# Two Teachers' Conceptions of a Reform-Oriented Curriculum: Implications for Mathematics Teacher Development

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<ul><li><p>GWENDOLYN M. LLOYD</p><p>TWO TEACHERS CONCEPTIONS OF A REFORM-ORIENTEDCURRICULUM: IMPLICATIONS FOR MATHEMATICS</p><p>TEACHER DEVELOPMENT ?</p><p>ABSTRACT. This paper describes two high school teachers conceptions of the coopera-tion and exploration components of a reform-oriented mathematics curriculum. Althoughthe teachers appreciated the themes of cooperation and exploration in theory, their concep-tions of these themes with respect to their implementations of the curriculum differed. Oneteacher viewed the curriculums problems as open-ended and challenging for students,whereas the other teacher claimed that the problems were overly structured. Each teacherattributed difficulties with students cooperative work to the amount of structure and direc-tion (too little or too much) offered by the problems. Discussion of such similaritiesand differences in the teachers conceptions emphasizes the dynamic, humanistic natureof curriculum implementation and gives rise to important implications for mathematicsteacher development in the context of reform.</p><p>By demanding changes in both the content and activity of mathematicsinstruction, recent reform recommendations in the United States challengea lasting tradition (Gregg, 1995; Richards, 1991). In light of the impressivedurability of traditional teacher-centered and procedure-oriented mathe-matics instruction, how do veteran teachers deal with calls for reform? Thispaper describes two secondary teachers experiences with reform recom-mendations in the context of their implementations of a set of innovativecurriculum materials. Focus is on the teachers conceptions of the meaningand importance of certain mathematics classroom activities, in particularcooperation and exploratory problem-solving.</p><p>Cooperation and Exploration in Mathematics Teaching and LearningCooperation and exploration are prominent themes both in the curriculummaterials implemented by the teachers (Hirsch, Coxford, Fey & Schoen,1995) and in more general documents that promote mathematics educationreforms (Mathematical Sciences Education Board [MSEB] & National? The research reported in this study was supported in part by the National Science</p><p>Foundation (MDR-9255257). The views herein are those of the author and do notnecessarily reflect those of the National Science Foundation.</p><p>Journal of Mathematics Teacher Education 2: 227252, 1999. 1999 Kluwer Academic Publishers. Printed in the Netherlands.</p></li><li><p>228 GWENDOLYN M. LLOYD</p><p>Research Council [NRC], 1989; National Council of Teachers of Mathe-matics [NCTM], 1989). Teachers are urged to establish mathematicsclassrooms in which students engage actively in exploration and cooper-ative work in order to help students develop rich understandings ofmathematics as a vibrant and useful subject.</p><p>Reform-oriented models of teaching and learning are supported bya broad base of empirical and theoretical literature about how studentsunderstand and learn mathematics. This literature makes a strong casefor classroom activities that give rise to genuine mathematical problemsfor students to resolve (Hiebert et al., 1996; Lampert, 1990; Schoenfeld,1992; P.W. Thompson, 1985; Yackel, Cobb, Wood, Wheatley & Merkel,1990). In contrast to traditional classroom activities that emphasize correctanswers and routinized solution methods, problem-centered instructioncapitalizes on opportunities for students to learn as they cooperate inthe solution process. This process can include accounting for surprisingoutcomes, such as when two alternative methods lead to the same result,justifying a solution method, or explaining why an apparently erroneousmethod leads to a contradiction (Yackel et al., 1990, p. 15). When studentswork in groups to communicate their ideas and questions, agree anddisagree among themselves, and negotiate joint theories and ideas, richmathematical learning can occur (Richards, 1991; Slavin, 1990; Voigt,1996).</p><p>Teachers Conceptions and ReformHow do teachers make sense of the themes of cooperation and explorationas they implement innovative curriculum materials? The role of teachersconceptions and classroom experiences in mathematics reform cannotbe overemphasized. An extensive body of research provides consistentevidence that teachers conceptions strongly impact instructional prac-tice (Brophy, 1991; Fennema & Franke, 1992; A.G. Thompson, 1992).Moreover, teachers conceptions have profound effects on their inter-pretations and implementations of reform recommendations and reform-oriented mathematics curricula (Cohen, 1990; Lloyd & Wilson, 1998;Romberg, 1997; M. Wilson & Goldenberg, 1998; S.M. Wilson, 1990).For example, in S.M. Wilsons (1990) case study of a teacher imple-menting a new curriculum as part of the California mathematics reforms,the teachers conceptions of appropriate instruction interfered with hisability to foster student inquiry in the ways intended by the reformcurriculum. Because reform-oriented pedagogies require that teachersreconceive their roles in mathematical activity and student learning,implementation of an innovative curriculum can pose significant chal-</p></li><li><p>TEACHERS CONCEPTIONS OF CURRICULUM 229</p><p>lenges even to the most committed teachers (Clarke, 1997; Smith,1996).</p><p>Reform documents and curriculum materials do not prescribe or definepractice for teachers, but rather offer visions orienting individuals andinstitutions toward collectively valued goals (Shulman, 1983, p. 501). Aricher understanding is needed of the relationships between teachers ownconceptions of mathematics teaching and the recommendations for changeoutlined in curriculum materials. The study reported in this paper inves-tigates how and why two teachers encountered particular successes anddifficulties as each implemented a set of novel curriculum materials for thefirst time. How did the teachers conceive of cooperation and exploration asthey implemented the curriculum materials in their classrooms?</p><p>RESEARCH METHODS</p><p>Curriculum Materials</p><p>The curriculum of the Core-Plus Mathematics Project attempts to supportteachers in enacting many recommendations of the Standards (NCTM,1989). Each year of the high school curriculum (Core-Plus Courses14) features algebra and functions, geometry and trigonometry, statis-tics and probability, and discrete mathematics. The curriculums units,each designed to guide approximately four weeks of student work, areorganized into several multi-day investigations that emphasize modelingreal-world situations, experimenting in order to develop and test theories,and debating with classmates. The materials direct teachers to organizethe classroom so that students can work cooperatively as they exploremathematical problems and ideas. Cooperative learning is complementedby whole-class discussions in which activities are introduced and summa-rized.</p><p>An example of a Core-Plus lesson may be helpful. The first lessonin a unit about probability, Simulating Chance Situations, begins witha description of Chinese government policies that restrict families toone child. Students are first asked to think of some alternative plans forcontrolling population growth. Then students are instructed to flip coinsto simulate an alternative plan, that each family may have two children,and to record their results in a frequency table. Students must decide howcoin flips can be used to simulate the births and to describe the possibleoutcomes for this particular plan. A variety of problems and questionsfollow, including: Use your frequency table to estimate the probabilitythat a family of two children will have at least one boy and Estimate the</p></li><li><p>230 GWENDOLYN M. LLOYD</p><p>probability that a family of two children will have at least one boy usinga mathematical method other than simulation. Next, students are askedto consider a different plan in which families continue having childrenuntil a boy is born, and to record data in frequency tables and histograms.This time, students predict responses to a set of questions prior to theirsimulation, and then compare predictions to actual results. Finally, studentsanalyze one of their own alternative plans proposed at the beginning of theinvestigation.</p><p>Although most Core-Plus activities are based in real-world contexts,such as the probability simulation described above, some engage studentsin exploring more abstract mathematical situations. For example, in a unitabout functions, an investigation titled The Shape of Rules asks studentsto generate tables and graphs associated with four sets of equations thatrepresent different types of relationships (linear, quadratic, exponential,etc.). Students are instructed to look for patterns across the three repre-sentations and summarize their ideas in statements such as If we see arule like . . . , we expect to get a table [or graph] like . . . . Further examplesand discussion of the Core-Plus curriculum can be found in a report by thematerials designers (Hirsch et al., 1995).</p><p>Participants, Site, and Context for the StudyThis study investigated the conceptions of two veteran high school mathe-matics teachers, Mr. Allen and Ms. Fay, in a public school district in theNortheastern U.S. where the Core-Plus materials were being field-tested.Empirical study of Mr. Allen and Ms. Fay began in 1994 and 1996, respec-tively, when each teacher implemented the Core-Plus materials for the firsttime. These two teachers were selected for study because (1) in contrastto most other teachers at the same school, Mr. Allen and Ms. Fay imple-mented the materials voluntarily, (2) each teacher communicated a desireto integrate more cooperation and exploration into his or her instruction,and (3) the teachers individual experiences appeared to have importantcontrasting elements.</p><p>At the beginning of the study, Mr. Allen had been teaching mathematicsfor 14 years and, by his own description, had largely adhered to traditionalclassroom practices. In the spring of 1994, he was invited by the chair ofhis mathematics department to participate in the field-testing of the Core-Plus materials. He said, If this is the way were going to go, I want tomake sure I have experience in it. During the 199495 school year, Mr.Allen implemented the Core-Plus Course 1 materials for the first time inone class of 32 ninth grade students. In the 199596 school year, he againtaught one Core-Plus Course 1 class of 32 ninth grade students. In 1996</p></li><li><p>TEACHERS CONCEPTIONS OF CURRICULUM 231</p><p>97, he taught one ninth grade Core-Plus Course 1 class and two tenth gradeCore-Plus Course 2 classes.</p><p>In the Fall of 1996, due largely to her interest in the Core-Plusprogram, Ms. Fay joined the mathematics faculty at the high schoolwhere Mr. Allen taught. Her previous employment included both 10years of classroom teaching and, most recently, a state government posi-tion in which she visited schools with innovative mathematics educationprograms. According to Ms. Fay, in her previous teaching she had alwaysused groups and . . . always had a project focus, and she wished for theCore-Plus materials to support her in developing and extending thesecomponents of her practice. Her commitment to the Core-Plus innovationwas one of the primary reasons that she was selected as a participant in thisstudy. During the 199697 school year, Ms. Fay taught with the Core-PlusCourse 2 materials in two classes of tenth-grade students.</p><p>Data Collection and AnalysisData sources consisted of teacher interviews, classroom observations, andfieldnotes. Because the two teachers joined the project at different times,the number and dates of interviews and observations varied considerably.Over a 3-year period between September 1994 and January 1997, Mr.Allen participated in 17 interviews (8 in Year 1, 7 in Year 2, and 2 inYear 3) and was observed 73 times teaching with the Core-Plus materials.This paper focuses primarily on data collected during his first two yearsof curriculum implementation. During the 199697 school year, Ms. Fayparticipated in five interviews and was observed 10 times in her Core-Plusclasses. Most interviews lasted approximately 1 hour, and all were audio-recorded and transcribed. All classroom observations were audio-recordedand approximately half were video-recorded. Fieldnotes were taken duringobservations, and written artifacts such as student work were collected andphotocopied.</p><p>In the interviews, teachers were invited to reflect on recent classroomevents, suggest goals for upcoming classroom activities, and respond toquestions about more general emerging themes. Data were analyzed duringand after collection (LeCompte & Preissle, 1993). Careful readings of tran-scripts and fieldnotes and creation of interview and observation summariesresulted in the identification of preliminary themes for subsequent focus ininterviews and observations. After the observation periods ended, morethorough review of data took place. The development of major themeswas aided by the use of taxonomic and thematic analytic strategies(Spradley, 1979). In the final stage of analysis, major themes were furthersynthesized within and across the data sources to illustrate important, and</p></li><li><p>232 GWENDOLYN M. LLOYD</p><p>often contrasting, aspects of the ways in which the teachers viewed theirimplementations of the Core-Plus curriculum.</p><p>THE CASE OF MR. ALLEN</p><p>Conceptions of ExplorationFrom the start of his teaching with the new curriculum, Mr. Allen posi-tively differentiated the Core-Plus problems from those found in traditionaltextbooks by pointing out ways in which they engaged students in sense-making activities. He described the Core-Plus activities as having anapproach that required students to develop informal ideas based on theirexperiences in contrast to the teacher or book feeding it to them (Int. 3,Yr. 1, 11/14/94). More specifically, in his second year using the materials,he indicated that the Core-Plus questions themselves required extensiveinterpretation by students:</p><p>The Core-Plus questions are a little more general not vague, but not as specific and sosometimes the students have to figure out exactly what it is they want them to do. Thatswhat they need to be able to work at and practice because thats what theyll use eventually the problem solving skills and attacking a problem, reading it, and struggling with it. (Int.2, Yr. 2, 11/10/95)</p><p>One characteristic of the Core-Plus questions that required more analysisthan traditional exercises was that they were different each time and theydont do the same thing over and over again. He contrasted problemsthat challenged students to think actively about mathematics with a moretraditional approach that enticed students to have the attitude Just tell methe steps I need to do and Ill do those, but I dont want to think about it toomuch (Int. 5, Yr. 2, 12/8/95). Further, Mr. Allen suggested that with tradi-tional problems and instructional methods, student understanding isntnearly what it is in Core-Plus.</p><p>Another reason that Mr. Allen thought the Core-Plus materials engagedstudents in sense-making was that the activities cent...</p></li></ul>

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