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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(Unpublished manuscript prepared for inclusion in the multi-media package "Introducing math teachers to inquiry: Framework and supporting materials to design professional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 recent literature on mathematics teacher education has identified a few "professional development practices" that have been effective in promoting reform informed by a constructivist perspective in school mathematics (e.g., Fennema & Nelson, 1997; Friel & Bright, 1997; Schifter & Fosnot, 1993). This essay will examine the nature and research basis of three of these professional development practices:
In addition, we will discuss another practice derived from our own professional development experiences, which we have characterized as:
We have selected to focus on these four professional development practices because we have come to consider their combination as very effective in promoting the changes in beliefs and practices called by an inquiry approach to mathematics instruction -- the focus of these materials. For each of these practices, in what follows we describe its characterizing elements, identify which "teachers' learning needs" it could potentially address, and examine theoretical and empirical results that can help us better understand the potential of such a practice. Since the value of these practices depends to a great extent on how they are used, some considerations to this regard will conclude the analysis of each practice.
This analysis was also informed by a preliminary identification of the needs that teachers engaged in school mathematics reform are likely to experience. This analysis is summarized in the companion essay entitled "Rationale and research evidence the learning needs of teachers engaging in school mathematics reform," which is also included in this CD-ROM. For the readers' convenience, the list of teachers' learning needs identified in this companion essay have been reproduced in Figure 1.
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Engaging participants in "experiences as learners".
A key element of several successful teacher education initiatives reported in the literature has been a set of experiences where the participating teachers become genuine "learners of mathematics" in situations intended to model a proposed instructional approach (e.g., Friel & Bright, 1997; Schifter & Fosnot, 1993).
Although these "experiences as learners" can obviously enhance the participants' own mathematical knowledge, they have been proposed in the literature primarily as an effective means to engage the participants in rethinking their conceptions of mathematics, learning and teaching (e.g., Brown, 1982; Simon & Schifter, 1991). Research has shown that teachers' mathematical and pedagogical beliefs are mostly formed as a result of their experiences as students in schools informed by a "transmission" approach (Thompson, 1992); therefore, personally experiencing alternative pedagogical approaches seems to be a pre-requisite to enable these teachers to challenge their existing beliefs and begin to consider alternative ones.
Simon's (1994) "Learning cycles" model sheds further light on the multiple roles that "experiences as learners" can play in a professional development or teacher preparation program. Building on key constructivist tenets, and Brousseau's (1987) didactical theory in particular, Simon suggests that teachers -- like any other learner -- can learn through cycles involving: (a) engaging in situations that engage them actively, raising cognitive dissonance and, thus, initiating new constructions of meaning; (b) sharing and discussing these constructions with a group, so as to come to some consensus and generalizations; and, (c) applying these generalizations to new situations, thus starting the learning cycle all over again at a higher level. This kind of cycle describes first of all how a good learning experience for teachers should be designed in order to enable them to learn some new mathematics most effectively. At the same time, Simon suggests that "experiences as learner" could also be considered as the catalyst for creating cognitive dissonance about the participants' views about math, learning and teaching, respectively. That is, by reflecting on these experiences as a group, participants can come to new understanding and generalizations about the nature of mathematics as a discipline, how people learn and what can best support such learning.
Further support for the value of engaging participants in "experiences as learners" also comes more indirectly from research on the learning of complex tasks. The model proposed by Collins and his colleagues (1989), in particular, identifies three key phases in this process:
"Experiences as learners" can play the role of modeling quite effectively when the task to be mastered is learning a novel approach to teaching mathematics. First, such experiences provide teachers with an opportunity to observe an expert (i.e., the teacher educator facilitating the experience) concretely demonstrate how mathematics can be taught in this way. Second, because teachers personally participate in this instructional experience in the role of learners, they are in a unique position to examine how their students may feel about the new approach and, thus, better evaluate the potential of the proposed approach and its possible outcomes.
Our analysis so far suggests that "experiences as learners" have the potential to meet several of the learning goals identified in Figure 1. This is further confirmed by empirical studies of their use in several successful professional development programs reported in the literature (see, in particular, Simon & Schifter, 1991).
Just because participants engage in an "experience as learners," however, does not automatically mean that such potential will always be realized. Rather, the power of these experiences to address specific learning goals will depend on the mathematical focus and content of these experiences, on the teaching practices modeled by the facilitator, and on the nature of the reflective sessions following them. Let us look more closely at what this means in the case of specific learning needs:
Examining video-tapes and stories of classroom instruction.
Narrative accounts, or "stories," of innovative classrooms experiences have a long tradition in education as both a concrete illustration of innovative pedagogical approaches and inspiration for teachers interested in improving their practices. Although this may have been less the case in mathematics than in other subject areas like literacy, we too have some notable examples of this kind of literature (e.g., Fawcett, 1938; Papy, 1970). More recently, video has attracted teacher educators' attention as an alternative media to convey images of alternative classroom instruction in an even more immediate way. At the same time, there is not much in the teacher education literature about how stories and videos could be used in professional development programs, nor about the effects of using these resources, as recently stated by Deborah Ball:
My experience with watching educators (and others) watch, talk about, and refer to videotapes of classroom lessons suggest that, despite the widespread enthusiasm for the medium, we know little about what people attend to and learn while watching tape. ... What can be learned from videotapes, under what kinds of circumstances, is worth investigation much more closely. (Ball, 1997, p. 98)
Despite this important disclaimer, stories and videotapes of classroom instruction seem to have at least the potential to play important roles in a professional development program introducing teachers to a novel pedagogical approach.
First of all, video-tapes and stories of instructional situations where the proposed approach was used can help participants develop images of what such an approach would look like in practice -- one of the key "learning goals" identified in Figure 1. Although "experiences as learners" could also contribute to developing such images, as discussed earlier, teachers need to see the same approach in action in instructional situations that are similar to their own classrooms (rather than involving a group of motivated adults):
Teachers who have never seen children engaged in a mathematical problem, or discussing mathematics, need to have opportunities to see what this can look like. These serve, in part, as existence proofs that such practice can happen in school. (Ball, 1997, p.97).
Detailed accounts of classroom instruction can also provide additional "models" of the proposed pedagogical approach as well as of specific teaching strategies. Furthermore, they can contribute to challenging the participants' views of mathematics, teaching and learning as a result of viewing/reading about alternative ways to approach the teaching of specific math topics and to organize classroom instruction.
Another valuable use of stories and video-tapes in a professional development program has to do with the goal of promoting teachers' understanding of the mathematical knowledge and thinking processes of their students. Video-tapes of students' problem solving sessions and clinical interviews have already proved to be a valuable means for examining students' mathematical thinking in several programs (e.g., Simon & Schifter, 1991; Carpenter & Fennema, 1992). In-depth accounts of classroom instruction could provide similar opportunities in an even more "natural context," provided that teacher educators select appropriate video clips to engage in this kind of analysis with the participating teachers.
As discussed in the case of the "experiences as learners," however, the potentials of using videos and stories identified so far can be fully realized only when these accounts of classroom instruction are sufficiently rich and thought-provoking and, furthermore, the participants engage in explicit and focused analyses of these experiences.
Engaging participants in supported "experiences as teachers."
Another key element characteristic of successful programs working towards instructional innovation and/or school reform in mathematics, are field experiences where the participating teachers are expected to put into practice what they learned in Summer Institutes or workshops (e.g., Friel & Bright, 1997; Schifter & Fosnot, 1993).
The value of having "supported experiences as teachers" has been suggested by research conducted in several areas. First, studies on teachers' beliefs have pointed out that the relationship between pedagogical beliefs and practices is not unidirectional (Thompson, 1992). Although it is rather obvious that teachers' beliefs will inform their practices, it has also been shown that experiencing "alternative practices" can be a powerful way for teachers to challenge existing beliefs -- especially as a result of witnessing how their own students demonstrate a higher level of learning and thinking than they had thought possible before! The importance of engaging in field experiences that involve both the planning of innovative instruction and the implementation of these lessons, and then reflecting explicitly on these experiences, is also emphasized in Simon's proposed model of teacher development. Doing so represents the last two "cycles" of his "Learning Cycles" framework (Simon, 1994).
At the same time, putting into practice novel instructional techniques or approaches to teaching mathematics presents a considerable challenge for most teachers, and many may fail in their first attempts towards instructional innovation unless they are appropriately supported. This consideration is consistent with the role given to "scaffolded practice" in the model suggested by Collins and his colleagues (1989) for the learning of complex task (as described earlier in our discussion of "experiences as learners").
While there is consensus on the need for "supported experiences as teachers," the mathematics education community is continuing to experiment with different models to provide appropriate as well as cost-effective scaffolds. The following practices, however, have found considerable support in the literature and have been widely employed in the last decade or so:
It is worth noting that most of these practices emphasize the need for participants to reflect on their classroom experiences, and to do so with others (experts and/or peers) -- thus meeting not only intellectual but also affective needs. All the practices we have mentioned in the previous list are also quite demanding in terms of time and energy of both participants and teacher educators -- in fact, their effectiveness seems directly proportional to the time devoted to them. Unfortunately, there have been few teacher enhancement projects so far that that could provide the personnel and funding needed to implement these practices as intended.
Using "illustrative units" as a common context for experiences as learners, accounts of innovative instruction and experiences as teachers
While building on the previous three practices, this last professional development practice was derived from our own experiences as teacher educators. As such, rather than relying on the existing mathematics education literature, our analysis of this practice will be based on the results of our own conceptual analysis and empirical work.
We have used the term "illustrative unit" to indicate a long-term instructional experience designed to provide a concrete illustration of the teaching approach(es) promoted by a professional development program, as well as a scaffold for the participants' first "experience as teachers" with such an approach. Participants are expected to commit to adapt and implement the illustrative unit in their classes as part of the program field experiences. Segments of the unit (appropriately modified so as to present a genuine learning experience for adult learners) provide the basis for prior "experiences as learners" taking place in the context of workshops preceding these field experiences. Detailed stories and/or videos of classroom implementations of the unit are then examined in these workshops. The participants' teaching of the illustrative unit is further scaffolded by means of instructional materials specially designed to support its planning and implementation.
Participants' feedback and external reviewers' evaluations have confirmed our own observations that providing an integrated context for the participants' "experiences as learners," "experiences as teachers," and use of classroom videos and stories, can further enhance the value of each of these separate practices (see Borasi, Fonzi, Smith and Rose, in press).
First of all, we believe that participants can benefit from "experiences as learners" that are not just limited to short activities, but rather reflect the complexity of an entire instructional unit -- involving a set of tasks that need to be carefully orchestrated and connected, homework assignments, and final reflections. Knowing that they are expected to teach a similar unit with their students later may increase teachers' motivation in participating in such a long term learning experience as part of a professional development program. Indeed, we heard very few complaints about the lengths of our "experiences as learners" from our participants, and they all took a very active role in these activities.
Having prior "experiences as learners" of a unit can greatly scaffold the teaching of a similar unit, as demonstrated in our programs. Several of our participants mentioned how their "experiences as learners" on Tessellation and Area greatly increased their confidence as they moved to plan and teach their first "inquiry unit", by providing not only a vivid "image" of the unit's contents and structure, but also a personal understanding of what their students might think and feel while engaging in it. School facilitators repeatedly noticed that teachers relied heavily on their memories of what they had experienced in the Summer Institute when planning their own implementations of our illustrative units. This preliminary "experience as learners" also often contributed to the participants' greater appreciation of their students' ability to think and problem solve -- as they were often surprised to find that the students in their classes could generate the very same issues and solutions as the adults in the Summer Institute.
The participants' teaching of an illustrative unit can be further enhanced by the opportunity to examine implementations of that same unit in instructional experiences more similar to their own classrooms. The stories or videos our participants examined did much to dispel their doubt that the units they had experienced as learners would work only with a responsible adult audience. By showing how the unit was implemented with school-age students, these accounts also illustrated some of the adaptations necessary for the success of the unit with younger learners.
There are yet other advantages of using illustrative units as the focus of the participants' "experience as teachers" -- especially when it is the first time the participants are asked to implement a new instructional approach. Instructional materials can be specifically created to support the planning and implementation of these units by "novice" teachers; in addition to various accounts of implementations of these units in different contexts, these materials may provide teachers with an articulation of the main goals of the unit and their rationale, possible assessment tools, and even some sample lessons plans and hand-outs that could be directly used or modified for the teachers' own class. Our participants were very appreciative of the support provided by these materials, as captured by the following quote:
"You did not have to create this whole thing from scratch, which was helpful too, because what you were wrestling with is a new approach. So, some of the materials were in hand at least. You don't have to start and create every single worksheet, every single project, every single diagram."
Having all the participants working on a few common units for their field experiences also presents considerable advantages for the teacher educator trying to support these experiences. First, being intimately familiar with the nature of the units being implemented, the teacher educators facilitating our programs were more effective in providing ideas and feedback in the planning stage as well as after classroom visits. Groups of teachers could also more easily collaborate on the planning of their units, and be supervised by the teacher educator in this crucial activity. We also found that sharing sessions on the field experiences were more easily orchestrated, and more meaningful for the participants, because everyone was familiar with the nature of the units implemented and thus could better relate to the experiences, results and feelings being shared by other teachers.
In sum, using illustrative units as an integrated context for experiences as learners, accounts of innovative classroom instruction and experiences as teachers has the potential to address all the learning needs of these three practices combined -- that is, enable teachers to learn more about specific mathematical content, the nature of mathematics, theories of learning and teaching, students' mathematical thinking, effective teaching strategies, as well as developing a need for changing their teaching practices, attending to the emotional aspects of engaging in instructional innovation and developing images of alternative classroom instruction. At the same time, it is also important to remember that realizing this potential will depend on the quality of the unit itself as well as the materials and experiences developed around it.
A good "illustrative unit" needs not only to provide a good illustration of the proposed instructional approach, but also to be sufficiently open-ended to be interesting and yet accessible for both students and adult learners. Furthermore, the instructional materials -- including stories and/or videos -- accompanying the unit have to be carefully developed in terms of both content and format. Finally, all that was said before about what it takes to develop good "experiences as learners," uses of stories and videos, and "experiences as teachers," continues to hold here as a condition for the successful use of "illustrative units."
Summary
In sum, we hope that this essay has contributed to clarify the rationale behind the practices required to implement the framework proposed in these materials. We hope that this, in turn, will also contribute to promote a better understanding of the role and rationale of the specific activities described in Section D of these materials.
References
Ball, D. L. (1997). Developing mathematics reform: What don't we know about teacher learning - but would make good working hypotheses. In S. N. Friel & G. W. Bright (Eds.), Reflecting on our work: NSF teacher enhancement in K-6 mathematics (pp. 77 - 111). NY: University Press of America.
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.
Brousseau, G . (1987). Fondements et methodes de la didactique de la mathematique. Recherches en Didactique de la Mathematique, 7, 33-115.
Brown, S. I. (1982). On humanistic alternatives on the practice of teacher education. Journal of Research and Development in Education, 15(4), 1 - 12.
Carpenter, T. & Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. International Journal of Educational Research, 17, 457-470.
Collins, A., Brown, J. S., & Newman, S. E. (1989). Cognitive apprenticeship: Teaching the craft of reading, writing and mathematics. In L. B. Resnick (Ed.), Knowing, learning and instruction : Essays in honor of Robert Glaser (pp. 453-494). Hillsdale, NJ: Erlbaum.
Fawcett, H. P. (1938). The nature of proof. The 1938 Yearbook of the National Council of Teachers of Mathematics. NY: Columbia University Teachers College Bureau of Publications.
Fennema, E. & Nelson, B. S. (Eds.) (1997). Mathematics teachers in transition. Mahwah, NJ: Erlbaum.
Friel, S. N. & Bright, G. W. (Eds.). (1997) Reflecting on our work: NSF teacher enhancement in K-6 mathematics. NY: University Press of America.
Papy, F. (1970-72). Les enfants et la mathematique. Brussels: Hachette.
Schifter, D. & Fosnot, C. T. (1993). Reconstructing mathematics education: Stories of teachers meeting the challenge of reform. New York: Teachers College Press.
Simon, M. (1994). Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics 26.
Simon, M. & Schifter, D. (1991). Towards a constructivist perspective: An intervention study of mathematics teacher development. Educational Studies in Mathematics 22.
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.
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