the reader can say. But we have become convinced, from studying these videotapes, and from working with teachers in going over analyses of these tapes, that both teachers and students have much to gain from making efforts toward this very fine-grained analysis of how students are actually thinking about mathematical problems and mathematical situations. This is a valuable and important form of assessment, and in the right context it can be carried out successfully.
Clearly, we argue for the importance of looking on a very minute level at how students think about mathematical situations, and at how teachers think that students think about mathematical situations. We see this as one of the most critical questions in mathematics education. It requires careful and insightful analysis--but even more, it requires teachers who will seek out the best possible data, even in instances where they may, given hindsight, wish that they had done something differently.
This is the spirit of the very best in science, seeking truth without placing blame. In our view, the real heroes of these studies are the teachers, people like Linda and Pat, who have worked so hard and so selflessly to get and to share the most complete possible data, even when they wish they had known earlier some of the things they only found out about later on.