Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
 Decisions

 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


VIDEO CLIPS: Internet access via modem may mean very long download times for video clips. If you are not on a fast line, you may want to read this paper without viewing the clips.



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Reflections

Stephen Weimar       [Biography]



As the Math Forum has experimented with ways of enhancing mathematics education through the use of the Internet, a key Forum strength has been its focus on community and its ability to take advantage of the unique possibilities that virtual communities afford.

The Internet offers some important new resources:

  • the ability to create online environments that cut across traditional communities, such that K12 teachers, students, parents, research mathematicians, teacher educators, publishers, and professionals in the workplace can support each other's mathematical interests. Such environments make possible tight feedback loops among the various constituencies.

  • a reversal of sorts in terms of who has access to the tools of publishing, and who can establish educational services. The so-called democratizing effect of the World Wide Web has meant that we, as teachers, have been able to respond to our own needs by building services such as the Problem of the Week and Ask Dr. Math that are immediately useful to our students and in our classrooms.

  • the integration of content and communication. Under the right circumstances, conversations can generate significant content, particularly when they involve individuals with developed expertise and interests. The Internet provides the ability to record, massage, and reorganize conversations within an environment structured according to the needs of the users. This new organization can then be extended to establish conversations around an published object, whether a video, a lesson plan, or a mentoring activity.

A result has been the emergence of a continuous learning environment for teachers, together with support for the increasing professionalization of mathematics education. Our videopaper represents an interesting moment in this effort.

Early in 1994, the Math Forum began to bridge research and practice through the Learning and Mathematics series, in which research summaries were produced to seed conversations on the geometry.pre-college newsgroup. This seeding approach was taken in order to meet the challenge in Internet discussion groups of creating focused, substantive conversations, rather than the flame wars or uneven exchanges that dominated many areas. Another form of grounding for conversations might result from a focus on actual classroom interactions, in hopes that, rather than broad ideological and theoretical concerns, issues of practice would dominate, informed where possible by research. Our videopaper represents an attempt both to establish a grounded conversation and to take advantage of the possibility for teachers themselves to be the initiators of the weaving together of research and practice.

At this point in the project, and in the context of the above remarks, I look forward to pursuing a number of open questions:

  • The first two stages of bridging research and practice are well under way, in that we have been reading research and reflecting on its implications for practice. We have also been conducting research through our practice, pursuing questions that have arisen out of our discussions. The next steps, in the form of a conference and the possibility of online conversations around the article, are to see what integration can take place through directly engaging other researchers and educators. In a sense, the videopaper is meant to create the context in which these communities mix.

    Having such a developed starting point for a conversation can be a hindrance in some circumstances. It also remains to be seen whether our paper is sufficiently developed to insert itself into the professional discourse of researchers. Certainly the in-person conference will be important in jump-starting conversations, just as our project workshops were critical for generating enough momentum to write the paper.

  • It's not easy to figure out the best Internet vehicles for eliciting and publishing the activities and thinking of teachers. Some have tried to create lesson plan exchanges, encouraging teachers to submit lessons. Others have established curriculum writing projects. For the majority of innovating teachers, neither of these approaches has seemed particularly efficient or well-targeted in supporting this effort.

    In contrast, the Math Forum creates opportunities for teachers to make use of projects and to interact with others in those contexts; we then facilitate the publication of resources out of those activities. This videopaper is one such project, in that we brought a group together and watched the interactions, trying to tease out what might be elements of a paper, and then facilitated the writing of it through weekly reflections on the classroom, reading articles together, and coming together for focused periods of collective writing. In the end, the Math Forum staff took responsibility for sewing together the contributions. In some other situations we have functioned in a similar manner, by providing a support team that facilitates product creation, rather than expecting busy teachers in classrooms to produce the bulk of the finished product. When is this process an effective one for both collaboration and professional growth? Is there a version of this that is scalable?

  • The next phase of this project, which we hope will involve online discussions, will provide another set of roles for teachers to play in the generation of resources. I am interested in continuing to evaluate the effectiveness of such vehicles in supporting teacher participation, and in productively eliciting the best ideas teachers can contribute as they move ahead in their own professional development.

    As a group I expect we will continue to discuss the relative merits of the various dimensions of this project, from workshops to online editing, probing whether we can improve on the writing and communication process we've been evolving. It's a challenge to integrate strategies for developing materials that are both immediately useful to those of us in the classroom and that have a broader use and achieve a compelling quality of their own. How will we extend the depth and coherence of conversations without losing the energy of the moment and honest connection to practice?

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