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Using small group discussions to gather evidence of mathematical power Anku, Sitsofe Enyonam 1994

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USING SMALL GROUP DISCUSSIONS TO GATHEREVIDENCE OF MATHEMATICAL POWERbySITSOFE ENYONAM ANKUB. Sc., University of Cape Coast, 1977M. Ed., University of Cape Coast, 1982A 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 standardJune 1994© Sitsofe Enyonam Anku, 1994THE UNIVERSITY OF BRITISHIn 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 I’( i-The University of British ColumbiaVancouver, Canada—Date i / LI iDE-6 (2188)ABSTRACTThe purpose of this study was to investigate, with or without prompts, students’small group discussions of their solutions to mathematical problems and to determinethe extent to which the students demonstrate mathematical power. Mathematicalpower was defined in terms of student assessment standards (SAS) and theirintegration. SAS, each of which has associated with it categories of mathematicalactivities, comprise communication, problem solving, mathematical concepts,mathematical procedures, and mathematical disposition. Other insights perceived tobe important from the discussions were also documented.Grade 9 students of the regular school program were used for the study. Therewere 18 students in the class but only one group of students comprising 2 females and2 males was the focus of the study. They responded to mathematical problemsindividually for 20 minutes and then used 40 minutes to discuss, in groups, theirsolutions to the problems. Also, they responded to questionnaire items. The groupdiscussions were video recorded and analyzed. Data were collected on 7 differentoccasions using 7 different problems over a period of 3 months. -Results of the study indicate that students demonstrated mathematical power tothe extent that at least one category of the mathematical activities associated with eachSAS was reflected by the small group discussions of students’ solutions tomathematical problems. Other results indicate that combining students written scriptswith students’ talk provides a better insight into the things about which students aretalking. Also, monitoring students and providing them with prompts while they work ingroups is useful in helping them accomplish tasks in which they are engaged. Finally,when students work in groups, they can shift their viewpoints consensually orconceptually to align their viewpoints with majority viewpoints.11TABLE OF CONTENTSABSTRACTTABLE OF CONTENTS iiiLISTOFTABLES viiLIST OF FIGURES viiiACKNOWLEDGEMENT xCHAPTER 1. INTRODUCTION 1Background 1Rationale for the Study 3Purpose of the Study 6Research question 6Assumptions of the Study 7Limitations of the Study 7Summary 9CHAPTER 2. LITERATURE REVIEW 10Background to Current Reform in Mathematics Education 10Basic mathematics era 11Modern mathematics era 11Back-to-basics era 12Problem-solving era 12Current Reform in Mathematics Education 13Curriculum standards 14Teaching standards 15Evaluation standards 16Problem solving 16Communication 16Reasoning 17111Mathematical concepts .17Mathematical procedures 18Mathematical disposition 18Mathematical power 19Underlying assumptions and recommendations 19Small Group Learning 20Theoretical Framework of the Study 22Constructivism 23Knowledge Through Social Interaction 26Complementarity of Perspectives 27Summary 27CHAPTER 3. METHOD 28Context of the Study 28My Background 28Teacher’s Background 29The School 30The Participating Class 31Procedures 32Small Group Formation 32The Problems 33Data Collection Techniques 34Prompts 35Data Analysis 36Credibility and Auditability 36Coding of Transcripts 37Interpretation of Data 38Summary 39ivCHAPTER4. ANALYSISAND RESULTS .40Research question 41Communication 41Summary of results pertaining to communication 53Mathematical concepts 54Summary of results pertaining to mathematicalconcepts 59Mathematical procedures 59Summary of results pertaining to mathematicalprocedures 64Problem solving 65Summary of results pertaining to problem solving... 72Mathematical disposition 72Summary of results pertaining to mathematicaldisposition 83Mathematical power 84Other insights perceived from the study 86Consensual shift 87Conceptual shift 92Summary of results pertaining to other perceivedinsights 95CHAPTER 5. CONCLUSION 96Summary of the study 96General discussion 98Significance of the results 103Implications for practice 103Results of Study 103Reflections 104VPossibilities for future research 106Final note 107BIBLIOGRAPHY 108APPENDIXA. Problems 1-7 118B. Student-questionnaire items 126C. Some points to note when you work in groups 127D. Frequency of excerpts reflecting categories of mathematicalactivities associated with SAS 128viLIST OF TABLESTable 4.01: Categories of SAS reflected by discussions of problems... 85viiLIST OF FIGURESFigure 4.01: December 10th, 1992 42Figure 4.02: February 10th, 1993 43Figure 4.03: March 1st, 1993 44Figure 4.04: Quincy, March 1st, 1993 45Figure 4.05: January 18th, 1993 47Figure 4.06: Jane, January 18th, 1993 48Figure 4.07: February 22nd, 1993 49Figure 4.08: February 10th, 1993 50Figure 4.09: Jane, February 10th, 1993 50Figure 4.10: December 10th, 1992 53Figure 4.11: February 10th, 1993 54Figure 4.12: February 10th, 1993 56Figure 4.13: February 22nd, 1993 57Figure 4.14: Paulina, February 22nd 1993 57Figure 4.15: January 18th, 1993 59Figure 4.16: March 1st, 1993 60Figure 4.17: Jane, March 1st, 1993 60Figure4.18: January28th, 1993 61Figure 4.19: February 10th, 1993 61Figure 4.20: February 15th, 1993 62Figure 4.21: January 28th, 1993 63Figure 4.22: January 18th, 1993 64Figure 4.23: March 1st, 1993 66Figure 4.24: Paulina, March 1st, 1993 66Figure 4.25: January 18th, 1993 68Figure 4.26: January 28th, 1993 69Figure 4.27: February 22nd, 1993 70viiiFigure 4.28: December 10th, 1992 72Figure 4.29: February 15th, 1993 74Figure 4.30: February 10th, 1993 76Figure 4.31: Paulina, February 10th, 1993 76Figure 4.32: January 18th, 1993 78Figure 4.33: March 1st, 1993 79Figure 4.34: March 1st, 1993 80Figure4.35: Januaryl8th, 1993 83Figure 4.36: December 10th, 1992 88Figure 4.37: December 10th, 1992 90Figure4.38: February22nd, 1993 92Figure 4.39: March 1st, 1993 95ixACKNOWLEDG EM ENTI wish to thank my wife Victoria and my two daughters Dzifa and Dodzi for theirsupport and understanding during the long periods I had to spend away from themworking on this study. I sincerely appreciate your tolerance. I wish to thank mydeparted parents for inspiring me by persistently reminding me never to give upclimbing the educational ladder. My brothers and sisters back home provided me withlots of encouragement and financial support and I am very grateful to you. Many goodfriends here in Vancouver supported me in several ways and I wish to thank them all,especially Seth and Emmanuel.I wish to thank Dr. Jim Sherrill who readily accepted to co-chair my thesiscommittee after my original thesis advisor, Dr. Doug Owens, left suddenly to anotheruniversity. Your readiness to meet with me at anytime despite your heavy scheduleand the promptness with which you dealt with my concerns are greatly appreciated.Also, I wish to thank Dr. Ann Anderson for co-chairing my thesis committee. To Dr.Jerrold Coombs, I say thank you very much for the intellectual stimulation you providedme as a member of my thesis committee and as an instructor. It has been a wonderfulexperience knowing you. Thank you also, Dr. Gaalen Erickson, Dr. Jim Gaskell, Dr.David Robitaille, and Dr. Tony Clarke for the various things you did for me while I wasin the department.I wish to thank the Vancouver School Board, the principal, and especially theteacher and her class, for making it possible for me to conduct this study. To you theteacher, you will always be remembered for the wonderful support you provided me.Finally, I wish to thank all my fellow graduate students, the technicians, and the clericalstaff of the department for their wonderful support. I appreciate it very much. Thankyou all.xCHAPTER 1INTRODUCTIONBackgroundRecently, the National Council of Teachers of Mathematics (NCTM) initiated amajor reform in mathematics education throughout North America. This major reforminvolves the provision of standards for curriculum and evaluation in K-i 2 mathematics(NCTM, 1989) and standards for teaching K-i 2 mathematics I 1991). Thestandards for curriculum and evaluation, and those for teaching, are the onesperceived by the NCTM as important if students are to attain “mathematical literacy”(NCTM, 1989, p. 2).The curriculum standards involve what mathematics students are to learn inorder to become mathematically literate (NCTM, 1989). These curriculum standardsinclude mathematics as problem solving, mathematics as communication,mathematics as reasoning, mathematical connections, algebra, functions,trigonometry, statistics, and probability, among others. There are suggestedmathematical activities that are associated with these curriculum standards and howclassroom instruction should be organized to help students make sense of thesemathematical activities. For example, it is suggested that “Classroom activities shouldprovide students the opportunity to work both individually and in small- and large-group arrangements.” (NCTM, 1989, p. 67). Consequently, suggesting small-grouparrangement as a format of classroom instruction within the on-going reform meansthat the NCTM recognizes that format as an important one through which students canmake sense of mathematical activities.The evaluation standards are categorized by focus, namely generalassessment, student assessment, and program evaluation (NCTM, 1989). Theseevaluation standards “propose changes in the processes and methods by whichinformation is collected” (NCTM, 1989, p. 190) regarding all aspects of the reform. Inparticular, student assessment involves students’ performance in those mathematical1activities that are associated with the curriculum standards, with emphasis on whatinformation to gather and how to gather the information. Standards suggested by theNCTM for student assessment involve students’ mathematical power, problem solving,communication, reasoning, mathematical concepts, mathematical procedures, andmathematical disposition (see NCTM, 1989, p. 189). To determine whether byengaging in mathematical activities students are providing information that is reflectiveof the student-assessment standards (SAS) listed above, the NCTM has suggested“written, oral, or computer-oriented” modes of assessment (NCTM, 1989, p. 192).Recommending an oral mode of assessment presupposes that the NCTM recognizesthe importance of interaction and the talk that can result when students interact (as canoccur in small groups) to make sense of mathematical activities.Also, professional standards for teaching mathematics that support the changesin curriculum and evaluation standards have been suggested by the N CTM (1991).The professional standards are categorized into standards for teaching mathematics,standards for the evaluation of the teaching of mathematics, and standards for theprofessional development of teachers of mathematics. Three of the teachingstandards relate to classroom discourse while the remaining three relate to classroomenvironment, worthwhile mathematical tasks, and analysis of teaching and learning(see NCTM, 1991, p. 19). Specifically, classroom discourse addresses issues thatinvolve the teacher’s role in discourse, the student’s role in discourse, and the tools forenhancing discourse. Classroom discourse, in terms of “the ways of representing,thinking, talking, agreeing and disagreeing,” is perceived as “central to what studentslearn about mathematics as a domain of human inquiry with characteristic ways ofknowing” (NCTM, 1991, p. 34). Furthermore, the NCTM recognizes that students’ talkshould not be only in response to the teacher, but that students must also talk with oneanother (see NCTM ,1989, p. 34), and one of the several ways students can talk withone another is through the small group format.2The NCTM (1991, p. 57) envisions “serious mathematical thinking” as thecentral focus of the classroom environment. They indicate it is important to create aclassroom environment that will permit students to represent ideas, think, talk, agreeand disagree. One means to foster this desired classroom discourse and helpstudents make sense of mathematics is to encourage students to work collaborativelythrough the small group format (NCTM, 1991). Also, mathematical tasks (or problems)should “facilitate significant classroom discourse” (NCTM, 1991, p. 25), which seemsviable through the small group format, among others. Furthermore, to analyze thelearning and teaching of mathematics, the NCTM (1989, p. 64) encourages teachers toobserve “students participating in small group discussions” so that they can gaininsights related to students’ understanding of mathematics. So, whether in relation toclassroom environment, mathematical tasks (or problems) or the analysis of teachingand learning mathematics, the importance of the small group format for promotingserious mathematical thinking has been recognized.Within the on-going reform in mathematics education, in addition to the NCTM(1989, 1991), other educators (Artzt & Newman, 1990; Bishop, 1988, 1985; Bishop &Nickson, 1983; British Columbia Ministry of Education, 1990; Cobb, 1989; Davidson,1989; Dees, 1991; Kroll, 1988; Vygotsky, 1978; Webb, 1991; and Yackel et al., 1990),have recognized the importance of small group format in promoting mathematicslearning. Based on the perceived importance of the small group format it was decidedto formally examine information from students’ small group discussions and thendetermine what evidence there is that such information is indicative of students’mathematical power as suggested by the NCTM.Rationale for the StudyThe importance the NCTM (1989, 1991) attaches to the small group format inpromoting mathematics teaching and learning has prompted the investigation intostudents’ small group discussions. In addition, students’ use of the small group format3to make sense of mathematics forms one of the important hallmarks of the revisedcurriculum in the Province of British Columbia (Ministry of Education, B. C., 1990;Robitaille, Schroeder, & Nicol, 1992). Furthermore, whereas several efforts are beingmade to examine students’ sense making of mathematics using the small group format(Kamii, 1984; Kroll, 1988; Smith & Confrey, 1991; Webb, 1991; Yackel et al, 1991),none has been directed specifically at providing evidence about students’demonstration of mathematical power as defined by the NCTM. An in-depthexamination of how students make sense of mathematics in one group as a case(Merriam, 1991; Yin, 1989) should provide mathematics educators with useful insightswhich can be used to influence the current reform.To examine students’ sense making of mathematics using the small groupformat and determine if information from the small group discussions is indicative ofstudents’ mathematical power, I have decided to focus on the student-assessmentstandards which the NCTM has recommended as criteria for judging whether or not“we have reached the Standards” (NCTM, 1989, p. 189). Gathering and analyzinginformation that is based on these student-assessment standards should provideeducators with insights about the contributions of the small group format to the ongoing reform in mathematics education.The seven focus areas of student assessment in K-I 2 mathematics listed by theNCTM are mathematical reasoning, problem solving, communication, mathematicalconcepts, mathematical procedures, mathematical disposition, and students’mathematical power. Although the NCTM places equal emphasis on all seven focusareas of student assessment, it recognizes the need for productive changes in thecurriculum and evaluation standards (see NCTM, 1989, p. 189). The NCTM definitionof students’ mathematical power encompasses the definitions of what constitutes theother standards of student assessment and their integration (see NCTM, 1989, p. 205).So, gathering information on the remaining six student-assessment standards shouldprovide evidence about students’ demonstration of mathematical power (see NCTM,41991, 21). In addition, the NTCM argues that “The goal of teaching mathematics is tohelp all students develop mathematical power.” (NCTM, 1991, p. 21). Therefore, thecurrent research investigated students’ demonstration of mathematical power.Also, in the present research, it was assumed that mathematical reasoning is amajor method by which students gain mathematical knowledge. (Notice thatmathematical knowledge involves all categories of mathematical activities associatedwith SAS.) Kline (cited in Clements & Ellerton, 1991) shares a similar view on the roleof reasoning in gaining knowledge when he argues that even though authority,revelation, experience, and experimentation are important sources of obtainingknowledge, the major method is reasoning. Furthermore, mathematics educatorsrecognize that mathematics is reasoning (NCTM, 1989) and for students to becomeautonomous in doing mathematics, they need to “gain confidence in their ability toreason and justify their thinking” (NCTM, 1989, p. 29). It follows then that mathematicalreasoning should involve “the kind of informal thinking, conjecturing, and validatingthat helps children to see that mathematics makes sense” (NCTM, 1989, p. 29).Many educators believe that when students work in groups in the mathematicsclassrooms, the kinds of informal thinking, conjecturing, and validating that students gothrough could be captured when they verbalize those activities (Artzt & Newman,1990; Kroll, 1988; NCTM, 1991). This is because talking, or oral discourse, orverbalizing forms an important medium of exchanging ideas as students work ingroups. Teachers of mathematics are encouraged to listen to students’ ideas, askstudents to clarify and justify their ideas orally, monitor students’ participation indiscussions, and urge students to talk in small groups (NCTM, 1991). Also,mathematics educators (NCTM, 1991) observe that “when students make publicconjectures and reason with others about mathematics, ideas and knowledge aredeveloped collaboratively” (p. 34). Consequently, mathematical reasoning throughtalk, should provide one basic source from which educators make inferences aboutstudents’ mathematical power. Thus, focusing on students’ reasoning during small5group discussions of mathematical activities should provide evidence about howstudents are meeting the student-assessment standards within mathematicseducation. So, to make inferences about students’ demonstration of mathematicalpower, the focus in this study is on students’ reasoning and arguments as the mainsource for deciding how students are meeting the student-assessment standards(SAS) of communication, problem solving, mathematical concepts, mathematicalprocedures, and mathematical disposition.Purpose of the StudyThe purpose of this study is to examine information from the discussions of agroup of students and determine if in the information generated by the students, onecan find the existence of data reflective of NCTM’s definition of students’ mathematicalpower. The focus of the study is on the information the students generate while theydiscuss their solutions to mathematical problems during student-student interactions.However, to a lesser extent, data were also gathered with respect to students’discussions following prompts from the researcher.Research questionSpecifically, the question addressed in this study is:To what extent is information from students’ small group discussions oftheir solutions to mathematical problems indicative of students’mathematical power?By information, I mean mathematical information that is related to the problemsthe students have solved. This mathematical information involves what a group of fourstudents said or wrote down individually as they engaged in student-studentinteractions to discuss their solutions to mathematical problems. The information alsoinvolves students’ individual responses to questionnaire items given to the students6each time they discussed their solutions to a mathematical problem. The problems forthe study relate to the mathematical topics covered by the students’ teacher over theperiod of the study. Furthermore, the mathematical problems are the types with whichthe students were familiar. Excerpts from the students’ discussions that are used toprovide evidence of students’ mathematical power are such that they reflect NCTM’sintended meaning of the categories of mathematical activities associated with eachSAS. Furthermore, the existence and integration of information indicative of SASprovide evidence of students’ demonstration of students’ mathematical power. In thisstudy, mathematical power is that which individual students demonstrated within thesmall group context.Assumptions of the StudyIn this study, it is believed that students in small groups will be able to reasonand engage in a meaningful discourse over mathematical activities. Also, Ihypothesize that while students discuss mathematical activities, there will beobservable events (Locke, Spirduso & Silverman, 1987) that will form the basis forevidence of students’ demonstration of mathematical power. That is, as studentsdiscuss their solutions to the problems used for the study, they will provide evidence oftheir ability to communicate mathematically, use mathematical concepts, and usemathematical procedures. Finally, students’ discussions will provide evidence of theirability to use mathematics to solve problems and of their mathematical disposition.Limitations of the StudySince an overlap in categories is possible whenever there is conceptualcategorization, one limitation to this study is the possible overlap of the categories ofmathematical activities associated with SAS. For example, evidence of students’ability to communicate mathematically can also constitute evidence of their use ofmathematical concepts. So, to minimize this limitation, it should be necessary,7sometimes, to use the same evidence as indicative of students’ ability regarding two ormore SAS.For this study where the problems are limited to the mathematical topics that theteacher taught the students, not all information that constitutes evidence that studentsmeet SAS will be provided by the students as they discuss their solutions to theproblems. Thus, the extent to which students information meets SAS depends on thenature of the mathematical problems. For example, students were not asked toformulate problems so the absence of evidence from the discussions indicative ofstudents’ ability to formulate problems should not be construed to mean that studentscannot formulate problems. Rather, it is the limitation of the problems that make itimpossible to make inferences about students ability to formulate problems. However,no single mathematical problem may permit students to generate enough informationto reflect all categories of mathematical activities associated with all of SAS. In regardto mathematical communication, for example, a single mathematical problem may notprovide information on all of what constitutes evidence of students’ ability tocommunicate mathematically. However, as observed by Lappan and Friel (1993), theimportant thing is that the problems can permit students to significantly talk about thesolutions and how they obtained those solutions.Another limitation to this study is the amount of time used for gathering data.The longer the data gathering period, the deeper the insights into the discussions ofthe members of the small group. Resources allowed seven data gathering sessions ofone hour per session, spread over three months. Even if more resources had beenavailable, an impending teachers’ strike would have terminated the data gatheringperiod. Nevertheless, as reported in chapter three, the amount of data collectedproved sufficient for the research question.8SummaryThe reform in mathematics education suggested by the NCTM provides themathematics education community with standards for curriculum and evaluation inschool mathematics. A major component of the evaluation standards involves student-assessment standards in mathematics as criteria for making decisions about students’demonstration of mathematical power. These assessment standards relate to problemsolving, communication, mathematical concepts, mathematical procedures, andmathematical disposition. To foster the attainment of students’ mathematical power,the NCTM has suggested some teaching standards for school mathematics. Three ofthe teaching standards relate to classroom discourse while the rest relate toworthwhile mathematical tasks, classroom environment, and analysis of teaching andlearning. Within all these standards, the small group format has been recommendedas important for fostering classroom discourse that will permit students to make senseof mathematics. However, no studies have been directed specifically at examining theextent to which information from the small group format is indicative of students’mathematical power. So, to provide such insight, I have decided to examineinformation from discussions of one group of students.9CHAPTER 2LITERATURE REVIEWIn this chapter, I present a brief review of the background leading to the currentreform in mathematics education. Then, I review the current reform, which has as oneof its recommendations, the use of small groups for promoting the teaching andlearning of mathematics. In addition, I review the literature on how educators areusing small groups to foster mathematics teaching and learning. Finally, I provide atheoretical framework for using small groups to gather information on students’demonstration of mathematical power.Background to Current Reform in Mathematics EducationThe apparent reforms in mathematics education over the last fifty years havebeen due mainly to changes in societal expectations of what graduates of themathematics curriculum should be capable of doing (Davis & Maher, 1993; Howson,Keite), & Kilpartrick, 1981; Kline, 1973; NCEE, 1983; NCTM, 1980, 1989; NRC, 1989;Willoughby, 1990). In trying to meet societal expectations, emphasis on what teachersshould teach within the mathematics curriculum has shifted several times, with newmaterials added and old ones removed from the mathematics curriculum. Whatteachers should teach shifted from basic mathematics (Howson, Keitel, & Kilpartrick,1981; National Commission on Excellence in Education [NCEE], 1983) through newmathematics (National Research Council [NRC], 1989), back-to-basics mathematics(Kline, 1973), problem solving (National Council of Teachers of Mathematics [NCTM],1980), and mathematical literacy, achievable through students’ mathematical power,as the main emphasis within the current reform in the discipline (NCTM, 1989, 1991).10Basic mathematics eraIn the 1950s, the major goal of mathematics education was for “training themind” (Kline, 1973, p. 9). The belief by educators at the time was that training of themind through numerous repetitions and memorization of mathematical rules andprocedures ensures the sharpness of the mind. Having attained a sharp mind,students could then solve problems is science, engineering, and other careers in life(Christiansen et al., 1986; Kline, 1973). Despite the emphasis on the sharpening ofthe mind, Kline (1973) observed that not only were students’ grades in mathematicslower than in other subjects, their dislike and dread for mathematics were widespread.Furthermore, he observed that “Educated adults retained almost nothing of themathematics they were taught and could not operate simple operations with fractions”(p. 15). In addition, not only did the military discover during World War II that they weredeficient in mathematics, but the Russians, instead of the Americans, were first to go tospace in 1957 (Howson, Keitel, & Kilpartrick, 1981; Kline, 1973, NRC, 1989). Thus,educators of the time became convinced that the mathematics curriculum during thatera, in terms of what it contained and how it was taught, did not prepare students tomeet societal expectations. Consequently, there were moves to reform the existingmathematics curriculum thereby giving birth to new or modern mathematics.Modern mathematics eraThe modern mathematics era (the ‘60s) was characterized by two mainfeatures, namely “a new approach to the traditional mathematics, and new contents”(Kline, 1973, p. 21). Familiar high school mathematics, including Euclidean geometry,was abandoned. The belief was that technology had made the learning of most of thetraditional topics obsolete and that abstract mathematics had become more importantas the basis for the development of modern science. Consequently, topics like theoryof numbers (including set theory), abstract algebra, linear algebra, projectivegeometry, topology, and calculus were emphasized. A new language involving set11theory was used to emphasize the “logic, the structure, and the unity of mathematics asa whole” (Kline, 1973. p. 18). Students were encouraged to be precise and rigorous intheir use of mathematical concepts. A deductive approach to learning these newtopics was emphasized by educators of the time. However, the modern mathematicsprogram was criticized for its excessive rigor and abstractness (Howson, Keitel, &Kilpartrick, 1981; Kline, 1973; Willoughby, 1990) and for the inability of its graduates toperform mathematical computations efficiently (Kline, 1973). Naturally, there weremoves to go back to the basics of skills development in mathematics.Back-to-basics eraThe mathematics curriculum for this era (the ‘70s) was informed by the beliefthat mathematical concepts arose from “physical situations or phenomena” and that“their meanings were physical for those who created mathematics in the first place”(Kline, 1973, p. 153). Furthermore, education (including mathematics education)should be broad rather than deep for elementary and secondary students so as toprovide opportunities for students to integrate their activities and interests and seemathematics as part of whole knowledge, but not as a different knowledge (Kline,1973). Thus, mathematics education should develop in students basic mathematicalskills that will enable them to function properly in several careers. Despite theemphasis on the development of basic mathematical skills, it was soon to be realizedthat students could still not utilize the accumulated mathematical skills to solve wordand real-life problems. So, around late ‘70s, there were moves by mathematicseducators to make problem solving the focus of mathematics education for the ‘80s.Problem-solving eraAs the previous reforms had tended to be slogan oriented, mathematicseducators considered problem solving as more than just a phrase or a slogan (Krulik &Reys, 1980). They considered problem solving as “the reason for teaching12mathematics’ (Krulik & Reys, 1980, p. xiv). Emphasis was placed on the developmentand the teaching of several problem-solving strategies, with Polya’s problem-solvingstrategies as one of the notable ones (Polya, 1957). The enthusiasm for problemsolving was great. There were attempts to define problem solving as a goal, process,and basic skill; suggest ways to pose problems properly; use pictorial language inproblem solving; suggest how to supplement and understand textbook problems, usecalculators to solve problems; and provide problem-solving experiences throughrecreational mathematics (Krulik & Reys, 1980).However, despite all these efforts at focusing on problem solving as the reasonfor teaching mathematics, the needs of the North American society, which are mainlysocial and economic, are not being met (NCEE, 1983; NCTM, 1989; NRC, 1989).Furthermore, Willoughby (1990) observes that the “real motivation for reform is achange in society itself” (p. 2). He argues further that:Never before has a change in technology made knowledge and understandingof mathematics so important to so many people. Never before has a change intechnology made the kind of mathematics people have been learning soobsolete. The technological revolution will not go away. We will not collect anddestroy all calculators and computers on some day in the future. The reformersmay die, but the reforms now taking place will continue to live. Those who fail tobenefit from these reforms will live less full and less productive lives than thosewho benefit from the reforms. Those societies that prepare people well for atechnological future will become better places to live. Those that don’t willwither. (p. 2)Such strong beliefs, among others, have compelled educators to initiate the currentreform in mathematics education, which encompasses all the reforms undertaken forthe last 50 years.Current Reform in Mathematics EducationThe current reform in mathematics education for students reflects society’sexpectations that schools produce a mathematically literate work force (NCTM, 1989).To meet society’s expectations, the NCTM has suggested some curriculum andevaluation standards of mathematics and what student activities are associated with13mathematics in such a curriculum (NCTM, 1989). Also, the NCTM has suggestedteaching standards through which teachers can facilitate the attainment of thecurriculum and evaluation standards (NCTM, 1991).Curriculum standardsThere are 13 curriculum standards for each of K-4 and grades 5-8, and 14 forgrades 9-12. The curriculum standards that are common to the grades includemathematics as problem solving, mathematics as communication, mathematics asreasoning, and mathematical connections. Whereas estimation is considered as onestandard under K-4, it is combined with computation under grades 5-8 and does notappear explicitly under grades 9-12. Topics involving numbers are grouped into threestandards under K-4 as number sense and numeration, concepts of whole numberoperations, and whole number computations. Under grades 5-8, topics involvingnumbers are grouped under two standards as number and number relationships andnumber systems and number theory; there are no number topics explicitly groupedunder grades 9-12. While there is geometry and spatial sense as one standard underK-4, it is simply geometry under grades 5-8, but separated into geometry from asynthetic perspective and geometry from an algebraic perspective under grades 9-12.There is measurement as one standard under K-4 and grades 5-8, but not undergrades 9-12.Other curriculum standards include patterns and relationships under K-4,patterns and functions under grades 5-8, and functions under grades 9-12. Fractionsand decimals form one standard under only K-4. Statistics and probability form onestandard under K-4, but form two separate standards under grades 5-8 and grades 9-12. Algebra as a standard appears first under grades 5-8 and then under grades 9-12.The remaining four standards under grades 9-12 include trigonometry, discretemathematics, conceptual underpinnings of calculus, and mathematical structure.14The mathematical activities associated with these curriculum standards indicateoverlaps across the standards and across grade levels. For example, under K-4, thestandard involving mathematics as problem solving requires that students “verify andinterpret results with respect to the original problem” (NCTM, 1989, p. 23). Similarly,the standard involving mathematics as reasoning requires that students “justify theiranswers and solution processes” (NCTM, 1989, p. 29). In these examples, the sameevidence can be used to indicate that students are either verifying and interpretingresults or that they are justifying their answers. Also, similar evidence can be used toindicate that students at K-4, grades 5-8, or grades 9-12 are verifying and interpretingresults with respect to the original problem. The presence of the overlaps suggeststhat in gathering information indicative of students’ achievement of the curriculumstandards, a holistic approach needs to be taken by mathematics educators.Teaching standardsThere are six teaching standards suggested by the NCTM for K-I 2. These aregrouped into four categories that are labeled tasks, discourse, environment, andanalysis. Under the tasks category, the teaching standard involves worthwhilemathematical tasks which are “the projects, questions, problems, constructions,applications, and exercises in which students engage” (NCTM, 1991, p. 20). Thediscourse category has three teaching standards. These are the teachers role indiscourse, students’ role in discourse, and tools for enhancing discourse. Discourseas used here “refers to the ways of representing, thinking, talking, and agreeing anddisagreeing that teachers and students use to engage those tasks” (NCTM, 1991, p.20). The teaching standard under the environment category involves the learningenvironment which is “the context in which the tasks and discourse are embedded”(NCTM, 1991, p. 20). Finally, under the analysis category, the teaching standardinvolves analysis of teaching and learning which is “how well the tasks, discourse, and15environment foster the development of every student’s mathematical literacy andpower” (NCTM, 1991, p. 20).Evaluation standardsThe evaluation standards are grouped into three categories. These are generalassessment, student assessment, and program evaluation (see NCTM, 1989, p. 189).The general assessment category comprises alignment, multiple sources ofinformation, and appropriate assessment methods and uses. The program evaluationcategory comprises indicators for program evaluation, curriculum and instructionalresources, instruction, and evaluation team. The student assessment category, whichcomprises students’ mathematical power, problem solving, communication, reasoning,mathematical concepts, mathematical procedures, and mathematical disposition, isreviewed more extensively because of its importance for this study.Problem solving. Problem solving refers to students’ abilities to usemathematics to solve problems. Students having the ability to use mathematics tosolve problems should provide evidence that they can:i) formulate problems;ii) apply a variety of strategies to solve problems;iii) solve problems;iv) verify and interpret results;v) generalize solutions (NCTM, 1989, p. 209).Communication. Students having the ability to communicate mathematicallyshould provide evidence that they can:i) express mathematical ideas by speaking, writing, demonstrating,and depicting them visually;16ii) understand, interpret, and evaluate mathematical ideas that arepresented in written, oral, or visual forms;iii) use mathematical vocabulary, notation, and structure to representideas, describe relationships, and model situations (NCTM, 1989,p. 214).Reasoning. Students who reason mathematically should provide evidence thatthey can:I) use inductive reasoning to recognize patterns and formconjectures;ii) use reasoning to develop plausible arguments for mathematicalstatements;iii) use proportional and spatial reasoning to solve problems;iv) use deductive reasoning to verify conclusions, judge the validity ofarguments, and construct valid arguments;v) analyze situations to determine common properties and structures;vi) appreciate the axiomatic nature of mathematics (NCTM, 1989, p.219).Mathematical concepts. Students having knowledge and understanding ofmathematical concepts should provide evidence that they can:i) label, verbalize, and define concepts;ii) identify and generate examples and nonexamples;iii) use models, diagrams and symbols to represent concepts;iv) translate from one mode of representation to another;v) recognize the various meanings and interpretations of concepts;vi) identify properties of a given concept and recognize conditionsthat determine a particular concept;17vii) compare and contrast concepts (NCTM, 1989, p. 223).Mathematical orocedures. Mathematical procedures generally meancomputational methods, even though they may include geometric constructions.Students having knowledge of mathematical procedures should:i) recognize when a procedure is appropriate;ii) give reasons for steps in a procedure;iii) reliably and efficiently execute procedures;iv) verify the results of procedures empirically (e.g., using models) oranalytically;v) recognize correct and incorrect procedures;vi) generate new procedures and extend or modify familiar ones;vii) appreciate the nature and role of procedures in mathematics(NCTM, 1989, p. 228).Mathematical disDosition. Disposition does not involve only attitudes, butincludes the “tendency to act in positive ways.” Aspects of students’ dispositiontowards mathematics include:i) confidence in using mathematics to solve problems, tocommunicate ideas, and to reason;ii) flexibility in exploring mathematical ideas and trying alternativemethods in solving problems;iii) willingness to persevere in mathematical tasks;iv) interest, curiosity, and inventiveness in doing mathematics;v) inclination to monitor and reflect on their own thinking andperformance;vi) valuing of the application of mathematics to situations arising inother disciplines and everyday experiences;18vii) appreciation of the role of mathematics in our culture and itsvalues as a tool and as a language (NCTM, 1989, P. 233).Mathematical Dower. Students’ mathematical power refers to “all aspects ofmathematical knowledge and their integration” (NCTM, 1989, p. 205). Aspects ofstudents’ mathematical knowledge include:i) ability to apply their knowledge to solve problems withinmathematics and in other disciplines;ii) ability to use mathematical language to communicate ideas;iii) ability to reason and analyze;iv) knowledge and understanding of concepts and procedures;v) disposition towards mathematics;vi) understanding of the nature of mathematics;vii) integration of these aspects of mathematical knowledge(NCTM, 1989, p. 205).Underlying assumDtions and recommendationsThere are many assumptions that underlie the current reform in mathematicseducation (NCTM, 1989, 1991). One of them is the belief that successfulimplementation of the reform should result in students acquiring mathematical power.In fact, the development of students’ mathematical power is seen as the goal forteaching mathematics within the current reform (see NCTM, 1991, p. 21). Another isthe belief that social interaction (Artzt & Newman, 1990; Bishop, 1988, 1985; Bishop &Nickson, 1983; British Columbia Ministry of Education, 1990; Cobb, 1989; Davidson,1989; Dees, 1991; KroIl, 1988; Vygotsky, 1978; Webb, 1991; and Yackel et al., 1990)is important for students’ construction of mathematical knowledge. So, to implementthe K-4 curriculum standards for example, the NCTM believes that children should beactively involved in doing mathematics “by interacting with the physical world,19materials, and other children’ (NCTM, 1989, p. 17). Similar beliefs have beenexpressed for the implementation of the curriculum standards for grades 5-8 (seeNCTM, 1989, p. 69) and for grades 9-12 (see NCTM, 1989, p. 124). Forimplementation of the teaching standards, the NCTM believes in the importance ofinteraction among children (see NCTM, 1991, p. 21). Also, to implement theevaluation standards, the NCTM believes in the importance of interaction amongchildren (see NCTM, 1989, p. 192)). Thus, interaction among children is consideredvery important for the implementation of the curriculum, teaching, and evaluationstandards.Arising out of these beliefs or assumptions is the NCTM recommendation thatsmall group formats form an important context within which educators implement thecurriculum, the teaching, and the evaluation standards (NCTM, 1989, 1991). Forexample, students are to be encouraged to engage mathematical activities in groups(see NCTM, 1989, p.8), to be taught in groups (see NCTM, 1991, p. 36), and to beassessed in groups (see NCTM, 1989, p. 192). Also, learning in small groups hasbeen recognized as one of the important hallmarks of the revised curriculum in theProvince of British Columbia (Robitaille, Schroeder, & Nicol, 1992). It follows then thatthe use of small groups to promote the construction of mathematical knowledge (andconsequently the development of students’ mathematical power) should be takenseriously by mathematics educators.Small Group LearningSmall group learning involves students collaborating in an intellectualendeavor to make sense of complex situations (Artzt & Newman, 1991; Davidson,1990; NCTM, 1989). This collaboration involves “talking, listening, explaining, andthinking with others, as well as by oneself” (Davidson, 1990, p. 4). To collaboratesuccessfully, it is important for students to observe some social norms that could guidetheir interactions (Davidson, 1990; Eichinger et al, 1991; Yackel et al, 1991; Webb,201991). Some useful social norms that Eichinger et al. (1991) identify include (1) theresponsibility of all students to contribute to the group and to seek to understand otherstudents’ ideas, (2) commitment to helping others contribute successfully to the group,and (3) tolerance for diverse cultural backgrounds and working styles. The use ofsuch social norms to guide classroom interaction is very useful in providing studentswith the opportunity to cooperate and benefit from the group interaction.Having students cooperate for the common good should not be construed tomean that individual accountability is abandoned. In fact, individual accountability isone of the seven major defining characteristics of small group learning that Davidson(1990) identifies. Recognizing and maintaining individual accountability, andconsequently individual ability, should promote small group learning since individualsform the group and their individual abilities influence the nature and outcome of thegroup interaction. An analysis of several studies involving group interactions (Webb,1991), indicates that individual students of high ability tend to give the mostexplanation to other group members. Findings from these studies also indicate that astudent with a particular level of ability may interact more actively and learn more insome group compositions than in others. Also, in many groups, the high ability andlow ability students tend to form teacher-learner relationships, while medium abilitystudents tend to participate less in group interactions than the highs and lows.However, in high-medium and medium-low ability groups, all students tend to beactive participants in group interactions. Thus, the individual’s ability has a significantinfluence on the outcome of the group interaction, and therefore the design of anystudy involving group interactions.Rau and Heyl (1990) indicate that “self-selection, random assignment, andcriterion-based selection are all possible” (p. 145) when forming small groups forlearning purposes. Also, the type of group dynamics taking place influences the typeof benefits the group members derive from the group. Each group member’s role,involving who is giving help, who is receiving help, whether help is given when21needed or not, are all important considerations. Also, there are relationships that existbetween gender and mathematics achievement. For example, boys tend to ask morespecific and direct questions than girls and this gender difference seems to be relatedto achievement (Webb, 1991). Webb also observes that there is higher maleachievement than female achievement within groups having higher male-female orfemale-male ratios, but for equal number of males and females, the achievement doesnot differ significantly. So, using the same number of males and females to formgroups can minimize gender differences in the achievement attained in the smallgroup setting. Also, Jungwirth (1991) observes that everything that happens in asocial interaction (student-student) is determined by the behavior of the individualparticipants and that mathematical competence on the part of students can beestablished during such interaction.Many educators recognize that social processes like peer regulation, feedback,support, and encouragement (Johnson & Johnson, 1985); kinds of help given orreceived by students (Webb, 1991); and group composition (Davidson & Kroll, 1991;Jungwirth, 1991; Yackel et al, 1991) influence mathematics learning. In fact, severalstudies (e. g., see Kroll, 1988; Webb, 1991; Yackel et al, 1991) show that studentsdemonstrate a better understanding of mathematics when they learn in small groups.Consequently, many educators use small groups to encourage mathematics learning.For example, Kamli (1984) observes that “when children confront the ideas of otherchildren for as brief as 10 minutes, higher levels of logical reasoning are often theoutcome” (p. 414). Also, Smith & Confrey (1991) found that peer interactions“enhance the development of logical reasoning through a process of cognitivereorganization induced by cognitive conflict” (p. 4).Theoretical Framework of the StudyIn this section, I discuss the constructivist’s perspective as a source ofknowledge generation. Then, I discuss Vygotsky’s perspectives on knowledge22generation through social interaction. Finally, I argue that constructivism andVygotsky’s perspectives on knowledge generation provide a useful theoreticalframework for investigating students’ small group discussions.ConstructivismThere are cognitive and methodological perspectives on constructivism, whichin simple terms, involve how the individual makes sense of things (Noddings, 1990).The methodological perspective assumes that human beings are knowing objects,they organize knowledge, and their behavior is purposive. The cognitive perspectiveassumes that individuals construct all knowledge. Of this cognitive perspective, oneschool of thought is that the instruments of construction are cognitive structures thatare innate while the other school believes that these cognitive structures aredevelopmental. Most constructivists in mathematics education hold the developmentalview (Noddings, 1990). Whichever perspective one holds, constructivists assert that itis the individual who has ownership of the knowledge he or she uses (Golding, 1990;Kamii and Kamii, 1990; Lerman, 1989; Lythcott & Duschl, 1990; Nodding, 1990; vonGlasersfeld, 1990). The individual’s knowledge is used to build interpretiveframeworks for making sense of the world (Schoenfeld, 1987).For example, when Yackel et al. (1990) asked second graders to solve9 + 11 = — in ways that made sense to them, the children offered varying solutionmethods that indicate different interpretive frameworks as illustrated below:Brenda: 9 and 9 is 18, plus 2 is 20.Adam: 7 and 7 is 14, so 8 and 8 is 16. 9 and 9 would be 18 so 9 +11 must equal 20Chris: 11 andil equals22. lOandil equals2l. 9andllequals 20.Jane: 11 and 9 more—12, 13, ..., 18, 19, 20. (p. 13).23Although the students in this study got the same solution as that expected bythe investigators, students’ interpretive frameworks do not always lead them to theexpected solution. For example, in a study by Ginsburg and Kaplan cited by Rowan etal. (1989), a first grader was presented with the numerical sentence 7 + 6 = 13 andasked if it was true; he responded positively. When he was presented with 13 = 7 + 6and then asked the same question, he responded that it was not true, and for it to betrue it must be changed to 7 + 6 = 13. Further probing showed that his notion of anumerical sentence being true is that the sentence must have two numbers joined byan operation on the left side of the equal sign and a single number to the right of theequal sign. These examples illustrate that students make use of different interpretiveframeworks in making sense of problems. Educators therefore need an assessmentframework that can capture how students, as autonomous individuals, make sense ofproblems and their solutions (Dreyfus, Jungwirth & Eliovitch, 1990; Vosnianou &Brewer, 1987).By autonomy, I mean the ability of the individual to make decisions on his or herown. The importance of autonomy as an aim of education has been recognized bymany educators (Wesson, 1986; Wenden, 1988; Weinstein, 1987; Lane & Lane, 1986;Haydon, 1983; Kamii, 1984). Haydon (1983) refers to autonomy as “some set ofqualities of mind and character which persons can in principle have despite externalconstraints” (p. 220). She posits further that autonomy is a matter of degree. Personsare more or less autonomous to different degrees in different aspects of their lives.Implied in this ability to govern and make decisions for oneself is the concept ofunderstanding. To make informed decisions while solving a problem, one needs tounderstand the problem and its context. Webb (1991) observes that: “The bestindication of students’ understanding is their ability to solve the problem on their own”(p. 369). However, students may get the answer (a product) to a problem withoutproviding any indication as to how (a process) they arrived at that answer. It willtherefore be inappropriate, I believe, to pass judgment on their understanding of24mathematics by relying solely on the answer to the problem. What educators need isan idea of what constitutes understanding and Pine (1988) shares a similar view byobserving that although one cannot fully comprehend the term “understanding” itself,mathematics educators must first have a viable model of understanding on which tomake sense of students’ work.There have been attempts to theoretically identify different kinds ofunderstanding (Pine & Kieren, 1989) to elucidate what it means to understandmathematics. For example, Skemp (1978) distinguishes between instrumental andrelational understanding, Buxton (1978) talks about formal or logical understanding,while Backhouse (1978) elaborates on symbolic understanding. Herscovics andBergeron (cited in Pine, 1988), in their attempt to categorize understanding, define fourlevels of understanding- intuitive understanding, initial conceptualization (procedural),abstraction (logico-physical), and formalization.It is important to note, however, that single categories do not describeunderstanding well, nor do such categories capture understanding as a process ratherthan as a single acquisition. Ohlsson (1988), Herscovics and Bergeron (1988), VonGlasersfeld (1987), and Pine & Kieren (1989), have made attempts at examiningunderstanding as a process. The view for considering understanding as a process isthat for all knowledge, as educators move nearer their goal, the goal itself recedes andsuccessive models that they create can be no more than approximations. So, inassessing students’ understanding, as in the present study, educators must be awarethat what they are assessing may only be approximations. Nevertheless, educatorscan make more prudent decisions if they consider all relevant factors that influencestudents’ mathematical understanding and use means that will permit gathering of avariety of information on such understanding.One such relevant factor which influences students’ mathematicalunderstanding is the social configuration of the mathematics classroom. Educatorsrecognize that social factors and their interrelationships influence students’25mathematical understanding when they work in groups (Artzt & Newman, 1991;Davidson, 1990; NCTM, 1989). However, educators observe also that individualconstruction does not have to conflict with concepts of social cognition (Smith &Confrey, 1991). Therefore, in dealing with the individual’s construction ofmathematical knowledge, the social context within which the knowledge constructiontakes places should be considered as important. The issue of social influence onmathematical understanding is explained by Vygotsky’s ideas on how socialinteraction between the student and the teacher (or more capable peers) can lead tohigher levels of generalization for the student. I address this issue next.Knowledge Through Social InteractionBasically, this perspective assumes that in order for an outsider to understandthe individual, one must first understand the social relations in which the individualexists (Vygotsky, 1978; Wertsch, 1985). Vygotsky argues that, internal processes(intrapsychologicat e. g, understanding) that result when an individual constructsknowledge, are related to external or social processes (interpsychological ; e. g.,competence) that result from the dynamics of the classroom setting. Also, a student onhis or her own, might be able to solve a problem only to a certain level without beingable to continue. But with adult guidance or collaboration with more capable peers,this same student might be able to solve the problem beyond the initial level. Thedifference in these two levels is what Vygotsky calls the zone of proximal developmentThe zone of proximal development defines cognitive functions that are yet tomature in the student and Vygotsky (1978) calls these functions “buds” or “flowers” ofdevelopment. The relevance of this zone of proximal development to assessingstudents’ understanding in mathematics is that it permits educators to “delineate thechild’s immediate future and his dynamic developmental state, allowing not only forwhat already has been achieved developmentally, but also for what is in the course ofmaturing” (Vygotsky, 1978, p. 87). Having knowledge of what is in the course of26maturing, I believe, should help in the selection of mathematical activities that willpromote the maturation. And having students discuss their solutions to mathematicalproblems should provide an opportunity for educators to gain an insight into what is inthe course of maturing.Complementarity of PerspectivesTo initiate and sustain verbal interactions among students, some form ofdiscourse is necessary. This discourse includes the way ideas are exchanged andwhat those ideas entail (NCTM, 1991). Throughout the discourse, the individual’sways of making sense of things (Golding, 1990; Kamii and Kamii, 1990; Lerman, 1989;Nodding, 1990; von Glasersfeld, 1990), are influenced by the social interaction thathelps the individual to make sense of those things (Cobb, 1989; Cobb et al, 1990;Smith & Confrey, 1991; Vygotsky, 1978; and Wertsch, 1985). Accordingly, whereindividual students interact to discuss their solutions to mathematical problems, Ibelieve it is important to consider both the individual’s ways of making sense of theproblems and the social interaction among the students which contributes to thegeneration of knowledge. Thus, the knowledge of constructivism and Vygotsky’s ideason knowledge generation, can provide educators with a useful theoretical frameworkfor assessing students’ demonstration of mathematical power in small group contexts.SummaryIn this chapter, I presented a brief summary of events leading to the currentreform in mathematics education. Then, while I discussed the current reform inrelation to the curriculum, teaching, and evaluation standards suggested by the NCTM,I emphasized the importance of the NCTM recommendation that small groups be usedto promote construction of mathematical knowledge. In addition, I reviewed smallgroup work and identified the need to investigate small group work to provideinformation on students’ mathematical power. Finally, I discussed a theoreticalframework within which to conduct small group investigations.27CHAPTER 3METHODIn this chapter, I describe how I conducted the study. To contextualize the study,I discuss my background, the teacher’s background, the school and the class thatparticipated in the study. Then, I detail the procedures used for collecting andanalyzing data.Context of the StudyMy BackgroundThroughout my school life in Ghana, I scored well on classroom tests andexternal examinations. Furthermore, as early as primary five (grade five in theCanadian school system), my good academic standing gave me the opportunity tolead small group discussions that the classroom teacher encouraged. After eachassignment, the teacher would put students into groups led by peers who did well inthose assignments. As leader, I gave explanations to students who could not do theassignments properly on their own. Then as a student, I realized from these groupdiscussions that I understood the assignments better after I tried explaining to thegroup members how I did those assignments.My participation in these group discussions influenced my learning style for therest of my school life. I always formed study groups and I was eager to contributemeaningfully to the group discussions. I studied hard privately and searched for newideas or ways of solving problems that I used in giving direction to the groupdiscussions. I also adopted the small group format when I was a mathematics teacherin the secondary school for nine years, and I used the same format when I taughtmathematics for three years at the Federal College of Education, Katsina, in Nigeria.28When I came to Canada, I realized that educators here place a lot of emphasison individual learning and instruction, at the expense of group learning andinstruction. My interest in the use of small group format for learning and teachingmathematics was rekindled when I read the Curriculum and Evaluation Standards forSchool Mathematics (N CTM, 1989), Professional Standards for Teaching Mathematics(NCTM, 1991), and Enabling Learners: The Year 2000 Document (British ColumbiaMinistry of Education, 1990) and realized these documents recommend the use of asmall group format for teaching and learning mathematics. When I had the opportunityto teach mathematics at the University of British Columbia during the summer of 1993and 1994, I once again used the small group format. Informed by my personalexperiences and success with this format, I decided to formally investigate students’small group discussions of mathematical activities.Teacher’s BackgroundTo investigate students’ discussions of mathematical activities while they workin small groups, I looked for a teacher who believed in and used a small group formatfor teaching and learning of mathematics. I met Ms Joanne Stansfield (pseudonym) ata British Columbia Association of Mathematics Teachers (BCAMT) conference held atthe University of British Columbia (UBC). Joanne had obtained a bachelor’s andmaster’s degrees in mathematics and started classroom teaching in 1978.Before I met Joanne, she had participated in several BCAMT workshops. Inparticular, a workshop on alternative teaching styles with a focus on cooperativelearning caught her interest and she has since then adopted cooperative teachingstrategies (Artzt & Newman, 1990; Davidson, 1990; NCTM, 1991; Yackel et al, 1990;Yackel, Cobb, & Wood, 1991) in her mathematics classroom. Her experience withusing cooperative teaching strategies suggested we would not disagree onorganization.29Joanne had just moved to a new school when I approached her to undertakethe study in her class. Joanne’s experience (and mine also) was that if norms ofcooperation are not firmly laid, students soon resorted to their individual learningstyles even though they were grouped together and expected to be workingcooperatively. So, she always tried to encourage students to observe the norms ofworking cooperatively (see Appendix C). Joanne transferred to another school whileshe was collaborating with me to lay a firm foundation for using cooperative learningstrategies in the class, so the study was abandoned for a while.There was a time lapse of about 6 months before Joanne started teaching in hernew school. When I asked whether I could continue my work with her, she readilyaccepted. I was curious to find out why she was still willing to work with me and shesaid that “working with other people, especially those researching, helps me keepabreast with current issues.” So, I followed Joanne to her latest school to continuewith my study. Here also, we had to collaborate to lay a firm foundation for usingcooperative learning strategies in the class. However, our experience workingtogether in the first school helped us to develop some trust for each other. Forexample, the principal of her latest school was hesitant in giving approval for the studyto take place because at the time, there were incidents of student molestation byintruders to the school. Joanne convinced the principal of the school that she hadworked with me before and I proved to be very responsible dealing with students. Theprincipal then gave approval for the study to take place in the school.The SchoolThe school structure is old and large and serves a population of about 1800students. There are three programs for students, the summit program, the flexprogram, and the regular program. According to Joanne, students in the summitprogram are those that teachers consider to be high achievers, those in the flexprogram are considered by teachers as highly motivated, and those in the regular30program are those remaining after the other two programs are full. The high achieversare those students who are very successful at completing their assignments and whoget high marks, while the highly motivated are those students that show eagerness tolearn but do not necessarily obtain very high marks. All other students are put in theregular program, including those who failed the previous grade but had to go on to thenext grade because space limitations do not permit students to repeat grades.The ParticiDatinçi ClassThe grade 9 students who participated in the study were in the regular program.There were 18 of them, 10 girls and 8 boys. According to Joanne, none of the studentshad failed grade 8 and some of them could have been in either the summit or flexprograms except for the fact that the two programs were full. Nevertheless, Joannelikes teaching students in the regular program because of her belief that “thesestudents seem neglected but they also have to experience mathematics as somethingmeaningful and interesting.” Because these students had not experienced smallgroup work prior to this grade, Joanne and I used our previous experiences to formallyintroduce these students to cooperating and learning in small groups.For about four months (that is from the start of the school year) prior to datacollection for my study, Joanne used the small group (or the cooperative) format for hermathematics lessons. She followed the mathematics topics as laid down in thecurriculum guide. She introduced the students to some guidelines (see Appendix C)for working together in groups and I assisted the students by responding to theirindividual and group difficulties as they adjusted to sharing their thoughts amongthemselves. (I visited Joanne’s class at least once every week during the fourmonths.) Some of the difficulties that arose include some of the students not wantingto share their ideas, some students dominating all the discussions, while at othertimes, some of the students would talk about other things not related to the topic underdiscussion. Sometimes when any one of the group members was found to be31disruptive or not contributing to the discussions, Joanne and I had to reallocate thatgroup member. Other times we had to encourage group members to be more tolerantof each other and their views as they discussed their solutions to problems provided inclass by Joanne. The mathematical problems were related to the topics taught andthey were taken from the mathematics textbook recommended for that grade. The aimat this stage was to get students used to discussing their solutions because, accordingto Joanne, it has not been their experience in previous classes to discuss theirsolutions to mathematical problems. No formal data were gathered at this stage.ProceduresSmall Grouo FormationFor this study, it was desirable to have group members who wouldcommunicate with each other and feel comfortable sharing their ideas together if Iwere to gather the sorts of information the group members generated as theydiscussed their solutions to the mathematical problems. Also, it was desirable to havegroup members who would validate their conjectures while others in the group try tomeaningfully criticize those conjectures. If a student was found not talking, that studentwould be replaced, since students’ talk was very vital for gathering information for thestudy. So, using Joanne’s rating of the students in terms of their ability tocommunicate together and with the familiarity I developed with the students byinteracting with them for about four months, I organized the 18 students in the classinto two groups of four and two groups of five.In order not to leave out any of the students while data were being gathered, allstudents in the class participated in the study. However, only one group of studentswas selected as the focus group for the study. Members of the focus group had stayedtogether during the four month period, prior to the data gathering, when the studentswere learning how to work in groups. It was the focus group’s discussions that were32analyzed to provide an answer to the research question. A combination of a student’smathematical ability as decided by performance in Joanne’s classes and the student’sability to talk with colleagues was used to decide the membership of the focus group.There were two high achievers (George and Jane) and two medium achievers(Paulina and Daniel) in the focus group. Another important consideration was tomaintain an equal number of males and females in the focus group so as to minimizedifferences in achievement that might be due to gender. The decision to balancemales and females in the focus group was informed by Webb’s (1991) observation thatfor equal number of males and females, achievement does not differ significantly whenstudents work in groups.Whenever there was the need to change the membership of the focus group,the following criteria were used: 1) mathematical ability, 2) ability to talk in a group,and 3) balancing of males and females. For example, when a female student had toleave the focus group to join the school’s basketball team, she was replaced byanother female of comparable mathematical ability and the ability to talk with the othermembers of the focus group. Again, when one male was found not to be talking duringthe discussions for the first three data gathering sessions, he was replaced by anothermale using the criteria listed above for deciding group membership.The ProblemsThe problems used for the study (see Appendix A) were developed by me ormodified from books or projects in which I participated. Although what constitutes aproblem varies for each student (Van de Walle, 1990), and that not all the problemscould provide information indicative of all the parts of SAS, what is important is that theproblem or task must have the “potential for students to engage in sound andsignificant mathematics as a part of accomplishing the task” (Lappan & Friel, 1993, p.525). Furthermore, the problem should provide the students the opportunity to havesomething to talk about. In that regard, I tried to use problem types with which the33students were familiar. While I tried to relate each problem to the topic Joanne wasteaching so as to make the concepts current for the students, I also provided thestudents with the opportunity to talk by asking them to explain their solutions or givereasons for solving the problems the way they did. During the study, the students didtalk most of the time.Data Collection TechniquesThere are several data gathering techniques that researchers use for qualitativestudies. These techniques include interviewing, participant observation, field notes,use of questionnaires, and video and audio taping (Borg & Gall, 1989; Fetterman,1989; Guba & Lincoln, 1991; Hammersley & Atkinson, 1991; Merriam, 1991; Patton,1987; Van Maanen, 1988). For this study, to gather information from the participants’perspectives (Hammersley & Atkinson, 1991; Patton, 1987), I video-recorded the focusgroup’s discussions of their solutions to the problems. The remaining groups’discussions were audio recorded. In addition, I collected all students’ writtenresponses to the problems. Furthermore, all students responded to questionnaireitems (see Appendix B)The study was conducted from December 10, 1992 - March 1, 1993. Therewere 7 data gathering sessions. (An impending teachers strike within the VancouverSchool District shortened my data gathering period. Nevertheless, the amount of datacollected proved sufficient for my research question.) For each data gatheringsession, the students attempted to solve the assigned problems individually within 20minutes and then later discussed the solutions they obtained with their groupmembers for 40 minutes. I urged the students to focus on explaining and givingjustifications for the solutions they obtained while they discussed their solutions.Occasionally, I gave students prompts either when they asked for help, or when I foundthey were stuck in their discussions. Except for the classroom teacher checking the34roll, sitting at her table, or occasionally moving around to ensure that the students wereon task, I was in complete control of the class during the data gathering stage.To ensure that all students in all the groups were on task, I also moved fromgroup to group (sometimes with Joanne) to listen to their discussions. However, Ispent most of the time listening to the focus group. I would stand at a distance fromwhere I could clearly hear the discussions of the focus group. When there was theneed for me to intervene (when students asked for help or I overheard a discussionthat required further explanation), I would quickly move to the focus group to do so. Irecorded into a notebook any other activities that I could not capture by the video orthe tape recorders. For example, I took note of students who had to leave during thediscussion stage for a basketball game. With about 10 minutes left in the discussions,all the students were to respond individually to the questionnaire items (see AppendixB) that I provided them. Occasionally, however, some of the students in all the groupsattempted discussing the questionnaire items col laboratively.PromptsAs stated in chapter 1, the main focus of the study was to gather evidence ofstudents’ mathematical power from student-student interactions within a small groupcontext. As such, prompting from the researcher was minimal. However, some of therealities of conducting classroom research involving students are that the studentsmight ask the researcher questions or the researcher may feel obligated to respond toactions of students.Thus, when students raised their hands or attracted my attention by banging onthe table for example, I approached the group to offer help. At times, the help was inthe form of prompts. That is, statements or questions that required them to explainwhat they did, how they did it, or why they did it the way they did. For example, whenstudents arrived at different answers after evaluating C = 1.80 + 0.75(m-1) for m = 5,my questioning (“So what do you do with the brackets?” and “So after the brackets35what did you do next?’) led students to explain their actions and provide reasons forthem.At other times, the help was in the form of “hints.” Hints were more prescriptiveand directed students toward a solution rather than prompts which question students’actions to encourage reflection. For instance, unlike above, a hint given to students forpart (b) of the same problem as above was “substitute $25.05 for ‘C’.” Studentsfollowed this advice and found a solution.Data AnalysisAll the focus group’s discussions were transcribed from the video tapes. Thetranscripts were then analyzed to provide answers to the research question. That is, Iorganized information from students’ small group discussions around SAS whichserved as key constructs (or events) (Fetterman, 1989; Guba & Lincoln, 1991;Hammersley & Atkinson, 1991; Merriam, 1991). For this study, the unit of analysis(Merriam, 1991; Yin, 1989) is the information students generated in 40 minutes, asthey discussed in small groups, their solutions to each of the mathematical problems.Any inferences or generalizations are not statistical, but rather analytical (Yin, 1989),and they are to “guide but not predict one’s actions” (Merriam, 1991, p. 176). Toprovide results that are “trustworthy”, efforts were made to ensure the “credibility” and“auditability” of the data and the results.Credibility and AuditabilityIssues of “trustworthiness” in qualitative studies include ensuring what Gubaand Lincoln (1991, p. 105) call credibility (validity) and auditability (reliability). Whilecredibility involves using multiple sources to confirm the data collected, auditabilityinvolves an outsider concurring that, given the data collected, the results make sense.However, Merriam (1991) argues that credibility and auditability are inextricably linked36and that ensuring credibility ensures auditability. So, for this study, I took several stepsto ensure the credibility of the data collected.For example, to ensure that all what the students said was properly transcribed,each video tape and transcript were re-examined together to further determine the“compatibility of the trancript” with the conversation it represented. Also, I listened tothe audio recordings of the other three groups of students in the class and found thediscussions similar to those of the focus group in terms of what the students weresaying. Furthermore, some of the responses to the questionnaire items corroboratedwhat the students in the focus group said and wrote down during the discussions. Thevideo recordings were given to my research advisor and Joanne (the classroomteacher) to view and they read my interpretations of the data. Both agreed with most ofmy interpretation of the data. In fact, Joanne even went through all the transcripts andshe agreed the transcriptions portrayed correctly her recol lection,students’ discussionsduring the study.Coding of TranscriDtsI went through the full transcript of each problem and coded portions of thediscussions as Cl, C2, ...; MP1, MP2, ...; MCi, MC2, ...; PSI, PS2, ...; and MD1,MD2,...; which represent categories of mathematical activities associated with SAS(see p. 16 - 19). Portions of the discussions involve some or all members of the group.For a particular problem, after coding portions of the discussions related to thatproblem, I sorted together all Ci, C2, ..., MP1, MP2,..., MCi, MC2, ..., and so on.(See Appendix D for the distribution of excerpts related to categories of mathematicalactivities associated with SAS.) I selected excerpts from these coded portions of thediscussions as examples that represent each category of mathematical activity. Forexample, in PS4, “PS” refers to “problem solving” while “4” refers to an excerpt thatreflects the “fourth” category of students’ mathematical activities listed under problemsolving, that is verify and interpret results. Then, from all portions of the discussions37coded PS4, I selected one excerpt that, in my judgment, best illustrates students’ability to verify and interpret results using the NCTM definition. Several examplesprovided by the NCTM (NCTM, 1989, 1991) of what constitutes students’ ability toverify and interpret results guided me in the selection of the excerpts. Transcripts forthe other problems were treated similarly.lnternretation of DataA combination of criteria was used to decide the extent to which informationfrom students’ group discussions are indicative of students’ mathematical power. Idescribed and interpreted the excerpts to provide insights which are related to SAS.For example, for an excerpt illustrating students’ ability to verify and interpret results, Iwould indicate whether the students checked that the answers they got for solving aproblem satisfied the initial conditions of that problem and whether they were able tomake sense of the answer to the problem. Also, I described the union of excerptsrelated to SAS during the 40 minutes that the discussions took place for each problemso as to provide a holistic picture of how students were integrating SAS. That is, foreach problem, I described the extent to which each SAS used for the study wasreflected in the discussions involving that problem. The extent to which studentsdemonstrate mathematical power is then provided in terms of the interpretations of theexcerpts relating to SAS and the union of those excerpts. What is important here is toprovide a holistic picture of students’ demonstration of mathematical power withinproblems and across problems.I also documented any other insights related to the study that I perceived to beimportant while I analyzed the transcripts. Whenever possible, the responses to thequestionnaire items by the focus group were used to collaborate what members of thegroup said during the discussions, or to provide group members’ opinions on theirparticipation in the study.38SummaryIn this chapter, I provided the context of the study by describing my background,the teachers background, the school in which I conducted the study, and theparticipating class for the study. Before gathering data, the students were exposed to4 months of teaching and learning using the small group format. There was a focusgroup of 4 students for the study, 2 males and 2 females. After trying to solvemathematical problems individually for about 20 minutes, they got together to discusstheir solutions. The discussions were video recorded. The transcripts of thediscussions were analyzed by providing evidence indicative of students’ ability tocommunicate mathematically, use mathematical concepts and procedures, usemathematics to solve problems, their disposition towards mathematics, and theirdemonstration of mathematical power.39CHAPTER 4RESULTS AND DISCUSSIONSIn this chapter, I present and discuss the results of the study in relation to theresearch question. The unit of analysis for the study is the information studentsgenerated in 40 minutes while they discussed, in a small group, their solutions to themathematical problems given them. The analysis is not to provide a synopsis or aprofile of any particular member of the group, but it is to provide an interpretation of thegroup discussions which consist of individual contributions. The main focus of thestudy was on the information students generated on their own during the discussions,but occasionally, when the students asked for help or were stuck in their discussions, Iexamined information they generated as responses to my prompts. From students’small group discussions, I provide examples of excerpts that reflect students’ ability tocommunicate mathematically, use mathematical concepts and procedures, and usemathematics to solve problems. Also, I provide examples of excerpts that are reflectiveof students’ mathematical disposition.Excerpts of students’ discussions that are used to provide evidence of theirabilities are presented verbatim. Comments in brackets“[ ]“ following students’discussions are my comments to highlight what was happening at the time. Students’written responses to the problems and their written responses to the questionnaireitems are used, whenever possible, to provide additional information that illuminatesthe answers to the research question. Also, I document any other insights I perceive tobe important while I analyzed the transcripts. It is important to note that students mightnot necessarily be aware that they demonstrated any particular abilities. Rather, it ismy interpretation of their discussions and the criteria provided in chapter three thatpermit me to make inferences about students’ mathematical power. Theinterpretations are guided by the definitions of SAS as provided by the NCTM (NCTM,401989). To maintain anonymity, pseudonyms are used for the classroom teacher andthe students.Research questionTo what extent is information from students’ small group discussions of theirsolutions to mathematical problems indicative of students’ mathematical power?By information, I mean mathematical information that is related to the problemsthe students have solved. This mathematical information from the discussions isrepresented by excerpts that are indicative of students’ ability to communicatemathematically and use mathematical concepts and procedures. Also, this informationis represented by excerpts that are indicative of students’ use of mathematics to solveproblems and their disposition towards mathematics. Furthermore, the existence andintegration of information indicative of SAS provide evidence of students’demonstration of mathematical power.CommunicationStudents’ ability to communicate mathematically was reflected by all threecategories of mathematical activities associated with mathematical communication.However, it was difficult to get any one excerpt to simultaneously reflect all themathematical activities associated with mathematical communication. One category ofmathematical activities where students demonstrated the ability to communicatemathematically involves expressing mathematical ideas by speaking, writing,demonstrating, and depicting them visually. There were 13 excerpts from thediscussions of all the seven problems suggesting that students in the group expressedmathematical ideas by speaking, writing, demonstrating, and depicting them visually;two excerpts from discussions involving problem 1, three excerpts involving problem 2,four excerpts involving problem 4, and one excerpt each involving problems 3, 5, 6,41and 7. Even here, while some of the excerpts involve speaking, others involve writing,demonstrating, or depicting mathematical ideas visually.Figure 4.01, for example, illustrates students’ attempts at expressingmathematical ideas by speaking. Students were speaking about the cost in dollars fortalking on the phone (problem 1, Appendix A). While Paulina was talking about part (i)of the problem involving how much to pay for speaking for five minutes, Daniel wastalking about part (ii) of the problem involving how much time is involved in $25.05 forspeaking. What is important here is that Paulina and Daniel were expressing themathematical idea of rate, in this case the cost in dollars per minute (or how manyminutes for a dollar). Even though Jane did not speak about rate, it is possible fromher response “I think it is true...” to Paulina’s question “Five dollars in ten minutes?1’that she was also referring to five dollars in ten minutes. The solutions of Paulina orDaniel may be wrong (as it turned out in this case), and all three of them may not knowthey were talking about rate, but the important thing is that they were talking about amathematical idea and that they provided an opportunity to whoever was listening tohelp them develop an understanding for rate as used in mathematics.Paulina: Five dollars in ten minutes?Jane: I think it is true...Daniel: Look, look, it’s ten minutes for twenty five bucks.Figure 4.01: December 10th, 1993The excerpt in figure 4.02 shows students speaking about perimeter and areaas they try to differentiate between the two and indicate how to find them. They spokeof perimeter as involving adding of sides and area as involving the multiplication ofsides. The way the students spoke about perimeter and area suggests a proceduralunderstanding rather than a conceptual understanding of these concepts. The excerpt42was from discussions of students’ solutions to problem 4 involving the lottery game(Appendix A).Jane: Okay, perimeter is adding all these sides and area is multiplying,right?Daniel: Goon!Paulina: No, perimeter...Jane: Perimeter is add, area is multiply.Daniel: Don’t get it? Go on!Paulina: Oh!Jane: This gives you one area, so you multiply these...Figure 4.02: February 10th, 1993The excerpt in figure 4.03 shows that while students discussed how they arrivedat their solutions to problem 7 involving a hockey game (Appendix A), they spokeabout the ideas of 1) guess and check, 2) using charts (systematic charting) to arrive atthe solution, and 3) the boundary conditions of the problem. Quincy’s affirmativeresponse to Jane’s question “Did you use the same charts?” indicates that he usedcharts to solve the problem. Paulina’s comment “Well, 10 games, 2 goals.. .20 goals in10 games, 50 goals and then 14 games left, right?” illustrates her awareness of theconditions to be satisfied in order to solve the problem. When Jane asked “So, inother words, you just kept on going?”, apparently she was, within the context of thediscussions, referring to Paulina continuously guessing and checking her solutions,while using the boundary conditions set up in the problem to guide her guesses.Daniel said he did not use guess and check to solve the problem but then talked about“4 times 6, 4 times 7, .. .“, a guess and check approach, which his script shows as theway he solved the problem. So, just listening to what was said might not be enough43for deciding what was done. Even though guess and check was used to solve thisproblem, it was hard to tell if the students saw guess and check as an authenticprocedure for solving the problem, from discourse alone.Jane: How did you do yours?Paulina: Well, 10 games, 2 goals.. .20 goals in 10 games, 50 goals and then14 games left, right? Then 14 games 50 goals and then I went 4 times 12equals 48, 3 doesn’t go to 50...[48 plus 3 doesn’t give 50].So 4 times 11... 44... it adds up to 50 but its not 14 numbers.So I went on with 4 times 10 , 4 times 9...Jane: So, in other words, you just kept on going?Paulina: I didn’t do systematic chart here...Jane: Did you use the same charts? [directing the question to Quincy].Quincy: Yes.Jane: Did you use the same charts? [directing the question to Daniel).Daniel: I just took...Jane: Guessed, right?Daniel: No. 4 times 6, 4 times 7,Quincy: That’s what we did.. .and got the answer.Figure 4.03: March 1st, 1993While responding to item 1 of the questionnaire where students were to indicatethe ideas they used in solving the problems, the most common ideas students referredto are addition and multiplication. This might be due to the fact that most of theproblems could be solved using addition and multiplication. However, there wereinstances when students identified some other ideas like factoring in solving problem5 and substitution in solving problem 2. Students might have used the idea of44factoring while discussing their solutions to problem 5 since factorize was used whileposing the problem, but the idea of substitution, at least for discussions involvingproblem 2, was first used by the students themselves. Also, the idea of systematiccharting was first used by the students in solving problem 7. Thus, for this study,students used some of these mathematical ideas without a direct influence from theresearcher.Checking students’ scripts showed that while students as a group spoke ofmathematical ideas, they sometimes wrote about them, demonstrated them, anddepicted them visually as individuals. A visual demonstration and depiction ofsystematic charting used in the discussion of problem 7 is illustrated in figure 4.04. Itcan also be inferred that a “systematic” guess and check was taking place. This isbecause, while maintaining 14 games as the number of games left to be played, thenumber of 3-goal games was systematically increased from one to 13, while thenumber of 4-goal games was systematically reduced from 13 to one. The total numberof goals for each 3-goal and 4-goal combinations was found and compared with therequired total of 50 goals. Once again, listening to students’ ideas and observing theirwritten scripts provided insights into how students were using those ideas.ry &iI c d/ 3 jq ,.3J ifdid /, 50. 5 i (ji dvtLZ>C4I q90d .. 1 liii II IiZzLrirT5Y.iiiq[5 1Figure 4.04: Quincy, March 1st, 1993.45For students to demonstrate the ability to communicate mathematically, theyshould also provide evidence that they can understand, interpret, and evaluatemathematical ideas that are presented in written, oral, or visual forms. In this study, themathematical ideas that students dealt with were presented by the researcher inwritten and visual forms. There were four excerpts from the discussions involvingproblem I and at least one excerpt involving each of the remaining problems, whenstudents demonstrated understanding and interpreted mathematical ideas presentedto them in written and visual forms. For example, figure 4.05 below is an illustration ofstudents’ understanding and interpretation of what it means to complete amultiplication table and find out the sum of the entries in the table (problem 2,Appendix A). Paulina’s comment “You can’t use those at the edge,” and Jane’scomment “Don’t add these.. .Add only those in the middle” go to show theirunderstanding and interpretation of what it means to find out the sum of entriesobtained from a multiplication table.Daniel was having some difficulty finding the sum of the entries, but withcomments from Jane and Paulina, he was able to successfully find the sum of entriesusing the multiplication table. Daniel’s success reflects what Vygotsky calls the “zoneof proximal development” in that with the help of the “more knowledgeable peers”Jane and Paulina, Daniel was able to achieve beyond his own “level of development.”Notice that even though it might seem that students were only making entries into amultiplication table, what was actually being tested was their knowledge andunderstanding of indices and how to manipulate them.46Paulina: So how much money did you get without the $50?Daniel: How much?Paulina: Yes.Daniel: 100, I don’t know.Jane: You had 99, right?Paulina: Let me see [collecting Daniel’s script].Daniel: You add up all the numbers, m and n.Paulina: Yeah.. .How did you get...?Daniel: You add up all the numbers, m and n.Paulina: Yes, but it’s supposed to be 99 if you add them up.Daniel: All of it?Paulina: Yeah, how did you get...? add again.. .try again. Oh Jane, he [Daniel].doesn’t know how to add yet!Jane: Oh, he adds these ones at the edge [Apparently Daniel was also addingthe column and row numbers to be multiplied].Paulina: You can’t use those at the edge.Jane: Don’t add these.. .Add only those in the middle. You add these...Figure 4.05: January 18th, 1993.Additional evidence of students’ understanding and interpretation of what itmeans to complete a multiplication table and find out the sum of the entries wasprovided by their written responses to problem 2. Figure 4.06 is an illustration of thisevidence from Jane’s script. Jane (as well as other students in the group) correctlysubstituted the values of m = 2 and n = 3 to find the amount of money to be won.47Is,q/Figure 4.06: Jane, January 18th, 1993.However, when the students were to simplify the sum of the entries first (for anadditional $50.00) before finding out the amount to be won, all of them could not dothe simplification properly, even though they recognized that the total amount to bewon if the simplification was properly done would be $149.00 (99 + 50). In this case,even though the students showed a clear conception of what the solution to theproblem would be, they had difficulty in manipulating the algebraic expressiongenerated by the sum of the entries in a manner that will preserve the meaning of theaddition of algebraic terms and the meaning of indices and then lead them to thesolution to this part of the problem. Apparently, lack of procedural knowledgehindered the solution of this part of the problem.Finally, for students to be communicating mathematically, they should provideevidence of the use of mathematical vocabulary, notation, and structure to representideas, describe relationships, and model situations. There were 15 excerpts from thediscussions indicative of such ability; one excerpt for solving problem 2, three excerpts48each for solving problems 3 and 7, and four excerpts each for solving problems 4 and6. Discussions of problems 1 and 5 did not provide any such evidence and there wasno evidence of students’ modeling of situations. An example of students’ use ofmathematical vocabulary to represent mathematical ideas and to describemathematical relationships is illustrated by figure 4.07. While the students werediscussing their solutions to problem 6 (Appendix A) where students were to decidethe larger of two algebraic expressions, they used “invert and multiply” and “reciprocal”which are mathematical vocabularies that represent mathematical ideas and describemathematical relationships. From students’ written responses it was clear that theyalso used mathematical vocabulary such as “decimals”, “combinations”, “exhaustingpossibilities”, and “variables”, all of which represent mathematical ideas and describemathematical relationships.Quincy: Now you put inverted multiply, you are supposed to put “invertand multiply.”Shawna: Is that the reciprocal thing?Paulina: What?Quincy: Invert and multiply.Daniel Yeah...Figure 4.07: February 22nd, 1993.An example of students’ use of mathematical notations to representmathematical ideas and to describe mathematical relationships is illustrated by theexcerpt in figure 4.08. Students were discussing problem 4 (Appendix A) and talkingabout ratio and how to represent it. Jane orally described ratio correctly as “.. .This,two dots, and that.” Her script (see figure 4.09) showed that she could represent ratioalso with a “slash” instead of a “double dot” and when Daniel responded to Jane’s49question “Does it matter if you write it this way or that way?” by saying that “They are allthe right answer”, apparently, he was assuring Jane that both ways of representingratio are correct. So, in addition to seeing how students represent notations, werecognize that they also “debate” its appropriateness.Jane: So then if it is ratio, it will be like....Paulina Is this ratio? [Asking Daniel].Daniel: Yes...Jane: .. .This, two dots, and that? Does it matter if you write it this way or thatway? What do you think?Daniel: They are all the right answer.Figure 4.08: February 10th, 1993An examination of Jane’s script (figure 4.09) shows that “this” was referring toone part of the ratio, “two dots” was referring to the symbol for ratio, and “that” wasreferring to the other part of the ratio.:f-&- Ji 4-+10f43o4- ,/Figure 4.09: Jane, February 10th, 1993.50There was an indication that the students continued to communicatemathematically when I used prompts to help them clarify some of their thinking andrefocus their discussions. For example, when students were having difficulty using theequation C = 1.80 + 0.75(m - 1) to figure out how much to pay for speaking for 5minutes (problem 1, Appendix A), the following discussion ensued (figure 4.10). Jane,Paulina, and Daniel added 1.80 and 0.75 first before multiplying the result by 4 (m - 1).They knew they had to evaluate (do the brackets) first what was in the brackets, m -1,but did not know they had to multiply the result of m -1 by 0.75 because, as put byPaulina, there should have been a bracket around 0.75. When I asked them what0.75(m -1) standing alone (without 1.80) would mean, they were quick to recognizethat they would have to multiply them together. Eventually, they solved this part of theproblem.Discussions involving the prompts I gave the students showed that theycontinued to express mathematical ideas through speaking. For example, they spokeof “do the bracket” as meaning evaluating what is within the brackets. They alsointerpreted mathematical ideas when they recognized that having brackets around m1 in the equation means they had to evaluate that first Also, the students usedmathematical vocabulary to express mathematical relationships when they referred toC = 1.80 + 0.75(m- 1) as an equation.So, students did not only continue to communicate mathematically when givenprompts, they also provided information on how they made sense of mathematicalideas. “Because of the.. .equation”, Jane, Daniel, and Paulina performed theoperations in the equation in the order in which they were presented, even thoughthey recognized that they had to evaluate m -1 before using it in the equation. Later,Daniel recognized that he had to first multiply four (from m -1 where m = 5) by 0.75before adding 1.80. Also, Paulina’s question “Why don’t we have the brackets aroundthe seventy five [0.75] like that.. .[showing it].. .and then.. .“ suggests that it is only whenthe equation is of the form C = 1.80 + (0. 75)(m - 1) would she have multiplied 0.75 with51m - 1 before adding on 1.80. Apparently, there is a lack of procedural knowledge ofhow to manipulate the equation so as to preserve the meaning of the equation.Furthermore, the use of prompts provided an opportunity for students to move beyondtheir own “level of development” to solve the problem.Sitsofe: Let’s get back to what you did.. .let’s look at it again.. .You said youadded this [1.80] to what?Paulina: Seventy five cents.Sitsofe: Why?.. .why did you do that?Paulina: Because of the .. .equation [whispered by Jane]Sitsofe: Is that what it is?.. .What do you have here? [pointing to the bracket partof the equation].Paulina: Brackets.Sitsofe: So what do you do with the brackets?Jane: You do [emphasis mine] the brackets...Paulina: But I did the brackets first...Sitsofe: Did you do the brackets first? [using their language].Paulina: Yeah!Sitsofe: So after the brackets, what did you do next?Paulina: That might be...Daniel: Four minutes times this [referring to 0.75].. .you do that one.Jane : Do you do that one first? Is that it?Sitsofe: You have to tell me.Daniel: Yes, yes!...Sitsofe: You have a reason for adding this one to this one first? [to Paulina].Paulina: Why don’t we have the brackets around the seventy five [0.75] like that.[showing it].. .and then...Sitsofe: What difference will it make to have the bracket around the 0.75?52Paulina: You know that you can multiply these two...Jane : By the way.. .then you add the one eighty [1.80].. .[Jane attempting tocomplete Paulina’s statement].Sitsofe: So you mean without the brackets around these, you wouldn’t know thatyou got to multiply this [0.75] by this Im-li?Paulina: Add these two [1 80+0.75] first, then you multiply by 4 [rn-i].Sitsofe: What about if you have something like this form? [2(5 -3)] If you want totake something out of the bracket, what do you do to the brackets? Whatdo the brackets mean?Paulina: You do this one first [referring to rn-i] then you multiply by this one[referring to 0.75].Paulina: Multiplication first.Sitsofe: So what does this mean [0.75(m-1)]? If you take out the brackets?.. .the0.75...?Paulina: Oh, now I got it.. .you have multiple.. .so you go in order.. .you domultiplication.Sitsofe: So what is the order now? Can you now go ahead?Paulina: Yeah, you understand, George?George: Yeah!Figure 4.10: December 10th, 1992Summary of results pertaining to communicationStudents’ ability to communicate mathematically within the group was reflectedby all three of NCTM’s categories of mathematical activities associated with suchability. Students’ discussions provided insights into how the students made sense ofmathematical ideas and debated the appropriateness of those ideas. Even thoughstudents were communicating mathematically, it was sometimes difficult to tell if the53students understood the ideas they were talking about. Also, combining students’written scripts with their discussions provided further insights into students’ groupdiscussions. Finally, prompts related to the discussions helped students to clarify theirthinking and they continued to communicate mathematically while responding to theprompts.Mathematical ConceotsAn indication of students’ ability to use mathematical concepts is that they canlabel, verbalize, and define concepts. Four excerpts were identified during thediscussions, 1 each while solving problems 2 and 5, and 2 instances while solvingproblem 2, when the students attempted either labeling, verbalizing, or definingconcepts. For example, in figure 4.11 below, Jane sought the reaction of othermembers in the group as she verbalized and defined area and perimeter in her ownwords. Daniel urged her to go on, apparently not disagreeing so far with what she wassaying. However, when Paulina’s comment of “No, perimeter.. .“ suggested adisagreement, Jane became emphatic and said “Perimeter is add, area is multiply.”Again, the group situation seemed to help Jane to clarify her thinking and becomemore assertive when she was challenged by Paulina regarding what perimeter andarea mean to Jane. Also, Jane’s verbalization of perimeter and area suggests aprocedural rather than a conceptual understanding of these mathematical ideas.Jane: Okay, perimeter is adding all these sides and area is multiplying, right?Daniel: Go on! [He was turning over a page of his script].Paulina: No, perimeter...Jane: Perimeter is add, area is multiply.Figure 4.11: February 10th, 199354Another indicator of students’ ability to use mathematical concepts is that theycan identify and generate examples and nonexamples. However, there was hardlyany evidence during the discussions that students identified and generated examplesor nonexamples. The study was to investigate the group discussions and identifyinformation related to SAS, given the problems whose solutions were to be discussedby the students. Notice that the students were not asked to specifically identify andgenerate examples and nonexamples. One might argue then that one should notexpect responses to questions not posed. However, that would mean not permittingand listening to what students can say during group discussions, even if notspecifically asked for. As shown in figure 4.12, it is possible (and desirable) to haveresponses to questions not directly posed as was the case when Paulina provided anexample of a concept using 2: 1 to represent ratio.Also, another indicator of students’ ability to use mathematical concepts is thatthey use models, diagrams and symbols to represent those concepts. Seven excerptswere identified throughout the discussions suggesting that students demonstrated thisability; one excerpt each while solving problems 1 and 7, two excerpts while solvingproblem 5, and 3 excerpts while solving problem 4. For example, Paulina used“double dot” as a symbol for ratio. While Jane was wondering if ratio is a fraction,Paulina suggested “No, ratio is two double dot one” [2: 1] as representing a ratio,apparently not recognizing her representation of ratio as also a possiblerepresentation for fraction. This part of the discussion is illustrated in figure 4.12 (seealso figure 4.09).55Paulina: How did you do that though?Jane: I don’t know how I can get the ratio.Paulina: Ratio?Jane Ratio is a fraction, right?Paulina: No ratio is....Jane: Because comparing...Paulina: .. .two double dot one.Figure 4.12: February 10th, 1993Another indicator of students’ ability to use mathematical concepts is that theyrecognize the various meanings and interpretations of concepts. Again, there werefew instances indicative of students’ ability in this regard throughout the discussionsand throughout the written responses to the problems and the questionnaire items. Apossible reason for the lack of students’ recognition of the various meanings andinterpretations of concepts, but which is hard to confirm from this study, is that oncestudents know a meaning of a concept, they might not want to search for othermeanings. One such instance is illustrated by the excerpt represented by figure 4.12.Here, one sees two different interpretations of “ratio” as a concept and the groupdiscussions provided the group members the opportunity to benefit from the twodifferent interpretations. However, in response to Jane, one realizes Paulina does notsee ratio as a fraction, but sees it as a representation of two numbers a and b in theform a : b.Students provided the same meaning of concepts using different words orexpressions, and as shown above (figure 4.1 2) they would not accept otherinterpretations. Figure 4.13 below illustrates how students used different words orphrases to describe reciprocal. Daniel referring to reciprocal or inverse multiply as56“Change around and multiply” and Quincy referring to the same concepts as “Threeover two divided by one over four becomes three over two multiplied by four over one”(figure 413) indicate understanding of these concepts. Except for Daniel whose scriptdid not indicate details of how he arrived at his solution, scripts of the other membersof the group reflect the meaning of the concepts as presented by Daniel and Quincy.Figure 4.14 illustrates how Paulina represented invert and multiply. In this case,similar information was obtained from the scripts as well as from the oral discourse.Shawna: Reciprocal inverse multiply.Daniel: Change around and multiply.Quincy: Three over two divided by one over four becomes three over twomultiplied by four over one...Figure 4.13: February 22nd, 19933/ 3 ± 1/ 21 xfiône -F +hm Iarq€rHhaj’rFigure 4.14: Paulina, February 22nd, 1993Finally, students should be able to translate any given concept from one modeof representation to another as an indication of their ability to use mathematicalconcepts. Students’ use of mathematical concepts should also be reflected by their57ability to identity the properties of a given concept and to recognize the conditions thatdetermine that concept It was difficult to identify excerpts from either the discussions,the written responses to the problems, or the written responses to the questionnaireitems indicating that students demonstrated this aspect of the ability to usemathematical concepts, except when Jane identified ratio as a fraction and having twoparts. Notice that even though students readily talked, wrote and drew, they weretranslating from one mode of presentation to another, rather than from one mode ofrepresentation to another.Giving prompts to students did not provide instances where studentsdemonstrated the ability to use mathematical concepts differently from the ability theyhad demonstrated on their own during the group discussions. The significant thing tonote however, is that giving of prompts apparently helped the students to clarify theirthinking and have a better conception of how to proceed in solving the problem. Forexample, in the equation C = 1.80 ÷ 0.75(m - 1) (problem 1, Appendix A), Paulina didnot know that she had to multiply the 0.75 by (rn-i) before adding the 1.80 becauseaccording to her, there was no bracket around 0.75. The prompts that followedapparently helped her to eventually solve this part of the problem successfully. Also,one striking situation when giving prompts provided some useful insight was students’continual use of “m two n” for “m squared n” even after prompts were given to correctthem (figure 4.15). This situation might suggest that compatibility (getting students todo things by convention), cannot be dictated. Students may adopt the conventionalway of doing things only if they find it helping them solve their problems.58Sitsofe: You normally call it “m two n”, right? That might confuse youat some stage. It is “m squared n.’Daniel: m four plus m two [when he was referring to m4 + m2].Paulina: Isn’t m two n and m two the same? [referring to m2n and m2 respectively].Figure 4.15: January 18th, 1993Summary of results oertaining to mathematical conceDtsStudents’ use of mathematical concepts was not generally wide spread. It wasreflected mainly by two of NCTM’s seven categories of mathematical activities that areassociated with the ability to use mathematical concepts. The two categories relate tothe ability to label, verbalize, and define concepts, and the ability to use models,diagrams and symbols to represent concepts. Use of students’ scripts in conjunctionwith the discussions provided additional insights into how students used mathematicalconcepts. Once again, the major impact of prompts was that they helped students toclarify their thinking while engaged in the group discussions and enabled the studentsto successfully solve the problems they were to solve, using mathematical concepts.Mathematical DroceduresAn indicator of students’ ability to use mathematical procedures is for them torecognize when a procedure is appropriate. Seven excerpts illustrating students’recognition of the appropriateness of a procedure were identified during thediscussions; one excerpt each involving problems 3, 4, 5 and 6, and three excerptsinvolving problem 7. Figure 4.16 provides evidence that while Paulina and Janerecognized “guessing” as an appropriate procedure for solving problem 7 (AppendixA), Quincy also recognized “systematic charting” as an appropriate procedure forsolving the same problem. Jane’s written script revealed that what she referred to as59“guessing” is actually “guess and check,” an appropriate mathematical procedure. Aportion of the script is illustrated in figure 4.17.Figure 4.16: March 1st, 1993)cZj,)AIC)Fiure 4.17: Jane, March 1st, 1993Another indicator of students’ ability to use mathematical procedures is whenthey give reasons for steps in a procedure and they reliably and efficiently executeprocedures. Only 2 excerpts were identified throughout the discussions, 1 eachinvolving problems 2 and 4, when students gave reasons for steps in a procedure.Figure 4.18 is an illustration of Paulina’s reason for selecting m3 first when shesimplified polynomials and arranged the results in descending order as she solvedPaulina:Jane:Paulina:Jane:Paulina:Quincy:So Jane, how did you do it?Well, I just guessed!That’s a good way of approaching it...I know.. .let’s go on...So, how did you do it? [asking Quincy].Systematic charting.;:2AJ OQQIDl41 5jj3I660problem 2 (Appendix A). All other students in the group agreed with her reason bynodding their heads. The other instance when a reason was given for steps taken in aprocedure is provided in figure 4.19. While the students were discussing how to findthe perimeter of the geometric figure in problem 4 (Appendix A), Paulina’s laststatement suggests that the reason she had to “add all these” (referring to the givenvalues of the sides of the geometric figure) was because she wanted to “get theperimeter.”Paulina: I chose m cubed first, because that has the highest degree.Jane: That’s right [others nodded their heads in agreement].Figure 4.18: January 18th, 1993Paulina: Four y plus eleven, isn’t that the perimeter?Jane: You got four y plus eleven, at least.Paulina: But you didn’t get that.. .[referring to Daniel].Daniel: I know how I got it [Laughing and apparently holding back someinformation from the group].George: Right?Paulina: I think this is right; you have to add all these to get the perimeter.Figure 4.19: February 10th, 1993Seven excerpts identified throughout the discussions illustrate when studentsreliably and efficiently executed procedures. There was one excerpt each involvingproblems 2, 5, and 7, and two excerpts each in solving problems 1 and 3. Figure 4.20illustrates how students reliably and efficiently used factorization and exhaustion of61possibilities to solve problem 5 (Appendix A). They found all possible sets of threefactors of 72. Then, they tested to find out those sets whose members add up to 14.The solution to the problem from their scripts showed that each solved the problemcorrectly, so apparently, they reliably and efficiently executed the procedure they used.Jane: I found every three number that can add up to 14...Paulina: Same here!.. .[pointing to her script]...all is 14...and then I times.Jane: .. .and then I added them together and I got 14, right?.. .and then Imultiplied them together.. .Agreed?Daniel: Yeah! [Quincy also nodded his head].Figure 4.20: February 15th, 1993Also, another indicator of students’ use of mathematical procedures is whenthey verity the results of procedures empirically or analytically and recognize correctand incorrect procedures. Eight excerpts were identified during the discussionsillustrating that students verified their results analytically; one excerpt each involvingproblems 1, 5, and 6, two excerpts involving problem 3, and three excerpts involvingproblem 7. There was no indication that they verified results empirically. For example,Shawna had to convince herself that “4802” was the same as “480 x 480” by using thecalculator to verify the procedure for solving parts b(i) and b(ii) of problem 3 (AppendixA). She had first used the square function on the calculator and then checked theresult by keying 480, the multiplication sign, 480, and then the equal sign. Figure 4.21below illustrates the discussion that took place.62Shawna: Wait, wait, wait.. . Four eighty times four eighty times two.Daniel: Write that down and you take the first one, then you take nine eightytimes four eighty and then subtract.Shawna: Four eighty to second power is different from four eighty times foureighty.. .oh no, oh no.. .1 was right [she used calculator to check].Figure 4.21: January 28th, 1993There were 12 excerpts identified throughout the discussions illustrating thatstudents recognized correct and incorrect procedures; one excerpt each involvingproblems 3, 5, 6, and 7, two excerpts involving problem 4, and three excerpts eachinvolving problems 1 and 2. One such instance is illustrated in figure 4.22. This waswhen students were discussing solutions to problem 2 part (b) (Appendix A). Studentswere to simplify the polynomial expression obtained from the entries of themultiplication table. (Notice that students were referring to m2n as m2n). WhenPaulina asked the group if it is a correct procedure to add “m2n” and “m2n” to get“m4n”, George’s emphatic “No” suggests that he recognized Paulina’s procedure asincorrect. An instance suggesting a recognition of a correct procedure is provided infigure 4.10. Paulina’s comment that “But I did the brackets first” suggests that sherecognized it as correct to evaluate what was in the brackets first while solving theequation C = 1.80 + 0.75(m - 1).63Paulina: Add these two.Jane: Yeah.Sitsofe: Add which two?Daniel: Add thosePaulina: Is m two n plus m two n equal to m four n? [m2n + m2n = m4n].George: No.Figure 4.22: January 18th, 1993Finally, students’ ability to use mathematical procedures should be reflected intheir appreciation for the nature and role ofprocedures in mathematics. For such anappreciation to occur, students should be seen, over a long period of time, to be usingwell known procedures, generating new ones, and extending or modifying familiarones (NCTM, 1989). Even though the study was over a three-month period, thediscussions, as well as their written responses to the problems and the questionnaireitems, do not indicate that these students generated any new procedures, extended ormodified familiar ones. However, evidence that they used some well knownprocedures like factoring, guess and check, and exhausting all possibilities, hasalready been discussed above. The effect of prompts on students’ ability to usemathematical procedures was similar to those already discussed undercommunication and mathematical concepts as these prompts provided opportunitiesto the researcher for explaining some of these procedures to the students (figure 4.10).Summary of results Dertaining to mathematical DroceduresStudents’ use of mathematical procedures was relatively wide spread. It wasreflected by four of NCTM’s seven categories of mathematical activities that areassociated with the ability to use mathematical procedures. These four categories64relate to the ability to recognize when a procedure is appropriate; reliably andefficiently execute procedures; verify the results of procedures empirically oranalytically, and recognize correct and incorrect procedures. Use of students’ scriptsin conjunction with the discussions provided additional insights into how studentsused mathematical procedures. The use of prompts provided an opportunity forstudents to clarify their thinking and it also provided the researcher an opportunity todo some explaining.Problem solvingTwo of the areas in which students’ ability to use mathematics to solve problemsshould be reflected are their ability to formulate problems and apply a variety ofstrategies to solve problems. For this study, students were not asked specifically toformulate problems and there was no evidence from the discussions and the writtenresponses to the problems and the questionnaire items that they did. Also, studentswere not asked to individually apply a variety of strategies to solve any of theproblems, and they did not. However, the small group context provided students theopportunity to become aware of the variety of strategies other members of the groupused for solving the same problems. Three excerpts were identified throughout thediscussions indicating that each student used a different strategy to solve a particularproblem. Some of these strategies used by students are factoring, guess and check,systematic charting, exhausting all possibilities, and substitution. Whether or not eachstudent in the group understood the other strategies he or she did not use for solvingthe problem was hard to decide from the discussions, except for statements like “Weall got the same answer” which suggests that the students recognize the otherstrategies as appropriate for solving the same problem. Figure 4.23 below is anillustration of one such instance when a variety of strategies (limited to 2 in this case)has been identified as being used within the group to solve a particular problem.When Jane said “I just guessed mine,” her script showed that she was actually65attempting guess and check (fig. 4.17) and when Paulina said “I used pureknowledge,” she was trying to systematically “exhaust” all possibilities by combining4-goal games with 3-goal games to get 50 goals in 14 games (fig. 4.24).Daniel: How did you get your answer?Jane: I just guessed mine.. .1 didn’t know the logical thing to write.Paulina: What?Jane: I guessed it!Paulina: I didnt. .1 used pure knowledge.Jane: Pure knowledge?Quincy: What?Paulina: What?Quincy: Look.. .the...Paulina: It isn’t true.. .[she interrupted Quincy].Jane: We all got the same answer [Paulina said the same thing].FiQure 4.23: March 1st, 1993q qc’ow-1ojarvisfIS 50 90&Sj_5 WonFigure 4.24: Paulina, March 1st, 199366Another indicator of students’ ability to use mathematics to solve problems isthat they are able to solve problems. Out of the seven problems for the study, thestudents were able to solve four on their own (problem 3 in a group, and problems 5,6, and 7 individually), but needed prompts from me before they were able to solve therest (problems 1, 2, and 4). Figure 4. 03 shows students’ discussions of how thesolved problem 7 (Appendix A), for example.It is important to note that even though all the students could not providecomplete solutions to problems 1, 2, and 4, they were able to provide solutions tosome parts of these problems. There were 4 excerpts that captured students’ partialsolutions to these problems, one excerpt each involving problems 1 and 4, and twoexcerpts involving problem 2. For example, when students were to find the amount ofmoney they could win in the lottery game (problem 2, Appendix A), they all got $99.00.However, they could not get the bonus prize of $50 because they could not simplify thepolynomial generated by completing the multiplication table, something they had to doto win the bonus prize. The students knew they had to simplify the polynomialexpression, and they could even speculate on what the final solution should be, butnot knowing how to simplify the expression prevented them from arriving at thesolution. Thus, lack of procedural knowledge had once more prevented the studentsfrom solving a problem. Students’ discussions reflected what they had written on theirscripts. An example of students’ discussions indicating that they had solved part of aproblem, while they had difficulty continuing, is provided in figure 4.25. When theycould not continue to solve the rest of the problem, Jane banged on the table toindicate her frustration. Jane’s act suggested to me she needed some help so I wentto the group only to find out that no one in the group could continue with solving theproblem. I realized then that the difficulty the students were having was with thesimplification of the algebraic expression with which they were dealing. Thus,monitoring the students helped me identify the difficulty they were having. However, it67was just about time to end that session for gathering data, so I could not provideprompts to students.Daniel: Okay, okay.. . ninety nine, ninety nine dollars, okay?Paulina: Okay [Jane also said the same thing].Jane: What are we trying to find? We are trying to find.. .[she paused and readover the problem again].Paulina: So, it’s ninety nine [she was confirming the $99 they got after checkingwith the calculator the addition she had performed].Jane: Yeah!Paulina: So we all agree it’s ninety nine dollars? [they all nodded their heads inthe affirmative].Jane: If you simplify, you get a bonus of fifty dollars...Paulina: Why don’t we try to figure that out, so that we get one forty nine dollars?[$149].Jane: Why can’t we just.. .you see, we don’t know what simplify.. .or at least, Idon’t.. .[she banged on the table in frustration].Figure 4.25: January 18th, 1993Another indicator of students’ ability to use mathematics to solve problems istheir ability to verify and interpret results of problems they have solved. Throughoutthe discussions, 20 excerpts that suggest students verified and interpreted the resultsof the problems they had solved were identified. There were two excerpts involvingproblem 6, three excerpts each involving problems 4, 5, and 7, and four and fiveinstances involving problems 2 and 3 respectively. Figure 4.26 illustrates how thestudents verified their solutions to parts b(i) and b(ii) of problem 3 (Appendix A).68Paulina used the calculator to verity that the results of 980x - 2x and 2x(490 - x) arethe same if x = 480.Similarly, figure 4.27 illustrates how students made sense of the solution theyobtained individually to problem 6 involving algebraic expressions by exchangingtheir scripts and checking how each arrived at the solution. When Paulina wonderedhow Quincy came by a 3.5 while solving the problem and how that helped him to solvethe problem, he explained he divided seven by two to get three point five. For the part(ii) of this problem, Quincy simplified the original expression of 21x4+ 7/2 x3 to get(21x)/3.5. Students’ responses to the questionnaire item asking them to providereasons for thinking they solved the problems showed that they said they either“checked” their solutions or “knew” they were right.Daniel: The same answer, the same answerPaulina: It’s the same answer, look.. .[she was showing her script to Shawna]Daniel: Same answer, as far as I knowPaulina: You have to use the calculator...Shawna: OkayPaulina: Two times four eighty times four ninety minus four eighty [usingcalculator]Daniel: You get the same answerFigure 4.26: January 28th, 199369Daniel: They are equal, equal [he was talking to Shawna].Paulina: I am doing okay.Quincy: Can I check? [talking to Paulina].Daniel: Look at that one!Quincy: Can we change scripts?Others: Yeah. [They changed scripts in clockwise direction].Paulina: Three point five?Quincy: Yeah. Seven divided by two is three point five.Figure 4.27: February 22nd, 1993Finally, students’ ability to use mathematics to solve problems should bereflected by their ability to generalize solutions to mathematical problems. However,there was no evidence from the discussions suggesting that students tried togeneralize solutions to the problems for the study. For example, even though thestudents were provided with the opportunity to comment on their solutions to problem3 where they could come up with general statements like “the value of an expressionis the same as the value of its factorized form11, they did not provide any suchgeneralizations. May be, teachers need to be providing students with the opportunityto be engaged with the new categories of mathematical activities that are beingemphasized within the current reform so that students become aware that thesecategories are valued by the mathematics education community.For the effect of prompts on students’ ability to use mathematics to solveproblems, evidence from the discussions suggests that the prompts helped thestudents to solve some of the problems which they could not solve on their own. Forexample, when the students were not sure of how to solve part (ii) of problem 1(Appendix A), the discussion illustrated by figure 4.28 ensued and the students finally70solved the problem. At first, I had wanted to encourage the students to go over theirwork and see if they could detect any mistakes they made or steps they missed.However, when I realized during the two-minute pause that they were not ready tocontinue the discussions unless I provided them with some vital clues (the studentsappeared to be saying we can’t continue on our own), I decided to suggest to them toreplace the C in the equation by the $25.05 to see if they could find m, the number ofminutes spoken. That provided a break through for Jane and she led the group tosolve the problem. Once more, prompts from a “more knowledgeable” person createdthe opportunity for the students to solve the problem.Jane: This is how we got a different value, we multiplied.Sitsofe: Yeah, but how come you got different answers?Jane: We used the equation.Paulina: It might be wrong.Sitsofe: That’s what you need to find out among yourselves if you missed anysteps along the way. You have to justify your [Daniel & George] solutionto your colleagues. This is what I did, and this the way I did it. And you[Paulina & Jane] also justify yours to them.George: How they got that answer? [looking up to me].Sitsofe: You question them, then find out. [There was a long pause (2 minutes)and students seemed not sure of what to do]. If you have a problem ofthis sort and you want to find the value of your m, you know the cost,okay... what is the cost? [Paulina and Jane pointed to a value on theirscript].It’s that much...If you replace this [C=25.05] by cost, can you find the value of the m?Jane: Yeah, I can do this...Sitsofe: What does that sign [=] mean?71Jane: Balance, it means this side is equal to this side, right?Sitsofe: Yeah, so if that’s what that sign [=1 means, and you are given the amountyou’ll pay, can you find out how many numbers.. . [The students set out tofigure out the solution]. Can you go ahead and find the value of m?Jane: Yeah.Sitsofe: Can you do that?Jane: Yeah.Sitsofe: Try and see, try and see, it might turn out.Figure 4.28: December 10th, 1992Summary of results rertaining to problem solvingStudents use of mathematics to solve problems was not wide spread. It wasreflected by two of NCTM’s five categories of mathematical activities that areassociated with the ability to use mathematics to solve problems. These twocategories relate to the ability to solve problems and to verh’ and interpret results.Use of students’ scripts in conjunction with the discussions provided additionalinsights into how students used mathematics to solve problems. The use of promptsprovided an opportunity for students to solve some of the problems they could notsolve on their own (either individually or in the group) and it also provided anopportunity to do some explaining.Mathematical disøositionStudents’ disposition towards mathematics should be reflected by theirconfidence in using mathematics to solve problems, to communicate ideas, and toreason. Seven excerpts were identified during the discussions, one excerpt involvingeach problem, when students demonstrated confidence in using mathematics to solveproblems, to communicate ideas, and to reason. For example, when students72discussed their solutions to problem 5 (Appendix A), they initially agreed that the set ofnumbers (8, 3, 3) should correspond to the ages of the three sons of the host.However, one student challenged that solution and the discussions that followed areillustrated in figure 4.29. Paulina and Quincy thought at first that the ages of the threeboys could not be (6, 6, 2) since, as Quincy put it, “It’s only one who goes fishing” andthat one must be the eldest among the three. Apparently, Paulina was not satisfiedwith the rejection of (6, 6, 2) as a possible solution to the problem when she cautioned“six, six, two, though.” Apparently, Quincy was supporting Paulina when he said therewas no problem [of having (6, 6, 2) as a possible solution]. His position was confirmedwhen he responded to Paulina’s question of whether there was no eldest son bysaying that there was. Then, when Quincy followed his response with the questionwhether the two boys of age 6 could have been born at the same time, Paulinaanswered by saying it would depend on what time means. Daniel contributed to thediscussions by suggesting that the time when the two boys of age 6 were born coulddiffer by 1 minute. Apparently, Paulina became more convinced that the three boyscould have ages (6, 6, 2), as well as (8, 3, 3). Thus, for example, knowledge ofmathematics and what time means, were confidently used to reason, communicateideas, and to provide a solution to the problem.Paulina: It can’t be six,.six,.two because all the...Quincy &Daniel: All the sons go fishing? [Both were talking simultaneously].Jane: There is nothing to do.Quincy: It’s only one who goes fishing.Paulina: six,.six,.two, though...Quincy: There is no problem.Paulina: There is no eldest son?73Quincy: There is! You think they both were born at the same time? Well, it’spossible...Paulina: It depends on what time means...Daniel: What about 1 minute?Paulina: Yeah, it could be six,.six,.two also [apparently convinced that even forage 6,.6,.2 one of the sons could be the eldest].Figure 4.29: February 15, 1993Also, students’ disposition towards mathematics should be reflected by theirflexibility in exploring mathematical ideas and trying alternative methods in solvingproblems. Throughout the discussions, seven excerpts were identified to illustratestudents’ flexibility in exploring mathematical ideas and trying alternative methods insolving problems. There was one excerpt each involving problems 1, 2, 4, 6, and 7,and 2 excerpts involving problem 3. For example, when the students discussed theirsolutions to the area and perimeter they were to find for problem 4 (Appendix A), theytried to differentiate between area and perimeter and then tried several ways ofcombining the dimensions of the geometric figure to get values for the area and theperimeter. After Jane wrote down an expression for the perimeter of the geometricfigure, she counted 7 terms altogether. Meanwhile, Daniel added up the terms and got6y + 10 and wanted to know if he could still simplify that [6y + 10]. Paulina also addedup the terms in the expression she wrote for the perimeter and got 4y + 11, differentfrom what Daniel and Jane got. Paulina was confident her expression for theperimeter was right and urged the group to move on to finding an expression for thearea. Jane indicated the area of the geometric figure should be the product of 2y - 6and 5 + y, after several trials. When she noticed that other members in the group weremultiplying all the given dimensions together, Jane cautioned that they could notmultiply everything together. For example, Paulina’s script shows how she multiplied74all given dimensions together to get the area (see fig. 4.31). Their discussions areillustrated in figure 4.30 below.Jane: How many terms did you get [for the perimeter]? One,two,.. .[she started counting the number of terms Paulina wrote down].Daniel: How much is six y plus ten [6y + 10]? [what he got after adding all theterms of the expression for the perimeter].Jane: I got seven.Daniel: six y plus ten [6y + 10].Paulina: Four y plus eleven [4y plus 11]. What are you doing? Clarifying things?Jane: Don’t worry.. .what is it? Four?Paulina: Why? You check! [Paulina checked]. Is it not five y plus four y [5y + 4y]?Daniel: I got...Paulina: Four y plus eleven [4y + 11], isn’t that the perimeter?Jane: You got four y plus eleven [4y + 11] at least.. .Yeah, yeah...Paulina: But you didn’t get that...Daniel: I know how I got it.George: Right?Paulina: I think this is right. You have to add all these to get theperimeter. Part (b) is right... .Now let’s do part (a).Daniel: How could (b) part be right?Jane: Cross that and that.. .and that multiplied by that equals area, right?[Looking at the written script shows Jane canceled- 4y andreplaced it by 2y - 6 , canceled 5y and replaced it 5+y and referred tomultiplying (2y-6) by (5+y)].Jane: Cause you can’t multiply everything together, right?Paulina: Then this has to equal negative four y then.. .because of this part.Jane: Look, this is equal to six. Yeah, this equal to six times five.75Daniel: Don’t you know that?Jane: Yes, this is 6 up here...Daniel: Yeah...Figure 4.30: February 10th, 19935-3D-Figure 4.31: Paulina, February 10th, 1993Another aspect of students’ disposition towards mathematics is reflected by theirwillingness to persevere in mathematical tasks, and by their inclination to monitor andreflect on their own thinking and performance. Eleven excerpts were identifiedthroughout the discussions that suggest students were willing to persevere inmathematical tasks; at least one excerpt each involving problems 3 to 7, and threeexcerpts each involving problems 1 and 2. Also, there were 13 excerpts throughoutthe discussions suggesting that students were inclined to monitor and reflect on theirthinking and performance. Of these 13 excerpts, there was one each involvingproblems 2, 5, 6, and 7; two involving problem 4, three involving problem 3, and fourinvolving problem 1.76After they solved part (b) of problem 2, the students were having difficultysimplifying the algebraic expression they got for the sum of the entries which wereobtained after completing the multiplication table. Students did not give up trying tosimplify the expression. Paulina was persistently providing encouragement to thegroup (especially Jane) by urging them to “try to figure that out” and by reminding thegroup that “that’s what we’ve been doing.” They tried to relate to what was previouslydone in class, they tried several ways of combining the terms of the algebraicexpression. They persevered in trying to solve this part of the problem. Jane alonewould have given up very early trying to solve this part of the problem, but the groupencouraged her to persevere. Apparently, trying to figure out what to do next, beingemphatic about a method of arriving at a solution or the solution itself being wrong,and agreeing and disagreeing among themselves are indications of their inclination tomonitor and reflect on their own thinking and performance. Thus, situations that makestudents argue among themselves intellectually provide the opportunity for them topersevere, monitor and reflect upon their thinking and performance. Figure 4.32illustrates students’ willingness to persevere in mathematical tasks, and theirinclination to monitor and reflect on their own thinking and performance.Paulina: So we all agree it’s ninety nine dollars? [they all nodded their heads inthe affirmative].Jane: If you simplify, you get a bonus of fifty dollars...Paulina: Why don’t we try to figure that out, so that we get one forty nine dollars?[$149].Jane: Why can’t we just.. .you see, we don’t know what simplify.. .or at least, Idon’t.. .[she banged on the table in frustration].Paulina: I do! If we have one of those questions.. .that’s what we’ve been doing.Jane: Come on, show me then? [She banged on the table in front of Paulina].Paulina: So I think we haven’t got.. .so we have to put all these m’s.. .Agreed?77[Quincy and Daniel nodded their heads].Jane: I did that.Paulina: And you have to add, like m three n [she was referring to m3n].Jane: I did that!Paulina: So what’s next?Jane: I put m two n equals three mnPaulina: How did you get that? How did you get 3mn?Jane: There are three that look the same thing.. .then same amount, exceptthat...Paulina: It’s wrong! It’s wrong! wrong!Jane: No, it’s the same quantity, except that it is different.Paulina: Yes, but you have to add this little thing, three mn.Jane: It’s the same thing!Paulina: It’s not the same thing!Figure 4.32: January 18th, 1993Finally, students’ disposition towards mathematics should be reflected bystudents’ activities that indicate they value the application of mathematics to situationsarising in other disciplines and everyday experiences. Also, disposition should bereflected in an appreciation of the role of mathematics in our culture and its value as atool and as a language and by their interest, curiosity, and inventiveness in doingmathematics. Throughout the discussions and the written responses to the problemsand the questionnaire items, it was difficult to find evidence to suggest that studentsdemonstrated these aspects of personal disposition towards mathematics. Onesituation which might suggest that students valued the application of mathematics inother disciplines and everyday experiences was when they discussed solutions toproblem 7 which involved hockey. While Janet was busy trying to figure out the78solution through guess and check, Paulina, Quincy, and Daniel were talking aboutbaseball, golf, and the Super Bowl. Jane felt the other three were not talking aboutmathematics so she banged on the table and said “We are talking math!” Quincyreplied that what they were talking about had to do with mathematics and Daniel andPaulina agreed with him. Apparently, the three wanted to impress it upon Jane thatsince they used mathematics to solve the problem involving hockey, their talk aboutbaseball, golf, and Super Bowl could also be mathematically valuable. Even ifPaulina, Quincy and Daniel were justifying an “off topic” conversation, justifying it theway they did suggests that they realize mathematics can be applied to those sportsactivities. Figure 4.33 is an illustration of the ensuing discussions.Sitsofe: If you say it didn’t work, what do you mean?Jane. When you multiply it, right?.. .The games and goals.. .lt’s supposed toequal to.. .not.. .well, it’s supposed to work out to...Sitsofe: Are you listening to the explanation? [Referring to other group members].Jane: Explanation? I don’t need one. [She thought I was referring to her].Sitsofe: No, I mean they listening to your explanation.Paulina: I like to watch Super Bowl.Jane: We are talking math!!! [Janet banged on the table].Paulina: Oh, sorry!Quincy: But that has to do with math [His discussions with Paulina and Daniel].Daniel: Yeah!Paulina: Yes, it’s hockey, hockey [She was pointing to her script].Sitsofe: Oh, I see!Jane: You are telling me you’re a Canucks fan?Paulina: Yes, of course!Figure 4.33: March 1st, 199379One instance which might suggest students’ appreciation of the role ofmathematics in our culture and its values as a tool and as a language was when theysuggested using the newspaper as a source for finding the solution to problem 7(Appendix A). Apparently, when Paulina, Jane and Daniel suggested a newspaper as“another way” of getting access to “the real scores” that can be used to solve theproblem, they were appreciating the role the newspaper can play in generatingmathematical activity. Apparently, they were also appreciating the role mathematicscan play in solving such a problem and communicating its results. A portion of thediscussions is illustrated in figure 4.34 below.Sitsofe: Could you have done it differently?Quincy: No!Sitsofe: You [Jane] said you guessed. You [Quincy] said you used some chart...Paulina: Systematic charting...Sitsofe: That’s fine. Could you have done it differently?Jane: Yeah, I could have done a chart. It would have been easier for me.Sitsofe: So apart from using the chart or guessing?Jane: Another way?Paulina: Go get newspaper and find out.Daniel:. The real scores.Jane: Yeah...Sitsofe: How could you do that from a newspaper?Jane: It has to be the truth but...Figure 4.34: March 1st, 1993Students’ interest or curiosity in doing mathematics was better captured by theirresponse to the questionnaire item asking them to indicate if they would like to see80group discussions of solutions to mathematical problems as part of their normalmathematics classes, except for Quincy who said “No” on one occasion, a typicalresponse from the students was “Yes, because it makes us understand mathematicsbetter.” Thus, group work provides a context that can make students interested orcurious in mathematics and lead them to understand mathematics better. Theoverwhelming interest or curiosity in group work may be due to the support studentsget from the group as they share their ideas. Despite the interest or curiosity, therewas no evidence to suggest their inventiveness in doing mathematics.Categories of students’ mathematical activities associated with dispositiontowards mathematics that students demonstrated when they discussed their solutionsto the problems on their own were also reflected in the discussions that followed theprompts I gave them. Students frequently showed tendencies “to think and to act inpositive ways” (NCTM, 1989, p. 233). There was evidence indicative of students’confidence to use mathematics to solve the problem, willingness to persevere insolving the problem, and monitoring and reflecting on their own thinking andperformance. I decided to find out if the students could solve part “c” of problem 3using the difference of squares, because this was presented in class by Joanne (theclassroom teacher) as a useful and fast strategy sometimes used to solve someproblems in mathematics. The students remembered the expression of the form x2-as representing the difference of two squares but could not factorize it properly. Theygot x2-y2 = (x - y)(x- y). Even though they recognized that the two negatives before yshould multiply to give a positive (which is different from the negative before y2, theydid not seem to resolve the impasse. Apparently, students memorized the factorizedform of x2-y2 without reproducing it correctly. So, they could not solve the problem.When I provided them with the acceptable factorized form of the expression, they wereable to solve the problem. Notice that students seemed to have applied onlyprocedural knowledge in this case, without any indication of conceptualunderstanding. However, they did not give up in the face of the difficulties they faced81but rather acted positively to solve the problem and to use the calculator to check theresults they got from factorization. Prompts once more helped the students to thinkand to act in positive ways to solve the problem. A portion of the discussions ofproblem 3 (Appendix A) reflecting students’ disposition towards mathematics whenprompts were given, is illustrated in ligure 4.35.Sitsofe: You were checking yourself to see if what you did was right. So if youwant to simplify...Daniel: You must change the sign.Sitsofe So what would you get if you want to factorize x2 - y2?Paulina: We don’t know [She was apparently speaking for the group].Sitsofe: You tried and did something. You only checked and found out that itdidn’t turn out to be this. You want to check and see if it comes back tothis...Paulina: Yes, I checked.Daniel: So if we multiply, do we change the signs?Shawna: Yes.Daniel: Negative, negative...Shawna: Yes, two negatives make a positive.Sitsofe: Why did you put the negative here?Paulina: I don’t know. Because people weren’t looking and they got this.. .that’swhat we did, though!Daniel: Yeah, why not?Sitsofe: So without the negative...?Paulina: Because...Sitsofe: If you are given that x2- y2 = (x+y) (x-y), can that help you do it?Paulina: Yeah.Sitsofe: How?82Paulina: I don’t know what you mean? Getting the answer?Sitsofe: Uhum!.. Do you find any similarity between this and that? [comparingx2-y2 with 642 - 362].Paulina: Yes, it’s the same, but x equals sixty four and y equals thirty six.Sitsofe: So how can you use that? If you want to factorize this and use this tosolve it, how can you do that?Paulina: Putting sixty four plus thirty six and sixty four take away thirty six?Sitsofe: Try it and see! [It was successfully done and the results checked usingthe calculator].Figure 4.35: January 18th, 1993Summary of results pertaining to mathematical dispositionStudents’ disposition towards mathematics was evident from the discussions. Itwas reflected by four of NCTM’s seven categories of mathematical activities that areassociated with mathematical disposition. Two of the four categories relate tostudents’ confidence in using mathematics to solve problems, to communicate ideas,and to reason and flexibility in exploring mathematical ideas and trying alternativemethods in solving problems. The other two relate to students’ willingness topersevere in mathematical tasks and inclination to monitor and reflect on their thinkingand performance. Use of students’ scripts in conjunction with the discussionsprovided additional insights into students’ disposition towards mathematics. The useof prompts provided an opportunity for students to further demonstrate their dispositiontowards mathematics.83Mathematical cowerStudents’ demonstration of mathematical power should be reflected by their:1) ability to communicate mathematically, 2) ability to use mathematical concepts, 3)ability to use mathematical procedures, 4) ability to use mathematics to solveproblems, and 5) disposition towards mathematics. In addition, students’mathematical power should be reflected by the extent to which students integrate allthese aspects of what should constitute mathematical knowledge. Appendix Dprovides, from the discussions of each problem, a distribution of excerpts that arereflective of SAS. Table 4.01 is a summary of Appendix D. It provides the number ofcategories, out of the total for each SAS, that has been reflected by discussions ofeach problem. For example, 2/3 in the row of “C” (communication) and in the columnof “Prob 1” (problem 1) means that there were excerpts from the discussions ofproblem 1 that reflect two out of the three categories of mathematical activitiesassociated with communication. Notice that 2/3 is not used to mean two out the threeequal categories; it is only used to mean that two of the three categories have beenreflected. It is important to remember that the group discussions of each problem tooka maximum of 40 minutes and that it is a holistic picture of students’ demonstration ofmathematical power that is being presented. Therefore, evidence that the discussionsreflect mathematical activities associated with any two or more SAS (union of excerptsrelated to SAS) should constitute evidence for integration.84Table 4.01Categories of Student Assessment Standards (SAS)Reflected by Discussions of ProblemsSAS Prob 1 Prob 2 Prob 3 Prob 4 Prob 5 Prob 6 Prob 7C 2/3 3/3 3/3 3/3 2/3 3/3 3/3MC 2/7 4/7 3/7 4/7 3/7 3/7 2/7MP 4/7 3/7 4/7 2/7 2/7 3/7 5/7PS 2/5 2/5 2/5 2/5 2/5 4/5 2/5MD 4/7 3/7 4/7 4/7 3/7 3/7 6/7Note:C = Communication, MC = Mathematical concepts, MP = Mathematicalprocedures, PS = Problem solving, MD = Mathematical dispositionFor the seven problems used for this study, evidence from the discussionssuggest that students demonstrated abilities reflective of all of SAS. There is a unionof excerpts reflecting categories of mathematical activities associated withcommunication, problem solving, mathematical concepts, mathematical procedures,and mathematical dispositions. Also, there is evidence that each SAS was reflectedthroughout the study. Communication was reflected by all three categories ofmathematical activities associated with it, namely expressing mathematical ideas byspeaking, writing, demonstrating, and depicting them visually; understanding,interpreting, and evaluating mathematical ideas that are presented in written, oral, orvisual forms; and using mathematical vocabulary, notation, and structure to representideas and describe relationships. Mathematical concepts were reflected mainly bycategories of mathematical activities that relate to labeling, verbalizing, and definingconcepts, and using models, diagrams and symbols to represent concepts.Mathematical procedures were reflected mainly by four categories of mathematical85activities. These categories involve recognizing when a procedure is appropriate;reliably and efficiently executing procedures; verifying the results of proceduresempirically or analytically; and recognizing correct and incorrect procedures.For problem solving, it was reflected mainly by two categories of mathematicalactivities associated with it. These categories involve solving problems and verifyingand interpreting results. Finally, mathematical disposition was reflected mainly by fourcategories of mathematical activities associated with it. These categories involveconfidence in using mathematics to solve problems, to communicate ideas, and toreason; flexibility in exploring mathematical ideas and trying alternative methods insolving problems; willingness to persevere in mathematical tasks; and an inclination tomonitor and reflect on their own thinking and performance.Now because categories of mathematical activities associated with all of SASwere reflected during the discussions involving each problem and because categoriesof mathematical activities associated with all of SAS were reflected throughout thestudy, results of the study indicate that students demonstrated mathematical power.The extent of students mathematical power demonstrated is provided by thedistribution of the categories of mathematical activities that are associated with all ofSAS (see Appendix D and Table 4.01).Other insights perceived from the studyWhile I was analyzing the data for the study, some insights became apparent tome. These insights involve students shifting their viewpoints as a result of thediscussions with the group, with or without prompts. Only shifts considered to be majorare discussed. A shift is major if it seemed to prevail at least during the discussions ofany of the problems. For example, a student who argued for the appropriateness of aparticular mathematical procedure and later accepted the inappropriateness of thesame procedure in light of evidence from the discussions is considered to have gonethrough a major shift in his or her mathematical reasoning and arguments. I discuss86below two types of major shifts that I observed during students’ discussions of theirsolutions to the mathematical problems used for the study. I call these types of majorshifts consensual and conceptual, respectively.Consensual shiftBy consensual shift, I mean a change in viewpoint to align with the viewpoint ofthe majority, seemingly because it is a majority viewpoint. Shifts in viewpoint due toconsensus could benefit the person whose viewpoint is shifting if the majorityviewpoint meets acceptable mathematical standards. However, shifts could bedetrimental to the person in the long run if the majority viewpoint does not meetacceptable mathematical standards. From the discussions throughout the study, threeexcerpts were identified that suggest a consensual shift occurred without it apparentlybenefiting the person undergoing the shift. Figure 4.36 below illustrates an apparentlynon-beneficial consensual shift in George’s initial viewpoint.While students were discussing their solutions to How many minutes will youspeak if you have to pay $25.05? (problem 1, Appendix A), Jane and Paulina agreedon 26 minutes (with Daniel nodding his head in support) as the time to speak if onehad to pay $25.05 for speaking on the phone. Earlier, George had obtained 31minutes as the time to speak for $25.05. Even though the procedures he used led himto get a better solution (he subtracted 1 from 32, using the expression (rn-i) to get 31minutes, without realizing that it is the m = 32 minutes that one should speak), heabandoned his own solution and went with the consensus of the group to accept 26minutes. George has undergone what I call a consensual shift and so long as themajority solution is not acceptable by mathematical standards, this shift in George’sviewpoint could be detrimental to him. Notice here that the students got the 2 differentsolutions on their own without prompts from me. So, George changing his mind to gowith the group suggests he was believing in the group more than in his own abilities.87Paulina: There’s problem with the second part. We are still on the second part.Four eighty for five minutes. We divided four eighty by five to get howmuch for a minute and then twenty five-o-five dollars divided by ninety sixcents...Sitsofe: Go ahead and see what you get and explain it to us.Paulina: Four eighty divided by five equals ninety six...Jane: Cents per minute.Sitsofe: Why did you divide by 5?Jane: To find out how much it is per minute.Paulina: Four eighty divided by five to find out how much it costs perminute.. .equals ninety six cents and twenty five-o-five divided by ninetysix equals twenty six minutes...Sitsofe: Do you think you could do it some other way?Jane &Paulina: I don’t know [In unison].Sitsofe: You got 31 minutes first, how did you get 31 minutes? Yours isdifferent from the 26 minutes. [Directed at George].George: I added seventy five cents every minute until I got twenty five dollars fivecents.Sitsofe: How different is yours from what they [Jane and Paulina] got? Do youagree with what they both got?George: Yes.Sitsofe: And you disagree with what you did first?George: Yes [nodding his head].Figure 4.36: December 10th, 199288Notice that there were several instances of consensual agreements, both fromthe discussions and the written responses to the questionnaire, that were based onstudents’ acceptance of a solution as correct because they all got the same solutionafter working individually. Apparently, no shifts are involved in such cases but it isimportant to note that students use consensus to accept the validity of a solution to aproblem. Even though seeking consensus is desirable, the danger could be that if allthe students use the same or similar faulty procedures to come to the same solution,there will not be any motivation to recheck how they arrived at the solution, since theyall got the same solution.Throughout the discussions of the study, four excerpts that suggest aconsensual shift that could benefit the person undergoing the shift were identified.Even though the majority viewpoint warranting the shifts in these four cases werebased on acceptable mathematical standards, it was hard to tell whether the shiftsactually benefited the person undergoing the shift. An example illustrating aconsensual shift that should benefit the person undergoing the shift is provided byfigure 4.37. After George changed his mind to accept the faulty solution of the group, Igave them several prompts that led them to solve the problem correctly. I asked themwhat the C and = in the equation stood for. When they identified C as standing for$25.05 and that they had to maintain a balance of the two sides of the equation whilefinding the value for m, they eventually solved the problem. Jane echoed the resultwhen she said “So, we spoke for thirty two minutes.” George did not come up with anysolution at this stage but he still agreed with the group, apparently maintaining hisposition of going with the group decision.89Paulina: We got to use thirty two to get thirty one [realizing that the time to speak ism but rn-i is in the equation, so if rn-i =31 then the time spoken shouldbe 32 minutes].Jane: So, we spoke for thirty two minutes.Paulina: Yes.Jane: You all agree? [She was asking for consensus from the group and theyall (including George) nodded their heads].Figure 4.37: December 10th, 1993However, there is an indication that when students have a viewpoint which theycan justify, they stick to it, even if a more knowledgeable person challenges thatviewpoint. Apparently, if students can self-validate (Anderson, 1993) their solutionsusing the rules and standards of mathematics, they become confident justifying thosesolutions through several means that are acceptable within mathematics. Forexample, after students had by consensus agreed to the solution to problem 6(involving students deciding between the larger of two algebraic expressions), theyconfidently justified the original solution despite the prompts I provided to challengetheir viewpoint of the original solution as being right. They substituted differentpositive and negative numbers and concluded that 6x = 6x would always be true forany given value of x. This is illustrated in figure 4.38 below.Sitsofe: You came up with 6x and 6x?Daniel: Yeah!Sitsofe: Why do you say they are the same?Quincy: Because they came out to be the same.Paulina: Because they are the same!...90Sitsofe: Yes, my question is 6x.. .and you don’t know the value of x. This is also6x and you don’t know the value of x, how do you claim they are thesame?Daniel: Because they are the same x.Quincy: Because it didn’t say that it was otherwise...Sitsofe: I see.Paulina: Yeah!.. .That’s right.Sitsofe: Do you think of any number for which this can be different? If you have avalue for x.. .can you think of a value for x which can make the twosolutions different?Paulina: No...Sitsofe: Think about it, I’ll be back.. .[students continued with the discussions].Quincy: Make it different by finding a value for x?Shawna: No, too bad!Paulina: We don’t know the value of x.Quincy: Value? Give them a value of x so that one of them is greater!Paulina: I thought we were finished.. .Okay, three over two.. .give me a number.Quincy: Has to do with a negative.Paulina: Why is it going to be negative?Daniel: Because we didn’t factorize.Quincy: Looks like.. . No, like what?Daniel: Constructing composite factors.Quincy: What?Daniel: Factorize... How do you factorize? [There was a pause while studentsworked individually].Paulina: Have we found the answer?Quincy: Yeah.Daniel: We are done!91Paulina: No, we aren’t!Sitsofe: Have you come up with something different [I was speaking to the group].Paulina: Yes...Sitsofe: What?Quincy: No matter what.. .the same!Sitsofe: No matter what? What did you try?Quincy: Negative.Paulina: Why not try positive number?Quincy: Cause they won’t change.Paulina: How do you know that?Quincy I say it won’t change, if you want to prove me wrong, go ahead!Paulina: No.. .Okay.Sitsofe: You think they will change.Daniel: No.Paulina: No, I said before it wouldn’tSitsofe: I see.. .So are you satisfied you solved the problems?Yeah [they all responded in unison].Sitsofe: So that you can explain it to anybody, right?Quincy: Yes.Sitsofe: I’ll get one of you to explain it to the whole class.Quincy: I will do it because I will get to be the teacher [after some of the groupmembers hesitated to offer themselves].Figure 4.38: February 22nd, 1993Concertual shiftBy conceptual shift, I mean holding a different conception from an earlierconception as a result of the group discussions. To be a major shift it must be seen to92prevail during the discussion of a problem or prevail over discussions of otherproblems. Any perceived conceptual shift only indicates that at least a differentcompeting conception is prevailing. If this ‘new” conception is compatible withacceptable conceptions within mathematics, then it is likely to promote mathematicslearning. On the other hand, if this new conception is not compatible with acceptableconceptions within mathematics, then the shift it brings about might be detrimental tothe person experiencing the shift, unless the “cognitive conflict” it creates in the personis resolved.The conceptual shifts observed for this study resulted in new conceptions thatare compatible with conceptions within mathematics. From the group discussions, fiveexcerpts suggesting this type of major shift were identified. An example of this shiftoccurred when Jane first thought that guess and check was not an appropriate methodfor solving problems: she said she guessed because she “didn’t know the logical thingto write”. Apparently, guess and check was not the “logical thing to write,” more sowhen Paulina’s script showed that pure knowledge” meant exhausting all possibilitiesbefore getting’ the solution (see figure 4.24). Later, Jane changed her mind during thediscussions and accepted guessing as an appropriate method when she reacted toPaulina’s remark that guessing was a good way of approaching the problem by saying‘1 know.. .let’s go on”. This was when the group was discussing their solutions toproblem 7 involving a hockey game. The discussions are illustrated in figure 4.39below.Jane: Every one got eight...Paulina: But there are other possibilities, probably?Quincy: No.. .[shaking his head emphatically].Jane: You are supposed to knock this one out!Paulina: I know.Jane: Did you guess? [She was asking Quincy].93Quincy: No.Jane: Did you guess? [She was asking Daniel].Daniel: No.Jane: I guessed.. .[lt looked like Jane was trying to find out if other groupmembers would accept guessing as authentic].Paulina: You copied! [She was referring to Daniel].Daniel: Yeah!Paulina: I got it for all.. .you can ask him [She was pointing to me].Daniel: You are right! How do you know? How do you know?Paulina: Because these.. .like the first one...Quincy: These vary.. .those too vary.. .and doesn’t.. . no one else...Daniel: How did you get your answer?Jane: I just guessed mine.. .1 didn’t know the logical thing to write.Paulina: What?Jane: I guessed it!Paulina: I didn’t.. .1 used pure knowledge [Apparently, she was implying thatguessing does not constitute pure knowledge].Jane: Pure knowledge?Quincy: What?Paulina: What?Quincy: Look.. .the...Paulina: It isn’t true.. .[I came to the group at this time].Jane &Daniel: We all got the same answer.Sitsofe: Same answer?Daniel: Yeah...Jane: It’s amazing we do.. .the same answer!Sitsofe: And did you come by the answer the same way?94Quincy: Yes.Daniel: Yeah.Jane: No.. .not sure.. .Sorry, I guessed.Paulina: We have to find out...Sitsofe: So, may be.. .you can find out how it was done differently.Paulina: So Jane, how did you do it?Jane: Well, I just guessed!Paulina: That’s a good way of approaching it...Jane: I know.. .Iet’s go on...Figure 4.39: March 1 St, 1993Summary of results Dertaining to other øerceived insightsTwo major types of shifts were perceived to have taken place as studentsdiscussed their solutions to the problems given them. These major types of shifts arelabeled consensual when the shift is to align an initial viewpoint with that of themajority, and conceptual when an initial conception is abandoned and changed to adifferent conception during the discussions. Four of the seven consensual shiftsperceived to have taken place from this study involved majority viewpoints that arecompatible with acceptable viewpoints within mathematics. The other threeviewpoints are not compatible with acceptable viewpoints within mathematics.Apparently, students do not change consensually if they have a solid grasp of an initialviewpoint. Finally, conceptual shifts observed from the study resulted in conceptionsthat are compatible with standard conceptions within mathematics.95CHAPTER 5CONCLUSIONIn this chapter, I provide a summary of the study, general discussions of theresults of the study, and the significance of the results of the study. Also, I discuss theimplications of the results of the study for practice, and finally, I make somesuggestions for future research in light of the results of this study.Summary of the studyThe purpose of this study was to appraise information from the discussions of agroup of students and to determine if the information indicates students’ mathematicalpower. Also, from the discussions, I documented any other insights that I perceived tobe important. The focus of the study was on the information the students generated ontheir own while they discussed their solutions to mathematical problems. To a lesserextent, data were also gathered with respect to students’ discussions followingprompts from the researcher. Grade 9 students in the regular school programparticipated in the study.A small group format was used for the study. There were four groups ofstudents, two groups of four students per group and two groups of five students pergroup. All the students were given problems based on the concepts their classroomteacher had taught them the previous week or two. They responded to the problemsindividually for about 20 minutes and then used about 40 minutes to discuss, in theirrespective groups, their solutions to the problems. In order not to leave out any of thestudents while data were being gathered, all students in the class participated in thestudy. However, only one group of students comprising two males and two femaleswas chosen as the focus group for the study and the group’s discussions were videorecorded and analyzed to provide answers to the research question. Students for the96study also responded to questionnaire items at the end of each problem session. Datawere collected on seven different occasions using seven different problems over aperiod of three months.Students’ discussions were analyzed around SAS as the key constructs. SAScomprises communication, problem solving, mathematical concepts, mathematicalprocedures, and mathematical disposition. Associated with each SAS are categoriesof mathematical activities. From the discussions of the solutions of each of theproblems used for the study, excerpts reflecting any of the categories of mathematicalactivities associated with any of SAS were identified. These excerpts were interpretedto provide evidence for students’ demonstration of mathematical power. Students’written responses to the problems and their written responses to the questionnaireitems were also used to illuminate the interpretation of the excerpts. Other insightsfrom the discussions that were perceived to be important were also documented.Results of the study indicate that through the small group discussions of.theirsolutions to mathematical problems, students demonstrated mathematical power to theextent that excerpts of the discussions reflected categories of mathematical activitiesassociated with SAS. Specifically, communication was reflected by all threecategories of mathematical activities associated with it, namely expressingmathematical ideas by speaking, writing, demonstrating, and depicting them visually;understanding, interpreting, and evaluating mathematical ideas that are presented inwritten, oral, or visual forms; and using mathematical vocabulary, notation, andstructure to represent ideas and describe relationships.Mathematical concepts were reflected mainly by categories of mathematicalactivities that relate to labeling, verbalizing, and defining concepts, and using models,diagrams and symbols to represent concepts. Also, mathematical procedures werereflected mainly by four categories of mathematical activities. These categoriesinvolve recognizing when a procedure is appropriate; reliably and efficiently executingprocedures; verifying the results of procedures empirically or analytically; and97recognizing correct and incorrect procedures. For problem solving, it was reflectedmainly by two categories of mathematical activities associated with it. Thesecategories involve solving problems and verifying and interpreting results.Finally, mathematical disposition was reflected mainly by four categories ofmathematical activities associated with it. These categories involve confidence inusing mathematics to solve problems, to communicate ideas, and to reason; flexibilityin exploring mathematical ideas and trying alternative methods in solving problems;willingness to persevere in mathematical tasks; and an inclination to monitor andreflect on their own thinking and performance.Other results indicate that combining students’ written scripts with students’ talkprovides a better insight into what students are talking about. Also, monitoringstudents and providing them with prompts while they work in groups is useful inhelping them accomplish tasks in which they are engaged. Finally, when studentswork in groups, they can shift their viewpoints consensually or conceptually to aligntheir viewpoints with majority viewpoints.General discussionIn this study, I set out to investigate students’ small group discussions and todetermine if there is information indicative of students’ demonstration of mathematicalpower. The importance the NCTM attaches to the use of small groups to promote theconstruction of mathematical knowledge and my own interest in the use of smallgroups motivated me to undertake the study. The design of the study wherebystudents discussed solutions after they attempted solving the problems individuallywas based on my belief that whenever a group of students reflect over mathematicalproblems with which they are familiar, they gain better insights into the problems. Thisbelief is consistent with evidence from research that “people who think about theirproblem solving after they have solved a problem are better problem solvers thanthose who don’t” (Willoughby, 1990, p. 43). So, for me, the important thing in this study98was to provide the opportunity for a group of students to talk about mathematicalproblems they have attempted to solve on their own.Also important is that the problems should contain something mathematicallysignificant to talk about, and they did. The major concern here was not to comparestudents’ discussions from one problem to the other, but to examine their discussions,given the problems. To that effect, what I did was to relate the problems of the study tothe mathematics content that the teacher, Ms Stansfield, taught the students. Forexample, problem 1 relates to the solving of equations while problem 2 relates tosimplifying polynomial expressions, the manipulation of indices, and estimation.Problem 3 relates to factorization and evaluation of binomial expressions and the useof the strategy of the difference of two squares to solve problems. Problem 4 connectsalgebra with geometry and relates to simplification of algebraic expressions. Problem5 relates to factorization while problem 6 refers to simplification of algebraicexpressions. Finally, problem 7 relates to the use of variables to solve problems.Notice that none of the problems as presented, and for that matter no singleproblem, would permit the students to provide information that reflects all categories ofmathematical activities associated with each of SAS. Consequently, only somecategories of the mathematical activities associated with each of SAS were reflectedthroughout the study by the students’ discussions. Nevertheless, students’discussions provided sufficient information to suggest that they demonstratedmathematical power. It was difficult to tell if students were aware of (or evenunderstood) some of the mathematical ideas that they talked about. Sometimes, themathematical idea was inferred from their talk as in the case of rate, while other times,they specifically mentioned the mathematical idea, as in the case of exhaustingpossibilities. Even though the problems were related to concepts like functions,indices, and polynomials, students did not identify these as some of the mathematicalconcepts they used. It would have been desirable to probe for further understanding,99but this was not possible since students were to be talking on their own most of thetime.Also, students’ demonstration of a particular ability might not imply that theyunderstand related abilities. For example, although the proper use of a mathematicalprocedure would suggest an understanding of the concept underlying the use of thatprocedure, a student might use a procedure properly with or without an understandingof the concept underlying the use of that procedure. It was hard to tell if the studentsonly reproduced “invert multiply” and the “reciprocal thing” as taught them by theirteacher or they understood the concepts behind them.Students were more inclined towards providing evidence for procedural usagerather than conceptual understanding. For example, students’ use of perimeter as“adding of sides”, area as “multiplying of sides”, and ratio as “this, two dots, and that”all suggest procedural usage. Also, there were instances, involving mainly algebraicmanipulations, when students’ demonstrated abilities suggest they did not have agood grasp of the concepts and the procedures involved. For example, anytime itcame to simplifying algebraic expressions (like m2n + m2n), the simplified versions thestudents got (m4n) were not acceptable within mathematics. Given that the studentssuccessfully solved most of the problems involving only numbers but could not simplifyalgebraic expressions, they apparently were having problems with algebraicmanipulations per Se.It was insightful to find students’ small group discussions providing informationabout categories of mathematical activities that are not traditionally emphasized. Forexample, students verified and interpreted the solutions they obtained to problems.Also, they shared information on strategies they used for solving problems, and theymonitored, not only their thinking and performance, but those of other members of thegroup. Students engaging in such activities is a positive sign that small groupdiscussions of mathematical activities can provide a context for demonstratingstudents’ mathematical power.100Major shifts were those that prevailed over the period of 40 minutes’discussions of a problem. There were more consensual shifts observed thanconceptual shifts. It was hard to tell if both types of shifts were taking place at the sametime. What would have been desirable is a consensual shift taking place alongside aconceptual shift and having these two shifts aligned to “new” viewpoints that are basedon acceptable standards of mathematics. Accepting majority viewpoint just becauseone is conforming to the “norm” is not enough, especially if the majority viewpoint isfaulty. Rather, I believe an acceptable conceptual shift is one that will make any shiftbeneficial to whoever is undergoing that shift. This should be so even if the shift is aresult of prompts from a more knowledgeable person like the teacher or someone fromthe peer group. Emphasizing conceptual shifts on the part of the person shifting isimportant if we are to avoid students shifting their reasoning because of the “authority”of the teacher or more capable peers. In any case, having the five conceptual shiftsthat occurred and four out of the seven consensual shifts that occurred meetacceptable mathematical standards suggests that these shifts could benefit thoseexperiencing the shifts.It was only through prompts that I had the opportunity to influence the sorts ofinformation students generated and the quality of such information. It was interestingto find how giving prompts provided useful teaching-learning situations. Students’“barriers” to solving the problems were identified and repeated questioning helpedstudents to clarify their thinking and refocus them to solve the problems. Apparently,giving of prompts made instruction an integral part of the data gathering process.Not all of students’ discussions were audible, and not all of the discussionsmade sense to an observer. Some information was lost because students weretalking in very low tones. Some information was also lost because students wereusing indefinite pronouns such as this and that and it was difficult to tell to what theywere referring. So, it was difficult to pass judgment on these aspects of the students’discussion. Sometimes, however, combining the discussions with what they wrote101down provided better insights into what they were discussing. Thus, the discussionsmake more sense if they are interpreted within the context in which the discussionstook place.Students used calculators to solve most of the problems. Even thoughcalculators could enhance students’ mathematical performance, information on howthey used the calculator was sometimes missing. One example where the order ofcomputation was important was the case for problem 1, involving speaking on thephone. When students did not follow acceptable order for carrying out thecomputations, they arrived at the wrong solutions and became frustrated for notsolving the problem. Other times, they became falsely confident as havingsuccessfully solved the problem, despite punching the wrong keys on the calculator.It was desirable for this study to have students talk. So, when before the fourthdata gathering session, I found out that George was not contributing appreciably to thegroup discussions, he was replaced by Quincy who contributed meaningfully to thegroup discussions. This changed the locus of interaction within the group. When Janeand Paulina were in the group, they did most of the talking with Daniel talkingoccasionally. George seldom talked. When Jane had to go for basketball training andcould not attend two of the data gathering sessions, Shawna replaced her and thelocus of the interaction shifted to Daniel, Paulina, and Quincy, with Shawna talkingless but more frequently than George had. So, sometimes changing the groupmembership may introduce more dynamism into group discussions. Notice howeverthat the reverse could also be true, but for this case it turned out that the new group ofstudents talked more. May be keeping to the criteria for changing group members wasthe contributing factor in this case.It was difficult to represent the discussions as sometimes taking placesimultaneously. However, for the purposes of this study, that did not matter because inwhatever order the students made their contributions to the discussions, the importantthing was that differences in reasoning resulted in discussions leading to the102resolution of the differences, with or without prompts. Likewise, similarities inreasoning, irrespective of who said what first, only helped the students come to thesame conclusions.Significance of the resultsThe results of this study are important in at least four respects. Firstly, theresults show that when students talk in small groups, they can provide informationindicative of students’ mathematical power. Since the attainment of students’mathematical power is the goal of the current reform in mathematics education, theuse of small groups as recommended by the NCTM, at least provides a context forgathering information on students’ demonstration of mathematical power. Secondly,students’ talk, when combined with their written scripts, provides better insights intowhat they are talking about. Thirdly, when students work in groups, they can shift theirviewpoints consensually or conceptually to align them with majority viewpoints.However, if they have a viewpoint that they can justify, they do not change it even inthe presence of an authority. Finally, monitoring students and providing them withprompts while they work in groups is useful in helping them to accomplish tasks inwhich they are engaged.Implications for practiceResults of the StudyEven though this is a “best case scenerio,” the results of the study suggestseveral implications for classroom practice. Since the small group discussionsprovided information indicative of students’ mathematical power, the result suggeststhat the small group context can be used to gather such information. As such,mathematics teachers are encouraged to use it as a context for gathering informationindicative of students’ mathematical power. Also, mathematics teachers are103encouraged to consciously provide for all categories of mathematical activities that areassociated with SAS if students are to meet the expectations of the reform. Limitingthe categories will limit the extent to which students develop mathematical power.Also, when teachers adopt the use of small groups to gather information indicative ofstudents’ mathematical power, they are encouraged not to focus only on students’ talk,since sometimes, combining students’ talk with their written scripts provides betterinsights into the subject of discussion.A classroom instructional process, which involves discussions of mathematicalactivities, may help improve students’ proficiency in mathematics because as studentsshift their reasoning consensually or conceptually as a result of group discussions,they tend to align themselves with viewpoints that are compatible with acceptableviewpoints within mathematics. For students to confidently align themselves withacceptable viewpoints, teachers need to encourage their students to self-validate(Anderson, 1993) their solution. This was evidenced in the study by students notchanging their solution when they could self-validate it. Thus, the ability to self-validate should provide the control element shaping the direction of the shifts.Finally, teachers need to monitor the group discussions so that prompts can begiven to challenge shifts not aligned with acceptable viewpoints within mathematics.Giving the appropriate prompts at the appropriate time means that the teachers areknowledgeable enough to detect students’ difficulties (and strengths) and know whatprompts to give to help clarify students’ thinking. Monitoring is also necessary ifteachers are to identify the “buds” or “flowers” that are “in the course of maturing”(Vygotsky, 1978, p. 87) and provide appropriate mathematical activities that willenhance the growth of those buds or flowers.ReflectionsHaving gone through this study, I have gained insights concerning difficultiesassociated with capturing students’ mathematical power through the student104assessment standards (SAS). I would like to share some of these insights with themathematics education community. The circular definition of students’ mathematicalpower makes it problematic when deciding what constitutes students’ mathematicalpower. For example, the NCTM considers students’ mathematical power as one of thestudent assessment standards and considers mathematical reasoning also as one ofthe student assessment standards. However, a category of mathematical activityassociated with students’ mathematical power involves mathematical reasoning also.Thus, conceptually, mathematical reasoning is presented as a subset of students’mathematical power and at the same time presented as of equal importance tostudents’ mathematical power which is a student assessment standard. Whatconstitutes students’ mathematical power is therefore difficult to determine and someconceptual clarification is needed.Talking about conceptual clarification brings to mind the difficulty I had decidingwhether the mathematical power demonstrated by the students in the small group wasfor the group or for the individuals in the group. During the discussions someparticular students seemed to talk frequently, but as responses to what other students,who seemed talk less frequently, said in the group. In either case, the talk reflected acategory of mathematical activity associated with one of the students assessmentstandards. So, was it the student who talked more frequently that demonstratedmathematical power or the one who talked less frequently? Or was it the whole groupthat demonstrated mathematical power? It was a difficult decision for me to take and Ifound myself “buying” into the idea that in the small group context, the individualdemonstrated mathematical power which was “mediated” by the group interaction. Bythat I mean there was some “group effect” on the individual’s demonstration ofmathematical power, and I am still grappling with how to determine the extent of thatgroup effect.Sometimes, deciding on which categories of mathematical activities particularinformation reflects was difficult because of the overlap of some of the categories105associated with SAS. Evidence that is indicative of a student’s ability “to apply avariety of strategies to solve problems”, for example, might also be indicative of thatstudent’s “flexibility in exploring mathematical ideas and trying alternative methods insolving problems.” However, these two categories of mathematical activities areassociated with two different SAS. Rather, instead of creating separate categories forsuch mathematical activities, efforts should be made to unity such categories so as toprovide a more holistic picture of students’ mathematical power.Possibilities for future researchSeveral issues raised by this study could become the focus of furtherinvestigations. Even though students were using some mathematical vocabularies,this study did not probe the extent of students’ understanding associated with the useof those vocabularies. Further investigations into students’ understanding associatedwith the mathematical vocabularies they use should be insightful.The study did not compare students’ discussions by problems. It should be ofinterest to investigate the type of problems that will permit students to provideinformation reflective of a wider range of categories of mathematical activitiesassociated with SAS. Should problems be open-ended, non routine? Or cantextbook problems provide similar information?For this study, the small group format provided a context for gatheringinformation on students’ demonstration of mathematical power. The focus was not onhow the small group format was contributing to the attainment of students’mathematical power. Information on the extent to which the small group format canfacilitate the attainment of students’ mathematical power deserves attention in furtherresearch.Also, George was moved to another group because he was not talking in thegroup he was in initially. A follow up investigation of George (or any student whose106group membership changes) to determine his oral interaction with members of thenew group can become the focus for further research.Finally, there could be further investigations into the effect of consensual shift onstudents’ understanding of mathematics and how students’ understanding ofmathematics influences how they align themselves with majority view points.Investigations into these issues could provide insights that would guide the use ofstudents’ discussions through the small group format for assessing students’mathematical power.Final noteThis study demonstrates in a small way that from small group discussions, therecan be observable events that reflect the categories of mathematical activitiesassociated with SAS. To continue with the current reform within mathematicseducation, teachers should be encouraged to take risks to identify classroom eventsthat reflect the seemingly rhetoric parts of the SAS. Teachers will need a lot ofguidance and encouragement, and I hope this study provides an additional source ofencouragement that it can be done. As reported in the March 1994 issue of theJournal of Research in Mathematics Education (volume 25 number 2, page 115):Perhaps the most obvious research-related response to the Standards is theidentification and clarification of the research base for the recommendationscontained in the document. The Standards document contains manyrecommendations, but in general it does not provide a research context for therecommendations, even when such a context is available.Research Advisory Committee, “NCTM Curriculum and Evaluation Standardsfor School Mathematics: Responses from the research community”JRME, 1988, 19, p. 339In line with the aspirations of the Research Advisory Committee, all that thisstudy sought to do was to provide a research context for using the small group formatto gather information indicative of students’ mathematical power.107BIBLIOGRAPHYAlberta Education. (1987). Problem solving in mathematics: Focus for the future. InSenior High School Monograph. Calgary: Alberta EducationAnderson, Ann. (1993). Assessment: a means to empower students? In Norman L.Webb & Arthur F. Coxford (eds.), Assessment in the Mathematics Classroom(1993 Yearbook) (103-110). Reston, Virginia: National Council of Teachers ofMathematics.Archbald, Doug A. & Newmann, Fred M. (1988). 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Small-group interactions as a sourceof learning opportunities in second grade mathematics. Journal for Research inMathematics Education, 22(5), 390-408.Yackel, Erna; Cobb, Paul; Wood, Terry; Merkel, Graceann. (1990). Research intopractice: Experience, problem solving, and discourse as central aspects ofconstructivism. Arithmetic Teacher, (4), 34-35.Yin, Robert K. (1989). Case study research: Design and methods. Newbury Park:Sage Publications.117Appendix A(Problems 1 - 7)Problem IName_____________________________________ Sex________Instruction: Solve the following problem on your own and discuss the solution withyour group members. Respond to the questionnaire as honestly as you can.Remember, this is not an examination.Problem: B. C. Telephone has a way of determining the cost in dollars of makingtelephone calls from cities in B. C. to other cities in Canada. The cost of a telephonecall from Vancouver to Calgary is determined fromC = 1.80 + 0.75(m - 1), where m is the number of minutes you speak.i). How much will you pay if you speak for 5 minutes?ii). How many minutes will you speak if you have to pay $25.05?118Problem 2Name_____________________________________ Sex________Instruction: Solve the following problem on your own and discuss the solution withyour group members. Respond to the questionnaire as honestly as you can.Remember, this is not an examination.Problem:a) Is 2m3 + 3m2 a polynomial? Explain your answer.b) A new type of lottery was introduced in Vancouver recently. A ticket costs$1.00 and it looks like the diagram below.V 2‘ m m nmnnmThe entries in the 9 boxes are obtained by completing the multiplication table.To win, you must pick without watching, any two digits from 0, 1, 2, 3, 4, 5, 6, 7,8, 9 in a container. The first number you pick is ‘m’ and the second number youpick is ‘n’. The amount of money you win is the sum of all the entries in the 9boxes.I) If you pick m = 2 and n = 3, how much will you win?119011ssen6inoAiosuosei8AIFJhIMse6iieqnoA9A!6hIMei1upue1woSflBAeqssen(I!o$osnuoqee6noicU!MnoALpnwM014nO6uipuieiojeqse!JueeiownseqAjdw.isnoAi:eoNProblem 3Name_____________________________________ Sex________Instruction: Solve the following problem on your own and discuss the solution withyour group members. Respond to the questionnaire as honestly as you can.Remember, this is not an examination.Problem:a) Without using a calculator:i) Find the value of x2 -6x +8 ifx= 5.ii) Factorize x2 - 6x + 8 and then find the factored value if x = 5.iii) Comment on your answers to (i) and (ii).b) Using a calculator or otherwise:i) Find the value of 980x - 2x if x = 480.ii) Given that the factored form of the binomial in (i) is 2x(490 - x), find itsvalue if x = 480.iii) Comment on your answers to (i) and (ii).C) Suppose your dad gives you and your sister a square plot of land of side 64meters. If your sister’s share is a square plot of side 36 meters, what area ofthe land is left for you?121Problem 4Name Sex_______Instruction: Solve the following problem on your own and discuss the solution withyour group members. Respond to the questionnaire as honestly as you can.Remember, this is not an examination.Problem:You have a plot of garden shown below with its dimensions measured inmeters.5a) Find and simplify an expression for the area of the plot.b) Find and simplify an expression for the perimeter of the plot.c) What is the ratio area/perimeter? Simplify your result.2g122Problem 5Name_____________________________________ Sex_______Instruction: Solve the following problem on your own and discuss the solution withyour group members. Respond to the questionnaire as honestly as you can.Remember, this is not an examination.Problem:At a recent Valentine Day party in Vancouver, the host came up with a puzzlefor the guests to solve. She told them: “I have three sons. The product of theirages is 72 and the sum of their ages is my car number. If my car number is VTM014 and my eldest son likes to go fishing with his father, what are the ages ofmy three boys?” Solve the puzzle.123Problem 6Name_____________________________________ Sex________Instruction: Solve the following problem on your own and discuss the solution withyour group members. Respond to the questionnaire as honestly as you can.Remember, this is not an examination.Problem:Which is larger and why (i) 3/2 x3 + 1/4 x2 or (ii) 21 x4 + 7/2 xExplain your solution.124Problem 7Name__________________________________ Sex_______Instruction: Solve the following problem on your own and discuss the solution withyour group members. Respond to the questionnaire as honestly as you can.Remember, this is not an examination.Problem:For the past hockey season, Vancouver Canucks scored 70 goals in the last 24games. In 10 of these games, the team scored 2 goals in each game. In theother games they scored 3 or 4 goals. The Canucks won only when theyscored 4 goals. How many games did the team win?125Appendix B(Student-Questionnaire Items)1. Name______________________ Sex_______2. Has the group discussion about the solution to the problem helped you tounderstand the problem and its solution better? In what way? Give specific examples.3. As you worked in groups, what mathematical concept(s) did you find important forsolving the problem?4. How did you use this (these) concept(s) you identified in (3) to solve the problem?Explain your solution.5. Were you able to identify the same concept(s) (as in 3 above) when you weresolving the problem on your own? If not, how did you learn about these concepts?6. Why do you think you have solved the problem? Explain.7. Would you like to see this type of group discussion of the solution to a mathematicalproblem as part of your normal mathematics classes? Why?126Appendix CSOME POINTS TO NOTE WHEN YOU WORK IN GROUPS•Ask questions.•Discuss ideas.•Make mistakes.•Learn to listen to other’s ideas.•Respect other members’ ideas.•Offer constructive criticism.•You must perceive that you are part of a team and that you have acommon goal.•Success of the group is the success of the individual.•Be prepared to talk with all members of the group.•AlI members of the group must contribute to the group activity.127PsiPS5Appendix DFreciuencv of ExcerDts Reflecting Cateaories of Mathematical ActivitiesAssociated with Student Assessment Standards (SAS)03 337Parts Prob 1 Prob 2 Prob 3 Prob 4 Prob 5 Prob 6 Prob 7 RowCl 2 3 1 4 1 1 1 13C2 4 1 1 1 1 1 1 10 38C3 1 3 4 4 3 15MCi 1 2 1 4MC2 1 1 2MC3 1 3 2 1 7MC4 2 1 1 1 5 29MC5 1 2 2 1 1 1 8MC6 2 2MC7 1 1MP1 1 1 1 1 3 7MP2 1 1 2MP3 2 1 2 1 1 7MP4 1 2 1 1 3 8 37MP5 2 3 1 3 1 1 1 12MP6 0MP7 1 1PS2 2 1 3PS3 1 2 3 1 1 2 10PS4 4 5 3 3 2 3 201281MD2 1 1 2 1 1 1 7MD3 3 3 1 1 1 1 1 11MD4 1 1MD5 4 1 3 2 1 1 1 13MD6 2 2Note. The columns represent the problems used for the study while the rowsrepresent categories of mathematical activities associated with SAS. An entry in a cellrepresents the number of excerpts from the discussions of a particular problem thatreflect categories of mathematical activities associated with each of SAS.MD1 1 1 1 1 1 1 7MD7432 2129

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