what levels?-160-Written tasksBelow* are some multiple-choice and open-ended questions. Pleaseanswer each question and provide your reasoning. Not only is your final answerimportant but also the method you used (or would use) to solve each of theseproblems.1. Birth sequence taskA couple plans to have six children. Which of the following orderlysequence of birth is most likely? (Boys = B, Girls = G).(a) BGBBBB(b) BBBGGG(c) GBGBBG(d) BGBGBG(e) All four sequences are equally likelyGive reasons for your choice.(Modified from Kahneman & Tversky, 1972 and Konold et al., 1993)*In the actual research the tasks were provided on separate sheets of paper.-161-2. Rectangular spinner taskIn the diagram above OA is a spinner. If it is spun a large number of timeson which number is it most likely to stop.(a) 1(b) 2(c) 3(d) 4(e) 5(f) The spinner will stop equal number of times in each case.Give reasons for your choice,3. Five-head taskA fair coin is flipped five times, each time landing with heads up. What isthe most likely outcome if the coin is flipped a sixth time?a) Another head is more likely than a tailb) A tail is more likely than another headc) The outcome (head or tail) is equally likelyGive reasons for your choice.(Adapted from Konold et al., 1993)-162-4. Lottery and car accident taskWhich one is more likely?(a) You will win a jackpot in BC lottery 6/49(b) You will be killed in a car accidentDescribe how you would solve this problem.5. Circular spinner taskIn the diagram below OA is a spinner. Four people are playing this game.Each person will pick two numbers from 1—8 and if the spinner stops at any ofthe numbers picked the person wins. If you are playing this game which twonumbers will you pick to have a best chance of winning? Why?-163-Appendix B - Pair-problem solvingBelow* are some multiple-choice and open-ended questions. Pleaseanswer each question and provide your reasoning. Not only is your final answerimportant but also the method you used (or would use) to solve each of theseproblems. Please speak aloud whatever you think so that your thinking can berecorded.1. Head-tail sequence taskWhich of the following orderly sequences is most likely to result fromflipping a fair coin six times? (Heads = H, Tails = T).a) HTHHHHb) J-ll-u-lrrrc) THTHHTd) HTHTHTe) All four sequences are equally likelyGive reasons for your choice.(Modified from Kahneman & Tversky, 1972 and Konold et al., 1993)2. Lottery and car accident taskWhich one is more likely?(a) You win a jackpot in BC lottery 6/49(b) You will be killed in a car accidentDescribe how you would solve this problem.*In the actual research the tasks were provided on separate sheets of paper.-164-3. Five-boy taskJohn and Tracy live in the interior of British Columbia. They have fivechildren, all boys. Both John and Tracy want to have a daughter. Since theyalready have five boys John and Tracy think that their sixth child would morelikely to be a girl. Do you agree with John and Tracy? Why, or why not?4. Circular spinner taskIn the diagram below OA is a spinner. Four people are playing this game.Each person will pick two numbers from 1—8 and if the spinner stops at any ofthe numbers picked the person wins. If you are playing this game which twonumbers will you pick to have a best chance of winning? Why?5. Bank teller taskA local bank has two teller windows open to serve customers. Thenumber of customers arriving at the bank varies between one and six customersper minute. Customers form a line, and the person at the front of the line goes tothe first available teller. Each teller services one customer per minute. If youwere the manager of the bank and you wish to make sure the average waitingtime is not more than 3 minutes, would you increase or decrease the number oftellers?(Adapted from Gnanadesikan, Scheaffer, & Swift, 1987, p. 27)-165-Appendix C - Interview protocol1. Head-tail sequence taskWhich of the following orderly sequences is most likely to result fromflipping a fair coin six times? (Heads H, Tails T).a) F1THHHHb) Hi-iHrrrc) THTHI-ITd) HTHTHTe) All four sequences are equally likelyWhy?(Modified from Kahneman & Tversky, 1972 and Konold et al., 1993)2. Lottery taskTwo students in a probability class have opposing views about selectingnumbers for a lottery 6/49. One student prefers to choose consecutive numberslike 1, 2, 3,4,5, 6. Other student thinks that chance of getting a sequence of sixconsecutive numbers in a lottery is smaller than getting a random sequence ofany six numbers. Both students come to you for an advice. How would youadvise them?(Adapted from Fischbein & Gazit, 1984)3. Board and marble taskA wooden board with different channels as given in the diagram belowwas shown to each preservice teachers.- 166 -When the preservice teachers observed the board and the channels thefollowing question was asked: “If a marble is dropped from the top of the mainchannel where do you think the marble will come out at the bottom?”(Adapted from Fischbein, 1975)4. Dice taskThree dice with each face numbered 1 through 6 were shown to eachpreservice teacher. He or she was told that all the dice would be rolled. Theperson who rolled the dice was the dealer and the person who placed bets wasthe player. The rules of the game were the following:“The player can choose any number 1 to 6. Suppose the player bets $1 forface 1. If face 1 turns up on any of the dice then he or she will be paid $1by the dealer. If face 1 turns up on two dice then he or she will be paiddouble the amount he or she bets, $2 in this case. If face 1 turns up onthree dice then he or she will be paid three times the amount he or shebets, $3 in this case.1 23 4-167-The player can go on doubling the bet each time he or she loses. Forexample, if the player bet $1 for the first time and lost he or she can bet $2for the second time, $4 for the third time, and so on. The player and thedealer each has a total of $100 to play.”When the preservice teacher read the rules the following questions wereasked: If you have an opportunity of becoming a dealer or a player which wouldyou choose? Why?5. Thumbtack taskEach preservice teacher was shown a thumbtack and shown two positionsof its landing on a wooden table, with the point down or with the point up asshown in the diagram below.4:7A BWhen the preservice teachers observed the thumbtack the followingquestion was asked: Which one (landing with the point down or point up) ismore likely? Why?(Adapted fromWilloughby, 1977, p. 106)6. Question about the relevance of probability to solve everyday problemsHow often do you use probability in your everyday lives? Do you thinkthat university or school knowledge of probability is useful in solving everydayproblems?168 -Appendix D - Extract from John’s interview transcriptsThe head-tail sequence taskHan: What’s your answer to this problem [shows the head-tail sequencetask]?John: (e).Han: Why did you say (e)?John: First of all throwing a head or a tail is equally likely. And the factthat going from the first trial to the second trial to the third trial orwhatever trial to the next trial they are independent. Therefore youcalculate equal probability for each of those.Han: Back in the written component you said something different [showshis response to birth sequence task], Do you remember this?John: Yes, I do [laughs]. I changed my mind.Han: Do you mean that even with 5 heads and I tail in one sequence and3 heads and 3 tails in the other sequence does not make anydifference?John: Urn-. I still have some doubt, I guess, obviously about choosingthat they are all equally likely. I still have to say that they are allequally likely, but I’m not quite sure if my answer is right. Thereare different ways you can look at it. I’m not sure which is the rightway. The mathematical way is heads and tails are equally likelyand each event is independent and you come to the conclusion thatthey are all equally likely. Or you could take a count of how manyheads and tails you get. For (b), (c), and Cd) [sequences HHHTIT,THTHHT, and HTHTHT] you got 3 heads and 3 tails. So those willhave the same probability. And ... if you think that in 6 throws of acoin you should get 3 heads and 3 tails is a fallacy. You could getanything really.Han: Do you think that all the sequences would be equally likely, if thetotal number of trials was 20 and in (a) I had 19 heads and 1 tail andsomewhere, in (c) for example, I had 12 heads and 8 tails or 10heads and 10 tails? [shows the examples of sequences HTHI-IHH...,HTHTHHHTHTHHH.. .1- 169 -John: Urn. No, I think it changes as the number of trials increases. Butthe thing is it’s a fallacy to assume that you get 10 heads and 10 tailsin 20 trials.Han: I understand that. But what about comparing those 19 heads and 1tail and 12 heads and 8 tails? Do you think that they’re equallylikely?John: No. Intuitively I’d not think so. I would expect that 12 heads and 8tails would be more likely. I’m trying to think how you cancalculate the probability of these sequences mathematically, but Ikind of forgot. [shows the head-tail sequence task] Intuitivelyspeaking, I’d still think that these three sequences (b), (c), and (d)[sequences HHH1Tr, THTHHT, and HTHTHT] are more likely than (a)[HTHHHH], that (a) is less likely. Urn-, but that’s intuitive. I’mthinking that any of these sequences are equally likely because thenumber of trials you’ve got is small, you only have six trials; andthis whole thing that the trials being independent and theprobability of heads and tails being equal, I just have to say they’reequally likely.Han: Even in 20 trials?John: If you extend this to 20 trials, I might change my answer. If youincrease the number of trials, I would go more with my intuitionand say those sequences have roughly half heads and half tails.What I would try to do is somehow reflect that in the mathematics.But right now I can’t quite figure out how to reflect.Han: Can you tell me the probability of getting this sequence [showssequence a]?John: 1/26Han: This sequence [shows the sequence b]?John: 1/26. They’re all 1/26.Han: If you look at these sequences their orders are different from eachother. Do you still think that they’re equally likely?John: Yes.Han: Which of the above sequences [shows a), b), c), and d)J is leastlikely?John: You’re asking me the same question in a different way.-170-The lottery taskJohn: [reads the question] One student wants to select consecutivenumbers. Another student thinks that the chance of getting asequence of six consecutive number is smaller than getting arandom sequence of six numbers. So one thinks the consecutive ishigher and the other thinks the random is higher. I’d say Urn-Han: No, the first student doesn’t say that it is higher. S/he just prefers.But the second student thinks that consecutive sequence hassmaller chance than the random sequence.John: I would probably talk to them about different ways of looking atthem. You could look at it mathematically and say that theprobability of getting a certain number is equally likely as gettingany other number. So in a sense it doesn’t matter which numberyou pick. You know you are still going to have the same chance ofwinning as picking any other numbers. Then again, you can look atthe past history of the numbers drawn from 6/49. You can keeptrack of what the winning numbers are and then look for trends orpatterns of the numbers that were chosen a lot. You can come upwith certain numbers that seem to be chosen quite frequently.Han: That’s interesting. “Luck” magazines publishes all the jackpotwinning numbers so far. But there is not a single consecutivenumber which has won the jackpot so far.John: I’d definitely look at that. That’ll cast some doubt on pickingconsecutive six numbers.Han: Not a single consecutive number has won the jackpot so far. Whatdo you think now?John: That’s actually crossed my mind. You know whenever I see thelottery winnings, I’ve never seen the consecutive numbers. Urn-,I’m getting into ethics here. I would have to agree with the otherstudent who thinks that the chance of getting a sequence of sixconsecutive numbers in the lottery is smaller than getting a randomsequence of any six numbers. Then again, if I’m advising thesestudents I would only present the information and let them come toa decision.Han: What’s your personal thinking?-171-John: A random sequence is more likely than choosing six consecutivenumbers. That’s my opinion.Han: What’s the reasoning for that?John: I’m just basing my decision on the fact that intuitively sixconsecutive numbers seem unlikely to me and by the informationyou’ve just provided it has never happened. My reasoning ispurely based on intuition. Mathematically speaking, any sixnumbers is equally likely, right? There’re mathematical ways oflooking at things and there is your intuition. Sometimes you havegot to separate the two or make compromises or whatever I don’tknow.The thumbtack taskHan: [shows a thumbtack] If I throw, or roll this thumbtack on this tablethere are two possibilities of landing: with the point down or withthe point up [shows the position on the table]. Which one (landingwith the point down or point up) is more likely?John: (b)Han: Why do you think so?John: Because, as it’s falling most of the weight is here [shows the tackpart]; it’s in the head. So the heavy part will go on the bottom.Han: What about bouncing factors?John: Oh! that’s a good question. Oh! dear, I’ve never thought of that one.[tries rolling the tack]. It [bouncing] can make anything. So I’llprobably just do like many trials, records and just base on that. I’llnot try to theorize it. [keeps on trying with the tack] I’ll not try toanalyze this bouncing behaviors, that’s absolutely ridiculous! Lookat that spinning! it’s too complicated. I’ll just keep track of theoutcomes over many times.Han: What about the heights? Does throwing or dropping from differentheights make the difference?John: I’ll just do whole bunch of trials with different heights and stratifythem.Han: How many trials do you need?John: Thousands, it’s just lots!-172-

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