perceptions of reform-based teaching and learning in a college mathematics class

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AMY ROTH-McDUFFIE, J. RANDY McGINNIS and ANNA O. GRAEBER PERCEPTIONS OF REFORM-BASED TEACHING AND LEARNING IN A COLLEGE MATHEMATICS CLASS ? ABSTRACT. This study investigated, from the students’, professor’s, and researchers’ perspectives, the effects of a reform-based introductory undergraduate mathematics course, and the efforts of a mathematics professor to teach such a course. The class had been designed for teacher candidates of middle school mathematics and science (Grades 4 to 8) but was open to all qualified students. We addressed the following research question: What perceptions about learning and teaching mathematics emerged through the participants’ experiences in a reform-based mathematics course? Results of the analysis of the data suggested that the teacher candidates and the professor took an important first step toward enculturation into a reform-based vision of mathematics learning and teaching. Implications for mathematics faculty and teacher education faculty interested in promoting reform-based mathematics are presented. Mathematics educators in the United States have proposed fundamental changes in teaching and learning. Documents of the National Council of Teachers of Mathematics [NCTM] (1989, 1991), the Mathematical Sciences Education Board [MSEB] (1995), the Mathematical Association of America [MAA] (Tucker & Leitzel, 1995), and the National Research Council [NRC] (1991) propose frameworks for change in mathematics education at all levels, from elementary school through college. Publications directed at college mathematics professors emphasize the importance of models of reform-based teaching for undergraduate students, especially teacher candidates (MAA, 1988; MSEB, 1995; NRC, 1991; Tucker & Leitzel, 1995). Because teachers tend to teach as they have been taught (Brown & Borko, 1992), teachers, including college professors, should model a type of teaching consistent with the reform documents (MSEB, 1995). Although all teachers serve as role models for future teachers, college faculty teach teacher candidates as they train for their careers; thus, college faculty should be especially concerned about modeling good teaching. “Unless college and university mathematicians ? A previous draft of this article was presented at the annual meeting of the American Educational Research Association, New York, April, 1996. The research reported in this paper was supported by the National Foundation under grant no. DUE 9255745. The opinions expressed do not necessarily reflect the views of the Foundation. Journal of Mathematics Teacher Education 3: 225–250, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Page 1: Perceptions of Reform-Based Teaching and Learning in a College Mathematics Class

AMY ROTH-McDUFFIE, J. RANDY McGINNIS and ANNA O. GRAEBER

PERCEPTIONS OF REFORM-BASED TEACHING AND LEARNINGIN A COLLEGE MATHEMATICS CLASS?

ABSTRACT. This study investigated, from the students’, professor’s, and researchers’perspectives, the effects of a reform-based introductory undergraduate mathematicscourse, and the efforts of a mathematics professor to teach such a course. The class hadbeen designed for teacher candidates of middle school mathematics and science (Grades4 to 8) but was open to all qualified students. We addressed the following researchquestion: What perceptions about learning and teaching mathematics emerged through theparticipants’ experiences in a reform-based mathematics course? Results of the analysisof the data suggested that the teacher candidates and the professor took an importantfirst step toward enculturation into a reform-based vision of mathematics learning andteaching. Implications for mathematics faculty and teacher education faculty interested inpromoting reform-based mathematics are presented.

Mathematics educators in the United States have proposed fundamentalchanges in teaching and learning. Documents of the National Councilof Teachers of Mathematics [NCTM] (1989, 1991), the MathematicalSciences Education Board [MSEB] (1995), the Mathematical Associationof America [MAA] (Tucker & Leitzel, 1995), and the National ResearchCouncil [NRC] (1991) propose frameworks for change in mathematicseducation at all levels, from elementary school through college.

Publications directed at college mathematics professors emphasizethe importance of models of reform-based teaching for undergraduatestudents, especially teacher candidates (MAA, 1988; MSEB, 1995; NRC,1991; Tucker & Leitzel, 1995). Because teachers tend to teach as theyhave been taught (Brown & Borko, 1992), teachers, including collegeprofessors, should model a type of teaching consistent with the reformdocuments (MSEB, 1995). Although all teachers serve as role models forfuture teachers, college faculty teach teacher candidates as they train fortheir careers; thus, college faculty should be especially concerned aboutmodeling good teaching. “Unless college and university mathematicians

? A previous draft of this article was presented at the annual meeting of the AmericanEducational Research Association, New York, April, 1996.The research reported in this paper was supported by the National Foundation under grantno. DUE 9255745. The opinions expressed do not necessarily reflect the views of theFoundation.

Journal of Mathematics Teacher Education3: 225–250, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

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model through their own teaching effective strategies that engage studentsin their own learning, school teachers will continue to present mathematicsas a dry subject to be learned by imitation and memorization” (NRC, 1991,p. 29). Thus, as a step toward change in instructional practice, teachercandidates need opportunities to experience reform-based teaching whilethey are students, so that they may construct a new model of what it meansto teach and learn mathematics.

Unfortunately, as Tucker (1995) noted, “The support that the NCTMStandardshave received from most college and university mathematicianshas not been matched by any significant change in the curriculum orteaching for prospective mathematics teachers” (p. 26). Further, existingbeliefs and perceptions about mathematics teaching and learning arebarriers to the teacher candidates constructing a new model. For example,faculty involved with the calculus reform movement have reported “alot of resistance and negative reaction from students who don’t want usto shake up their comfortable relationship with ‘math,’ no matter howdistasteful that relationship may be” (Solow, 1995, p. 11). In a theoret-ical analysis of the research on prospective teachers’ conceptions, beliefs,and practices, Smith (1996) found that “Future teachers have extensiveparticipation in the practice of teaching by telling” (p. 391) and thus haveestablished certain expectations for mathematics instruction. Courses thatemphasize creative problem solving and conceptual understanding ratherthan mimicry and drill do not meet the expectations of college students(Solow, 1995). As a result, the students have difficulties in adjusting tocooperative learning, writing, and projects in their mathematics classes(Tucker & Leitzel, 1995).

Given these barriers, is it possible for teacher candidates to construct areform-based model of mathematics teaching and learning? Moreover, if itis possible, under what circumstances does this construction take place?

BACKGROUND AND FRAMEWORK

The Maryland Collaborative for Teacher Preparation

The Maryland Collaborative for Teacher Preparation (MCTP) isaddressing the need for design, implementation, and documentation of areform-based teacher education program. The MCTP is a National ScienceFoundation funded project with the mission to develop, implement, eval-uate, and learn from an interdisciplinary mathematics and science, upperelementary/middle school teacher preparation program consistent withcontemporary goals for reform in mathematics and science education.

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MCTP involves college faculty from mathematics, science, and educa-tion departments who collaborate in developing and implementing theprogram. In designing the courses and field experiences, the principalinvestigators developed a number of basic principles to guide the facultyparticipating in the MCTP program:

1. Teacher candidates should be actively involved in the learning of math-ematics and science through instruction that models practices that theywill be expected to employ in their teaching careers.

2. Courses and field experiences should reflect the integrated nature ofmathematics and science so that prospective teachers can develop anunderstanding of the connections between mathematics and science.

3. Courses and experiences of all teacher candidates should focus ondeveloping their ability to use modern technologies as standard toolsfor problem solving.

Research on Perceptions of Teaching and Learning

Schifter and Fosnot (1993) studied practicing teachers who participated inSummerMath, a workshop for teachers interested in implementing reformgoals in their elementary mathematics classes. One of the key premisesof the SummerMath program was that, “If teachers are expected to teachmathematics for understanding [as defined in the reform documents] theymust themselves become mathematics learners” (Schifter & Fosnot, 1993,p. 16). Similarly, theProfessional Teaching Standards(NCTM, 1991) callfor such experiences when they state, “If teachers are to change the waythey teach, they need to learn significant mathematics in situations wheregood teaching is modeled” (p. 191). In other words, teachers need tobe enculturated into viewing mathematics as a way of knowing and intoviewing learning as constructing knowledge through shared mathematicalmeanings and practice.

Gee (1990) presented two forms of enculturation when he made thedistinction between acquisition and learning: According to Gee, acquisi-tion “is a process of acquiring something subconsciously by exposure tomodels, a process of trial and error, and practice within social groups,without formal teaching” (p. 146). Learning, on the other hand, “isa process that involves conscious knowledge gained through teaching(though not necessarily from someone officially designated a teacher) orthrough certain life-experiences that trigger conscious reflection” (p. 146).

Gee (1990) further contended that, “Acquisition must (at least,partially) precede learning; apprenticeship must precede ‘teaching’ (in thenormal sense of the word ‘teaching’)” (p. 147). Here Gee linked acquis-

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ition to apprenticeship. The notion of apprenticeship is also discussedby Lave and Wenger (1991); however, they preferred the term “situ-ated learning” (p. 31). Lave and Wenger emphasized the importance ofsituated learning as “learning by doing” (p. 31). Furthermore, educa-tional researchers emphasized the value of metacognition in learning (e.g.,Flavell, 1981; Schoenfeld, 1992). Flavell (1981) defined metacognition as“knowledge or cognition that takes as its object or regulates any aspect ofany cognitive endeavor” (p. 37). In the context of this study, the object ofthe learning was the mathematics content; therefore, for the teacher candid-ates, an important metacognitive aspect of learning was the relation of themathematical content to the students’ future teaching of mathematics.

To better understand the process of enculturation for teacher candidatesin the context of mathematics teaching and learning, we need to examinethe perceptions held by teacher candidates and professors in reform-basedmathematics classrooms. Brown and Borko (1992) stated that existingresearch

provides limited evidence about the design and implementation of good mathematicsteacher education programs. . . . Careful documentation of the experiences of teachers insuch programs and the resulting changes in their knowledge, beliefs, dispositions, thinking,and actions will provide further insight into the process of becoming a mathematics teacher(pp. 235–236).

Although some research on college students’ learning of specific mathem-atical concepts is available (e.g., Tall, 1990), less research is available thatexamines teacher candidates’ and professors’ perceptions of learning andteaching mathematics in reform-based classrooms. We shall briefly discusswork in this area that is relevant to our research.

Civil (1993) conducted a study on teacher candidates enrolled in amathematics education course that emphasized mathematical and pedago-gical connections and that focused on the goals set forth in theStandards(NCTM, 1989, 1991). The participants were near the end of their under-graduate preparation and were preparing for student teaching. Civil foundthat the teacher candidates in her study had definite beliefs about teachingand learning mathematics based on their previous experience as students.She stated that in the beginning of the semester, all teacher candidatesbelieved that “their role as teachers was to tell the children what to do”(p. 84). Civil saw growth in the teacher candidates over the course of thesemester as they became reflective learners and began to change their ideasof what it meant to do mathematics. However, she found that the soon-to-be student teachers’ primary focus was “the course content in termsof how they would teach these ideas to students, rather than to work onunderstanding the ideas themselves” (p. 97). This focus lead Civil to ques-

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tion whether this one experience toward the end of the teacher candidates’preparation was enough to affect lasting change once they began teaching.

In a similar study, Wilson (1994) investigated the impact of a course thatintegrated mathematical content and pedagogy by examining one teachercandidate’s (“Molly”) understanding of function throughout a unit on func-tions. He found that prior to the start of the unit, Molly viewed mathematics“as a collection of ‘concrete’ procedures to be applied to isolated contextsto obtain correct answers to well-defined problems” (p. 361). By the endof the unit, Molly’s understanding of function had grown immensely.Moreover, Molly stated that her experiences had caused her to change herviews on teaching as well. However, Wilson argued that Molly’s views didnot reflect a radical shift. He contended that Molly only was interested inadopting surface features of reform-based teaching such as using computersimulations and other hands-on activities, and she held on to her belief thatunderstanding the meaning of mathematical procedures was not importantfor her students.

Frykolm (1995) examined teacher candidates during their studentteaching experience. The teacher candidates in Frykolm’s study had beenin a mathematics education program which emphasized theStandards(NCTM, 1989, 1991). He reported that although the student teachersseemed to affirm the goals and content of theStandards, they werenot successful in implementing theStandardsin the classroom. Frykolmsuggested that one reason for the lack of implementation was thatthe student teachers “had very few [reform-based] teaching models toemulate” (p. 19), and thus, they modeled their teaching practices afterthe traditional teaching practices of their cooperating teachers. Frykolm’sfindings imply that student teachers need to be placed with cooperatingteachers who are successfully implementing theStandards. In addition,teacher candidates need more (or better) experiences as learners in math-ematics classrooms beyond the education they receive in their methodscourses. In other words, given that the student teachers in Frykolm’s studylacked models for reform-based instruction, perhaps learning about theStandards in methods courses was too late in the process of preparingteacher candidates to bring about enculturation. Instead, there seems to bea need for professors of mathematics content courses to provide reform-based experiences early in the undergraduate program in addition to theexperiences provided in methods courses.

Objectives

In our study we examine the perceptions of five MCTP teacher candid-ates and their mathematics professor as participants in a reform-based

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college mathematics classroom. Specifically, we address the followingresearch question: What perceptions about learning and teaching math-ematics emerged through the participants’ experiences in a reform-basedmathematics course?

This study contributes to the existing research by: (1) examining teachercandidates in the first semester of their undergraduate experience in thecontext of a mathematics content course; and (2) expanding the scantresearch available, especially at the undergraduate level, that depicts theefforts of a mathematics faculty member who was essentially a pioneerin reform-based mathematics teaching on his campus. We maintain thepremise held by Schifter and Fosnot (1993) in that we believe that teachercandidates must have opportunities to experience mathematics as learnersin reform-based classrooms. Moreover, we gain from the research of Civil(1993), Wilson (1994), and Frykolm (1995) by heeding their warningsthat the later part of the undergraduate experience may be too late to firstintroduce the ideas of reform and that single courses may not accomplishthe depth of understanding the reform seeks.

We are aware of research on teachers’ beliefs that argues that professedbeliefs often differ from beliefs in practice (e.g., Richardson, 1996). There-fore, teacher candidates’ expressed beliefs and teaching and learningphilosophies as aligned with reform goals do not ensure future reform-based practice. However, part of the process of becoming a teacher isdeveloping a teaching and learning philosophy. Thus, we believe it isimperative to document and interpret early experiences of the teachercandidates in reform-based mathematics classes. Our study complementsstudies on teacher candidates’ later experiences in method classes, instudent teaching, and during their induction years. We hope to furtherunderstanding of how efforts are being undertaken in all facets of math-ematics teacher preparation to enact the vision of reform and to determinewhere future research and practice should focus.

Framework

In our study, we viewed learning on three relevant levels:

1. In observing students in the process of learning, we looked for evid-ence of the extent to which the students, as active learners, constructedknowledge as they encountered information, ideas, and activities in theclassroom.

2. In observing and interviewing the professor, we looked for evidenceof the extent to which he constructed an image and an understandingof reform-based practice.

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3. In analyzing the data and writing the study, we co-constructed withthe participants an understanding and interpretation of the participants’experiences.

To further describe our views of the social aspects of teaching and learning,we turn to the ideas of social constructivism. The fundamental premise ofsocial constructivism is that knowledge and understanding are constructedwhen individuals engage socially to solve problems or perform tasks(Driver, Asoko, Leach, Mortimer & Scott, 1994). Learning is the processby which individuals are introduced to a culture by more skilled members(Driver et al.). The ideas espoused by social constructivists further developthe notion of enculturation as part of the learning process.

As we interviewed participants and observed classes we looked forincidents in which the process of enculturation was or was not taking place.For example, did the professor perceive his role as one of an experiencedmember of the class with the responsibility of structuring tasks to supportlearning? Did the teacher candidates indicate that they were involved ina process of enculturation to the practices of reform-based teaching andlearning?

METHODOLOGY

Context of Study

Setting. The research setting was an undergraduate mathematics classroomat a large state university. The mathematics course, taught in Fall semesterof 1994, was an introductory mathematical modeling course that required amathematical background of high school mathematics through the secondyear of algebra. The mathematics course was open to all undergraduates.

The class studied was not bound to the established syllabus of theexisting course because it was field testing reform-based instructionalstrategies such as active learning through meaningful problem solving andcollaboration. The course had been developed and was taught by Dr. Taylor(pseudonym), a university mathematics professor. The three-credit coursemet three days a week; one session was 110 minutes long, the other twowere 50 minutes long. The two- to three-week long units included: (a)introduction to mathematical modeling, (b) geometric models, (c) linearmodels, (d) exponential and power models, and (e) models of chance (anintroduction to probability). Typically, at the beginning of a unit, smallgroups of students would collect data, or alternately, Dr. Taylor providedthe students with real data (e.g., the population of wolves in Alaska).Thestudents would then develop mathematical models to represent the data.

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By the end of the unit, the students would have generalized their initialmodels to other situations. In addition to explaining their ideas in writtensentences, the students used symbolic, numeric, and graphical represent-ations throughout the course. Also, the class utilized graphing calculatorson a regular basis. The professor assessed his students through individualand group assignments, journals, in-class exams, a portfolio, and a finalexam with a performance assessment component.

Participants. Among the 21 students participating in the study, 5 wereMCTP teacher candidates and 16 were non-MCTP students. Dr. Taylor,also a participant, had over 20 years of teaching experience, and hisinvolvement in mathematics education reform extended beyond his associ-ation with MCTP. Dr. Taylor’s class was selected because he was strivingto implement substantial changes in teaching and learning of mathematics.In fact, the course was Dr. Taylor’s first attempt to teach a class that fullyincorporated the goals of MCTP and the reform movement. The MCTPteacher candidates were first-year undergraduates and were between 17and 19 years old. Four teacher candidates were women, and one was a man.Although the focus was on Dr. Taylor and the MCTP teacher candidates,the other students in the class were also informants. These students had awide range of academic majors and concentrations including elementaryand secondary education, business, and journalism.

Because the MCTP teacher candidates were in their first semester, noneof them had previously taken an MCTP or an education course. However,they were all concurrently enrolled in an MCTP science course (eitherphysics or chemistry) and in a one-credit MCTP seminar taught by Dr.Taylor and by an MCTP science professor. The purpose of the seminarwas to enhance connections between the mathematics and science and todiscuss issues related to their future teaching of these subjects.

Data Collection

Because the study involved an in-depth examination of a phenomenon,we selected a case-study format (Goetz & LeCompte, 1984; LeCompte,Millroy & Preissle, 1992; Stake, 1995) as the most promising strategy tounderstand the perceptions of faculty and students. We were interested indescribing and interpreting the experiences of the participants in a reform-based mathematics class rather than testing for specific outcomes. Theprimary data collection for the study was bound by the experiences ina one-semester mathematics content class, and the study did not aim toreport on the entire undergraduate experience in the MCTP.

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Interviews. At the beginning and the end of the semester, we conductedindividual, semi-structured interviews with Dr. Taylor and with each ofthe five MCTP student participants (see Table I for the interview proto-cols). The student interviews lasted approximately 20–30 minutes, and Dr.Taylor’s interviews lasted approximately 45 minutes. The interview ques-tions were based on MCTP project goals. We were interested in hearingthe perceptions of Dr. Taylor and the MCTP students as to whether thegoals were emphasized in the class. Informal interviews with Dr. Taylorand the MCTP student participants were conducted prior to and followingeach class observation.

At the end of the semester, all of the students in Dr. Taylor’s class wereinterviewed in a group setting. Dr. Taylor had asked for the whole-classinterview because he wanted to be informed about the students’ evaluationof the class and his instructional strategies. This interview lasted approx-imately 50 minutes and consisted of a free-flowing discussion among theclass members. Although Dr. Taylor was not present at the group interview,the students were informed that he was interested in their perceptions ofthe class and that their comments would be summarized and reported tohim anonymously. Questions asked in the discussion included: “To whatextent did the course help you to see the usefulness of mathematics?”; “Towhat extent did the course focus on problem solving?”; “To what extentdid the course focus on practicing procedures?”; and “Do you believe youhave improved your ability to communicate ideas in mathematics?”

An initial draft report was given to a consultant to the MCTP researchgroup. The consultant suggested that more data be collected from the non-MCTP students in the class. Thus, we asked Dr. Taylor for copies ofthe University’s student course evaluations, and we then interviewed Dr.Taylor again for his thoughts on these evaluations. Also, although the classinterview and the evaluations provided some insights into the non-MCTPstudents’ perceptions of the class, we recognized that we also neededindividual interviews with some of these students. During the followingfall semester, we were able to locate and interview two of the non-MCTPstudents and obtained their retrospective views on the course.

Observations. In addition to interviews, one to two times each month ofthe semester I (first author) observed the class and took field notes toobtain data on Dr. Taylor’s and the MCTP students’ actions in the processof teaching and learning. As is reflected in the reporting of the findingsbelow, we used these observations to validate the perceptions revealed bythe participants in the interviews.

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TABLE I

Participant Interview Questions

Teacher Candidate Interview #1

1. What does it take for a student to be successful in mathematics?2. What do you expect of a good math teacher?3. What does it take for a student to be successful in science?4. What do you expect of a good science teacher?5. Can a student do well in both mathematics and science? If yes, what are the charac-

teristics of such a student? If no, please explain why you do not think they can dowell in both.

Teacher Candidate Interview #2

(Note: As part of being involved in MCTP, the teacher candidates were aware of the goalsof MCTP courses (e.g., to emphasize reasoning, logic, and understanding). Since the parti-cipants knew of these goals, we wanted to hear their perceptions as to whether these thingswere being emphasized in the class.)

1. Has the instruction in [Dr. Taylor’s] class helped you make connections betweenmath and science? Please give examples.

2. To what extent has this class involved the application of technologies?3. Has the instructor made significant attempts to understand your understanding of a

topic before instruction? Please describe anything your instructor has done to seehow well you understand a topic before beginning instruction. Did the tests reflectthis emphasis?

4. To what extent has this course stressed reasoning, logic, and understanding overmemorization of facts and procedures?

5. Do you think the teaching you experienced in this course models the type of teachingthat you believe should be done in grades 4–8? How? Why?

6. Did your instructor explicitly encourage you to reflect on what you learned in thisclass? Please give examples.

Faculty Interview Protocol(Used for both interviews – with verb tense changed for secondinterview.)

1. To what extent is the instruction in this class planned to highlight connectionsbetween mathematics and the sciences?

2. To what extent will this class involve the application of technologies?3. To what extent will you make significant attempts to access you students’ prior

knowledge of a topic before instruction? What techniques will you use?4. To what extent do the tests and exams of this course stress reasoning, logic and

understanding over memorization of facts and procedures? Would you providecopies of these materials?

5. In what ways do you think your teaching in this course models the type of teachingthat you believe should be done in grades 4–8?

6. To what extent will you explicitly encourage your students to reflect on changes intheir ideas about topics in your course? Can you give an example? What techniquesdo you anticipate using?

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Survey. On the first day of the semester, prior to the start of instruction, weadministered the MCTP survey instrument: Attitudes and Beliefs aboutthe Nature of and the Teaching of Mathematics and Science (McGinnis,Kramer, Shama, Watanabe & Graeber, (1997). For this study, the surveywas used to obtain an impression of our participants’ beliefs about theteaching and learning of mathematics before they were influenced by theirprofessor, their MCTP mathematics course, or their participation in theMCTP.

Data Analysis

We analyzed the data by analytic induction: we searched for patternsof similarities and differences between the professor’s and teachercandidates’ perceptions (Bogdan & Biklen, 1992; Gee, 1990; Goetz &LeCompte, 1984; LeCompte, Millroy & Preissle, 1992). We eventuallydecided to report the findings in four categories: I. Doing Mathem-atics in Traditional Courses; II. Doing Mathematics in a Reform-BasedCourse: Emphasizing Concepts; III. Doing Mathematics in a Reform-Based Course: Communication and Collaboration; IV. The Voices ofNon-MCTP Students.

Wading through the data. As a first step, we read through all of theinterview transcripts and observation field notes in order to develop aglobal perspective of the data. Next, we made notations in the margins tosummarize what seemed to be the main ideas of the participants. Fromthese notes, we made our first attempt to categorize the participants’comments and our field notes. Using our initial categories, we placed directquotations from the transcribed interviews in any of the categories thatseemed appropriate. Often a single quotation would appear in more thanone category.

Where the quotations in a category warranted, we formed sub-categories. At the point that all of the salient quotations were placedin categories and sub-categories, we had pared down over 200 pages ofinterview transcripts and field notes to 26 pages. We again read throughthe data to assess the credibility of the categories formed and to collapsecategories or divide a category into sub-categories. Further, we re-namedthe categories to better capture to ideas presented by the participants. Fromthis next iteration of categories, we selected from among all the accruedquotations a few that seemed to best represent the participants’ views.

Delving into an interpretation of the data. Once we had organized andsynthesized the data, we turned to the analysis phase, asking, “What does

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this mean for the preparation of teacher candidates?” It became apparentthat the students had begun to become enculturated into a way of teachingand learning mathematics that was very different from the one they wereaccustomed to, and this lead us to the notion of “A first step” (as it laterappears in the discussion). Also, in regard to Dr. Taylor, the notion of“pioneer” assisted us in understanding his growing uncertainty about theprocess of promoting a transformation of his students’ image of the natureof mathematics even as the data suggested his reform-based teaching wasachieving promising results.

Member checking. Finally, we distributed a draft of the manuscript toselected participants for their reactions. Although we had discussedvarious theories with them throughout the analysis and writing, this wasthe first opportunity for participants to review our work in written form.

FINDINGS

Our analysis of the data indicated that Dr. Taylor and the teacher candidatesperceived vast differences between traditional instruction and Dr. Taylor’sversion of reform-based teaching and learning. We turn to the survey datato give some measure of the students’ (both MCTP teacher candidates andthe other students) beliefs about mathematics teaching and learning priorto the start of the class. Although we did not intend to use this data forrigorous statistical analysis, the responses revealed some clear patterns inthe students’ beliefs. Seven items from the MCTP Survey are germane tothis study. The items and the frequencies of responses to each item aregiven in Table II.

As is evident in the clustering of the students’ beginning-of-the-yearresponses to items 13, 15, 18, and 20; most of the students believed thatmathematics was a dynamic discipline involving a search for the patterns.They also believed that it was important to know why an answer wascorrect as well as being able to find a correct answer. However, it seemedthat the students’ expressed beliefs reflected some dissonance in their viewof mathematics. Namely, although they believed the discipline of mathem-atics to be dynamic and involve understanding why, they also believed thatlearning mathematics in school involved step-by-step practice of proced-ures with the practice of such procedures occurring before mathematicalproblem solving was attempted. This emphasis on procedures and rulesis consistent with the aspects of traditional instruction that theStandards(NCTM, 1989, 1991, 1995) recommended to change.

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TABLE II

Frequencies of Responses to Relevant Items in the MCTP Survey

Response Choices:

A Strongly Agree

B Sort of Agree

C Not Sure

D Sort of Disagree

E Strongly Disagree

Item Frequency of responses

A B C D E

12. Students learn mathematics best by figuring outfor themselves the ways to find answers to math-ematics problems.

3(2) 4(1) 5(1) 4(0) 2(1)

13. Mathematics is a constantly expanding field andits study involves searching for patterns.

7(3) 6(0) 5(2) 0(0) 0(0)

14. Truly understanding mathematics requires specialabilities that only some people possess.

2(0) 6(1) 2(0) 4(2) 4(2)

15. Before students spend much time solving math-ematics problems, they should practice computa-tional procedures.

6(2) 4(2) 8(1) 0(0) 0(0)

17. Mathematics is made up of unrelated topics likearithmetic, algebra, geometry, and measurement.

0(0) 4(1) 4(1) 4(0) 6(3)

18. Success in mathematics requires lots of practicein following rules and implementing step-by-stepprocedures.

8(2) 9(3) 0(0) 1(0) 0(0)

20. Knowing why an answer is correct is as importantas getting the correct answer in math.

13(5) 4(0) 1(0) 0(0) 0(0)

Note: The first number listed is the total number of participants (both MCTP and non-MCTP) who selected the given response. The number contained in the parentheses is thenumber of MCTP teacher candidates who selected the given response.

In interviews conducted later in the semester, both Dr. Taylor and theteacher candidates expressed a clear image of what they thought teachingin grades 4 through 8 should be. This image of ideal mathematics teachingwas consistent with the teaching and learning that they experienced inDr. Taylor’s class. Three categories emerged from the data in regardto the participants’ perceptions of traditional teaching and learning, andteaching and learning in Dr. Taylor’s class. In addition to these categories,we have included an additional category that examines the non-MCTPstudents’ views on the course. This last category is separated from the

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others because these data reflect perspectives of non-MCTP students inthe course. For each category we first relate the teacher candidates’ andprofessor’s perceptions, based on the interview data. We then provide ourperception to more fully explicate our understanding of the data.

I. Doing Mathematics in Traditional Courses

The participants perceived that traditional mathematics courses focusedon following procedures and memorizing formulas. An analysis of theirperceptions and of those of Dr. Taylor and the researchers follows.

Teacher candidates’ perceptions.For many students, doing mathematicsmeant repeated practice of exercises without regard to understandingthe mathematical concepts involved. Julie, for example, related her priorexperiences with mathematics as consisting entirely of procedures withoutunderstanding:

Before when I would have math classes, . . . the way it was set up, it didn’t really requirethat I had to really understand it. It’s just that I had to be able to mimic what the teacherdid; I just had to be able to follow the steps and just do it without understanding what I wasactually doing (Interview, December).

Also, Kevin discussed the lack of interest he felt and observed fromother classmates when mathematics was taught as formulas to memorize:

[Typically in mathematics classes] they stress memorizing formulas and things like that,or they’d give you the formula and then you’d have to go home and do 20 like that forhomework . . . . I’ve had classes where you sit down and people will fall asleep, and theteacher was goin’ on talking (Interview, December).

In addition, Heidi described the lack of active participation found inmost mathematics classes:

My math classes were always, you sat at a desk with your book, and you had examplesto do, and the teacher would write on the board,. . . and Imean, that was math, and that’swhat you expected from math. You sit and listen to the teacher (Interview, December).

Dr. Taylor’s perception. In an interview after the course was completed,Dr. Taylor discussed the mathematics department’s existing version of thecourse as an example of a typical mathematics class.

A traditional version of this course, . . . and that’s probably not much different from anyother math course, I think the goals of the course, the actual goals, not the advertised goals,are to train students in a pretty mechanical, procedural approach to a fairly limited and welldefined set of procedures . . . . So a typical math class is, “I will explain something toyou,describe as clearly as I can how to do the next set of tasks, and give you some examples ofhow to do the next set of tasks, and then you do some more, and then I’ll check you.”

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Thus, Dr. Taylor perceived the emphasis of traditional mathematics coursesto be limited to training students to perform established procedures.

Researchers’ perception. During the study, a consistent referent we heardfrom all five MCTP teacher candidates was their prior conception of math-ematics teaching and learning as dry, rule-based, and consisting of a setof procedures. The first-year college students explicitly stated that theirpredominant vision of mathematics instruction was based on their recenthigh school mathematics instruction. They believed they needed to mimicthe procedures presented by their mathematics teacher to obtain correctanswers to the problems given. The teacher candidates were accustomedto doing large sets of similar mathematics problems without understandingthe meaning or purpose of the problems. Dr. Taylor’s description of whathappened in traditional mathematics classrooms was consistent with whatthese teacher candidates portrayed regarding their experiences in priormathematics classes. Therefore, we decided to include this category as away to acknowledge how prevalent a similar vision of traditional math-ematics instruction is among the participant teacher candidates and theprofessor of mathematics in this study. This category also helped us to laterunderstand the consonant reactions of the participants to the mathematicsexperiences in Dr. Taylor’s class.

II. Doing Mathematics in a Reform-Based Course: EmphasizingConcepts

In contrast to the previous category, the participants perceived Dr. Taylor’scourse as focusing on understanding mathematical concepts and strategies.Instead of primarily following procedures and memorizing formulas, thestudents saw themselves as investigating meaningful problems.

Teacher candidates’ perceptions. Julie explained how the course emphas-ized concepts and understanding the significance of mathematics overmemorization of facts.

[In this course] the emphasis was on concepts. It was a lot of understanding just in general,like knowing how things work – more than just a memorization of facts – just under-standing what we were doing and not just kind of following what he said to do, and what thebook said to do. . . . You have to do a lot more thinking about the bigger picture (Interview,December).

Beth compared the focus on understanding in Dr. Taylor’s course to anemphasis on memorizing empty facts that she experienced in previousmathematics courses.

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[Dr. Taylor’s course] has definitely been more of understanding of how to solve theproblems as opposed to the memorization of facts and stuff (Interview, December).

Dr. Taylor’s perception. Early in the semester, Dr. Taylor discussed hisplans for teaching and learning in the class. He emphasized that the coursewould not focus on procedures without understanding when he said, “Ithink that one thing that we [will not do] is a lot of procedural routines”(Interview, September).

Dr. Taylor also described what he considered to be important learningfor the students in his class: learning based on reasoning, connections, andmeaningful problems. He stated,

[The students should] be able to explain [methods of problem solving]. . . [it’s] not goingto be just memory of a fact, it’s going to be understanding of a whole way of reasoningabout a problem. . . . We’re trying to help students . . . make the connection between thereal object and the mathematical representation or the mathematical model of it . . . . We’retrying to have the course problem-based in a sense that the mathematical ideas will beencountered first in looking at the context of working on a problem of some kind ratherthan “here’s how we’re gonna do today’s problems.” It’s trying to embed the mathematicsin problem-solving activity . . . It’s more an applied problem; . . . more making sense of areal situation and patterns in data (Interview, September).

After the course had ended Dr. Taylor was asked to review the abovequotation and to discuss the extent to which he believed he achievedthese goals. He confirmed that he believed he was able to embed themathematics, but he went on to share these thoughts:

The whole transformation of the way students look at the nature of mathematics isn’t clear;it’s a messy process. They are somewhere very different, and getting them somewhere elseis not going to be just, “I’ll pick you up here and drop you there.”

Dr. Taylor discovered that instruction focused on reasoning and problemsolving is complex and difficult. It is less direct and predictable thantraditional approaches.

Researchers’ perception. This category emerged out of our interest injuxtaposing the participants’ perceptions of typical mathematics classeswith what they perceived were the foci of Dr. Taylor’s class. As with thefirst category, we perceived that the participants seemed to speak with onevoice when they described the areas of emphasis in Dr. Taylor’s class.All of the teacher candidates viewed Dr. Taylor’s class as different fromwhat they were used to in mathematics class. They recognized that thecourse was focused on concepts, understanding, and learning meaningfulmathematics. The teacher candidates’ perceptions that the course emphas-ized concepts and understanding were relatively consistent with whatDr. Taylor envisioned in planning the course. Furthermore, Dr. Taylor’s

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description of what he considered to be important in mathematics teachingand learning was consistent with what we observed in his class. However,the students’ perception that they were being asked to do more thinkingand understanding did not reflect all the sophistication embedded in thecourse approach, such as producing mathematical models and problemsolving prior to learning a specific procedure. This lack of sophisti-cation was consistent with Dr. Taylor’s reported struggles in creating thistype of learning environment and with his observation that the students’shift in perspectives on learning mathematics was neither rapid noreven.

III. Doing Mathematics in a Reform-Based Course: Communicating andCollaborating

In encouraging his students to develop and apply their reasoning skillsin solving problems, Dr. Taylor was continually questioning students’thinking and facilitating collaboration within groups of students. Thestudents realized that this instructor assumed a role different from alecturer. An analysis of their and Dr. Taylor’s views of the reform-orientedapproach follows.

Teacher candidates’ perceptions. Kevin explained how Dr. Taylor wouldask questions in an effort to engage students in thinking about a problem:

The teacher will come around and sort of direct you in a certain direction, or ask you morequestions, get you thinking more. It seems, that you’re sort of widening your focus on mathinstead of running a single process, and you will learn that process, but you also, along theway, you know, sort of pick up this other stuff. And you’re not just copying things off theboard (Interview, October).

Julie stated that Dr. Taylor’s questions would help to re-direct theirthinking if they were having difficulties approaching a problem:

[Dr. Taylor] would step in and kind of guide us the right way, maybe asking us questionsin different ways so that we can see in a different way what he’s trying to get across, andthat way remember it because we understand it (Interview, December).

Kevin also discussed how Dr. Taylor used group work to facilitate theprocess of generating ideas and strategies for problem solving:

[Dr. Taylor] gives you a problem that you have to solve, and you get together with otherstudents and you all try to solve the problem together, so you’re coming up with all thesedifferent ideas of ways to conquer this problem (Interview, October).

Dr. Taylor’s perception. Dr. Taylor explained that his intention in teachingwas not to tell students information and what to do to solve a problem, but

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instead, it was to let the students attempt to solve the problem. Accordingto Dr. Taylor, what was important for him to do was to “get them thinking”not necessarily to arrive at a specific answer.

In addition, Dr. Taylor expressed his interest in incorporating teachercandidates’ collaboration and communication in facilitating their learningof mathematics:

[I am] asking students to collaborate with each other and to work cooperatively. Quite oftenasking students to present . . . to communicate their ideas in writing, submitting write-upsabout their solutions to a problem or talking, sharing what their group has come up withorally in class (Interview, September).

Researchers’ perception. As the study progressed, we observed that theMCTP teacher candidates increasingly became accustomed to Dr. Taylor’sapproach to teaching and seemed to welcome his involvement in theirlearning. In several instances, the MCTP teacher candidates discussedthe actions of Dr. Taylor and what he did as a teacher to create thelearning environment. All of the teacher candidates, in one way or another,mentioned that Dr. Taylor acted as a facilitator or guide to learning asopposed to a lecturer who delivered information and facts to students. Thenotion that Dr. Taylor was always “walking around” and “asking ques-tions” to guide learning was prevalent in the teacher candidates’ commentsand in the researcher’s observations of the class.

Throughout the semester long class, we were impressed by Dr. Taylor’sextensive use of cooperative learning groups and other strategies thatrequired his students to actively communicate among themselves and withhim about mathematics. Dr. Taylor presented the students with a problemthat would stimulate discussion and some form of data collection as a basisfor reasoning through a problem. Rarely were the students given problemsthat had a single correct numerical answer. Whereas Dr. Taylor viewedhis interventions as attempts to promote thinking, students interpreted thequestions as guides to prompt “the right way.” Students had seeminglyviewed learning mathematics as including the need to understand process,but perhaps did not fully appreciate the role they were expected to pursuein finding their own way toward a solution – not just understanding theprocess of solution.

IV. The Voices of Non-MCTP Students

After the course was over and significant analysis and writing of the manu-script had been done, we were prompted to check and see if those studentsnot participating in the MCTP held perceptions of efforts made to enactedreform-based practices in Dr. Taylor’s class that were similar to or different

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from those of the MCTP students. Overall we found that the non-MCTPstudents held similar views for the above categories. However, unlike theMCTP teacher candidates, the non-MCTP students revealed their initialfrustrations with Dr. Taylor’s different role and expectations. As theycame to understand the course goals, these feelings were replaced with anappreciation for Dr. Taylor’s approaches. The interviews and course eval-uations indicated that the students had a favorable reaction to the course asindicated by their comments.

Non-MCTP students’ perceptions. In the whole-class interview at theend of the semester, Karen related the following views on traditionalmathematics instruction and instruction in this class:

We had to actually explain what we were doing and not just do a step-by-step thing andmemorize a routine, but we actually know what we were doing (Class Interview).

Justin also emphasized that this class focused on understanding theconcepts by connecting the mathematics to the real world. Tiffany revealedher struggles early in the course to adapt to the different approach whenshe explained,

At first, what really got me annoyed with this class is, . . . it was kind of hard for no one togive you an answer when you’re kind of sick of dealing with it [a problem]. But it kind oftaught you that you had to problem solve on your own (Class Interview).

Derrick described the emphasis on communication when he added,

You had to have an explanation . . . . In most math classes, . . . it’s easy to say, “Well, whatdid you get for this?” And if you get the right answer, the teacher thinks you know what’sgoing on. But in this class it . . . forces you [to] stay with the pack and just do it allyourselfbecause you had to have the explanation. . . . And it wasn’t just answers, it was concepts(Class Interview).

Jason described how it was frustrating in the beginning of the semesterto adjust to Dr. Taylor’s expectations of explaining solutions:

It’s really frustrating [to have to explain your work] if you know the answer. You just writedown the answer and you just think that’s what he wants. That’s not what he really wants.He wants the reasoning behind the answer (Class Interview).

Bill discussed the role of group work in learning to communicatemathematical ideas when he stated:

The group work when one person sort of catches onto the idea and has to explain to therest of the group, I think that definitely improves your ability to communicate becausethat’s the best way to learn. The best way to prove that you know something is to deal withconvincing other people and know they understand (Class Interview).

In regard to Dr. Taylor’s role as a facilitator and guide, Michelle stated,

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TABLE III

Summary of Students’ Course Evaluation: Selected Items (College of Human Resourcesand Education at West Virginia University, 1994)

Item Item Stand.

mean dev.

1. The overall quality of this course was 4.4 0.6

2. In general, this instructor’s teaching was 4.5 0.6

11. The instructor’s interest in students learning the content was 4.8 0.4

12. The class sessions, as far as being interesting, were 4.2 0.9

13. The instructor’s encouragement for student discussion was 4.8 0.4

15. The respect the instructor showed toward students was 4.5 0.9

Notes: Scale of 1 to 5, with 1 poor and 5 excellent. Most of the items are paraphrased orabbreviated from the original form.

I think a lot of what he [Dr. Taylor] was trying to do . . . was show us that we should tryand understand why something is, rather [than] just figuring it out and just knowing it. Itwas figuring out why, . . . and that’s what we had to do with all the problems throughout theclass (Class Interview).

Greg reinforced this notion when he described the kind of teacher thatDr. Taylor wasnot: “He never really said, ‘Here’s the equation, figure outthis from this equation’ ” (Class Interview).

Later, we contacted Marie and Bob, two non-MCTP teacher candidates,for individual interviews. In regard to Dr. Taylor’s role in the classroom,Bob recalled that Dr. Taylor was “not a lecturer. [He was] one of the classmembers instead of the boss.” Also, Marie emphasized that the instructorwas more a facilitator than a teacher.

After being asked for her general recollections about the course, Marieencapsulated the major themes of the course:

I remember a lot of cooperative learning and group work. Learning math more in a hands-on nature as opposed to just sitting there doing problems, and we worked as a group. . . .You’d think up the formulas by experimenting, and we used more real life kind of problemsas opposed to problems from books that you can’t apply.

Students’ evaluations of the class. The course evaluations (College ofHuman Resources and Education at West Virginia University, 1994) wereanother source of information on the students’ view of the course and Dr.Taylor (see Table III). Eighteen students in the class completed the eval-uation. In the first part of the form, the students rated the course and theinstructor on a scale of 1 to 5 where 1 represents “poor” and 5 represents“excellent.” Of the 19 items that rated either the class or the instructor,

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the mean rating was 4.38 with a standard deviation of 0.72. More thanhalf of those responses were 5’s. An additional item asked students tochoose their preferred setting for learning. The choices given were Lecture,Lecture/Discussion, Group Activity, Programmed Instruction, and Indi-vidual Projects. Thirteen of the students (72%) selected Group Activity,followed by four (22%) who selected Lecture/Discussion, and one (6%)who selected Individual Projects. Notably, lecture, a typical feature ofa college mathematics course (Prichard, 1993), was not selected by anystudent.

Finally, the students were provided space for written comments onparticularly “valuable aspects of the course and instructor” and on“problem areas with the course or instructor.” These comments were eitherbrief (one to two sentences), or the student left this section blank. Elevenstudents chose to write a comment. Of the comments offered, a pattern ofresponses emerged that can be summarized by two categories: Dr. Taylorwas able to motivate and interest students, and the students liked groupwork. For example, in regard to motivating students, one student wrote,

I think he did an excellent job teaching math concepts [and] relating [them] to life andother subjects. This is the first time I have enjoyed the content of my math class and feltlike I was learning valuable information.

Also, in writing about group work, another student commented,

[Dr. Taylor] tried to get us to teach ourselves by means of personal reflection and groupwork. I liked that aspect a lot.

Researchers’ perception of Non-MCTP students’ comments. The com-ments offered by the non-MCTP students in the whole-class interview andin the course evaluations were consistent with and further validated ourinterpretations. The non-MCTP students had a positive reaction to the classand the instructor, and most preferred and recognized the value of grouplearning. The students appreciated Dr. Taylor’s efforts to motivate theirinterest in mathematics. However, the non-MCTP students revealed someinitial feelings of frustration as they were learning about Dr. Taylor’s verydifferent expectations for their work. These feelings dissipated during thesemester. Interestingly, the MCTP students did not express these frustra-tions. This difference in the groups might be due to the fact that the MCTPstudents enrolled in the course knowing that it would incorporate reform-based approaches, even though they had not experienced these approachesbefore. In contrast, the non-MCTP students only knew that this course waslabeled an “experimental” version of the regular course in the universitycourse listings.

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In general, the uniformity of the MCTP and Non-MCTP studentresponses on the evaluation was important to ascertain. There was a possib-ility that by virtue of being a part of the MCTP, the five MCTP teachercandidates may have been pre-disposed to look for and appreciate certainreform-based aspects of this MCTP course. However, an analysis of theresponses from both MCTP and non-MCTP students indicated that parti-cipation in the MCTP program did not seem to influence those students’perceptions of the class to the extent that they saw or experienced thingsthat others did not. Thus, Dr. Taylor’s actions and behavior were inter-preted similarly by both MCTP teacher candidates and the non-MCTPstudents.

DISCUSSION

The preservice teachers’ experience in a reform-based mathematics classoccurred very early in their teacher preparation sequence. Thus, unlike thestudents involved in Civil’s (1993) study, the MCTP students were notfocused on how to teach mathematics. Whereas Wilson (1994) focusedon a student’s depth of understanding of a mathematical topic, our studywas concerned with preservice teachers’ grasp of the reform-based natureof the course. We inferred from their descriptions of the course that theMCTP students did not view the strategies and goals of the course withthe same degree of sophistication as the instructor. Although the datawould suggest that the students’ understanding of the course goals andstrategies were somewhat beyond the “surface level,” their uneven andincomplete progress in understanding the professor’s goals and strategieswere apparent. Whereas Frykolm (1995) contended that more than onecourse is needed to have preservice teachers reach the point of being ableto implement the goals and strategies of a reform-based classroom, theauthors interpret this study as noting that more than one course is neededjust to have the preservice teachers understand the goals and strategies. Theauthors view the present study as the documentation of one early interven-tion designed to help the preservice teachers experience and build a morerobust understanding of reform-based instruction. The chief remainingquestion is: Will experiences such as what Dr. Taylor’s students had,combined with further content courses along with educational courseworkand field experiences, enable these teacher candidates to meet the goals forreform in their teaching? The answer to this question is the long-term goalof the MCTP research program.

The experiences of the teacher candidates and the professor haveimplications for teacher education programs interested in preparing teacher

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candidates to achieve the standards for teaching and learning set forthin the reform documents. A major implication gained from this study isthat the college students who experienced a reform-based mathematicsclassroom early in their undergraduate program completed afirst stepinachieving the vision for reform of mathematics education: constructingan initial model of mathematics teaching and learning which embraces theideals of the reform movement. The students’ prior conceptions and exper-iences of mathematics instruction were that mathematics teaching andlearning was procedural and rule-based. However, being in a classroomin which reform-based teaching was modeled and where students wereengaged in active learning through meaningful problem solving andcollaboration enabled the students to construct a new model of math-ematics teaching and learning. A process of enculturation similar to thatwhich Schifter and Fosnot (1993) strive for in their program for prac-ticing teachers was evidenced in the perceptions voiced by Dr. Taylor’sstudents.

The question then becomes: What level of enculturation was achieved?Referring back to Gee’s (1990) definitions of acquisition and learningand Lave and Wenger’s (1991) notion of situated learning, it seems thatwhile Dr. Taylor’s students may have beenlearning mathematics, theywere acquiring ideas about the teaching and learning of mathematics.The students were being exposed to Dr. Taylor’s model of teaching andlearning, and it was in the natural setting of teaching and learning: aclassroom. Formal teaching about mathematics occurred; however, formalteaching about the teaching and learning process was not present. Dr.Taylor’s students were enculturated into the ideas of reform-based teachingand learning by experiencing them as students. They were “learning bydoing” from the perspective of students. What had not yet taken place isthe “teaching” of how to become a reform-based teacher.

We are convinced that this initial experience is not sufficient forpreparing teacher candidates. In accordance with the findings of Borko,Eisenhart, and colleagues (Borko, Eisenhart, Brown, Underhill, Jones &Agard, 1992; Eisenhart, Borko, Underhill, Brown, Jones & Agard, 1993),the teacher candidates in Dr. Taylor’s class believed that further educa-tional coursework and field experiences were necessary before they wouldbe prepared to “do the things that [Dr. Taylor is] doing now” (Beth,Interview, December) in their own teaching. The findings from this studysuggest that one content course taught from a constructivist perspective isnot sufficient in preparing teacher candidates to meet the goals for reform.The program studied is consistent with Gee’s stated notion that acquisitionmust precede learning. It seems that the phase of enculturation into the

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social practices associated with reform-based teaching is an important andhelpful step in the process of preparing teacher candidates to incorporatereform-based practices into their future mathematics teaching. Of course,the question of how many and what types education courses are necessaryremains unanswered.

Another major implication is that professors teaching a reform-basedclass in which they model reform-based teaching practices consistent withthe reform documents should anticipate taking on many aspects of apioneer venturing into new territory. Dr. Taylor believed that as a math-ematics professor whose class included teacher candidates he needed tobe especially concerned with modeling good teaching if the cycle ofineffective mathematics teaching practices was to be broken. Thus, heneeded to teach in innovative ways consistent with reform-based teachingstrategies to promote mathematical learning of content, while presenting anew image of a MCTP mathematics teacher to his teacher candidates. Thedilemma he faced was to enact this new role without appeal to an acquis-ition apprenticeship (Gee, 1990). He did not have the opportunity to firstserve as an apprentice but had to enact his vision of good teaching based onhis understanding of the teaching strategies advocated in the reform docu-ments. In contrast to his teacher candidate students, as a content specialistDr. Taylor was enculturated into the practice of reform-based teaching andlearning by experiencing it as ateacher. Although he was learning-by-doing in his situated practice (Lave & Wenger, 1991) he was neverthelessnot in isolation. Fully aware of the central insight from social construct-ivism that learning is culturally shaped and defined in a social community(Schoenfeld, 1992), Dr. Taylor went beyond learning about implementingreform-based teaching practices as guided solely by the reading of reformdocuments. He valued the perspectives of his students and the researchers.These data suggested to him that his pioneering of reform-based teachingpractices in his mathematics classes achieved some promising resultsfor all of his students, including the targeted MCTP teacher candidates.However, while drawing encouragement from this information that thereform-based teaching strategies were effective, he came to believe that theprocess of transforming his students’ view of mathematics was a complexand “messy process” (Dr. Taylor, interview) with many remaining unclearaspects.

Both the teacher candidates and Dr. Taylor took an important firststep in realizing the vision for reform-based mathematics education. Theteacher candidates acquired an understanding of what it means to be alearner in a reform-based class, and Dr. Taylor was enculturated into the

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practices of reform-based instruction by experiencing the complexities ofteaching and learning in this environment.

ACKNOWLEDGMENT

We would like to express our appreciation to Catherine Brown who advisedus on an earlier version of this paper.

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Department of Teaching and Learning, Amy Roth-McDuffieWashington State University Tri-Cities,2710 University Drive,Richland, WA 99352,e-mail: [email protected]

Department of Curriculum and Instruction, J. Randy McGinnisUniversity of Maryland, Anna O. GraeberCollege Park, MD 20742,e-mail: [email protected] (McGinnis’s)e-mail: [email protected] (Graeber’s)