A report on the March 2006 IM&E Workshop on Teacher Education, by Mark Saul

There are meetings you go to that are like museums, a series of frozen tableaux, illustrating the work of this or that project.

There are meetings that are like sporting events, with different sides competing for the prize: the attention of an audience or a consensus of agreement.

This meeting was very different: neither a set of exhibits nor a contest. It was the slow emergence of a community, of a group of people from varying backgrounds bringing their resources to bear on a significant problem: the preparation and continued education of teachers of mathematics.

The observations below, of themes that emerged during this conference, is intended as a ``reader's guide'' to the materials posted on this website. But it also contains notes about comments made in workshops, question sessions, and in corridor discussions, fleeting comments that reinforced some of the points brought out in more formal contexts.

One theme that emerged almost immediately was the effort required when a mathematician and an educator collaborate on a course---the time and effort required is more than that required by either one of them, acting alone. Each input is doubly scrutinized, from a content point of view and from a methods point of view, and tensions between these points of view resolved. The result, when things go well, is a much higher quality course, or a much more highly refined set of materials, than might have been produced by people from just one of these two communities.

The return on this investment, when it pays off, is a higher quality of direct interactions with teachers. Each partner in the collaboration derives credibility from the presence of the other, and teachers find themselves with models for instruction and content which are validated from more than one point of view.

Along with the budgeting of instructors' time, participants often remarked on the budgeting of class time. One group noted, to start the conversation, that they had planned a two hour session on the theme of `invert and multiply'. The actual work took 12 hours.

Mathematics courses are typically highly structured and compressed. The task of assimilating the material, rather than being part of the structure of the course, is often left to the student. In the education tradition, on the other hand, the assimilation of content is part of the structure of the course itself, and so considerable contact time is spent on hands-on work with projects, with discussions, with activities related to the mathematics the students will be learning. This difference in how courses are conducted can lead to tension between mathematicians and educators who are teaching the same course, and each must make concessions to the tradition represented by the other.

Another theme that kept emerging was the prevalence of vertically-integrated course content for teachers. As the mathematical point of view becomes integrated into teacher preparation and enhancement, it becomes natural to look ahead and see how the topics discussed in a particular course will unfold as students learn more: how arithmetic will grow into algebra, patterns into functions, sets and measurement into logic and later probability. The creation of these connectionsin the minds of teachers taking cannot be taken for granted, and is part of the art of teaching the course.

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