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Introducing computers into the Mathematics classroom: a case study Garfinkel, Sandra Lea 1992

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INTRODUCING COMPUTERS INTO THE MATHEMATICS CLASSROOM:A CASE STUDYBySANDRA LEA GARFINKELB.Sc., The University of Manitoba, 1977Teacher Certification, The University of Manitoba, 1978A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF ARTSinTHE FACULTY OF GRADUATE STUDIES(Department of Mathematics and Science Education)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992© Sandra Lea Garfinkel, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of  161-4yitd7Y,S7 ^0e,ActThe University of British ColumbiaVancouver, CanadaDate apiAl 30. )99-1DE-6 (2/88)ABSTRACTThis research study is concerned with the introduction ofcomputers into a mathematics classroom. The studyinvestigates the teaching of a graphing unit in theMathematics 11 curriculum in a computer-enhanced setting andis framed within four research questions. The studyinvestigates the features that emerged in a computer-basedmathematics classroom, whether the teachers and studentsadapted to such a setting, and the advantages anddisadvantages experienced by both groups.Two teachers and their students were involved in thisstudy. The mathematics classroom was equipped with 16networked computers and a teacher workstation outfitted withan overhead projection device. The computer was used by theteacher as a demonstration tool and by the students as an aidto completing assignments. The students usually worked insmall groups of two or three. Over a period of two weeksthree mathematics classes were observed. The data collectedduring this period consist of detailed descriptions of eventsand people, as well as analytic ideas and personalimpressions.At the conclusion of the two week observation period, 11classes had been observed and ten salient features emerged inthe analysis of the data. The site and the software seemed tohave the greatest influence on the introduction ofiicomputers into the mathematics classroom. Cooperativelearning also played a major role in the computer-basedclassroom. It was also shown that teachers require in-servicetraining in the use of computers; they need to developmaterials and tests to correlate with the software and thecurriculum; and they need teaching assistants to helpimplement and monitor computer related activities.This study illustrated the first steps towards fullyintegrating computers in a mathematics classroom. Theteachers modified their teaching format slightly withtraditional teaching methods dominating most of the lessons.They realized the potential of using computers and remainedchallenged to develop their use. Integrating computers in themathematics classroom is a realistic goal; this research studysuggests that the process can only proceed one step at a timewith careful planning and clear goal setting with regard toteaching and evaluation objectives.iiiTABLE OF CONTENTSPAGEABSTRACT^  iiTABLE OF CONTENTS^  ivLIST OF TABLES  viiLIST OF FIGURES^  viiiACKNOWLEDGEMENTS  ixCHAPTER1 THE PROBLEM ^  1Background  3Educational Significance of the Study^ 5Statement of the Problem^ 11Research Questions^  12Method of the Study  12Limitations of the Study^ 142 REVIEW OF THE LITERATURE  16Educational Innovation and the Challengesof the Changing Mathematics Curriculum ^ 17Potential of Computers in theMathematics Classroom^ 17New and Traditional Programs^ 19Implications for Teacher Training^ 21Changing Roles for Teachers andStudents^  22Cooperative Learning^  24Teacher Decision-making  25Summary  29The Effects of Computers on ClassroomInstruction In General^  30Cooperative Learning  31The Integration of Computers InClassrooms^  33Summary  41The Effects of Computers in MathematicsClassrooms^  42Computer Use and Availability^ 43Achievement and Attitudes  46Cooperative Learning^  49Summary^  55Summary of the Review of the Literature ^ 553 METHOD AND PROCEDURE^  57The Case Study Method  57The Site^  58The Teachers^  62The Students  63The Course and the Software^ 64Data Collection and Analysis  674 RESULTS^  69Introduction^  69A Chronological Log of the Data^ 7070758389929699104106109111Summary^  1135^SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS... 117Summary of Observations^  117The Site^  117The Teachers  120The Students  122The Course and the Software^ 123The Research Questions ^  125Salient Features  126How the Teachers Adapt  129Observation ^1Observation ^2Observation ^3Observation ^4Observation ^5Observation ^6Observation ^7Observation ^8Observation ^9Observation ^10Observation ^11How the Students Adapt^  130Advantages and Disadvantages^ 130Conclusions^  131Recommendations for Practice^ 132Recommendations for Future Research^ 133BIBLIOGRAPHY^  135APPENDIX: Student Worksheets^ 142viLIST OF TABLESTABLE^ PAGE1. A chronological log of the observations^ 1152. Salient features of computer use^ 116viiLIST OF FIGURESFIGURE^ PAGE1. The 8x5 timetable^ 592. Site plan^ 60viiiACKNOWLEDGEMENTSI would like to thank Marcia Johnson for reviewing myresearch proposal and giving me the incentive to get thispaper started.I would like to thank Wayne Gatley, Peter Kerr, and DougSuper for their interest, encouragement, and assistance beforeand during data collection.I would like to thank Tom Schroeder, Ann Anderson, andJanice Woodrow for their input during the editing process.Finally, I would like to thank Joe Leal for his expertediting assistance, his patience and support throughout.Thank-you to Jeremy Leal for giving me the reason to finish.ixCHAPTER 1THE PROBLEMThis study investigates the introduction of computertechnology into a mathematics classroom. The UnderachievingCurriculum (McKnight et al., 1987) suggests that "technology,including calculators and computers, should find anappropriate place in the mathematics classroom. The use ofsuch technology will promote the upgrading of the mathematicalcontent of the curriculum as well as assist in the effectiveteaching of that content" (pp. 116-117). The 21st century isquickly approaching, and technology has taken a giant leapinto the classrooms of British Columbia. New instructionalaids have arrived in the form of computers of varying types,sizes, and speeds. Along with computers have come videodisks, CD-ROMS, hyper-media, and many other confusing, yetexciting technologies. Ragsdale (1988) states that these newmedia will reshape societies, their governments,bureaucracies, and institutions.In this age of the Information Revolution, many of theneeds our society creates are easily met by the computer.Technology plays a dominant role in our day to daytransactions, yet the relationship between most teachers andcomputers is a vicarious one. That is, most teachers haveonly heard or read about the overwhelming advances of computertechnology in Western society; few have experienced them. Onthe other hand, many teachers do have and will have the1opportunity in this decade to experience the technology of themicrocomputer.Yet the argument rages on within the educationalcommunity as to the value of computers and accompanyingsoftware. For example, some teachers fear that the adoptionof a new application-based curriculum will increase thedemands placed on them in terms of flexibility and creativity(Fey, 1984). Pea and Sheingold (1987) observe that the broadscale changes at the classroom level, much less than atsocietal level, appear to be slow to emerge and elusive tostudy. The National Council of Teachers of Mathematics (1991)urges strongly for mathematics teaching reform based ondeveloping students' mathematical power, which is defined asthe ability to reason and explore logically, to solve non-routine problems, and to connect mathematics with otherintellectual activities. Although the controversies supportand oppose the adoption of technological changes, theprofessional standards document (National Council of Teachersof Mathematics, 1991) argues that a classroom environment mustbe created that fosters these new ways of thinking about andapproaching mathematics. Technology is only one new tool thatcan help students pursue such mathematical investigations.Upon reading this and other studies, the reader must decideabout the value of computers, about whether teachers' fearsare justified, and about the implications of mathematicsteaching reform.2BackgroundThe decision to investigate the use of computertechnology in the classroom evolved from a keen personalinterest in computer technology for the past eight or nineyears. Initially, this interest was manifested in the form ofsupport for the teaching of computer literacy in the secondaryschools and in the teaching of computing studies and computerscience courses. The success of these early endeavors wasgauged by the steady increase in enrollment in these electivecourses and in the rise in the general level of computerliteracy in the student population. However, the maximumeducational impact of the computer (as asserted by Howson &Wilson, 1986; Ragsdale, 1988; Heid, Sheets, & Matras, 1990;Fullan, 1991) lies in its potential as a tool to enhancestudent learning not as an object of study itself. In orderto actualize this potential, computer technology must beintegrated with other means of instruction in standardeducational settings.Mathematics was chosen as the curriculum area for thestudy for several reasons. In the early 1980s, computingscience was under the jurisdiction of the mathematicsdepartments in many secondary schools. Thus the researcher,having access to the computers used for computer science, wasable to experiment with a single computer in her mathematicsclassroom. The software which was available for use on theApple Ile computer was limited, but the author maintained her3interest, as she could see the potential for demonstration,especially for the teaching of graphing techniques in algebra.She had attended many presentations and workshops on the useof specific pieces of software, but she was curious about howthe use of the computer and the software could be incorporatedinto the regular classroom routine. How could the computer,or computers if one was fortunate to have access to more thanone, be included in the normal classroom routine? How couldthe computer be used effectively other than being a reward forthe students finishing their assignment early? How could thecomputer be used for the weak students so that they are notpenalized by having to finish in-class assignments at home?Would the teacher be able to finish the curriculum if time wasallotted for computer use? These questions always weighedheavily on the mind of the author.In the fall of 1989, Lord Byng Secondary School inVancouver, B.C. entered into a joint study with IBM Canada touse IBM hardware and a software package called the MathematicsExploration Toolkit (MET) (Wicat Systems Inc., 1988) in anetworked setting. IBM provided 16 PS/1 computers networkedto a central fileserver and Lord Byng Secondary was licensedto use the MET software. The MET software is a computer-basedalgebra system which has the capacity to perform symbolicmanipulation in algebra and calculus, as well as the capacityto graph various equations. The researcher was a teacher atLord Byng Secondary at that time and saw the joint study as anexcellent opportunity to conduct a study of the use of4computers in the mathematics classroom. The researcher wasinterested in observing various aspects of introducingcomputers into a regular classroom setting, from the technicalaspects of setting up the hardware, to the physicalarrangement of the classroom and its furniture, to themanagement of student movement and activities within thatsetting.The researcher felt strongly that if she could observeand document what actually transpires in a computer enrichedmathematics classroom, other teachers would be encouraged toadopt worthwhile technological changes in their classrooms.It is the challenge of the unknown which inhibits many peoplefrom using the computer. The researcher felt that if teacherscould have some grasp of what to anticipate and know thatothers have experienced the same challenges, they would notabandon their endeavors with the computer in frustration. Theresearcher found from prior personal experience that one mustdevelop a tolerance for unanticipated events when working withcomputers for teaching and learning purposes.Educational Significance of the StudyRagsdale (1988) feels that the school is the place wherethe future of computers will be predicted and shaped. "It maybe that not only is the ultimate effect of computers not yetknown, but major effects of their use have not even beenanticipated" (p. 14). Papert (1980) describes his microworlds 5as incubators in which certain kinds of mathematical thinkingcan hatch and easily grow. In computer-assisted instruction(CAI), for example, a child acquires a sense of mastery over apowerful piece of technology. Programming and CAI are onlytwo of the applications of computers in the classroom. Otherapplications are games, simulation programs, and drill andpractice tutorials. The potential for education and adventureis mind boggling. The emphasis in the field of computermathematics education is on promoting better understanding ofthe mathematics topics. In School Mathematics in the 1990s Howson and Wilson (1986) state that the rapid developments inthe applications in mathematics brought about by the computerrevolution offer a rich source for experiments anddevelopments with innovative content within a new model. Fey(1984) states that the computer presents new challenges thatdemand higher-level analytical thinking.Mathematical concepts new to school mathematics curriculawill eventually emerge in school curricula because of the newmethods of writing and calculating offered by computers andinformatics. Computers are not only a new tool formathematical research and teaching, but also a source of newareas of educational research. With graphic representations,computer simulations, and symbolic manipulation,interpretation of data will be easier. Students are beginningto be able to exercise processes of exploration and discoverywith the infusion of current mathematics software. The authorpredicts that wide screen displays and new video projection6techniques will open new possibilities for teachers to expand"chalk and talk" techniques. "This in turn will require thatteachers relax their control over the ways in which studentsreceive and process information" (Baird, 1986, p. 49).Driscoll (1988) refers to experts who say that students needto become more involved in doing rather than learning aboutmathematics. He feels that if the underachieving curriculum(as described in McKnight et al., 1987) is to move beyond thecurrent model of teaching, class periods dominated by lectureand silent student practice can no longer be the rule.Fey (1984) asserts that the assumptions on which thetraditional mathematics curriculum is based will be changedaccording to the promised and demonstrated capabilities ofcomputers and information technology. Some of the mostprominent capabilities are numerical calculations ("numbercrunching"), symbolic calculations, graphics, databases, andnetworking. What seems likely to succeed is a sequence ofgradual changes (Fey, 1984). This study will observeprimarily the use of one piece of software, MET, which doessymbolic manipulation and has limited graphic capabilities.In this study only MET's graphing features were observed.The National Advisory Committee on Mathematical Education(NACOME) (1975) recommended implementing computers inclassrooms but with the stipulation that computers must gobeyond the traditional computer-assisted instruction orcomputer-management systems, and that computer literacy mustbe "hands-on." The present research study is significant in7that it is a case study of what actually occurs in amathematics classroom while computers are used to teach aparticular algebra unit. Frustration results when the willexists to experiment with new teaching methods and tools butthe support and experience are lacking. Battista (1988)cautions that if we are not careful, teachers who are naiveand negative about the use of computers in mathematics maybecome frustrated and eventually reject the use of thistechnology as a tool for teaching and learning. This studywill share the experience of two teachers at a similar entrylevel, that is, the two teachers in the study had little or noprior experience in teaching mathematics using the computerand the selected software.One of the concerns that has been raised since computersstarted to find their way into mathematics classrooms is howthe curriculum will change if students are to be prepared forthe technological age. As computers decrease in cost, theirpurchase becomes feasible for many school districts.Gawronski (1982) suggests that the potential for curriculumchange is increased for two reasons: the relatively low costof the microcomputer, and as more computers are beingpurchased for business and personal use, the public willexpect the schools to be teaching about their uses. Today,literacy courses are very common, if not present in everysecondary school in the country, but integration of computersin the teaching of other subjects is not nearly so widespread.8The opinions in the literature on the topic of curriculumchanges are very similar. Taylor (1981) writing on the topicof calculators and computers, supports a total revision of themathematics curriculum because of the proliferation ofcomputing power. He believes the curriculum needs developmentin the areas of problem solving, consumer applications,estimation and approximation, and mathematical modeling. Morerecently, Heid, Sheets, and Matras (1990) identified twochallenges: the creation and adoption of a curriculum thatutilizes computers and a changing role for students andteachers. The literature often expounds the virtues of thecomputer and bursts with enthusiasm, but unfortunately it isdifficult to find teachers using computers for instruction.There is a lack of empirical research to support thesuggestions that have been made. Day (1987) found that themost frequent use of computers has been to reinforcecomputational skills. This finding contradicts the wish tode-emphasize calculations and the desire to promoteapplications and problem-solving. Cuban (1986) defendsteachers by saying that it is unrealistic to expect them toincorporate technological innovations automatically.Hatfield (1983) defined instructional computing as "anyapplication of a computer involving direct contact by eitherthe student or the teacher, which serves the goals andfunctions of instruction" (p. 44). This definition describesthe computer environment in which the current research studytakes place. The study relates the experiences of both the9teachers and the students involved. The teachers used onecomputer for demonstration and discussion. The studentsworking alone or in small groups completed the assignedlessons using computers. This study concentrates on the waysin which the teachers may organize their instruction and theirclassroom, and how they can adapt their use of the computer tothe current mathematics curriculum with no implications forcurriculum change.In summary, this research study is significant in that itdoes not require a new curriculum in order to use the computeras a tool in the mathematics classroom. It is a study of bothteacher use and student use of the computer to teach and learna traditional algebra unit in a novel way. It is a practicalstudy because it describes the slow introduction of thecomputer, that is, the computer is not used throughout theentire school year in the mathematics course. The teachershave instead decided to try their hand at one particular unitwhich they suspect is appropriate to be taught and learnedwith the aid of a computer. Finally, this study iseducationally significant because so few teachers are usingcomputers for instruction and because of the scarcity ofresearch in this area. Much of the research into the use ofcomputers in secondary school mathematics focuses onachievement gains in test scores, but little is known aboutthe difficulties involved with adapting teaching and learningstyles. Students are accustomed to very traditional classroomsettings, and the mere presence of a computer implies aspecial situation.10Statement of the ProblemThe purpose of this study is to discover some of thefactors involved when computers are introduced into atraditional mathematics classroom. The Research AdvisoryCommittee of the National Council of Teachers of Mathematics(1990) notes that technology allows the emphasis of differentparts of the traditional curriculum and the de-emphasis ofothers in order to introduce topics new to the curriculum andto reorganize instruction. This study investigates the use ofthe computer in teaching a single topic, the graphing offunctions, that is already part of the Mathematics 11curriculum.It is based upon Fey's advice that "...without aprecipitous break from tradition, teachers can begin preparingtheir students for work in the technical environment of thefuture by modest changes in class activity" (Fey, 1984, p.93). The researcher was interested in documenting her ownexperiences as an instructor and observing and describing acolleague in the same setting. The students in the studyoften worked in pairs, thus the investigator observed how thiscooperative arrangement influenced class activity.11Research OuestionsWhat salient features emerge in the mathematics classroomwhen the computer is used to teach a graphing unit?How do the teachers adapt the use of the computer to thegraphing topic and the already established classroomroutine and setting?How do the students adapt to the computer-based learningactivities they experience?What advantages and/or disadvantages do the teachers andstudents experience when using the computer?Method of the StudyThe case study method of inquiry has been chosen for usein this study. Stake (1978) states that the perspective ofeveryday life is better when reporting to lay audiences.Since one of the goals of this study is to assist practisingmathematics teachers to implement computers in their classroomand many of these teachers are inexperienced computer users,the case study method is deemed appropriate.The research site was a secondary school in Vancouver,British Columbia. Three Mathematics 11 classes were observedover a period of two weeks. The researcher was the teacher in12one of the three classes. The mathematics classroom contained16 IBM PS/1 computers networked to a fileserver, and it alsohad a stand-alone computer outfitted with an overheadprojection device at the front of the room for demonstrationpurposes.The researcher chose to observe the Mathematics 11classes while they were studying the unit on graphing offunctions, which is chapters 6 and 7 in their textbook. Inanother teacher's classes, the researcher was a passiveobserver while the teacher presented his lesson, and was aparticipant observer when the students worked on theirassigned tasks at the computer. The researcher took fieldnotes as she observed passively and as she circulated aroundthe classroom chatting with and observing the students. Thefield notes were typed up each evening to produce a thoroughdescription of the activities during the mathematics classes.The researcher conducted informal interviews with the teacheron a daily basis and recorded his comments in point form inthe field notes. These interviews were also typed from pointform into a more detailed account.The researcher also used her own Mathematics 11 class forresearch purposes. She took field notes as the students wereworking at the computer on their assignments. Each eveningthese notes were typed in greater detail than the point formtaken on site. As well, the researcher kept a journal of herperceptions of her own teaching and her personal experience ofteaching a lesson with the aid of a computer.13At the beginning of the study, the researcher explainedto the students her dual role as teacher and investigator.The students were given the option of not participating in thestudy by declining to answer the researcher's queries of themwhile she circulated around the room and chatted with them.The method of the study will be described in greater detail inChapter 3.Limitations of the StudyThe research study was limited to one pre-selectedresearch site. As well, at this one site, one particularpiece of software was used predominantly and at one gradelevel. As explained earlier, this site was chosen because ofthe joint study into the use of the MET software. This was avery convenient situation in that under the sponsorship of theIBM company, the availability of hardware, software, andservice technicians would greatly encourage and assist in theimplementation of the computer in the mathematics classrooms.The study was limited to two teachers, one being theauthor. These two teachers were teaching Mathematics 11 inthe computer laboratory at the time of the study. Thisfacilitated access to the computers.This study is also limited to one instructional unit inMathematics 11, the graphing of functions. This unit waschosen because the researcher had some previous experienceusing the computer to teach graphing techniques in14mathematics. In the previous school year, the other teacherin the study had collaborated with the district's mathematicsconsultant in developing some lesson materials, which wereguided explorations using the MET software. However, theopportunity to test and refine those materials was notavailable prior to the study.15CHAPTER 2REVIEW OF THE LITERATUREThis study is concerned with the process of introducingcomputers into the mathematics classroom. The literaturereview focuses on the process of integration and other relatedissues. The Research Advisory Committee of the NationalCouncil of Teachers of Mathematics report on mathematicseducation reform (1990) states that technology allows one toemphasize different parts of the traditional curriculum andde-emphasize others, to introduce topics new to thecurriculum, and to reorganize instruction. The currentliterature not only focuses on the use of technology in theclassroom, but calls for research into the ways teacherschoose to reorganize their instruction and the curriculum. Atpresent, the use of computers in the mathematics classroom isan option for teachers, and the uses vary from computer-assisted instruction to games. The literature review isdivided into three sections: (a) educational innovation andthe challenges of the changing mathematics curriculum, (b) theeffects of computers on classroom instruction in general, and(c) the effects of computers in mathematics classrooms.16Educational Innovation and the Challenges of the ChangingMathematics CurriculumEducational innovation is a broad reaching term whichdescribes classroom methods that have not been part of atraditional educational setting. These innovations are notnecessarily completely new, but are now more widely acceptedin educational communities. Computers are a good example ofan innovation that is more accepted now than it was in earlieryears in that computers have been used in schools since the1970s even though their widespread use has not been feasiblebecause of the high cost of the hardware and the poorperformance of the software.The discussion of the use of computers as an educationalinnovation in mathematics is presented in six sections: (a)potential of computers in the mathematics classroom, (b) newand traditional programs, (c) implications for teachertraining, (d) changing roles for teachers and students, (e)cooperative learning, and (f) teacher decision-making.Potential of Computers in the Mathematics ClassroomThe potential of computers in the classroom is excitingand at times overwhelming, but "sometimes innovations arerationally solid on the basis of sound theory and principles,but they turn out not to be translatable into practice withthe resources at the disposal of the teachers" (Fullan, 1991,17p. 130). Fullan presents a very cautious view of computer usein the classroom and warns that even where good intentionsexist, the gap between the benefits promised and thosereceived remains very wide. Hofmeister (1984) shares asimilar view, H... [A]s our cultural experience withtelevision indicates, great potential does not guarantee wiseuse" (p. 4).Mathematics classrooms were among the first to introducecomputers because many mathematics teachers had computerprogramming experience. The use of computers in mathematicsclasses from the late 1970s through the early 1980s includedthe teaching of programming skills in languages such asFortran and BASIC and it also included drill and practiceprograms. These applications do not illustrate the potentialof present day computing.We must remember the fate of drill and practiceprogrammed learning and not make the same mistakes again.... If we develop lessons whereby data can illustrate,motivate, and help students remember the traditionalconcepts of algebra, then we will have tapped thecomputer's resources (Krist, 1981, p. 55).The potential of the computer in mathematics education is inits ability to motivate students to think about thepresentations and to ask questions in order to sharpen theirproblem solving abilities.18New and Traditional ProgramsThe comparison of new and traditional programs goesbeyond the introduction of new tools such as computers. Theintroduction of a new program must also consider new orrevised teaching techniques, approaches, materials, andbeliefs. Heid, Sheets, and Matras (1990) discuss howevaluation techniques, time allocation, and classroomactivities change in a computer enhanced curriculum ascompared to the traditional curriculum.The three factors which affect the evaluation techniquesare that the evaluation of a wider range of mathematicalabilities will take place, student learning is more visible ina computer enriched problem solving environment, and newevaluation objectives must be developed. Teachers will haveto consider assigning group marks as well as individual marks.The teacher must keep in mind that some students will havecomputers at home and they may also have access to thesoftware so that they may develop an unfair advantage attesting time in that they will have had more time to practicethe computer related tasks.In discussing time allocation, Heid, Sheets, and Matras(1990) state that it appears that the length and content ofclass discussions is unpredictable. The usual formula ofintroduction/class discussion/guided practice does not workwhen applied in a computer laboratory. If the traditionalformula is used, there is not enough time for computer19activities, and students are forced to try and use thecomputers at lunch time or after school; thus those studentswith computers at home would definitely have an advantage.Geisert and Futrell (1990) also discuss the time factor notingthat "the cost-effectiveness of a computer hinges on how manyhours a day it helps in the teaching/learning process" (p.173). Also with regard to time allocation, absentees andslower paced students are difficult to deal with in alaboratory setting. Heid, Sheets, and Matras (1990) suggestthat an alternative is multi-day goals rather than having tofinish an assignment in a single day.The classroom lessons in the computer-based program mustbe developed so that they interface with the computeractivities and with the prescribed textbook. For moststudents, the textbook is the only resource, other than thenotes they take in class, that they will be able to take homewith them. The lessons must include not only the subjectmatter but software and hardware features and problem-solvingstrategies as well. In a traditional classroom, lessonactivities are centered on classroom notes and the textbook,while in a computer-based classroom lesson activities mustalso take into account features of the software, hardware, andrelated problem-solving strategies.20Implications for Teacher TrainingIn many settings teachers are thrust into computerenriched classrooms with little or no prior training and withlittle opportunity to have completed lesson planning or goalsetting with respect to computer use. Maddux (1988) describeswhat he calls the "Everest Syndrome" which refers to educatorswho use computers simply because they are there. People whosuccumb to this syndrome believe that children will benefitfrom computers merely by being exposed to them. "In schools,this may result in computer implementations that over-emphasize what hardware can be made to do, rather than whatchildren using computers can be empowered to do" (Maddux,1988, p. 5).Empowerment is supported by Munday, Windham, and Stamper(1991) who call for pre-service teacher preparation."Teachers must be given the training needed to empower notonly themselves but also their students through technology"(p. 31). Flake (1990) spent four years interviewing anddocumenting the experiences of pre-service secondarymathematics teachers and then compared the attitudes of thefirst year's group and the last year's group. In their firstyear these student teachers were resistant to using computers.They did not see teachers out in the field using computers intheir classrooms. These pre-service teachers had anopportunity over the course of the four years to use variouspieces of mathematics software and slowly the computers21acquired meaning for them as an instructional tool. Flakefound that in order for the computers to achieve meaning thestudent teachers needed to: a) use the computers often enoughso that they reached a level of immersion, b) experiencelearning in a meaningful way, using computers in this case toimprove their skills in mathematical proofs, and c) work withother students and observe them learning with the computer.Thus teachers require pre-service as well as in-servicetraining. The training will raise their confidence andenlighten them to the potential of computers. The trainingwill help teachers assume new roles and help them to assisttheir students to adopt new roles.Changing Roles for Teachers and StudentsAs computers are integrated in mathematics classrooms,the roles of teachers and students are apt to change.Teachers must learn to act as facilitators of cooperativelearning as well as being learners, and teachers must learn tobecome effective catalysts for student-directed learning.Corbitt (1985) provides cautious advice in stating thatteachers "must recognize that individualized technology-enhanced learning is not synonymous with independent study"(p. 248) and suggests that teachers will become learners.When asked a question by a student in a computer setting, theteacher, in Corbitt's opinion, must be prepared to answer, "Idon't know, yet." In using the computer for exploration, the22student may pose a whole range of questions that the teacherhas not yet considered.Heid, Sheets, and Matras (1990) suggest that teachersmust become technical assistants, collaborators, andfacilitators. These teachers will now be faced with a widerrange of student problems to solve. Many computer settingsare laboratory settings where students must share computerwork stations; thus the students will have to learn to worktogether in these laboratory settings which will be differentfrom science laboratory settings, for example. This impliesthat the students too must become collaborators. They mustaccept greater responsibility for their own learning, and theymust find new ways of assessing their understanding. Thestudents will be forced to think about their findings becauseanswers to computer activities cannot be found at the back ofa textbook.Computer based learning (CBL), which is the use ofcomputers to enhance learning, is problematic. According toOlson (1988), when teachers supervise CBL, their influence ischallenged by such factors as looser class arrangements andmore disruptions. Teachers must diagnose and remedy problemsquickly, and this is often not feasible. As well, in CBL thestudents ask many unanticipated questions and teachers arerequired to be flexible in their responses. Thus, Olson feelsthat computers must fit into classroom routines if they are tobe useful to teachers.23Cooperative LearningAs shown in the previous discussion, the teacher's rolechanges with the introduction of computers into the classroomto include that of a facilitator of cooperative learning. Thestudents must learn to work together in groups of two or morein order to share computer facilities. As well, task relateddiscussions among groups of cooperating students will oftenguide the learning. Heid, Sheets, and Matras (1990)acknowledge the benefits of cooperative learning and statethat the teacher needs to be able to assess the work of thestudents at the computer stations and must be able toencourage successful group arrangements.Lampert (1985) realizes that a clear distinction betweentasks related to social organization and tasks related toinstruction is unachievable. It is not possible in schools toseparate social problems from subject-matter knowledge. Inthe computer laboratory/classroom this is very true; thecomputer classroom tends to be a very social environmentbecause students must usually share equipment; they work inclose proximity to each other; and they are moving from desksto computer stations or from classroom to laboratory. Thus,in order to apply cooperative learning techniques in theclassroom, the innovative teacher is challenged to plancomputer activities in conjunction with cooperative learning.24Teacher Decision-makingThe successful use of a computer in the classroom dependson varied teacher decisions and many planning aspects. Theintroduction of computers broadens the range of decisions tobe made, the first decision being the commitment to actuallyuse the computer. There is no global authority on howcomputers should or can be used in the classroom, thereforeteachers are obliged to exercise more responsibility for whatgoes on in the classroom. "The teacher, responding to theneeds of the pupils, appears to be able to do almost anythingthat professional judgement deems correct" (Eggliston, 1979,p. 2). Some of the problems that arise when attempting to usecomputers in the traditional classroom are hardwaremalfunctioning, equal time allocation for all students at thecomputer, and the realization that the lesson could have beenbetter taught using traditional methods. Lampert (1985)indicates that a great deal of work is required to managethese problems.Some further opinions are offered on the obligationsteachers have to make critical decisions (Corbitt, 1985;Ediger, 1988). Corbitt (1985) discusses the need for teachersto make informed decisions about questions that relate tocertain aspects of the mathematics curriculum, such as whereand how to use technology, how to use computer-managedinstruction, and how to use visual displays to make the25transition from concrete experience to abstract mathematicalideas.Ediger (1988) perceives that administrators and teachersneed to study the philosophical implications involved inutilizing modern technology. There must be relevant purposesinvolved when using the computer in the classroom. As shownearlier (Maddux, 1988), many teachers in the field have nothad pre-service training in the classroom use of computers,and they do not have the luxury of time to study thephilosophical implications. The first decision required ofthe teacher is that of actually deciding to use the computerfor instructional purposes. Once that decision has been made,many other decisions follow in the course of actuallyproceeding with the implementation.The decision-making techniques as described in theliterature (Eggliston, 1979; Corbitt, 1985; Lampert, 1985;Lampert, 1986; Russell and Munby, 1991), evolve over timethrough direct experience. "The teacher is not onlyconfronting choices about how to use means to arrive atdesired ends, but continuously redefining what those ends canbe" (Lampert, 1986, p. 82). Teachers often make spur of themoment decisions and continuously weigh changing evidence.Many of the dilemmas in a computer setting cannot beanticipated and teachers are faced with making the decision torespond to the problem with a solution or to defer a solutionuntil they have had time to ponder a solution further.26Reframing is an important process for guiding thedecision-making process. Russell and Munby (1991) describethe process of refraining which "involves 'seeing' or 'hearing'differently, so the process of perception is a unified processin which observation is interpretive" (p. 165). Reframingreveals new meanings in theory and new strategies forpractice. Teachers learn from experience and this experiencenaturally leads to refraining.Shavelson, Winkler, Stasz, and Feibel (1985) described astudy where teachers have made decisions on how to usemicrocomputers in their classrooms based on their own personalbackgrounds. The study will be described in detail because ofits similarities to the current research study; it is a casestudy, the computers are used in mathematics, and the teachershave decided to use computers because of their personalinterests. Their research was a naturalistic field study ofpublic school teachers who were nominated as "successful" intheir microcomputer based mathematics or science teaching.Shavelson et al. adopted a theoretical framework referred toas teacher decision-making. Teachers' plans are the majorfocus of this conceptualization. The process of formulatingand evaluating plans involves four steps:1. Integrating information about the students, subjectmatter, and teaching environments.2. Monitoring the on-going activities.3. Maintaining the flow of activities or activating aroutine for handling unplanned events.274. Evaluating outcomes of instruction in order toimprove teaching.The data collected in the study were obtained frompersonal semi-structured interviews with 40 elementary and 20secondary teachers. The observational and interview noteswere transferred onto an extensive questionnaire. Sixteenvariables were identified as ways of characterizing teachers'methods for integrating technology in the classroom. These 16variables were then clustered into four groups: those thatreflected the teacher's instructional goals; variables thathighlighted features of the curriculum; those that indexedinstructional activities; and variables which indicated someform of evaluation or perception of success. Six of thevariables characterizing instructional use were then utilizedin a cluster analysis of the data.The researchers found that the teachers varied greatlywith respect to their goals, with respect to the degree towhich they used microcomputers instructionally and integratedthem with other classroom activities, and with the extent towhich they varied their mode of instruction. Another findingwas that all teachers were similar in that they did not usethe computer to motivate the students or for managementpurposes. All the teachers attempted to allocate equalcomputer time to each student, approximately one hour perweek. As well, most teachers used drill and practice, five ofthe 60 teachers using it exclusively. Teachers with a sciencebackground tended to use the drill and practice activities,28whereas teachers with substantial software knowledge tended touse a variety of methods. The researchers also found thatminority and low-ability students received mainly drill andpractice.This study showed that the patterns of microcomputer usevaried among the teachers from enrichment, to exclusivelydrill and practice, to an orchestration of various methods.Although the teachers were chosen for the study because theywere nominated as successful in microcomputer use, successfulin this context can mean that they simply had a microcomputeravailable for their use and accessed it when they could. Theauthors admit that there was not a large group to choose from.It is interesting to note that the researchers found thatdistrict and school policies did not influence use; rather,patterns of use were related to differences in subject-matterbackgrounds of the teachers and the composition of theirclassrooms.SummaryThe literature concerning educational innovation and thechanging mathematics curriculum presents many challenges toteachers and to students. The challenges include changingroles, new cooperative learning settings, and an emphasis onthinking and problem solving skills. Teachers will have tomake decisions on how they will adapt the curriculum anddevelop new curricula to usher in the innovative educational29methods. Teachers will also realize the need for newresources.The literature on decision-making has shown that teachersconstantly have to make decisions, sometimes on the spur ofthe moment, and that they have to accept responsibility forthose decisions and live with the consequences. Problem-solving involves the skill of making informed decisions afterweighing the choices. Reform depends on the teachers beingcritical and on their being able to reframe their experiencesand evaluate the alternatives. Computer use allows theteachers and the students to experiment with new techniquesand to experience education in new ways. The next sectionwill discuss some of the literature dealing with the effectsof computers on classroom instruction in general.The Effects of Computers on Classroom Instruction In General This section of the literature review will concentrate onthe use of computers in classrooms in general, notspecifically mathematics classrooms. An emphasis will placedon case studies. Trollip and Alessi (1988) cite sixcircumstances in which the computer will improve the learningenvironment: a) When the material is difficult to teach usingother media, for example, graphing of complicated equations;b) when practice is needed; c) when there are safety factors,for example, a science experiment; d) when costs are high, forexample, using simulations for training; e) when motivation is30required; and f) when there are logistical difficulties, forexample, studying whale migration. It has been shown (Heid,Sheets & Matras, 1990) that the use of computer technologypromotes small group activities. The discussion will begin bypresenting further claims (Hawkins & Sheingold, 1986; Male,1990; Demana & Waits, 1990) regarding the benefits ofcooperative learning and research supporting those claims(Feldman, Fish, Friend, & Bastone, 1991; Johnson, Johnson, &Stanne, 1986).Cooperative LearningCooperative group work is often necessary in a computer-based learning environment because of the small number ofavailable computers. Cooperative learning requires thatstudents share equipment and collaborate in solving problemsor completing assigned tasks. Hawkins and Sheingold's (1986)research noted that two kinds of changes were observed in acomputer-based learning environment: the ways in whichteachers interact with students in learning situations and theincreasing emphasis on students' collaborative learning. Theypoint out that group work raises questions of how to monitorthe progress of individual students and when to intervene instudent-based activities.Feldman, Fish, Friend, and Bastone (1991) found thatstudents paired at the computer demonstrated enhanced task-oriented social behaviours. They surmised that this could be31because there was less reliance on instructions from theteacher. They felt that there may be more social assistanceamong peers and more self-reliant behaviour because studentsget so many opportunities to work together when using thecomputer. Gender was not a factor in any of the socialbehaviours observed.Johnson, Johnson, and Stanne (1986) investigated theimpact of computer-assisted cooperative, competitive, andindividualistic learning situations on achievement, task-related oral interaction among students, relationships amongstudents, and attitudes towards computers. In general, thestudents working in groups, in this case computer groups,completed more worksheets, correctly answered more questions,scored higher on the final exam, accumulated significantlymore gold (F(2,64) = 28.72, p<0.01), expressed more task-related statements (more to other students, less to theteacher), and nominated more females as future work partnersthan the students working in the other two learningsituations. The students in the cooperative and competitivegroups liked computers more than the students in theindividualistic groups. The results of this study show thepositive effects of the cooperative process, once again, astrong statement in support of cooperative learning.Male (1990) outlines five components required for asuccessful computer lesson:1. Assignment to teams and team preparation.2. Establishing positive interdependence.323. Direct teaching of social skills.4. Ensuring individual accountability.5. Processing.The first component, team assignment, ensures a heterogeneousmix. Male suggests assigning a group grade to provide forstudents depending on each other. There must be a reason forstudents to work together; otherwise they will workindividually. Some of the various social roles suggested arekeyboarder, reporter, praiser, checker, summarizer,encourager, and timekeeper. To ensure individualaccountability, individual quizzes or collecting work atrandom should be attempted. The processing component involvesaddressing issues about how well the groups are functioning.These five components are part of the general cooperativelearning model and are not specifically designed for computerlessons. Demana and Waits (1990) support Male's model withtheir finding that the greatest benefits come from interactivetechnology that is under student and teacher control, thatpromotes exploration, and that enables generalization.The Integration of Computers in Classrooms The integration of computers in classrooms is by no meansa simple process. The literature on the integration processhighlights promises, concerns, successes and failures. Hill,Manzo, Liberman, York, Nichols, and Morgan (1988) make a pleafor computer integration. They see the computer as a "non-33threatening, non-judgemental device which stimulates minds andwhets perceptions" (p. 48). Maddux (1989) on the other handmakes a plea for caution. He believes that the calls forintegration are premature, though they are appropriate for thelong term.Vannatta (1981) describes the evolution of computers forinstructional purposes in the Indianapolis Public Schools(IPS). The three goals of the IPS were to use computers forteaching computer literacy, for practice and reinforcement ofmathematics skills, and for computer programming forinterested students. Although according to today's standardsthese goals are not considered computer integration, Vannattaidentified four areas of concern which are certainly prevalenttoday. These concerns will be presented in detail since theyrelate closely to many of the salient features found in thecurrent research study.The first concern was with regard to administrativedifficulties. An administrator would have to deal withtraining teachers, scheduling, providing instruction regardingthe use of the facilities, and coping with skepticism andpassivity on the part of the staff. It was found that teachertraining, discussed earlier as one of the implications ofeducational innovation and one of the administrative dutiesmentioned above, was a second area for concern. Teachers hadreceived poor college training in the use of computers, sosummer courses were arranged as well as on-site training. Theproblem with on-site training was that the qualified trainers34were often the computer administrators. Those people wereusually over-worked, which is understandable, considering alltheir responsibilities.The third area of concern was the location of thecomputers. They were situated in either classrooms,laboratories, or media rooms. In the classroom situation, theclassrooms were usually mathematics classrooms. Though theyprovided immediate availability, students were denied accesswhile other classes were in session. Security in thissituation was good as long as the teacher kept the doorlocked. The laboratory provided a fixed accessible location,although an aide was required in order to provide security andaccess to the students. The media center tended to encouragebroader use, but classroom demonstrations were more difficultto arrange.Service and maintenance were the fourth area of concern.There was some controversy about service contracts versus on-call service. In-house service is the ideal solution thuseliminating the dispute over who pays for the service.Five case studies will highlight other aspects of theintegration of computers in classrooms (Olson, 1986; Watson,1990; Plomp, Pelgrum, & Steerneman, 1990; Sheingold, Kane, &Endreweit, 1983; Bresler & Walker, 1990).Olson (1986) describes the "teach yourself" routine aspart of "doing" computers as a subject. In the cases hestudied, teachers were experimenting with a new subject andwith new ways of teaching that subject within the existing35constraints of the curriculum and available resources. Thestudents would take turns using the computer while the teachermaintained whole class instruction. There was only onecomputer, and it was part of the classroom reward structure.It is common for teachers to do two things at once but inthis case they found it difficult to teach two classes at thesame time. The "teach yourself" routine works well in othercontexts, for example, in library situations or when eachstudent has a work station. In the case of the stand-alonecomputer, the programming was difficult and not user-friendlythus the students needed help constantly. The students posedmanagement problems that could not be dealt with quickly.Their peers did not assist; they just fixed the problem. Eventhough the routine did not work as well as expected, theteachers were going to continue with this way of teaching.The teachers perceived using the computer as being modernand an expression of what kind of teacher they were. Theyenjoyed observing the students' pleasure of having a computerin the room. Olson says that "as teachers work their waythrough the expressive dimensions of their task, it is likelythat instrumental issues will receive more attention" (Olson,1986, p. 138). The teachers obviously valued the computer asa symbol to be used expressively by them. Olson states thatthe teachers felt that they could ignore the minordifficulties in achieving computer literacy for the short termbecause they were not as important as the expressive process.36Watson (1990) conducted a case study of 11 classrooms andteachers. Four of the classes used one microcomputer, oneclass used two, and six classes used eight to ten machines.Six of the 11 classes took place in specialized computerrooms. The levels of activity and interest were observed, andteachers and pupils reported that there was not enough time tocomplete the exercises. Watson supports a preference forusing stand-alone computers in subject based classrooms. "Itis possible that the establishment of specialist computerrooms will threaten the very innovations that they are meantto herald" (p. 33). The shift from regular classrooms to acomputer room proved to be a negative factor and did notencourage a sense of integration. Though Watson sees thestand-alone system as an easily managed resource, he does notanticipate the difficulties that Olson (1986) describes in hisstudy.Plomp, Pelgrum, and Steerneman (1990) conducted a casestudy of computer use in three schools and followed up with atelephone survey to validate their findings. They selectedthree junior secondary schools which could be considered asleading schools. To correct for possible bias, they followedup their case study with a telephone survey among a sample ofleading schools. Using interviews and questionnaires, theycollected information concerning computer facilities and usesin schools, and the factors influencing implementation.The data collected in the study included factorsfavouring implementation and factors constraining computer37use. Some of the implementation factors reported byprincipals and computer coordinators were reason to startusing computers, policy, facilities, organization, trainingand support. The data from the telephone survey (16 schoolswere surveyed), identified the factors constraining computeruse. Some of the factors were lack of hardware and software,poor quality software, too few computer using teachers, andlack of support and inservice training. Though many computeractivities were taking place, "real" integration was nottaking place. The leading schools did not progress much pastthe grass roots development. "One may suspect that animportant part of the disappointments when introducingcomputers in schools is due to paying insufficient attentionto factors which play a crucial role in implementingeducational change" (Plomp, Pelgrum, & Steerneman, 1990, p.159).In the case study conducted by Sheingold, Kane, andEndreweit (1983) the emphasis was on the social and politicalcontexts that influenced the classrooms as well as whathappened in them. They wanted to identify problems and issuesas well as generate an empirically based research agenda.Fifty-one observations were made across three sites. In theelementary schools, the computers were found in media rooms,in resource centers, and in hallways. These locations avoidedthe challenge of integrating computers in classrooms andcurricula. The researchers found that each school assimilatedthe microcomputers to their own goals, needs, and ways of38operating. The social skills were relatively obvious, butalthough the teachers felt that there were gains in socialskills, no one seemed to know what the children were learningfrom the computer interactive experiences. Perhaps thecomputers were meeting the expressive needs of the teachers asfound by Olson (1986). Bresler and Walker (1990) also addressthe issue of expressive teaching acts, noting that "[e]venwhen an innovation meets people's expressed needs, it maystill not succeed unless it fits in with the patterns by whichthey run their lives as students and teachers" (p. 71).Integration of computers in classrooms requires changesin actual uses of computers, in teaching approaches, and inbeliefs. Bresler and Walker (1990) found an ideal setting forcomputer integration. They present the case study of anintroductory music class at a private university in a high-tech industrial community. They collected their data over aten week period. Seventeen students were observed usingcomputers over approximately 13 sessions. They then conductedfollow-up interviews with 14 of the computer users, six non-users, and an instructor and a teaching assistant.The setting proved to be ideal because it offeredadequate hardware and appropriate software; there was a closematch between the content of the software and the goals of thecourse; there was a commitment from the institution to supportthe course; the instructor supported the use of computers andknew how to use them for the subject (music) of instruction;and the students were interested. Despite these favourable39aspects, the attempt to integrate the computer in this musiccourse (ear-training/theory) failed. Bresler and Walker(1990) outline some of the barriers in terms of theinstructor, the students, and the program designers.The researchers fault the instructor and the factorswhich relate directly to the instructor for the lack ofsuccess. The instructor did not clearly state the goals ofthe course. He did not reconsider the non-computer aspects ofthe course in order to implement the computer component.Evaluation was based on pencil and paper tests. In terms ofthe pedagogical elements, the instructor showed a strongpreference for being the center of attention. He would havehad to relinquish his control by allowing students to move tothe computers. He considered it a chore to ensure that allstudents worked at the computer. While the computer softwarewas student controlled, the students were passive participantsin the music theory class. None of the other teachers usedcomputers for instruction, thus the instructor had no peersupport, and there was a lack of external incentives for theteacher to volunteer extracurricular time to develop thecomputer course. "As mature adults with responsibilities forjobs and families, teachers have little free time for learninga new system. Efforts should be made to relieve matureindividuals from competing responsibilities and to provide anextended period for mastering the new technology" (Bresler andWalker, 1990, p. 71).40The students found the software boring and repetitive andsimply not a lot of fun. Since they had to sign up forcomputer time, and the tests were based on the text materialsanyway, they did not spend their voluntary time at thecomputer. The program designers did not give explicitguidance on how integration should be done. The programmanual did not include tutorial information nor did it stateexplicit goals. This case study once again underscores theneed to rethink and reorganize instruction as it relates tothe curriculum, and since the teachers are ultimatelyresponsible for the successes or failures, they must be givenample opportunity to develop materials and to reshape theirtraditional role.SummarySuccesses and failures to integrate computers in aclassroom routine have been documented in the literature. Theresearch outlines some of the variables and features ofcomputer use in classrooms which have implications forpractising teachers. The major role that cooperative learningserves in computer integrated classrooms is noteworthy. Thecurrent case study provides further insights into many of thesame features presented.41The Effects of Computers in Mathematics ClassroomsThe recent literature concerning the effects of computersin mathematics classrooms contains mostly suggestions aboutthe possibilities of computer use and relies on predictionsrather than facts. The studies which have been conducted arein many ways contrived in that they portray specializedsituations, rather than realistic classroom situations. Forexample, Ernest (1988b) used a small group of only 12students, McCoy's (1991) subjects consisted of high abilitystudents, Martin (1987) conducted her research at the BankStreet Institute, a school designed specifically for researchpurposes, and the school in the study by Zehavi (1988) boughtcomputers with substantial support from the parents, thusthere was political pressure to use them and the teachers hadto ensure that all students had equal access to them. Thepositive results obtained in these studies were mostly inrespect to students' attitudes and motivation.Like the more general studies, the literature regardingcomputer integration in mathematics classrooms at thesecondary and college levels also highlights contributingfactors such as computer use and availability, achievement andattitude, and cooperative learning. Once again, the groupwork model is seen as an exemplar used in most computerintegration situations.42Computer Use and AvailabilityComputer use varies from situation to situation, based onavailability and accessibility factors. As well, theteacher's personal experience with using computers will affecttheir use. "Using computer-augmented instruction is useful insituations in which there are time-consuming activities whichare not central to the objective of the lesson but arenecessary for attaining that objective" (Hassler, 1986, p.186).Tall (1987) presents an opposing point of view. Hereports on the findings of the Teaching Sub-committee of theMathematical Association in the United Kingdom. He found thatthose teachers who used computers more often than others feltinhibited by the difficulties in setting up the computers, thetime it took the students to start using the software in groupwork, and the general difficulties of administering its use.In one school cited in the report, where the computers werenetworked, although there were more computers and the studentsdid not have to line up to use them, there were new skills theteacher had to use to manage the networked system. Thecurriculum had to be planned so that the time that wasallocated for use of the computer room was put to its bestadvantage. Often the system was used so infrequently that theteachers forgot the technical skills and lost confidence.Thus they never developed the expertise to make them want touse the system more often.43These are useful suggestions offered by Tall (1987) andserve as a reference point when considering the studies oncomputer use and availability. The computer tasks must beplanned so that the students get on with their work right awayand the teacher is allowed the freedom to move around the roomto observe and give individual attention. It appears thatthose teachers who are computer literate are using thecomputers successfully. It is these teachers who must helpothers to gain confidence.The software programs used in Damarin, Dziak, Stull, andWhiteman's (1988) study were integrated within the regularmathematics classroom by being placed on three computers inthe classroom. Students were allowed to use the computers asan alternative to beginning the next assignment or instead ofasking the teacher for help during worktime at the end of theclass. The assumption implicit in the study was that thecomputer-based instruction, in this case drill and practice inestimation skills, could be added to the classroom withminimum disruption to the regular routines.Over the eight week period, students were rotated threeto a computer, such that each student received four hours ofcomputer-based instruction. With this approach, the teachercould continue to help individual students with ongoingcurriculum and use the computer to reinforce estimationskills, not a curriculum topic, while not sacrificing timefrom current work.44Martin (1987) looked at district procedures and how theyinfluence classroom change. The first phase of her researchfocused on technology and how it was seen to influence whatwas done in the classrooms, and what in turn influenced itsuse. The 16 teachers of grades four, five, and six and tenstaff developers were given a week long training session inusing the Voyage of the Mimi series. The teachers werepleased with the software and lesson materials because theyallowed an entry point for everyone. The teachers' effortswere highly reinforced by student attentiveness.Martin (1987) considered the demands on the teachers,such as utilizing new technologies in new ways and approachingsubject matter in new contexts, as manageable, evenrevitalizing. The microcomputer was for all the teachers anovel teaching tool and an unfamiliar medium. Though it wasnot evident in the study, Martin suspects that as teachersbecome more comfortable with the materials they will becomemore open-ended in their lesson arrangements.McGivney (1990) describes the use of computers in anintroductory college course catering to students who had had anegative experience with mathematics in the past. There were27 students in the course, and they used a computer laboratorysituated in a room different from the classroom. Theinstructor saw the need for interesting laboratory problemsand found the writing of such material enjoyable but very timeconsuming. He enlisted the aid of a laboratory assistant inorder to manage the course. In general, the student45evaluations of the course were positive, but some criticismswere: a) crowded laboratories, b) lack of help in thelaboratory outside of class time, c) the need for softwaredemonstrations before the laboratory sessions, and d) theabsence of answers to the assignment. Managing classroomtime, resources, and activities is a concern of many teachers,as it was for the teacher in McGivney's study. Unfortunately,this aspect of computer use is not emphasized in theliterature.Achievement and AttitudesMany of the studies of computer use in mathematicsclassrooms are quantitative and have investigated achievementand attitude. In the Zehavi (1988) study, where students wereensured equal access to the computers, the software wasdeveloped to build on students' understanding of points andtheir corresponding coordinates. The students had to identifywhich points fit a given rule, or which rule fits a givenpoint. The seventh grade experimental group of 78 studentsdemonstrated intuitive understanding of how to solve linearequations and inequalities. The retention study done eightmonths later, when the students were in the eighth grade,showed significant positive results (p<0.05).Short term positive results were obtained by Tall andThomas (1991). They found that computer experiences resultedin significant improvement in understanding in the short and46medium term, that is up to 18 months, but if the experienceswere not continued, their effects diminished in the face ofthe overwhelming influences of more recent experiences. Theyalso found evidence among students in the experimental groupof versatile thinking where "global, holistic processingcomplements local, sequential processing" (p. 125). Inindividual interviews, these students were able to offerreasons for their thinking, to discuss processes withouthaving first to carry them out, and to take a global view ofthe problem rather than relying on processes.Ganguli (1990) investigated the use of the computer, atthe college level, essentially as an electronic chalkboard.He found that the students in the experimental group werebetter able to form pictures of algebraic functions after thelesson was taught with computers; thus they performedsignificantly better on the comprehensive final exam, althoughthere were no significant gains with testing in the shortterm.Many of the studies in the literature on computer use inmathematics take place either at the elementary level or thecollege level. This may be due to the constraints of highschool schedules. For example, MacGregor, Shapiro, andNiemier (1988) conducted a study at the college level wherestudents generally received four hours per week of classroominstruction in elementary algebra and an additional one hourin the computer laboratory. Their study showed positiveattitudes with no significant differences in terms of47achievement as measured by test scores. High school studentstypically receive three hours per week of classroominstruction.High ability students were used as the experimental groupin the study conducted by McCoy (1991). The students used acomputer program called the Geometric Supposer once every twoweeks. They worked in small groups on activities from themanual or adapted from their textbook. The group whichreceived the Supposer treatment scored significantly higher onthe final exam than students who did not use the GeometricSupposer computer program. The difference in posttestachievement scores was found in application and higher orderquestions. There was no significant difference on lower levelknowledge and comprehension questions.Using a small group of only 12 students, Ernest (1988b)found that the gain scores were highest with respect tomotivation. He predicted that for a full class, the demandson the teacher would increase four fold.In researching mathematical problem solving skills,Ernest (1988a) used two groups, not assigned randomly, acomputer room group and a classroom group. The computer roomgroup worked in pairs at computers; the classroom group satclose to a single computer and watched a teacherdemonstration. He found that the two CAI modes seemed to haveno significant overall effect on student achievement. Infact, the modes seemed to be ineffective.48Dalton and Hannafin (1988) studied the use of computersto deliver remediation in computation skills. The subjectswere given the "revised math attitude scale," a 20 item Likerttype survey. It was found that students performed best whenthe delivery system employed for the remediation was differentfrom the system used for the initial instruction. The resultsof their study suggest that traditional and computer-baseddelivery have valuable roles in supporting instruction butthey are most valuable when complementing each other. Daltonand Hannafin claim that the issue in computer-assistedinstruction is how to vary instructional methods andtechnologies meaningfully and effectively.Cooperative LearningThe next two studies conclude the review of theliterature. They are relevant to the current research studyin that they are also case studies exploring the mathematicaltopic of functions in a computer laboratory environment.Sheets and Heid (1990) observed teachers and students using anexperimental curriculum called Algebra with Computers. Thiscurriculum focused on the concept of functions and exploredways in which functions arise naturally in a variety of real-world settings. They discuss the group work that emergedduring the implementation of this mathematics curriculum.Most of the classes were held in a standard classroomwith the occasional use of a microcomputer and a large screen49monitor. The teachers and students also had access to alaboratory with 15 microcomputers. The students were given aproblem (to plan a talent show) and an outline of the tasks tobe completed, but they were not given any guidelines on how toorganize themselves into groups in the laboratory. In theearly stages of the laboratory experiences, the teachersfunctioned as technical assistants and as task masters. Thestudents seemed to choose their roles based on differences intheir personal styles. Initially, they paid more attention toproducing an individual report rather than a group report andhurriedly input the data.The students did not hurry through the follow-upactivities. They deliberated and shifted their attention tothe mathematical ideas. It was noted by the observers thatthe stronger students often offered assistance to the weakerones. The students pooled resources by forming teams from thecomputer pairs. "The computer labs provided a learningenvironment where engaging in pair and small-group decisionmaking and utilizing fellow students as legitimate resourcesfor learning increasingly became the order of the day" (Sheets& Heid, 1990, p. 274). Groups seemed to share strategies, andeach pair remained on task until the end of the class period.The teachers found it effective to move from group togroup making problem-solving hints they picked up from othergroups. The teachers shared strategies with students who wereapparently having difficulty in identifying the functionrules. Sheets and Heid (1990) saw this computer environment50as a promising move toward a classroom environment fosteringmore student initiative, responsibility, and commitment totask accomplishment. They also noted that the nature of thesoftware, the type of problems assigned, and the developmentof concepts prior to the computer laboratory work allcontributed to successful small-group environments. TheAlgebra with Computers curriculum illustrates the evolution ofgroup work, the unfolding of individual work styles, and theshift in teacher and student roles.Lynch, Fischer, and Green (1989) developed and field-tested curricular materials to achieve three basic objectives:(a) to develop students' understanding of concepts andabilities to solve problems before they master conventionalsymbol-manipulation techniques, (b) to make the concepts offunction and relation the central organizing themes fortheory, problem-solving, and techniques in algebra, and (c) togive students realistic applications. The authors found that"finding the right combination of challenge and support forstudents in laboratory sessions is not an easy task" (p. 691).They call for new classroom organization and new student-teacher interaction patterns.The lessons consisted of an introduction, an exploration,and a homework assignment. Each significant idea was usuallyintroduced in the context of a realistic situation. Theintroduction was then followed by related explorations insmall groups. The homework assignments focused on analysisand interpretation of the function tables, graphs, rules, or51other results of the computer activities. They used a varietyof software to complement the textbook, and the use ofcomputers was required on most tests and quizzes. Thesestudents were required to read and write in mathematics class.There were two components to the class; students worked aloneon the non-computer component and in the small groups on thecomputer component. The laboratory was across the hall fromthe classroom, so the teacher monitored both rooms by movingfrom one to another.In the first year of the project, the teachers haddifficulty asking good probing questions. The class periodoften ended before closure on an idea had been achieved.The researchers also observed that the discussions increasedthe verbal facility of students. The students graduallybecame more flexible and willing to take risks in problem-solving situations. "The design and management of aneffective laboratory lesson is clearly a valuable skill themathematics teachers should acquire" (Lynch, Fischer, & Green,1989, p. 692). This computer-intensive algebra curriculumillustrates new ways of organizing, delivering, and assessingmathematics instruction.The preceding two case studies (Sheets & Heid, 1990;Lynch, Fischer, & Green, 1989) illustrate the need formathematics instruction to focus on concepts and applications.Proper use of symbolic manipulation software with applicationproblems will facilitate this change of focus (Swadener &Blubaugh, 1990). Bollinger (1989) discusses facts about52symbol manipulation programs and their relation to high schoolmathematics. He emphasizes that their incorporation into highschool instruction is impractical because of their initialdesign for researchers, the high cost of purchasing computers,and because of the lack of a historical precedent for amachine doing symbolic computation as opposed to calculatorswhich have been used since the early 1970s by business,scientists, and engineers.The impression gained from reading published reports onusing computer algebra systems in undergraduate education isthat such use is part of isolated experiments rather than partof a rapidly growing trend. Bollinger (1989) agrees that itis easy to find predictions but difficult to find illuminatingfacts.The push to incorporate symbolic mathematical systems inalgebra is questionable because we are not sure of therelationships between procedural knowledge and skills andthe understanding of algebra. The mathematics educationcommunity needs to determine the effects on mathematicalunderstanding of the use of symbolic mathematics systemsor any other device that carries out procedures beforeembracing them (Bollinger, 1989, p. 14).The opinions in the literature assert some claims andraise some questions regarding computer algebra systems.Bollinger summarizes some of the claims:531. They help with conceptual development. The timesaved in calculations can be used to think about relevantmathematical concepts and to interpret output.2. There is an increase in learning efficiency.3. Students are introduced to an important futureresearch tool.Some of the questions that arise are:1. What skill proficiency is needed to interpret theoutput?2. Will the students actually spend more timeinvestigating concepts if the machine does all the numbercrunching?3. How will the curriculum content change to account forthe use of symbolic manipulation programs?4. Some students may use the computer at home tocomplete assignments. How would students from low incomefamilies receive equal treatment?Bollinger's paper, though published in 1989, was actuallywritten in 1987 before the appearance of the MET software.Although he raises some important questions, recentdevelopments and purchases by high schools of computers andsoftware make the symbolic manipulation program a viablechoice for many educators. For many at this time, thegreatest influence may be found in the use of these programsas a toy, one that can explore intriguing patterns. If theyare used for their computational abilities, they will force achange from what has been traditionally emphasized. There54will be no algebra errors in long calculations. The mistakeswill occur at the idea stage usually by a false assumption ora missing detail. This will force the students to think aboutproblems at a deeper level thus promoting betterunderstanding. The teachers will also need an inquiring mindwith a vision towards the future.SummaryThe literature illustrates that the effects of computersin the mathematics classroom are varied. The computer usesrange from individual CAI with minimum teacher intervention(Damarin et al., 1988) to computer laboratory situations withthe inherent management challenges for the teacher (McGivney,1990). The cooperative learning effects appear to be the mostinspiring. In this context, problem-solving is the focus ofmost computer enhanced mathematics classes. The use ofsymbolic manipulation programs will enhance the problem-solving process and bring with them the promise of a new focusfor computer enriched mathematics classrooms.Summary of the Review of the LiteratureThe literature review highlights innovation and thechanging curriculum, and the effects of computers inside theclassroom. The changing roles of teachers and studentspresent unique challenges, as do the many decisions the55teachers must make. The literature supports the new roles forteachers and students, the need for the development of newteaching skills, and the new resources that must be availablefor teachers (Heid, Sheets, & Matras, 1990; Olson, 1988;Corbitt, 1985; Krist, 1981).Computers are often found in cooperative learningsettings with the emphasis on problem solving skills. Hencecooperative learning is a predominant theme in each section ofthe literature review. The opinions and studies in theliterature outline the potential of the computer andilluminate the salient features of a computer enhancedlearning and teaching environment.The literature contains many opinions about the potentialof the computer as well as some concerns about the hazards ofnon-prudent use. The research into computer use in theclassroom has not shown conclusive evidence that computer useis superior to traditional classroom methods. The opinions inthe literature make predictions about the potential and leavethe reader with a sense that the future of educationalcomputing is yet to be determined and shaped.56CHAPTER 3METHOD AND PROCEDUREThis chapter will present a discussion of the case studymethodology. As well, the site, including teachers, students,and software, will be described. The procedure used and themethod of data collection and analysis will be outlined.The Case Study MethodThe case study, sometimes referred to as field researchor ethnography, was chosen as the method of research. Thischoice was influenced by the work of Olson (1988). In hiscase studies, Olson's goals were to witness and to understandboth the "devils" and the joys of innovation. "Case studiesare useful because they help us see first what practicessignify, and they allow us to form judgements about the pointof those practices" (Olson, 1988, p. viii). Olson sees hisaction research as a way of testing curricular ideas inpractice. The tentative ideas of how the computer mightfunction are made concrete. Action research allows teachersto become the designers of what the innovation is to be,through their analysis of their experience. When thatexperience is reviewed it becomes the basis for newexperience. This case study includes one aspect of actionresearch in that the author is also one of the subjects of thestudy.57Since the purpose of this study is to discover thefactors involved when the computer is introduced into thetraditional mathematics classroom, the case study method wasdeemed most appropriate. The case study method is generallyused when studying contemporary events. "If a case study isabout a new technology, for instance, observations of thetechnology at work are invaluable aids to any furtherunderstanding of the limits or problems with technology" (Yin,1989, p. 91).Day (1984) believes that if teachers are activelyinvolved in research then more learning will be promoted byteachers and researchers. He also feels that if researchersare to achieve success, they must observe teachers in theirnatural settings. As Olson (1984) puts it, "[w]hat teachersknow is in the practice not only in the cognitions thataccompany practice" (p. 37). Bresler and Walker (1990) havethe same concern. They refer to the notion of transferabilitywhich is "the extent to which the case study facilitates thedrawing of inferences by the reader that may haveapplicability in his or her own context or situation" (p. 67).The SiteThe research site was a public secondary school inVancouver, British Columbia. The over 900 students receivethree hours per week of instruction in each of eight subjects.The classroom in which the case study took place was used58solely for teaching mathematics. The classroom was used everyhour of the five hour (block) day (Figure 1).Monday Tuesday Wednesday Thursday FridayA F C H EB G D A FC H E B GD A F C HE B G D XFigure 1. The 8x5 timetableThe teacher assigned to the classroom was Mr. Crosby(pseudonym), and he taught six classes there; the researchertaught two classes in this classroom. Thus the classroom wasused continuously throughout the day with the exception ofbefore and after school, and lunch hour. Mr. Crosby wasusually in this classroom at 8:00 a.m. (classes actually beganat 8:50 a.m.), and he was very often available during lunchtime and after school as well, offering tutorial help tostudents.The classroom (Figure 2) was equipped with tables andchairs, and conventional desks. The teacher's lecture areawas set up along the south wall. It consisted of a table onwhich sat an IBM model 50 computer and keyboard. There was nomonitor attached to the teacher's computer station as it wouldhave made it difficult for the students at the very front ofthe classroom to see the teacher. Beside this table was a59N DOvif,>v----'^ - •- -N. ITFK EYBC'.1L` •^Li .HEATL_^R ELi Li If ^ L[1]^LE, I^II^IIHAIRS am_DESKS WITH CHAIRSfl /ThLThrf t 1TE fZE,HE L^---«CJFILE g_POWER SOURCE I- DOOR^—BOAR DS91 E. L.. YEE.SCREENtrolley with an overhead projector and an overhead projectiondevice connected to the computer. There was very little roomfor the teacher's books and notes. As well, the computerequipment and overhead projector required the use of powercords and extension cords which plugged into the south wall,about one metre behind the teacher. On this wall were whiteboards for writing. A projection screen was suspended fromthe ceiling about midway along the south wall.9.77 SckE: ;soN9 . 2 4'-• rnFigure 2. Site PlanThe tables, chairs, and desks were arranged width-wise inthe classroom. There were enough desks and chairs to60accommodate 35 students sitting in close proximity. Therewere windows with blackout curtains along the north wall.Below the windows along this wall, the heating registers werelocated. Built-in book shelves were situated along the westwall of the classroom. The 16 computer workstations werelocated along the west and north perimeter of the classroom.Each workstation contained a computer, monitor (black andwhite), and keyboard. There were two printers, one networkedto all the workstations, and one connected to a singleworkstation. There were only a few extra chairs in theclassroom, so most students had to move their chairs from thetables in the main area of the classroom to the computerstations. The fileserver was located at the beginning of thebanks of computers. The main power switch was on the wall atthe entrance to the classroom. It had a light indicator andwas operated by a control key. The entrance door to theclassroom was wired to the main security system of the school.The computer equipment was marked with school identification,but had no individual alarms.The computers were purchased with school funds inJanuary, 1991. Previous to that, they were in the school onloan from IBM Canada as part of a joint-study between IBMCanada and the Vancouver School Board, directed towardobserving the use of The Mathematics Exploration Toolkitsoftware, also an IBM marketed product.IBM Canada accepted responsibility for servicing duringthe loan period (September, 1989 until December, 1990). After61that, maintenance and servicing was the school'sresponsibility. It was assumed by the teachers andadministration that the district service personnel wouldservice the equipment when necessary.The TeachersAs mentioned above, there were two teachers involved inthe study, Mr. Crosby and the author. Mr. Crosby had threeyears of experience teaching secondary mathematics but hadlittle computer experience and certainly no formal training incomputer use. He had received some technical training theprevious year (as part of the joint-study agreement) on how touse the computer network, but was sometimes overwhelmed by thetechnical aspects. He received less than ten hours oftraining on the Mathematics Exploration Toolkit, the softwareused in the study. He did not have a computer at home but wasvery keen to try innovative ways of teaching mathematics.There was a computer science teacher at the school whoprovided invaluable assistance to Mr. Crosby. In fact, inSeptember, 1990, she introduced a network management course atthe school. This was an elective course offered to studentswho had the prerequisite Computer Studies 11 course. After afew months in the course, the network management students wereable to provide technical assistance to Mr. Crosby. They alsoprovided assistance to the researcher during the presentstudy.62The researcher had taken two courses on computerprogramming at university and had five years experienceworking with, and teaching about computers. She had givennumerous workshops to colleagues on the use of computers. Aswell, she had seven years of Mathematics teaching experience.The researcher had just completed four years (part-time) ofgraduate courses at the University of British Columbia with amajor in mathematics education and computer studies education.She had limited experience teaching mathematics withcomputers, although she had evaluated numerous pieces ofmathematics software over the course of the previous sixyears, and she kept up-to-date with current literature.The StudentsThe students observed were all enrolled in Mathematics11. Three classes of mathematics were observed, the author'sclass (block A), and Mr. Crosby's two classes (blocks C andE). About 40% of Mr. Crosby's students were known to theresearcher as she had taught them mathematics in previousyears. The students in general had had very little exposureto computers in a mathematics class, though many of them hadexperience using computers either at home or at school in dataprocessing or computer studies classes.There were 36 females and 33 males in total in the threeclasses. Block A had 10 females and 9 males, block C, 9females and 14 males, and block E, 17 females and 10 males.63The researcher explained prior to the commencement of thestudy her dual role of teacher and researcher.The Course and the SoftwareMathematics 11 (British Columbia Ministry of Education,1988) is one of three mathematics courses offered at the Grade11 level in the school. It is the most challenging and isrequired as a pre-requisite for university entrance. Itincludes five topics: algebra (27%), relations and functions(25%), geometry (13%), trigonometry (25%), and data analysis(10%). The percentages refer to estimated instructional time.The Mathematics 11 Curriculum Guide states that it is expectedthat students use calculators throughout the course. Incasual conversations with teachers, it was found that it wasdifficult for teachers to cover the content of the course inone school year (approximately 100 hours). As a result, mostteachers did not teach the data analysis section.The textbook used for the course was Mathematics 11 (Kelly, Alexander, Atkinson, & Swift, 1989). In designing thecase study, the researcher decided that the teaching of therelations and functions topic would be the most appropriate toobserve. This topic consisted of teaching many graphingskills. In Chapter 2, the literature included an example(Zehavi, 1988) of using the computer for teaching graphing atthe secondary level, and there were many commercial softwarepackages available at reasonable cost to teach that skill.64Mr. Crosby and the researcher decided to use the computer toassist in the teaching of chapters 6 and 7 of the textbook.Chapter 6 covers the quadratic functions, and chapter 7 coverstransformations of relations. The teachers estimated that itwould take 16 hours to cover these two chapters without theuse of the computer.The intended learning outcomes (ILOs) of the relationsand functions unit from the curriculum guide are as follows:11.23 Graph functions expressed as rules:a) constant functionsb) linear functionsc) powersd) reciprocal functionse) discontinuous functionsf) absolute value functions11.25 Graph and recognize the following relations:a) parabolab) circle11.26 Show general changes in graphs resulting from changesin the defining equation:a) translationb) dilationc) reflection11.27 Determine the equation of a relation given its graph.11.28 Graph quadratic inequalities in two variables.65The textbook (Kelly, Alexander, Atkinson, & Swift, 1989)was organized such that chapter 6 introduced only theparabola, with each section introducing the three generalchanges, the translation, dilation, and reflection. Theexercises in each section included drawing graphs anddetermining the equation given the graph (ILOS 11.25, 11.26,11.27). Chapter 7 then introduced the other relations witheach section applying the general changes to those graphs(ILOS 11.23, 11.25, 11.26, 11.27).The Mathematics Exploration Toolkit (MET) permits thecomputer to be used as a function plotter, calculator, andsymbolic manipulator. It is also possible to generate a tableof values using MET. It is useful for teachers and studentsfrom algebra through introductory calculus. It provides on-screen help menus and a comprehensive tutorial. Teachers canprepare lessons by using defer sequences. These are sequencesof commands which can be saved and executed at a later time.This allows the teacher to prepare a "slide show," forexample, of graphs or steps necessary to solve an equation.The manual includes teaching ideas, sample presentations, andstudent worksheets related to each application, called guidedexplorations. MET requires MS-DOS computers with 512K RAM,and is in colour. It costs approximately $350 for a singlecopy, $1200 for a school package, or $2300 for a local areanetwork (LAN) package.66Data Collection and AnalysisThe data for this study were gathered during the periodof January 28, 1991 to February 5, 1991. Within the weekprior to observation, the researcher explained the nature andpurpose of the study to the students. Students could opt notto participate by simply declining to answer questions.The data were collected by means of direct observation.In Mr. Crosby's class the researcher took a seat at the backof the classroom and took field notes as Mr. Crosby deliveredhis lesson. When Mr. Crosby was finished presenting hislesson, the researcher circulated throughout the classroom.The field notes were jotted down during the observations andfrom them a detailed description of the events that tookplace, the people, the conversations among people, and theconversations with people was typed up each evening. Thedetailed notes also contained previously forgottenobservations, analytic ideas, and personal impressions.The data collection in the researcher's own classconsisted mainly of mental notes, as she was only able to jotdown notes when the students were actually working at thecomputer or at their desks. These notes also became writtenrecollections. In this situation the researcher was at adisadvantage because there was insufficient time to takenotes, but she did have the opportunity to observe realityfrom the "inside," and she had the advantage of being veryfamiliar with the site and the students. Thus the data67collection consisted of looking, listening, and askingquestions.The most typical mode of qualitative data analysis is anarrative report (Miles & Huberman, 1984). The data gatheringand data analysis may run concurrently, and the analysisrelies heavily on description (Yin, 1989; Wiersma, 1986;Lofland, 1971). "The final stage of analysis occurring afterobservation has ceased becomes, then, a period for bringingfinal order into previously developed ideas" (Lofland, 1971,p. 118). The researcher found that, as she was typing up herfield notes each evening, she was adding comments on herobservations. Her experience with computers and studentsinfluenced her analysis of the data as she observed it. Sherecognized phenomena reported in the literature as shewitnessed them in Mr. Crosby's classes and in her own class.Thus the analysis of the current research study will be in theform of a descriptive case study report, with the fourresearch questions guiding the analysis.68CHAPTER 4RESULTSIntroductionThis chapter presents the case study report, thedescriptive narrative which will serve as the analysis of thecurrent research study. The results will be presented as achronological log of the observations in which the field noteshave been condensed. Following the presentation of eachobservation or teaching session, the data will be analyzed interms of site, teachers, students, curriculum, hardware andsoftware, and any other features which emerge due to theanalysis.In total, eleven Mathematics 11 sessions were observed,six sessions taught by Mr. Crosby and five sessions taught bythe researcher. In most of these sessions, the computer wasused by the instructor and/or the students. In teaching therelations and functions unit, the researcher spent eightsessions (teaching hours), and Mr. Crosby used 12 teachinghours, both taking less than the 16 hours they had eachprojected. This could be due to the computer use, or in theresearcher's case, due to the fact that she combined chapters6 and 7 from the textbook. Both teachers began teaching theunit at the same time, but the instructional period wasinterrupted by a crossgrade examination, the review classes69for the said examination, and a teacher's strike that lastedeight teaching days.A Chronological Log of the DataIn order to impart the full flavour of what is involvedin teaching mathematics with a computer, a capsule account ofthe field notes is included with this report. In preparing todocument the events as observed, the researcher considered twooptions: the choice of organizing the data such that fieldnotes of Mr. Crosby's classes were presented, followed by thenotes relating the researcher's teaching sessions; or thesecond option of relating data as they occurred. The secondoption was deemed the most appropriate as the purpose of thecase study was not to compare teaching styles but to documentthe specific aspects involved in the introduction of computersto the mathematics classroom.Observation 1Block A Lesson: Expansions, Compressions, and ReflectionsThe lesson began on time, at 8:50 with theory on theboard. Nineteen students were present, ten females and ninemales. The theory began with the listing of the six basicfunction equations which were f(x) = x 2 , f(x) = V3E, f(x) =x3 ,= 2x , f(x) = ^f(x) = lx11/x, f(x)^ . The letter A was70introduced as a coefficient to those equations, for example,y=ax2 . In general, the teacher stated that if a>l, theequation indicated a vertical expansion. If 0<a<1, a verticalcompression was indicated, and if a<0, the graph reflected inthe x-axis.The teacher then put the equation y=-2x 2 on the board andshowed that according to the theory, this indicated a verticalexpansion by a factor of 2, and a reflection in the x-axis.One student in particular could not understand how a factor of-2 indicated a vertical expansion since a was not greater than1. The teacher explained that first expansions orcompressions are considered by looking at the absolute valueof A, then, reflections are indicated by the sign of thecoefficient.At this point, the students were asked to go to thecomputers. There were 13 terminals available, which meantthat six students would need to work with partners. In fact,only ten of the terminals were used, and the students workedin groups of two or three, of their own choosing.The students were instructed to graph y=x 2 and then tograph y=2x 2 . The teacher used the computer station and theoverhead projection device to illustrate the graphs. Thestudents were then asked for their observations. It seemedthat everyone noticed that the second graph was narrower thanthe first. The teacher pointed out that this illustrated avertical expansion. This concept was difficult for thestudents to grasp.71The teacher then asked the students to compare the pointson the new graph to the basic equation. Not one student wasable to make a correct comparison, and no one ventured to evenguess; the students remained silent when called upon to give aresponse. The resolution on the computer screen and the scaleof the grid were not very clear. The students had no idea howthe graph expanded by a factor of 2. The teacher theninstructed the students to graph the reflection y=-2x 2 . The"flip" of the graph was obvious to them but the students didnot appear to understand how the 2 was affecting the basicgraph of y=x 2 .The teacher felt that she had done a poor job ofintroducing the theory, so she then used the computer graphwhich was projected on the white board to illustrate thefactor change. She highlighted the point (1,1) on the basicequation. Using the same x-value of 1, she then drew an arrowto the corresponding point on the expanded equation. Shecontinued to do this for four different points, and thestudents were able to see that the y-values were doubling, orincreasing by a factor of 2. At this point the lesson endedwith the bell.Discussion.The teacher felt that using the computer actuallyhindered her presentation of expansions and compressions. Thestudents were able to see the obvious, the reflection of the72graph, which is just a mirror image of the original graph, ora "flip" over the x-axis. The students could not see theexpansion or the stretching of a graph until the teachermechanically showed them the transformation of the points onthe board. The board technique worked because the teacherexplained and showed the expansion whereas the computertechnique only showed, without an explanation, a graphdifferent from the basic graph. The difference between abasic graph and an expanded or compressed graph can be subtlebecause of the non-linearity of the expression. Though theteacher used the computer image on the white board of thegraph, she could have just as easily used an overheadtransparency. This illustrates that the teacher chose to usea traditional board presentation because she did not see a wayof highlighting the relationship between the points of thegraphs by using the computer. In general, a visualpresentation of compressions and expansions using a computeris usually more difficult to orchestrate than the presentationof translations and reflections. A traditional boardpresentation is easier because the students can follow thesequential construction of the graphs as opposed to aninstantaneous display on the computer screen.The teacher reflected on what she was doing and abortedher plan to use the computer in time to save the lesson,though there was not enough time to reinforce the concept withstudent practice. The student practice would not haveincluded computer activities, but drawing a table of values73and plotting the points on paper with pencil. Lynch, Fischer,and Green (1989) observed that in the first year of theirproject, the class period often ended before closure on anidea had been achieved. As well, the teacher decided not touse the computer just because it was there, but to teach thelesson the best way she knew how.It should be noted that the students had used thecomputer and the MET software earlier on in the year. Theymade a smooth transition from seatwork to computer stationsand needed no instructions on using the software. Thesoftware was user friendly, meaning it was simple enough forthe students to use without instructions from the teacher,even after not using it for a period of time. Also, therewere only 19 students in this class, a relatively small class,and there was ample room for everybody at computer terminals.The students preferred to work in pairs, with the odd personchoosing to form a group of three rather than work alone.The description of the site in Chapter 3 indicated thatthere were 16 computer terminals, yet the data indicate thatthere were only 13 terminals available. The reason for thisdiscrepancy is that one of the work stations was needed tooperate the fileserver, and two of the work stations were notoperating. Attempts to arrange for servicing were foiled,since IBM was no longer responsible for maintenance followingthe conclusion of the joint-study, and requests made to thedistrict repair department were not answered. The teacher74eventually lost interest since 13 machines was a generousnumber for a class of 19.This first observation illustrates three features ofintroducing the computer to the mathematics classroom:1. This teacher did not have to use valuable class timeto instruct the students on how to use the hardware orsoftware. The students worked well in groups of two or three.2. This particular use of the computer did not assistthe teacher in meeting her lesson objective. When thestudents seemed confused, the teacher decided to revert towhat amounted to a traditional method which seemed to be moreeffective.3. The teacher had the added responsibility of arrangingfor the servicing of equipment.Observation 2 Block C Lesson: Review of the parabolaThe class began at 11:10 with eight females and 14 malespresent. Mr. Crosby talked about the symmetry displayed bythe graph of a parabola. He sketched a general parabola(y=x2 ) on the white board and then asked the students tospecify the equation of the axis of symmetry. His questionwas an open question and not directed at any specific student;two or three students called out the correct answer.75The students were then asked to sketch the same graph,y=x2 , in their notebooks. They were reminded that theyrequired graph paper in order to draw accurate graphs. Aswell, they were instructed to generate a table of values. Theteacher made a table of values for this equation on the boardand then sketched the graph so that the students could comparehis graph with their notes.The next example was y=x2+3. Again the teacherconstructed a table of values and showed that the graph wasmoving three units upwards because each of the y values hadincreased by 3 units. He then sketched the graph on the boardand highlighted the vertex and showed that it had moved threeunits up. The teacher continued with one more example on theboard, and the students worked individually drawing tables ofvalues and graphs in their notebooks. By the last exampleonly a few students were working on their own; they appearedto be waiting for the teacher to show the answer on the board.The teacher said that more board examples were required beforethey would get an opportunity to finish the previous day'sexercises at the computer (the researcher was not present asthis was before the study began). The teacher did one moreexample, y=x2 -2.The students were then instructed to complete theircomputer exercises (Student Worksheet #1) from the previousday, working in the same groups. There was a textbookassignment for those students who had finished. The studentsrushed over to the computers with the exception of four76females and one male who stayed at their desks. All thestudents worked in pairs, except for one group of threefemales. There were no male/female groups. The teacher gavesome brief instructions about using the software and somespecial keys. The MET program began with a choice of eitherdoing a tutorial or going directly to the program. Despiteclear instructions on the screen and the teacher telling themnot to do the tutorial, some students entered the tutorial anddid not know how to get out.The researcher moved around the room with the teacher andanswered questions about the use of the software. Thequestions asked by the students were mostly about thesoftware, rather than about the task at hand. The task was tocomplete a worksheet requiring the students to indicate thevertex of a graph and to sketch the graph. One studentindicated to the researcher that he had done three or four ofthe questions using the computer then noticed a pattern, andcompleted the worksheet without the use of the computer.(This student left his partner earlier on and went to work byhimself at another computer.)Three more groups finished their worksheet but did notreturn to their desks to do the textbook assignment. Thesestudents remained at the computer and entered all kinds ofdifferent equations to see the effects on the graph. Afterabout ten minutes at the computer, eight groups had finishedtheir task and all remained at the computer and experimentedwith entering various equations. The five students who77initially stayed at their desks did not appear to be havingdifficulty with the textbook assignment which emphasizedconcepts developed by the computer exercises.When the bell rang to indicate lunch hour, three studentsdid not leave their work at the computer immediately. Whenall of the students had left the room, the researcher noticedthat someone had altered the software so that it printed arude message when the user chose to "quit."Discussion.In this lesson, the teacher did not use the computer atall for his presentation. Instead he used a table of valuesand hand generated graphs to illustrate his point. Hisstudents had had an introduction to the computers in the classbefore this one and seemed very eager to get back to theircomputer exercises. The computer was used to practice aconcept already taught, and it seemed to achieve that goalsuccessfully. The students worked eagerly and most finishedtheir task quickly. They seemed to enjoy working with thecomputer. In fact, some used it as an excuse not to return totheir desks to what the students perceived as the drudgery ofwritten exercises.The initial difficulty experienced by the students whowere lost in the tutorial presented some frustrating moments.They all wanted assistance at the same time and did not readthe instructions on the screen which indicated which key to78push to get back to the program. Despite the fact that thesoftware itself was easy to use, the teacher should havereminded the students that they would first be given thechoice in the main menu to use the tutorial. The researcherassisted the teacher by helping those students. The classroomwas very crowded and it was difficult to get from one cornerto the opposite corner quickly. The researcher, acting as anassistant to the teacher, made the teacher's job much easier.She assisted by one group of computers while the teacherassisted by the other.As the students began experimenting with the variousgraphs they became curious, and questions were posed to theteacher. It was difficult for the teacher, and theresearcher/assistant, to quickly study the graph on theirdisplay and indicate to them what they were seeing. In fact,with all the demands on the teacher, it was difficult tomonitor what each student or group of students was doing.This is evident by the fact that one student had the expertiseto alter the software. This is a good example of the loss ofteacher influence as described by Olson (1988).The teacher felt even more powerless because now that therude message was on the screen; he did not know how to correctthe deed and had to ask the computer science teacher who inturn had to ask one of her computer "whizzes" in the networkmanagement class to correct the software and tighten thesecurity access to the fileserver. The teacher did not havethe time to master the network management skills. This is79similar to what Tall (1987) observed in that if the system isused infrequently important skills are forgotten. In thisstudy, the little training that the teacher did receive 16months earlier was forgotten. His training did include theadministration of the security of the network. Since he hadforgotten some of those features, he allowed certain studentsaccess to the password which would enable them to tamper withthe software. The data from this observation illustrate thelooser class arrangements and the encouragement the studentsreceive to try things out for themselves. Unfortunately, noteverything they try for themselves is positive (likeinstructing the computer to print rude messages).Heid, Sheets, and Matras (1990) discuss the decisions theteacher has to make regarding how much to explain about theprocess they (the teachers) were demonstrating and how much toleave to discovery. In this observation, the teacher usedabout 60% of the hour explaining, and the students seemed toget bored; perhaps he should have left more time fordiscovery. The worksheet was prepared by the researcher; infact the researcher prepared all the worksheets in advance andcorrelated them to the textbook material and the curriculum.This was a time-consuming task and yet another responsibilityof the teacher wishing to introduce computers to themathematics classroom. There were very few questions from thestudents about the computer task, therefore the time spentdesigning a worksheet is worth every minute if theinstructions and purpose are clear to the students.80As suggested above, the teacher perhaps should haveincluded more relationships for the students to discover. Indoing the worksheets, the computer was used more as a checkingtool. The explorations came after the assigned work wascompleted as the students entered various equations andobserved their graphs. Since the teacher had not organized anactivity around these explorations, it is doubtful that muchwas gained. A useful activity would have been to write downan equation and describe it using a few sentences. Theteacher had not designed an activity to capitalize on theexploratory nature of the computer. Regardless, the studentsexperimented and were excited to be using a computer.The students remained on task until they completed theirworksheets, and some even stayed after the bell. Thecomputer, being a novel mode of learning, seemed to capturethe interest of the students. The cooperative group workseemed to be effective from the perspective that the studentsremained on task throughout the class, and it was noted thatgirls tended to work with girls, and boys worked with otherboys. Though the students were instructed to take turns askeyboarders and recorders, it was difficult to determinewhether the students actually rotated positions. As well, itwas difficult to assess the understanding level of eachindividual student. Though the worksheet was completed, itwas difficult to determine if the work was copied. Neitherteacher planned to incorporate a computer component in theirunit test.81Eight features stand out in the second observation:I. The computer is a novel approach to teaching whichcaptures the students' attention and enthusiasm, and lendsitself to experimentation.2. Because of the exploratory nature of many computerexercises, the teacher may not have answers readily availableto questions. This can be very intimidating to a teacher asshe/he relinquishes some of her/his influence.3. The computer room should be planned in such a waythat the teacher can see all the workstations and move freelyamong them.4. Some students are more technically advanced than theteacher in their knowledge of computer systems, which can beintimidating to the teacher.5. Lesson materials must be developed which correlatethe computer lessons with the curriculum and the textbook.6. The teacher received very little training in networkmanagement skills. Since the system was used so infrequently,the skills were easily forgotten.7. In using the computer to teach the graphing offunctions, the students should be allowed to explore therelationship between the graphs and the equations before beingtold what the correlation is.8. Security of computer hardware and software is verycritical. Passwords should not be shared with students.82Observation 3 Block E Lesson: Translations of the graph y=x 2There were 27 students present in this afternoon class,17 girls and 10 boys. The lesson began with Mr. Crosbyinforming the students that they would have an opportunity towork with the computers. He told them that they would beworking in pairs and that they would be able to choose theirown partners, but before the computer activities, he said,they would sketch some graphs in their notebooks using a tableof values.The students were asked to sketch the graph of y=x 2 usinga table of values. The teacher sketched the graph at the sametime as the students on the white board, then asked thestudents for their observations. One student said that it waspurple (the teacher had used a purple pen). Another studentsaid that it was symmetrical. The teacher used this lastresponse to discuss the axis of symmetry.The teacher did another example on the board as thestudents worked individually doing the same example in theirnotes. He then instructed the students to do some exercisesfrom their textbook which required matching equations tographs. The students did not appear to have difficulty doingso. Finally, the teacher put the equation y=(x+2) 2 on theboard and asked the students to describe the translationwithout sketching the graph. There were many incorrect83responses before one student correctly identified thetranslation of two units to the left. Another student askedif there was any easy way to find that out.The teacher then used the overhead projection device toshow the students how to enter the MET program. He listedsome of the important keys on the board. About two minuteslater the students chose partners and went over to thecomputers. The teacher distributed one worksheet per group(Student Worksheets #2 and #3, which were printed back to backin their original form), and the students rushed over to getstarted. Many of the groups did not read the instructions ontheir screens and began with the tutorial rather than theactual program. The researcher helped those students byshowing them the instructions on the screen that indicatedthat the F9 key needed to be pressed.The computer work areas were crowded. The teachercirculated to make sure that all students had found a partner.One quiet student remained at his desk and the teacher foundhim a partner to work with. The teacher encouraged thestudents to use all of the terminals, but one terminalremained unused. The groupings were as follows: there werethree groups of two males (MM); there were four groups of twofemales (FF); two groups of three females (FFF); one FFMgroup, one FMM group, and one male worked by himself.The teacher continued to circulate around the room. Heencouraged one of the girls in the FFF group to use the freecomputer station but she declined. It appeared that in the84groups, one student did all the typing while the othersobserved and filled in the worksheet. One female student lefther group (FMM) and returned to her desk (this student hadcome to class ten minutes late). A particularly weak studentseemed to be especially enjoying the computer activity andremarked: "I wouldn't be able to do this without thecomputer."One female student complained that she could not workwith other people and left her group (FFF) to sit at her desk.The researcher pointed out to her that there was an unusedcomputer but she said, "forget it!" Otherwise, the groupsworked through their worksheet exercises and stayed at thecomputer when finished and explored other equations. Theteacher circulated and observed the students. At one point heassisted one group to use advanced features of the software.They "zoomed-in" by creating a window on the screen whichamplified the display. By doing this they could accuratelyread the coordinates of the vertex. The bell rang to indicatethe end of the school day, but two groups remained at thecomputer, exploring.Discussion.The teacher spent considerable time (about 25 minutes) inclass sketching graphs using tables of values. This isnecessary because the graphing software does not teach orreinforce the skill of drawing graphs. What it does reinforce85is recognizing graphs, given an equation. The students seemedto have no difficulty recognizing graphs when assigned thetextbook exercises. The table of values is a valuable toolfor recognizing the relationship between the x and y values,the domain and the range. The MET software is capable ofgenerating a table of values for a given function but thestudents seemed to need the practice of plugging values intothe equations and finding the range on their own.The students had difficulty recognizing the translationson the board. They could see that the graph moved, but theycould not specify how it moved, for example, that the originalgraph slid 4 units up, or 2 units to the left. In this case,the computer was able to generate graphs quickly, so thatafter seeing a few at their own workstations, the studentswere able to recognize the patterns and the relationshipbetween the equation and the graph. Once again it was evidentthat the students were reluctant to read instructions on thecomputer screen, and some found themselves in the tutorialdespite the teacher having taken the time to explain thesoftware features to them. This was a relatively large class,and the researcher's assistance was appreciated in showing thestudents how to get into the program. This was not the firsttime this class had used the computer, and the teacher hadtaken the time to explain some of the key commands.It was also evident in this class that boys tended tochoose boys as partners, and girls tended to choose girls.The researcher decided to watch the division of labour closely86within the groups and observed that the students did notrotate duties. One person typed the whole time, the otherswatched and recorded. The teacher was kept very busy first ofall ensuring that everyone was using the computers, andsecondly watching that everyone was on task. Besidesreminding the class periodically to rotate duties, there wasnot much he could do to guarantee that they did. In somesituations, it is appropriate at a given time, perhaps halfway through the work period, to tell everyone to changeduties. The difficulty with this method is that itdiscourages the assignment of duties based on personal workingstyles. The students remained on task and discussed the taskcontinuously and seemed to enjoy working with the computer.In this study, it seemed that the students who could typeusually did the typing, and those with the neatest handwritingusually recorded the information on the student worksheets.The teacher distributed one worksheet per group, and thestudents were to hand it in for marking when complete. Therewere no criteria for individual accountability as the teacherplanned to assign a group mark. There were no consequencesfor the student who left her group because the teacher was toobusy helping other students to notice that she had notcompleted the work and her name remained on the worksheet withthe group.Despite the minor difficulty of students not reading thecomputer screen and entering the tutorial, the software wassimple to use. The MET software is very powerful, though the87teacher gave the students only the basic commands necessary tocomplete the graphing. As the students became curious or asthe teacher saw the opportunity, he would show individualgroups some of the advanced features, for example, how to"zoom-in" or how to change the colour of the display (thoughthe monitors were black and white, it was possible to obtaindifferent shades of grey). Once one group started doingsomething different, the neighbouring groups would becomecurious and share strategies and discoveries. Thus there wasno need for detailed instruction on the use of the software.Once again it was evident that the student computeractivities (Student Worksheets #1 and #2) replaced a textbookassignment and the computer was used as a tool for checkingthe correctness of their answers. But even in the checking,the students were required to interpret the graphicrepresentations to determine if their answers were correct.Since the teacher spent considerable time sketching graphs onthe board, the worksheets allowed the students time topractice the skills taught. The explorations began when theworksheets were finished. The students enjoyed creatingpatterns on the screen with the various shadings, and theexplorations were more of an artistic nature than of amathematical nature.The groups of two seemed to be more effective than groupsof three for mainly physical reasons. The computer stationswere very close thus in a three person group, one person satbehind the other two and really was not part of the activity.88Also, in these graphing activities, there was no need for thedistribution of social roles (as described by Male (1990)).The only roles needed were keyboarder and recorder. Thecomputer activities did not require for example, timekeepers,checkers, or praisers as larger group activities would.Eight features are evident in the third observation:1. The students did not change roles or duties while ontask.2. Most students remained on task.3. The teacher was kept very busy moving about throughthe crowded classroom.4. The software was easy to use. No formal instructionwas necessary. The teacher showed new features to individualstudents as the opportunity arose.5. The students explored new features and shareddiscoveries with other groups.6. Two person groups seemed more effective than threeperson groups.7. The was no individual accountability for the workhanded in for marking.8. The worksheets were used for practice. The computerwas used essentially as a checking tool.Observation 4 Block A Lesson: Expansions, Compressions, Reflections89This lesson reviewed the concepts taught the previous day(documented in Observation 1). The author began by writingthe equation y=ax2 on the board and then proceeded to outlineits features, that is, the vertex, the equation of the axis ofsymmetry, and the direction of opening. She then continued bydiscussing vertical expansions, compressions, and reflectionsin the x-axis. One student interrupted to ask if they weregoing to learn how to do the graphs on paper, not on thecomputer. The teacher answered that she would be assigningsome textbook exercises that required drawing graphs on paper.Three equations were written on the board and thestudents were instructed to complete the corresponding tableof values. The equations were y=x 2 , y=2x2 , and y=0.5x2 . Theteacher allowed the students a few minutes to complete theirtables, then she filled in her tables on the board. Thestudents were now able to see how the y-coordinate increasedor decreased by a factor of A. A student asked how they coulddraw the graphs easily by themselves without using thecomputer. The teacher replied that they would have togenerate a table of values and draw the graphs on graph paper.The teacher then used the computer and overheadprojection device to graph those same equations. She labelledthe graphs on the white board. She felt that this lesson wasmuch more effective than the previous day's lesson. Theteacher then proceeded to demonstrate other functions: thesquare root, the cubic, the reciprocal, and the absolute value90function. Textbook exercises were assigned requiring thestudents to draw graphs using a table of values.Discussion.The computer was used in this lesson only in theteacher's presentation and was a time-saving device. Theteacher realized that the computer could not help her teachhow to draw a graph, or help the students learn how to draw agraph, thus there was no computer activity planned for thestudents. The teacher felt that the students needed thepractice in mechanically drawing and completing a table ofvalues, and then plotting their ordered pairs on graph paper.The teacher showed the emergence of the table of values on theboard, and used the computer to draw the graph in order toshow the students that the graph contained the points in thetable (and infinitely many more). Thus the computer had verylimited use in delivering this lesson.Some of the students, on the other hand, had the maturityto realize that mechanically drawing graphs for themselves wasan important skill they needed to master. Despite the factthat the computer was a useful tool and the students enjoyedusing it, the student who interrupted the teacher to ask aboutthe drawing of graphs, voiced a very valid concern. Theteacher was not able to conceive a way of using the computerto assist the students in drawing graphs. With a differentsoftware program and a different data entry device, for91example a mouse, it may have been possible to actually draw atable and enter values. In fact, if there was a softwareprogram available which allowed the user to click on or pointto the coordinates on a grid, and then joined the points toform a graph, a valuable computer lesson could be planned.Thus, salient in this observation is the fact that:1. This teacher chose a traditional teaching methodbecause a way of incorporating the computer into the lessonwas not conceived.Observation 5 Block E Lesson: Translations, Expansions, Compressions ofthe ParabolaThe teacher began the lesson by asking the students, "whohas brought graph paper?" Not one person replied in theaffirmative. He then proceeded with a review of thehorizontal and vertical translations, moving into the reviewof vertical expansions and compressions. The teacherinstructed the students to draw a table of values and sketchfour graphs: y=2x2 , y=2(x-3) 2 , y=2(x+2) 2 -4, y=0.5x2+1. Allbut four students appeared to be following the teacher'sinstructions. The teacher sketched three of the four exampleson the board.After about 15 minutes of review, which included thesketching of the above examples, the teacher told the students92that they could go over to the computers. There was nodiscussion of how changes to the equation related to changesin the graph. The students were expected to check their workwith the sketches on the board. Students were instructed topick up one copy of the worksheet (Student Worksheet #4) fortheir group. Although they could choose different partners,the students formed the same groups and began work immediatelyat the computers.With 27 students present, the computer work area wasquite crowded. The groups preferred sitting side by side,even the three person groups, thus one computer station wasnot used since there was not enough room for even one personto sit at it. There was one faulty computer station with anon-functioning keyboard, at the end of one table, so theresearcher replaced the broken keyboard with the one from theunused computer. Thus a group of two girls went over and usedthis station. Only four students had to sit behind theirpartners rather than beside.The students were busy entering equations and specifyinginformation about their graphs. The teacher had not yetintroduced the concept of reflections, but the studentsobserved that some of the graphs "flipped" over the x-axis.The worksheet required that the students specify the vertex,concavity, axis of symmetry, and vertical compression orexpansion for given functions. They were then to check theirwork by graphing. As mentioned above, the students enteredthe equations first, then filled in the specific information.93They did not follow instructions and filled the worksheetafter graphing the equations. When the students finished theassigned worksheet, they explored other equations. Two groupsexperimented with shading various parts of the coordinateplane, thus creating designs. One group tried to superimposetwo parabolas to form a circle. Every group finished theassignment before the bell rang.Discussion.The teacher had intended to reinforce the graphing skillsby asking the students to draw graphs using a table of values.But many of the students did not have graph paper and wereprobably just drawing sketches in their notebooks. Althoughhe knew that the software would not provide practice in thatskill he continued with the lesson as planned. It is possiblethat the teacher wished to continue with the lesson and sendthe students over to the computers because the researcher wasin the room.The computer exercises related to identifying how thetransformed graph related to the basic graph. Although thecomputer was intended to be used as a checking tool forStudent Worksheet #4, five of the 12 equations included areflection, a transformation that the teacher had notintroduced yet. Though the researcher noticed that some ofthe students commented on the graph flipping over the x-axis,94the teacher did not make mention of it in this class or inObservation 8, which was the next time this class met.The computer area seemed more crowded. It could havebeen improved if the two non-functioning work stations wereremoved completely and the others moved over. The teacher didnot have the technical expertise to disconnect cables andconsidering that the computers were not used that often, itprobably was not worth the time and effort. The students didnot seem to mind the crowded conditions because many of themwould have been just as happy to sit back and watch one persondo all the work. Again, there was no individualaccountability. It appeared that the students enjoyed thecomputer activity as it was a change from the regular routineand it certainly encouraged looser class structure, that is,the students could talk and work with their friends. Thus,the students seemed to enjoy the computer situation ratherthan the work or the learning.Only two of the groups experimented with creatingdesigns. Though the computer affords the opportunity toexplore, the majority of students are content to just completethe assigned work. It would be an even greater task for theteacher to design explorations beyond the curriculum, andthere usually is not enough time in the curriculum foroptional explorations. It is possible though, that by usingthe computer to teach certain topics, time will be saved,freeing the teacher to present optional topics. Then, onceagain, the development of materials to guide the explorations95requires time, which the teacher using innovative methods hasin dwindling supply.The researcher acted as an assistant to lessen thecrowding. She would have liked to rearrange the physical set-up completely, but it was not her place to do so. Even in herdual role as teacher and researcher, she was a guest in hercolleague's classroom and used much discretion when makingsuggestions.The following features must be considered whenintroducing computers into the mathematics classroom:1. The site must be planned to facilitate grouplearning.2. This teacher did not use the computer to teach aboutthe drawing of graphs. The teacher felt most comfortableusing traditional methods.3. Follow-up exploration materials or worksheets shouldbe designed and ready for use.4. A technical assistant should be on call in theschool, in order to help the teacher when problems arise suchas replacing or repairing non-functioning equipment.Observation 6Block A Lesson: Translations, Expansions, Compressions ofthe Parabola96The lesson began with the author writing the generalequation y=a(x-p) 2+q on the board. She asked the students forall the information they could provide from this generalequation. Together they established that given the generalequation, the following information could be obtained usingthe a, R, and g: the coordinates of the vertex, the directionof opening, the maximum or minimum, the equation of the axisof symmetry, the transformations (translations, expansions orcompressions, reflections), and the intercepts.The teacher handed out the worksheet (Student Worksheet#4), one per student, which required that the students specifythe coordinates of the vertex, the equation of the axis ofsymmetry, the direction of opening, the basic equation, andthe nature of the vertical expansion or compression. Theywere to check their answers by graphing the function on thecomputer. The students went directly over to the computersand began working, with the exception of three male studentsand one female student who generally displayed a negativeattitude, with her comments about using the computer being noexception. This student left the classroom to go to thewashroom. The remaining 18 students had no difficultiesentering the software and worked diligently.The student returned from the washroom and went over toan unused computer and actually enjoyed doing her assignment,much to the amazement and joy of her teacher. One malestudent left the computer after five minutes saying that hecould fill in the worksheet at home. The other students who97remained at their desks, completed the worksheet without usingthe computer. The teacher had a difficult time makingcomputer use mandatory since the work could be completed withpaper and pencil only.Discussion.The teacher, in reflecting after the lesson, admittedthat using the computer for the given exercises was not themost efficient method of completing the assignment. Theworksheet instructions said "complete the chart then check bygraphing the function." She realized that she should haveinstructed all of the students to work at their desks, andwhen the worksheet was completed, to go over to the computersto check their work. The computer for most students was anovelty and a break from the routine lecture/guidedpractice/assignment model, though the teacher generallymaintained this style of teaching. The students who chose towork at their desks were bright students who did not needreinforcement from the computer activity nor the novelty ofusing the computer as a checking tool. Those students did notneed to be motivated either. The teacher did not have afollow up activity prepared which would have challenged thosestudents who did not wish to work at the computer.It is evident in this observation that the computerpresents a challenge to the teacher. The teacher is chargedwith the task of tapping the computer resources so that she98provides tutoring and reinforcement for the middle and lowability student and enrichment for the higher abilitystudents. The teacher realized that there was no benefit inusing computers just because they were there. Though thestudents received immediate feedback by checking the computergraphs as opposed to turning to the back of a textbook, theystill had to read the graph for the vertex and axis ofsymmetry and make an interpretation for the vertical expansionor compression.Therefore the predominant feature here is:1. Teachers are challenged to plan activities and modifyteaching styles in order to take advantage of the computer'spotential.Observation 7 Block C Lesson: Translations, Expansions, Compressions ofthe ParabolaMr. Crosby used his computer and the overhead projectiondevice to show the graph of a parabola. He reviewed thekeyboard and some of the keys which could be used. He thenshowed some of the software commands that could be used. Forexample, TABLE generated a table of values given the lower andupper domain. He generated a table of values for a domain of-3 to +3. He noticed that some of the domain values were99rational and admitted that he did not know why. Upon furthercontemplation, the teacher realized that the TABLE commandgenerated eleven ordered pairs each time it was invoked, thushe tried -5 to +5 and saw that the domain values wereintegers.The teacher reviewed some of the transformations of thegraph using his computer and the overhead projection device.At one point he assumed that he had generated a new graph(since his back was to the screen), but when he turned around,the previous graph still appeared. He quickly realized thathe had forgotten to press the ENTER key.The teacher put three more examples on the board andasked the students to describe the transformations. As theteacher lectured, one student sat at the back tapping on oneof the keyboards. When the worksheets (Student Worksheet #4)were distributed, all of the students except one went over tothe computers. All of the workstations were used by the 22students, working individually or in groups of two. Oncefinished, the students entered equations spontaneously, somewith very large integral exponents. The researcher showedsome of the groups another piece of software called GreenGlobs (Dugdale & Kibbey, 1986). This software challenged thestudents to enter a correct equation to describe a givengraph.100Discussion.This observation highlights the importance to teachers ofbeing familiar with the software before using it with a class.As well, it shows the difficulty in using a computer fordemonstration if the teacher does not have a monitor. In thisparticular observation, the teacher had forgotten that theTABLE command automatically generates 11 ordered pairs. Manypieces of software, MET included, include so many featuresthat it is difficult to remember them all. As well, thevarious commands and their results are easily forgotten if thesoftware is used infrequently. On the other hand, if manydifferent software programs are used frequently, it is easy toconfuse commands and features.Teachers are typically seen as authorities in theirsubject matter. This authority is constantly challenged whenusing the computer unless teachers have spent many hourstesting and using the various features of the software. Thechanging role of teachers and students is emphasized here.The teacher is no longer the sole authority in the classroom.Mr. Crosby did not immediately realize how the TABLE commandworked and realized the solution after about two minutes ofreflection.When addressing a class, the teacher usually attempts tobe visible to all of the students and positions himself sothat he can see the entire class. The teacher makes an effortnot to turn his back when writing on the board and to be in101tune with the subtle interactions between himself and thestudents, and between student to student. The computer usedas a demonstration tool introduces some physical constraints.In this observation, the teacher was typing commands at thekeyboard with no monitor in front of him to verify theresults. He had to physically turn around to look at theboard behind him, and noticed then, that the wrong graph wasdisplayed. He re-entered the correct commands and obtainedthe desired result.The teacher in this observation was not flustered by theerrors he made at the keyboard, nor was he embarrassed when hecould not recall exactly how the TABLE command worked. Thiswas evident by his easy manner and relaxed rapport with thestudents. He accepted his new role as a learner, learningabout computers on the job. He could have deferred theproblem to another time, or could have ignored it completely.He established an atmosphere in the classroom where thestudents were not concerned about the teacher's lack ofknowledge and were challenged to assist him to find thecorrect command or the accurate method of instructing thecomputer. Even if the teacher had spent hours and hourstesting the software, the nature of computer exploration issuch that unexpected situations are always arising. It isvery difficult to anticipate many outcomes. This can becompared to geometry proofs, for example, or the nature ofproblem solving in general. Both these applications requirecareful thought before suggesting a solution.102The MET software did not really challenge the students inthis particular assignment. The students diligently completedthe task of graphing the equations and identifying the vertex,axis of symmetry, maximum or minimum, and intercepts. Theygraphed the equations first rather than use the computer tocheck their answer as the worksheet instructed. Theresearcher introduced the Green Globs program to the studentswho were finished in order to sustain their interest and tochallenge them further in graphing techniques. The teacherwas not familiar with this program at this time, but quicklylearned how to use it with the students. Once the researchershowed one group how to use it, the others quickly followedsuit, facilitated by the user-friendly nature of the software.It was noted that a student at the back of the room wastapping on the keyboard. This fact is significant in that oneof the major security problems in the computer laboratory wasprotecting the keyboards from tampering. The caps on the keyscould easily be removed, and the students often removed andrearranged them. Somehow, no one was ever caught in theprocess, but it was suspected that it was done by the studentswho sat at the back of the classroom while the teacherlectured. One suggestion for avoiding this is providingcovers for the keyboards, but simpler still the students couldbe trained to slide the keyboards beside the computer or toput them on top when not in use.Five features are evident here:1031. Teachers and students must accept their changingroles if computer implementation is to be successful andunstressful.2. It is difficult for one instructor in a computerlaboratory environment to monitor all on-going activities.For example, the researcher assisted by providing activitiesfor those students who had finished their work.3. Quality software makes computer implementation muchmore effective and simple to introduce as shown by the use ofthe Green Globs program.4. Teachers must implement strict security measures tosafeguard the computer equipment.5. The site should be planned so that the teacher has amonitor at his computer station.Observation 8 Block E Lesson: Transformations of the ParabolaThe researcher arrived five minutes after the class hadstarted. Mr. Crosby was reviewing the concept of expandingand compressing the graph. The researcher sat down at theback of the room and noticed that the computers had not beenturned on yet, (this was the first class of the day). Theresearcher turned on the fileserver and waited a few minutesbut the computer would not boot (turn on). She tried a numberof times, checked for loose connections, and then asked the104teacher if he knew that the network was down (not working); hedid not.The researcher left the room to find a student from thenetwork management class (Donald, pseudonym). Donald was ableto boot the computer after trying a few different things. Theteacher was continuing with his review, following a similarroutine as in previous classes, sketching graphs using a tableof values. The researcher walked around and booted all theindividual workstations, and by the time all the computerswere up and running there were only ten minutes remaining inthe period. The teacher assigned work from the textbook.Discussion.Mr. Crosby was able to carry on with his lesson withoutthe use of the computer. He used the board to draw hissketches and then assigned work from the textbook. This isanalogous to planning to show a film and having the projectorbreak down. Teachers are trained to improvise when faced withunexpected events. The earlier observations show that thecomputer was never planned as an integral component of thelesson. The important point here is that the researcher,acting as an assistant, spent approximately 40 minutesattempting to get the computers to work (checking, findingDonald, watching Donald, booting the individual workstations).If the researcher had not taken the time to boot the network,105the teacher probably would not have found the time to do sohimself, with a full teaching load on that day.Thus, the one feature noted here is:1. A technical assistant is critical to the maintenanceof a computer laboratory in a school.Observation 9 Block A Lesson: Review of Transformations: All FunctionsThe researcher began the lesson with a review of all thetransformations. She placed an example on the board, andasked the students to list all the transformations from thebasic equation. The teacher, using the overhead projector anda graph transparency, graphed the equation. Next, the teacherintroduced the students to the Green Globs program using theteacher workstation and the overhead projection device to showthe students how the program worked. This took approximatelyten minutes.The students were instructed to go over to the computersin pairs or by themselves and use the Green Globs program toidentify the equations that matched the given graphs. Theywere told to do 20 questions then move on to the gamecomponent. All of the students went over to the computers andall of the workstations were used. At first they found thatidentifying the equations was a challenge. Of the 19 studentspresent, one group of two and one girl working by herself did106not get to the game. Two of the male groups attempted a morechallenging game called Tracker.The students were relaxed and were enjoying themselves.Three groups continued playing even after the bell rang.Discussion.Illustrated here is the use of software which was simpleand fun to use. The Green Globs program was introduced to thestudents by the teacher in a matter of minutes. It has higheducational value and a relatively low purchase cost ($89).The Mathematics Exploration Toolkit is also very usefuleducationally, but for its purpose in this case study, it wasnot cost effective. The Green Globs program had the addedfeature of a game component, which kept the students' interestlong after the assignment was finished. The motivation andenrichment features of the game made classroom managementeasier for the teacher.The teacher was not asked any questions, and the studentswere challenged by the software. The teacher did not spendany lecture time guiding the students through activities whichrequired stating the equation to match a given graph. Thesoftware was used to guide the activity. This illustrates amodification of the teaching style observed earlier. Theclassroom setting in this observation was ideal in that theteacher could circulate amongst the students, monitorprogress, and ask probing questions to further challenge them.107Thus software selection is an important aspect of implementingcomputers in the classroom. The author feels that most of thequality software is at the elementary level. How then doesthe secondary teacher become aware of quality software and whoprovides the funds for its purchase?In the present study, the researcher had been involved inthe evaluation of software for many years and kept up-to-datewith software reviews. She attended conferences annually,where software was often demonstrated. Green Globs, forexample, proved to be such a popular program that a sitelicense was negotiated by the district and the program is nowavailable to all schools in the district at a cost of only$10. The other teacher in this study, Mr. Crosby, was notfamiliar with the available mathematics software and relied onadvice from the district mathematics consultant and fromcolleagues.The school in this study was fortunate to be part of thejoint study between IBM and the school district; thus therewere funds available for the purchase of software. The intentwas to make the computer classroom/laboratory available to allmathematics classes. Mr. Crosby and the author were given theopportunity to recommend software for purchase based onsoftware reviews and review of the software catalogues.Unfortunately, there was not enough software in the commercialmarket which correlated with the mathematics curricula ingrades eight to twelve. This raises the issue of professionaldevelopment and release time from teaching duties to learn108about new techniques and software. The discussion of thischaracteristic is beyond the scope of the current study.Thus the above discussion stresses software features:1. Quality, easy to use, simple to learn, challengingand entertaining software is very important for successfulcomputer implementation.2. This teacher allowed the software to guide thelearning and altered her style of presenting board examplesbefore assigning computer activities.Observation 10 Block E Lesson: ReviewMr. Crosby was absent and there was a substitute teacherin his place. The researcher introduced herself to thesubstitute. The teacher told the students that they couldspend the first 20 minutes using the Green Globs program, andthen they were to review for the crossgrade exam scheduled fornext week.The researcher noticed that the computers had not beenturned on, so she booted the system which took about tenminutes. The students indicated to the substitute teacherthat they did not know how to use Green Globs, so theresearcher gave a quick demonstration using the computer andthe overhead projection device. The students were very109attentive and rushed over to the computers as soon as theresearcher finished her presentation.Although the students were instructed to use thecomponent of the program where they were to enter the equationwhich matched the given graph, one group went immediately tothe Green Globs game. The students worked well in theirgroups and continued playing the game after completing thegraph matching exercises, until the end of the period(although they were given a review worksheet for the nextweek's exam and were told to spend only 20 minutes at thecomputer).Discussion.One of the drawbacks of being innovative and tryingtechniques that others are not familiar with is that, whenteachers are away, there is unlikely to be anyone to taketheir place and carry on. In this observation, the researcherfilled in for the teacher (with the substitute's permission).This stresses the importance of having technical or staffassistants on site to assist a substitute. These assistantsshould receive training at the beginning of the year in usingall of the available software. Another alternative istraining student assistants or monitors whose duties couldinclude turning on and off the equipment.This class was the large class of 27 students, but theyposed no management or discipline problems for the substitute.110The students were familiar with the routine of findingpartners and moving over to the computers. They were alsoengaged in an activity which they enjoyed and which kept themon task for the entire period. It was unrealistic to expectthat they would stop playing the Green Globs game and do areview sheet of algebra questions when it was seen in earlierobservations that it was difficult to get some groups to leavethe computer when the bell rang.The feature that dominates this observation is:1. Good quality software will motivate and entertainstudents.Observation 11 Block A Lesson: Review for Crossgrade ExaminationThe researcher wanted to make certain that the studentsknew how to graph their equations on paper. She handed out ashort quiz where the students were required to draw the graphof y=-2(x+3) 2-1 using a table of values. After the quiz wascollected the teacher distributed a review worksheet that thestudents were to complete at their desks. The reviewworksheet was for the crossgrade examination and includedtopics covered earlier in the year; it did not include thepresent topic, the graphing of functions.Three females asked if they could use the computers. Theteacher agreed. The three worked separately at the computers,111using the Green Globs program. Two of the students chose tomatch equations to the graph, the third played the game.Discussion.The teacher was bothered by the contradiction of usingthe computers in class, but not allowing their use on tests.She knew she was just starting to learn about incorporatingthe computer in the regular classroom, and decided to make ita personal goal to develop a test with a computer componentfor next year. The other difficulty was that the pendingcrossgrade exam was a cumulative examination given to allstudents in Mathematics 11 in the school, but not all studentsin the course were exposed to the computers. Thus an equityissue arises: if it was deemed that the computer enhanced theteaching and learning of mathematics, would those studentsexposed to its use have an advantage over the other students?And, if it was found that some students had an advantage, cancomputer use be mandated and teachers coerced to use them?This raises political and ethical questions, beyond the domainof the present study.The teacher in this observation was bothered by anotherinternal conflict. The objective of her lesson was thatstudents practice skills learned in the term by completing aworksheet. Yet, three students wanted to use the computersinstead. The teacher was pleased with their enthusiasm andwanted to encourage computer use (especially among girls).112She decided to allow them to use the computers and remindedthem to complete their review at home. The only consequencewould be that the teacher would not be available to assistthem if they had difficulty with any question. The teachermade herself available after school on the day before the examso the students could get tutoring if required. The questionstill remains, once the computer is introduced to the lessons,how can it be incorporated into the assessment process?Thus the dominant feature here is:1. Assessment instruments must be developed that includea computer component in order to evaluate the skills learnedon the computer.SummaryThe chronological log of the observations is summarizedin Table 1. The observation period was a very interesting onefor the researcher. By observing Mr. Crosby and reflecting onher own teaching daily, she was able to make slightimprovements with each class. For example, in Observation 7,while watching the students work, it occurred to theresearcher to introduce the Green Globs program. Therefore,in her next class (Observation 9) the researcher included theintroduction of Green Globs in her lesson plan. She was ableto anticipate some of the questions and difficulties inadvance. In turn, she and Mr. Crosby collaborated in lessonplanning and frequently discussed their experiences informally113after class. The documenting of the field notes each eveningforced her to reflect on her teaching and Mr. Crosby'steaching, thus building the next lesson on the strengths ofboth teachers. The researcher has studied the literature oncooperative learning among students but the experience of thecase study emphasized for her the advantages of cooperativeteaching.The chronological log presents two teachers' attempts tointroduce computers into their mathematics teaching. Theliterature presented many views on innovation and thepotential of computers in the mathematics classroom but theobservations reveal that little innovation took place. Inmost of the observations, the teachers decided to proceed withtheir lectures without using the computer, and introduced thecomputer when they felt comfortable doing so. The studentsused the MET software primarily to check their work and forsome very general explorations.The features which have been outlined in the abovediscussions appeared repeatedly (Table 2) in many of theobservations. Though the mathematics teaching (graphing offunctions) occurred with minimal change over the two weekperiod, there were ten salient features which influencedteacher and student routines. These features were the mostsignificant for the researcher and will be summarized in thefinal chapter.114Table 1A Chronological Log of the ObservationsObservation Block Date Lesson Text ILO Assignment1 A 01.28.91 y = ax2 6-4 11.25a none11.26b,c2 C 01.28.91 review of parabola 6-2/6-3 11.25a Worksheets #1-33 E 01.28.91 y = (x-p)2 + q 6-2/6-3 11.25a Worksheets #2,#34 A 01.29.91 y=ax2 6-4 11.25a Text: pp.209/24911.26b,c5 E 01.30.91 y = a(x-p) 2 + q 6-5 11.25a Worksheet #46 A 01.31.91 y = a(x-p) 2 + q 6-5 11.25a Worksheet #47 C 01.31.91 y = a(x-p) 2 + q 6-5 11.25a Worksheet #48 E 02.01.91 y = a(x-p) 2 + q 6-5 11.25a Text: pp.211-2129 A 02.04.91 Review Transform- 6-5 11.26a-c Use Green Globsations all Fncns 7-5/7-6 11.2710 E 02.04.91 Review none11 A 02.05.91 Review 7-5/7-6 Use Green GlobeQuiz Review115Table 2Salient Features of Computer UseFeature ObservationQuality Software 4, 7, 9, 10Method (Traditional vs computer) 1, 3, 5, 6Assistants 1, 5, 7, 8,Motivation/Exploration 2, 3, 4, 11Teacher role 2Site 2, 4, 5, 7Lesson Materials 2, 5Professional development 2Cooperative learning 1, 3, 4, 5Accountability/assessment 6, 1110116CHAPTER 5SUMMARY, CONCLUSIONS, AND RECOMMENDATIONSThis final chapter of the case study report on theintroduction of computers to the secondary mathematicsclassroom will summarize the results reported in Chapter 4.The research questions posed in the first chapter will beaddressed, and recommendations for practice and futureresearch will be made.The researcher reported on her observations in threeMathematics 11 classrooms, her own, and Mr. Crosby's twoclasses. The computer was used to supplement the teaching ofa unit on graphing relations and functions. The data indicatethat three intended learning outcomes (as described in Chapter3) were addressed in the observation period, ILOs 11.25a,11.26, and 11.27. The summary of the field notes is includedin Chapter 4 so that the readers can apply them to their owncontext, as suggested by Bresler and Walker (1990). The twoteachers were actively involved in the research thus promotingtheir own learning, as described by Day (1984), andestablishing the teacher as a learner.Summary of ObservationsThe SiteThe site, in terms of studying the introduction ofcomputers into a mathematics classroom, was nearly ideal. The117mathematics classroom and the computer laboratory were one inthe same. There were 16 workstations, sufficient toaccommodate all the students in groups of no more than two.The computers were networked so that the teacher did not haveto worry about distributing and managing software. Theteacher had a computer workstation outfitted with an overheadprojection device. There were funds available for thepurchase of software. All of the above conditions seem idealfor establishing a computer enriched environment, and theyare, almost. The deficiencies of the site did not seem tohinder the teachers in the study but are included in thespirit of recommendations for those readers planning their owncomputer site and the necessary resources.The classroom was not designed to accommodate computers.The room was very crowded, as reported in the data. Thecomputers on one side were close to the heating registers,thus exposed to dust and direct heat. They were sitting onconventional tables, approximately 2.5 metres in length, anddid not accommodate groups of students comfortably.The black and white monitors were adequate, but colourdisplays would have been more effective, especially forshowing the contrast between a basic graph and a transformedgraph. It was mentioned in Chapter 3 that the teacher'sworkstation did not have a monitor, thus it was difficult forthe teacher to see the results of his demonstrations. Had hebeen supplied with a lower table specifically designed for acomputer, he would have been able to use a monitor, and the118students at the front of the classroom would not have had anydifficulty seeing over it.The size and shape of the classroom were also restrictingin that when sitting at the computers the students had theirbacks to the teacher. This arrangement discouragesinteractive demonstrations, where the teacher demonstratessome feature, and the students immediately try it forthemselves.It is the opinion of the author that the lack of aservice or repair arrangement is a serious oversight. It isunwise to purchase computer equipment without establishing whowill be responsible for its repair. The other featureregarding the site, which the author perceives to be of aserious nature, is the inattention to security. The room wasleft unlocked in the teacher's absence, and the computers werenot secured to the tables or interconnected in some fashion todiscourage theft. As well, the keyboards were a prime targetfor bored or mischievous students. These are the physicalsecurity weaknesses.The integrity and protection of the software must also bemaintained. Mr. Crosby, being a novice computer user, was notaware of some of the potential hazards such as softwaretampering or computer viruses. The access to the fileservershould have been limited to teachers only, and even theadvanced network management students should not have beengiven the password to access the fileserver.119The Teachers The teachers in this study were not frightened of usingcomputers; if anything, they worked hard to try to tap thecomputers' resources. They realized that integrating thecomputer in their mathematics teaching was a challenge, andthis study documents some of those challenges.The two teachers did experience some of the problems towhich Lampert (1985) referred, for example, hardware notfunctioning or realizing that the lesson could have beentaught better using traditional methods. The teachers wereflexible and worked around the site drawbacks discussed in theprevious section. They heard in workshops of the greatpotential of the computer in displaying graphs and promotingexploration but they still remained challenged to develop wiseuse of the computer. They began their teaching unit expectingto use the computer for all of the lessons. Instead, theygraphed quite a few examples on the board or on the overheadprojector because they could not conceive of ways of using thecomputer to demonstrate the relationship between the pictureof the graph and the actual values.The teachers did not really set detailed goals withrespect to computer use. They were learning "on the job" whatsome of those goals could be. They did realize, however, thatthey had to improve on their worksheet materials and had touse those worksheets to promote learning by exploration. Thecurrent study supports Lynch, Fischer, and Green's (1989)120assertion that "finding the right combination of challenge andsupport for students in laboratory sessions is not an easytask" (p. 691).The teachers were disappointed that they did not have thetime to explore other capabilities of the computer and thesoftware, but they hoped to do so in the future. Collis(1988) emphasized that teachers do not have the time to finduses for school computers. The two teachers in the study feltobligated to use the computers more than they did, and alittle bit guilty that such a computer enriched classroom wasnot being put to its most cost effective use. The literaturesupports the realization that the changes in teaching andlearning have to be gradual and that it is unrealistic toexpect teachers to incorporate technological innovationsimmediately (Cuban, 1986; Fey, 1984). This highlights theurgency of professional development activities.The teachers in the study remained undaunted by thenegative aspects of computer implementation. They were stillenthusiastic about the challenges that were involved inattempting to implement computers in the mathematicsclassroom, primarily because of their knowledge of thewidespread use of computer technology in industry and researchand the rapid advances and developments in both hardware andsoftware. They would have greatly benefitted fromprofessional development activities which presented materialsthat they could incorporate into their teaching. It isimportant that the leaders in computer applications in the121curriculum meet and exchange ideas and techniques relevant toparticular subject matter. The difficulty is that very fewteachers consider themselves leaders, and the teachers in thecurrent study are no exception. The pooling together of thegradual changes that each individual makes would create a richresource of computer applications in mathematics.The StudentsThe students in the present study adapted to the alteredroutine in their mathematics classes very quickly. Theyusually rushed over to the computers when given the cue fromthe teacher, and as noted in the observations, they generallyremained on task. The students worked together readily andpresented few discipline problems. They did not abuse thelooser classroom structure, the transition from seatwork tocomputer work went smoothly, and they generally remained ontask.The researcher predicted at the onset of the study thatthere would be more conversations to monitor. She found thatthere was very little idle chatter, and the conversations wereusually task related. Fey (1984) wondered whether teacherscould prepare their students for working in a technologicalenvironment by making only modest changes in class activity.It was inferred that the computer was a highly motivatingfeature in the mathematics classroom which was evident fromthe observations of the students. It was a change of routine,122and the students were able to work with their friends. Sincethere was no evaluation attached to the computer exercises,that is, no test or exam, the students viewed the computeractivities as fun activities, especially when the Green Globsprogram included an actual game.The Course and the SoftwareThis study relates two teachers' experiences teaching theunit on graphing relations and functions, and it is suspectedthat there are probably other topics in the Mathematics 11course that lend themselves to computer use. The teachers hadpredicted that they would require about 16 teaching hours tocover the entire unit, learning outcomes 11.23, 11.25-11.28.The researcher observed ILOs 11.25a, 11.26, and 11.27 asreported in Chapter 4. This is about 60% of the materialincluded in the unit, and the teachers used on the averageeight teaching hours. It is suspected that the entire unitwould have required about 13 teaching hours, using thecomputer. A follow up study would be required to determine ifthere actually is a time saving.Two software programs were used during the observationperiod, the Mathematics Exploration Toolkit (MET) and GreenGlobs. In designing the study, in consultation with Mr.Crosby, it had been intended to use only MET, but it wasconcluded that MET was not challenging enough for the chosengoal of teaching about the graphing of functions, thus the123program Green Globs was introduced to the students. MET is apowerful program with its symbolic manipulation capabilitiesand the defer sequence programming feature. Both teachersinvolved in the study felt that they needed much more practicebefore they became proficient in its use and before they couldadapt it to other topics in the curriculum.The teachers were not able to correlate ILOs 11.25a and11.26 with the MET software. These ILOs required that thestudents graph and recognize the parabola and, show generaltransformations from a basic equation. The teachers used theoverhead or board to illustrate the graphing techniques andthe effects of the a, R, and g in the general equationy = af(x-p)+q. The MET software was primarily used forchecking the correctness of the answers. Though using thecomputer was a novel way of checking answers, MET also allowedthe students the opportunity to create artistic patterns bysuperimposing graphs and altering the shading which somestudents used. However, when compared to Green Globs MET heldvery little appeal for the students.Green Globs was introduced because it could give thestudents practice in determining the equation of a relationgiven its graph (ILO 11.27). As well, it could be used as achecking tool, and it included a game which required that thestudents enter equations to intersect points (green globs)randomly placed on the screen. Green Globs was easy to learnand use and could have been used exclusively to teach ILOs11.25a, 11.26 and 11.27.124The defer sequence programming feature of MET offersteachers an opportunity to create their own presentations. Itwould be useful for teachers interested in using MET to meetand develop presentations and guided explorations. Theteachers in this study did not get an opportunity to explorethis feature of MET.The Research QuestionsFour research questions were posed in Chapter 1:What salient features emerge in the mathematics classroomwhen the computer is used to teach a graphing unit?How do the teachers adapt the use of the computer to thegraphing topic and the already established classroomroutine and setting?How do the students adapt to the computer-based learningactivities they experience?What advantages and/or disadvantages do the teachers andstudents experience when using the computer?125Salient FeaturesThe salient features which emerged from the study ofintroducing the computer to a mathematics classroom aresummarized below. In the assessment of the features, somerecommendations for practice are included.1. Quality softwareQuality software ensures that the teacher will not haveto spend many instructional hours teaching about its use. Italso facilitates better management of the computer learningenvironment. Both of the software programs used in the studywere user-friendly in that the students used them with verylittle direction from the teacher. The chosen software shouldmatch the curriculum objectives and promote student learning.2. Traditional methods versus computer-useThe teacher must be flexible and constantly needs to makedecisions about when and when not to use the computer. Thesedecisions are based on experience, and this study shows thatwith limited experience using computers in a mathematics classthe teachers relied on the more familiar traditional boardmethods and used the computer mainly for the studentassignments.3. Technical/Staff AssistantsAssistants are required in a computer laboratory as theyare in science laboratories. The teacher usually does nothave the time to become a technical expert, and someone needs126to be on-site to assist with the laboratory when the teacheris away. An in-school assistant should be on-call full timeto maintain the network and service the machines. Districtsshould provide resource people who could be available toassist with software difficulties and to offer advice on howto correlate software with curriculum content.4. Motivation and ExplorationThe computer tends to be a motivating device, awakeningstudents from complacency or lethargy. The students tend toremain on task and tend to have the desire to explore deeperinto the topics being studied. It is possible, however, thatthe novelty of using a computer in a mathematics classcontributed to the high degree of motivation observed.5. Teacher RoleThe changing role for teachers and students challengesthe teacher's traditional role of provider of information.The teacher may not always have the answers readily availableand may not have the same technical expertise as somestudents. The teacher must not only be prepared to answer, "Idon't know," but must also be prepared to learn from on thejob experiences.6. SiteThe physical site must be planned carefully to facilitatethe ease of movement, group learning, vantage points, andsecurity.1277. Lesson materialsThe lesson materials must be self-explanatory so thatthey make use of the computer's potential as well as correlatewith the curriculum. The tasks should accommodate more thanone ability level.8. Professional DevelopmentTeachers need release time to train in the use ofcomputers in mathematics and in technical aspects relating toa computer environment. Teachers need to find more uses forthe computers in mathematics class so that the computers areused more often and are cost effective. Teachers need to usethe computers more often so that they do not forget theirskills.9. Cooperative learningThe students teach each other about the software, andthey share discoveries. It was found that groups of two weremore effective than groups of three because it was moredifficult for three students to see the monitor clearly.10. Accountability and AssessmentThe teacher needs to pre-set guidelines for individualaccountability and needs to develop an instrument (quiz,activity, etc.) to assess individual students on theircomputer activities in order to determine if the computerlesson is actually effective.128How the Teachers AdaptThe study shows that the teachers adapted to theobstacles inherent in the computer environment. They movedabout the crowded classroom, helping and observing students.They chose traditional teaching methods or their usual methodof teaching and used the computer when they felt comfortabledoing so. The teachers did not really adapt the computer totheir teaching style or to the curriculum, but they did use itto project certain graphs on the white board.The teachers encouraged the students to work in groups.In the larger classes it was necessary to work in groups oftwo or three. Worksheets were distributed as assignmentsthough the teachers did not adapt their evaluation proceduresas no group marks were assigned. In fact, no assessment tookplace on the computer activities. Although the computer wasused by the students mainly for practice and checking, it wasevident from the observations that the teachers werechallenged by the physical computer environment and did notadapt the computer to the curriculum as much as they hadhoped. It was only with the introduction of the Green Globsprogram that the teachers relinquished some of their controlover lesson presentation to the computer and the software.Perhaps the two week observation period was too short a timeto expect adjustments to teaching style and presentation.129How the Students AdaptThe students had the opportunity to work in groups, in amore relaxed atmosphere than the traditional classroom. Theydid not seem to take advantage of the looser classroomarrangements and remained on task during their assignedactivities helping each other. In fact, some studentscontinued their work on the computer after the bell rang toend class.Many of the students explored some of the additionalfeatures of the MET software without prompting from theteacher. The students used the Green Globs program withoutany difficulty and learned the game rules very quickly. Thestudents seemed to adapt their work habits to the computerenvironment quite quickly, but the study was not able assesshow student learning was affected.Advantages and DisadvantagesThe advantage of using the computer to teach and to learnis that it is highly motivational for the students. Itfacilitates cooperative learning and provides self-guidedactivities which allow the teacher the freedom to circulateamongst the students and monitor their progress. Given theproper software, the students can explore topics in greaterdetail than presented by the teacher and choose a level ofdifficulty appropriate to their own personal skill ability, as130shown through the use of the Green Globs software.Unfortunately, the relatively short duration of this study didnot permit the opportunity to evaluate these advantages.The software can also pose a disadvantage in that it isnot all inclusive. For example, in this study, it could notteach the manual skills required to draw a graph. Anotherdisadvantage is that a computer environment creates more workfor teachers in that they have to prepare lesson materials tocorrelate the curriculum with the software and they have tocontend with technical and mechanical problems in theoperation of the hardware and the software. The site itselfcan also be an inconvenience, in that the computer sites inmost schools are not usually designed specifically to suitcomputers, but rather the classrooms are adapted after thefact.ConclusionsIntegrating computers in the mathematics classroom is arealistic goal. This research study illustrates that theprocess of implementation must proceed one step at a time withcareful planning and clear goal setting with regard toteaching and evaluation objectives. This study shows thatthere is a substantial amount of effort required inimplementing computers in the Mathematics 11 curriculum.Since effort is relative to the individuals involved, theresearch could not determine if there were more successes or131difficulties. The teachers in this study were notdiscouraged; in fact, they were challenged to attempt toimprove on their computer enriched unit. They also hoped tofind other areas in the mathematics curriculum in generalwhere they could implement the computers.The computer provided a challenge to the teachers in thestudy, as does any innovation. However, to initiate theinnovative process teachers need the desire and curiosity.Once that process commences, the challenges and the successesare the fuel which drive the innovation. The teachers in thisstudy introduced the computer into their teaching routine, butfor the most part used the traditional method of lecture andassignment. The observations illustrate the "idea takinghold" as described by Ellis (1990) and that even with agradual and limited introduction of computers into themathematics classroom, many factors must be considered. Thesefactors were framed in the discussion of the researchquestions.Recommendations for PracticeIt is recommended the teachers create an environment in acomputer setting where students must accept greaterresponsibility for their own learning, and where thesestudents must find new ways of assessing their understanding.The classroom lessons in the non-traditional program must bedeveloped so that they interface with the computer activities132and with the prescribed textbook. The lessons must includenot only the subject matter but software and hardware featuresand problem-solving strategies as well. The teacher wishingto integrate computers in the mathematics classroom must havea vision of the future and make decisions on whether thecomputer is actually an effective alternative or a complementin the classroom.The responsibility for security should be at two levels.The administration should be responsible for establishing theguidelines; the teacher's task would then be to enforce them.It should be noted that enforcement is difficult when manyteachers use the same computer laboratory, and in that case astaff assistant should be responsible for enforcement. In thecase of maintenance, again the administration should beresponsible for establishing a contract or entering into anarrangement with the appropriate district department or anindependent contractor.Though it has been stated that teachers working in acomputer environment have very little time to spare, theyshould make every effort to search for quality software andevaluate available software before introducing it for use inthe classroom.Recommendations for Future ResearchThe emergence of the ten features listed earlier is thepredominant result of the current research study. Two of133those features are recommended for future inquiry. Thestudents appeared to be motivated, and they displayed a verypositive attitude. This could have been a result of thenovelty of the computer experience. Would these studentsremain motivated over the long term, after the novelty woreoff?The other feature which is recommended for future studyis the evaluation instrument. In this case study, there wereno tests or quizzes on the computer activities. What type ofinstrument would be appropriate for a computer integratedmathematics course?The graphics calculator has become very popular sincethis study was carried out. In fact, at the school in thestudy thirty graphics calculators are available for use at allgrade levels (appropriate for grades 10-12). Some studentseven own their own. Future research will indicate whetherthey are effective in teaching graphing techniques, and acomparison of computer use and graphics calculator use wouldbe pertinent.Qualitative research has highlighted many of thecomponents of a computer integrated mathematics class. Thisstudy has explored the potential of the computer in themathematics classroom, and it has raised some importantchallenges for teachers. Future quantitative research willhelp to determine the effectiveness of some of the techniquesillustrated in the present study.134BIBLIOGRAPHYBaird, W. (1986). 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Use the graph tocomplete the table.(Create your own equation for 10 and 11)I.^Function Vertex Axis of Sym. Dir. of Translation from y = x2e.g. ,y = (x-3)2 + 2 (3.2) x-3 = 0x = 3right 3 units, up to 21. y=x2 - 22. y = (x - 2)23. y = (x + 5) 24. y = (x + 7) 2-35. y = x2 + 56. (7.4)7. left 1 unit, up 2 units8. down 3 units9. (-8,0)10.11.142Student Worksheet #2Graph the lollowing equations and complete the table.Function 1 Coordinates of a"Significant Point"Translation Type of Function1.^(a) y = X 2 (0,0)- - quadratic(b) y = x 2 + 3 3 up quadratic(c) y = (x-2) 2(d) y = (x+5) 2 - 62.^(a)^y =-1_,x(1,1) - -(b) y = _1_x+3x(c) y = _1+ 3(d) y = a_ + 6x-23.^(a) y =^x (0,0,) - -(b) y=^x + 8(c) y =^x - 5(d) y =^x + 2 - 54.^(a)^y = I x^1 (0,0,) - -(b) y= I x - 4 1(c) y=lx+41(d)^= I x -^21-2143Student Worksheet #3Function Coordinates of a"Significant Point"Translation Type of Function5. (a) y = x3 (0,0) --(b) y = (x + 3) 3(c) y = x3 - 6 .(d) y = (x-2) 3 + 56. (a) y = 2x (0,0)(b) y = 2 1( + 5(c) y = 2x+3(d) y = 2x-1 _ 8144Student Worksheet #4x - p = 0 is the axis of symmetryy = a(x - p) 2 + qcoordinates of vertex are (p.q)congruent to y = ax2concave up if a > 0concave down if a < 0vertical expansion if a > 1 or a < -1Vertical compression if -1 < a <Complete this chart then check by graphing the function.Function Vertex Concavity Axis ofSymmetryCongruentto:Verticalcompressionor expansione.g..y = -3(x+2)2 + 1 (-2. 1) down x + 2 = 0x = -2y = -3x2 verticalexpansion1. y = .5(x-1) 2 - 22. y = -1.5(x+3) 2 + 53. y = (x - 7)2 - 24. y = -(x-2) 2 - 55. y = 5.2(x-3) 2 + 76. y =—Ux+3.21 2 -6.751. y = -(x+2.5) 2 + 32. y = -2.5(x-1.6) 2 - 33. y-5 = 3(x+2)24. y = 3x2 - 75. y-2 = x26. y = 3-2x2145

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