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Writing to learn mathematics: student journals and student-constructed questions Menon, Ramakrishnan 1993

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WRITING TO LEARN MATHEMATICS: STUDENT JOURNALS ANDSTUDENT-CONSTRUCTED QUESTIONSbyRAMAKRISHNAN MENONB.A., University of Malaya, 1973B.Sc., Univesity of London, 1974M.A., University of Northern Iowa, 1983A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY(CURRICULUM AND INSTRUCTION)inTHE FACULTY OF GRADUATE STUDIES(Department of Mathematics and Science Education)We accept this dissertation as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1992© Ramakrishnan Menon, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of MFMCS 4S SCIENCE GI. (CUitt./COCuM sg I Nriku cri 0A4)The University of British ColumbiaVancouver, CanadaDate telp^( et, 1 19 3DE-6 (2/88)ABSTRACTThe purpose of the study was to investigate the mathematicallearning indicated by using student journals (SJ) and student-constructed word problems (SCQ) in a mathematics class where theteacher initiated the procedures and conducted the lessons. The classwas a multi-level Grade 5 and Grade 6 class but only the Grade 6students were the focus of the study.Three times a week, towards the end of their mathematics lessonstudents wrote in their SJ in response to teacher prompts. Once a week,the students also prepared SCQ, in groups and individually. These SCQwere collected, edited, typed and redistributed to the class by theteacher, and used as class exercises. Six students were interviewed(video-taped) three times each over the 15-week period of study.Records of these interviews, classroom observations and a teacherinterview were all used to complement the analysis of SJ and theindividually-prepared SCQ.Although the SJ did give insights about students' mathematicalknowledge, students' oral explanations indicated that they understoodmore than what was written in their SJ. Hence, the lack of ability tocommunicate through written words in the SJ was not an indicator ofstudent's mathematical understanding.In contrast, the SCQ indicated students' knowledge of fractionsbetter, both (a) implicitly, through the type of question asked, informationgiven, and the accompanying solution and (b) explicitly, through the typeof fraction relationships, the amount of detail, and the number of stepsand operations involved. The SCQ also revealed that students tended toiibase their word problems on (a) their own experiences and interests,(b) an assumption of shared knowledge between the reader and writer ofthe word problem, (c) numbers which made computation easy and (d) thediscrete model of fractions rather than the region model.The results of this study indicate that the SCQ assistedmathematical learning in a classroom context but that the value of the SJneeds to be reconsidered.iiiTABLE OF CONTENTSABSTRACT^ iiTABLE OF CONTENTS^ ivLIST OF FIGURES viiiACKNOWLEDGEMENT^ xiCHAPTER 1. INTRODUCTION^ 1Background^ 1Rationale 3Purpose of the Study^ 8Research Questions^ 8Overview of Study^ 9CHAPTER 2. REVIEW OF LITERATURE^ 10Writing and learning^ 10Writing to learn mathematics^ 13Mathematics Learning and Language Learning^ 16Mathematics Learning and Pedagogy^ 23Mathematics as a Social Enterprise 29Journals^ 32Journals in Language Arts^ 32Mathematics Journals 33Writing explanations of mathematicaldefinitions, procedures or solutions^ 34Writing students' feelings about, and attitudestowards, mathematics^ 36Writing summaries 38ivSome Cautions about Mathematics Journals^40Student-constructed questions (SCQ)^ 42Significance of the Study^ 46Conclusion^ 49CHAPTER 3. METHOD^ 50Context of the study 50My Background^ 50The Previous Project 51The Teacher's Background^ 52Linda's Growing Professional Development^53Linda's Commitment^ 57The Present Research Setting^ 57Student Participants^ 59Teaching-Learning Environment^ 61Procedure^ 65Data analysis 68Analysis of SJ^ 68Analysis of SCQ 69Conclusion^ 70CHAPTER 4. RESULTS AND DISCUSSION^ 71Research question #1 and results 71Part-whole (Common Fractions)^ 72Equivalence and Renaming (Common Fractions)^78Decimal Fractions and Inter-relationship betweenCommon and Decimal Fractions^ 87vAddition of Common Fractions^ 96Word Problems (Decimal Fractions) 100Rule for Placement of Decimal Point in DecimalMultiplication^ 103SCQ and Mathematical Knowledge^ 108Summary of results pertaining to research question#1 ^ 117Research question #2 and results^ 121Theme 1use of student experience and interest^ 122Theme 2assumption of shared knowledge^ 125Theme 3The use of numbers which made computationeasy^ 128Theme 4use of discrete rather than region models^ 132Summary of results pertaining to research question#2^ 134Research question #3 and results^ 135Students' View of the Role of SJ in MathematicsLearning^ 135Clarification of ideas^ 138Feedback^ 140Students' View of the Role of SCQ in MathematicsLearning^ 141To get a better understanding^ 142viPractice in solving problems^ 142To provide questions at an appropriate level.^ 143Teacher's View of the Role of SCQ in MathematicsLearning^ 144Teacher's View of the Role SJ in MathematicsLearning^ 147Summary of Results Pertaining to Research Question#3^ 150CHAPTER 5. CONCLUSION^ 151Summary of study 151General discussion^ 152Limitations of the study 156Implications for practice^ 157Possibilities for future research^ 160BIBLIOGRAPHY^ 164APPENDIX 178Interview 1^ 178Interview 2 179Interview 3^ 180LIST OF FIGURESFigure 4.01: Jackie, January 7^ 72Figure 4.02: Mike, January 7 73Figure 4.03: Sam, January 7^ 73Figure 4.04: Tammy, January 7 74Figure 4.05: Tommy, January 17^ 75Figure 4.06: Shaun, January 17 75Figure 4.07: Tanya, January 17^ 76Figure 4.08: Tommy, February 12 79Figure 4.09: Jackie, February 12^ 80Figure 4.10: Maureen, February 12 80Figure 4.11: Maureen, February 12^ 81Figure 4.12: Darlene, February 12 81Figure 4.13: Sam, February 12^ 82Figure 4.14: Tommy, February 17 83Figure 4.15: Jackie, February 17^ 84Figure 4.16: Shaun, February 17 84Figure 4.17: Tommy, January 21 ^ 87Figure 4.18: Darlene, January 21 88Figure 4.19: Tanya, January 21 ^ 88Figure 4.20: Tommy, February 19 90Figure 4.21: Jackie, February 19^ 91Figure 4.22: Mike, February 5 92Figure 4.23: Darlene, February 5^ 92Figure 4.24: Tommy, February 26^ 94Figure 4.25: Tanya, February 26 94Figure 4.26: Darlene, February 26^ 95Figure 4.27: Tommy, March 5 97Figure 4.28: Maureen, March 5^ 97Figure 4.29: Shaun, March 5 97Figure 4.30: Tommy, March 13^ 100Figure 4.31: Mike, March 13 101Figure 4.32: Tommy, March 25^ 103Figure 4.33: Mike, March 25 104Figure 4.34: Maureen, March 25^ 104Figure 4.35: Darlene, March 25 104Figure 4.36: Darlene, January 7^ 109Figure 4.37: Jackie, February 10 109Figure 4.38: Mike, February 19^ 109Figure 4.39: Tanya, January 24 111Figure 4.40: Tanya, February 6^ 112Figure 4.41: Anne, February 10 112Figure 4.42: Anne, March 23^ 113Figure 4.43: Tommy, January 27 113Figure 4.44: Tommy, February 10^ 114Figure 4.45: James, January 24 114Figure 4.46: James, March 4^ 115Figure 4.47: Jackie, January 24 115Figure 4.48: Jackie, February 10^ 116ixFigure 4.49: Anne, March 2^ 122Figure 4.50: Shaun, February 19 122Figure 4.51: Jackie, March 3  ^ 123Figure 4.52: Tommy, February 7^ 123Figure 4.53: Jackie, March 2 123Figure 4.54: Sam, March 4^ 124Figure 4.55: Shaun, January 27 125Figure 4.56: Shaun, February 10^ 126Figure 4.57: Maureen, March 31 126Figure 4.58: Mike, March 4^ 128Figure 4.59: Tanya, February 6 129Figure 4.60: Tommy, March 4^ 129Figure 4.61: Tommy, February 10 129Figure 4.62: Mike, February 19^ 131Figure 4.63: Jackie, March 31 133Figure 4.64: Anne, February 6^ 133xACKNOWLEDGEMENTI wish to thank my wife Sutamati, for ensuring that I could devotemost of my time to my studies by taking care of myriad family and non-academic matters. But for her assistance I would not have been able tocomplete my studies in the time that I did. I also wish to thank my sister-in-law, Radha, who made it possible for me to pursue my studies here inCanada, by being my financial guarantor. To my dear departed youngerbrother, Gopal, who helped me right from my undergraduate days, I owea debt of gratitude that is impossible to repay.I wish to thank Dr. Douglas Owens, who, in addition to being mysupervisor, helped me find accommodation and also provided financialsupport in the form of a research assistantship. Dr. Ann Anderson wasexemplary as a member of my thesis committee by providing invaluableand timely feedback on my drafts, and was never too busy to help me, inspite of her busy schedule. Dr. Jim Anderson, the other member on mythesis committee, not only provided prompt and valuable feedback, butwas extremely helpful in providing me with resources to enhance mystudy. To Dr. John Willinsky, Dr. Gaalen Erickson, Dr. Jim Gaskell, Dr.Ken Clements and Dr. Nerida Ellerton I wish to express my appreciationfor their constant encouragement and support.I wish to thank the Richmond School Board, the principal and staffof the school, and especially the participating teacher and her class fortheir help in conducting this study. To my fellow students, and especiallyto Sandra Crespo who helped video tape some of the interviews, I wishto express my heartfelt gratitude. Finally, I wish to thank the clerical andtechnical staff of the department, especially Diana Colquohon, for theirpatience and help.xiCHAPTER 1. INTRODUCTIONBackgroundJust as the fashion world sees ever changing styles and fabrics,mathematics curricula seem to pass through many fads. For example,the "modern mathematics" of the late 1950s was followed by the "back tothe basics" mastery learning movement, which was itself soon replacedby a "problem solving" focus in school mathematics in the 1980s(Clements, 1989). Currently, "constructivism" (for example, Cobb, 1986)and, to a lesser extent, "writing to learn mathematics" (Connolly & Vilardi,1989) seem to be in vogue.This current interest in writing to learn mathematics was sparkedby the "writing across the curriculum" movement (Fulwiler, 1987) whichowed much to the earlier established theoretical links between writtenlanguage and learning (Emig, 1983; Luria & Yudovich, 1971; Smith,1982; Vygotsky, 1962). Prior to this burgeoning interest in writing to learnmathematics, many scholars and educators perceived language andmathematics as having little in common. For example, Snow (1965)reflected the feeling of the times when he differentiated scientific culture(as embodied in mathematics and natural sciences) from humanisticculture (as embodied in literature and the arts). He commented thatmembers of the scientific community rarely communicated with membersof the literary community, or vice versa and each were unaware anduncaring of the other's achievements. He lamented the lack ofcommunication between these two cultures saying:I believe the intellectual life of the whole of western society isincreasingly being split into two polar groups. . . . Literaryintellectuals at one pole--at the other scientists, and as the most1representative, the physical scientists. Between the two a gulf ofmutual incomprehension. (pp. 3-4)However, it is now becoming more common not only to talk about thesimilarities and differences in language and mathematics learning (Zepp,1989), but also to explore how writing can assist both languagedevelopment (Emig, 1983; Smith, 1982) and mathematical development(Ellerton, 1989; Ellerton & Clements, 1991). Writing to learnmathematics, then, seems a step in the right direction to bridge the long-perceived dichotomy between language and mathematics, and hencebetween the two cultures Snow identified.In addition to the links between mathematics and languagedevelopment, many scholars perceive learning as recursive and notsimply the accretion of discrete pieces of knowledge (Woditsch &Schmittroth, 1991). Such a view emphasizes that the process of learningis as important as the product of learning. As instances of this, we havephilosophers (Dewey, 1933; Rorty, 1969); linguists and languageeducators (Atwell, 1990; Austin, 1975; Berthoff, 1981; Britton, Burgess,Martin, McLeod, & Rosen, 1975; Elbow, 1981; Graves, 1983; Kirby &Liner, 1988; Macrorie, 1988; Moffett, 1981; Pimm, 1987; Searle, 1969;Shaughnessy, 1977); social historians (Havelock, 1963; Ong, 1982);psychologists (Bruner, 1986); anthropologists (Geertz, 1973); andscientists (Kuhn, 1962).Connolly (1989) sums up the importance of process in learning, bydrawing from social constructivist (Ernest, 1991) and socio-psycholinguistic (Gawned, 1990) perspectives of the learning process,saying:First, . . . knowledge is socially constructed within a community, notdiscovered raw in nature by individual intellects; and second, that2the agents of construction are the symbol systems through whichpeople "make meaning"--musical, mathematical, graphic, kinetic,but most important, verbal. (p.4)Applied to the idea of writing to learn, the views above imply that onelearns to develop and clarify concepts through the process of writing(both simultaneously with, and antecedent to conceptualization) ratherthan illustrating clearly-learned concepts through writing as a product(subsequent to conceptualization). Hence, approaches to learningmathematics through writing seem to be promising, especially as theyare related to the well-established writing-learning link as well as theinterest in process-learning.RationaleRose (1989) attests that "despite the apparent disparity betweenthe fields of writing and mathematics, a review of recent literature revealsa growing interest in the relationship of writing to teaching mathematics."(p. 17). Indeed, many studies on the theme "writing to learnmathematics" have been undertaken over the past decade and varioussuggestions have been made about the types of writing which mightassist mathematics learning.These suggestions include summaries, questions, explanations,definitions, reports and word problems (King, 1982); freewriting (Burton,1985; Countryman, 1987; McMillen, 1986; Mett, 1987; Sachs, 1987);letters (King, 1982; Kennedy, 1985; Schmidt, 1985); admit slips (Schmidt,1985); journals (Clarke, Waywood, & Stephens, 1992; McMillen, 1986;Nahrgang & Petersen, 1986; Powell, 1986; Rose, 1989; Schubert, 1987;Shaw, 1983; Vukovich, 1985; Waywood, 1991); writing problems(Ellerton, 1988) and story maths (Ellerton & Clements, 1992).3Most of the suggestions about the types of writing which mightassist mathematics learning have come from researchers who haveconducted studies at secondary or tertiary institutions in North America,where the researchers themselves instructed the students involved in thestudy. So it is puzzling why Waywood (1991) states that most reportedresearch on writing to learn mathematics has focused on the elementaryschool level. It is not clear whether he was referring to Australian studieswhich may not have been reported in North American journals. Althoughhe states that there were only two exceptions to the focus of suchresearch on elementary schools, my readings indicate otherwise.Whether or not results from studies on writing to learn mathematicsin other countries such as Australia apply to Canada, there is a need forCanadian data because far too often mathematics educators have triedto implement the latest "trend" without considering its suitability for theirown situation (Clements, 1989). In spite of the paucity of Canadianschool-based research on writing to learn mathematics, especially inelementary schools, there are a few studies in this area. For example,Wason-Ellam (1987) reports on a Canadian grade one class usingjournals to learn mathematics but her analysis was predominantly on thelinguistic, not mathematical, features of the journal entries. She reports,for instance, on compositional features of beginners' writing such as"beginnings, patterns of organization, and closures" (p. 11).Regardless of the type of writing or level of research, very oftenpast research in schools was "someone else's agenda." There is aconcern that the approach succeeded because of the luxury of time andresources available to the researcher, who has access to commoditiesthat are normally in short supply to classroom teachers. Thus, there is a4need for such research to be complemented by research where only"normal'" classroom resources are employed.In this study, I investigate two types of writing to learn mathematics,namely student journals (SJ) and student-constructed questions (SCQ).By SJ, I mean students' in-class, individually written responses to theteacher's prompts, which mainly ask for explanations and clarifications.My use of the word "journal" is similar to that of Schubert's (1987), whereshe talks about mathematics journals for fourth graders as regular in-class written responses to teacher prompts. Similarly, Kirby and Liner(1988), use the term "class journal" to include in-class written responsesto teacher prompts in language arts classes. I use the term SJ to focuson what students write about mathematics, not around mathematics.Most of the previous studies on journal writing have given prominence tothe affective aspect of learning through exploratory and expressive(Britton et al., 1975) writing. I am not denying the importance of affect onmathematics learning, but affect is not the focus of my study.By SCQ, I mean word problem(s) constructed and written out bythe students either in groups or individually. I am using the term SCQinstead of the more common term "problem-posing," because I want toemphasize the centrality of students' direct involvement in the activity, aswell as to emphasize the use of written words in these problems.Moreover, SCQ connotes a student constructing mathematics, ratherthan writing about mathematics.I chose to study both SJ and SCQ in order to tap different aspectsof mathematics learning: the SJ, to indicate how students communicatetheir mathematics understanding to the reader through predominantlytransactional writing (Britton et al., 1975), and the SCQ to reveal the5depths of their mathematics knowledge through expressive writing (DelCampo & Clements, 1987). Britton uses the word "transactional" to meanwriting such as that used to inform or explain (writing that is "directedtowards external objectives" according to Bereiter & Scardamalia, 1987,p. 184). For Del Campo and Clements (1987), the word "expressive"indicates writing that is a result of the writer's own constructions, basedon personal experiences and referents and can be either to exploreone's feelings or to explain or both. For example, explaining in the SJwhy 0.8 # 3/5 would entail Britton's transactional writing as the emphasisis on informing or explaining something. Writing a word problem aboutcommon and decimal fractions using the writer's experiences andpersonal referents would be an instance of Del Campo and Clements'expressive writing.The two aspects of mathematics learning alluded to(communication through written words and knowledge of mathematics)are similar to, but not identical with, what Ellerton and Clarkson (1992)refer to as children writing about mathematics and children writingmathematics. While journal writing and problem posing in mathematicsclasses have been studied separately before, they have not been studiedtogether in an elementary mathematics class. For example, Burton(1985) studied affect through the use of journals in mathematics methodscourses at the college level and Ellerton (1989) studied mathematicsword problems prepared by secondary school students.My choice of both SJ and SCQ as appropriate vehicles for writingto learn mathematics is also in line with the National Council of Teachersof Mathematics (NCTM) Curriculum and Evaluation Standards (1989)where communication of mathematics as well as problem posing and6problem solving are stressed. In addition, the writing engenderedthrough the SJ and SCQ might encourage students to "fish in the river" oftheir minds (Kirby & Liner, 1988, p.58) to reflect on and communicate themathematics they have learned.Moreover, in studying both, I hope to better understandmathematics learning in specific topics such as common and decimalfractions. In short, by choosing both SJ and SCQ, I hope to explore howstudents use written language (mainly words) to communicatemathematical understanding and ideas (through SJ) and how creatingtheir own questions (through SCQ) might indicate their understanding ofcertain mathematical topics.There are several reasons for selecting student journals as one ofthe two innovations to be implemented in this study. First, many reportsof journals in mathematics have emphasized the affective domain (likestudents' feelings), and have neglected the cognitive aspect (like reportson specific mathematical gains). Second, where the journals did notemphasize feelings and attitudes, they were likely to be influenced by theprocess writing approach (for example, Graves, 1983), resulting in eitherthe narrative genre overshadowing mathematics content or the genrebeing used peripherally in the mathematics class, "after the 'real'mathematics has been done" (Marks & Mousley, 1990, p.121). Forexample, research has shown that students sometimes write stories withnumbers substituted for names of persons but without relating themathematical associations among the numbers (Del Campo & Clements,1987, p. 19).Other than the research issue discussed above, the use of SJrelates to a pedagogical issue: the implementation of an innovation in7the classroom. Since most students and teachers have already usedjournals in subjects other than mathematics, they might not feeluncomfortable using them in mathematics, too.I now turn to why I chose SCQ. First, my previous experience withSCQ (where SCQ were used intermittently in a Grade 6 class) wassufficiently positive to lead me to believe that the potential of SCQneeded to be explored. Moreover, SCQ, in providing opportunities forstudents to create and write their own problems, are providingopportunities to construct and record personal mathematical meanings(Del Campo & Clements, 1987). In addition, I feel that SCQ might beable to indicate what mathematics has been learned, as it would bealmost impossible to construct and solve questions withoutunderstanding mathematical content.Purpose of the StudyThe main purpose of the study was to investigate the mathematicallearning revealed through the use of SJ and SCQ in a mathematics classwhere the teacher herself initiated the procedures and conducted thelessons. In addition, I wanted to investigate the perceptions of thestudents and the teacher about the usefulness of writing in amathematics class and also to investigate whether there were anypatterns discernible in the word problems constructed by the studentsResearch Questions Specifically, the questions addressed in this study were:1. What mathematical knowledge do students reveal through theirwriting?82. What are the salient and recurrent features (themes) of theSCQ, if any?3. What are the roles of SJ and SCQ in mathematics learning,according to (a) students? and (b) the teacher?Overview of StudyThis study focused on the mathematical learning of Grade 6Canadian children in a multigrade (Grade 5 and 6) class where theteacher initiated two types of writing tasks (SJ and SCQ) as part of themathematics class. The teacher was willing to participate in the studymainly because she wanted to try some different approaches to a topic(fractions) she has found difficult to teach .Data for the study were obtained through what was written in theSJ and SCQ, together with individual student interviews, an interviewwith the teacher and my notes on classroom observations. Such data,together with the thick description (Geertz, 1973) provided in chaptersthree and four, encapsulating both the context and details of the studyenable conceptual (Yin, 1984) or naturalistic generalisations (Stake,1978) rather than statistical generalisations to be made.9CHAPTER 2. REVIEW OF LITERATUREIn this chapter I first review the literature on the use of writing as atool for learning in general. Then I discuss current literature in variousfields which, directly or indirectly, seem to support the use of writing tolearn mathematics. Finally, I discuss previous studies on the use ofjournals and student-constructed questions in mathematics lessons.Writing and learningFor many years, writing was thought to be a result of carefully-planned, well-conceptualized ideas. According to Emig (1983), thesebeliefs were based on three major assumptions: (a) there is a dichotomybetween planning and writing, (b) writing never precedes planning and(c) writing is the recording of the already formulated conception. Shepoints out the falsity of these assumptions by providing evidence fromwriters who agree that "planning is a tentative or sketchy affair andwriting itself is the major act of discovery of how they think and feel" (p.16).The philosopher Ryle (1949), too, agrees that it is fallacious tobelieve that for an operation to be intelligent, it has to be guided by aprevious intellectual operation. That is, Ryle denies that we always planeverything that we subsequently do successfully.Emig (1983) says writing "connects the three major tenses of ourexperiences to make meaning" by "shuttling among past, present, andfuture" using the "processes of analysis and synthesis" (p. 129). Sheargues that writing is a unique mode of learning because it is10multirepresentational, in that it uses Bruner's enactive, iconic andsymbolic modes of representation concurrently or at least contiguously.Moreover, when writing, the hand, eye and brain are all working inconcert.Emig (1983, p. 129) draws some useful correspondences betweenlearning and writing, saying that both give self-provided feedback,generate and connect concepts and are multirepresentational,integrative, active, personal and self-rhythmed. Others who echo similarbeliefs about writing and learning are Elbow (1981), Luria and Yudovich(1971), Smith (1982), Vygotsky (1962). More recently, support for suchbeliefs come from the educationists Boyer (1987) and Fulwiler (1987),the futurists Naisbitt and Aburdene (1985) and writers such as Zinsser(1988). All those cited above emphasize that the very act of writingengenders thinking.Writing has gone from an emphasis on a finished, polishedproduct to writing as a process-product, where exploratory or tentativeconcept-building is encouraged, especially in informal writing. Thisbuilding up of concepts through language and writing is emphasized byboth Barnes (1976) and Smith (1982). Smith, in discussing howlanguage and writing help learning, says that writing helps us find outwhat we know and think because "language creates as well ascommunicates" (p. 67).The "whole language" movement (e.g. Shanahan, 1991) exhibitssome of these process-oriented approaches. In this approach, childrenare encouraged to find their own voice by learning language (however"unpolished") through actively using it, rather than learning just the rulesabout how to use grammatically correct language. The advocates of this11approach believe that the very process of using language (with feedbackand guidance) brings about facility in using it correctly and appropriately.The supporters of writing to learn also point out that becausewriting forces one to focus on certain aspects of the deluge ofexperience, one has to examine these aspects in greater detail, ratherthan leave them in the nebulous wash of unattended perceptions. AsPimm (1991) puts it, "externalizing thought through spoken or writtenlanguage can provide greater access to one's own (as well as for others)thoughts, thus aiding the crucial process of reflection, without whichlearning rarely takes place" (p. 23). Similarly, Keith (1990) says,"Frequently we may think we understand something when we onlyrecognize it; we confuse familiarity with understanding. This becomesobvious when we have to explain it in writing" (p. 6). In other words,writing helps in the clarification and generation of concepts and thereforeis critical to the process of learning.In pointing out the relationship between writing and learning,Mayher, Lester, and Pradl (1983) say:Writing's capacity to place the learner at the center of her ownlearning can and should make writing an important facilitator oflearning anything that involves language. Writing that involveslanguage choice requires each writer to find her own words toexpress whatever is being learned. Such a process may initiallyserve to reveal more gaps than mastery of a particular subject, buteven that can be of immense diagnostic value for teacher andlearner alike. And as the process is repeated, real and lastingmastery of the subject and its technical vocabulary is achieved. (p.79)In short, the proponents of writing to learn believe that writing helpsthinking in two ways: one, by the process of writing, one's thinkingbecomes clearer; and two, because it is a more permanent record,12writing allows one to look more closely at what is written and revise whathas been written.In spite of the benefits of writing to learn, it seems fitting to takenote of some of the disadvantages which have been pointed out. Forexample, Pimm (1987, p.117-118) cautions that (a) expression throughthe written medium can be both time-consuming and arduous, (b) there isreluctance to write when there is no clear reason why one should write,(c) the problem of writing legibly and clearly makes one lose sight of whatone was trying to express, (d) writing can act as a brake when one iscaught up in a rush of ideas and (e) it is difficult to write from thestandpoint of an uninformed reader or to make explicit assumptionswhich are self-evident to the writer.Moreover, not all types of writing function to clarify and generateconcepts. For instance, Mayher, Lester, and Pradl (1983) claim that"writing that involves minimal language choices, such as filling-in-blanksexercises or answering questions with someone else's language--thetextbook's or the teacher's--are of limited value in promoting writing orlearning" (p.78).While the cautions against an uncritical acceptance of writing tolearn are timely, there seems to be sufficient support for using writing asa learning tool.Writing to learn mathematicsInitially, it might look odd to contemplate writing as a tool for thelearning of mathematics especially the use of everyday language in asubject usually associated with precise and abstract mathematicalsymbols. This view is not surprising given that "most children are very13good at learning and using language--they make remarkableachievements in this domain before they commence schooling and in theabsence of formal instruction--while very few children take so readily tomathematics" (Durkin, 1991, p. 4). Such a contrast in language andmathematical facility could be because everyday language can becomprehended even without knowing the meaning of every word. Incontrast, mathematics is more difficult to comprehend because it useswords which are non-redundant and have densely-packed meaning.Mathematics educators like Connolly (1989), Rose (1989) andEllerton and Clarkson (1992) have documented the rapid and increasedinterest in writing to learn mathematics in recent years. Indeed, more andmore educators perceive the similarities and differences in language andmathematics learning and believe that writing can help develop bothlanguage and mathematics learning.While it would be useful to have a comprehensive model linkingwriting to mathematics learning, such a model has not been articulatedyet. However, a model of learning that seems to link writing tomathematical problem-solving, is the Adaptive Control of Thought (ACT)model resulting from the application of computer science, informationprocessing theory and linguistics (Anderson, 1983). In this model,learning is associated with two types of memory: declarative memory,where knowledge, such as propositions, spatial images and temporalstrings are stored; and production memory, where skills based on theconstruction and addition of procedures originating from otherprocedures and propositions are stored (Kenyon, 1989, p. 75).According to Kenyon, there seems to be a relationship among theACT learning model, the three (recursive) phases of writing--namely the14prewriting or planning phase, the composition stage and the rewritingphase--and the mathematical problem solving approach. When facedwith a problem, a search is initially executed to access propositions fromdeclarative memory and procedures from production memory. So, whenone writes at the prewriting or planning phase, one is attempting tounderstand what is being asked and what are the attendant conditions ofthe problem. A memory search (from both types of memories) forstrategies and similar problems take place at this phase of writing. Then,during this exploratory, prewriting phase, possible strategies are plannedand some strategy is selected from both types of memories.In the second phase, composition, the writing is more organizedand cohesive in order to execute the strategy selected. This phase isanalogous to the actual solving of the problem when a certain approachis implemented, for example, by using some known procedures. If thestrategy selected does not lead to the desired result, phase one isrepeated, similar to the rejection of a procedure that did not lead to asolution and a search for another procedure. In the third phase,rewriting, the writing resembles that of the transactional mode as there ismore clarity and focus because of an expected audience. This phase issimilar to checking the reasonableness of the solution of a problem afterwhich it may be either rewritten in order to communicate the solution tosomeone else or an attempt at a more elegant solution is made. Hence,the ACT model of learning seems to link writing, learning andmathematics learning, albeit more specifically to mathematical problemsolving.Mildren (1992) points out that just as learning language throughexpository essay writing involves "topic choice, planning and structuring15a text, organising information, drafting, revising, and editing" (p. 34),learning mathematics through problem-solving involves "defining theunknown, determining what information one already knows, designing astrategy or plan for solving the problem, reaching a conclusion and thenchecking the results" (Bell & Bell, 1985, p. 212). Hence, he too supportsa link between writing and mathematical problem-solving.According to Botstein (1989), a technologically-advanced societyneeds citizenry who are mathematically literate to maintain and improveitself. Because the mathematics on which technological devices arebased is seldom derivable from daily experience, "the bridge betweenthe technical and specialized worlds of modern mathematics and scienceand daily life and experience must be constructed out of ordinarylanguage" (Botstein, 1989, p. xv). One way to build this bridge seems tobe by expressive, exploratory writing in mathematics.But just as there were cautions about the use of writing to learn, soalso there are cautions about writing to learn mathematics. Most of theconcerns about writing to learn mathematics have been about the trivialtype of mathematics that seem to be reflected in students' writing inmathematics (Caughey & Stephens, 1987; Ormell. 1992; Pengelly,1990). However, there is sufficient empirical and theoretical evidencecontradicting these concerns. In the following sections, I elucidate someof these theoretical and empirical bases for supporting writing as a toolfor learning mathematics.Mathematics Learning and Language LearningAccording to Ernest (1991), mathematical knowledge begins withlinguistic knowledge. He says, "Natural language includes the basis of16mathematics through its register of elementary mathematical terms,through everyday knowledge of the uses and interconnections of theseterms" (p. 75). As well, van Doren (1959, cited by Talman, 1990, p. 107)indicates that "language and mathematics are the mother tongues of ourrational selves." Laborde (1990), too, states that "the functions oflanguage in the context of mathematics classrooms are those that havebeen recognized for a long time in the development of thought:Language serves both as a means of representation and as a means ofcommunication" (p. 53). Research from various parts of the world(Hughes, 1986; Clarke et al., 1992; Boero, 1989; Laborde, 1990; Stigler& Baranes, 1988; Rose, 1989) attest to language factors affectingmathematics learning.While language in the mathematics classroom can represent andcommunicate mathematical ideas, and mathematics as language mightbe a useful notion, "mathematics is not a natural language in the sensethat English and Japanese are" (Pimm, 1987, p.207) and languageteaching methods may not be directly transferable to the teaching ofmathematics (Ellerton and Clements, 1991, p. 125). Nevertheless, I willargue in this section that there are some parallels between the learningof a language and that of mathematics and that everyday language bothhelps and hinders mathematical learning. I would like to emphasize that,unless otherwise stated, I use the word "mathematics" to mean "schoolmathematics."While both Layzer (1989) and Pimm (1987) view mathematics as aforeign language, I think it might be more useful to view it as a secondlanguage. There have been many different views about what constitutesa foreign language and what constitutes a second language (e.g.17Richards, 1978; Stern, 1983) but my use of these two terms is clarifiedbelow.To me, a foreign language can be construed as one that isseldom, if ever, used in everyday discourse, while a second language isone that has to be used, at least some of the time, to function moreeffectively in the society in which one lives. The second language,moreover, can be thought of as learned second, after the learning of thefirst language. For example, English is a second language while Frenchis a foreign language to many recent Canadians of Asian origin.Mathematics, specifically school mathematics, is somewhat like asecond language in that it has to be learned to function more effectivelyin society. For example, the functionally numerate person has to makesense of percentage discounts in stores, simple statistics and graphs innewspapers. While children generally acquire (Krashen, 1981) theirmother tongue or first language "naturally" without explicit instruction,mathematics as a second language has to be approached slightlydifferently and has to be learned . While Krashen's use of the term"learning" is restricted to the learning of procedures, I am using the termin a broader sense, to encompass whatever knowledge is gained fromexplicit instruction.I say "slightly differently" because of some similarities between firstlanguage and second language learning. For example, just as a childmoves from holophrases to complete sentences through aninterlanguage (Corder, 1981) in first language acquisition, so too a childmoves through the phases of using natural (but possibly mathematicallyimprecise) language to the more sophisticated register of themathematician.18Currently, the approach to second language learning is basedmore on communicative competence (Hymes, 1972; Widdowson, 1990)than on the memorisation of rules of syntax. That is, how to uselanguage appropriate to the context is more important than being able toparse a sentence into clauses. Similarly, communicating mathematicalideas is considered an important component of mathematicalcompetence (NCTM Standards, 1989).In comparing second language instruction with mathematicsinstruction, Borasi and Agor (1990) argue that many of the approaches tosecond language instruction (for example, the "Delayed Oral Production"of Postovsky, 1977; "Silent Way" of Gattegno, 1976; and "CounselingLearning/Community Language Learning" of Curran, 1976) might beusefully modified and applied to mathematics instruction.For example, by delaying the actual plotting and drawing ofgraphs by the students and instead asking students to interpret a givengraph (p. 13), the teacher is directing students to use mathematicsmeaningfully rather than concentrating on the mechanics of plottingaccurately--similar to using language in context, rather than learning,say, the rules of grammar. Another example is the problem-posingapproach of Brown and Walter (1990) who argue that even with limitedfactual mathematical knowledge and partial understanding, one canengage in creative learning in mathematics. For instance, in the case ofthe Pythagorean theorem, a student-generated question on what wouldhappen if the theorem did not deal with a right triangle (Brown & Walter,1990, p. 45) would open up many possibilities for mathematicalinvestigation through student initiative and responsibility and allow forinformative discussion. Such an approach, according to Borasi and Agor19(1990, pp. 15-18), is similar to the Silent Way and CounselingLearning/Community Language Learning approaches to secondlanguage instruction.Mathematics is usually taught in a context-reduced manner, whilesecond language teaching emphasizes context-embeddedness(Cummins, 1981). As Spanos, Rhodes, Dale, and Crandall, (1988) say,"the pedagogical advantages of incorporating real situations in aninteractive framework, a common practice in most English as a secondlanguage (ESL) curricula, seem to be absent in the traditionalmathematics curriculum" (p. 232).When children write in mathematics using their own words andexperiences they are embedding their learning in a context and arecommunicating their mathematical ideas, both to themselves and toothers. Such writing therefore helps move the learner from embeddedthought (embedded in the context) to "disembedded thought"(Donaldson, 1984)--thought which abstracts the concept from the context.Hence, the route to abstract mathematical symbolisms via formalmathematical language has to be paved by everyday language.However, linguistic ambiguities, especially lexical ambiguities,abound when language from an everyday context is used in amathematics context (Pimm, 1987; Durkin & Shire, 1991). Accordingly,writing that promotes discussion of ambiguous words might resolve theseconflicting roles of everyday language (of helping and hinderingmathematical learning) by fostering an awareness of how words used ineveryday language can have different meanings in mathematics. Suchwriting may also act as a window to the mathematical thinking of thechild.20There are factors other than lexical ambiguity associated with thetransfer from everyday language to the mathematical register. One suchfactor is the linguistic structure of natural language. According toLaborde (1990), "linguistic features of natural language can affect thetransition of a situation from natural language into an algebraicstatement" (p. 61).This linguistic factor is one of the reasons many tertiary studentscannot translate relationships expressed in everyday language intocorresponding mathematical expressions (Clement, Lochhead & Monk,1981; Mestre & Lochhead, 1983; Rosnick & Clement, 1980). Theexample most cited by these researchers is the following problem: Thereare six times as many students as professors. If S represents studentsand P represents professors, write a mathematical relationshipconnecting S and P. Most students write 6S = P rather than S = 6P as asolution to this problem. One explanation for this confusion is that thelinguistic structure of the problem statement, where the expression "sixtimes as many students" precedes the word "professors," could have ledto the left-to-right translation of the problem.Another factor associated with the transfer from everydaylanguage to the mathematical register is the difference in levels oflanguage required as one progresses from the colloquial to themathematical (Freudenthal, 1978, pp. 233-242). New concepts areexemplified by exemplary or demonstrative language first. Laterrefinement leads to relative language and finally to functional language.Examples of these levels of language follow:1. demonstrative (pointing out instances, without explanations) :half is like this part here.212. relative (using words to indicate relationship or procedures):when something is cut into two equal parts, each part is called a half.3. functional (generalisations or relationship betweenrelationships): The common fraction 1/2 is the same as the decimalfraction 0.5 because one out of two equal parts is equivalent to five out often equal parts.The levels suggested by Freudenthal seem to form a continuum rangingfrom personal referents to more abstract ones. Therefore, if children areencouraged to write in mathematics, their initial exploratory writing usingtheir own words might lead them from demonstrative language to relativeand functional languages later.Even though facility in sophisticated mathematical language is thefinal objective, initial exploratory writing in mathematics, using personal,everyday language can help the learner construct meaning. Havingsomething written down allows a re-vision, which in turn helpsreconstruct meaning. From the point of view of language learning, eachconstruction may be perceived as an interlanguage (rather than an erroror misconception) leading to the appropriate mathematical register.What I have discussed so far suggests that certain aspects oflanguage learning might be usefully linked to mathematics learning andthat everyday language might assist in learning mathematics. In short,just as written language and learning seem to be linked, writing and thelearning of mathematics might also be linked.22Mathematics Learning and PedagogyIn this section, I discuss research that supports the notion thatwriting in mathematics, especially informal exploratory writing, ispedagogically sound.Many students seem to view mathematics with anxiety. To allaysuch anxiety, Tobias (1989), suggests that "getting students to writeabout their feelings and misconceptions would relieve their anxiety andunlearn models and techniques that were no longer useful to them" (p.50). Even though she does caution that her techniques might haveworked because her students were already "predisposed to verbalexpression" (p. 50), I believe her point about relieving student anxiety iswell taken, and writing might very well act as a catharsis.LeGere (1991), too, concurs that writing does seem to diminishstress and anxiety about getting the "correct" answer, and allows risktaking. Morrow and Schifter (1988) have this to say about the "anxiety-reducing" role of writing in mathematics:Turning to a more familiar and often more comfortable mode, suchas writing, can provide a sense of security to a math-anxiousstudent. Alternating among various modes of discourse (writing,talking, drawing, and symbolic representation) builds bridgesbetween formal mathematics knowledge embedded in students'everyday experiences; insistence that mathematically validthought is restricted to rule-governed manipulation of strings ofsymbols keeps the mathematics insulated from their personalknowledge. (p. 380)Mathematics educators have recently started emphasizing studentownership of their learning (e.g. Ellerton & Clements, 1991). Becausewriting to learn mathematics entails students using their own languageand ideas to express mathematical concepts and principles, someownership of learning seems to be engendered.23According to Nahrgang and Petersen (1986) and Wilde (1991),writing in mathematics promotes the learning of mathematics. Wilde(1991) has shown that, even for ESL students, writing aboutmathematical ideas initially in their first language, and then translatingthese ideas (also in written form) using their limited knowledge ofEnglish, does benefit their mathematical learning.Associated with ownership of mathematics learning is the idea ofempowering the learner. Buerk (1990) states that many students viewmathematics as truths external to themselves, derived from theindisputable authority of the text book, teacher or mathematician. Theybelieve that "mathematics is something they can have no ideas about" (p.79) and that they have a sense of "powerlessness in its presence" (p. 78).Students seldom realise that mathematicians struggle tounderstand, clarify and verify mathematical ideas just as studentsthemselves do. Moreover, mathematics is often presented as objectivetruths, unsullied by subjectivity or human struggle. Brookes puts thispicturesquely ". . . the most neglected existence theorem in mathematicsis the existence of people" (1970, p.vii, cited in Pimm, 1987, p. xvii). Byencouraging exploratory writing in mathematics which itself entailsuncertainty, students are empowered and given dignity as learners(Hoffman & Powell, 1989; Worsley, 1989).Writing to learn mathematics finds support in the currenttransactional-interactive model of teaching. In this model, teachers arefacilitators and their role is to provide situations that give rise todiscussion and learning. As Charbonneau and John-Steiner (1988) say,"the teacher will be more a catalyst to learning rather than a presenter ofa body of facts and figures" (p. 98).24Such models are actually not new. For example, Piaget's notionof cognitive dissonance, Vygotsky's zone of proximal development, andBruner's discovery learning all implicitly encapsulate the interactivemodel. More recently, we have the 4 MAT System (McCarthy,1986) andother suggestions that emphasize learning through reflection andinteraction. Since writing in mathematics is closely associated withreflection on what is written and interaction with both what is written andthe expected reader, writing seems to be an appropriate vehicle forinteractive learning and teaching.For instance, Keith (1990) says, when talking about calculuscourses, that "to keep mathematics alive as a subject in college, we mustbe prepared to create a more vigorous, interactive classroomenvironment" (p. 6). One way to achieve this, according to her, is to givewriting assignments in the calculus course. Powell and Lopez (1989)agree that such writing in mathematics courses allows crystallisation andgeneration of mathematical concepts through constant reflection andrevision.Another facet of reflective learning can be viewed through the ideaof active involvement in mathematics through doing mathematics. But, asPimm (1987) suggests, it is not just doing, but thinking about doing, thatcounts. Thus, this reflection, which is aided by writing, allows for greateractive involvement in mathematics learning (Azzolino, 1990; Rose, 1989;LeGere, 1991).Another pedagogical benefit from writing in mathematics is that itgives feedback on the learning-teaching enterprise, by helping assessand monitor both learning and teaching (Powell, 1986; Powell andLopez, 1989; Sachs, 1987) . Such feedback, according to Countryman25(1987), also allows for diagnosis and remediation, whether teacher- orpupil-initiated. Certainly, writing mathematics, using mainlymathematical symbols with minimal explanatory words (as in finding themaximum or minimum points using differentiation), can also give suchfeedback. For example, the student might have written all theappropriate steps to show that a maximum point exists, but may not haveany idea why the "tests" seem to work and under what conditions thesetests might have to be modified or extended. Or in the case of fractions,students may be able to add fractions using common denominators, butmay not be aware why they need to do so. However, such writing(predominantly mathematical symbols) cannot identify students who mayonly be using procedural knowledge (Hiebert, 1986) by manipulatingsymbols, apparently logically and sequentially, to arrive at a correctsolution without substantive understanding of the mathematical conceptsor principles involved.Even if they get an incorrect answer, and subsequently exhibiterror patterns in their work--which might be a basis for remediation--sucherror patterns can only be detected by careful construction of a diagnostictest. With the use of writing in mathematics, not only does the teacher nothave to prepare and administer the diagnostic test but also idiosyncraticmisunderstandings can be attended to, as long as students can articulatetheir thoughts in writing.With the emphasis on conceptual rather than procedural learning(Hiebert, 1986), elementary school students seem to have a poor recallof mathematical facts and procedures. While learning concepts areimportant, it would seem that students should benefit if certainmathematical facts and procedures had become automatised. This26supposed benefit of automaticity might seem at odds with writing to learn,but I will show why such automaticity is actually helpful to students writingto learn mathematics and conversely, how such writing will helpautomaticity.Because writing has a communicative function (even to oneself), itresults in further clarity and better retention of mathematical concepts(Stempien & Borasi, 1985). Such retention of concepts leads to beingable to recall and use them in further writing, without having to re-thinkthe concepts, thereby allowing the short term memory (from theinformation processing point of view) to work more efficiently to enhanceproblem solving processes.Another pedagogical concern is the problem of attending toindividual needs of the students. Because writing, unlike speaking, canbe done simultaneously by the whole class, every individual is activelyinvolved. Each student can write about his or her own perceptions,difficulties, solutions and feelings. While speaking allows for articulatingone's difficulties, not every student would risk making mistakes in front oftheir peers. Writing is more private and allows for a confidential dialoguebetween the teacher and the student. Responses from the teacher makeit a personal learning experience as well, because students know thatthe teacher hears and cares (Thompson, 1990; Watson, 1980). In short,writing in mathematics is like individualising instruction in a group setting(Bemiller, 1987).Another concern of mathematics teachers is to help studentsbecome better problem solvers. One major difficulty is that students donot know where, or how, to start. Bemiller (1987), and King (1982) found27that when students were stuck with a problem, writing out their thoughtsoften helped them resolve the problem by themselves.Additional support for writing as a means of assisting studentsbecome better problem solvers is given by Bell and Bell (1985) whocompared ninth graders with and without a writing component during theteaching of mathematics problem-solving. The experimental group--whoused writing to analyze information needed to solve the problem anddescribe the process used to solve the problem--outperformed thecontrol group in a posttest on problem solving.Helping weaker students to develop and learn mathematics is alsoa concern of mathematics teachers. According to Evans (1984), writingin mathematics does help the weaker mathematics students. Shecompared two groups of fifth graders on units of geometry andmultiplication, using three types of writing for the experimental group:explanations on how to do something, definitions, and explanation oferrors on homework and tests. Even though the control group, unlike theexperimental group, comprised some gifted students and also scoredbetter in the pretests, the experimental group performed better on bothunits in the posttest.Pallmann's (1982) study also included an experimental groupusing writing (in their remedial college mathematics course) to explain intheir own words how, for example, they worked out problems andinterpreted graphs and algebraic laws. Results of her study indicate thatthough there was a mean gain for the experimental group on the posttest,this was not statistically significant. However, the fact that this group hadmore of the weaker students made it educationally significant.28In summary, pedagogical theory and practice suggest thefollowing benefits for writing in mathematics classrooms: more studentownership of learning, less student anxiety, individualised learning in agroup setting, monitoring and diagnosis of learning, and enhancedachievement in tests on problem solving.Mathematics as a Social Enterprise Views about the nature of knowledge have inspired manyphilosophical debates and driven many practices. For example, thoseviewing knowledge essentially as dualistic (Perry, 1970), would relyheavily on authority (be it the text book, teacher, mathematician or logicitself) and believe in answers to mathematical questions as eitherabsolutely right or wrong. Such views would emphasize correctprocedures and exclude conscious guessing (Lakatos, 1976) andtentative procedures, like exploratory writing in mathematics.In contrast, those who hold a relativistic outlook (Perry, 1970),acknowledge multiple approaches and solutions to mathematicalproblems relative to evaluative criteria in a given framework. Many ofthose who hold such a view of mathematics would say that mathematicsis a social enterprise. For example, Wittgenstein (1956) states thatmathematics is a human practice, depending crucially on the consensusof a community and is very much a socialisation process. Similar viewswere expressed by Restivo (1981) and by Lakatos (1976). Lakatos, forinstance, points to the fallibility of mathematics and views mathematics asa human enterprise, relying on conjectures, criticisms and corrections.Radical constructivists like von Glasersfeld (1987), who believethat knowledge is constructed by the individual, would be considered29Relativistic. So would social constructivists such as Cobb, Yackel andWood (1992), Ernest (1991) and Steffe (1989) who believe thatmathematical knowledge is a result of the individual interacting in asocial setting.(For an elaboration of the differences between radicalconstructivism and social constructivism, see Ernest, 1991, p. 71). Asocial constructivist perspective supports the notion of mathematics as asocial enterprise because social constructivists believe that there is noreality outside the learners' own experiential interpretations and thatmathematical knowledge is actively constructed by the learners andconsensually validated by the community (Bloor, 1984; Ernest, 1991).When children write mathematics using their own experientialreferents and discuss their writing with peers or the teacher (thisdiscussion could be oral or in the form of written responses), they willcome to realize that mathematics is also subject to consensual validationand shared meaning (Lampert, 1988; Pirie, 1991). This is in keepingwith the view that mathematics "is an ever-changing field that exists notas an abstraction but as a piece of linguistic and cultural fabric"(Scholnick, 1988, p. 86).In Lampert's study (1988) on how to make mathematicsclassrooms reflect what mathematicians actually do, she emphasizes therole of negotiation of meaning in the social context of the class. Sheclaims that her fifth grade students "learned to participate as activemembers of a community of discourse about mathematics" (p. 465), citingboth Bauersfeld (1979) and Steiner (1987). She argues that"mathematics learning is a process of social as well as individualconstruction" (p. 473). The increasing interest in ethnomathematics anda corresponding decrease in eurocentrism ("Western mathematics is30best") and findings from studies of mathematics in different cultures(Bishop, 1985; Cocking & Mestre, 1988; Harris, 1989) also indicate thatmathematics is socially and culturally bound.Mellin-Olsen (1987) also views mathematics as a socialenterprise, arguing that activity is a means of survival in society, wherethe individual acts on society and in turn is acted upon by society.Language and mathematics can then be perceived as thinking andcommunicative tools for changing the individual and society, especiallyby resolving problems in society by the use of mathematics andlanguage.Mellin-Olsen gives examples of how students identified andsolved problems affecting their neighbourhood by using bothmathematics and language (for example, by verbalising their ideas andthen by writing letters to people in authority, giving reasons based onmathematics and statistical data) to implement change. By allowingchildren to use whatever mathematics they felt suitable to support theircase, teachers helped children to see the relevance of mathematics.And by accomplishing something, largely by themselves, they also felt asense of ownership and autonomy.Earlier, I discussed the relationship between learning andlanguage. Hence, using language by writing and subsequently sharingone's writing with others, should help with the renegotiation ofknowledge in a social context. Because writing to learn mathematicsencourages reflection, interaction and renegotiation of meaning, suchwriting should be able to help reinforce socially constructedmathematical knowledge.31In summary, I argue that the view of mathematics as sociallyconstructed supports the approach of writing mathematics to learnmathematics, because (a) language itself is a social enterprise and (b)use of everyday language in writing to learn mathematics, will, afterhelping construct and reconstruct idiosyncratic mathematical concepts,be subjected to a critical audience (peers or the teacher), therebypromoting communication, validation and renegotiation of sociallyconstructed mathematical meaning.JournalsIn this section I first discuss the use of journals as a learning tool inlanguage arts and then discuss in more detail the use of journals as alearning tool in mathematics specifically. For purposes of this review, Iam using the word journal to include a range of writings, from "diary-likeentries to focused (i.e. assigned) entries such as summaries of lecturesand discussions of problems" (Sipka, 1990, p. 12).Journals in Language Arts According to Fulwiler (1987), the increased attention to informalwriting in the late 1960s made journal writing more accepted ineducation. He lists some cognitive activities associated with such writing.Among such activities are speculations, revisions, questions,observations and synthesis of ideas. Other advocates of journal writingin language arts include Atwell (1990); Kirby and Liner (1988); Macrorie(1988); and Martin et al. (1976). Kirby and Liner (1988), for instance,found journals an effective tool for improving fluency in written English.32They felt that a journal is effective because "it's a private, protected place,. . . to explore" (p. 58). Also, according to Carswell (1988),The journal writing activity was certainly a success in both my andmy students' estimation. The journal clearly enhancedcommunication between the instructor and the students, appearedto provide some therapy, stimulated conceptualization andreconceptualization, and generated more than occasionalenjoyment. (p.112)However, there are some problems associated with the use of journalsas a pedagogical tool. Autry (1991) points out that the journal "is theproduct of two contradictory genres--the commonplace book and thediary" (p. 74) where the former may be a rehearsal for an intendedaudience and the latter is for private consumption. Accordingly, journalentries may not reveal spontaneous and sincere struggles to learnbecause of the intended audience, unless a risk-taking and trustingatmosphere has been established.Anderson (1992), too, while acknowledging the role of journalwriting to enhance learning, warns of some problems associated withtheir use, such as overuse of the genre, ethical issues, aversion towriting, problem of grading, lack of analysis, lack of synthesis, lack ofgrowth in writing ability, and writing for the teacher.Mathematics Journals In this section, I review literature on the use of mathematicsjournals. Journal writing at all levels involved some or all of thefollowing: (a) writing explanations of mathematical definitions,procedures or solutions, (b) writing students' feelings about, and attitudestowards, mathematics and (c) writing summaries. Examples of how thesetypes of writing were used at different age levels and what benefits were33derived are given next. Note that in most studies, more than one type ofwriting was used, but in the examples below, I am highlighting only onetype of writing.Writing explanations of mathematical definitions. procedures or solutions Watson (1980) asked her second-year algebra classes to writeexplanations in their journals about specific topics that had been justtaught and at other times to write without her specifying the topics. Shereports that journal writing encouraged student reflection and also servedas a two-way communication between herself and her students.Vukovich (1985) reports similar benefits about the use of weekly journalsin a basic mathematics program for college students but in addition shereports that the journals enabled the teacher to identify students whoneeded help.Selfe, Petersen and Nahrgang (1986) asked their college studentsto explain in their journals concepts studied in their analytic geometryand calculus courses. Although their study used an experimental designand multiple measures, quantitative measures did not give a clearindication of whether journal writing affected mathematical learning.However, analysis of the qualitative data (open ended questions andjournal entries) enabled them to conclude that students haddemonstrated an understanding of mathematics concepts. In addition,they suggest that using students' own words enabled students to viewmathematical ideas less abstractly.Bemiller (1987) used what he called a mathematics workbook forhis college students. He used both expressive and transactional writing(Britton et al., 1975). For the three types of transactional writing Bemiller34assigned, students did directed writing during class time, conceptualwriting outside of class and mini-reports on assigned problems. Hereports that writing in journals enabled students to clarify and concretisetheir mathematics as well as allowed for individualised learning. Hestates that the journals allow the instructor to monitor both the "academicprogress and personal growth" (p. 366) of the student.Mett (1987) used daily journals to help her students learnbusiness calculus. She asked her students to explain, as if to aprecalculus student, concepts such as average and instantaneousvelocity. She reports that writing such explanations prepared studentsfor, and helped students to understand, more difficult but relatedconcepts such as differentiation. She states that student-teachercommunication was improved through these journals.Schubert (1987) worked with Grade 4 students and used journalsfor a unit on fractions. Students wrote in their journals an explanation ofspecific mathematical concepts, principles or operations in their ownwords . Not only did her students find word problems less intimidating,but their use of appropriate mathematical terms improved considerably.When she compared the mathematics test scores of students who hadused journals in their mathematics class with those who had not, evenafter a year, she found that the mean score of the former were better thanthose of the latter. She found the student journal entries and herresponses "a way for me to keep up with the life of each of these fourthgrade children in a sympathetic, understanding way" (p. 356).35Writing students' feelings about. and attitudes towards. mathematics Brandau (1990) worked with elementary school teachers andstudent teachers. She used journals to explore students' feelings aboutmathematics, as well as students' feelings and emotions about solvingmathematical problems. This she did by asking them to solve problemsof their choice and write down their feelings as they attempted to solvethese problems. A number of problems had no solution given, either byher or by the textbook from which the problems were chosen. Shereports that students' confidence in doing mathematics problemsincreased and that the instructor received feedback about studentdifficulties.Powell and Lopez (1989) used journals for college-level studentsenrolled in the developmental mathematics course (designed formathematically-underprepared college students). Students wrote whatthey felt about their learning of mathematics in general or on themathematics course itself. They also wrote about discoveries they hadmade about mathematical patterns, relationships and procedures. Theyreport that journals provided an effective two-way communicationbetween the instructor and the students, that students developed moreconfidence in their own ideas, and that students moved from expressiveto transactional writing.Rose (1989, 1990) used journal writing for college students takingher calculus course. Her students were required to write at least as manyone-page entries as there were calculus classes attended. She reportsthat students wrote more about their feelings about mathematics and thecourse and less about mathematics itself. Yet such writing encouragedstudents' reflective thought, provided a diagnostic tool as well as36feedback on the course for the teacher, and allowed a more personalizedapproach to the teaching-learning interaction.Tobias (1989), in studying reasons for mathematics anxiety, usedjournals for peers (from fields outside mathematics) and students (fromsecondary and tertiary levels) to explore negative feelings aboutmathematics by using "divided-page" exercises (pp. 51-52). The left-hand page was for the feelings (for example, struggles and false starts)as they worked through problems and the right-hand page was for theactual working of the correct solution. She reports that such writingallowed (a) students to move from unfocused "emotional detritus" (p. 52)to ways of identifying and overcoming their misconceptions anddifficulties in mathematics and (b) instructors to get valuable feedback.Burton (1985) refers to students in preservice and in-servicemathematics education courses and recommends recording personalreactions to memorable lessons (either particularly difficult or interestinglessons) in students' and instructors' journals as soon as possible afterthe lesson, preferably towards the end of a lesson. According to her,such immediate recording of reactions by students and instructors can beused for reflection later on, as "the expressed emotions of the experienceare relived and the cognitive aspects recalled" (p. 41).Stempien and Borasi (1985) required their preservice teachers tokeep weekly journals and share their writings with their peers and theinstructors. The student teachers found that keeping journals about theirfeelings about mathematics, the mathematics education classes and theirexperiences in the practicum (student teaching) and sharing theirwritings encouraged communication and made them aware thatmathematics was not controversy-free. The instructors found the journals37provided valuable feedback on their teaching and also helped establishbetter rapport with the students.Shaw (1983) suggests using daily logs for junior high schoolstudents to record their feelings about the mathematics lessons and towrite a review of the daily events in these logs. She reports that suchlogs allow the teacher to get to know the students better and to receivefeedback on the teaching. In addition, she states that students becomemore aware that what they write has to be understandable to everyone,as they are required to take turns reading their entries to the class.Writing summaries Clarke, Waywood, and Stephens (1992) conducted a 4-yearlongitudinal study involving about 500 junior secondary school girls (allthe students from Grades 7 through 11 from an all-girls school inAustralia). The students wrote in their journals after every mathematicslesson as a major part of their homework. They were asked to write asummary of the main ideas of the day's lesson, together with somecomments and questions. Clarke et al. report that the journal entriescould be classified under the categories (a) narrative, (b) summary and(c) dialogue. The narrative category refers to students recounting,without evaluative or thoughtful questions, what transpired in the lesson.The summary refers to a synthesis of knowledge useful for the learner interms of examination preparation or content mastery by recognizing andsequencing important ideas, and the dialogue refers to creating andrecreating knowledge through posing and attempting to answerquestions leading to more sophisticated knowledge. They report that the38summary category was most prevalent, implying a utilitarian viewtowards the learning of mathematics by most of the students.McMillen (1986) studied college students and reported the use ofdouble-entry journals, where students wrote summaries of their readingson one page, with their reaction to the readings on the facing page. Shereports that such journals help students understand the mathematicalcontent better as well as encourage them to be more critical. Indeed,students could even synthesize their notes to create their own textbooks.King (1982), in describing transactional writing tasks for highschool and college level students in mathematics, regards makingsummaries of mathematics lessons as helpful for students to identify,focus and synthesize their mathematical learning. Johnson (1983) statesthat even rewriting an unclear page or paragraph of an algebra orcalculus textbook or a word problem can be perceived as writing asummary, as it entails similar skills to those suggested by King.Talman (1990) required college students to keep weekly journals,containing three kinds of writing: summary of topics covered during theweek, a report of students' own relevant activities during the week, andan analysis of the week's work. He also expected an analysis of thesolution to at least one problem not solved in class, and in the weekfollowing an examination, part of the analysis was an analysis of theexamination and their performance in it. He reports enhancedconfidence in their own mathematical abilities and better performance inmathematics problems and tests.39In summary, using journals in mathematics classes seems to havethe following benefits:1. Enhanced mathematical performance (McMillen, 1986;Nahrgang & Petersen, 1986; Selfe, Petersen & Nahrgang, 1986; Powell,1986; Schubert, 1987; Talman, 1990).2. Ability to monitor and clarify thoughts (Bemiller, 1987; Buerk,1990; Burton, 1985; Clarke et al., 1992; Kennedy, 1985; McMillen, 1986;Mildren, 1992; Rose, 1989, 1990; Shaw, 1983; Waywood, 1991).3. Ability to solve problems by themselves (King, 1982; Powell,1986; Watson, 1980).4. Greater confidence in ability to do mathematics (Brandau,1990; Buerk, 1990; Powell, 1986; Powell & Lopez, 1989; Schubert, 1987;Talman, 1990).5. Enhanced student-student interaction and cooperative learning(Stempien & Borasi, 1985; Vukovich, 1985).6. Improved student-teacher rapport (King, 1982; Mett, 1987;Powell & Lopez, 1989; Rose, 1989, 1990; Vukovich, 1985; Watson,1980).7. Feedback to instructors (Brandau, 1990; Powell & Lopez, 1989;Rose, 1989, 1990; Stempien & Borasi, 1985; Tobias, 1989).8. Reduced student anxiety about mathematics (Powell & Lopez,1989; Tobias, 1989).Some Cautions about Mathematics JournalsIn spite of the optimistic reports on the benefits of mathematicsjournals, there are some dissenting voices. For example, Mildren (1992),investigated the use of journals among 120 Australian children from40Grades 4, 5 and 6 over a period of between six and ten months. He usedthe headings "What I did" (to obtain student reflection), "What I learnt" (toobtain knowledge about student knowledge and beliefs) and "Problemsand difficulties" (to obtain knowledge of attitudes) in order to initiate theentries. Mildren reports that none of the entries included reasons forstatements and very few attitude statements were made. He inferred thatsuch results could be reflecting classroom practice where justification forstatements were not asked and student attitudes towards mathematicswere not considered important. In short, he says that the classroommathematics program was text-centred rather than student-centred (p.88).Clarke, Waywood, and Stephens (1992) caution that self-reporting of the value of journals by students may not be indicative oftheir benefits. Waywood is presently working on ways to corroborateinferences about the nature of students' mathematical writing. I believethat one of the reasons for being unsure of the veracity of students'statements is a lack of rapport between the teacher or researcher and thestudents (for example, the students may be afraid of writing downnegative feelings about something they perceive the teacher values).Moreover, when marks are given for journal entries, as Clarke et al.(1992) did, there could be a tendency to be more careful and lessspontaneous.OrmeIl (1992) suggests that journal writing might not reveal theextent of, or student involvement in mathematical learning. For example,he criticizes the examples of journal entries in Mildren's (1992) study,saying that the entries had "virtually no sense of internalised meaning"(p. 230).41Some researchers look to quantitative studies in journal writing togive evidence of mathematical learning through journal writing. Selfe etal. (1986) embarked on just such an enterprise, but they did not find any"hard" evidence, one way or the other. They warn that not only arequantitative measures about journals enhancing mathematics learningdifficult to come by, but such measures have to be viewed with caution,because the questions answered by such quantitative measures may notbe answers to the substantive questions raised in the first place (p. 202).According to them, they obtained more valuable answers from thequalitative analysis employed, and they suggest that a combination ofquantitative and qualitative measures be employed in future research.For teachers, one of the main drawbacks about the use ofmathematics journals (or other journals, for that matter), is the amount oftime required to respond effectively to such writing. Among thesuggested solutions are: random collection of a limited number of journalentries every day (Schubert, 1987); using journals only for classes thatare not too large (Talman, 1990); using peer responses and discussion(Kirby & Liner, 1988); and requiring only short, in-class writing (Powell,1986).Student-constructed questions (SCQ)The literature on problem solving in general, and word problemsin particular, covers a wide range of factors associated with the difficultiesstudents and adults find in trying to solve these problems. For example,some variables commonly studied are the number of words in a problem,the presence of cues or key words, the semantic structure of the problemand order of presentation of the given numbers (Carpenter, 1985; De42Corte & Verschaffel, 1991; Krutetskii, 1976; Riley, Greeno, & Heller,1983). In these studies, the emphasis has been on the solving of theproblems, rather than on students constructing their own problems to useas a learning tool. Because of this emphasis, there has been very littleresearch on the use of SCQ in mathematics or the effectiveness of SCQas a learning tool in mathematics.A notable exception to the problem-solving focus has been theproblem-posing emphasis of Brown and Walter (1990), who organised awhole graduate course around problem-posing, considering problem-posing as "a worthy activity in its own right " (Pimm, 1987, p. 205).According to Brown and Walter (1990), "essentially no understandingcan take place without some effort at problem generation" (p. 139). Also,generating a problem reduces mathematics anxiety "because posing ofproblems or asking of questions is potentially less threatening thananswering them" (p. 140).According to Willoughby (1990, p. 56), the process of formulatingproblems, even as a member in a group, helps students understandproblems created by others. He also states that in industry and in real lifesituations, 'The fact that the mathematician knows an answer is ofabsolutely no interest unless the mathematician can communicate theanswer to someone who will use it. . . . Communication is, in manyrespects, the most important part of mathematics" (p. 55). Such viewswere echoed years ago by Einstein and Infield (1938) about sciencewhen they saidThe formulation of a problem is often more essential than itssolution, which may be merely a matter of mathematical orexperimental skill. To raise new questions, new possibilities, to4 3regard old problems from a new angle, requires creativeimagination and marks real advances in science. (p. 29)The largest study on problem-posing (and subsequent solving) inmathematics has been the investigation by Ellerton (1989). She asked10,500 secondary school students in Australia and New Zealand to makeup a difficult mathematics problem, and then solve it. She found that theyused problems similar to the ones encountered in class, and rarelyreferred to real world contexts.King (1982) asked secondary and college level students to writeword problems of their own to demonstrate their understanding ofmathematical concepts. According to her, writing their own wordproblems tended to enhance communication as they had to write clear,specific and complete instructions. A prerequisite for suchcommunication was a good understanding of underlying mathematicalconcepts.Johnson (1983) asked students in his basic algebra class torewrite story problems or construct their own problems. By rewriting storyproblems, students became aware of key words and relationships. Bywriting their own problems, students became aware of the contents of thequestion as well as how to solve the questions. Abel (1987) andAckerman (1987) conducted studies with college students, similar to thatof Johnson (1983) and found similar benefits (such as awareness ofmathematical relationships).In Kennedy's (1985) study, middle school students weresometimes supplied data on which they were to prepare questions. Atother times, both data and questions were prepared by the students.Sometimes he also asked them to prepare problems with insufficient44information. When the questions were put forward for class discussion,clarifications were sought and given, thus promoting critical thinking andlearning.Stempien and Borasi (1985) report that when students were givensome situations where they had to relate numerical data to mathematicaloperations or principles, they chose to share experientially-embeddedcontexts relating to real-life applications. Such practice seemed toincrease confidence in tackling word problems from textbooks. Theirfindings about context-embedded questions contrast with Ellerton's(1989) study.Hodgin (1987) reports that her elementary school studentsconstructed their own problems based on a picture or cartoon. Afterwriting the question, discussion and evaluation (in pairs) took place.Then, after appropriate revision if necessary by the students themselves,the questions were displayed on the class bulletin board. Studentsseemed to enjoy this activity. Moreover, they learned to applymathematical concepts by constructing these problems.Instead of SCQ for classroom exercises, some researchers haveencouraged the use of student-constructed tests. For example, Clarke,Clarke, & Lovitt (1990) have used tests made up partly of student-constructed test items duly edited. Students seemed to find this a veryeffective revision strategy and they also felt a sense of participation in theassessment process.45Significance of the StudyPrevious researchers have suggested a number of benefits forlearners arising out of using writing in mathematics classrooms. Amongthe suggested benefits are: (a) encouraging students to express theirown experiences and language allows for more autonomy andownership of learning, (b) individualising learning in a group settingbecause every child is involved at his or her level of understanding andpace, (c) monitoring and diagnosing children's mathematical progressand (d) developing an interest in mathematics.These suggested benefits seem to emphasize the affectivedomain. In contrast, I am investigating the possibility of specificmathematical (cognitive) benefits through writing to learn. The two typesof writing tasks (the SJ and the SCQ), focus on specific mathematicalknowledge and the extent to which such knowledge can becommunicated. By focusing on these two writing tasks, I am alsoaddressing criticisms about the trivial mathematics that seems to bereflected in student journals (e.g. Caughey & Stephens, 1987; Ormell.1992; Pengelly, 1990). Because this study also emphasizescommunication of mathematical ideas through written words, it shouldenhance knowledge about language factors affecting the learning ofmathematics. Moreover, the SJ and SCQ (as used here) have neverbeen used together in previous studies, and should thus be able toprovide data to form a broader picture of how different writing tasks in theclassroom might help the learning of mathematics.There has been considerable research on journal writing atsecondary and tertiary levels, but not many studies have been conductedat the elementary school level. Results of studies conducted at46secondary and tertiary levels might not be applicable to students at theelementary school level (for example, because of differences in writingexperiences and levels of language proficiency) and so this study shouldaugment existing knowledge about writing to learn mathematics at theelementary school level.Previous studies were usually conducted and reported by theresearcher as teacher (e.g. Rose, 1989; Selfe et al., 1986; Stempien &Borasi, 1985) at the tertiary level. In this study, the researcher (who doesnot teach the class) reports on an elementary classroom teacher whoinitiates and conducts the mathematics lessons using writing as a routinecomponent of the mathematics lesson.While Brown and Walter (1990) and Einstein and Infield (1938)talk about problem-posing in a more global manner (for example, bychanging certain initial or boundary conditions), I confine myself in thisstudy to word problems involving common and decimal fractions.Another difference from previous studies on problem posing is that theword problems constructed by the students are used as classroomexercises in this study. In contrast, Ellerton's (1989) study is aboutstudents constructing problems for the researcher to analyze and not forstudents to attempt as class exercises. Although Ellerton's sample islarge, she studied only one question per child. This study uses a smallsample but investigates a range of questions over a 15-week period.Moreover, this study involves Grade 6 children, in contrast to Ellerton'ssecondary school students. Even where the students' problems wereused in the classroom (e.g. Abe1,1987; Kennedy, 1985; King, 1982), theywere at secondary and tertiary levels and were used peripherally and47sporadically and not centrally, or as an ongoing component of themathematics lesson over a 15-week period, as in this study.Whereas King (1982) dealt with secondary and college levelstudents and topics, this study focuses on elementary school studentsand topics. This study also has students generating their own wordproblems, but the students are much younger and do not have as muchexposure to word problems as the students in the studies described byJohnson (1983), Abel (1987) and Ackerman (1987). In this study, too, thefocus is not on the discussion generated by the SCQ as in Kennedy's(1985) study but is on the mathematical learning reflected throughstudents' written questions. Unlike the study by Stempien and Borasi(1985) where numerical data are given by the teacher for students to usein their problems, in this study generally the numbers in the SCQ arechosen by the students themselves. While Hodgin's (1987) study reportson problems constructed by elementary school children, these problemswere based on pictures and cartoons, in contrast to the SCQ in this studywhich are based on the teacher's verbal prompts such as "write a wordproblem involving common and decimal fractions."Teachers may find this study significant for the following reasons:1. The study documents an attempt to encourage students to takemore responsibility for their own learning and to become activeparticipants in the learning of mathematics by generating their own ideasand constructing their own meanings rather than treating mathematics asa spectator sport.2. Valuable information on students' understanding of, anddifficulties in, mathematics might be obtained by examining whatstudents write in their SJ and SCQ.483. Some ideas on implementing change in the mathematics classmight be obtained by examining how SJ and SCQ were initiated in thisstudy.4. Teachers might become more aware of discrepancies instudent- and teacher-perceptions on the usefulness of certaininnovations and try to address these discrepancies.ConclusionIn this chapter, I have tried to relate writing to learning, first bybriefly surveying the evidence that writing supports learning in generaland learning mathematics in particular. Then, I gave theoretical andempirical support to the notion of writing to learn mathematics based onthe literature in various disciplines and from various perspectives Finally,I reviewed the literature on how journals and student-constructedquestions seem to help the learning of mathematics, and at the sametime attempted to clarify how this study seeks to extend those previouslydone.49CHAPTER 3. METHODIn this chapter, I discuss how I conducted the study, by describingthe context of the study and the procedures for collection and analysis ofdata. In order to contextualize the study, I first describe my ownbackground and the teacher's and then explain my basis for selectingthis particular group of students for study. Then I detail the proceduresfor collecting and analyzing the data.Context of the studyMy Background On the basis of good scores on school and public examinations, Ihad been considered a good student in my school days. It was onlywhen I began my undergraduate studies in mathematics that I realisedthat I owed much of my success in examinations to a good memory andprocedural knowledge (Hiebert, 1986). So, after my undergraduatestudies, I decided to pursue an M.A. in mathematics education ratherthan in mathematics per se as I became interested in why many childrendid not seem to be learning mathematics successfully, even though theyseemed to succeed at other school subjects. As well, because languagelearning fascinated me, I pursued an advanced diploma in Teaching ofEnglish as a Second Language (TESL). In addition, I have had morethan 12 years of experience in working with high school students andpreservice elementary and junior high school teachers who have had anegative attitude towards mathematics but a positive attitude towardslanguage. These interests in language and mathematics education have50led me to pursue doctoral studies, with a focus on how using writtenlanguage might assist the learning of mathematics.Next, I describe that part of a previous research project whichinvolved the particular teacher who subsequently agreed to help me inthe present study.The Previous ProjectThe project, funded by the Social Sciences and HumanitiesResearch Council of Canada (SSHRCC, Grant No. 410-90-1369) andconducted by Dr. Douglas Owens of the University of British Columbia(UBC), lasted from January 23 to March 22, 1991.The project supported a substitute teacher for the cooperatingteacher, Linda, to be released periodically for half day planning sessionswith us. We developed tasks in the form of worksheets for class use.Unlike the usual use of worksheets, where pupils tried to complete asmany exercises as possible, these worksheets were to be taken asstarting points for class discussion. For example, students would work onone problem, or a series of related problems, either individually or ingroups and then Linda would lead a class discussion on their solutionsor difficulties. Manipulatives such as fraction strips, flats (10 x 10 cm gridpaper), longs (10 x 1cm strips) and cubes (wooden 1 cm cubes) wereused to introduce, develop and relate common and decimal fractionconcepts and operations.Students also wrote in their mathematics "learning logs" (Linda'sform of journal), usually once a week, (but sometimes once every twoweeks) about how they felt about the mathematics lessons and tried toanswer the teacher's question about "how can you help me teach you?"51Linda responded to all the journal entries. Also, once a week studentsprepared their own questions (and solutions) in groups and Lindaselected student-prepared questions for group and class discussion.Lessons were video taped, as were four interviews with sixchildren (who were recommended by Linda as representative of a rangeof mathematical ability). Linda and the researchers evaluated the projecttowards the end of July, 1992, after we had viewed the video tapes.The Teacher's BackgroundLinda, the teacher, started teaching in 1966 but had noopportunities to attend in-service mathematics courses during her earlyteaching career. She reached a turning point in her teaching outlookwhen, in the summer of 1990, she attended a professional developmentcourse on whole language which had one session on mathematics.Even though the course made her more aware of her teachingphilosophy, most of the mathematics part of it was "just filed away"without any connection to her teaching practices in mathematics.The second turning point came after she attended a professionaldevelopment course in the Fall of 1990 on the implementation of theYear 2000 document, a major curriculum change in British Columbia. Atthis point she felt that she should change her teaching to try to establish aclassroom environment conducive to risk taking.However, she was acutely aware that she needed moreconfidence in teaching mathematics, especially the topic of common anddecimal fractions. So when she was approached to take part in theproject on students' understanding of common and decimal fractions, sheagreed enthusiastically "because division with decimals was the most52difficult concept I tried to teach my Grade 6 class last year." But shemade it clear that she would need a lot of support from us (the projectsupervisor and me). In spite of her interest in division of decimalfractions, by the time the project was completed, it turned out that weincluded decimal multiplication, not decimal division.Linda's Growing Professional DevelopmentInitially, Linda viewed us as experts and herself as a novice. Also,her early lack of confidence made her very dependent on our ideas onwhat would be suitable as lessons and worksheets. However, she didsuggest that she needed an overview of the lessons before getting intospecific lessons. During the planning stages she wrote down whichconcept was to be the focus, which worksheet would go with whichlesson and so on. We prepared all worksheets but before using anyworksheet, Linda examined it thoroughly. Throughout the project sherarely used the textbook as most lessons were based on the worksheets.These worksheet exercises were meant to initiate discussion whereastextbook exercises were usually practice exercises.During the lessons she consulted with us on anything about whichshe was unsure. She said that she was confident of her proceduralknowledge but was unsure of some of the different ways ofconceptualizing and relating common and decimal fractions. Forexample, she said that she had "never thought of 0.2 x 0. 3 as 2/10 x3/10" and just placed the decimal point in the answer 0.06 procedurally,by using the rule of counting the number of places after the decimal point.As the lessons progressed, Linda became less and lessdependent on us. For example, at one stage we asked her whether she53would like to look at the worksheet for the next day and she replied thatshe did not feel she needed to go over it with us as closely as she usedto before. According to her, there were two crucial points when shestarted becoming more confident of herself: one, when she saw whatchildren could do and learn while preparing their own questions (duringthe fifth lesson) and the other, when she saw the children's excitementand involvement when they tried to find their own rule for placing thedecimal point in the product of two decimal fractions.She found students highly motivated while preparing questions fortheir classmates to answer. They also asked and answered more difficultquestions than they were accustomed to. In trying to find the rule for theplacement of the decimal point, children were so involved that they losttrack of time. Students were encouraged to use, as Linda put it, a"think/pair/share strategy" where they tried to solve the problemthemselves and then compared and justified their procedure and solutionwith a partner.She also had to respond to unexpected and unplanned situations.Instead of being dismayed by this, she began to enjoy "learning together"with the pupils. She became more and more comfortable with childrensetting the pace and direction of the lessons. In short, instead ofaddressing "your (the researcher's) agenda," she did not mind being"sidetracked" by her students, and began to assume more ownership ofthe project as the days passed. Linda had this to say about the lessonson multiplication of decimal fractions in her report on the project:My teaching method underwent radical change during theselessons. In the past when I "taught" this concept, I would presentand explain the algorithm, guide the practice and assign someindependent practice. This time I did not tell the students how to54multiply decimal fractions. They were asked to struggle and messaround with the problems and try to make a little sense out ofmultiplying decimals I was never sure what direction the lessonwould take or what the outcome might be. One student reported inher log that this search for the rule made her frustrated and angryat first, but excited and proud when she was successful. The quietmurmur of Oft was quite wonderful.A key factor in Linda's growing confidence was her excellent rapport withthe class and her willingness to cultivate a risk-taking environment,without fear of children or of herself making mistakes. Her de-emphasison getting the right answer also helped pupils to feel comfortable about"messing around," without having to worry about the answer or aboutcompleting the problem. Often she celebrated different answers bysaying "Aha, we have someone with a different answer, let's hear this!" orshe even expressed disappointment when there was no disagreement!Because of her participation in the project, she said:1. She increased her confidence in teaching mathematicsconceptually.2. She would never separate the teaching of common fractionsfrom decimal fractions: these topics would be taught together.3. Student journals and student constructed questions had goodpotential as learning tools.4. Her experience in the project encouraged her to participate insimilar projects and share her experience with colleagues.5. She could study problems in her teaching-learning situationmore systematically and collaboratively, if not with university researchers,at least with other teachers.6. Student interviews were informative, and viewing videotapes ofsuch interviews would help plan future lessons. Alternatively, such55interviews should somehow be incorporated into the teaching-learningprocess.7. Spending a longer time discussing rather than telling, helpedstudents develop mathematics more meaningfully.8. Manipulatives not only helped develop mathematical concepts,they became a "communicative tool, after learning the concepts" by theiruse to explain and clarify concepts.Her professional development continued to grow, even after theproject was over. Below are some activities indicating her continuedprofessional growth and empowerment.1. She taught division of decimal fractions herself--a topic she hadno confidence in teaching prior to her involvement in the project--withminimal support from us.2. She shared her project experiences at a workshop, becauseshe felt like a "born again mathematics teacher."3. She enrolled for a Master's program in Curriculum andInstruction, specifically to improve her ability to undertake classroom-based research that might help her become a better teacher.4. One year later, she attended a mathematics education courseand gathered data from her class as part of her project for the course.5. She conducted regular, short, in-class interviews ("oral"journals) with her students during journal writing time.6. She video taped some of her mathematics lessons to getfeedback and plan for future lessons.7. She agreed to participate in data collection for my doctoralstudy.56Linda's CommitmentIt is important to recognize that this teacher had very good rapportwith her class, and that could have contributed to the students' enhancedunderstanding of common and decimal fraction concepts studied in theresearch project. Moreover, she felt that she had developedprofessionally as a result of her involvement with the project. Indeed, shesaid she was positively "evangelical" about her experiences with theproject, and wanted to get involved in other projects that might enhanceher professional development. In other words, I had a very committedteacher who was willing to try some innovative approaches.The Present Research Setting While actively involved in the research project mentioned earlieras a research assistant investigating middle school children's conceptsof common and decimal fractions, I piloted the use of student journals(SJ) and student-constructed questions (SCQ). The SJ used previouslywere designed to provide feedback to the teacher on student's difficultiesand feelings about mathematics. Moreover, they were used as and whentime permitted or when the teacher thought it appropriate to use them. Ifound that using the SJ to get feedback once a week or so was notbeneficial and neither were the student entries allowing me to gaugetheir understanding or communication of their understanding ofmathematics. Hence, I decided that for this study I would attempt a moresystematic approach to the SJ (such as three times a week, with specificteacher prompts focused mostly on mathematics than on feelings aboutmathematics).57The SCQ in the previous study were prepared in groups and wereused mainly to initiate discussion, but again they were not used routinelyin the class. Students evidenced enthusiasm in preparing the SCQ, butsometimes they tended to give problems that they themselves were notsure how to solve. Moreover, certain students dominated the discussionand not every group member contributed to the preparation of the SCQ.So I decided that for this study I would attempt SCQ once a week, withboth group and individually-constructed SCQ, but with only the latter tobe used for data analysis. Hence, from the previous project experiencesin a Grade 6 class for a period of about 2 months and then in a Grade 4and 5 (combined) class, for another 2 months, I became more aware ofhow to refine and use SJ and SCQ and also gained more confidence inthe viability and significance of my proposed study. The Grade 6 teacher,Linda, also offered to help me for this study.In this study, Linda was teaching a Grade 5 and 6 combined class.At the beginning of the study she told me that she had yet to establish therapport she had with her previous Grade 6 class. At the end of the study,she said that she still felt her rapport with her previous class was muchbetter. She thought that one of the reasons for her lack of rapport hadsomething to do with the different mix of students. For example,previously she had four or five "bright" students who acted as "sparks"and "catalysts" for initiating and leading discussions whereas in thepresent class there was only one such student. In addition, she had noESL (English as a Second Language) students the previous year.Linda had agreed to participate in the study because of hercontinued commitment to try to improve her students' mathematicallearning while working within the usual constraints of time and topic58coverage. Moreover, based on her experience in the previous project,she felt that the SJ and SCQ might be beneficial for the learning ofmathematics.It should be emphasized that the learning log used previously wasmainly to give Linda feedback on her teaching whereas the SJ used herefocused on how the students could communicate mathematical ideasthrough written explanations. The SCQ in the previous project wereused to initiate discussion, and were used sporadically, whereas those inthe present study were used as class exercises (with and withoutdiscussion) on a regular basis.In the present study, Linda started with the same topics that shehad taught in the previous project, namely common and decimalfractions. Linda and I agreed that because she had taught these topics inthe previous project, she could use her prior experience to advantage,and perhaps be more systematic than before in the use of SJ and SCQ.Linda was free to discuss with me (before and after classes)possible approaches to instruction and types of resources that might bemade available for her. I did not teach the class, but observed thelessons and kept notes of salient aspects of the lessons and discussions.Student Participants Some students had exposure to journal writing in a mathematicsclass before, but their experience was so minimal (about three or fourtimes a year) as to be negligible. Neither had any student ever usedSCQ in previous classes. As well, none of the students had Linda astheir teacher before.59All students in the combined Grade 5 (n = 15) and 6 (n = 12) classtook part in the mathematics lessons, but I chose to focus on the Grade 6children (6 girls, 6 boys) because, according to the teacher, most of herGrade 6 children have had sufficient experiences with language andwriting for the purposes of this study. Moreover, four of the 15 fifthgraders children were ESL (English as a Second Language) studentswho had to attend extra English lessons which were scheduled duringsome of the mathematics periods.I used the SJ and SCQ data from 11 Grade 6 children in the class,but selected six of these students (3 girls, 3 boys) for individualinterviews. (One student returned to class after one and a half months,on March 23, three weeks before the end of the study, so I did not includehim in the data analysis.) My sample selection for the interviews can bethought of as reputational-case sampling (Goetz & LeCompte, 1984).The interview data were meant to clarify and complement the SJ andSCQ data, as well as answer research question number three. Thechoice of the six Grade 6 children was dependent on a number of factors,such as the following:1. Their writing in the SJ and SCQ. I wanted a range of SJentries, from those by students who were not very fluent in expressingtheir mathematical ideas in written words to those who were fluent. Forthe SCQ, I wanted a range of word problems from brief word problemsinvolving one or two operations to long, multi-step word problems withextraneous information.2. A range of mathematical and writing abilities as identified bythe teacher, based on her observations of, and interaction with her60students, such as students who were considered good at writing butweak in mathematics and vice versa.From these criteria, I believed that the interview data drawn fromthe six students would be sufficient to analyze and interpret the unit ofanalysis. Note that this is a case study where the unit of analysis is aphenomenon, an approach (using SJ and SCQ in a mathematicsclassroom) and not a particular child and this unit of analysis is what Iwould like "to be able to say something about at the end of the study"(Patton, 1980, p. 100).Teaching-Learning EnvironmentTo give a better picture of Linda's approach and her rapport withthe students, I describe the teaching-learning environment. Lindagenerally used heterogeneous groups of four to five students in hermathematics classes, and changed group members every month. Sheused a number of criteria in deciding the membership of any group, andtried getting a good mix of (a) assertive and non-assertive students, (b)girls and boys, (c) language ability and (d) mathematics ability.She used mainly discussion and manipulatives to teach commonand decimal fractions. When students were given exercises fordiscussion in groups, she insisted that "everyone in the group has to beable to explain." In order to emphasize the relationship betweencommon and decimal fractions, she encouraged the reading of decimalfractions in common fraction notation (e.g. 0.3 was to be verbalised as"three tenths"), by asking all the other students to literally hold their handsout and point at anyone who read 0.3 as "zero point three," just as the"point" was read. The students, including the one who had read "point",61seemed to treat this action as a jocular reminder of how to read thedecimal fraction.Linda used "good" questions (Clarke et al., 1990) effectively. Notethat for Clarke et al. (1990), a question does not necessarily end with aquestion mark and a good question enables students to learn and theteacher to know more about the student as well as allows for more thanone correct answer. For example, she asked "in your groups, find out asmany different names or equivalent fractions for 1 +6 ." Generally sheused such "questions" to encourage exploration and a variety ofanswers.Sometimes she got student attention by saying "it's 'can I trick younow' time" and giving them some challenging questions. She constantlyreminded students that she wanted to "see your thinking," or "explain sothat it makes sense." Some expressions she used in class were: I don'tunderstand that; give me an example. Who are confused? Good, peoplewho are confused will learn something. You are stuck? Good, thosesitting next can ask some good questions to help those stuck. I'minterested in how you think, not just what the answer is. Convince me,justify your answer.Most of the expressions she used were designed to promote anon-threatening, risk-taking atmosphere in the classroom. Sheemphasized to students that explaining their thinking, even if they had ananswer they were unsure about, was more important than just getting thesolution correct. That her encouragement was not in vain was evidencedby students who wrote that they learned "by making mistakes," and thatlearning through mistakes was a priority in their list of "important things inthe learning of mathematics." I did not see any evidence of a student6 2afraid to voice his or her tentative solutions, or being embarrassed atgiving a wrong answer. Just as in her previous class, Linda celebratederrors and disagreements, saying "good, we have some disagreementhere; let's see why."In spite of her slight dissatisfaction with class rapport, I saw signsto the contrary. For example, on April Fool's Day, she told the studentsthat cutbacks in funding had forced the school to lay off some teachers.As a consequence, their class was to be dissolved and the students wereto be distributed over a number of other classes. (All this was said with astraight face!) The students reluctantly packed their bags and books togo to their new classes, unhappy to leave their friends or Linda. ThenLinda said "I've got something important to say. I hope all of you arepaying attention. Are you ready for this? April Fool's Day!" The studentslaughed at being fooled (but relief was written on their faces at not havingto leave their present class) while some first muttered good-naturedly,and then directly to her "We'll get you for this!" Indeed, Linda agreed thatthey could play an April Fool's Day trick on her, as long as it was notdangerous or messy (in the sense of dirtying the classroom). All in all,Linda's trick was taken in good humour, and the class settled back totheir mathematics lesson without undue delay.Another example showing good rapport was a questionconstructed by the students. They were to make up a word probleminvolving multiplication of the numbers 32 and 0.125. Two studentsmade up the following problem and brought it for Linda's comments:"Mrs. Lomax eats 32 chocolate bars a year. If each chocolate bar makesher 0.125 lb heavier, how much weight does she gain in a year?"Neither the students nor Linda (Mrs. Lomax) were uncomfortable with the63problem, a sign of the rapport between her and the students, indicatingthat the children did not mind joking with, and about, their teacher.That they liked her teaching approach was evidenced by studentcomments (during interviews) such as:Mrs. Lomax tells us to write down how we think, what goes on inour brain. Teacher last year just told us to write down the answer.I like it this way.She doesn't give us work out of the book as often; she makes it funfor me. I don't have as much pressure as I used to. I feelcomfortable in her class.She lets us figure out a lot of things ourselves, and I like that.I think Mrs. Lomax is the best person I have had to teachmathematics because the previous teacher used only thetextbook, and there were people crying because they couldn't dodivision.In addition to good rapport between the students and the teacher, Lindawas also firm in that she made sure that students did involve themselvesactively in learning tasks. For example, she checked that students hadprepared their SCQ. The SCQ were collected and edited by Lindabefore distributing them to the class as typewritten exercises (with namesof those who prepared the question written next to the relevant problem).These SCQ, together with a few teacher-prepared problems, became thesource of exercises for the students. A few of these questions wereattempted every day, first individually, and then through groupdiscussion. Sometimes students were given a choice of doing thequestions individually or in groups, but either way, they had to explainhow they obtained the solution. The students seemed very animated••64during the preparation of the SCQ. Students commented on theirenjoyment of the SCQ in their journals as well as during interviews.The SJ were used on alternate days (three times a week) in themathematics class. Students wrote in their SJ in response to specificteacher prompts such as "Explain why you agree or disagree that 0.8 =3/5." Most of the time students wrote in their SJ towards the end of themathematics lesson (the last five to nine minutes). All individually-constructed SCQ were also written in the SJ, but students were givenmore time to do this (ten to twelve minutes) than when explanations ofmathematical concepts were required.Very few textbook exercises were used for the duration of thestudy. One reason for this was that the teacher was attempting to usestudent-generated questions for discussion and as exercises. She alsowanted to see how students would fare on standard textbook exercises,even though they followed an approach that minimised the role of thetextbook. She found that the students thought that the textbook exerciseswere "no big deal"--that is, they had no difficulty with these exercises.ProcedureThe study covered a period of 15 weeks, from January 7, 1992, toApril 14, 1992. I was in the class for three days a week, namelyMondays, Wednesdays and Fridays, except for the Spring break fromMarch 16 to 20, 1992. I observed the mathematics lessons, but did nottake part in teaching or other in-class activities such as group and classdiscussions. However, I kept notes of salient and significant events thattook place. For example, I noted what happened in the mathematicslessons (e.g. discussion, the teaching-learning environment, use of65manipulatives, type of journal entry initiated, and interesting or insightfulquestions).The students wrote in their SJ for an average of seven minutes alesson (usually towards the end of the lesson) for the three days a week Iwas there. The teacher responded in writing to the entries once a week,with responses such as "How did you decide this?" and "Good thinking,Shaun." I did not respond to the students about their journal entries but Ikept track of the entries in three ways: (a) by reading the entries as theywrote, while walking around the class; (b) by collecting and skimmingthrough the SJ as soon as they had finished writing, and returning themto Linda within an hour; and (c) by collecting the journals and perusingthem before the next journal entry was due. In all three instances, I keptnotes about salient points I found in the SJ.Although Linda and I discussed the journal entries informally and Isummarized these discussions for my records, I did not offer, on a regularbasis, any suggestions on what entries might be suitable. However,once or twice, when Linda asked for some suggestions on how to makethe journal entries more interesting or mathematically relevant, I gave hersome suggestions. For example, I suggested a possible way to get thestudents to write about what they thought mathematics was like. At times,Linda was worried that I was not getting the data that I wanted, but Iassured her that I was quite satisfied, because what I wanted to knowwas what could be done to assist students' mathematical learning in amathematics classroom with SJ and SCQ implemented by a practisingteacher rather than a researcher from outside the school.The students also wrote SCQ once a week, on Fridays, in groupsof three or four and discussed their questions both in groups and in class.66I kept notes on these discussions, as well as copies of each group'squestions (both edited and original versions). In addition, each person inthe group was asked to write a question in their SJ, either similar to ordifferent from the group question. I examined and analyzed theseindividually-constructed questions (which were in the SJ) of the Grade 6students. I also looked through the SCQ prepared in groups but did notanalyze them because these SCQ were prepared by both Grade 6 andGrade 5 students (and the latter were not the focus of my study).I initially interviewed (and videotaped) nine of the Grade 6 childrenon February 5, 1992. I did not interview all 12 of the Grade 6 studentsbecause two were absent and one was hearing-impaired. I interviewednine rather than my target of six because it was too early to use any of thecriteria suggested earlier for the selection, and also because of possibleattrition. Then, on March 4, 1992, together with the teacher I selected thesix Grade 6 students based on the student writings in SJ and SCQ (usingthe criteria mentioned earlier) and interviewed them. Each studentinterview took about twenty minutes. There were three interviews perstudent, the first within four weeks into the study (with the nine students),the second within eight weeks into the study and the other towards theend of the study (April 14, 1992). Patton's (1987) suggestions aboutqualitative interviewing guided these semi-structured interviews whichwere designed to elaborate on SJ entries and SCQ, and to indicateanswers to some of the research questions. Sample questions from allthree interviews are in the appendix.I interviewed Linda once (audio-recorded) on April 30, 1992, twoweeks after the end of the study (because she was not free earlier), tofind out her perceptions regarding the use of SJ and SCQ. But I also67kept notes of my informal discussions with her throughout the study andused these data to supplement data from the interview. Linda wanted tokeep a journal of her own but could not because of her busy schedule.However, she did keep notes of topics taught and exercises given,including notes of lessons on the days I was not there. Her notes,together with mine, as well as the SJ, SCQ and the interview (students'and teacher's) transcripts served as a form of triangulation of data for thestudy.Data analysisAnalysis of SJ Analyses of the individually-constructed SCQ, students' writtenexplanations in the SJ, and their verbal explanations in the interview,coupled with the notions of conceptual and procedural knowledge inmathematics (Hiebert, 1986) helped answer research question #1 (Whatmathematical knowledge do students reveal through their writing?)In addition to prompts requesting explanations of mathematicalconcepts, some of the prompts for the SJ were about the role of SJ andSCQ in the mathematics class. Responses to such prompts, togetherwith interview data, were analyzed using the constant comparativemethod (Glaser & Strauss, 1967), inductive analysis (Patton, 1987) andnotions of convergence (Guba, 1978) as guides for the emergentcategories, in order to answer research question # 3 (What are the rolesof SJ and SCQ in mathematics learning, according to (a) students? and(b) the teacher?)If communication of mathematical knowledge is an indication of amathematical competency (NCTM Curriculum and Evaluation Standards,681989) and if communication plays a crucial part in the growth ofmathematical knowledge (e.g. Ernest, 1991), then it would beappropriate to include students' use of language to communicate theirknowledge of mathematics (fractions, in this instance) under researchquestion #1 (What mathematical knowledge do students reveal throughtheir writing?) Hence, I attempted to identify students' use of language tocommunicate their knowledge of fractions, using Freudenthal's (1978,pp. 233-242) levels of language. According to Freudenthal, newmathematical concepts are first articulated in words by usingdemonstrative language (just pointing at or giving examples), followed byrelative language (showing an understanding of relationships), andfinally by functional language (showing relationship betweenrelationships, or making generalisations). I also analyzed the interviewtranscripts for evidence to support Freudenthal's classification. One ofthe reasons for choosing the SJ as a writing task in this study was to findout how students use words to communicate their mathematicalknowledge to others, and so Freudenthal's levels of language provide anindication of this communicative ability (although there were no researchquestions per se on levels of language).Analysis of SCQFor an analysis of the SCQ, I looked for salient and recurrentfeatures (themes) and knowledge of fractions evidenced by theindividually-prepared SCQ. For example, were the SCQ becomingprogressively more difficult in terms of the number of steps involved intheir solution? These analyses helped answer research questions #16 9and #2 (What are the salient and recurrent features (themes) of the SCQ,if any?)ConclusionIn this chapter, in order to contextualize the study, I first describedmy background, the teacher's background, the reason for choosing theGrade 6 students for the study, and the teaching-learning environment. Ialso emphasized that the SJ and SCQ of all the 11 Grade 6 studentswere used for data analysis, but that the interview data from six of theGrade 6 students were used to complement data from the SJ and SCQand to answer research question number three. Then I described theprocedures for data collection and analysis.70CHAPTER 4. RESULTS AND DISCUSSIONIn this chapter, I discuss results of this study in relation to theresearch questions. Before going any further, I wish to emphasize thatthe "case" in my study is the use of SJ and SCQ as part of a mathematicsclass. The work of the eleven Grade 6 students together with theinterview data from the six Grade 6 students are just instances of thecase "within which issues are indicated, discovered or studied so that atolerably full understanding of the case is possible" (Adelman, Jenkins, &Kemmis, 1983, p. 3).Below each research question, I interpret the Grade 6 students'written responses relating to that research question. I also includeinterview and other relevant data from my study, where necessary. ThenI synthesize information from the eleven instances of the case, togetherwith information from the interviews, and summarise the findings inrelation to the research question. Note that in using students' responses,I attempt to present words they used, and avoid correcting spelling orother errors, unless clarity of meaning necessitates such corrections. Toensure anonymity, pseudonyms are used for the students and theteacher.Research question #1 and resultsResearch question # 1: What mathematical knowledge dostudents reveal through their writing?By mathematical knowledge I mean knowledge about commonand decimal fractions such as part-whole, equivalence and renaming.Even though the research question is on mathematical knowledge, I will71juxtapose my discussion of mathematical knowledge with the use oflanguage to communicate mathematical understanding, as language isinextricably linked with thought (Vygotsky, 1962) and hence tounderstanding. Admittedly, mathematical knowledge andcommunication of mathematical knowledge can be seen as distinct fromeach other but the prevailing view of mathematics as socially-constructedknowledge (e.g. Bishop, 1985,1988; Ernest, 1991; Lampert, 1988)implies that communication is vital for the growth of such knowledge.Indeed, the NCTM Standards (1989) emphasizes that communication ofmathematical ideas is an essential component of the mathematicscurriculum. For discussing the communication of mathematicalknowledge, I will use the levels of language suggested by Freudenthal(1978, pp. 233-242). According to him, such communication progressesthrough three levels of language: demonstrative (pointing out instances,without explanations), relative (using words to indicate relationship orprocedures) and functional (generalisations or relationship betweenrelationships).Part-whole (Common Fractions) To the prompt "What do you understand by fractions, for example,the fraction 3/8? Show 3/8 on the pink strip of paper," students typicallywrote as in Figures 4.01 and 4.02.All the pieces have to be equal. 3/8 of the parts you colour. It's a 8strip and the three parts are called 3 8ths.Figure 4.01: Jackie, January 7.72Fractions are mainly pieces of something. Take 3/8 for instance. Itsimply means 3 parts of 8 parts. So 5/8 would be 5 of 8 and 6/7would be 6 of 7.Figure 4.02: Mike, January 7.Possibly because they had used such fraction strips in recent lessons,the students had no difficulty shading in 3 out of 8 equal parts on the pinkstrip of paper provided. The entries indicate that students had the notionthat fractions had something to do with equal parts and parts of a wholeand that there was some sort of comparison involved, namely acomparison of the part to the whole. This part-whole concept wasillustrated by diagrams such as those in Figure 4.03 and Figure 4.04:Nei,x4 I II^C 0 ronAlatis^C ra.c.I.  I co 01 S—a--41^.1^C% -Fid,i;or,^Hit.^i--4,-. 1T.,,,7 flaFigure 4.03: Sam, January 7.73gainICNOCOCIVICO^If" -56‘45.-1• (,1-.0.+ .^Know-1-0.c_1-1 0 rl -3^4,00000Not ecralFigure 4.04: Tammy, January 7.The first entry illustrates a region model (geometric shapes divided intoequal-sized parts) and this model was used predominantly by thestudents in explaining basic fraction concepts. The second entryillustrates a discrete model (where a number of elements in a subset iscompared to the total number of elements in the whole set) shown by thecircles and a region model shown by the rectangle. Indeed, the studentwho had the second entry was the only one who illustrated herexplanation with a discrete model (as well as a region model). Even she,however, drew equal-sized circles, although the number of circles ratherthan the size of the circles determines the fraction represented by thediscrete model (e.g. 3/8 of 8 students is 3 students, irrespective of thesize of the students). Students' explanatory diagrams in the SJ indicate74that, for them, the part-whole concept is a comparison of equal parts to awhole, whether the parts are from a whole region or from a total numberof objects. Since fraction concepts are usually taught using regionmodels (where the parts are equal), it seems natural that most studentsuse region models to represent fraction concepts and even if they usediscrete models, equal-sized parts are used.The part-whole concept also seems to be influenced by the imageof a physical representation, as exemplified in Figures 4.05 and 4.06, inresponse to the prompt "Is half always equal to a half?"No, because if you have a medium pizza, and a large pizza, thenyou cut out them in half, what you have is 2 larger pieces from thelarge than the medium.Figure 4.05: Tommy, January 17.No, because if one whole can be bigger than the other whole, so ifthe wholes are different, the 1/2 will be different too.Figure 4.06: Shaun, January 17.From Figure 4.05 and 4.06, students seem to know that the fraction "1/2"is indeed a relationship between a part and a whole. On the other hand,the part-whole relationship seems to be perceptually bound. Forexample, when they said that a half is not always equal to a half, theyusually based their explanations on images such as two different-sizedrectangles (or circles) indicating different sized wholes and comparinghalf of one rectangle with half of another rectangle. No student argued7 5that because the fraction 'half' was a relation of a part to a whole, in thatsense, as long as that relationship held good, a half is always a half.One student (Tanya), however, did seem to suggest that a half haddifferent symbolic representations but retained its value (see Figure4.07).I think 1/2 is always equal to one half no matter what number youhave you can still make it one half like 4/8, 3/6, 6/12.Figure 4.07: Tanya, January 17.But even she thought she had misunderstood the question and said thata half is not always equal to a half when Linda drew two different-sizedrectangles and asked her to compare the halves in the two rectangles. Itmust be admitted that Linda did not point out the possibility of perceivingthe symbol "1/2" as representing a number disembedded from concretereferents. Anyway, the crucial role of the whole seemed well-understoodby all students.In terms of levels of language for communicating the part-wholerelationship, three students (whom the teacher had perceived as weak inmathematics), exemplified demonstrative language by drawing adiagram and writing that the diagram represented the fraction as in theresponses shown in Figure 4.03 and Figure 4.04.Five students used the relative level of language, for example, byrelating the fraction to pieces of an object, as in Jackie's response(Figure 4.01). Three students used the functional level of language, forexample, by giving some sort of generalisation, as shown by Mike's76response (Figure 4.02). Mike's explanation indicates a generalisationthat fractions are a relationship between parts and wholes, without,however, explicitly stating that the parts have to be equal in size.It must be noted that when I say that students used demonstrativelanguage, I am saying that they used words not so much to explain butrather to point out instances or examples. For example, if a student wereto explain what an even number is, the response "Two, four and six areeven numbers" would be an example of the use of demonstrativelanguage whereas the responses "a number that ends in 2, 4, 6, 8 orzero is an even number" and "an even number is any number which isdivisible by 2" would be examples of the relative and functional levels oflanguage, respectively. Furthermore, in saying that students used thedemonstrative level of language, I am only indicating the level oflanguage used to communicate the student's mathematical knowledge,and am not implying that use of demonstrative language indicates a lackof conceptual knowledge.In summary, the SJ indicated that students generally used theregion model to explain the part-whole relationship of fractions. Sincemost teaching of fraction concepts makes use of region models, it seemsnatural that such models dominate students' explanations of fractions.The equality of the parts was an important aspect of this explanation,even in the only instance when discrete models (which did notnecessitate such equal-sized parts) were used. It seems to be the casethat while students recognized the importance of the whole and therelationship of the part to the whole, they had difficulty disembedding theabstract concept of fractions from perceptual cues such as the size of thepart (e.g. when they explained why a half is not always a half). It is77instructive that even though early (whole) number concepts areembedded in images--such as 2 apples, 2 books and 2 people-- moststudents can abstract the `twoness' of different sets of two objects byperceiving the number 2 as referring to the numerosity of a set ratherthan the type of objects in the set. Students find such disembeddednessmore difficult in the case of the part-whole relationship in fractions, asevidenced by their perceptually bound explanations in the SJ. Such adifficulty could stem in part from the way fractions are taught (for example,by limiting the type of manipulatives used to region models). In terms ofcommunicating the part-whole relationship through the SJ, the studentsseemed to indicate differences in communicative competence consistentwith Freudenthal's (1978) levels of language.Equivalence and Renaming (Common Fractions) To probe students' understanding of equivalent fractions andrenaming, they were asked to "explain how would you rename 1/2."Three responses are given in Figures 4.08, 4.09 and 4.10. There seemsto be a slight difference in the written explanations by Tommy and Jackie(the best in mathematics and the weakest in mathematics, respectively,according to Linda), see Figures 4.08 and 4.09. For example, thoughTommy wrote 16 equivalent fractions, starting from 1/2 and successivelymultiplying both the numerator and denominator by two, and Jackiewrote fewer equivalent fractions, Tommy had two different ways ofgenerating equivalent fractions (two "rules") while Jackie had only onerule. Judging from the type of fractions generated, Tommy seems to befollowing an arithmetic algorithm whereas Jackie seems to be selectingspecific fractions which are easily recognisable as equivalent to a half78(for example, 2/4, 3/6, 4/8 and 5/10) or are obtainable by following apattern (for example, the fractions 3/6, 30/60 and 300/600 can begenerated by affixing zeroes). Moreover, even though Jackie's use of theword "imagine" might indicate spatial partitioning, she drew only onediagram and interview data indicated that she could not partition aregion, mentally or on paper, into a large number of equal parts. I wouldinfer from these entries that both Tommy and she used the diagram forillustrating rather than for generating equivalent fractions. Anotherdifference was in the type of language used. While Tommy used afunctional level of language to express a generalisation, for example, bywriting "I will choose an even number and divide it by two," Jackie used arelative level of language in trying to relate the equivalence to the size ofthe piece by writing "because if you imagine cuting it in half agin it still ishalf." Their explanations seem to emphasize procedural rather thanconceptual knowledge.1/2, 2/4, ...32768/65536. I will choose an even number and divideby 2. I could also multiply both top and bottom by the samenumber.■E■Figure 4.08: Tommy, February 12.791/2, half, 2/4, 3/6, 5/10, 4/8, 30/60, 300/600. 1/2 is = to 2/4because if you imagine cuting it in half agin it still is half, but 2/4.Figure 4.09: Jackie, February 12.1/2 = 2/4, 3/6, 6/12, 9/18, 12/24, 4/8; My procedure that happens inmy head is you say start from the least number that is equal to 1/2.Then you think of a number that could be split in half like 2/4 s or12/24.22412 j2IBM 24Figure 4.10: Maureen, February 12.Maureen's explanation (Figure 4.10) also shows the use of an arithmeticalgorithm, but she seems to have made a generalisation in that shewrites "start from the least number that is equal to 1/2." Her verbalexplanation complemented by her use of two partitioned rectangles80I S^1319 IC^g,C--.S7r.kA/ 4'7,ro4._ 7a4e)_^_^Iasi.^'Wt.^rt,TACLACOIindicate that she is using both relative language and functionallanguage. She seems to reflect conceptual and proceduralknowledge.Further examples of students' knowledge of equivalence andrenaming are shown (see Figures 4.11, 4.12 and 4.13) when studentswere asked to justify whether 8/15 was to the left or to the right of 1/2 on anumber line:Figure 4.11: Maureen, February 12.Figure 4.12: Darlene, February 12K1vi.N 15 people. ‘A A d /a- b0dwou.14^el- ell•*11 )57 eX c‘konfleear n eat i^14 I-^/nor^ 4 \0 0,AC.Z..=4.(s^if^a.4^fe nd^f0^Siia ^e_^50to^56...)-1f'ee'l=je-11 1111 1J1=4).1-4-It iftii 44411 76 9Figure 4.13: Sam, February 12.Figures 4.11 and 4.12 relate 8/15 to 1/2, with no explicit renaming ineither one. The first compares the fractions with pieces of pie and alsouses a number line (with the mark next to 1/2 unlabelled) with the wholerenamed as 15/15 and the half to indicate 8/16 (as confirmed duringclass discussions). Maureen uses everyday language (such as "piecesof pie") in her verbal explanation and then uses her explanation toanswer the question which referred to a number line. Perhaps she ismore comfortable using everyday language to explain the concept inwords, before transferring her answer to the number line, rather than usethe number line directly to answer the question. The second excerpt hadno diagrams but I gathered from classroom discussions that Darleneperceived halving 15 as equivalent to dividing 15 by 2 and also that theexpression "how number can be half" implied fractions equivalent to half.In the third excerpt, Sam does not explicitly state any relationship82between 8/15 and 1/2, but like Maureen, indicates sharing something,and uses circles, rectangles and a number line as illustrations. Sam usesrelative language by making a meaningful comparison between the sizeof a fraction with the size of a person's share and the number of personsgetting a share. Indeed, all the three excerpts mentioned can be said tobe using the relative level of language because they attempt to indicatemeaningful relationships. Moreover, all indicate that they are aware thatfractions could be written differently but nevertheless have the samevalue, thereby evidencing conceptual knowledge.Other instances of SJ entries indicating the extent of students'understanding of fractions through renaming was shown in answers tothe prompt "In your journal, I want you to order these: 1/2, 3/4, 3/8, 5/8,1/4. Then write how you made the decisions. You can explain withdiagrams. I want to know how you think." Three excerpts are shown inFigures 4.14, 4.15 and 4.16:(0 3_2_ 7--•^--77^, 1^7(111117Z 71 '1 LEA dizt (4w74t^ctek, cjie,e,fta V!,eAani9at^cztjgrteltd -r-,^u;Z. (Ana.,caei -,'&2//1/./ (2.777,,ItaqAt") cAz01940, czo- 8 . ,fa^!=Gcrixi^.^,Z,11-ttr)t,i•L-th-exit) CA11444ed^‘Ath cLiCgrit.t'-4'4 xti-t2aztl 44caq 41.6f)v VAL- 2:nd^It4a4neAl. Lund Z UY4..ttiU difviv ICr ititt4V4-. Am'01e6V- I/2^(Ala/^2j::A. 17 ,Z16,77 /11.‘07^th01/71 „(41/11_ Fed9-idz<7<3.0 236 1/1Aikixi cis,^clbadvAraz ups- dxylnied^u<1 /ft 1,7,3/4Figure 4.14: Tommy, February 17.83Ordes■ge Fra‘ctior\Sove"(Stcp Wes—aOrkr }hest^L.tttc-A.^Sree*Oawitch btu ho Smi tsc.Figure 4.15: Jackie, February 17.J tte. dat, 3K=ept. othe.beco^-tit@ piectodaati ka4 cribi, 3 a-0,07,Jizi,44,^1.4 klig• semi), ore ,tieco/im^combbecame UtirL' 64,geir awm,91iink tArl arit be TY* en beazzae,TV* Lilatn, anzi fec LA"°^A g^42, 'mot en be ca/uor, § 7,./3 1 g/ifliflWirr7 640-1,^cons lad 6)wiii, 7Ritok ekth^bify.e& ov,,befalac ik0 bflgeFigure 4.16: Shaun, February 17.WO.K3M11111111,1184Tommy (Figure 4.14) seems to have understood equivalent fractions tothe extent that he can order fractions by renaming and comparing theresulting equivalent fractions. For example, he renamed 1/4 to itsequivalent 2/8, and 1/2 to its equivalent 4/8 and compared the resultingfractions 2/8 and 4/8. He did not use diagrams to explain his reasoning.As well, the "rule" for renaming fractions was not taught by the teacher.Rather, many "rules" were attempted by students, and Tommy was one ofthe few who could justify his rule verbally and pictorially. Hence, I couldconclude that when he wrote "All fractions had eight groups so whichevernumber had the highest top number when changed was the largest,"Tommy had abstracted and generalised the concept that as long as thedenominators are equal, the numerators indicate the magnitude of thefractions. Although he did not use any specific mathematicalterminology, I would say that he was operating at the functional languagelevel, as he could make generalisations.In the second excerpt, Jackie drew same-sized units/wholes for1/2, 3/4, and 1/4 to show 3/4 > 1/2 > 1/4 but she had two biggerrectangles to represent the wholes for 3/8 and 5/8 (her rectangles for theeighths were double the size of her rectangles for the fourths, see Figure4.15). So, even though she showed 5/8 > 3/8, she had changed thewhole (for the fourths and eighths), and it was difficult to see how to orderthe fractions. From her statement "witch one has the biggest pecies" andthe diagram, I infer that she considered 5/8 the largest and 1/4 the least.But her diagram gives no indication of whether she considered 3/8 equalto, less than or larger than 3/4.She seemed very dependent on her perception of diagrams. Theconcept of a common "whole" on which to base comparisons of fractions85was not evident. Though she articulated relationships through diagrams(not entirely accurately), thereby indicating some conceptual knowledge,the fact that she did not use words to elaborate on these relationshipsindicated that she was using the demonstrative level of language.Just as in Figure 4.14, in the third excerpt (Figure 4.16), too, nodiagrams or number lines were used for explanations. Even withoutdiagrams, I gathered that it was clear to Shaun that 5/8 was more than ahalf, and that 5/8 was 1/8 bigger than a half. Hence he evidencedconceptual knowledge. The interview data also indicated that he usedvisual imagery rather than overt diagrams to "see" fraction relationshipssuch as the ones described in the excerpt above and that he had nodifficulty drawing the appropriate diagrams when asked to do so. Fromhis statement (see Figure 4.16) "5/8 is 1 eight more than 1/2 and lessthan 3/4," I concluded that he was using the relative level of language, ashe could relate the fractions 5/8, 1/8, 1/2 and 3/4.In summary, students explained equivalent (common) fractions byusing region models. In addition, renaming was used throughout togenerate equivalent fractions and to order common fractions. As well,the majority of students explained correct symbolic procedures forrenaming common fractions and used relative language to communicatetheir understanding of renaming and equivalent fractions. Theirconceptual knowledge seemed evident when they related everydayexperiences like sharing pieces of pie to explain the order of magnitudeof fractions. Their awareness that fractions could be written differently butnevertheless have the same value is also indicative of their conceptualknowledge.86This difference between the form (e.g. the actual symbols used torepresent half) and substance (e.g. the equivalence of the fractions) inmathematics has a parallel in language where the surface and deepstructure of an utterance are differentiated, with the same form used toconvey different meanings or different forms used to convey the samemeaning. For example, while the English expressions "How are youtoday?" is just one of many equivalent forms of greeting or openinggambits in conversation, the mathematical symbols 2/4, 5/10, and 31/62are all equivalent to the fraction 1/2. Where communicative competencein language is indicated by being able to use different expressions toconvey the same meaning, conceptual knowledge of fractions isindicated by the use of renaming.Decimal Fractions and Inter-relationship between Common and Decimal Fractions Some SJ excerpts demonstrating student understanding ofdecimal fractions and the inter-relationships between common anddecimal fractions are shown (Figures 4.17, 4.18 and 4.19) in response tothe prompt "What does 3.1 mean? Explain with words and a diagram."3.1 is another way of saying 31/10 or 3 1/10. 3.1 means 3 wholepieces and 1/10 added on.INAMIRMEMMIFigure 4.17: Tommy, January 21.87corm.. Caziwrs^xeci mffinimte37fat 03.1 means 3 whole units and 1 tenths of a whole unit.NEEMENEVERLEMEMENEEMENMENEENNERISM■■111■■11111111■Figure 4.18: Darlene, January 21.3.1^42/11011.1Ar coat. l* 1 VIIi 0 ' 34h . . •^At- taco(.  ickev^43-ti... cdurcalx,,ipit- 4 - zz,f1) intte- ca pie-ear. 'S aLAW_Me-- kcsataiLe4, -Int, ,a1-C. 4./..e. ',At. Iciei-oe.4. , ,4tr-x.att. 'Leaks_ 4letai 4 ke 1 titaAL ia ( LavAdet4-rzt -dif— 'eitt 4(4). 'Figure 4.19: Tanya, January 21.The first excerpt (Figure 4.17) shows that Tommy can represent thedecimal fraction 3.1 as the common fraction 31/10 and the mixed number3 1/10. In relating his explanation to his diagram, he has indicated that 3wholes is equivalent to 30/30. His flexibility in representing the decimalfraction using various symbols and showing their inter-relationship88evidences the use of a relative level of language and also demonstrateshis understanding of common and decimal fractions.In Figure 4.18, Darlene (supposedly a weak mathematics student,according to the teacher) demonstrates her understanding of the decimalfraction 3.1 by using relative language and relating it to a figure (onceagain a region model) which is very similar to that given by Tommy.Hence, like Tommy, she too evidences conceptual knowledge andutilizes relative language to communicate in words her knowledge aboutthe inter-relationship between common and decimal fractions. But unlikeTommy, she does not indicate different common fraction notations for 3.1,possibly because she felt she had answered the question adequately, asshe was asked to explain the meaning of 3.1 "with words and a diagram."Hence, though she was supposed to be weak at mathematics, sheseemed to be at the same level of conceptual knowledge as Tommy, atleast for this topic.In the last excerpt (Figure 4.19) the student explains herunderstanding of 3.1 through the use of circles (representing cakes) withcandles drawn in each of the sectors, instead of shading the sectors asstudents usually do for the region model. In a sense she was using adiscrete model (31 candles) but her explanation that "you divide eachcake up into 10 pieces," resembles the language used with the regionmodel. Unlike Tommy and Darlene who use a mathematical context(rectangles) to illustrate 3.1, Tanya relates 3.1 to an everyday context(cakes and candles), but like them, she, too, uses relative language.Both the words and diagrams demonstrate students understood thedifferent ways of writing 3.1, as either a decimal or common fraction and89also that the students could move from the decimal fraction to thecommon fraction notation.Further SJ entries (Figures 4.20 and 4.21) indicating students'knowledge of decimal fractions and an awareness of the relationshipbetween common and decimal fractions, are given in response to thepromptFind out who received the gold medal for the downhill skiingchampionships, if the top three won in the following times:7:37.59, 7:31.57 and 7:39.80. write how you thought it out. Alsowork out how much faster was the gold medal winner compared tothe one who got second. (Feb. 19)rturfrI. 11192A -^7:5759 7:31. r3-39.iv ^TN 3-avid^fit. s aerie Ti t^►S kwir^7: 3 P-SZ, radte‘te-j s :37-5`1 7:37.51 1:3 1 ,C1 O: 4 (-a -L Figure 4.20: Tommy, February 19.In the excerpt in Figure 4.20, Tommy subtracted and got the answer"0:06.02 seconds faster." He looked at the minutes column first, saw itwas the same, then examined the other digits systematically. He alsoused his everyday experience to decide that the polarised comparative(Lean, Clements, & Del Campo, 1990) lower/higher was not compatiblewith slower/faster, and that lower was related to faster rather than slower.He used an efficient strategy for ordering these decimal fractions, and90communicated well his knowledge of decimal fractions through hiswriting.^- Y.Y)- 5%0) &..coc-c. NA:1'z^30131.54^ t14-44‘. 42maile5-t norim^- eiv:5 tr_coiN► )sx- 3. A^.ao11'_tafi+ Portersw.5c1 trl*in 441e_rrikictle,Figure 4.21: Jackie, February 19.Jackie (figure 4.21) seemed to have a grasp of decimal fractions.Admittedly, her explanation is very terse, resembling telegraphic speech.However, she too used a procedure similar to Tommy's to order thewinners correctly. When she wrote "Big second," she meant the numberof seconds was large, referring to the time taken to complete the race.She had ignored the minutes (because all are the same, that is 7minutes), just as Tommy had done. Where Tommy has made hisexplanation more explicit, her explanation has assumed, implicitly,certain shared knowledge between herself and the teacher.To me, her understanding of decimal fractions in this context wasno less than Tommy's understanding. The only difference seemed to bethat Jackie was writing in a "think aloud" way, to solve the word problem,and not trying to explicitly explain her solution to someone else. That is,Tommy seemed aware that he was writing for an audience but Jackiewas not. Given that communicating clearly involves awareness of91audience and that the teacher had repeatedly reminded the students toexplain "as if you are explaining to someone else," Jackie's seeming lackof awareness of audience is indicative of a lower level of communicativecompetence compared to Tommy. According to Vygotsky (1962), verbalexplanation is about 7 years ahead of written explanation. On the otherhand, even though Jackie's explanation lacks the coherence expected ineveryday prose, it could be argued that, by using such a parsimoniousexplanation, she is communicating as a mathematician would--that is, byconcentrating on the bare essentials and disregarding superfluouswords. Hence, her lack of fluency in using everyday written prose tocommunicate should not be taken as a lack of mathematicalunderstanding.Further evidence of the students' facility to relate common anddecimal fractions by renaming is shown in Figures 4.22 and 4.23, whenthey were asked to explain which was larger, 4 3/10 or 4.03.4 and 3/10 is bigger than 4.03 because 4 3/10 is also 4.30, andthat is more than 4.03. Just like 4.03 can also be changed to 43/100 which is smaller than 4 3/10.Figure 4.22: Mike, February 5.I think 4 3/10 is bigger than 4.03 because 4 3/10 means 4 wholeunits and 3 of a tenth. 4.03 means 4 whole units and 3 onehundredths. What I means 3/10 is bigger.Figure 4.23: Darlene, February 5.92The first excerpt (Figure 4.22) indicates that Mike can relate common anddecimal fractions. He has given two explanations, one by comparingdecimal fractions (4.30 and 4.03) and the other by comparing commonfractions (4 3/100.and 4 3/10). Although he did not use diagrams in thisinstance, he is able to use words to describe different representations ofthe same mathematical concept.In the second excerpt (Figure 4.23), Darlene's explanation of what4 3/10 means and what 4.03 means showed a clear understanding ofthese symbols. Although she did not explain clearly that 3/10 is biggerthan 3 hundredths, her understanding of it is implicit in her writing "Imeans 3/10 is bigger." Whereas Mike has used more symbols, Darlenehas used less, but both explanations reflect similar understanding of therelationship between common and decimal fractions.More evidence of students' ability to relate common and decimalfractions is given in Figures 4.24, 4.25 and 4.26, in response to theprompt "Jason thinks 0.8 = 3/5. Do you agree or disagree? Explain,prove, convince me. Whoever reads this should understand yourthinking."93b13 4 4^_ Al....L I^0, ..^ .J IF: C&.4 iti al.,44-  Jo___uict_fko__j,Leskw,;katz^- 11vvir icar, _7_141.___aztiz_iera,mAte/ ..^il it.^ 01 Y 1 T lir iii 1^1 ,i;44 watt Az;n^-St Eli 1,4.1A. ' 40___&42Xli_d tAILLisi, g,t__LtaIGA. vurvuli Lh1/ 10 y‘littcle . Ago I „<ilown-.24°^ 'Aill'iwittz•a_^0.1?^',LeoFigure 4.24: Tommy, February 26.^13.1. Amu, -hail&^ildhAveZtu letAA),Cdeatdel -one. )44- to (414141(r r e, 4:# 4r_■lati fie. I0 ,dam c A14A14.^Ad) 1.47tt^veA.A1-1 ,<Lle.,^.^jr45,e,e.,-61.4^4(1==14-t/W_t4)^,s/,iiiFigure 4.25: Tanya, February 268 G^eTITaNairyNo Lk) .„..i..1Figure 4.26: Darlene, February 26.All the three excerpts indicate students can relate common and decimalfractions through renaming. For example, 0.8 is renamed as the commonfraction 8/10, the common fraction 3/5 is renamed 6/10 and the resultingcommon fractions 8/10 and 6/10 are compared, implying a knowledge ofinter-relationships between decimal and common fractions.The explanation in the first excerpt (Figure 4.24) is more detailedthan in the other two, but is still not very explicit (for example, "5ths wouldbe 2 by 2"). It also demonstrates a combination of the demonstrative andrelative levels of language: demonstrative, because of statements like"you can see 8/10 is larger," and "this is 10ths," and relative, because ofan attempt to relate fractions to everyday experience such as the sizeand number of pieces. However, on discussing further with Tommy, Ifound that he could make explicit what was implicit in his writtenexplanation (such as "5ths would be 2 by 2" meant that 2 of the spacesmarked in tenths was equivalent to a fifth). Hence, I infer that he has anunderstanding of the inter-relationships between common and decimalfractions. The second excerpt (Figure 4.25) relates fractions to everydayexamples (such as granola bars) and could be said to demonstrate the95relative level of language. The third excerpt (Figure 4.26) indicates thedemonstrative level of language, as it points to instances without muchexplanation using words. Although Darlene seemed to have difficulty inusing written words to explain what she meant, from interview data Ifound that she could explain orally (and with the help of diagrams) why0.8 # 3/5. She was helped in her oral explanation by the fact that shecould point at parts of the diagram during her explanation, demonstratingthat she understood the relationship between common and decimalfractions, although she was much less articulate in using written words.In summary, students demonstrated conceptual understanding ofthe relationship between common and decimal fractions, and usedrelative language to communicate such understanding. Where studentsused demonstrative language, further discussions and interviewsrevealed that they, too, understood common and decimal fractionsrelationships, once again indicating that demonstrative language cannotbe equated with lack of mathematical understanding.Addition of Common Fractions Students also used renaming to add fractions withrelated denominators. Figures 4.27, 4.28 and 4.29 showthe prompt "Explain to Adam, who was absent, how to dodifferent, butresponses tothis^3 „i  ;if. + e-.96First I would probably draw a diagram.This is 28This is 141/4 could also be cut into eighths. So 1/4 could also be 2/8. Takethe 2 top numbers and add them up. So 2 + 3 = 5. Put the answerof the 2 top numbers on top of the 8 to make 5/8.Figure 4.27: Tommy, March 5.Well first what you could do is you could draw fraction strips:4^So the answer is 5/8 because if you put those in 8'sthen it would be 5/8.Figure 4.28: Maureen, March 5.To make it easier to add we will rename 1/4 to 2/8, now its easierto add because the bottom # is the same. So 2 + 3 = 5 and thebottom # stays the same = 5/8.figure 4.29 , : Shaun, March 5.97Tommy (Figure 4.27) explains using relative language and also indicatesconceptual and procedural knowledge by using two separate fractions(3/8 and 1/4) and an algorithm to get the sum of 5/8. Maureen's (Figure4.28) referent for "those" in the expression "if you put those in 8's" is notvery clear. She seems to expect the reader to grasp her meaning fromthe diagram that 1/4 can be renamed as 2/8. In other words, she resortsto demonstrative language, but her diagrams indicate her conceptualknowledge of addition of these fractions. In contrast to the other two,Shaun (Figure 4.29) uses words and symbols but no diagrams in hisexplanation, and it seems that he uses procedural knowledge.Even though the addition was related through diagrams or wordsto equivalent fractions, from subsequent interviews I found that students(other than Tommy) could not add other fractions where one denominatorwas not a multiple of the other. Indeed, for the addition of 1/3 and 1/2,five of the six interviewees did not try renaming the fractions. Rather, theyhad the two fractions represented by shaded parts in one rectangle (evenShaun, who usually did not resort to diagrams) and estimated the sum tobe either 2/3 or 3/4. So, even though they could add by first renamingthe fractions when one denominator was a multiple of the other (as in 1/4and 3/8), they could not extend such renaming to fractions withoutcommon multiples (as in 1/3 and 1/2), thus demonstrating they neededperceptual cues to support and illustrate renaming and equivalentfractions. Even Shaun, who had said "now its easier to add because thebottom # is the same" for his procedural addition of 1/4 and 3/8 (Figure4.29), resorted to a diagram for 1/3 + 1/2. Hence I infer that the relativelevel of language was used by most students in communicating theirknowledge of addition of common fractions.98Tommy, on the other hand, used diagrams, words and renaming(to obtain fractions with common denominators) to add fractions. Forexample, during the third interview, he stated that 1/2 and 1/4 could notbe added without renaming as "they are different items." He went on tosay that "if you have someone who doesn't understand, they might givethe answer as 2/6," and he explained how 2/6 was obtained as well aswhy 2/6 was wrong. He also could add 1/3 and 1/5, both by renamingusing symbols and by drawing diagrams. Although not warranted fromthe one entry used (Figure 4.27), I was able to conclude from theinterview and from listening to earlier discussions in class that he wasoperating at the functional level of language in communicating about hisknowledge of addition of fractions, as he could explain the need to usecommon denominators. In general, however, he evidenced conceptualand procedural knowledge of addition of common fractions.In summary, students can add fractions where one denominator isa multiple of the other, by using renaming with or without diagrams forillustrative purposes. From my observations, interview data and SJ, Ifound that no Grade 6 student added numerator to numerator anddenominator to denominator, perhaps reflecting Linda's de-emphasis onsymbol manipulation and emphasis on understanding mathematicalconcepts and relationships. Linda used word problems to initiatediscussion on addition of fractions She encouraged students to justifytheir solutions (whatever the solution) and to use manipulatives anddiagrams. She did not teach a rule for finding common denominators nordid she give any exercises involving only numbers, such as 1/2 + 3/8.Other than Tommy, no student had articulated the notion of commondenominators for adding fractions or used the functional level of99language to communicate their knowledge of addition of fractions (moststudents used relative language).Word Problems (Decimal Fractions) Written explanations of how to solve word problems involvingdecimal fractions are shown in Figures 4.30 and 4.31. They were inresponse to the teacher's question "Mr Campbell drove about 95 km anhour for 2.5 h and then 60 km an hour for 1.25 h. How far did he drive?"95-`^i°uc .5' 6^X 2 Le' (A ES -12^.s.11)^•1744S^2fr ^kzri4 f-1/^ // ..4 ^It 7. c km. ./J 4/4/), F=`,°ila _Mai/ kz . c Az7,eW Creec ) b(0 4:r1-^ks,ac. So47^X o u if^e•C.. /4. d./A. Ar/^(.1.4 re237-‹ 0,2( /-5- 6741 4111 ._g92.tct.1^.117_^,fract.k___LTb a .^ejr..._// a C7.1Z4 4 14  4=4-oi/Y  a Lin cam.^Ici.e._;‘ is-74h^?j71 4- 7r ^0A.707.4^c/r oc.A.C.^3 I 2 - ( km •Figure 4.30: Tommy, March 13.100Figure 4.31: Mike, March 13.All the students (except Tommy) operated with mathematical symbols,without words of explanation, as typified by Figure 4.31. Even so, theyare able to indicate conceptual knowledge, as they use relationshipsamong speed, time and distance as well as that between common anddecimal fractions. Indeed, they were answering the question asked, justas many mathematicians would do, without superfluous words.However, in the sense that they are computing without an accompanyingexplanation in words, I would infer that they were using demonstrativelanguage. That they used demonstrative language is not surprisingbecause they were more interested in arriving at a solution to theproblem (as that is what was asked) than in trying to communicate to theteacher by using words. Moreover, the assumption seemed to be that thesteps that they showed, though symbolic, would be self-explanatory tosomeone like the teacher.101Tommy (Figure 4.30) was the only one to explain in words what hedid in order to arrive at his solution. Just like the other students, Tommy,too, inter-relates different aspects of the problem, indicating hisconceptual knowledge. In addition, he seems to be putting his thoughtsto paper--a kind of "thinking aloud"--and because he makes these"relationships between relationships" (for example 60 km an hour isequivalent to "a km each minute" and "0.25 is left . . . 1/4 of an hour is15/60") explicit by using words, I infer that he was using the functionallevel of language.It is instructive to note that this question was part of a class test tobe done in the SJ, and so it should not be surprising that getting asolution was more important for the student than giving an elaborateexplanation to the teacher. However, according to the teacher (and Iobserved this happening too), even in solving non-test word problems inclass, students generally tended to concentrate on the computationsneeded to arrive at the solution, as in the second excerpt (Figure 4.31).Only when they were explicitly asked to explain their reasoning (andreminded repeatedly to do so) did they make attempts to use words inaddition to symbols. On the other hand, when they were asked topresent their solutions in front of the class, the students tended to explainand justify more articulately "in response to an explicit request" or in anattempt to "communicate aspects of their mathematical thinking that theythink were not readily apparent to others" (Cobb, Wood, Yackel, &McNeal, 1992, p. 577). It is not surprising that such classroom discourseinvolves negotiations of meaning, and is in fact expected by classroomroutines. But (a) the lack of practice in writing for peers, (b) the lack of anawareness that writing in a mathematics class for an audience other than102the teacher is legitimate, and (c) the way students have been generallytaught mathematics all these years all seem to explain in part why thefunctional level of language is seldom used in solving word problems.In summary, students evidenced conceptual knowledge in solvingword problems involving decimal fractions. However, they seldom usedwords to explain their solutions. Rather, they tended to show only thecomputational steps, unless reminded explicitly to do so. It looks as ifstudents have a difficult time reconciling old habits of getting an answeras quickly as possible with the present teacher's emphasis on writtenexplanations of their solutions.Rule for Placement of Decimal Point in Decimal Multiplication Just as in the addition of common fractions, no rule was given tothe students for multiplication of decimals. Instead, they were to find outa suitable rule by themselves first without, then with, calculators to helpthem, by attempting some word problems given by the teacher. Whenasked the rule for placing the decimal point when multiplying decimals,some of them wrote as shown in Figures 4.32, 4.33, 4.34 and 4.35.I think the rule is if you are multiplying 2 numbers with so and sotenths times so and so tenths, the answer will be so and sohundredths. If your multiplying 2 numbers that both have so andso hundredths, the answer would be so and so ten thousandths.Example 2.56 x 2.56 = 6.5536 [He showed the vertical form ofmultiplication]. If you have one number with hundredths and onewith tenths the answer would be thousandths. [Gave theexamples 4.2 x 3.8 = 15.96, 0.67 x 1.38 = 0.9246 and 0.02 x 0.3 =0.0061Figure 4.32: Tommy, March 25.103When you multiply with numbers on both sides of the decimal, theanswer can be a whole number, but when it is only the right side, itcan't be a whole. [He elaborated by computing the following: 4.2x 3.8 = 15.96 (correct), 0.67 x 1.38 =.09246 (incorrect), and 0.02 x0.30 = 0.60 (incorrect)].Figure 4.33: Mike, March 25.My rule is that all you have to do is just answer the whole questioninstead of putting it wherever you want. [She exemplified her ruleby multiplying 0.2 by 0.7 to get 0.14, ignoring decimal points in thepartial products]. If that doesn't work then be logicall. Or countdigits to the right of the decimal. [Then she used her "counting"rule on the following: 4.2 x 3.8 = 15.96, 0.67 x 1.38 = 09.246(incorrect) and 0.02 x 0.3 = 0.006].Figure 4.34: Maureen, March 25.I think the rule is conected to the estimation. What I mean is thatthe digets you get in your estimation thats how many digets youwill have before the decimal. Heres an example:question 3.64 x 8.8, estimation 4 x 9 = 36, so answer is 32.032Figure 4.35: Darlene, March 25.Figure 4.32 shows that Tommy was using the functional level oflanguage as he made a generalisation about the placement of thedecimal point involving decimal multiplication. He seemed to have nodifficulties in placing the decimal point correctly and his procedure wasbased on a pattern he had found. But when asked why he obtained anumber (0.06) less than the numbers he started with (0.3 x 0.2), he104replied "Because not multiplying by whole numbers." So, even thoughhe used the functional level of language in his SJ to communicate hisknowledge of decimal multiplication, and he seemed to have a correctprocedure for placing the decimal point, it seems that he (just like theother students I talked to) treats decimal numbers as obeying rules ofmultiplication different from that of whole number multiplication. Hence,his conceptual base for multiplication of decimals does not seem verystrong, even though he has discovered some patterns in decimalmultiplication.In spite of his facility in moving from common to decimal fractionnotation and vice versa, he did not use renaming and common fractionsto justify his generalisation that multiplying tenths results in hundredths.The reason for not comparing decimal fraction multiplication withcommon fraction multiplication is not surprising, given that he had notcome across such multiplication before in class, even though he couldanswer verbal questions such as "What is a third of six tenths?" and"What is a third of zero point six?" To me, Tommy seemed to have aquicker grasp of decimal multiplication, even though he seemed to havestarted with conceptual knowledge not very different from his classmates.I say this because during interview three, when I asked him to think aboutthe symbol "x" as representing "of" as in "1/2 of 0.4" being equivalent to"1/2 x 0.4" he said that he could now see why the product could becomeless, unlike in whole number multiplication. In contrast, the otherstudents did not seem to understand why the product of decimal fractionscould sometimes become less than the numbers they started with, eventhough I used the same analogy.105The examples in Figure 4.33 indicate that Mike felt that if one ofthe multiplicands were less than one, then the product would be lessthan one. His rule depended on the type of numbers that were beingmultiplied: If the numbers were greater than one (e.g. 4.2 x 3.8), then oneset of rules applied; if the numbers were less than one (e.g. 0.02 x 0.30),then another set of rules applied. While it is true that he conjectures--andconjecturing is mathematical--his rules are number specific and notgeneralised: he sees no contradictions in changing rules to fit particularsituations rather than perceiving rules as a generalisation with a wideapplicability. So he, too, has a conceptual base for decimalmultiplication, albeit not a very strong one.The language used in Figure 4.33 seems closer to the relativelevel of language than the functional because he attempts to relate hisrules to specific sets of numbers rather than to numbers in general.However, it has lexical ambiguities (Durkin & Shire, 1991). For example,when Mike used the word "numbers" in "multiply with numbers," hemeant non-zero digits, but when he used "number" in "whole number," hemeant that the digits to the left of the decimal point represented wholenumbers, not that the product itself was a whole number. Such ambiguityin the use of the word "number" could hinder the abstraction of themathematical concept of number, as "number" seems to mean onlywhole numbers to Mike. It is possible that such an abstraction is madeeven more difficult because teachers seldom explicitly point out thatfractions are numbers, too (just as Linda did not).In Figure 4.34, Maureen shows two rules for the placement of thedecimal point, and two levels of language: (a) by counting the totalnumber of digits to the right of the decimal point in the multiplier and106multiplicand, she is using the demonstrative level of language as she isjust stating a procedure without any justification; and (b) by using anestimation to get reasonable answers (which, according to her verbalexplanation later, was what she meant by the word "logical'," and her firstsentence, too), she is using the relative level of language, as she isrelating the answer to her everyday experience and "common sense." Inother words, Maureen perceived the placement of the decimal point indecimal multiplication as dependent on the type of decimal fractionsinvolved. However, judging by the placement of estimation before her"counting" rule in her explanation (Figure 4.34) her counting rule seemedto be an alternative only when estimation or logic did not "work" (agreewith the calculator-computed answer).In Figure 4.35, Darlene, too, uses the relative level of language bystating a meaningful approach (estimation) to locating the decimal pointin the product of decimal fraction multiplication. Both Maureen's andDarlene's strategies had their advantages. For instance, Darlene'sstrategy has the advantage that it applies to all instances of decimalfractions as long as estimation skills are good. But she did havedifficulties with numbers close to zero, such as 0.1 x 0.2 and 0.02 x 0.3,when her initial estimations resulted in zero for both the products. On theother hand, Maureen had another strategy to fall back on if one failed.Moreover, her rule was also applicable to all cases, except that she hadto be careful about numbers like 0.5 x 0.2 which would result in 0.10 butwith the ever-present danger of discarding the zero in the hundredthsplace because of rules like "trailing zeroes can be ignored," resulting inconfusion as to the placement of the decimal point. But because107Maureen was prepared to resort to an arbitrary (but possibly pattern-engendered) rule, she might do so in other topics, too.In summary, students obtained rules for placement of the decimalpoint in decimal multiplication through conjectures, estimation andfollowing patterns, based on word problems. For example, in order tofind the amount of tax to be paid for a pair of jeans costing $30, if the taxwere 7%, students realised that $210 and $21 would be unreasonable,and so they obtained $2.10 and attempted to find a pattern for othersituations involving decimal fraction multiplication. Although most ofthem used estimation and obtained reasonable solutions, they haddifficulties in estimating answers for multiplying numbers like 0.02 x 0.3.None of them could satisfactorily explain why the product of decimalnumbers could sometimes be less than the numbers with which theystarted. Overall, they had some conceptual understanding of decimalmultiplication but had better procedural knowledge. As well, most ofthem used either demonstrative or relative language in their writing tocommunicate their knowledge of decimal multiplication.SCQ and Mathematical Knowledge So far, I have only discussed journal entries. I now examine someof the SCQ briefly but postpone more detailed discussion to a latersection (e.g. when discussing results pertaining to research questions #2and #3). Following are some examples of SCQ indicating students'knowledge of common and decimal fractions.108Someone ate 3/8 of a pizza. You and your friend are going toshare the rest. How much of the pizza will you get?Figure 4.36: Darlene, January 7.Caroline went to the mall. She brot $100.00. She went to "Off TheWall" and spent $32.00 and 1/4 of the money went to thegovernment. How much does the store keep? Answer: The storekeeps $24.00.Figure 4.37: Jackie, February 10.1050.coGoaceL- okck r ersov, clot a vvietefT. out cf IP, S3Litowleohe: re Kam ect soliv‘f cyf tke^1,\Ao wo‘s1-Pvt:3 was igAIrd ? Can 7orA resnotwie019cAiv1 (A a l'Efe r evCt Wc•‘f1 Arw.5wer ria10 - 10) to la4K. 10iToFigure 4.38: Mike, February 19.The question shown in Figure 4.36 was constructed after Linda hadasked them to write about where fractions are used in the students' daily109life. Many of the students had given the pizza as an example of fractionsin daily life. Moreover, the region model had been exemplified bygranola bars and pizzas in previous classes (before Grade 6, withteachers other than Linda) and by fraction strips in the present class. So,it was not surprising that Darlene--supposedly a weak mathematicsstudent, according to the teacher--constructed a "pizza problem."Although Figure 4.37 has superfluous information like $100.00and the name of the store, such information draws on the student's ownmeaningful experiences and situates the word problem in a realisticcontext. The mathematical operations involved in the SCQ aremultiplication and subtraction, with the former as yet untaught. Eventhough the student was supposedly weak at mathematics (according tothe teacher) and had no knowledge of the multiplication algorithm forcommon fractions, she had no difficulty in explaining (verbally) how shesolved her SCQ, indicating her awareness that a fourth of 32 is the sameas dividing 32 by 4.In Figure 4.38, Mike has chosen all except one fraction withmultiples of 10 in the denominator. Even so, he did not seem to have anydifficulty in converting 2/2 to 10/10, or 20/50 and 60/200 to fractions with10 in the denominator. He admitted that while preparing the question, hetried a few fractions before choosing numbers which he could rename intenths. A week before this he had explained both algorithmically andthrough illustrative diagrams in his SJ how to rename fractions so as toobtain equivalent fractions, indicating that he had both procedural andconceptual knowledge of renaming and equivalent fractions. Thisexcerpt (Figure 4.38) seems to give further support that he understandsequivalent fractions and how they relate to ordering fractions. His110solution has no verbal comments, once again indicating what I saidearlier (p. 106) about students' habit of not writing elaborate verbalexplanations of solutions to (decimal fraction) word problems.While the previous examples of SCQ give an indication of thestudents' knowledge of common and decimal fractions, the followingexamples of the SCQ give an indication of the growing competence ofthe majority of students to write questions that not only made sense tothem, but were mathematically (but not necessarily computationally)more sophisticated.Consider one student's SCQ written about two weeks apart(Figures 4.39 and 4.40) :1400.^ #41, areelf-,i4A-^/24.1tea.,^Actor__ aut.:4 ,ctie_.11teXxv 1417-x) 4- laza) -Wow- Jima^taen.0'lit/C.6ga . Lela _CY ../1/t1.0e, 115#PLiee.4&Mc^evrei tt pi.LeFigure 4.39: Tanya, January 24.111Stepheny wants to watch her favorite show at 5.30 p.m. It's a 1/2hour show. In the morning she gets up at 10 a.m. She is at the icearena an hour later. She gets off the ice at 2/3 from regularskating time which is 3 hours. She eats lunch for 1/3 of an hour.Then she goes shopping for twice as long as she went skating. Ittakes her 1/2 an hour to walk home. How much of her show willshe get to watch?Figure 4.40: Tanya, February 6.Tanya had gone from a fraction question (Figure 4.39) involving a regionmodel together with the operations of addition, subtraction and wholenumber multiplication to one that involved no explicit region models butdealt with intervals of time, fractions (including equivalent fractions) andmultiplication of fractions and whole numbers (e.g. 2/3 of 3 hours).Moreover, the second SCQ (Figure 4.40) had a wealth of realistic detailsas well as a temporal sequence that had to be followed in order to solvethe problem.Another student wrote these SCQ (Figures 4.41 and 4.42) about sixweeks apart:Winning Spirit had 76 baseball bats and 38/76 were wood and13/76 were metal. How many baseball bats are left over.Figure 4.41: Anne, February 10.112Jack made $0.25 an hour and he worked 6 hr a day. He workedfor 7 days. How much did he make? [To which the teacherresponded "Jack was underpaid!"]Figure 4.42: Anne, March 23.Anne's SCQ progressed from a word problem (Figure 4.41) that lookedsuperficially like a fraction problem with addition and subtraction of wholenumbers, to a word problem (Figure 4.42) involving multiplication ofdecimal fractions (for example, in her solution accompanying thequestion, she wrote "0.25 x 6 h = $1.50, $1.50 x 7 = $10.50"). Admittedly,she uses only one operation, and she uses a problem involving money,which might not have needed a knowledge of decimal fractions for asolution. But given that she was weak at mathematics and was asked towrite a word problem "involving multiplication of two numbers with atleast one of the numbers a decimal fraction less than one," she didprepare a question satisfying the conditions.Figures 4.43 and 4.44 show Tommy's SCQ over two weeks.You buy a box of Upper Deck Hockey cards. Every box youreceive 36 packs. Each pack you get 12 cards. You are missing1/6 of cards in each pack. How many cards are you missing, andhow many cards will you have now?Figure 4.43: Tommy, January 27.113There was a pie eating contest that was held at Stanley Park.There were 4 contestants in the contest. Tom ate 4 pies, and 1/3of another. It took Tom 1:54.03 to pass out. Exactly :24.04 laterRoxanne passed out eating 1 pie more than Tom. 1.54.44 laterJake passed out in 1.00.00 later eating 4 more pies thanRoxanne. This declared Jeremy the winner. Jeremy didn't stop,he was trying to beat the record. The record was 13 pies eaten in6.04.00. Jeremy ate 12 pies and 2/3 of another in 5.45.35. Howmuch pie was eaten, in how much time? Counting their timestogether.Figure 4.44: Tommy, February 10.Figure 4.43 shows a word problem involving multiplication andsubtraction of whole numbers with the fraction 1/6 which could be treatedas dividing by 6. Figure 4.44 is not only a longer question but involvesaddition of common and decimal fractions. Tommy has incorporatedboth common and decimal fractions in the second SCQ, possiblybecause he has learned decimal fractions in addition to commonfractions. On the other hand he has shown an increase in his knowledgeof fractions by including addition of fractions, whereas his first SCQ couldhave been solved using just operations on whole numbers.Another instance of the improvement in the SCQ is shown inFigures 4.45 and 4.46, which took place over a five week period.There were 160 parking spaces at the mall. The parking lot was1/4 full. How many parking spaces were left?Figure 4.45: James, January 24.114I had to walk back and forth to work every day for one week andthe walk was 63.12 metres each way. I also had to walk to thegrocery store twice that week. The grocery store was 32.15 meach way. I also walked to the doctor's office which was 67.94 m.How far did I walk that week? For every metre that I walked Igained 0.05 kg. How much weight did I gain that week?Figure 4.46: James, March 4.Figure 4.45 indicates an SCQ with two operations (division andsubtraction) whereas Figure 4.46 shows an SCQ that is multi-step, withaddition and multiplication of decimal fractions. While it is unrealistic toexpect someone to gain weight by walking, other details in the questionseem realistic enough to indicate everyday experience. Moreover, thesecond SCQ (Figure 4.46) has more mathematical information in it aswell, compared to the first SCQ. So I infer that there is an improvement inthe SCQ, both in terms of mathematical knowledge and backgrounddetails.As the study progressed, even the "weaker" students wrote SCQthat were rich in realistic, everyday details . For example, the weakeststudent, Jackie, went from the question in Figures 4.47 to that in Figure4.48 in just over two weeks.We had 200 salmon eggs. 1/4 died, how many were still alive?Figure 4.47: Jackie, January 24.115Caroline went to the mall. She brot $100.00. She went to "Off TheWall" and spent $32.00 and 1/4 of the money went to thegovernment. How much does the store keep?Figure 4.48: Jackie, February 10.As can be seen, the later question (Figure 4.48) has a richer backgroundof details (for example, use of "mall," "brot $100," the name of the store"Off The Wall," and "money went to the government") than the earlierquestion (Figure 4.46), which seemed rather bare. While it could beargued that the richer background has nothing to do with themathematics involved, it must be remembered that such backgroundsituates mathematics in a meaningful context and could make problemsmore amenable to solution (e.g. Ellerton & Clements, 1991; Laborde,1990; Mason & Davis, 1991; Spanos et al., 1988). As well, though theoperations involved are the same (division and subtraction), in thesecond question, the fact that the $100 was not needed in thecomputation could point to not only everyday experience, but also couldbe thought of as providing extraneous information. Such extraneousinformation has to be recognized as unnecessary for the solution andthat too speaks of a level of mathematical sophistication.In summary, initially, students wrote word problems without manywords and the solutions needed few steps, possibly indicating both theirunfamiliarity with SCQ and the level of mathematical understandingabout common and decimal fractions as well as the number of topicstaught. Later on, students embellished their word problems withbackground (and sometimes even superfluous) information and required116multiple steps for their solution, demonstrating familiarity with the SCQ aswell as more understanding of common and decimal fractions, especiallywhen they included both common and decimal fractions in their wordproblems. Students themselves embellished their questions withbackground and Linda neither encouraged nor discouraged theirembellishment. Because each SCQ had to be handed up with thesolution, students' understanding of the problem was demonstrated bythe accompanying explanation or solution as well. Where there was onlythe numerical solution with no accompanying explanation to the solution,students' understanding was demonstrated through their verbalexplanation of the solution to the group, class or researcher. Overall, theSCQ showed a slight, but noticeable improvement in students'mathematical knowledge about common and decimal fractions, with allthe students making longer questions, but with the weaker studentsgiving more non-mathematical background and the better studentsincluding more topics, operations and steps.Summary of results pertaining to research question #1 Initially, students tended to communicate their mathematicalknowledge orally by using demonstrative language when grappling withcommon and decimal fraction concepts and progress to the relative andfunctional levels of language as they developed better understanding ofthese concepts. The findings of this study about oral communicationseem to confirm Freudenthal's (1978) view that communication inmathematics progresses through three levels of language(demonstrative, relative and functional). In writing, however, while allstudents moved from the demonstrative to the relative level of language117for a limited number of concepts (such as the part-whole concept), themajority of the students used relative language when writing about theother common and decimal fraction concepts, and sometimes used bothdemonstrative and relative levels of language. A minority alsosometimes used the functional level of language. In short, there was nonoticeable progression in the levels of written language, althoughstudents seemed to understand fraction concepts better. Indeed, therewere instances of students using demonstrative language in the SJ butshowing a good level of understanding through their pictorial and oralexplanations that seem to indicate that the levels of language used forcommunicating mathematical knowledge (a) vary according to the modeof communication (say, oral versus written), (b) the fraction conceptsbeing learned and (c) are not unambiguous indicators of a student's levelof mathematical understanding or knowledge. Hence, the findings of thisstudy indicate that Freudenthal's levels of language aboutcommunication in mathematics need to be re-examined where writtencommunication is concerned.Most of the students used diagrams to complement their written,verbal explanations. Where their writing did not give satisfactoryexplanations, follow-up interviews revealed that they understood muchmore mathematics than initially indicated by their writing. As Linda, theteacher, said " sometimes they do understand, they just don't know howto say it in words"--which agrees with Freudenthal's (1978) statementthat "most of us understand more language than we can speak" (p. 234).Only one student in this study could articulate fraction conceptsand relationships clearly through written words. My observations of classdiscussions and the interview data where students satisfactorily118explained fraction concepts confirm that the majority of students in thisstudy were more articulate speaking than writing about fraction conceptsand relationships. This finding contradicts Loban's (1976) study whichindicates that language proficiency in writing catches up with that inspeech by the age of twelve. A possible reason for the difference in oraland written proficiency is in the mode of communication itself: where oraldiscourse lends itself to immediate feedback from the audience, writingdoes not (Bereiter & Scardamalia, 1987). For example, when studentstry to explain verbally why 8/15 is larger than 1/2, any lack of clarity in theexplanation is immediately brought to the attention of the speaker whocan then modify the explanation accordingly. In contrast, in writing aboutthe relative magnitudes of 8/15 and 1/2, the writer may be makingassumptions about shared knowledge between the writer and the readerwhich might make the import of the writing unclear to the reader.Interestingly, those students perceived as poor writers by theteacher (in other subject areas such as Social Studies and LanguageArts) seemed to be also poor at writing in mathematics. So it would seemthat the difficulty in writing was not a function of the content or topic(fractions, in this instance) but of something more generic such as writtencommunication. Thus transactional writing (Britton et al., 1975), as usedin the SJ for this group of students, was not as effective as verbal andpictorial communication about mathematics. That is not to say that the SJdid not reflect students' mathematical knowledge at all. On the contrary,the SJ did give insights about students' mathematical knowledge. It isonly that verbal explanations demonstrated students' mathematicalknowledge more clearly, possibly because of an audience who gavefeedback so as to allow for immediate clarification. In contrast , the SCQ,1 19which required the writer to give solutions, showed the students'mathematical knowledge implicitly (through the type of question asked,information given, and the accompanying solution) and also theirimprovement in their fraction knowledge explicitly (through the type offraction relationships, the amount of detail, and the number of steps andoperations involved).Although most students were not fluent in expressing themselvesin written language, they did demonstrate conceptual knowledge(Hiebert, 1986) and relational understanding (Skemp, 1976) of commonand decimal fractions by using relative language (Freudenthal, 1978),that is, "where objects are described by their relations to other objects" (p.237) either in their SJ or orally. For example, Shaun used mathematicalsymbols frequently in his SJ to explain his method of solution rather thandiagrams or everyday language and so he might be construed (as theteacher did) as using procedural knowledge (Hiebert, 1986). Most of hisSJ entries indicated that he used such abbreviated explanationscorrectly (e.g. he used "rules" such as "multiply top and bottom by 2" forequivalent fractions) but he could explain his reasoning when asked todo so verbally and by referring to diagrams, thereby demonstratingconceptual knowledge of common and decimal fractions and the use ofrelative language as well. Hence a preponderance of symbols in writingabout fractions is not indicative of a lack of conceptual understanding.Neither is a lack of everyday prose as evidenced by demonstrativelanguage (for example, both Tommy and Darlene used demonstrativelanguage in justifying that 0.8 # 3/5, see comments on Figures 4.24 and4.26) an indicator of lack of conceptual understanding.120In contrast to Shaun, the writing of a student like Mike indicatedthat he was not very familiar with mathematical terminology andsometimes obtained wrong answers (like misplacing the decimal point).However, he used everyday language in his journal to relatemathematical concepts and principles (such as the interchangeability ofcommon and decimal fractions). In addition, his self-constructedquestions, though computationally not very difficult for students of grade6, did demonstrate a grasp of relationships between common anddecimal fractions, thereby evidencing conceptual knowledge (Hiebert,1986). Overall, I can say that this group of students did reveal conceptualand procedural knowledge of mathematics as well as growth in suchknowledge through their writing.Research question #2 and resultsResearch question # 2: What are the salient and recurrentfeatures (themes) of the SCQ, if any?Four themes emerged from the student-constructed questions(SCQ). They were:1. The use of student experience and interest as the context of theSCQ.2. The assumption of shared knowledge between the reader andwriter of the SCQ.3. The use of numbers which made computation easy.4. The use of questions reflecting the discrete model of fractionsrather than the region model.121Theme 1: use of student experience and interestThe SCQ reflected students' daily, out-of-school experiences or in-school experiences such as other subject areas currently being studied.For example,some students used their shopping experience in theirSCQ, as in Figure 4.49.A regular Bulls parka costs $175 and the regular Bulls jacket costs$75. There was a half off sale. Brian bought 1 parka and onejacket. Jason bought 2 parkas. How much money did Brian andJason each pay? How much did the store collect?Figure 4.49: Anne, March 2.Others drew upon topics currently being learned in other subject areas.For example, the students were learning about whales in Social Studies,and they wrote SCQ related to whales (Figure 4.50).Vancouver Aquarium has a weekly budget of $100, of which 0.5will go for the killer whales, 0.44 for belugas, and the left overmoney for the dolphin. 1. Find out is 0.5, 0.44 in $. 2. Find outhow much goes for the dolphin.Figure 4.50: Shaun, February 19.Others wrote on recent or ongoing experiences and activities in whichthey had been involved. For example, the students were responsible fora salmon tank in the classroom. They took turns to feed the salmon andmeasure the temperature of the water but some of the salmon died. AnSCQ indicating their recent involvement in salmon rearing is given inFigure 4.51.122We had 200 salmon eggs. 1/4 died, how many were still alive?Figure 4.51: Jackie, March 3.Some of the students wrote SCQ reflecting their interest in games andsports, such as ice hockey and gymnastics. Two examples of SCQshowing these interests are given in Figures 4.52 and 4.53.The Winnipeg Jets were on a 4 game road trip. Through allgames, the Jets scored 24 goals. 1/3 were against the VancouverCanucks, 1/2 were on the Sharks 1/8 were on the Nordiques.How many did they score against the Canadians?Figure 4.52: Tommy, February 7.Girls Valte Univer bars floor Blanceing BeamsJackie 35.40 35.80 33.70 35.90Maryon 33.90 34.50 32.95 35.40How much more did Jackie get in all events?Figure 4.53: Jackie, March 2.Some students wrote SCQ based on specialised interests. For example,only two students--the father of one was a pilot--were very interested inaircraft and an SCQ related to aircraft is shown in Figure 4.54.123An airplane was flying at 50.2 m at 12.00 a.m. It flew for 4 hoursand gained 1.4 m of altitude each hour. What time would it be andwhat altitude would the plane be at?Figure 4.54: Sam, March 4.The SCQ in Figures 4.49 to 4.54 illustrate that students seem to drawupon their interests or ongoing daily in-school and out-of-schoolexperiences in which to situate their SCQ. Moreover, recent or ongoingactivities (e.g salmon rearing) and what is being discussed in othersubject areas (e.g. whales in Social Studies)--which generally did notinvolve any fractions--seem to play a part in the type of SCQ beingconstructed. It looks as if students can, given the opportunity, write SCQwhich are meaningful to them, basing them on their experiences, ratherthan limiting the SCQ to what is learned in the mathematics class. Suchexperientially-based SCQ are not surprising, given that how anindividual constructs meaning is dependent "on the context and on themathematical content underlying the formulations" (Laborde, 1990, p.62).Another factor that might have encouraged students to look toother sources of ideas for the SCQ was the fact that frequent textbookexercises were not part of the practice in this class. Instead, groupdiscussions formed the basis for the SCQ initially, after which both groupand individual SCQ were collected, edited and distributed by the teacheras class exercises. (I am not including examples of SCQ resulting fromgroup discussion, as the groups comprised both Grade 5 and Grade 6students, whereas my focus is on the Grade 6 students.) I would124speculate that these group discussions helped students clarify ideas forthe preparation of individual SCQ.Theme 2: assumption of shared knowledge Time and again the SCQ indicated that the writer of the questionassumed that the reader was privy to all the information the writerpossessed, even though such information was not explicitly stated in thequestion. Furthermore, such unstated information was often crucial to thedetermination of the answer. Examples of such SCQ follow.In band there are 60 people that play different instruments. 29/60are clarinets, 10/60 are flutes and 10/60 are saxaphones. Howmany trumpets are there? (The answer is in sixtieths) Answer:the amount of the trumpets is 11/60.Figure 4.55: Shaun, January 27.The first assumption in the SCQ (Figure 4.55) is that the reader knowsthat there are only four different kinds of musical instruments in this band.The second, non-crucial assumption is that the reader is aware that thesymbols expected by the writer in the answer ("in sixtieths") can bereconciled with the reader's expectation of a whole number as theanswer. Indeed, even though students found the computation easy,there was disagreement and discussion on this particular issue ("Howmany trumpets are there?" as being indicative of a whole number answerrather than a fraction) when solutions were attempted.Another example of a question which assumed shared knowledgeis in Figure 4.56.125In Vancouver Aquarium there are 10 belugas. 4/10 are youngbelugas. 0.5 of the whole herd are adults. How many babybelugas are there? Answer: There is 1 baby beluga because 4 + 5= 9, 10 - 9 = 1.Figure 4.56: Shaun, February 10.In Figure 4.56, Shaun assumes that the reader will know that "young"belugas are not equivalent to "baby" belugas. Without such anassumption, the reader might work out the problem by assuming that"young" and "baby" mean the same, and get a solution different from theone intended by Shaun.Yet another example of an SCQ assuming shared knowledge isthe one in Figure 4.57.A boy went to a roller ring. People borrowed 5/20, then gave back4/20, then borrowed 1/20, then how much do they have left?Figure 4.57: Maureen, March 31.Evidently, Maureen assumed that the reader was aware that she waswriting about skates, and that she had given the number of pairs ofskates (20) to start with. Without assuming such shared knowledge, thereader would have been unable to solve the above SCQ.In the examples of the SCQ given in Figures 4.55 to 4.57,students, irrespective of whether they were considered weak, average orgood at mathematics by the teacher, seemed to be unable to view theSCQ from the reader's perspective. A partial reason for providing126insufficient information in the SCQ could be that the development ofdecentration from egocentrism (e.g. Barnes, 1976; Piaget & Inhelder,1969) takes time and the grade 6 students here are still too egocentric tobe aware of the need to provide information which seems obvious tothem. It may also be the case that students feel they know their readers(their classmates) well enough to take such shared knowledge forgranted. Another reason, which I have alluded to earlier whendiscussing students' written and oral explanations in the SJ, is that thelack of an audience militates against self-monitoring for clarity becauseimmediate feedback in the form of verbal or non-verbal cues is absent.Yet another way to interpret the assumption of shared knowledgein the SCQ is to base the reasons for such assumptions on Bereiter andScardamalia's (1987) knowledge telling and knowledge transformingmodels of writing. Knowledge telling here refers to a model of writingwhich makes use of "readily available material from memory" (p. 29),whereas knowledge transforming refers to a writing model where "re-processing of knowledge" (p. 7-8) takes place through metacognition.While admitting that writing down something from memory might itselfhave a "knowledge-transforming effect" (p. 29), they maintain that writersworking from a knowledge telling model present content that "is salient inthe mind of the writer but not necessarily sufficient or relevant for thereader" (p. 345).Even though they caution that the two models they propose aremodels that "refer to mental processes by which texts are composed, notto texts themselves" (p. 13), I would infer that these SCQ reflect themental processes characteristic of the knowledge telling model of writing,as they evidence assumption of shared knowledge. The SJ indicate that127the students in the study did not explain mathematics concepts clearly inwriting, thereby revealing themselves as "novice" writers.According to Laborde (1990), "the necessity of communicating amessage to someone requires an awareness that what is obvious orknown to the speaker is neither automatically clear nor necessarilyknown to the listener or reader" (p. 54). An absence of such awarenessin the SCQ would seem to indicate that the students had providedinsufficient or vague information and therefore had failed tocommunicative effectively, as was evident a number of times whengroups discussed how to solve the SCQ prepared by others.Theme 3: The use of numbers which made computation easy Most of the SCQ, even if they were multistep problems, containednumbers that were easy to compute. Some examples of SCQ withnumbers that allowed uncomplicated computing follow.#1 movie is 108.55 min. long. #2 movie is 90.2 min. long. #3movie is 98.551 min. long. Use addition and predict which two,put together, would take the longest, in a 2 for 1 movie special?Which two would be shortest?Figure 4.58: Mike, March 4.128Stepheny wants to watch her favorite show at 5.30 p.m. It's a 1/2hour show. In the morning she gets up at 10 a.m. She is at the icearena an hour later. She gets off the ice at 2/3 from regularskating time which is 3 hours. She eats lunch for 1/3 of an hour.Then she goes shopping for twice as long as she went skating. Ittakes her 1/2 an hour to walk home. How much of her show willshe get to watch?Figure 4.59: Tanya, February 6.A new couple just moved in there new house. Their new livingroom is exactly (square) 6.82 m 2 .(for each wall). The coupledecided to put their 1.75 m piano on one side. On another sidethey put their 1.52 m sofa. Across they put their 0.84 m televisionset. Beside their sofa they put their 1.02 m loveseat. Attached to(Part of one wall in) the living room was a 2.04 m (log) fire place.How much (wallspace) was left of the living room along the sides?Figure 4.60: Tommy, March 4.There was a pie eating contest that was held at Stanley Park.There were 4 contestants in the contest. Tom ate 4 pies, and 1/3of another. It took Tom 1:54.03 to pass out. Exactly :24.04 laterRoxanne passed out eating 1 pie more than Tom. 1.54.44 laterJake passed out in 1.00.00 later eating 4 more pies thanRoxanne. This declared Jeremy the winner. Jeremy didn't stop,he was trying to beat the record. The record was 13 pies eaten in6.04.00. Jeremy ate 12 pies and 2/3 of another in 5.45.35. Howmuch pie was eaten, in how much time? Counting their timestogether.Figure 4.61: Tommy, February 10.In Figure 4.58, though the decimal fractions look complicated, thecomputation involves addition and subtraction, with no regrouping across129units of time such as seconds and minutes (Mike's solution showed108.550 + 98.551 = 207.101 min). Note that the expression "2 for 1movie special" is familiar to students, and such realistic contexts for theSCQ (as pointed out earlier), seem to aid students in its solution.The second question (Figure 4.59) is remarkable because it isseemingly convoluted, but on closer inspection proves to be areasonable reflection of some students' routine for a weekend. Althoughthe context is realistic enough, because of the sequence of actions, onehas to be careful about computing each step. However, the computationitself is not difficult, as fractions such as 1/2, 1/3 and 2/3 are easynumbers for the students to deal with (both according to the students andthe teacher).In Figure 4.60, the words in parentheses were added on by theteacher to make the problem easier to read and comprehend. All thestudents attempted this problem with the aid of diagrams. But once thediagram was in place, the computations were not difficult. Although itwas a long question, once a mathematical model--in this instance adiagram representing the data--was used, the actual computation posedno difficulties for the students.The fourth SCQ (on pie eating, Figure 4.61) had both common anddecimal fractions based on a rather involved "story" and used ratherunconventional and inconsistent notation for the time periods, with noindication whether hours, minutes or seconds were involved. (From theaccompanying solution, it seemed that 1:54.03 referred to 1 minute and54.03 seconds.) In spite of the common fractions in the SCQ, thecomputation involved only adding decimal fractions and convertingseconds to minutes.130A glance at the SCQ presented so far under all the themes, as wellas some SCQ I will discuss under the next theme, will reveal the extent towhich students chose numbers which were easy to compute with. How,then, did students come to choose such numbers?From observations of group discussions and talks with thestudents and teacher, I gathered that there were basically four ways bywhich they came to choose such computationally easy fractions: (a) bysheer coincidence or "luck", (b) by first choosing the fractions and thendeciding the whole numbers which were divisible by the denominator ofthe chosen fraction, (c) by first choosing the whole numbers and thendeciding the fraction whose denominator was a factor of the wholenumber, and (d) from personal interest and experience.Although (d) above did feature in a number of questions (e.g.Jackie said because she was interested in Rhythmic Gymnastics, sheused that as a basis of one of her questions), by and large, (b) and (c)above were the most frequent ways of choosing such easy-to-computenumbers. For example, Mike admitted that while preparing the SCQshown in Figure 4.62, he tried a few fractions before choosing numberswhich he could rename in tenths, indicating that such choices weremainly a result of trial and error.Each person got a mark out of 10. But someone renamed some ofthem! Who was first? Who was third? Can you rename themagain in a different way? Given: 4/10, 20/50, 10/100, 2/2, 60/200.Answer: 2/2, 4/10 = 20/50, 60/200, 10/100; 10/10, 4/10 = 4/10,3/10, 1/10Figure 4.62: Mike, February 19.131In summary, students prepared SCQ which had numbers that were easyto compute with, even though the questions were long and required anumber of steps for their solution. Generally, students tended to use trialand error to get these "easy to compute" numbers, though there weresome instances of students using personal experience (as in scoring ofevents they had taken part in, like Rhythmic gymnastics and skating).Where the SCQ were long, a diagram was sufficient to model thesituation and assist in arriving at the solution. In other words, the SCQseem to provide the students an opportunity to mathematise (Mason &Davis, 1991), for example, by taking into account all the informationgiven, understanding what the question was about, and translating theirmental imagery of the problem into mathematical models such asdiagrams and other forms of representation before actually carrying outthe computation.Theme 4: use of discrete rather than region models Most mathematics teachers would agree that the region model(such as a rectangular or circular region divided up into a number ofparts) is the one very frequently used in the teaching-learning offractions, almost to the total exclusion of other models, such as the set ordiscrete model (like a fraction of a number of people). But the SCQ hereshow a surprising amount of the seldom-emphasized discrete model.Before speculating on some reasons for this high frequency of discretemodels in the SCQ, let me give two examples of SCQ which used thediscrete model of fractions (Figures 4.63 and 4.64):132Their are 5000 computer discs. The night befor last, someonestole 2/3 of the discs. How many are left?Figure 4.63: Jackie, March 31.Tiffany had 200 cookies. 3/4 had chocolate chips and 1/4 wereplain. Tiffany ate 2/50 of the chocolate chip cookies and 1/50 ofthe plain cookies. How many cookies did Tiffany eat? How manyare left?Figure 4.64: Anne, February 6.Figure 4.64 was typical of many questions reflecting a discrete modelSCQ on fractions, where a countable set of objects was given and a partof the number of objects had to be computed. The first SCQ (Figure 4.63)was one of the few examples where the numbers did not result in wholenumber solutions. Even so, the computer discs are countable, andhence represent a discrete set. While there was nothing to preventstudents drawing a region like a rectangle and letting it represent thewhole (5000), the student who constructed the question indicated thatshe expected a solution using a discrete model.There were many other SCQ using the discrete model for fractions(such as those referring to a fraction of a number of flowers, trucks, cards,instruments, skates, balloons, basketballs and so on). Instead of givingfurther examples of SCQ using the discrete model for fractions, suffice itto say that slightly over 85% of the SCQ, especially those involvingcommon fractions, used the discrete model.133This finding does not seem consistent with Ellerton's (1989) whereproblems constructed by students tended to be the type of questionsprovided by teachers and textbooks. (An exception did surface duringinitial SCQ involving decimal fractions when students tended to useunrealistic situations such as "15.27 people were blond" until the teachergave examples of decimal fraction word problems in sports.) Eventhough the teacher in the present study did provide some exercises fromthe textbook and from teacher-generated questions, most were questionsarising out of group discussion in class. The non-threateningatmosphere, the de-emphasis on correct answers and encouragement tojustify their "thinking" could have contributed to students' going beyondthe usual classroom model of regions to using the discrete model whichis more closely related to students' daily (non-mathematics classroom)experiences.Summary of results pertaining to research question #2Overall, the SCQ revealed that students seemed to constructexperience- and interest-based questions which assumed sharedknowledge and were computationally simple, but were neverthelessmulti-step problems. Almost all the questions were rich in detailsreflecting everyday experience--details considered extraneous to themathematical information needed for solutions to these questions, butnevertheless details that served to situate the questions in contextsmeaningful to the student. The SCQ were also generally more complex,in terms of the steps or number of operations required to arrive atsolutions, compared to standard textbook word problems on the sametopics.134Research question #3 and resultsResearch question #3: What are the roles of SJ and SCQ inmathematics learning, according to (a) students? and (b) the teacher?Students' View of the Role of SJ in Mathematics Learning I start with Tommy, the only Grade 6 student who articulated bothorally and in writing that he did not find any benefit in writing journals in amathematics class. Right at the outset, however, I have to say that I havegiven him more "voice" because (a) he was the only one who had anegative attitude towards the SJ, (b) he was the most articulate, bothorally and in writing, and (c) he was present throughout the study andkept his SJ entries current. If some of the students have not been givenmuch voice, it is because they were either not very articulate or had anumber of absences or both.In an interview, when asked whether SJ in a mathematics classwere useful, Tommy said "All you're doing is what's in your head, itdoesn't help in any way" although "writing down is much easier becausethere is no one there to stop and say they don't understand" (March 4).He wrote the following in response to what he thinks about writing inmathematics:During Grade 6 I have changed my mind constantly, right now Ithink writing in my math journal is a waste of time kind of. It's justlike when you (Mrs. Lomax) don't like it wasting your time. We canjust tell you in person what we think. With the time we use writingyou can give a slow understanding lesson. When this is over, youonly have a quick 10 minute review. I don't know what this is for,all I know is it is for Mr Rama. (Tommy, April 10)To which the teacher responded "Tommy, I wish all the students feltconfident enough to say in person what they think." Tommy was referring135to journal writing and not writing questions in the mathematics classroomin the above excerpt since in another journal entry he indicated that heliked to write his own questions, stating "Yes, I like to make my ownquestions. I like to because I put in every thing I have learned into thequestion." (March 13)In answer to my question on whether writing should be done inmathematics classes, Tommy replied (during an interview) that "itsounded odd to do writing in mathematics classes when we're onlysupposed to usually write numbers and explain how you got the answer"and that it was a waste of time becausewhen we speak, let's say I write something down and I can talk itout when I write it, and someone times me and I see when I'mdone and say, I memorize it, I can speak out before writing.(Tommy, April 14)From these excerpts , it is clear that Tommy has maintained that he sawno advantages to using SJ in a mathematics class. From the interviewdata, it seems that Tommy feels that writing takes a longer time thanspeaking, and therefore less time is given for the teacher to teach. Healso did not see any purpose for this activity. His comments seem toecho Pimm (1987) who remarked "Many adults and children alike arereluctant to write things down, particularly in situations where there is noclear reason why they should" (p. 118). But in spite of his negativeattitude towards SJ, he "obeyed" the teacher's injunction to write--anindication of the asymmetrical power relationship existing in theclassroom, however well-intentioned or "democratic" the teacher mightbe. In spite of his stated aversion to writing in the mathematics class, hewrote the most, perhaps because of his fluency in writing. Moreover,most of his writing indicated that he was aware that he was supposed to136explain to an audience. As well, he sometimes wrote as if he werethinking aloud or having a conversation with an absentee audience.When I spoke to the teacher about Tommy's attitude towards SJ,she had this to say:You see, the thing with Tommy is that he doesn't really need this tosolidify his understanding, because he has such goodunderstanding, so maybe in his case it's not as useful, because heis doing it for himself in his mind. (April 30)From the teacher's comments and from my own interaction with Tommyover the period of study, I feel that he is one of those who did not benefitfrom the use of SJ in this study. He was very articulate, and did not seemto feel the need to write something down and look over it, to clarify ideas.He seems to typify students capable of reflecting on and monitoring theirlearning without having to put everything down "on paper" as seems tobe the assumption underlying writing as a metacognitive tool. Therelative ineffectiveness of SJ for Tommy could be because most of thewriting done in SJ in this study was not the exploratory type. Rather, itwas a transactional type of writing, and since he could explain orally justas well as he could in writing, the type of SJ used here did not seem tohave any particular advantage for him. Another reason could be that heperceived mathematics as something to do with numbers (recall hisstatement that he felt it "odd" to do writing in mathematics classes when"we're only supposed to usually write numbers . . . ") and he preferred tosolve problems that made him "think" (according to what he told meabout what he likes about mathematics questions). So, for him, whilethinking about and writing down (and even orally explaining) a solutionto a word problem would constitute a valid mathematical task, writingabout it in words would be considered not a worthwhile or valid137mathematical task. Exploratory writing (like "what would happen if ?"), onthe other hand, might have satisfied his liking for challenging questionsand led to a more positive attitude towards writing in mathematics.What the other students felt about the role of SJ for mathematicallearning can be categorised under the headings (a) clarification of ideas,and (b) feedback. Some excerpts exemplifying each of these categoriesare given next.Clarification of ideasI think you learn more when you write. Like, say, I don't know, sayMrs. Lomax asks us a question and we actually write it down in ourbooks and solve it and everything, like it just helps, like you canexplain how you do it and everything. (Jackie, April 14)If she talked to you, you might not fully understand, but if you wroteit yourself, you'd understand and probably the others wouldunderstand. (Mike, April 14)In spite of her stated support for the SJ in mathematics classes (as shownin her excerpt), Jackie's written explanations were rather terse andseemed to depend on diagrams, with very few words of explanation (e.g.she just drew diagrams when asked to order fractions). Jackie typifiedthe students who stated that SJ assisted in mathematical learningthrough clarification and explanation of ideas but provided little overtevidence of such a role for SJ in their writing, possibly because theywere writing for themselves. On the other hand, their oral explanationsshowed more elaboration of their mathematical understanding, as didtheir SCQ.From the second excerpt (Mike's)--and after requesting forclarification from Mike--I gathered that "she" referred to the teacher and"others" referred to his classmates, implying that an advantage of writing138student explanations in the SJ would be that such explanations would bemore understandable than those of a teacher to other students in theclass. In other words, he seemed to be saying that writing in the SJserves two purposes: (a) writing for oneself helps one understand, and(b) other students who read the SJ can also understand, possiblybecause of shared experiences, difficulties and language. His SJ didprovide evidence of his understanding (e.g when he explained what afraction meant), but there was no evidence of other students'understanding his explanation, as the SJ were meant to be read by theteacher. One reason for his support of the SJ as a learning tool formathematics could be that he generally liked to write (according to theteacher and also from interview data).These two excerpts indicate that the students view writing in theSJ as a means to clarify ideas and to promote understanding ofmathematics. Such a view is in keeping with that of proponents of writingto learn mathematics, that the very act of writing forces one to grapplewith, and clarify, mathematical ideas (e.g. Kenyon, 1989; Keith, 1990;Pimm, 1991). However, while the SJ did reflect students' mathematicallearning, most of the written explanations were not as clear as their oralexplanations. So, while it may be true that they learned through writingin the SJ, they seemed to be learning just as well, if not better, throughoral discussion and explanations. Hence, though SJ might be a usefullearning tool for clarifying ideas in mathematics, not all students who saythey benefit in such a way from the SJ seem to be able to show it in theirwriting (for example, their written explanations towards the end of thestudy did not seem to be very different from those at the beginning of thestudy).139FeedbackFor instance, one of our math classes, she said, try to put downyour answer and explain how you thought, and even to me itshowed me how I think about, in words, how I think and how Iprogress, even myself, not just my teacher. (Mike, Feb. 5)Helps one understand things, because when I think something, Iwrite something in there and Mrs. Lomax corrects it and I figure outwhat I did wrong. If you think something the wrong way theteacher could fix it. (Shaun, March 4)Well, that's good. Well, sometimes she can, if you write somethingand she asks you a question, you can talk to her later, and even ifyou are confused or something, she can help you. Because if youare just thinking about it, you might not be able to do it in yourhead, because you have, maybe, to write it down or draw a pictureor something. (Maureen, March 4)I sort of like it because it's sort of different, get your ideas out andwhat you think about it. I guess it's sort of easier because you candraw what you are thinking and write down what you're thinking.(Jackie, March 4)The excerpts under the heading "Feedback" seem to indicate thatstudents viewed the role of SJ as providing feedback from the teacherabout their mathematical learning. The type of feedback referred toseems to differ. For example, there is feedback from the text to the writer,as exemplified by Mike, who was the only one who said that writing, evento himself as reader, helped him clarify ideas. The other type of feedbackimplied is the feedback from the teacher to the writer, as indicated by thesecond and (first part of the) third excerpts.When I pointed out during an interview that the teacher could helpeven if the student were to talk rather than write about their mathematics,Shaun saidBecause, if you just spoke to the teacher, she wouldn't rememberwhat you said. (Shaun, April 14)140When asked to elaborate, he said that it would be impossible for theteacher to remember what everyone in the class said to her, but shecould remember if they had put everything in writing. I infer that he wasemphasizing two points here: one, that SJ provided feedback from theteacher to the student; and two, the relative permanence of writtenexpression as opposed to oral expression allowed for more concertedattention.This last point seems to be what Jackie, and to some extent,Maureen, too, are making when they talk about writing it down or drawinga picture to "see" their thoughts. In the sense of easier accessibility totheir thoughts, such writing seems to encourage interactive writer-textfeedback. Mike, too, seemed to agree that writing was a more permanentrecord allowing for easier accessibility when he said "thoughts, justthinking in your head, you can't etch it in stone in your brain" (April 14).But where Jackie and Maureen implied that writing allows their thoughtsto be "seen" on paper, he said that writing helped him "make pictures inmy mind" (March 4).Students' View of the Role of SCQ in Mathematics LearningAs to the role of SCQ in mathematics learning, there seemed tothe following categories: (a) to get a better understanding, (b) practice insolving problems, and (c) to provide questions at an appropriate level.Excerpts of SCQ follow under each category.141To get a better understandingMakes you think more. When you write, you have to have asolution, so you are writing for everyone else, but you are alsowriting for yourself and answering. So you have to do two things,not one thing. (Tommy, March 4 Interview)Writing my own problems help me because when you'requestioning yourself for the answer or trying to think up a goodproblem. (Maureen, March 13)I think when I write my own questions it helps me understand howto make problems and get my own answers. (Darlene, March 13)The excerpts imply that preparing SCQ cannot be done without anattempt at understanding the mathematics involved, when a solution hasto be worked out, too. Hence, preparing SCQ made them understandmathematics better because in doing so, they had to think about both thequestion and the solution. Although the excerpt by Tommy indicates therole SCQ play in understanding mathematics quite clearly, I haveincluded the other two excerpts to show how the less articulate studentshave stated the same view.Practice in solving problems In the future, you may get a similar question and it will be a loteasier. (Tommy, Feb. 5 )It will help you get a review of it, practice stuff. (Maureen, March13 Journal Entry)It helps because you know how you do it in your own question, soyou can solve somebody else's question, too. (Shaun, Feb. 5)From the excerpts, it is clear that the students perceive that the SCQ canserve as review and practice material. In trying to solve their ownquestion, and by the practice obtained in doing so, they also found iteasier to solve other students' word problems.142To provide questions at an appropriate level. I know how people think when they make up questions. I knowhow other 11 year olds think. (Mike, March 13)I learn more solving other students problems because they areabout something that we know and understand. (Shaun, March13)I think solving students problems are better because, one studentmay put all their knowledge in one question and that would makeyou think real hard on finishing it, while a teacher puts an easyone, and gradually make it harder. So in 100 questions theteacher makes, it would be the same as 1 of the students hardestproblem they can make. (Tommy, March 13 )The excerpts indicate that, according to the students, the SCQ are betterable to provide questions at a level appropriate to the students thanquestions by the teacher. The reason for the appropriateness is that theSCQ are prepared by peers.In summary, except for Tommy, all the other students feel that SJhad a number of roles to play in the mathematics classroom. Among theones the students mentioned were: to clarify thoughts, to help expressoneself, to understand better, to help teachers assist students havingdifficulties, and to help remember and review. All these roles Iencapsulated under the two categories of "clarification of ideas" and"feedback." And some of the roles for SCQ, according to the students,were: to get a better understanding (as preparing and solving questionsneeded clearer understanding of concepts), to give practice in solvingproblems (as every question had to be accompanied by an answer), andto provide questions at an appropriate level (as the questions wereprepared by their peers).143Teacher's View of the Role of SCQ in Mathematics Learning ,The teacher's views recorded below were mostly obtained throughan interview on April 30, 1992 (some were obtained through discussionwith her and also through what she had written in her assignment for hernight class). According to the teacher (Linda), she initially felt positiveabout SJ and SCQ, mostly because of her experience with them in theprevious project. She changed her mind about the SJ approximatelyhalf-way into the study, but not about the usefulness of the SCQ. Shesaid:I think that the SCQ definitely are motivational, the kids really likedwriting them and they liked solving each other's questions and Ithink that they have potential for showing the teacher what thekids' conceptual understanding is like. Think you can use them assort of an evaluation of the understanding.When asked to comment specifically on whether SCQ helped themathematical learning of the students, this is what she said:I think that I got more from the SCQ in terms of their mathematicallearning. I could know from what they have written, who had apretty good grasp of this and who didn't. And it was easier to seewith the questions when there had been a change in theirmathematical learning. Like Chris is a good example of that.Because at first he wrote problems that had no fractions in themand then he whoops, in one week, wrote a really good interestingproblem about fractions that shows that he had gained a whole lotof understanding about what these really meant.And in terms of equivalence, that was really obvious, too. So thatwhen the Shaun's band problem, when he had 12 sixtieth of theband played the clarinet, some sixtieth played this and that, hehad a good understanding of what the fractions meant, but it wasonly if he had the fractions with the same denominator as thenumber in the whole.But it was interesting to see which students were ready to look atthat, they all thought that it was an okay problem to solve, but that itwas easy. And it was interesting to work on that to see which kidscould see how to make that more difficult and they could see thatthe factors of 60 and they could be rewritten and all of those144things. So I really like the potential of what that reveals, and alsothat you can work with that in the terms of the context, to see whichkids you can shove along a little bit further in their understanding.One reason for her viewing SCQ as useful was that she could seestudents' involvement and interest in the SCQ, as evidenced by theirdiscussion and willingness to work on the SCQ without being aware ofthe time passing. For example, sometimes the students were soengrossed in their SCQ, they looked surprised when the mathematicsperiod came to an end. At other times, students were eager to presenttheir solutions to the class, which were usually done with transparenciesand overhead projectors, something which was usually done by theteacher. In other words, other than the novelty of the SCQ and thepresentation, they had an opportunity to take some responsibility for theirmathematics learning. Moreover, they had some sense of ownership ofthe exercises as their SCQ were used for class exercises.As a teacher, Linda realised the value of the feedback sheobtained through what they wrote, as their SCQ mirrored theirunderstanding of the topic being taught. The SCQ also revealed theimprovement of the students' knowledge of fractions, as she pointed outabove about Chris and Shaun. She was also amazed at the range ofquestions students seemed capable of, revealing the SCQ as a potentialresource for meaningful exercises (this she told me during an informaltalk with her during the course of the study).When I asked her how she felt about the frequency of SCQ, andwhether she would use them for other topics as well, she replied:in terms of SCQ in math, I was really happy about writing once aweek and then, so the next week, so that when that routine washappening, that worked really well. They would write, I wouldpublish and then we would work on those. And one week thathappened that there were a lot of really valuable problems to145solve and we could do one a day and on weeks where there wereseveral that were of similar kind we could chunk them and say doone of this kind, and then we could talk and talk about problemsolving more generally than just this particular problem. So oncea week worked pretty well for that, I think that was okay, but I don'tthink we'd wanna do it for every topic in math.Her reservation about using SCQ for other topics stemmed in part fromher experience when students were asked to prepare SCQ on decimalfractions. According to her, initially they prepared unrealistic questionswhich included information such as "15.27 people were blond." Sherealised that the students' difficulties arose from a lack of meaningfulsituations in their everyday experience where they had to use decimalfractions. She said "They know about 1/2 a chocolate bar and 1/4 of apizza but they have no reason to know about 12.3 metres of anything!"So, after she gave examples of word problems from sports, studentsgenerated many interesting SCQ on decimal fractions, with many of themdifferent from the type of examples Linda had given. Anyway, she saidthat she would only use SCQ for topics which she felt students couldrelate to their own experiences.In addition to what she said in the interview about the role of SCQ,from my discussions with her and from her night class assignment, Igathered that the SCQ allowed her to give increased attention to problemsolving, communication and reasoning. As well the SCQ helped heraccommodate the diverse needs of the students, and was a source ofinformation for evaluation. Overall, she felt that the SCQ was a valuableteaching-learning tool for mathematics.146Teacher's View of the Role SJ in Mathematics Learning ,As to the SJ, she had this to say:In terms of the journals, I've been thinking about why I'm not asenthusiastic about the journals and I think that's because theproblem writing was my project and the journal writing was yours,and I, whether I realised that, viewed it that way, and the kids viewit that way, so that they view writing in their journal as somethingwe are doing for you, for your research. They don't view it assomething they are doing for themselves or even something theyare doing for me, so I think that it has to be clearer, the purpose ofit to the kids: we are doing this because I believe that it will helpyou to understand this better, and I also think that it has to beclearer in my mind what the purpose is of the question I am askingthem: I am asking them this because I want to know this fromthem, not just what are we going to write in our journal today, ohwe'll write a question about how we feel about something, wellthere wasn't , it wasn't purposeful enough for my teaching andtheir learning. We shifted that purpose for "this is for Mr. Rama'sresearch."I think the crucial point she was making is brought out by her expressionthat "it wasn't purposeful enough for my teaching and their learning. Weshifted that purpose for 'this is for Mr Rama's research'." Her view seemsto parallel Stubbs' (1980) statement that "a general principle in teachingany kind of communicative competence, spoken or written, is that thespeaking, listening, writing or reading should have some genuinecommunicative purpose" (p. 115). The SJ as used here did not seem tohave this genuine communicative purpose.A reason for her waning enthusiasm for the SJ was that she wasinvolved in a number of time-consuming activities. For example, her timecommitment to her ongoing project (an assignment for her night classwhich she was doing for credit at the university focused on "The use ofstudent written problems to promote mathematical power") did not allowher to give as much time as she might have liked for the SJ. She alsohad a student-teacher to supervise and she was involved in a number of147professional development courses. In addition, she had her normalteaching load.She elaborated on the apparent lack of success of the SJ, sayingI think that as a teacher, I need to think more clearly about what isit that I am trying to find out when I ask them to write in theirjournal. Is it for them or for me? What are they getting out of it?What is it that I am gonna look for, when I look at this? Ya, itwasn't, the purpose wasn't clear enough in my mind or their mind.That's basically why I don't think that was as successful as SCQ.One of her reasons for assuming lack of success for the SJ was thatsometimes she had to think really hard to initiate appropriate entries forthe SJ, so much so that she felt it was rather artificial to push forsomething when learning seemed to be going on reasonablysuccessfully in her mathematics classes. Some of the entries were veryshort and unclear as to the intent of the writer. In other words, thestudents were not fluent writers. She commented on her difficulty inassessing student understanding of mathematics from entries in the SJ.Was it "the language that is not clear or the math they are confusedabout?" According to her, the SCQ gave her better information aboutstudents' mathematical understanding. Moreover, she seemed to learnmuch more about student understanding from classroom discussionsand presentations by the students than from the SJ.She felt that in order for the SJ to be more successful, she had tospend more time and effort on her responses to students entries, as wellas plan what she hoped to get out of asking students to write in their SJ.And she just could not afford the time and effort involved, especiallywhen the students seemed to be learning pretty effectively with one of theinnovations, namely the SCQ and she had so many other competingcommitments.148When asked about the frequency of the SJ, she saidWhen I first started doing learning logs, it was done on Friday and Iused to have the topics from the week, and some I wanted them tospecifically to comment on and then there was choices what doyou think of this or that and how is this or that going, but I think thatthat was, somewhat useful as a sort of a summary of the week andit was a good way to report to parents and it did go home andcome back and all that stuff, but I think that is different than thiskind of learning log where it is specific to the subject and today'slesson and it has to happen right now and it's part of the lesson.So, I am not sure about how frequently. Not too often is the bigfear.While she could see the benefits of writing summaries once a week (as inher previous project), she was doubtful about the benefits of SJ entrieson specific topics done on alternate days , as was done here. Overall,she felt disappointed with the use of SJ in the mathematics classroom.She felt that the way she used SJ did not seem worth the time and effort,and for it to be more successful she needed to be really aware of specificpurposes for using them. In other words, she felt she and the studentsdid not assume ownership of the SJ, as they did of the SCQ. But shewas convinced that the SCQ were really motivating, and helped thestudents improve their understanding of specific mathematics topics, andthe teacher to get feedback on student understanding.I shared both her misgivings about SJ and her enthusiasm for theSCQ. My misgivings were mainly due to the wide disparity in students'levels of discussion and verbal explanations, compared to their writingsin the SJ. While students were articulate during the oral discussions,they were rather stilted in their writing, or seemed unable to expressthemselves clearly, except for a small minority.However, perhaps we (Linda and I) were unduly pessimistic, giventhat the majority of the students felt that SJ did assist in their mathematics149learning. Even if the students were trying to be accommodating, there issome evidence from their SJ that they were useful, at least insofar asidentifying misconceptions or partial understanding as well as focusingon certain concepts, principles and relationships in mathematics.Moreover, if "one use of written language is to externalize thought in arelatively stable and permanent form, so it may be reflected upon by thewriter, as well as providing access to it for others" (Pimm, 1991, p. 20-21),the SJ did serve a purpose. Even so, the findings of this study about thebenefits of SJ (as used here) are that these benefits do not seem to beobtainable only through the SJ or that these benefits are superior tothose that could have just as easily been obtained through verbalexplanations and discussions.Summary of Results Pertaining to Research Question #3 ,Students were generally positive about the role of SJ and SCQ inthe learning of mathematics. But while not all agreed about the positiverole of SJ, the teacher and students were unanimous about the benefitsof SCQ. In contrast, the teacher did not feel that the SJ were verysuccessful.150CHAPTER 5. CONCLUSIONThis chapter is divided into five sections: summary of the study,general discussion about the results of the study, limitations of the study,implications for practice, and possibilities for future research.Summary of studyIn this study, I investigated the mathematical learning revealedthrough the use of student journals (SJ) and student-constructed wordproblems (SCQ) in a mathematics class where the teacher initiated theprocedures and conducted the lessons. The class was a multi-levelGrade 5 and Grade 6 class but only the Grade 6 students were the focusof the study.Three times a week, towards the end of their mathematics lessonstudents wrote in their SJ in response to teacher prompts. Once a week,the students also prepared SCQ, in groups and individually. These SCQwere collected, edited, typed and redistributed to the class by theteacher, and used as class exercises. Six Grade 6 students wereinterviewed (video-taped) three times each over the 15-week period ofstudy. Records of these interviews, classroom observations and ateacher interview were all used to complement the analysis of SJ and theindividually-prepared SCQ.Although the SJ did give insights about students' mathematicalknowledge, the SCQ was a better indicator of students' knowledge offractions as they showed the students'1. mathematical knowledge implicitly (through the type of questionasked, information given, and the accompanying solution) and1512. improvement in their fraction knowledge explicitly (through thetype of fraction relationships, the amount of detail, and the number ofsteps and operations involved).The SCQ also revealed that students tended to base their wordproblems on1. their own experiences and interests2. an assumption of shared knowledge between the reader andwriter of the word problem3. numbers which made computation easy and4. the discrete model of fractions rather than the region model.The results of this study indicate that the SCQ assisted mathematicallearning in a classroom context but that the value of the SJ needs to bereconsidered.General discussionIn this study I set out to investigate the mathematical learningrevealed through the use of SJ and SCQ in a "normal" mathematics classwhere the teacher herself initiated the procedures and conducted thelessons. The results of the study indicate that students demonstratemathematical knowledge about common and decimal fractions throughtheir SCQ and SJ, but that their ability to communicate such knowledge,especially through the SJ, is not necessarily an indicator of theirmathematical knowledge. Indeed, as evidenced by discussions and oralexplanations, some students seemed to have more knowledge thanindicated by their writing in the SJ, so much so that it could be said thatability to communicate mathematical knowledge through written words152seems to be a sufficient but not a necessary condition for possession ofsuch knowledge. Use of diagrams to complement explanation in wordsseemed to be a common feature of the SJ as well as of oral explanations.Many benefits of the SJ were documented in this study (such asproviding feedback for the students) but such benefits could have beenobtained as well, if not better, through group and class discussions aswell as oral explanations, according to my observations throughout thisstudy. Indeed, the benefits derived through the SJ do not seem worth theamount of effort put in by the teacher (to read and respond) and thestudent (to write).Most journal usage in previous studies had encouragedexploratory language and feelings about mathematics but here the SJwere used specifically to communicate the student's understanding of amathematical concept or principle in written language. Perhaps this typeof SJ, where written explanations of concepts and principles were thefocus, is not suitable for this age level. I found that most of the studentscould express themselves better orally than in written form. This findingis consistent with what others have found about student writing inmathematics at higher levels. For example, Miller (1992), in discussingbenefits of writing in some high school algebra classes, said that"students' written responses do not always accurately portray theirunderstanding" because "at this stage the student was better able tocommunicate mathematical knowledge verbally than in writing" (p. 338).While it is true that Linda's busy schedule (such as supervising astudent teacher and attending night class) and gradual disenchantmentwith the SJ (such as lack of clear purpose for the SJ) might havecontributed to what she perceived as lack of success of the SJ, perhaps it153was too much to expect her to try to implement two innovations (SJ andSCQ) concurrently for the duration of almost three months.The students' and teacher's perceptions about the usefulness ofthe SJ were different. Whereas the former stated a number of benefits(such as being able to clarify ideas) for the SJ, the latter was doubtful ifthe SJ could reveal whether the student's difficulty was with language orwith mathematics. Perhaps the very novelty of the SJ might have ledstudents to accept the SJ as beneficial, just as the teacher herself hadbeen interested in SJ earlier. In spite of the students' stated views of theSJ as contributing to their understanding, I did not find much evidence ofthis in the writing of the less articulate students, though it is certainlypossible that such understanding is covert and may not be revealedthrough their writing.The success of the SCQ might at least in part be attributed toLinda's extra motivation in the assignment she had chosen for her nightclass. The assignment was entitled "Classroom Inquiry" and shefocussed on SCQ as a way to "promote mathematical power." Linda'sinterest in and attention to the SCQ could have been "caught" by thestudents. Even if the students were not influenced by Linda's feelingsabout the SCQ, they seemed to enjoy the SCQ, perhaps for the novelty,sense of ownership, or the student-student interaction they engendered.The discussion that evolved during attempts to solve the SCQ served toinform the teacher about her students' current state of mathematicalknowledge and gave an opportunity for the students to ask forclarification of unclear questions, unstated assumptions, or to evaluateunrealistic problems. Whatever the reasons, the students and teacher154were unanimous about the benefits of SCQ as a teaching-learning toolfor mathematics.Among the four themes identified in the SCQ, the fourth theme ofthe predominance of discrete models in their SCQ seemed at variancewith the students' use of the region model to explain fraction concepts inan interview context. While more than 85% of the SCQ were based onthe discrete model, almost all the explanations during interviews werebased on the region model--possibly because basic fraction conceptssuch as equivalence and ordering were involved. As well, perhapsstudents associated interviews with the formal context of classrooms(where the region model was dominant), whereas the SCQ were moreexperience-oriented and student-owned.Both SJ and SCQ did make use of class time--with the SJgenerally taking less time--but the time expended for these two activitiesseemed to be viewed differently: the SJ seemed to be something thathad to be done but the SCQ seemed to be something the studentswanted to do. From my observations in class, I could witness thestudents' enthusiasm (e.g. animated discussion) during the SCQ, butsuch enthusiasm was not evident when they were asked to do their SJ. .Even though students did not evidence enthusiasm for the SJ, theytreated the SJ as part of the class routine.The SJ required the teacher to think about appropriate prompts,the students to write (individually, with hardly any discussion), and theteacher to read and respond. In contrast, the SCQ (prepared both ingroups and individually) did not need too much extra time or effort by theteacher because the teacher did not have to prepare questions cateringto individual needs and interests. She only had to edit the SCQ and155group them in order to optimize learning during the discussion of theirsolutions by groups and individuals. For the students, their enthusiasmin preparing the SCQ seemed to make them overlook the effort neededto prepare the SCQ.Overall, in the context of this study, the SCQ seem an effectiveteaching-learning tool that can be integrated into the normal routine of amathematics class. The SCQ also seem to reflect the importance ofstudent experience and ownership of learning. In contrast, the SJ, asused here, do not seem to have any decided advantage over verbalclassroom discussion and explanation by the students and whateverbenefits derived from the SJ seem disproportionate to the time and effortrequired.Limitations of the studyOne of the main limitations of the study was the difficulty indeciding the extent to which written language could reflect mathematicalknowledge. While this limitation was less severe in the SCQ (because ofthe emphasis on asking rather than explaining), the nature of the tasksinvolved in the SJ--explaining mathematical concepts and principlesusing written language--made written proficiency imperative. Classdiscussions and follow-up interviews revealed students could explainthings better orally than in writing. If it were not for these discussions andinterviews, I might have concluded that lack of written proficiency mightindicate lack of mathematical understanding. Now, however, I infer thatwritten proficiency was a sufficient but not a necessary criterion formathematical understanding.156Another limitation of the study was that while the teacher hadmade the SCQ part of her own agenda, she did not seem as committedto the SJ. She had admitted her lack of enthusiasm for the SJ about half-way into the study and later said she felt the SJ had not seemedpurposeful enough for her. Perhaps her view that SJ was not verysuccessful might have altered if she had been as enthusiastic about theSJ as she had been about the SCQ.A further limitation of the study was that it did not take muchaccount of the student-student interaction during the preparation of theSCQ. The language students used during such interactions could haveprovided data on the extent and type of mathematical discourse thatresulted in the SCQ. Such information could have contributed to anawareness of conditions influencing the production and quality of SCQ.Even though I knew that classroom discourse was an important factor ina teaching-learning situation, I did not want to make that a focus of mystudy, just as I did not want to focus on affect, in spite of its importance. Inretrospect, it looks as if I have erred on the side of caution, as by trying tobe cautious about being drawn into factors such as classroom discourse--which have potential for complete studies in themselves--I have stayedtoo much at the fringes to inform my own study advantageously.Implications for practiceIt is a truism that unless the teacher sees a purpose for aninnovation and is committed to it, it will not be worth the time or effort ofthe teacher. In this instance, it seems that the teacher's views about theusefulness of the SJ did not correspond to the students' view. Becausethe students indicated they did find benefits for the use of SJ, the teacher157might try using SJ for a period of time before deciding to continue ordiscontinue their use. Even if the teacher and students wanted to use theSJ, the students' written proficiency would have to be taken into accountbefore attempting the type of SJ used here. Where student proficiency isin doubt, they should be encouraged to use diagrams and oralexplanations to complement their written explanations. As the findingsindicate, students seem to use diagrams to complement theirexplanation, even without being asked to do so. And, instead of thewritten journal being the culminating task, it could be used to initiate oraldiscussion. Perhaps some examples by the teacher--either her own orthose from other students--might also prove helpful for students not usedto journals or for those unsure of what is expected of the journal entries.The evidence for the benefits of SCQ seems indubitable. From thefirst theme of student experience and interest, it would seem appropriateto encourage students to prepare SCQ as part of the routine of themathematics lesson. Since teachers are always looking for ways toindividualize and vary exercises for the students, I would suggest thatSCQ seem suited for these purposes.The assumption of shared knowledge, the second theme of theSCQ, could be used to good advantage in a classroom situation. Forexample, such SCQ could be used to initiate discussion of ambiguities inquestions, other ways of looking at questions, the artificial or realisticaspect of a question, and justification of arguments in the solutionprocess.The third theme of the SCQ, the use of numbers which makecomputation easier, can make teachers aware that a word problem neednot have complex computations to make it mathematically adequate. In158other words, teachers can use SCQ to identify student understanding ofmathematics principles, even without including complex manipulation ofnumbers. It is well known that many students have proceduralknowledge (Hiebert, 1986) and can manipulate numbers by using analgorithm without understanding the mathematical concepts involved.Hence, just as a child who can recite the number names need notnecessarily understand cardinality and ordinality of numbers, so astudent who can obtain correct answers for fraction questions need notunderstand fraction concepts. A move away from equatingcomputational competence to mathematical competence (such asestimating, problem-posing and problem solving) through the SCQ mightprove pedagogically more defensible and optimal, especially with theavailability of calculators and computers.The fourth theme, the use of discrete rather than region models,should make teachers pause and reflect on the type of models theyusually use for fractions. In spite of the predominance of the regionmodel in previous classroom exercises and manipulatives used duringthe present study, students chose the discrete model. Their choiceshowed that left to their own devices, students tend to revert to familiarexperiences to situate their problems and make the problems meaningfulto them. Perhaps such a choice is also indicative that discrete sets aremore amenable to the operation of division (e.g. a fourth of 24 is viewedas 24 divided by 4)--which might be how students view basic fractionconcepts and operations.I would also speculate that if teachers believe students shouldstart from what they know, and have some responsibility and choice fortheir learning, then SCQ and SJ might have a role to play in the159mathematics classroom. Furthermore, if one aspect of mathematicalcompetence is communicative competence--knowing how tocommunicate effectively according to context-dependent cues--thenwriting in mathematics by using SJ and SCQ should be encouraged inthe mathematics classroom.Possibilities for future researchA number of issues raised by this study could be useful startingpoints for further research. For example, the SJ used here attempted togather information about students' communication of their understandingof mathematical concepts and principles. The students wrote to and forthe teacher, and they were dependent on the teacher to tell them howwell they had succeeded in their task. There were two facets of this task.One was writing proficiency. The other was external accountabilitythrough written explanation to the teacher.To compensate for the lack of written proficiency, students couldbe encouraged to explain orally and draw diagrams complementing theirverbal explanation. An analysis of the student diagrams and oralexplanation could provide data on students' understanding ofmathematical concepts and principles. In addition, rather than requiringonly explanations, they could be asked to question themselves or theteacher about things that were puzzling them. In questioning themselvesor the teacher, students are attempting the use of metacognition to aidtheir learning. Thus a study comprising both these facets--explanations(drawn diagrams and verbal explanation) and questions puzzlingstudents--could expand our knowledge of students' communicative160competence in mathematics as well as their control and internalization ofmathematics learning.In this study, the issue--whether lack of writing proficiency mighthinder the expression of mathematical understanding through writtencommunication--kept coming up. A study comparing mathematicsjournals of students in different grade levels on similar mathematicstopics might indicate how language background and writing proficiencymight influence communicative competence in mathematics.It is instructive that one student--who was the best in mathematicsin that class and who could also articulate well in his SJ--did not see anyadvantages in using SJ in a mathematics class. Perhaps there arestudents in other classes who might also find SJ not beneficial and whomight, like Tommy, prefer oral to written explanations in mathematics. Ifso, how can teachers address such an issue, especially if they believe inthe SJ and also in allowing students in general and such students inparticular, more choice in, and responsibility for, their learning ofmathematics? What are the wider implications to using writing as a toolfor learning mathematics? While this study has raised these questions, Ihave not attempted to address them. Perhaps future research mightaddress these concerns.Another area of research could be an analysis of discourse inmathematics classrooms. For example, Pimm (1987) has studiedteaching gambits as a factor in initiating verbal communication in themathematics classroom. Because the student-teacher and student-student interactions influence mathematics learning in the mathematicsclassroom, the two aspects of discourse analysis--production of text andinterpretation of text--should reveal information about how students are161helped to produce text and how students and teachers interpret textrelated to mathematical learning.A related aspect is whether the SCQ prepared in groups woulddiffer from those prepared individually. For example, were the SCQsimilar in quality? What if students prepared SCQ individually first andthen in groups (that is, the reverse of what was done in this study)?These, and other such questions, might prove a fruitful area for researchabout SCQ.The SJ, as used here, focussed on how students attempted tocommunicate their understanding of fraction concepts and principles.Previous studies have shown that attitudes toward and difficulties inmathematics affect students' mathematical competence. So, for thoseinterested in gaining a more comprehensive picture of studentunderstanding of mathematics, they might want to use a modified form ofthe SJ used here together with Clarke's (1987) IMPACT procedure.Such a combination might be useful as the IMPACT procedure allows forinformation about attitudes and difficulties to be obtained withoutrequiring a high level of written proficiency, and a modified form of the SJallowing for lower language proficiency might be incorporated into anexpanded version of the IMPACT procedure.For example, if students seem to prepare SCQ based on recentexperiences in school, teachers might wish to consider how suchexperiences could be incorporated into the mathematics classroom tocomplement or provide an alternative to teacher-generated mathematicsexercises. Suppose the student changes the unit or whole whencomparing the magnitude of fractions--for example, by drawing twodifferent-sized rectangles to compare 2/3 and 3/4. 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Mathematics education for a changing world.Alexandra, VA: Association for Supervision and CurriculumDevelopment.Wittgenstein, L. (1956). Remarks on the foundations of mathematics.Oxford: Blackwell.Woditsch, G.A., & Schmittroth, J. (1991). The thoughtful teachers guideto thinking skills. NJ: Lawrence Erlbaum Associates.176Worsley, D. (1989). The dignity quotient. In P. Connolly & T. Vilardi(Eds.), Writing to learn mathematics and science ( pp.276-282).NY: Teachers CollegeZepp, R. (1989). Language and mathematics education. Hong Kong:API Press.Zinsser, W. (1988). Writing to learn. NY: Harper & Row.177APPENDIX(Sample interview questions)Interview 1 (Wed., Feb. 5, 1992)1. When you hear the word "Christmas," what picture comes to yourmind?2. When you hear the word "mathematics," what picture comes to yourmind? Explain.3. Is there anything different about this year's math lessons? What isdifferent?4. Do you think writing in the math journal/notebook helps you learnmaths? How does it help?5. Do you think writing your own questions helps you learn maths? Howdoes it help?6. Give me an equivalent fraction for 1 fourth. Explain how you got youranswer.7. Alex and Maureen share a chocolate bar. If Alex eats 1 half of thechocolate bar and Mercedes eats 1 third of it, how much of the chocolatebar has been eaten? Explain how you worked it out.178Interview 2  (Wednesday, March 4, 1992)1. What is a fraction? How would you explain to your younger brother orsister (who does not know about fractions) what a fraction is?2. You have been doing quite a lot of journal writing in mathematics.Earlier you told me it helps you learn mathematics. Do you still feeljournal writing helps you learn mathematics? Explain.3. What about writing your own questions? How do you feel about itnow? Do you prefer to work alone or in groups when preparingquestions? Do you think your questions are better now than before?Explain.4. You seem to have got a rule for equivalent fractions. What is yourrule? Does it always work? What helped you in getting the rule?(discussing, journal writing, preparing own questions)5. How do you compare and order fractions?179Interview  3 (April 15, 1992)1. Do you think that learning math is important? What would be 2important things that you should learn in math?2. How do you learn math best? How should math be taught ?3. In math classes, usually we do sums. But you have been doing quitea lot of writing in addition to doing sums. Do you think writing should bedone in math classes? What advantages do you see for writing in mathclasses? What disadvantages? How do you think writing helps youlearn math?4. Earlier you had said "math is like . .." Explain . Complete "fractionsare like ..."5. How do you add common fractions, like 1/2 + 1/4; 1/2 + 1/3?6 How would you multiply decimal fractions?7. Why do you think multiplying decimal fractions sometimes makes theanswer smaller?8. Do you think my being present for your math classes affected the wayMrs. Lomax taught you? The way you learned math? Did you feelcomfortable, uncomfortable, or whatever, because of my being in theclassroom?180

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