involving preservice secondary mathematics teachers in data analysis

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Involving preservice secondarymathematics teachers in dataanalysisMelvin R. Wilson aa University of MichiganPublished online: 09 Jul 2006.

To cite this article: Melvin R. Wilson (1995) Involving preservice secondary mathematicsteachers in data analysis, International Journal of Qualitative Studies in Education, 8:4,345-356, DOI: 10.1080/0951839950080403

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QUALITATIVE STUDIES IN EDUCATION, 1995, VOL. 8, NO. 4, 345-356

Involving preservice secondary mathematics teachersin data analysis

MELVIN R. WILSONUniversity of Michigan

An adaptation of the constant comparative method of qualitative data analysis (Glaser & Strauss, 1967;Strauss, 1987) is illustrated using data from a study of three preservice secondary mathematics teachers. Datawere collected using interviews, observations, and written artifacts. Data analysis stages included one inwhich research participants assisted in the coding process by categorizing statements they had made inprevious interviews. This involvement simplified the data analysis process, motivated ongoing analysis duringdata collection, and helped in identifying accurate and meaningful information about the preserviceteachers' beliefs. Findings illustrate how these benefits were achieved as well as how theory emerged fromdata during analysis.

Introduction

The purpose of this article is to help bridge the gap between research theory and practice. Idescribe the data analysis procedures of a recent study wherein I adapted ideas from the constantcomparative method (Glaser & Strauss, 1967; Grove, 1988; Lincoln & Guba, 1985; Strauss,1987). Specifically, I illustrate how involving research participants in the data-coding process cansimplify data analysis procedures and help generate an interpretation that produces rich andvaluable information.

My interest in exploring mathematics teachers' views is based on my experiences as amathematics teacher and teacher educator. I have come to believe that it is more important forstudents to value and use mathematics than to master "fundamental" skills. Students shouldunderstand relationships among important mathematical concepts and procedures so they canuse them to solve meaningful problems and communicate important ideas. I believe that teacherswill be less successful in helping students achieve these goals if they limit their teaching practiceto lecturing and requiring students to practice mathematical procedures. To help students buildmeaningful understanding, I believe teachers must engage students in nontraditional learningactivities such as cooperative projects and discussions of meaningful (sometimes not well-defined)problem situations.

My experience with preservice secondary mathematics teachers in various contexts hastaught me that many do not possess meaningful mathematical understanding, nor do theynecessarily share my beliefs about what constitutes effective teaching. To examine in more depthhow preservice teachers understand and view mathematics and mathematics teaching, Iconducted the study reported here.

Guiding each phase of the study, including the design of data collection instruments, thechoice of analysis procedures, and the interpretation and reporting of results, was the assumptionthat knowledge and beliefs are relative, personal, and actively constructed by individuals.Therefore, a case study design was used; the cases investigated involved three preservice teachers.

0951-8398/95 $10·00 © 1995 Taylor & Francis Ltd.

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346 MELVIN R. WILSON

Such a design was consistent with the goal to obtain a detailed, in-depth understanding of theparticipants' views and to represent those views in a way that reflected the participants'perspectives.

My rationale for utilizing the constant comparative model in analyzing data was that itsuggests ways to allow theory to emerge from data. One might argue that Glaser and Strauss(1967) were not interested in specific theories (e.g., what preservice teachers think aboutmathematics). Nonetheless, systematic data analysis procedures, such as those suggested by theconstant comparative method, assist researchers in generating theory that is grounded in data.This is the case whether the theory is focused on a topic such as teachers' views of mathematicsand mathematics teaching or on a general theory such as teachers' beliefs more generally.

Since the primary goal here is to describe the methodology utilized and not the results of theinvestigation, the order of the following sections is reversed from that of traditional researchreports. To assist the reader in understanding my findings resulting from data analysis, a samplingof results is provided before the methodology is described. Most of the examples that followinvolve Barbara, one of the three preservice teachers studied.

Results

Participant

Barbara is a 20-year-old university junior who grew up in a suburb of a large U.S. city. She isa bright, articulate, and friendly person who anticipates with considerable eagerness the prospectof becoming a teacher. She reflected, "I've always wanted to be a teacher. I like school. I likethat atmosphere. I like going in and having interaction with people every day and knowing thatevery day is going to be different and that I'm going to be teaching them [students] somethingnew." Barbara has always excelled in school, particularly in mathematics. Her high schoolmathematics background included advanced-placement calculus, which enabled her to begin herstudy of mathematics at the university with a second calculus course. She performed well inuniversity mathematics as well as in other academic areas. At the time of the study her cumulativegrade point average was 3.6 out of 4. Barbara has limited teaching experience. She has servedas a counselor in a summer camp for elementary-age children. During the previous semester,she participated in a practicum that involved observation and limited teaching in local middleand high schools. She has also done some tutoring of friends and fellow students.

Barbara's views about mathematics and mathematics teaching

Throughout the data-collection period, Barbara communicated two separate and seeminglyinconsistent views about mathematics. One view depicted mathematics as an open, challenging,and sometimes ambiguous subject to be explored. The other view portrayed mathematics as a fixedset of rules and procedures to be mastered. As interesting as these themes are independently, theyare even more interesting in their relationship to one another.

Mathematics as a way of thinking. Barbara has always been good at mathematics and, as aconsequence, enjoys it. One of the things she has always liked about it is that, in her view, thereis less ambiguity in mathematics than in other subjects. Although the exactness of mathematicsappeals to her, Barbara admitted that much of mathematics is not well defined. She contrastedher high school mathematics experience, where everything was well defined, to her university

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INVOLVING TEACHERS IN DATA ANALYSIS 347

experience. She came to realize, "It's not that way, it's not the numbers, it's a bunch of ideasthat fit together; and it's so much: a lot of it you can't visualize, it's just out there somewhere,at least for me it is, it's just floating out there."

Barbara insisted on several occasions that mathematics is "a way of thinking." For example:"You're not really trying to find an answer, you're trying to get the idea of it, the thinking thatwent in behind it." She also said, "I think [the way of thinking] is more important than beingable to solve for x." While reflecting on this statement during one interview, she classified it asbeing very important to her conceptions of mathematics.

Mathematics as a set of correct rules. In contrast with her professed belief about mathematics as a wayof thinking, many of her statements about mathematics teaching reflected a more narrow or"dualistic" (Perry, 1970) orientation to mathematics. Her views of school mathematics and ofmathematics teaching were particularly narrow. Throughout the study, Barbara emphasized thatteachers should teach from a "high school point of view," which she implied to mean was a viewthat mathematics is a well-defined system of rules. For Barbara, the correctness of mathematicalrules is determined largely by textbook and teacher authority rather than student understandingor context. For example, she made it clear that she feels teachers should "pull out the basic stuffthat everyone else is getting from textbooks," even though she acknowledged that openapproaches to teaching might have advantages in providing students with rich perspectives ofmathematics. Her statements about potential uses of technology in teaching were consistent withthis dualistic view. She characterized calculators and computers as being fun and interesting butperipheral to understanding the "basic" mathematics found in textbooks. On another occasion,while discussing the relative value of alternative methods for solving systems of equations, sheemphasized that it would be important for teachers to teach multiple methods, but only becausestudents need to "have all those skills."

Discussion. By considering Barbara's experience as a student of high school and universitymathematics, one can better understand why she communicated two different and seeminglycontradictory sets of beliefs about mathematics. The "exactness" of mathematics has alwaysappealed to Barbara. She speaks fondly about her high school algebra courses where she could"solve for x" and her university calculus courses where she could always "check herself in theback of the book." She speaks less affectionately of other university courses that portrayedmathematics as more "abstract" and "undefined." She described her experience in a universitymathematics course as a "nightmare" in which she "never had a clue" as to what the teacherwas talking about. Her experience in the mathematics education course (where this studyoccurred) was even frustrating at times. In relating her feelings about these experiences, Barbaramentioned that many of her friends had similar frustrating experiences in high school and hadconsequently given up the study of mathematics. Barbara's ability to work through frustrationsseparated her from many of her former high school classmates.

Barbara's university mathematics experiences have not all been negative. She describedseveral university courses that were "total theory" but were interesting to her. Barbara feels thather experiences in these courses gave her a broader perspective about the nature of mathematicsand have taught her to consider it as "a way of thinking." This view of mathematics, for Barbara,is not consistent with her central belief about not confusing or discouraging students. Forexample, Barbara believes it would not be appropriate for teachers to give students problems towhich there are no solutions or for which the teacher does not know the solution. She says, "Youcould lead them completely in the wrong direction and get them more confused. That's not whatyour job is. You're supposed to be teaching them, not confusing them." She explained why shefelt many students do not succeed in mathematics: "I think some just get discouraged; I think

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it all stems from an earlier experience, if you get discouraged early on, you are never going tosucceed at it." Barbara wants mathematics to be a positive, nondiscouraging experience for herstudents. Although probably not consciously planning to do so, this suggests that she may shieldher students from what she perceives are the most difficult aspects of mathematics. As aconsequence, she may communicate a narrower view of mathematics to her students than theone she personally holds.

The brief description of some of the results of the study will serve as a backdrop to thedescription of the methodology, particularly the analysis procedures utilized.

Design

Triangulation. (Denzin, 1970), or the use of multiple research methods to study the samephenomenon, was a primary consideration in designing the study. Across-method triangulationinvolved the use of several distinct methods, including interviews, observations, a written task,and artifacts. The use of various interview types and several observation types accounted forwithin-method triangulation.

Research site

The study involved three preservice secondary mathematics teachers enrolled in an undergrad-uate mathematics education course at a large American university. The purpose of the courseis to help preservice teachers determine content for the secondary school mathematics curriculumand demonstrate competency in this content. The course precedes a field-based methods courseand student teaching. The class in which this study took place met for two hours, two (andsometimes three) times per week, for 10 weeks.

Data sources

Written instrument. All 20 preservice teachers enrolled in the course were asked to complete awritten instrument. Preservice teachers were asked to describe verbally their thinking whilesolving the mathematics problems posed in this instrument. Their comments were audiorecordedand transcribed. Following preliminary analysis of the written and taped responses, fourpreservice teachers were invited to participate in a more open-ended and in-depth investigation.Included were two preservice teachers whose responses demonstrated a relatively strongunderstanding of the mathematics tested and two who demonstrated a weaker understanding(Barbara was in the first group). All four preservice teachers agreed to participate, although onlythree completed the data-collection process.

Interviews. Each of the three preservice teachers who completed the study participated in sevenformal interviews. These interviews varied in purpose, structure, and format. Each lastedapproximately one hour and was audiorecorded and transcribed. The first two interviewsengaged participants in mathematical problem-solving activities. The third interview includeda discussion about potential uses of calculators and computers in teaching mathematics. The foursubsequent interviews focused more directly on participants' beliefs about mathematics andmathematics teaching. The questions for two of these "beliefs interviews" consisted primarily ofepisodes (Brown, Brown, Cooney, & Smith, 1982) that describe hypothetical situations designed

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to stimulate in-depth discussions about mathematics and mathematics teaching. Following arethree of the episodes used during interviews with Barbara:

1. You are the department head of a large high school and one of your responsibilities is toobserve and evaluate other math teachers in your department. You observe a particular lessonin which the teacher demonstrates several methods of solving systems of equations, includinggraphing, elimination, and substitution. The students complain that it is confusing to see so manydifferent ways. The teacher is quite perplexed at students' reactions. What advice could you givethe teacher in this situation?

2. Have you ever been confused or frustrated in mathematics?3. What are some adjectives you would use to describe mathematics?The final two interviews were open ended. I prepared outlines before the interviews and

referred to them during conversations with the preservice teachers. In one interview, teacherswere engaged in a clustering activity (Cooney, 1985) in which they were asked to organize andelaborate on statements they had made in previous interviews. The final interview was conductedto probe prominent themes identified during previous interviews and observations. Anotherpurpose of this final interview was to bring closure to the research process by encouragingparticipants to comment on issues they believed to be interesting and important.

Teacher/student interview. Each preservice teacher planned and conducted a short interview withan undergraduate student to assess the student's understanding of a specific mathematical topic(function). I met individually with the preservice teachers for about 20 minutes each as theyplanned the interviews. I also observed the actual interviews. During the interviews, preserviceteachers focused on their students' responses; I concentrated on the preservice teachers' questionsand the nature of their reactions to student responses. I also conducted debriefing sessionsimmediately following the interviews during which the preservice teachers were asked to describeinformation gained from the interviews as well as why they asked specific questions. Each of thesemeetings (planning session, interview, debriefing session) was audiorecorded and transcribed.

Class observation and artifacts. As an outside observer, I joined the university mathematics educationcourse in which the preservice teachers were enrolled (Becker & Geer, 1982) and observed allcourse meetings. Most class meetings were split equally between whole-class discussion andlecture and small-group activities. All three of my participants were in the same small group,hence I was able to observe them during both whole-class and small-group activities. Data fromclass observations supplemented and clarified information gained through interviews andprevious observations. During the observations, I took careful fieldnotes and occasionallyconducted brief, informal interviews with the participants. Additional data sources includedcopies of written work (e.g., assignments and tests) completed by the participants for the course.Because this study focused on the conceptions and experiences of the participating preserviceteachers (and not the course instructor), no formal interviews were held with the instructor,although I met with him periodically during the term to coordinate curriculum plans and obtainparticipants' written work.

Data analysis

Glaser and Strauss (1967) recommend a constant comparative data analysis approach forgenerating what they refer to as "grounded theory" from social settings. However, they cautionthat this approach should be viewed as a set of guidelines or "rules of thumb" as opposed to "rules"

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to be indiscriminately applied to obtain valid results. In analyzing data from this study, I usedguidelines suggested by Glaser and Strauss to generate theory about the mathematical views ofpreservice teachers. By systematically analyzing data during data collection, I was able to makeappropriate adjustments in data-collection instruments and to look for evidence that conflictedwith emerging theory as well as evidence that supported that theory. This study extended theconstant comparative method in that the research participants were actively involved in dataanalysis. Data analysis progressed through several stages that involved generating categories andcoding data according to those categories, integrating categories, and identifying categories thatbecame "theoretically saturated" (Strauss, 1987, p. 21).

Open coding

Initial coding of interview data occurred immediately after each interview and involved readingthe transcript, writing memos about the overall tone of the interview, and generating possiblequestions for subsequent interviews. For example, as I read the transcript of one of Barbara'sinterviews, I noted that in describing an observation at a local school, she indicated dissatisfactionwith the teacher's approach. Consequently, I designed questions for the next interview thatinvestigated this point. Her responses to questions in the subsequent interview not only reinforcedand clarified some of my initial interpretations but also revealed that I had not properlyunderstood her criticism of the teacher discussed in the previous interview. Whereas I thoughtshe was describing how she would teach differently from the regular teacher, her intent was tocommunicate how one of the teachers at this particular school had done things differently fromthe other teachers.

Fieldnotes from class observations as well as assignments and tests were analyzed in similarways. For example, as I initially reviewed fieldnotes from a classroom observation, I observedthat Barbara experienced some frustration during one of the course activities. During our nextinterview, I gave her an opportunity to comment on that activity. She described how she andother students initially felt frustration with the task but worked through this frustration by meetingas a small group outside class to work on the task.

A second round of coding involved a line-by-line analysis of interview transcripts andobservation fieldnotes in which I identified major categories and themes emerging from the dataas well as less prominent subthemes. For each transcript, as I identified categories, I sketchedrough diagrams such as the partial schema in Figure 1 of Barbara's third interview. The schemadiagrams summarized the content of the interviews as well as illustrated ways in which the variouscategories were related. The diagram in Figure 1 summarizes some of Barbara's thoughtsconcerning potential instructional uses of computers. The diagram shows that Barbara's "couldbe useful" category is related to her "possible uses" category. Further, her "possible uses" categoryis related to her "demonstration," "side activity," and "reinforce basics" categories. Excerpts ofthe actual interview transcript from which the diagram in Figure 1 emerged are contained in theappendix. The rough schema diagrams helped me during later stages of analysis to definecategories, see relationships among categories, and merge categories within and across datasources.

Teacher participation in analysis

The sixth interview engaged preservice teachers in a clustering activity. This activity was designedto let participants elaborate on their previous statements and to organize them in ways that

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demonstration

sideactivity

computers

could be usefulnot very »familiar with hindrance for me

reinforce /basics not used exposed

much in h. s. some Geometer's sketch pad

at home / geoi

Dr.Davis'stuff

fungeometricsupposer

programming inan algebra class

computer \lab w/book c a n t remember much

Figure 1. Partial schema of Barbara's third interview-statements about computers.

represented how they understood the various ideas to fit together. After analyzing transcripts ofthe first five interviews and identifying prominent themes in belief systems, I selected about 40statements for each teacher and typed them on 4 X 6-inch cards. Included for each teacher werestatements that reflected the major themes identified for that teacher. Where possible, statementswere selected that represented "discrepant cases" (Erickson, 1986, p. 140), that is, statements thatseemed to contradict the themes. Also identified were neutral statements or statements that didnot appear to fit any of the themes. Following are three statements made by Barbara that illustratethis range.

Theme: Mathematics is a way of thinking.Consistent: You're not really trying to find an answer, you're trying to get the idea of it.

The thinking that went in behind it.Negative: I still like being exact... where you came up with an answer and you knew

it was right.Neutral: I think certain advanced classes could really look at proofs and understand

[them].

During the clustering interview, participants were asked to sort their 40 statements intocategories using whatever criteria they wanted. After sorting the statements, the teachers wereasked to provide a one- or two-word heading for each category and a brief sentence to describewhat the collection of statements expressed. They were also asked to identify which statementsin each group most strongly represented their beliefs about that category. The results of one suchclustering are shown in Figure 2.

The clustering activity provided many benefits. Allowing participants to participate in theanalysis process helped me better understand their perspectives on how their beliefs were relatedand which ones were most important to them. The activity also gave participants the opportunityto elaborate on previous statements and clarify questions I had about their meanings as well asto qualify or disagree with statements they had made previously.

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Statements clustered together:• They may be able to look at [geometry] a little differently than most people

if they are seeing it in art or something. That's fine, as long they're still ableto pull out the basic stuff that everyone else is getting from the textbooks.

• I think certain advanced classes could really look at proofs and understand[them].

• I think [an algebraic proof] would be more concrete, it would be a betterproof than this demonstration [picture].

• I think the constructions . . . might help [high school students].• I want [students] to know what they are doing before they play with the

machine [calculator].• I probably wouldn't rely solely on it [computer]. I'd want them to know the

basics without the computer and maybe use that as a demonstration oractivity on the side.

Heading: TEACHINGDescriptive statement: These statements describe how I would teach.

Figure 2. One category generated by Barbara during the clustering interview.

Selective coding

The previously described phases of analysis constituted what Strauss (1987) refers to as "opencoding" (p. 28). Selective coding involved defining categories by identifying quotations from theinterviews and examples from fieldnotes and written work that pertained to each category. I alsoidentified categories that shared common subcategories or themes in an effort to mergecategories. For example, for Barbara, I noted that both her calculator and computer categoriesshared the subcategory of "uses." She also described a similar range of potential instructionaluses for both calculators and computers. Consequently, I merged the calculator and computercategories into a single category called "technology." I also noticed that during the clusteringinterview Barbara placed her statements about calculators and computers in the cluster she called"teaching," thus I included the "technology" category as a subcategory of her "teaching"category. As I defined categories, I merged them both within and across data sources andidentified how the merged categories fit together and were related to a common "core category"(Strauss, 1987, p. 18).

Figure 3 shows a grand schema diagram summarizing the beliefs about mathematics andmathematics teaching communicated by Barbara. The diagram evolved through a synthesis ofthe intermediate schema diagrams (e.g., Figure 1) created and revised throughout the open- andselective-coding processes. The outer clusters of categories in the diagram are those suggestedby Barbara in the clustering interview. Although considerable overlap exists among categories,I kept the four categories separate to preserve Barbara's interpretation of how the ideas arerelated. The core category, "Prevent Confusion and Discouragement," is not one that Barbarasuggested explicitly; rather, it emerged as I observed common themes among clusters ofcategories.

Selective coding occurred between the clustering (sixth) interview and the final (seventh)interview. One purpose of the final interview was to check my models of the participants' views.Rather than suggesting my interpretations explicitly to the participants, I asked them indirectquestions. For example, to further investigate Barbara's core category (Prevent Confusion andDiscouragement), I asked her to describe circumstances, if any, in which she believed it mightbe appropriate for a teacher to confuse students. This is another example of how, throughout

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My opinions-Math and teaching

teachers should know answers

students' point of view is important

help students understand

confusion causes hate

emphasize basics

some kids never adapt

basic classes—do not spoon feed

math has part in everything

not too focused on one method

About me-Things I like

I like exact answers

important to know mathematics

I want the gift of knowing

always wanted to be teacher—like school

MAT 510 nightmare

more comfortable w/ algebra than geometry

PreventConfusion

andDiscourage-

ment

Teaching—How and what

basic stuff from book

proofs ok for advanced h.s.

constructions help h.s. students

understand before calc/computer

technology uses—demonsration, side activity

Mathematics-How I describe it

way of thinking

helps thinking processes

thinking more important than solving

rules-some things not allowed

innate-you just know it

abstract—just 'out there'

undefined

complex—ideas that fit together

Figure 3. Grand schema describing Barbara's views of mathematics and mathematicsteaching.

my analysis, I checked and cross-checked my hypotheses to make certain that I reported onlythose categories and themes that were supported by the data.

Importance of the study

Since most teachers are the primary mediators between the subject and learners, it is importantto understand how teachers think about the subject they teach. Information about the natureof mathematics teachers' views about mathematics and mathematics teaching can be a valuableresource in improving the quality of teaching by providing feedback for mathematics teachersand teacher educators.

The results of the study are not the main focus of this article. Its primary purpose is to describea specific example of how participant involvement in data analysis can be effectively employedto help illuminate the conceptions held by the persons studied. Involving participants in thecoding of their own data helped in at least three ways: (a) it motivated me to conduct meaningfulanalysis of the data during the data collection period, (b) it made data analysis easier and lesstime consuming, and (c) it made interpretations less subjective.

Ongoing analysis

Ongoing analysis is an essential component of constant comparison (Glaser & Strauss, 1967) aswell as most other qualitative analysis procedures (e.g., Goetz & LeCompte, 1984; Miles, 1984).Qualitative data analysis requires a tremendous investment of time and effort, and time is at a

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premium while one is involved in collecting data. Every researcher faces the temptation of lettingthe data "sit" until after data collection has been completed. But ongoing analysis is essentialbecause it makes final analysis easier and also because it improves the quality of subsequentdata-collection procedures. One benefit of participant involvement in data analysis was that itmotivated me to analyze data during the data-collection period. In planning for later interviews,I read transcripts of previous interviews. To prepare effectively for the clustering interview, Ineeded to conduct a careful and detailed analysis of data collected throughout the first half ofthe data-collection period. I had to generate categories and find quotations from the interviewsthat represented those categories, and I had to find neutral and contradictory statements as well.Had I not been forced to become familiar with the data, I might have made the potentially fatalmistake of waiting until after data collection to conduct a detailed analysis.

Saving time

The inductive process of qualitative analysis is extremely time consuming; data analysis generallytakes at least as much time as data collection. Further, "More 'rigorous' data analysis procedurescannot balance a less rigorous design" (McGee-Brown, 1988, p. 96). That is not to say thatsystematic analysis procedures cannot assist researchers in managing and using their time. In thepresent study, involving participants in data analysis did save me some time. Part of this benefitwas realized because I systematically analyzed data on an ongoing basis during data collection,which I left less of the analysis to be conducted at the end of the study. However, havingparticipants code their own data added further to this benefit. Before the clustering interview,I struggled with how to make sense of Barbara's two seemingly incompatible views ofmathematics. As I analyzed Barbara's classification of her own statements, it became clear to methat her two views were not at all incompatible but were intimately related to a central beliefconcerning teaching- not to confuse students. Even if the same theory describing Barbara's viewsabout mathematics and mathematics teaching had emerged from the data without Barbara'sdirect input in coding the data, it would have required far more time and effort on my part.

Reducing subjectivity

In qualitative studies, all data must pass through the mind of the researcher. It is no wonder thata common criticism of qualitative research is its potential subjectivity. Some individuals rightlyask, "Do the conclusions of this study represent an accurate portrayal of the participants'perspectives, or are they unfairly biased by an outsider's point of view?" As Phillips (1990) pointsout, "there is no escape from subjectivity" (p. 20). On the other hand, all interpretations are notequally subjective, and many suggestions have been offered concerning how to "tame" (Peshkin,1988) the subjectivity inherent in qualitative studies. Triangulation (Denzin, 1970) is a commonlycited design approach employed to "validate" conclusions. Extended time in the field andsystematic data analysis procedures, such as those involved in the constant comparative method,offer other ways to account for subjectivity. In this study, involving participants in the analysisprocess not only contributed to a systematic analysis but also served as a means for providingtriangulation, of a sort. While I was the only researcher formally involved in data collection andanalysis, this study involved a type of researcher triangulation because multiple (informed)individuals actually coded the data.

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Closing comments

Obtaining an emic perspective is a goal of many qualitative studies. In the words of Franz Boas(1943), "If it is our serious purpose to understand the thoughts of people, the whole analysis ofexperience must be based on their concepts, not ours" (p. 314). Involving participants in dataanalysis can certainly help generate theory that offers such a perspective. After all, who knowsbetter how to make sense of an individual's statements than the individual? On the other hand,participants have their own prejudices and biases and may be too close to a given situation orhave competing interests that prevent them from seeing (or admitting to) issues that should beconsidered. In interpreting data and generating theory, I do not advocate that researchers relycompletely on information gained during participant analysis. Emergent theory should not bebased too heavily on either the participant's or the researcher's viewpoint. It should representa balance between thoughtful input from knowledgeable insiders and outsiders.

Acknowledgments

An earlier version of this paper was presented at the American Educational ResearchAssociation Annual Meeting, San Francisco, April 1992.

This paper is based on the author's doctoral dissertation completed at the University ofGeorgia in 1991 under the direction of Thomas J. Cooney. The research was supported by theNational Science Foundation (NSF Grant No. TPE-9050016). The conclusions and recommen-dations expressed here are those of the author and do not represent official positions of the NSF.

The author would like to thank Tom Cooney for his guidance throughout the study and MaryJ. McGee-Brown for her assistance in planning the study.

Appendix: Excerpts from an interview with Barbara

I: Barbara, what were your perceptions of the activity that we did [in class] last Thursday withthe computer, with the Geometer's Sketch Pad?

B: I think it would be very useful in the classroom. I liked using it. I think it's fun. It's a littlefrustrating if you don't know what you're doing. I'm not very familiar with computers, soI was starting from square one and had to figure the whole thing out. But I thought it wasinteresting. I've used it before.

I: You have used the Geometer's Sketch Pad?B: Something similar, I've used the, is it Geometric Supposer?I: Uh huh.B: On the Apple He.I: On the Apple II.B: Right, I liked it. I think it's neat.

I: When you were in high school did your teacher ever use the computer?B: I had one algebra class that we used the computer, and it was basically programming. And

I couldn't tell you a thing about it 'cause we did it once a week, and our teacher was nottrained to use the computer. They just sent us to the computer lab with a book. But I thinkit would be, if you had a teacher that knew how to use it. I think it would be a useful tool

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in the classroom if you knew what you were doing. But like I said, I didn't; so it'd be more

of a hindrance.

I: So do you have any kind of visions about the kinds of things you might do with the computer

as a teacher?

B: I probably wouldn't rely solely on it. I would probably start, I'd want them to know the basics

without the computer and maybe use that as a demonstration or something or activity on

the side.

I: You're thinking specifically about that particular program, the Geometer's Sketch Pad, or

are you thinking in more general terms?

B: I'm not as familiar with too many. I have an article from, I think it was, Dr. Davis. He was

president over here or something, and he wrote a program that was used in the Augusta

school system.

I: I'm not familiar with it. Tell me about it.

B: I have this whole book on it. It was the actual, the presentation of his program.... Only I

don't understand it 'cause I don't have the program in front of me, so it's just a bunch of

words. But that's the only other one I've ever seen.

I: I noticed that you and some of the other students felt pretty comfortable with what was going

on in class [with the computers].

B: We've all been. Yeah, we've all been exposed to computers to some extent or computers

at home like with the mouse and stuff like that. But we had to pick up on it, but still for

a while, it's like, what are we doing? How to get things selected pressing that button, that

was fun.

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involving preservice secondary mathematics teachers in data analysis

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