Open-Ended Tasks as a Trigger for Mathematics Teachers' Professional Development.

The vision expressed in the NCTM's Curriculum and evaluation standards [NCTM, 1989] and Professional teaching standards [NCTM, 1991] encourages most mathematics teachers to teach mathematics differently than the ways in which they were taught as students [Leinward, 1994]. It is suggested that teachers' conceptions of teaching are strongly influenced by the way in which they themselves have learned the subject matter [Kagan, 1992; Stofflett & Stoddart, 1994]. Thus, it is particularly important to create for them learning situations in which they engage in powerful learning of mathematics, followed by reflection on the processes that take place. In this spirit, Clarke [1994] discusses the need to "model desired classroom approaches during in-service sessions to project a clearer vision of the proposed changes" [p. 38, ibid.]. In designing such learning situations for mathematics teachers we face two conflicting considerations regarding the choice of content to be used: On the one hand, selecting an advanced mathematical topic which is new to the teachers can be powerful in terms of their own learning experience, however, they may remain with the feeling that such experience is only possible with high level mathematics that appears richer than most parts of the secondary curriculum. In such cases we often hear teachers claim that implementing such ideas could only be done on special occasions or for special students, not as an integral part of the ordinary curriculum that must be "covered". On the other hand, selecting familiar topics from the secondary mathematics curriculum may not create real learning experiences for mathematics teachers who already know and even teach these topics. In this paper we suggest a way to bridge between these two conflicting arguments. We propose a tested approach to creating rich and powerful learning situations for teachers (as well as for students) by modifying standard tasks based on familiar mathematical content from the secondary curriculum and turning them into open-ended ones with multiple correct answers. Socha [1991], inspired by Judah Schwartz, refers in a general manner to the kind of approach presented and analyzed here. In addition to the role that such problems play in teacher development processes, the proposed approach to generating such openended problems can easily be used by teachers for implementation in their own classrooms, because it is quite straightforward and can be applied to many standard tasks available in common textbooks. Evidence about the power of such modified tasks was collected in in-service workshops for secondary mathematics teachers within the framework of a professional development program aimed at introducing innovative approaches to teaching mathematics. In this paper one task is deeply analyzed to illustrate the potential of this approach. Many other tasks of the same nature were used with similar results (see a sample of tasks in the Appendix). The analyzed task deals with quadratic and linear Junctionsa part of the standard curriculum which teachers are both familiar with and often teach.