Exploring Changes in Pre-service Elementary Teachers' Mathematical Beliefs

JENNIFER E. SZYDLIK, STEPHEN D. SZYDLIK and STEVEN R. BENSON

EXPLORING CHANGES IN PRE-SERVICE ELEMENTARYTEACHERS’ MATHEMATICAL BELIEFS

ABSTRACT. Research literature (e.g., Thompson, 1992) suggests that teachers’ beliefsabout the nature of mathematics provide a strong indicator of their future teachingpractices. Moreover, current reform efforts (e.g., NCTM, 2000) ask teachers to lead mathe-matical explorations that allow their own students to construct mathematics. Understandingprospective teachers’ mathematical beliefs and the circumstances under which these beliefsmight be changed is therefore critical to teacher educators. In this paper we describethe culture of a mathematics content course for prospective elementary teachers that isdesigned to provide participants with authentic mathematical experiences and to fosterautonomous mathematical behaviors. Using both survey and interview data, we exploredparticipants’ beliefs about the nature of mathematical behavior both at the commencementand at the completion of the course. We argue that the participants’ beliefs became moresupportive of autonomous behaviors during the course. We report that students attributedchanges in beliefs to specific classroom social norms and sociomathematical norms thatincluded facets of work on “big” problems with underlying structures, a broadening in theacceptable methods of solving problems, a focus on explanation and argument, and theopportunity to generate mathematics as a classroom community.

KEY WORDS: autonomy, elementary education, mathematical beliefs, problem solving,sociomathematical norms

INTRODUCTION AND BACKGROUND

Hersh (1986) writes, “One’s conception of what mathematics is affects one’s conceptionof how it should be presented. One’s manner of presenting is an indication of what onebelieves is most essential in it . . . The issue, then, is not, What is the best way to teach?but, What is mathematics really all about?” (p. 13).

Mathematics is a human activity; problems are encountered, interpretedand refined, assumptions are made and clarified, and arguments arecreated. Halmos (1985) explains, “When you try to prove a theorem, youdon’t just list hypotheses and start to reason. What you do is trial anderror, experimentation, guesswork” (p. 321). On the other hand, whileinsight comes from exploration, conviction in mathematics arises throughdeduction and consistency. In its rigorous form, mathematics is basedon fundamental assumptions and validity arises from those assumptionsthrough proof. Tall (1992) writes, “Advanced mathematical thinking –

Journal of Mathematics Teacher Education 6: 253–279, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

254 JENNIFER E. SZYDLIK ET AL.

as evidenced by publications in research journals – is characterized bytwo important components; precise mathematical definitions (includingthe statement of axioms in axiomatic theories) and logical deductionsbased upon them” (p. 495). Reason, logic, and internal consistency aresources of conviction for the mathematical community. Moreover, the actof producing new mathematics is inherently creative; problem refinement,exploration, and arguments come from within the community, not froman external authority. Thus, arguably, an authentic mathematical experi-ence consists of exploration, sense making, and creating arguments usingdefinitions and logical deduction.

The literature suggests that prospective elementary teachers see mathe-matics as an authoritarian discipline, and that they believe that doingmathematics means applying memorized formulas and procedures to text-book exercises (Carpenter, Linquist, Mattews & Silver, 1983; Ball, 1990;Shuck, 1996). In fact, traditional mathematics classroom norms appearto foster this belief. They do not provide authentic mathematical experi-ences or promote autonomy; instead, they encourage students to rely onan external authority (their instructors or textbooks) for the determinationof mathematical validity (Schoenfeld, 1989; Frid & Olson, 1993). Green(1971) asserts that a classroom emphasis on rules and memorized proce-dures would be better called indoctrination, and he distinguishes teachingas the process of helping an individual to know based on rational evidence.Cooney, Shealy & Arvold (1998) write, “An indoctrinated view of mathe-matics minimizes the impact of rationality in favor of memorization. Thisview constitutes the antithesis of considering mathematics as a humanendeavor” (p. 311).

That elementary education students hold beliefs inconsistent withautonomous behavior is particularly worrisome given that teachers’ beliefsabout the nature of mathematical behavior influence the way they conveymaterial to their students (for a synthesis of this literature see Thompson,1992). For example, Thompson (1984) found that a teacher who viewsmathematics as a collection of facts and rules to be memorized andapplied is more likely to teach in a prescriptive manner, emphasizing rulesand procedures conveyed by the teacher. On the other hand, a teacherwho holds a problem-solving view of mathematics is more likely toemploy activities that allow students to construct mathematical ideas forthemselves. If we are to educate future teachers who are able to leadmathematical explorations and to allow their own students to constructmathematics, that is, if we are to help them teach in a way consistentwith current calls for reform (NCTM, 2000), we must help them seemathematics as a discipline that makes sense.

MATHEMATICAL BELIEFS 255

Unfortunately, pre-existing beliefs about teaching, learning, and subjectmatter tend to be tenacious and resistant to change (Lerman, 1987; Brown,Cooney & Jones, 1990; Pajares, 1992; Foss & Kleinsasser, 1996). Kagan(1992) observed that pre-service teachers tend to leave their universityprograms holding primarily the same beliefs with which they arrived, withmany of their initial biases having in fact grown stronger throughout theirprogram. She asserted “If a program is to promote growth among novices,it must require them to make their pre-existing personal beliefs explicit;it must challenge the adequacy of those beliefs; and it must give novicesextended opportunities to examine, elaborate, and integrate new informa-tion into their existing belief systems. In short, pre-service teachers needopportunities to make knowledge their own” (p. 77).

Yackel & Cobb (1996) have suggested that cultural and social processesare integral to mathematical activity and have posited that the culture ofthe mathematics classroom is central to the development of mathematicaldisposition among students and to bringing about change in mathematicalbeliefs. This perspective of the classroom as a culture is described byBauersfeld (1993):

“[T]he understanding of learning and teaching mathematics . . . support[s] a model ofparticipating in a culture rather than a model of transmitting knowledge. Participating in theprocesses of a mathematics classroom is participating in a culture of using mathematics,or better: a culture of mathematizing as a practice. The many skills, which an observer canidentify and will take as the main performance of a culture, form the procedural surfaceonly. These are the bricks of the building, but the design for the house of mathematizingis processed on another level. As it is with cultures, the core of what is learned throughparticipation is when to do what and how to do it” (p. 4).

Yackel and Cobb propose that within this culture there is a reflexiverelationship between beliefs and classroom norms: the beliefs that studentsbring to the classroom will interact with the agenda brought by the teacher,and together the students and teacher will negotiate norms and taken-as-shared meanings. The student beliefs will influence the classroom normsand those norms, in turn, will influence the beliefs of students.

Sociomathematical norms are those that provide access to mathematicaldistinctions. For example, the expectation that students respond to eachother’s ideas is a social norm. What counts as a different mathematicalsolution or argument, a relevant mathematical comment, and an elegantsolution are sociomathematical norms (Yackel & Cobb, 1996). Similarly,the level of rigor expected in justifying one’s claim and the acceptedform of an argument are sociomathematical norms. These norms informstudents when it is appropriate to contribute to the classroom discourse(Is their idea different? Is it relevant? Is it more elegant or efficient than


what has been previously shared?), and how to make it (Is the contributionvalid? Rigorous? Convincing? In an accepted form?).

In this work, we describe a classroom for elementary education studentsin which participation in the classroom culture meant engaging in authenticmathematical behaviors (in the sense we described previously), and weexplore participants’ beliefs about the nature of mathematical behavior,both at the commencement and at the completion of the course, and reporthow students attribute changes in beliefs to specific classroom social normsand sociomathematical norms. By autonomous mathematical behavior, werefer to that behavior that involves sense-making rather than memorizationor appeals to authority. Within the class community, we focus on sociallynegotiated meaning and on promoting community autonomy (as distinctfrom reliance on an outside authority) rather than on the autonomy of indi-viduals. In fact we assume “. . . the development of individuals’ reasoningand sense-making processes cannot be separated from their participationin the interactive constitution of mathematical meanings” (Yackel & Cobb,1996).

The classroom culture we attempt to nurture was inspired in part byclassrooms described by Lampert (1988); Wilcox, Schram, Lappen &Lanier (1991); Civil (1993). These educators were also concerned withcreating classroom communities in which the mathematical behaviors ofexploring, conjecturing, and reasoning are the norm. All attempted tochallenge authoritarian assumptions held by the students about the natureof mathematical behavior and to convince them that mathematics madesense. As Lampert (1988) writes, “I assumed that changing students’ideas about what it means to know and do mathematics was a matterof immersing them in a social situation that worked according to rulesdifferent from those that ordinarily pertain to classrooms, and then respect-fully challenging their assumptions about what knowing mathematicsentails” (p. 470). Like Lampert, we hoped that the culture of such a classwould nurture student beliefs and would make them powerful practitionersof mathematics. Furthermore, we hoped to attribute changes in beliefs tospecific norms.

Our effort focused on establishing two primary mathematical beliefsthat support autonomous behavior: 1) mathematics is a logical andconsistent discipline as opposed to a collection of facts, and therefore2) mathematics is something that can be figured out as opposed to some-thing that must be handed down by an authority. We attempt to documentand make sense of changes in student beliefs about these and relatednotions in the context of the social and sociomathematical norms thatmight influence (and be influenced by) these beliefs. We use the term


norms broadly to encompass the expected ways of participating in theclassroom culture.

The remainder of the paper is organized in five sections. In the first,we provide a description of the culture of the mathematics classrooms inwhich our intervention took place. As part of this description, we attemptto detail explicitly the social and sociomathematical norms of that culturevia quotes from transcribed videotape. In the second section, we describethe methodology of this study. In the third section, we describe partici-pant responses, at the start of the course, to a questionnaire and structuredinterviews about the nature of mathematical behavior and autonomy. Inthe fourth section, we explore changes in students’ beliefs based on thefinal interview data, and we report students’ attributions of changes intheir beliefs, focusing on both social and sociomathematical norms ofthe classroom. In the final section, we reflect on the study and provideconcluding remarks.

THE NUMBER SYSTEMS CLASSROOM

Overview

Number Systems is the first of three required mathematics courses forelementary education majors at our university (a public, comprehensive,undergraduate institution in the United States). The content of the courseis typical of many first mathematics courses for prospective elementaryteachers. It focuses on sets, whole number and integer operations, modelsfor operations, algorithms, rational numbers and their operations, logicand number theory. The class meets for an hour three times each weekfor 14 consecutive weeks. The participants in the study were 93 studentsenrolled in three sections of the Number Systems course. Each of thethree sections consisted of approximately 30 students. Typically, thestudents had completed three years of high school mathematics preparationincluding two years of algebra and a year of geometry. The first author ofthis article was the instructor for the three sections. She is a mathematicianand a mathematics educator.

The course was generated almost entirely by the conversations thatarose out of community work on a set of demanding problems, each ofwhich had an underlying mathematical structure that formed a part of thecourse content. Furthermore, the problems generally allowed for a varietyof problem solving strategies. As a typical example, consider the (perhapsfamiliar) Locker Problem:


The lockers at Martin Luther King Middle School are numbered from 1 to 100. Onemorning a teacher opens all of the lockers. Then another teacher closes every second locker(that is, those numbered 2, 4, 6, . . .). Then the first teacher changes every third locker(numbers 3, 6, 9, . . .): if it is open, she closes it and if it is closed, she opens it. Then thesecond teacher changes every fourth locker, and so on. At the last stage, one of the teacherschanges every 100th locker. Which lockers are open at the end of all this and why?

The open lockers are those showing perfect square numbers. Why?Each locker is changed once for each factor its number has, and the perfectsquares are exactly those numbers with an odd number of factors (thesquare root factor will have no “partner”). Hence they will be the onlyopen lockers. This mathematical structure can be discovered by collectingdata, solving a smaller version of the problem, considering several specificcases, or by logical considerations.

During a typical class meeting, students worked for 20 to 30 minutes ona problem, like the one above, in small groups of three or four students. Theclass then convened in a large semicircle for a discussion of their findings,strategies, solutions and arguments. In these discussions, the instructoremphasized the necessity of mathematical justification; complete solu-tions required logical arguments. However, the class was designated as themathematical authority. The instructor declined to give the final word onthe correctness or the completeness of any solution and there was no text.

In both small and large group discussions, the instructor saw herprimary roles as that of a motivator, scribe, challenger and guide. Inthese roles, she often engaged consciously in the following behaviors:intent listening, feigned (and sometimes real) confusion, skepticism andsilence. The goal was for the essential mathematics and underlying struc-ture of each problem to be revealed, but by the students, not the instructor.We caution that this did not mean that the instructor contributed littleto the discourse or withheld information needed to solve a problem. AsCobb, Yackel & Wood (1992) argue “[t]he conclusion that teachers shouldnot attempt to influence students’ constructive efforts seems indefensible,given our contention that mathematics can be viewed as social practiceor a community project” (pp. 27–28). When a student made a conjecturethat the instructor could not immediately evaluate, she acted as one of thecommunity, joining in the attempt to argue or create counter-examples. Asstudents described mathematical ideas, the instructor gave those ideas thestandard names and assigned notation. There were a few occasions whenthe instructor suggested a known (to her) approach or organized the dataon the board in such a way as to reveal structure or suggest an argument,and, on three occasions, the instructor presented some mathematics (e.g.,a proof that the square root of two is irrational). However the spirit of theclass was consistently one of community inquiry.


Though, generally, the class had discovered a solution to a courseproblem by the end of class, this was not always the case, and frequentlythere remained further mathematics to be uncovered. On six occasionsthroughout the semester, students were asked to produce written reports ontheir problem-solving efforts. These assignments provided opportunitiesfor further reflection and discussion. In the reports, students were requiredto focus their mathematical thinking by describing the problem, discussingthe strategies they used to work on the problem (including those that leddirectly to a solution and those that did not), providing a solution, and,finally, arguing that their solution was complete and valid. In some casesthese reports were produced as group projects.

Classroom Norms

We structured the course with the intent of establishing social and socio-mathematical norms that might foster autonomy. The norms identified(based on a priori design of the course, the video tape, and reflections ofthe instructor) within the classroom culture included the following:

Social Norms

1. The environment of the classroom is informal and respectful. Studentsare not afraid to contribute their ideas.

2. Doing mathematics is a community effort.3. In a whole class discussion, students share ideas with each other as

opposed to just the instructor. Students respond directly to each other’sideas.

4. Solutions and arguments come primarily from the students and not theinstructor.

5. The role of the instructor is to guide the discussion and to provideencouragement.

Sociomathematical Norms

6. The content of the course is generated by big problems that haveunderlying structures. Doing mathematics means working to uncoverand describe the structure.

7. A variety of solution strategies (including trial and error, physicalmodeling, data collection, pattern seeking, and consideration of aparticular example), and arguments (including algebra, logic, exhaus-tion, and revealing diagram) are valued. An argument or solution isdifferent if it uses a different strategy, reveals the structure in a differentway, or reveals a different aspect of the structure.


8. Complete (if informal) deductive argument is required for all mathe-matical claims, and all claims are subject to scrutiny.

9. The inquiry does not end until the given problem is solved andunderstood. To understand a problem is to see the underlying structure.

10. Elegant solutions are those that most directly reveal the underlyingstructure thus providing a powerful explanation for an observed patternor solution.

11. Students are expected to reflect on the entire problem-solving process,using precise and careful language, often in writing.

Students may have experienced some of these norms, such as the socialnorm of an informal but respectful classroom, in other courses. Many ofthe norms, however, were foreign to the students, and negotiating them inthe classroom culture initially caused them discomfort. Establishing andmaintaining the norms required a conscious effort and consistent rein-forcement on the part of the instructor. In fact, though not every classactivity provided an opportunity to reinforce every identified norm, thevery structure of the course continually supported these norms.

We illustrate the classroom culture with two episodes from a sectionof a Number Systems class. Although the classes were not observedon a regular basis, this typical session was videotaped about halfwaythrough the semester. In this particular period, the “big problem” wasto find the number less than 1000 that has the most factors. Underlyingmathematics included prime factorization, the fundamental theorem ofarithmetic, theorems about numbers of factors, and rules of exponents. Theepisode highlights the classroom culture and the ways the instructor andstudents reinforced the norms.

Episode 1: [All the students have formed a large semi-circle opening toward a black board.The instructor is in front of the board with chalk in hand. Amy, Sarah, and Cari are workingtogether in an attempt to argue that (xn)m = xnm. They have asked the instructor to write(xn)m = (x x x x . . . x) (x x x x . . . x) . . . (x x x x . . . x). Underneath each group of ‘x’sthey have had her write “n of these” and under the entire right hand expression they haverequested that she write “m of these ‘n’s.” John and others expected to see n ‘x’s followedby m ‘x’s.]

Instructor: “Is this clear?”John: “Nope.”Instructor: “What’s not clear John?”John: “What’s not clear to me is that you’re taking the n . . . The first part, like see thatlittle part right there, that’s clear. The second part is not clear because it’s not n. That’sm.”Amy [and others]: “That’s not m.”John: “It should be.”Tara and Dawn [agreeing with John]: “Yes. Yes”Sarah: “It doesn’t matter what it is.”


Cari: “It’s representing the same thing.”John: “No. If they’re different numbers, then they are going to be different . . . youcan’t call that just anything. I don’t think you can call it n.”Sarah: “Another way of writing ‘to the n power’ is whatever number of ‘x’s.”Amy: “Would you rather see it like using numbers?”John: “Yeah, I would. You can’t call a 3 a 2.”Amy, Sarah, Cari: “No. No. No.”Sarah: “Those ‘x’s from the brackets, its representing n ‘x’s.Kaila: “But aren’t you saying n and m are the same thing here?”[Several students are talking at once. The instructor interrupts the confusion to call onTina who has her hand up.]Tina [siding with John]: “I think that we had a miscommunication in our group becauseI don’t agree with that [indicating the board]. What I was trying to say is that if youmultiply those two exponents, the second group would be ‘m of these.’ ”[Others agree][Cari wants to propose a different notation. She asks that the instructor write (xn)m =(xn)(xn) . . . (xn) with “m of these” under the right hand expression.]Sarah: “That’s the same thing. Those ‘x to the n’ are repeating x, n times.”John: “Write (23)2 equals (2 × 2 × 2) × (2 × 2). So what they’re saying is 2 × 2 is 2. . .”Several: “No!”Amy: “That’s not it. What it should be is (2 × 2 × 2) × (2 × 2 × 2). It is 2 to the third,twice.”Instructor: “Which one is right?”Cari: “The top one [referring to Amy’s suggestion].”John: “Oops. I guess, yeah. Yeah, that is right!” [Laughs][General Laughter]John: “Say it’s not squared. If it’s 2 cubed to the third, would there be three groups ofthem then?”Several: “Yes!”

This episode illustrates the following social norms: First, the class wasfocused as a community on understanding an argument; second, it was anenvironment where students were not afraid to contribute their ideas; andthird, ideas came from the students and not the instructor. In fact, in thisepisode the instructor contributed no mathematical ideas. Fourth, studentsresponded directly to each other; note how little the instructor spoke duringthe exchange. Furthermore, we see that students were focused on providinga deductive argument and that mathematical claims were subject to criti-cism and scrutiny. Finally, we see that the inquiry continued until Johnunderstood.

Episode 2: The students have returned to the problem of finding the number less than 1000that has the most factors. 720 = 24325 is the best candidate so far, with 30 factors. Studentsare in the large group discussion semi-circle. The instructor is at the board.

Instructor: “Did anyone do better [than 30 factors]?” [5-second pause] “No? No onegot better. Can you argue why this will be the one with the most factors?” [15-secondpause] “Big silence. No?”


Cari: “I think it’s all about exponents.”Instructor: “Ok, what do you mean by that?”Cari: “When you use that formula, you use the number of exponents. When you havehigh exponents, you are going to get a bigger number of factors.”Sarah: “Because you want to have more exponents that you’re multiplying.”[Instructor clarifies these comments and suggests using a fourth prime in the product.Several students say that this will not work.]Cari: “No, it will be over 1000.”Sarah: “Well, the next prime will have to be 7.”Instructor [to another student who said “no”]: “Why do you say no, Dayna?”Dayna: “Because I tried it, like I took the same exponents and made . . . [5 secondpause] . . . Can I think for a minute?”Instructor: “Yeah, you can. Cari, did you want to say something?”Cari: “If you put a seven, I don’t think it’s going to work because you’ll have to belowering the exponents on the other ones.”Instructor [noting that time is running out for this class period]: “How many peoplethink we can do better than 30?” [several students nod or raise their hands] “How manyare pretty sure 30 is the best we can do? I see several ‘pretty sure’s.’ I’m going to leavethis (until tomorrow) . . .” Several: “NO!” [we hear groaning and the sound of a handslamming on a desk]

In this episode, we see that classroom discourse was focused on thestructure that underlies this “big problem”. It is a sociomathematical normthat doing mathematics means searching for the structure and making argu-ments based on it. Note that the sociomathematical norm that mathematicalclaims are subject to scrutiny extended to the instructor; her suggestion (touse a fourth prime) was not automatically accepted as a correct way toproceed. Note also that there was an expectation among the students thatthe inquiry should not end until the problem is understood, and severalpeople were disappointed to end the class period in confusion.

METHODOLOGY

On the first day of class, the 93 participants completed a ten-item, ten-minute questionnaire (see Table I for the list of items) with Likert Scaleresponse options. The questionnaire was adapted from work by Szydlik(2000) and Frid & Olson (1993) on describing students’ sources of convic-tion in mathematics. Its design was based on two fundamental constructs:1) Mathematics is logical and consistent, and therefore 2) Mathematicsis something that can be figured out. We assume that these beliefs arecentral to the fostering of autonomous behaviors. A diagram showing our(partial) view of the relationships among the questionnaire constructs isshown in Figure 1. Note that while all of the constructs are expressednegatively in the diagram, some are framed positively on the questionnaire.


Figure 1.

Arrows point to beliefs that are logical consequences of another belief.The instrument was used to provide a rough assessment of the nature ofstudents’ mathematical beliefs at both the start and end of the NumberSystems course, to provide a protocol for interviews, and to identify thosestudents with beliefs consistent, and those with beliefs inconsistent, withautonomous behaviors.

Each student was awarded a score on each item of the question-naire. Positive scores indicated beliefs about mathematics supportingautonomy and negative scores indicated beliefs that suggest dependenceon an external authority. For example, students were given the optionto “Strongly Agree”, “Agree”, be “Not Sure”, “Disagree”, or “StronglyDisagree” with the item: “In mathematics I need to memorize how todo most things”. Responses were coded –2, –1, 0, 1, and 2 respectively.Using the same response options, Item 2: “In mathematics everythinggoes together in a logical consistent way” was scored 2, 1, 0, –1, and –2respectively.

Each student was assigned a cumulative score based on the sum of thescores on the ten individual items. In the first two weeks of the semester,


eight students chosen at random from each of the upper quartile, middlehalf, and lower quartile of the cumulative scores were invited to participatein structured interviews. (The middle half of the students was containedin a sufficiently small range that, rather than interview eight from eachquartile, we chose to interview a total of eight from that half.) In thatinterview, students were given the opportunity to clarify and expand oneach questionnaire response. For example:

Interviewer: Item [6] says, [“I am interested in knowing how mathematical formulasare derived or where they come from.”] And you said that you [disagree]. Why did youanswer like that?Student: [response]Typical Probes: What do you mean when you say [whatever they said]? How did youinterpret that question? Can you tell me more about what you were thinking about[whatever is confusing]? Anything else?

The typical interview lasted approximately fifteen minutes. All inter-views were audio-taped.

The remainder of the beliefs data was collected in the last two weeksof the semester. All the students again completed the questionnaire andthe same 24 students were invited to complete a final interview. Twenty-two students accepted the invitation and two declined (one of them hadwithdrawn from the class). In the final interview, students were again askedto clarify and expand on their questionnaire responses. At the end of thefinal interview, students were asked two additional questions: 1) Is thereanything about this class that has changed your view about mathematics inany way; and 2) What is it about the way the class was run or structuredthat allowed you to see [whatever is was they said had changed].

All interviews were transcribed and each transcript was analyzed bytwo researchers and was coded based on several a priori themes. Theseincluded the role of an authority in mathematics (Questionnaire Items7, 8, and 9), the role of memorization in mathematics (QuestionnaireItems 2 and 9), distinctions and expectations about the nature of mathe-matical arguments and truth (Questionnaire Items 1, 3, 4, 5, 6 and 10),student expectations of classroom culture, sociomathematical norms, andthe process of changing beliefs. Based on the initial interview, we classifieda student’s beliefs as either supportive of autonomous behavior, mixed, orsupportive of non-autonomous behavior. The final interviews (this timeincluding the student responses to the two additional open-ended ques-tions) were analyzed using the same methodology. Students were againsorted into the three categories. In the two cases where there was disagree-ment between the researchers as to which category a student belongedbased on the final interview, we placed the student in the lesser autonomouscategory.


STUDENTS’ MATHEMATICAL BELIEFS AT THECOMMENCEMENT OF THE COURSE

Initial Questionnaire Responses

In this section we will give a brief report of the questionnaire data collectedfrom the large sample (N = 93) and discuss some of apparent contradic-tions among student responses in light of student explanations of theirquestionnaire responses in the initial interview. In the following discus-sion, survey data is reported for the large sample, whereas student explana-tions are always based on statements made by the 24 interview participants.Students’ cumulative scores on the initial questionnaire ranged from –14to 10 out of a possible range of –20 to 20 points. The median score waszero, and half of the students scored between –3 and 3.

The questionnaire data revealed that students generally agreed thatmathematics is logical and consistent and also agreed that, in mathematics,they need to memorize how to do most things. The initial interview datasuggest at least two reasons for the apparent inconsistency. First, severalstudents interpreted logical to mean “precise” or “rigid”. For example,Stephanie explained that mathematics is logical and consistent because,“In math there is a certain way you have to do everything”, and Cyndisaid “. . . there are different formulas used for every problem but there isonly one precise, exact answer”. Second, for some students the responseto the first item was hypothetical; interview participants said mathematicsis logical and consistent although they do not personally understand it.For example, Jenni explained, “I’m sure it does [go together in a logicaland consistent way], if you know what you are doing . . .”. The responseto the second item was practical. Students said that they must memorizeformulas, procedures, or template problems in order to work new prob-lems. As Tanya explained, “I have to have a couple of example problemsin order for me to even attempt to try the next, or another example, withoutgiving up”.

Students were divided on whether mathematics is something theycan usually figure out for themselves; 44% expressed agreement, 38%expressed disagreement, and 18% said they were not sure. The interviewparticipants indicated that this division is not as strong as it appears.Because these students may have interpreted this item in the context oftheir prior school mathematics culture, “figure things out” likely meantthey could do a problem on their own after they have been shown howto do a similar one or when they are given a formula. As Ann explained,“They show you how to do the problem and give a couple examples withthe numbers in them, you know, actually do them. That helps. So if I


have examples, I can figure things out”. Terrence agreed, “. . . as longas it’s pretty black and white on the chalkboard or in the textbook, it’sjust a matter of following what’s been told to you”. If the interviewedstudents are representative of the larger sample, then the vast majority ofstudents actually disagreed that they could “figure out” mathematics at thecommencement of the course.

Prior school norms also appeared to influence student responses toItem 7 (“If I’m given a problem that is quite a bit different from theexamples in the book, I can usually figure it out myself”) and Item 8 (“Ihave to rely on the teacher or textbook to tell me how to do the problems”)in similar ways. Several of the students who were interviewed indicatedthat they could not imagine being asked to do a problem significantlydifferent from those in the text or having a teacher who did not first showthem how to do similar problems. For example, although Shannon agreedwith Item 7, she explained her answer this way: “Usually the books willhave like a step to step process. They show you how to do it, and so if Ido the problem exactly how they do it in the book, then I can usually getthe answer right”. A summary of student responses to the initial question-naire can be found in Table I. However, in light of the above discussion,the reader is cautioned that this table provides only a rough measure ofstudents’ beliefs.

Classification of Interview Participants Based on their MathematicalBeliefs

Our reservations about the validity of the questionnaire convinced usto focus primarily on the interviews rather than questionnaire scores asa measure of students’ beliefs. While the questionnaire score was anaccurate measure of beliefs for some students, it was not for others, and,as described above, some participants interpreted items in ways that wedid not intend or they simply were able to express themselves more fullyduring the interview. Based on the initial interview, we classified students’beliefs as either: non-autonomous (11 students), mixed (10 students), orautonomous (3 students). To give the reader a sense of student beliefs atthe onset of the course, we will describe a proto-typical example of eachtype of interviewed student.

Case 1, Mandy. (cumulative score of –8 on the initial questionnaire)Mandy exhibited primarily non-autonomous beliefs. Overall, she appearedto interpret the questionnaire items as intended and was consistent inher responses. She explained that mathematics must make sense to somepeople, but it does not to her personally: “Maybe to some people whounderstand it more clearly, everything would make sense and there would


TABLE I

Initial survey responses: (N = 93)

Initial Survey Responses Mean σ

SA A NS D SD

1. In mathematics everything 8 54 10 20 1 0.52 0.96

goes together in a logical 8.6% 58.1% 10.8% 21.5% 1.1%

and consistent way.

2. In mathematics I need 14 57 6 16 0 –0.74 0.92

to memorize how to 15.1% 61.3% 6.5% 17.2% 0.0%

do most things.

3. In mathematics I can 1 40 17 32 3 0.04 0.98

usually figure things 1.1% 43.0% 18.3% 34.4% 3.2%

out for myself.

4. Mathematics has never 7 25 5 46 10 0.29 1.19

made too much sense 7.5% 26.9% 5.4% 49.5% 10.8%

to me even though I

can often get the

right answer.

5. Drawing pictures or 16 53 14 9 1 0.80 0.88

imagining real physical 17.2% 57.0% 15.1% 9.7% 1.1%

situations is a main

thing that helps me

do mathematics.

6. I am interested in knowing 11 33 26 20 3 0.31 1.04

how mathematical formulas 11.8% 35.5% 28.0% 21.5% 3.2%

are derived or where they

come from.

7. If I’m given a problem 0 28 32 28 5 –0.11 0.90

that is quite a bit different 0.0% 30.1% 34.4% 30.1% 5.4%

from the examples in the

book, I can usually figure

it out myself.

8. I have to rely on the 5 46 18 22 2 –0.32 0.97

teacher or the textbook 5.4% 49.5% 19.4% 23.7% 2.2%

to tell me how to do

the problems.


TABLE I

Continued

Initial Survey Responses Mean σ

SA A NS D SD

9. I learn mathematics best 31 47 7 8 0 –1.09 0.87

when someone shows me 33.3% 50.5% 7.5% 8.6% 0.0%

exactly how to do the

problem and I can

practice the technique.

10. In mathematics if I know 2 35 37 17 2 0.19 0.84

a few concepts I can 2.2% 37.6% 39.8% 18.3% 2.2%

figure out the rest.

always be a right answer, but I sometimes don’t understand why certainanswers are considered right”. Because she does not make sense of mathe-matics, she feels forced to memorize how to solve problems. She admitted:“if I don’t understand it, the only way to get it right is to memorize it . . .”.She says she relies on examples to use as templates for working problems.

As with the vast majority of the surveyed students, Mandy agrees thatshe learns mathematics best when someone shows her exactly how to dothe problems and she can practice the technique. “If somebody shows mehow to do it, then I know I’m doing it right and I won’t question why I’mdoing it”. For Mandy, mathematics is a collection of rules and proceduresthat come from an outside authority. It is this authority that decides whethereach procedure is valid. She cannot personally decide and indeed has littleinterest in doing so. When asked if she was interested in how the formulaswere derived, she responded that, while perhaps people who were good atmath might want to know this, she personally does not have a “strong urgeto know”. Ten other students who were interviewed (eight women and twomen) had interviews similar to that of Mandy.

Case 2, Tara. (cumulative score of 4 on the initial questionnaire) Taraexhibited a mix of beliefs. She agrees that mathematics is logical andconsistent and she claims that she has personally made sense of mathe-matics on at least some occasions. She explains, “. . . when you figureout a problem . . . there’s a reason why – why it is going together”. Inparticular, she recognizes that knowing where formulas come from givesinsight in mathematics, and therefore it is important to her to see the under-lying logic and derivations. However, she does not think she could create


derivations on her own and expects to be shown by a teacher or textbook.In other ways she is much like Mandy, agreeing that the “easiest” way tolearn mathematics is to be shown a technique and then practice. She lacksconfidence in her ability to figure things out on her own; she is “not sure”whether she could do a problem that is quite a bit different from examplesin the book, but thinks she could do a problem if shown how to solve asimilar one first. She explains, “. . . if it’s a problem you’ve already had inthe past you know how to do it and you don’t need a textbook or a teacherto say how to do it”. Nine other of the students who were interviewed (fivewomen and four men) provided responses to questionnaire items similarto those of Tara.

Case 3, Donald. (cumulative score of 11 on the initial questionnaire)Donald exhibited primarily autonomous beliefs. He strongly agreed thatmathematics is logical and consistent and he disagreed that he needs tomemorize things: “There is way too much to memorize, you’ve just got toknow little bits and pieces and how to string them together”. Althoughhe said he is not interested in how mathematical formulas are derived,his theme of making connections is prevalent throughout his interview.Because mathematics reflects reality, drawing pictures helps him work outproblems. “Most of the time it is usually about something that exists in thephysical world, drawing a picture helps you visualize it. That just makesit a lot easier to actually figure it out”. Donald is confident in his ability todo mathematics and likes to figure out problems on his own. “If somebodyshows me how to do it and I just practice the technique, I won’t rememberhow to do it the next time. But if I figure it out myself it will be mine, and Iwill know it for sure”. Two other of the students interviewed (both women)exhibited beliefs similar to that of Donald.

CHANGES IN STUDENT BELIEFS

As mentioned earlier, because student questionnaire responses were some-times inconsistent with their verbal explanations, we elected to focus ourdiscussion of changes in beliefs primarily on the qualitative data. We donote, however, that the quantitative data is consistent with qualitative data,showing a statistically significant shift (3 points) in the cumulative averageon the questionnaire in the direction of more autonomous beliefs (seeFigure 2) for the large sample (N = 84). A summary of the quantitativedata for the final questionnaire can be found in Table II.


Figure 2. Distribution of cumulative student initial and final survey scores.

TABLE II

Final survey responses: (N = 84)

Final Survey Responses

SA A NS D SD Mean σ

1. In mathematics everything 14 53 4 12 1 0.80 0.93

goes together in a logical 16.7% 63.1% 4.8% 14.3% 1.2%

and consistent way.

2. In mathematics I need 6 34 6 35 3 –0.06 1.12

to memorize how to 7.1% 40.5% 7.1% 41.7% 3.6%

do most things.

3. In mathematics I can 4 42 17 19 2 0.32 0.96

usually figure things 4.8% 50.0% 20.2% 22.6% 2.4%

out for myself.

4. Mathematics has never 3 17 3 49 12 0.60 1.08

made too much sense 3.6% 20.2% 3.6% 58.3% 14.3%

to me even though I

can often get the

right answer.


TABLE II

Continued

Final Survey Responses

SA A NS D SD Mean σ

5. Drawing pictures or 28 49 5 2 0 1.23 0.66

imagining real physical 33.3% 58.3% 6.0% 2.4% 0.0%

situations is a main

thing that helps me

do mathematics.

6. I am interested in knowing 4 27.5 18.5 33 1 0.01 0.98

how mathematical formulas 4.8% 32.7% 22.0% 39.3% 1.2%

are derived or where they

come from.

7. If I’m given a problem 3 39 21 18 3 0.25 0.95

that is quite a bit different 3.6% 46.4% 25.0% 21.4% 3.6%

from the examples in the

book, I can usually figure

it out myself.

8. I have to rely on the 2 27 14 39.5 1.5 0.14 0.97

teacher or the textbook 2.4% 32.1% 16.7% 47.0% 1.8%

to tell me how to do

the problems.

9. I learn mathematics best 21 38 11 14 0 –0.79 1.01

when someone shows me 25.0% 45.2% 13.1% 16.7% 0.0%

exactly how to do the

problem and I can

practice the technique.

10. In mathematics if I know 5 53 19 6 1 0.65 0.75

a few concepts I can 6.0% 63.1% 22.6% 7.1% 1.2%

figure out the rest.

Changes Expressed by Interview Participants

Based on their final interview explanations, we classified six of the 22students as holding non-autonomous beliefs, ten students as having mixedbeliefs, and six students as showing primarily autonomous beliefs. (Note:Two students we had classified as having primarily non-autonomousbeliefs at the initial interview did not participate in the final survey orinterview.) A total of five students changed categories from the initial tothe final interview, all in the direction of more autonomous beliefs.


As a group, the students who were interviewed described three specificways in which their beliefs changed. First and foremost, students said thatthe course had allowed them to see mathematics as a more sensible discip-line (As Gwen said, “It’s a lot more logical than I thought it was beforeeven”); now they know there is an underlying reason why formulas andprocedures work the way they do. Tanya explained, “I feel like I know alot . . . I feel like I know, since I know why [things] happen, and I dig muchdeeper on how they’re discovered . . . and I have a better understanding ofmath as whole. It made me think so hard some days . . . I never thought Icould think that hard”. Sam said, “Before mathematics was just numbersand . . . all I cared about was getting the right answer. I never really thoughtabout it and I never really did that well in math. I guess [now I] really wantto know why, why things work and why they came out the way they are. . . now I’m doing a lot better because I know why it’s happening . . .”.Nancy agreed, “When I was in high school, I would just get an answer andthen I just figured that was the answer. But this math class taught me why. . . like, what is the reason behind the answer . . . and why things workout like they do . . .” Sarah explained, “Usually through high school andelementary you’re just [using] numbers and spitting numbers out . . . butthis, you actually see the theories and stuff . . . I didn’t realize any of thesetheories. I appreciate math a lot more because it’s not so dry anymore,there’s actually some meaning behind it”. Thirteen of the 22 interviewedstudents volunteered a comment like those above when asked how theirview of mathematics had changed.

Students indicated that they are now aware that mathematics is a humancreation and they can be a part of making mathematics themselves. Annexplained it this way: “I think I’m not as intimidated by [mathematics].By the way you gave us the problems and had us figure it out and told usthat ‘Okay, even if you think the way you’re doing it is stupid, if it works,it’s just as good as any other solution’. I realize I can figure things outfor myself. I don’t have to be told by the teacher”. Ann’s interpretationof the instructor’s encouragement needs to be placed in the context of aclassroom where it was the norm for students to solve problems in waysthat made sense to them rather than in ways they perceived were expectedby the teacher. While a mathematician might argue that not all solutionstrategies are equivalent in terms of elegance, efficiency, and their abilityto be generalized, it is important to appreciate Ann’s realization that astrategy that makes sense to her is better that an elegant solution that shedoes not understand.

Terrence said that he has seen for the first time that there are manydifferent approaches to problems. He explained:


“. . . you [the instructor] would say, ‘Okay, here’s a question, now everybody come up withan answer’. And when I sat down to think it out, I would think, ‘this is the most logicalway to do it’ – and all of a sudden, there’s about at least a dozen different ways of doing it.And as they’re all going up [on the blackboard], I’m thinking, ‘Wow, this is a way I wouldhave never thought of – but it’s so easy’. It’s just amazing that there’s all these differentpeople in the class thinking a totally different way – probably thinking that this is the waythat it is done . . . [The class] has changed my whole attitude as far as school, as far as theway I will teach”.

Six of the interviewed students made similar comments to those made byAnn or Terrence.

Finally, several students expressed more confidence in their own abili-ties to solve problems and more interest in mathematics in general. Samsaid, “I don’t think of [mathematics] so much as the enemy anymore”.Barb explained, “I’m more interested in [mathematics] because I know alot of why things work the way they do”. Six students made commentssimilar to those made by Sam and Barb.

Interviewed Students’ Attributions for Changes in Beliefs

The structure of the Number Systems course was a conscious attemptto establish social and sociomathematical norms in the classroom thatencourage more autonomous student beliefs about mathematics. Indescribing the culture of the classroom, we identified eleven such norms(see p. 259). In the last question of each final interview, the student wasasked if there was anything about the way the class was structured thathelped change his or her view of mathematics (if indeed that view hadchanged). A typical probe by the interviewer on this question took thefollowing form:“What is it about the way the class was run or structured that allowed you to see mathe-matics in this different way? Was it the problems used? Was it the group work? Was it thediscussion? Is there some specific aspect that you could point to?”

We were able to match all of the students’ attributions with “goal norms”of the class. Table III provides a summary of the student attributions.

Not surprisingly, the norm most frequently cited by students was thesocial norm that mathematics was done as a community. They said thatworking together helped give them confidence and allowed them to seethe ideas of peers. Ann explained, “You didn’t have to figure it out all byyourself. Everybody would contribute what they thought”. Jane said, “Noteveryone knows as much as everyone else, so you get to show each otherways of thinking” and Nancy explained that she liked “having differentpeople in class give different views of it . . . because then it’s not likethere’s only one reason this works, there’s more than one reason and there’sdifferent ways to find it”.


TABLE III

Student attributions for changes in beliefs about mathematics

Social/Sociomathematical Norm # of studentattributions

1. Respectful classroom environment. 5

2. Students work as a community. 14

3. Students interact with each other in large group discussion. 3

4. Solutions come from students. 7

5. Instructor acts as a guide. 2

6. Content generated by “big” problems. 11

7. Variety of strategies and arguments valued. 7

8. Complete deductive argument required. 1

9. All problems must be solved and understood. 1

10. Elegant solutions reveal structure. 0

11. Students reflect on the problem-solving process. 4

Eleven students stated that the sociomathematical norm of having thecourse content generated by big problems with an underlying structure(Goal Norm #6) helped them to see the sense in mathematics. Studentssaid that the problems were deep and challenging enough to allow for realthinking. Ben liked this aspect: “I guess it was because you didn’t hand usa book and say, ‘Go and do problems 1 through 15 for Wednesday’. Yougave us a big problem, like a story problem, and it was more challengingand interesting because it kind of had other stories behind it”.

Seven students cited the sociomathematical norm that a variety ofstrategies and arguments were valued as helping them to see that peoplethink about mathematics in many ways (Goal Norm #7). As Nancyexplained, “Having different people in the class give different views ofit, different examples from yours so that it was easier to understand. Causethen it’s not just like there’s only one reason this works, there’s more thanone reason and there’s different ways to find it”.

Seven students said the social norm that they (and not the instructor)were the ones who decided whether a strategy worked or a solution wasvalid (Goal Norm #4) changed their beliefs about mathematics. Gwennoted, “You never ever, ever gave us the answer . . . and so that I reallyliked a lot” and Bob explained, “there’s really no book to rely on, or youknow, you don’t go around and give the answers right away. You let us pickat [the problems] for awhile, and that’s helped to give me confidence”.

Students also appreciated the informal yet respectful nature of theclassroom (Goal Norm #1). Five students identified this norm as an


important aspect of the course. As Bob stated in his interview, “I thoughtthe discussions were very, very open. You know, the floor was open andyou could say anything you wanted to even if it was wrong . . . You didn’tmake anybody feel like their answer was dumb even if it was wrong oranything like that”. Ann expressed similar feelings: “It was good the waythat you really tried to convince us that any way of solving the problemcame up with was ok even if it was kind of dumb . . . just the way youmade sure everyone knew nothing was too stupid to say and raise yourhand”.

Several students remarked on the effect of reflecting on the problem-solving process (Goal Norm #11). Steve noted, “I liked the way that wehad one sheet, usually the one problem a day. We worked in groups andthen we had the discussion and that helped. With some of the problems, weactually had to say, ‘Ok, what are the steps for doing this?’ ” Jane was evenmore explicit in her attribution: “. . . and it was the papers that we had towrite, those assignments where you had to explain, you know, what’s thequestion about, why did you get the answer, why is that the only answer.It was really good that way. At first it seemed like, oh my gosh, I can’tdo this. But it was really helpful. I liked that”. Four students attributedchanges to this particular sociomathematical norm.

Not all of the classroom goal norms were widely identified by thestudents (in particular, Goal Norms #8 and #9 were only mentioned byone student each, while Goal Norm #10 was not mentioned at all). Thisis not surprising, however, since the student attributions for changes intheir mathematical views were free responses. The interviewer did notprovide a complete list of the goal norms; rather, the students were invitedto contribute their own thoughts. It is reasonable to expect that studentswould tend to attribute clearly observable norms more often than moresubtle norms such as emphasis on complete deductive reasoning.

While the students responded overwhelmingly that the course had beenan experience that made them more powerful in doing mathematics, thiswas not the case for one student. Sarah expressed a loss of confidence inher ability to solve problems, and this seemed to be a direct consequenceof her experience of the course: “. . . being in class, there’s a lot of stuffthat I thought I would be quicker on. But a lot of these problems just didn’tcome to me right away – there were very few of them actually. I had toreally work at them harder. I had to listen to you, or my classmates gotthem before I did. So . . . this class wasn’t really logical and I had a reallyhard time figuring them out for myself unless I spent a lot of time”. Sarahbegan the course with beliefs supportive of non-autonomous behavior andended it with even less confidence.


CONCLUDING REMARKS

The study lends additional support for the adoption of the NationalCouncil of Teachers of Mathematics (1989, 2000) process recommenda-tions regarding problem solving, reasoning and proof, and communication.Our analysis suggests that a classroom focusing on problem solving usinga variety of strategies, reflection on the process of problem solving,and engagement in the process of exploration, conjecture, and argumentcan help students develop mathematical beliefs that are consistent withautonomous behavior. Specifically, a classroom culture in which studentssolve challenging problems without external assistance and where mathe-matical conviction is determined based on logic and consistency can alterstudent beliefs about mathematics, at least in the context of the new culture.Many students in the Number Systems course expressed mathematicalbeliefs that were more consistent with autonomous behavior at the endof the course than they did at its commencement, and many were able tocommunicate changes in their beliefs in an insightful manner. Studentssaid they saw – many for the first time – the meaning in mathematicsand gained confidence that they can construct that meaning themselves.This result is particularly important for prospective teachers who will beexpected to help their future students make, refine, and explore conjecturesbased on evidence and make arguments to support or refute those claims(NCTM, 2000).

Students in our study attributed changes in their beliefs to three primaryaspects of this classroom culture. First, the course was built around care-fully constructed and very challenging problems – each with an underlyingmathematical structure or lesson. Students began to have faith that eachproblem would yield this structure or story (why it worked) and weremotivated to uncover it. Second, the course instructor provided almost noassistance in the problem solving aspect of the course and no answers wereprovided for problems. The only way for the class to understand a problemwas to figure it out. The only way to know they were correct was to find aconvincing argument. Third, the community work on the problems madethe process less frustrating for students, allowed them to see the ways inwhich their peers did mathematics, and showed them that problems couldbe solved in more than one way.

Did the Number Systems experience fulfill Kagan’s (1992) condi-tions for changing beliefs (making personal beliefs explicit, challengingthe adequacy of those beliefs, and providing extended opportunities toexamine, elaborate, and integrate new information)? We assert that it did.In part, it was the methodology of our study that helped students makebeliefs about mathematics explicit. All participants, but particularly those


who participated in the interviews, had the opportunity to reflect on theirmathematical beliefs through their work on the questionnaire items, as wellas through course assignments. The classroom culture showed studentswhat authentic mathematical experiences look like (often revealing theirmathematical beliefs to be inadequate), and provided opportunities forreflection on the process of doing mathematics through group work andwritten problem reflections. In the future, the course may be improved bymaking these processes even more transparent to the students.

There must be two qualifications to this work. First, one of theresearchers was the instructor for the course. It is certainly possible thatstudents (either unconsciously or consciously) made statements in the finalinterviews in order to please their teacher. Even if this is true to someextent, however, we assert that the students could not have made (and didnot make) such insightful statements regarding the nature of mathema-tical behavior at the commencement of the course; they simply did notknow enough about what mathematics was supposed to be about. This issomething that they learned from the class, and that alone is an achieve-ment. Second, students’ initial interpretations of terms and phrases suchas “understand”, “figure it out”, and “problem solve” were made in thecontext of prior mathematical experiences. This means that in the initialsurvey “understand” possibly meant “recall the procedure to use”, “figureit out” possibly meant “get the right answer”, and “problem solving” mayhave referred to doing more problems just like a given example. Statementsprovided by the interviewed students suggested that the meaning of theseterms changed as their beliefs about these ideas changed. This makes itmore difficult to compare initial survey and interview responses with finalsurvey and interview responses.

Our data supports the assertions of others (Schram et al., 1988; Wilcoxet al., 1991; Civil, 1993; Yackel & Cobb, 1996) who advocate focusingon classroom culture as a means of affecting beliefs and adds weight toviewing this avenue as a powerful agent of change. This is an area that weintend to investigate further. We are now interested, for example, in how“non-traditional” classroom norms are established. Specifically, what arethe interactions related to specific themes or content (Bauersfeld, 1994)that form the mathematical habits of the students? What is the teacher’sway of being in the classroom that might help create a new culture? Infuture research, we will attempt to document how the Number Systemscourse culture is established and nurtured, and we will attempt to measureautonomous behavior directly, both by interviewing participants and byobserving ways in which the students work problems.


ACKNOWLEDGEMENTS

This work was supported in part by a Faculty Development Grant from theUniversity of Wisconsin Oshkosh. We would like to thank the reviewersand editors of JMTE for their thoughtful advice on revisions of this paper.

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JENNIFER E. SZYDLIK

Mathematics DepartmentUniversity of Wisconsin Oshkosh800 Algoma Blvd.Oshkosh, WI 54901, USAE-mail: [email protected]

STEPHEN D. SZYDLIK

University of Wisconsin Oshkosh

STEVEN R. BENSON

Education Development Center

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Exploring Changes in Pre-service Elementary Teachers' Mathematical Beliefs