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Strengthening one’s knowledge of mathematics. Experiences-as-learners are ideal for strengthening teachers’ knowledge of mathematics. However, the nature and extent of this learning depends on the duration and design of the learning experience. For example, immersion experiences (as shown in Illustration 2) expose teachers to mathematical tools and applications used in business, not in the traditional school curriculum. By seeing what mathematical knowledge and skills are really needed to solve real-life problems, teachers may begin to question what their students should learn. Consequently, they may rethink the goals of the mathematics courses they teach.
Teachers can also learn something new about topics that are currently in the K-12 curriculum, as shown in the area inquiry in Illustration 1. There are many benefits to doing so, since teachers – even those who have taken several college-level mathematics courses – often lack the deep conceptual understanding of mathematical topics in the K-12 curriculum that are necessary to implement reform lessons. As reported earlier, several teachers in the inquiry on area had never questioned the significance of using squares as units when measuring area, nor had they really understood what area formulas are or where they come from. However, the mathematical insights these teachers gained might not have been achieved at the same level without the reflection and discussions that followed the learning experience itself. Follow-up reflective discussions, such as the “What I have learned” analysis that followed the inquiry on area, are critical to challenge participants’ views of mathematics as a discipline and their perceptions of themselves (and their students) as learners of mathematics.
Understanding the pedagogical theories that underlie school mathematics reform. Experiencing mathematics as learners has also the potential to help teachers understand better the pedagogical theories that inform current reform efforts. As Simon’s (1994) model of learning cycles suggests, this kind of professional development activity not only provides an experiential basis for new learning approaches but also stimulates teachers to reflect on, and inquire further about, the theories of learning and teaching on which these approaches are based. To ensure a thorough understanding of learning theories, however, personal reflections need to be augmented by specially designed follow-up readings and/or presentations, something that was missing in our illustrations.
Understanding students’ mathematical thinking. Because experiences-as-learners focus on the teachers’ learning, they are not an ideal vehicle to pursue an understanding of students’ learning and thinking processes. However, these experiences do help teachers become aware of their own – and other adults’ – mathematical thinking and problem-solving strategies. This awareness can be eye-opening for many teachers, and it can inspire them to examine their students’ thinking in the future.
Learning to use effective teaching and assessment strategies. Experiences-as-learners are especially appropriate for modeling effective teaching practices, at least when the facilitator has the expertise to do so. As we argued earlier, modeling is a critical part of learning complex tasks (Collins, Brown, & Newman, 1989). To be most effective, modeling should not stop with the expert performing the novel task in front of the novice. Rather, it should be accompanied by explicit reflection on the teaching practice that was demonstrated so that participants can recognize and internalize its key elements. We believe, therefore, that a focused follow-up reflective session is necessary to help teachers identify the teaching practices modeled and to analyze the implications for mathematics instruction (as shown in Illustration 1).
Becoming familiar with exemplary instructional materials and resources. Depending on the content of the mathematical learning experience, experiences-as-learners may or may not help participants become familiar with exemplary instructional materials and resources. Teacher educators who want to introduce participants to an exemplary curriculum series or to a replacement unit that teachers will be expected to implement later in their classes need to select mathematical tasks from these materials and adapt them for an adult audience. This is what happened in the inquiry on area we featured in Illustration 1, and it is a practice used in many projects designed to support the implementation of NSF-funded curricula.
Understanding equity issues and their implications for the classroom. By doing mathematics in a group, teachers are inescapably confronted with the diversity in learning styles and approaches that exist. This is especially the case, though, when the mathematical task is open-ended and there are opportunities to share different solution processes. The experience can be especially powerful when the group is highly diverse and the implications of the differences are addressed explicitly. However, it is our experience that given an appropriate mathematical task, any group of learners will produce enough diversity in responses to begin a conversation. Facilitated experiences-as-learners are also ideal for modeling strategies for differentiated instruction based on diverse learning needs and, then, discussing participants’ reactions to these strategies.
Coping with the emotional aspects of engaging in reform. Coping with the emotional aspects of engaging in reform is not a central goal of engaging teachers in experiences as learners of mathematics. Nevertheless, using this kind of professional development experience early in a program can be instrumental in creating a bond among participants and engendering a “community of learners” that can offer emotional support as the participants undertake instructional innovation in their classrooms later on. It is also important to recognize that for some elementary and special education teachers just engaging as learners in a mathematical task may evoke painful memories of failure and raise anxiety levels. Acknowledging and addressing these feelings within the context of an experience as learners may help these teachers overcome their fears, thus mitigating emotional obstacles to their individual efforts at instructional innovation later on.
Developing an attitude of inquiry toward one’s instructional practice. As teachers critically analyze the experience they participated in as learners, they begin to appreciate the power of reflecting on instructional practice. These reflective sessions can also model ways for teachers to structure their own reflections to make the process more productive. Therefore, experiences-as-learners can be valuable in addressing this teacher learning need, provided that the follow-up reflective sessions are designed to achieve that goal.

Summary

Our analysis shows that activities in which teachers become learners of mathematics can be a powerful way to accomplish multiple professional development goals, especially when they are thoughtfully designed and led by a capable facilitator. Any variation within this type of professional development experience can promote the learning of new mathematics and challenge teachers’ beliefs about what students should learn and how. These experiences can also help teachers develop a vision for school mathematics reform, examine pedagogical theories and effective teaching practices and become aware of diversity in approaches to problem-solving and learning styles. However, we caution that these benefits depend on whether a facilitator carefully models novel teaching strategies and orchestrates focused reflections on these experiences. The length of the activity, the complexity of the tasks, the design of the format, and the structure of the follow-up reflection may also determine the extent to which this kind of professional development experience can meet various kinds of teacher learning needs.

Suggested follow-up resources

If you are interested in learning more about exemplary professional development materials that can help teacher educators plan and facilitate mathematical experiences-as-learners, we recommend the following resources:

Corwin, R.B., Price, S.L., and Storeygard, J. (1996). Talking mathematics: Resources for developing professionals. Portsmouth, NH: Heinemann.
This multi-media package is intended to support teacher educators in planning professional development for elementary teachers to help them promote and facilitate in their classes the kind of mathematical discourse recommended by the NCTM Standards. A main component of the proposed professional development program are experiences-as-learners where the teachers engage in a number of mathematical problems, chosen because they are mathematically rich and “engaging” yet accessible to elementary students. The materials include a facilitator guide, videotapes providing images of elementary classrooms engaged in mathematical discourse and a book for the participants. The facilitator guide provides considerable support for setting-up and facilitating the suggested experiences-as-learners.
Friel, S.N., and Joyner, J.M. (Eds.). (1997). Teach-Stat for teachers: Professional development manual. Palo Alto, CA: Seymour.
This manual is intended to support teacher educators interested in replicating the 3-week summer institute developed and field-tested by the NSF-funded Teach-Stat project. This program was designed to prepare elementary teachers to teach statistics and at its core has a carefully-designed series of experiences where the teachers themselves learn statistics in the inquiry-oriented way they are expected to encourage in their students. The manual provides valuable directions and support about how to plan and implement the summer institute.
Fonzi, J., and Borasi, R. (2000). Orchestrating math experiences for teachers. (videotape + facilitator’s guide) (available from the authors).
This 50-minute videotape features the mathematical inquiry on area described in Illustration 1. The accompanying guide provides additional information about and a commentary on this mathematical learning experience and a rich set of questions to help teacher educators use the illustration to design similar mathematical learning experiences for teachers.
Fonzi, J., and Borasi, R. (2000). Promoting focused reflections on learning experiences. (videotape + facilitator’s guide) (available from the authors).
This 40-minute videotape features excerpts from three reflective sessions that followed the inquiry on area featured in Orchestrating math experiences for teachers and another inquiry on the topic of tessellations. Taken together, the three sessions illustrate complementary ways to focus and structure follow-up reflections, a critical component of effective experiences as learners. The accompanying guide offers additional information about and a commentary on the illustrations and questions to help teacher educators analyze what it takes to successfully design and facilitate this kind of reflective session.
Borasi, R., and Fonzi, J. (in preparation). Introducing math teachers to inquiry: A framework and supporting materials for teacher educators. (multi-media package) (available from the authors).
This multimedia package supports mathematics teacher educators who want to implement a professional development program to begin the process of school reform. It shows teacher educators how to design experiences as learners that introduce teachers to an inquiry approach to teaching mathematics. The package contains two 2-hour-long videos, each featuring an experience-as-learners. The CD-ROM included in the package contains a detailed set of artifacts from these experiences and suggestions for implementing similar ones.
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