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Additional essays on mathematics teacher education (C2)
(Unpublished manuscript prepared for inclusion in the multi-media package "Introducing math teachers to inquiry: Framework and supporting materials to design professional development" [Borasi & Fonzi, 1998]; this essay was based on an invited talk entitled "Understanding teacher change towards mathematics reform: Research results and working hypotheses", given by the author to National Science Foundation program officers in Arlington (VA), May 1997.)
Introduction
The design of any professional development initiative intended to promote school mathematics reform must take into consideration the needs that teachers engaging in school mathematics reform are likely to experience. These needs should determine the goals of the professional development and, thus, the choices teacher educators make to most effectively achieve these goals. Although a review of the mathematics teacher education literature did not uncover a widely-accepted "compendium" of these teachers' learning needs, it revealed a high level of consensus about the recognition that teachers need to:
In what follows, I will examine each of these needs separately, with the goals of better articulating what it means, making explicit its research base, and briefly noting its implications for the design of professional development. This analysis will build not only on the literature on mathematics teacher education, but also on contributions coming from research on teacher education, teachers' beliefs, and learning, more generally.
Teachers' need to learn more mathematics
Shulman's research identified "subject matter knowledge" and "pedagogical content knowledge" as key variables influencing teachers' decisions and behavior in the classroom:
"Prior subject matter and background in a content area affect the ways in which teachers select and structure content for teaching, choose activities and assignments for students, and use textbook and other curriculum materials." (Shulman & Gross man, 1988, p.12).
Thus, developing teachers' mathematical background should be deemed a desirable goal of any professional development program. However, programs trying to respond to the most recent calls for school mathematics reform need to address quite different mathematics than has traditionally been the case. The sought-after reform is informed by different instructional goals that include not only areas of mathematics that have so far been neglected in the traditional curriculum, but also a new emphasis on the understanding of mathematical "big ideas" and on developing "ways of thinking" characteristic of mathematics. These are aspects of mathematics learning that most teachers (even those certified to teach secondary mathematics) have not sufficiently developed in their previous mathematical training -- as suggested by the research studies on teachers' knowledge of mathematics reported in Fennema & Franke's (1992) review of research in this area.
Teachers' need to learn about the nature of mathematics as a discipline
Research on teachers' beliefs (as reviewed in Thompson, 1992) has also noted the impact on curriculum decisions and instructional practices of the views that a teacher holds about: the nature of mathematics as a discipline; what constitute legitimate math procedures, results and justifications; what are desirable goals for school mathematics and acceptable outcomes of math instruction.
These views are rarely made explicit, as readings and discussions about mathematics as a discipline are notably absent from school mathematics and even college-level mathematics courses. Yet, most teachers develop strongly held beliefs about mathematics as a body of absolute truths, with little room for creativity or personal judgment, as a result of their experiences as students in traditional mathematics classes. This means that, as teachers, they are likely to value right answers over tentative conjectures, the correct application of standard procedures over developing personal approaches to solutions, and learning a body of facts and algorithms over improving problem solving and reasoning skills.
Since these views conflict with those informing the most recent calls for school mathematics reform (as argued, for example, in Borasi [1996]), professional development programs designed to promote such reform will have to provide the participants with opportunities to critically examine their views of mathematics as a discipline and offer alternative perspectives.
Teachers' need to learn about theories of teaching and learning
Similarly, research has shown that most math teachers (and even perspective teachers!) have strongly-held beliefs about teaching and learning, which translates more concretely in views about: student and teacher's roles; desirable instructional approaches; students' math knowledge; how students learn; the role and purposes of schools (Thompson, 1992). These beliefs have once again developed as a result of their own experiences within the traditional "transmission paradigm," which is informed (however implicitly!) by the following views of knowledge, learning and teaching:
Once again, we can expect that in most cases this set of beliefs will need to be critically examined and challenged in order to enable teachers to put into practice recommendations about mathematics instruction such as those articulated in the NCTM Standards, which are based on views of knowledge, learning and teaching informed by a constructivist perspective (e.g., NCTM, 1989, 1991). In order to do so, teachers will need to be exposed to these alternative theories and examine the research supporting them.
Teachers' need to learn about students' mathematical thinking
Research on Cognitive Guided Instruction (CGI) has provided both theoretical arguments and empirical evidence in support of the fact that math teachers should know more about the mathematical knowledge their students have, and the learning processes they use, with respect to specific mathematical contents (Carpenter & Fennema, 1992). For example, realizing that children often develop their own procedures to solve simple arithmetic problems before they enter school can be an eye-opener for many elementary teachers, and challenge how arithmetic operations are currently introduced. Furthermore, knowing what these child-constructed procedures actually are is crucial to develop instructional experiences that capitalize -- rather than override -- the informal mathematical knowledge children come to school with.
Learning about students' mathematical thinking is especially important within any constructivist-based instructional approach, as such knowledge is necessary if teachers are expected to design instructional experiences that help students build on their existing knowledge (e.g., Confrey, 1991). Thus teachers will need to engage in experiences designed to teach them how to uncover students' thinking.
Teachers' need to develop images of alternative classroom instruction
Developing an understanding of the kind of mathematics, teaching and learning that is promoted by a novel pedagogical approach is certainly necessary, but not sufficient, for teachers to see how this approach would translate into classroom practice. Rather, teachers will also need to develop images of how instruction would look in mathematics classrooms informed by these alternative views -- that is, what would be typical activities and tasks, what kind of learning environment and classroom practices are established, and what are students' reactions to such tasks and practices.
This is especially the case when the instructional innovation considered represents a radical departure from the traditional way of teaching mathematics. Indeed, this need has been identified in several studies reporting on professional development initiatives aimed at promoting school mathematics reform (e.g., Borasi, Fonzi, Smith & Rose, in press; Friel & Bright, 1997). Professional development programs will thus need to include detailed stories and videotapes, or actual classroom observations, in order to provide such images.
Teachers' need to become familiar with and adopting effective teaching practices
The recommendations articulated in the NCTM Standards (NCTM, 1991) call for teaching practices -- such as cooperative learning or "writing to learn" (see Koehler & Grouws [1992] for a comprehensive list) -- that are not currently used by many mathematics teachers, especially at the secondary level. In fact, some mathematics teachers may only know about the existence of the more popular among these teaching practices! To be empowered to effectively use these practices in their classrooms, the literature suggests that teachers should: become aware of as many as possible of these strategies, so as to develop a rich repertoire they could refer to when designing learning experiences for their students; learn how to evaluate the strengths and limitations of each of these practices, especially with respect to their own instructional goals and audiences; personally experience these teaching practices in order to fully evaluate their pedagogical potential and implications.
Teachers' need to attend to the emotional aspects of engaging in instructional innovation
Several reform projects have identified the important role played by the emotions -- both positive and negative -- that inevitably accompany efforts at changing one's teaching practices (e.g., Clarke, 1994; Ferrini-Mundi, 1997). This should not come as a surprise, since several studies of learning and problem solving have led us to realize that learning and change most often involve strong feelings of stress and frustration, as well as accomplishment (e.g., McLeod, 1992). Professional development programs should recognize and explicitly address these affective elements. Indeed, Weissglass has suggested that "any reform that does not provide methods for people to systematically and profoundly address their feelings, emotions and values related to reform will be inadequate" (Weissglass, 1993, p. 3).
Teachers' need to develop ownership of reform goals and agendas
Finally, research on professional development efforts has also shown that program outcomes, and the extent of teacher change in particular, correlates with the level of teacher participation and effort in the program, which in turn depends on individual teachers' identification with the reform goals and agendas (e.g., Clarke, 1994; Loucks-Horsley, 1997). This should not come as a surprise. Given the time, energy and emotional stress involved in instructional innovation, it is not likely that teachers will engage in it unless they personally feel the need to improve their teaching practices.
Yet, this need for ownership needs to be balanced with the recognition that initially the participating teachers may have a "potentially limited" vision of their needs and goals in terms of instructional innovation (Ferrini-Mundi, 1997). Thus, a professional development program should strive not so much to meet the participants' perceived needs, but rather to create a genuine need for the reform the program is trying to promote, while at the same time developing an agenda that takes into consideration the participants' perceived needs and actual constraints.
Summary
Although this list of "teachers' learning needs" may not be exhaustive, we believe that it has identified several complementary dimensions that teacher educators need to consider when designing professional development intended to promote school mathematics reform. At the same time, we expect that different subsets of these needs will be emphasized in each specific program, depending on its goals and audience.
References
Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Norwood, NJ: Ablex.
Borasi, R., Fonzi, J., Smith, C., & Rose, B. (in press). Beginning the process of rethinking mathematics instruction: A professional development program. Journal of Mathematics Teacher Education.
Borasi, R. & Siegel, M. (1992). Reading, writing and mathematics: Rethinking the basics and their relationship. Subplenary lecture delivered at the International Congress in Mathematics Education, Quebec City, Quebec, Canada.
Carpenter, T. & Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. International Journal of Educational Research, 17, 457-470.
Clarke, D. (1994). Ten key principles from research for the professional development of mathematics teachers. In D. B. Aichele & A. F. Coxford (Eds.), NCTM Yearbook, Professional development for teachers of mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Confrey, J. (1991). Learning to listen: A student's understanding of powers of ten. In E.von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 111-138). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Fennema, E. & Franke, M. L. (1992). Teachers' knowledge and its impact. In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147-164). NY: Macmillan Publishing Co.
Ferrini-Mundi, J. (1997). Reform efforts in mathematics education: Reckoning with reality. In S. N. Friel & G. W. Bright (Eds.), Reflecting on our work: NSF teacher enhancement in K-6 mathematics (pp. 113 - 132). NY: University Press of America.
Friel, S. N. & Bright, G. W. (Eds.). (1997) Reflecting on our work: NSF teacher enhancement in K-6 mathematics. NY: University Press of America.
Koehler, M. & Grouws, D. A. (1992). Mathematics teaching practices and their effects. In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 115-126). NY: Macmillan Publishing Co.
Loucks-Horsley, S. (1997). Teacher change, staff development, and systemic change: Reflections in the eye of a paradigm shift. In S. N. Friel & G. W. Bright (Eds.), Reflecting on our work: NSF teacher enhancement in K-6 mathematics (pp. 133 -150). NY: University Press of America.
McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575-596). NY: Macmillan Publishing Co.
National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (NCTM). (1991). Professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.
Shulman, L.S. & Grossman, P.L. (1988). Knowledge growth in teaching: A final report to the Spencer Foundation. Stanford, CA: Stanford University.
Thompson, Alba. (1992). Teachers' beliefs and conceptions: A synthesis of research. . In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127-146). NY: Macmillan Publishing Co.
Weissglass, J. (1993). The social and psychological dimensions of educational change. Santa Barbara, CA: University of California, Center for Educational Change in Mathematics and Science.
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