Teaching Mathematics for Understanding: Case Studies of Four Fifth-Grade Teachers

Teaching Mathematics for Understanding: Case Studies of Four Fifth-Grade TeachersAuthor(s): Richard S. Prawat, Janine Remillard, Ralph T. Putnam and Ruth M. HeatonReviewed work(s):Source: The Elementary School Journal, Vol. 93, No. 2 (Nov., 1992), pp. 145-152Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1001697 .Accessed: 27/05/2012 02:25

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to TheElementary School Journal.

http://www.jstor.org

http://www.jstor.org/action/showPublisher?publisherCode=ucpress

http://www.jstor.org/stable/1001697?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp

Teaching Mathematics for Understanding: Case Studies of Four Fifth-Grade Teachers

Richard S. Prawat Janine Remillard Ralph T. Putnam Ruth M. Heaton Michigan State University

The Elementary School Journal Volume 93, Number 2 ? 1992 by The University of Chicago. All rights reserved. 0013-5984/93/9302-0003$01.00

Abstract

This article describes the rationale for and the methods used in the 4 case studies of mathematics teaching discussed in the 5 articles that follow this one in this issue. Building on earlier research, we have examined how calls by the California Mathematics Framework for change in teaching are influenced by the resources teachers bring to teaching. Focusing specifically on mathematics teaching in elementary schools, we use narrative, descriptive, and in-depth interview techniques to elicit possible relationships between teachers' knowledge of mathematics and beliefs about the nature of mathematics teaching and learning, and their mathematics instruction. Each of the 4 fifth-grade teachers whose cases are described in the articles in this issue reflects the complex nature of this relationship. We hope this research will prove useful to teachers and policymakers interested in reforming instruction.

A growing body of research points to the

important relationship between teachers'

knowledge and beliefs and their classroom

practice (Clark & Peterson, 1986). A good example of this line of inquiry is the recent research examining how teachers' subject- matter (content) knowledge influences instruction (Brophy, 1991). According to Ball

(1991), earlier research failed to demonstrate much of a relationship between these two variables. This earlier research, however, relied on indirect measures of teachers' subject-matter knowledge, such as the number of college-level courses taken in a

particular domain. More recent work is based on direct assessments of teachers'

knowledge, and as a result we can now document the long-suspected relationship between knowledge and teaching effective- ness. For example, Steinberg, Haymore, and Marks (1985) followed four secondary

146 THE ELEMENTARY SCHOOL JOURNAL

mathematics teachers during their first year of teaching. The two teachers who had the surest grasp of mathematics were best able to explain during instruction why certain mathematical procedures do or do not work. Furthermore, compared to those with a less secure command of the content, the

knowledgeable teachers tended to stress more important ideas and to be less didactic in their instruction.

Teachers' content knowledge appears to be just as important in elementary schools. Thus, Stein, Baxter, and Leinhardt (1990) were able to demonstrate how a fifth-grade teacher's fragile grasp of mathematics content led to an overemphasis on rules or procedures at the expense of what might be considered more "meaningful" content. Stein and her colleagues' concern, they ex-

plain, is that the heavy emphasis on rules

may contribute to student understanding that is "structurally weak"; that is, it is un- connected to the concepts constituting a

deeper, more mathematically correct un-

derstanding of a particular topic. Stein et al. also point out in their case study that it is

extremely difficult to separate a teacher's knowledge of subject matter from his or her views about the nature and purpose of that

knowledge. These views appear to play a

key role in influencing teachers' practice. Using a case study approach similar to

those cited above, Thompson (1984, 1985) documented a relationship between junior high school teachers' views about mathematics and their classroom practice. Thompson found that the teacher who most deviated from traditional practice, in a "conceptual" sense, was also the one with the most dynamic view of the discipline. Like most mathematicians, this individual viewed the field as one that was continually changing. Teachers who view mathematics dynamically, as a subject about which students can reason or conjecture, may orga- nize instruction in different ways than teachers who view the discipline as a fixed or static body of knowledge-a finished product, as it were. It is also worth noting

that the apparent relationship between teachers' knowledge and beliefs and their classroom practice extends well beyond subject-matter variables. Of equal impor- tance, according to several studies, are teachers' views of learning and learners, and of the teaching process as a whole (Pe- terson, Fennema, Carpenter, & Loef, 1989; Roth, 1987).

One thing already seems evident from this relatively new line of inquiry: The resources teachers bring to teaching-their knowledge, skills, and beliefs-affect their actions in a number of ways. Furthermore, the effect of these resources may be especially salient during times of change-a no- tion supported by a large body of research on student learning that demonstrates how

prior knowledge can influence subsequent action and performance. This possibility is examined in the four case studies reported in this issue. Thus, the case studies focus on teachers' existing knowledge, skills, and beliefs-their "prior knowledge"-and how those may have shaped their response to calls for change in classroom practice. We observed and interviewed the teachers we describe as part of a larger study examining relationships between state policies and classroom practice (Cohen et al., 1990). The state of California was selected for study because of the concerted effort underway within that state to change instruction so that it is more consistent with the ideas presented by a number of educational reformers.

The focus of this research is mathematics teaching, in part due to our interests as researchers and in part because of the prom- inent role that mathematics educators have

played in the overall reform movement. These educators have argued that the traditional approach to mathematics must be replaced by an approach that emphasizes "teaching for understanding." To facilitate this change, an organization of mathematics educators-the National Council of Teach- ers of Mathematics (NCTM, 1989)-has specified curriculum standards in mathematics that would change the nature of

NOVEMBER 1992

CASE STUDIES 147

what is taught and learned in most classrooms. These standards are based on the

following assumptions about good mathematics instruction: First, the instruction should be conceptually oriented, which is to say, it should focus on the important ideas that help define various mathematical

approaches to situations-emphasizing, in

particular, the important relationships between these ideas. Second, mathematics instruction should actively involve children in the doing of mathematics, encouraging all students to explore, discuss, and apply ideas. Third, the mathematics curriculum should emphasize mathematical thinking and reasoning in students as opposed to the

prevailing focus on correct answers. Finally, the authors of the NCTM document argue that problem solving should permeate the

elementary school curriculum. The vision of mathematics instruction

outlined in the NCTM document is at var- iance with what one is likely to observe in most classrooms. Studies indicate that individual seatwork predominates in elemen-

tary mathematics, consuming approxi- mately 75% of the time typically allotted to this subject (Denham & Lieberman, 1980). Furthermore, the content is primarily low level (Peterson, 1988). As Stodolsky (1988, p. 7) explains, "Arithmetic is the heart of the mathematics curriculum in American el-

ementary schools, and skill in computation is the primary goal.... While many math educators are proponents of problem solv-

ing and analysis, most instruction is geared to algorithmic learning." The focus in early mathematics continues to be on eighteenth- century arithmetic, which puts it at odds with the sort of practice called for by reform-minded mathematics educators. Thus, if teachers are to move toward a more dis- cussion- and problem-oriented approach to teaching, they are going to have to alter instruction in important ways. Further com- plicating the picture, according to many reformers, is the fact that teachers must also change the ways they think about teaching

and learning. This last point needs further elaboration.

One important difference between past and present reform efforts relates to the scope of the changes called for in education.

Today, many reformers insist that teachers do more than change the way they teach.

They are also expected to change their views about a number of things, including students, the teaching-learning process, and the nature of subject-matter knowledge (Prawat, 1992). The basis for this argument is twofold. First, there is a close connection between a teacher's practice and how he or she views that practice (see above). Ac-

ceptance of this claim can be attributed, in

large part, to the cognitive revolution in

psychology. Second, and relatedly, there has been a change in the way many reformers view teachers; instead of being seen as passive receivers of innovation-the as- sumption behind earlier attempts to de-

velop "teacher-proof curricula"-teachers

increasingly are being viewed as important agents of change (McDonald, 1988). This explains the growth in rhetoric-if not ac- tual practice-aimed at getting teachers to function less as instruments of policy and more as collaborators in the reform movement. All of this adds a unique dimension to the current effort.

According to reformers, there is another, more important reason why teachers must change their views about teaching and learning. The impetus of reform itself has been attributed to an important change in viewpoint: Thus, it is asserted, teaching for understanding involves a dramatic change in focus, putting students' own efforts to understand at the center of the educational enterprise. In such a focus, the emphasis is less on the production of correct responses and more on the reasoning that underlies responses. This call for a change in viewpoint distinguishes the current reform movements from the last major educational reform effort 30 years ago. At that time, most disagreements in educational philos- ophy were handled by variations on a fa-


miliar theme: Curriculum reform emphasized either subject-centered intellectual objectives (i.e., "teach the discipline") or more general, child-centered competencies. Reforms based on the latter objectives were aimed at enhancing individual development (i.e., inquiry and decision-making processes, especially in science and social studies) (Prawat, 1992).

Regardless of the particulars, reformers agree that achieving the desired goals in education will place greater demands on teachers and students alike. According to Cohen (1988, p. 255), "Teachers who take this path must work harder, concentrate more, and embrace larger pedagogical re- sponsibilities than if they only assigned text chapters and seatwork." He adds that students must also assume more responsibility for what happens in the classroom: "It is, after all, their ideas, explanations, and other encounters with the material that become the subject matter of the class" (1988, p. 256). Changing practice in this way will likely require new kinds of knowledge and skill on the part of teachers and students.

The four cases presented in the articles that follow examine some of the issues raised to this point. The teachers described in these cases are attempting to come to terms with the latest calls for change in mathematics instruction. For the most part, they are doing this on their own and in a domain (i.e., mathematics) that many individuals regard as challenging, if not daunting. All of this should be kept in mind as important contextual information when reading the cases. Additional contextual information is presented in the next section of this introductory article. In the final section, we describe our research methods.

Mathematics Reform in California In 1985, California adopted a new Mathe- matics Framework (California State Depart- ment of Education, 1985). Although the vision of mathematics teaching and learning presented in this document was consistent with views emerging elsewhere, the use of

the document as an instrument for change is noteworthy and unique. Unlike earlier frameworks, which were educative or ad- visory, the 1985 document was assigned a more important function. It served as a standard against which mathematics textbooks were to be judged. Given the Framework's commitment to "teaching for understanding" in mathematics, it is not surprising that standard textbooks were found wanting.

Needless to say, the rejection of these texts captured the attention of publishers. Faced with the possibility of being removed from the state-approved list, several textbook companies altered their standard texts, producing "California editions" that incor- porated some-but not all-of the desired new content and methods (i.e., more emphasis on problem solving, estimation, the use of manipulatives in teaching concepts and procedures, etc.). The Framework is also being used to revise the state achievement tests. The effects of these changes, however, have yet to be felt by schools and districts.

California's Mathematics Framework is but one example of the nationwide effort to reform mathematics education. Like the NCTM document, it calls for fundamental change in what teachers are being asked to teach (e.g., increase the amount of problem solving and estimation) and how they are to teach it (e.g., provide more opportunity for questioning and responding). Like all such documents, the Framework can be interpreted in various ways. It is not surprising that those who have studied this document have carried away differing views about what it means based on their own experience, knowledge, and beliefs. To further complicate the issue in the present study, few if any of the teachers we visited had examined the document. For the most part, their sense of what the document had to say was based on indirect information ob- tained either through the district's newly adopted textbooks or through various district or school in-service activities. The latter may or may not have been designed with the Framework explicitly in mind. Thus, the

NOVEMBER 1992

CASE STUDIES 149

process of change described in our cases, at least as it relates to the Framework, is one in which individual interpretation plays an

especially important role. The sense that teachers make of this particular call for

change undoubtedly has been heavily influenced by their prior knowledge and beliefs. By examining how the knowledge and beliefs of the teachers in our cases interacted with their practice and with various mes-

sages for change, we hope to learn more about the nature of these relationships and, ultimately, about how to assist other teachers to change their teaching.

Method Three varied school districts were selected for the study: a large urban district, a moderate-sized suburban district in a metropol- itan area, and a smaller district adjoining the urban district. The two larger districts were considered by state informants to be actively involved in changing mathematics teaching in elementary schools, whereas the smaller district was described as taking a more traditional approach to mathematics education. Two schools were selected from each district-one serving a population of

predominantly low socioeconomic status (SES) students and the other students of moderate to high SES. Within each of the six schools, we focused on the classrooms of two second-grade and two fifth-grade teachers. Teachers were selected to ensure a range of experience and availability for study at each of the two prearranged times. (Many schools in California operate on a year-round schedule, with only three- fourths of the students and teachers present at any given time.) By limiting our sample of 24 teachers to two grades, we were able to limit the range of mathematical topics we were likely to encounter, allowing compar- isons across teachers at the same grade while still permitting contrasts of lower- and upper-elementary mathematics teaching. The subset of four cases presented in this volume was selected because it illus- trates and helps elucidate the complex na-

ture of the relationships between teachers' knowledge and beliefs and their responses to various calls for change in mathematic education.

We visited each teacher twice during the 1988-1989 school year, the first year teachers were using the newly adopted mathematics textbooks, spending a week in each district in December 1988, and again in March 1989. During each visit, teachers were observed twice during the times nor- mally set side for mathematics; the duration of their lessons ranged from 30 to 90 min- utes. Teachers were interviewed on a range of issues. The interviews, which averaged 3 hours, probed teachers' familiarity with the Mathematics Framework and related state policies, as well as their interpretation, opinion, and use of the newly adopted textbooks. Our purpose was to learn about the teachers' experiences in the midst of the statewide initiative and to gain an under-

standing of their perspectives on teaching mathematics for understanding. We in- quired into teachers' goals for mathematics

teaching, how they viewed teaching that subject for understanding, how they as- sessed student learning, and other factors

influencing their mathematics teaching. We also asked questions designed to assess teachers' beliefs about, and understandings of, mathematics, teaching, learning, and students. The format was often similar to that used by researchers in the National Center for Research in Teacher Education (NCRTE, 1989) in that questions involved hypothetical situations consistent with each teacher's grade and topic domains. Thus, we asked second-grade teachers about subtraction with regrouping, fifth-grade teachers about the addition and subtraction of fractions. The questions for each topic included: What do you think is important for students to learn about this? Are there things that kids find difficult in learning this? We also showed examples of students' correct and incorrect solutions to various problems and asked teachers to comment on what they would do if their students de-


veloped those solutions. To learn about teachers' thinking on a topic and their use of the textbook, we asked teachers to examine and assess a textbook lesson, ex-

plaining how they used or modified these

pages when they taught subtraction or fractions and why. Through these questions we tried to assess each teacher's knowledge in

ways close to his or her own practice. Before each observation, we conducted

brief preobservation interviews with teachers to find out what they were planning for the lesson and how it fit into previous work.

Additionally, we conducted extensive post- observation interviews that focused on what teachers were trying to teach, how and

why they did what they did, and what teachers thought the students got out of the lessons. We asked specific questions about decisions teachers made during the lesson and what influenced these decisions. The lessons were audiotaped, and each observer used the tape and detailed fieldnotes to con- struct a narrative summary and analysis of each lesson. These summaries included information on what students were saying and doing in the class, the mathematics topics being taught, the instructional representations teachers and students used, and a characterization of the major "parts" of the lesson.

Our lesson analyses were shaped by a series of analytic questions about various dimensions of mathematics instruction from a collaboratively developed observation guide. The questions increased the breadth of our observations, alerting us to dimensions that may have escaped analysis. The dimensions addressed by the analytic questions included the mathematical content of the lesson, the use of representations and tools, what the students were expected to do, and the nature of the discourse in the classroom. We tried to analyze the nature and character of factors related to each of these dimensions, using examples from the lesson to explicate and support our claims. For example, we considered the degree to which a teacher emphasized underlying

mathematical meanings in contrast to me- chanical procedures; who introduced the various representations and mathematical tools; how representations and tools were used and related to the content being taught; the kinds of discourse encouraged and the role that discourse played in instruction; the degree to which students were

given opportunities to reason, verbalize, justify, and explain their thinking in contrast to there being a press for right answers; and the "intellectual space" provided by the teacher and the mathematical tasks that were posed. The lesson narratives and anal-

yses served as summaries of the lessons that could be discussed with other researchers in the group. The issues that emerged from the first set of analyses informed our sub-

sequent inquiry. One of the most important issues to surface early related to the relation between the teaching we observed and what teachers knew and believed about that

teaching. Each of the four teachers whose cases are described in this issue reflects the

complex nature of this relationship. In addition to the procedures described

above, we also attempted to gather important contextual data by interviewing various district personnel involved in shaping the

elementary mathematics curriculum and how it is taught. Included were individuals

responsible for staff development, curriculum, and testing, some of whom worked

directly with teachers in their classrooms, and some of whom were several steps removed from the classroom (e.g., superintendents and assistant superintendents). We also interviewed state policymakers and individuals involved in the writing of the Mathematics Framework. Through these interviews we sought to understand how the Framework was being interpreted by state, district, and classroom personnel, in terms of both the kind of mathematics teaching it portrays and the role it plays in shaping mathematics education in the state, district, and classroom. These interviews were cru- cial in helping us understand the larger con- text that surrounded our case study teachers.

NOVEMBER 1992

CASE STUDIES 151

Our immediate focus in this set of cases, however, is on the four fifth-grade teachers we visited-their classroom practice, and how it was influenced by their knowledge and beliefs and the messages they had re- ceived about changing the way they taught. The cases should not be taken as complete portraits or analyses of these teachers. We did not spend enough time with the teachers to develop comprehensive cases of them as teachers. The stories of these teachers, however, do illustrate issues that we saw as

important for other teachers we visited. We also believe the cases are useful, more gen- erally, for any teachers making major changes in their teaching practice. We begin with Valerie Taft, a teacher who has not seen the Framework, but who is using a new textbook adopted by her district. (All names are pseudonyms.) Her instruction as she teaches about averages sheds light on her beliefs about mathematics and how it should be taught, as well as the difficulties she encounters because of limited knowl-

edge about certain aspects of that topic. Sandra Stein, in contrast, is quite familiar with the Framework. Her use of resources

designed to align teaching with the Frame- work highlights the difficulty of teaching mathematics for understanding and the

challenges faced by textbook developers and in-service program organizers whose intentions may be thwarted by teachers' limited understanding of the subject matter. Unlike Valerie and Sandra, Jim Green and Karen Hill work in districts where an em-

phasis on basic skills and standardized tests greatly influences their practice. Jim's belief that conceptual understandings are inher-

ently connected to the learning of procedures affects his practice and his interpretation of the reform effort. Karen, more so than the others, is cast as a teacher in the midst of change. She has made changes that seem, on the surface, to have altered her practice-but that also appear to have had little influence on her beliefs about the nature of mathematics and the learning process. In presenting these cases, we do not

intend to evaluate or judge the teaching we observed as good or bad; rather, we hope to illustrate the complexity of the teachers'

practice as it has emerged for us in our study of how teachers make sense of their teach-

ing in a climate of change.

Note

This research was sponsored in part by the Center for the Learning and Teaching of Ele- mentary Subjects (cooperative agreement G0098C0226) and the National Center for Re- search on Teacher Education (grant G008690001), Michigan State University, Col- lege of Education, and funded by the Office for Educational Research and Improvement, U.S. Department of Education (grant R117P80004). The opinions herein are those of the authors and do not reflect the position, policy, or endorse- ment of the office or the department. The con- versations of our research group, including Deb- orah Ball, David Cohen, Nancy Jennings, Penelope Peterson, and Suzanne Wilson, con- tributed to the thinking reflected in this article.

References

Ball, D. L. (1991). Research on teaching mathematics: Making subject matter part of the equation. In J. Brophy (Ed.), Advances in research on teaching: Vol. 2. Teachers' subject matter knowledge and classroom instruction (pp. 1-48). Greenwich, CT: JAI.

Brophy, J. (Ed.). (1991). Advances in research on teaching: Vol. 2. Teachers' subject matter knowledge and classroom instruction Green- wich, CT: JAI.

California State Department of Education. (1985). Mathematics curriculum framework for California public schools. Sacramento: Au- thor.

Clark, C. M., & Peterson, P. L. (1986). Teachers' thought process. In M.C. Wittrock (Ed.), Handbook of research on teaching (3d ed., pp. 255-296). New York: Macmillan.

Cohen, D. K. (1988). Educational technology and school organization. In R. S. Nickerson & P. P. Zodhiates (Eds.), Technology in education: Looking toward 2020 (pp. 231-264). Hillsdale, NJ: Erlbaum.

Cohen, D. K., Peterson, P. L., Wilson, S., Ball, D. L., Putnam, R., Prawat, R., Heaton, R.,


Remillard, J., & Wiemers, N. (1990). Effects of state-level reform of elementary school mathematics curriculum on classroom practice (Fi- nal report, U.S. Department of Education, OERI Grant No. R117P8004). East Lansing: Michigan State University, Center for the Learning and Teaching of Elementary Sub- jects and National Center for Research on Teacher Education.

Denham, C., & Lieberman, A. (1980). Time to learn. Washington, DC: U.S. Government Printing Office.

McDonald, J. P. (1988). The emergence of the teacher's voice: Implications for the new reform. Teachers College Record, 89, 471-486.

National Center for Research on Teacher Edu- cation. (1989). Study package: Tracking teachers' learning. East Lansing: Michigan State University, College of Education.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Peterson, P. L. (1988). Teaching for higher-order thinking in mathematics: The challenge for the next decade. In D. A. Grouws & T. J. Cooney (Eds.), Perspectives on research on effective mathematics teaching (Vol. 1, pp. 20- 26). Hillsdale, NJ: Lawrence Erlbaum Asso- ciates; Reston, VA: National Council of Teachers of Mathematics.

Peterson, P. L., Fennema, E., Carpenter, T. C., & Loef, M. (1989). Teachers' pedagogical content beliefs in mathematics. Cognition and In- struction, 6, 1-40.

Prawat, R. S. (1992). Teachers' beliefs about teaching and learning: A constructivist per- spective. American Journal of Education, 100(3), 354-395.

Roth, K. J. (1987, April). Helping science teachers change: The critical role of teachers' knowledge about science and science learning. Paper presented at the annual meeting of the Ameri- can Educational Research Association, Washington, DC.

Stein, M. K., Baxter, J. A., & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graph- ing. American Educational Research Journal, 27, 639-663.

Steinberg, R., Haymore, J., & Marks, R. (1985, April). Teachers' knowledge and structuring content in mathematics. Paper presented at the annual meeting of the American Edu- cational Research Association, Chicago.

Stodolsky, S. S. (1988). The subject matters: Classroom activity in math and social studies. Chicago: University of Chicago Press.

Thompson, A. G. (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127.

Thompson, A. G. (1985). Teachers' conceptions of mathematics and the teaching of problem solving. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 281-294). Hillsdale, NJ: Erlbaum.

NOVEMBER 1992

Documents

Teaching Mathematics for Understanding: Case Studies of Four Fifth-Grade Teachers