Mark Saul

Many participants, both mathematicians and educators, noted how much they learned from the teachers they worked with. First, they learned about the actual life of a teacher, the feeling of working with children, the reward system of a school environment. Teachers could also tell their instructors what might be difficult for their students, what might be easy, and what new tacks students are likely to take. At the same time, participants noted a tension between this idea and the need to push the envelope, to try out new ideas and examples with the students, to get them to achieve on a higher level than the teacher might have thought possible.

A goal mentioned often by participants was that of creating `teacher leaders'. This phrase had different meanings in different contexts. Perhaps the group from Portland State articulated one view best: they want to increase the achievement of all students, not just students of teachers who come to the institute. One way they can do this is by using teachers who do attend as `turnkeys' or models for other teachers. If such teachers can change the culture of a school, the effect can be very powerful.

The field of teacher education is missing, especially on the middle and high school levels, a canonical set of experiences that pre- or in-service teachers should have.

The CEMELA group presented an innovative use of algebra tiles that led students to conjecture and prove a theorem about perimeter. Such an experience combines the exploration of a pedagogical technique with a model of how mathematics is developed.

``Pattern problems'' were the subject of significant discussion. Educators and mathematicians discussed the reasons why they would or would not use these problems, how they would phrase them, in what contexts they might be most useful, and why some mathematicians had objections to them in certain contexts.

One participant, from New Mexico, warned of the danger of treating reform curriculum materials as if they were just a basis for traditional pedagogy. He did not want his teachers to take these materials home and `figure out how to lecture using them'.

The Boston University team commented that if participants go to an instructor with certain questions, they don't get answers to those questions. Instead, they get more questions. They also commented that nothing in their program gets discussed in `lectures' until at least three days after participants have had a chance to discover things on their own. The group attributed this tradition to the Arnold Ross style of pedagogy.

This group also introduced the notion of `low threshold, high ceiling' problems: problems which could be stated and worked on with very little background, but whose solution could entail serious work and lead to significant results.

The group from Boston University offered a taxonomy of mathematics for teachers, which draws som subtle distinctions:

- Knowing mathematics as a scholar: its structure, history, and applications
- Knowing mathematics as an educator: what students must know and how they come to know it.
- Knowing mathematics as a mathematician: how one goes about doing mathematics
- Knowing mathematics as a teacher: how do you communicate all this to students in a classroom.

The group from CEMELA gave a particularly vivid example: they had teachers attend a lesson taught entirely in Mandarin, by a native Chinese speaker. This experience enabled the teachers to access the feelings of students who are struggling both with the mathematics and the language of instruction. Having this experience, rather than reading about it, was found effective pedagogically in bringing to life discussion about work with this population of students.