This document addresses issues raised during the Research on Teaching Mathematics Conference, examining them in view of the central tenets of the Institute for Research on Teaching at Michigan State University. It explains why teacher judgment is the focus of IRT research and discusses the concept of "the wisdom of the practitioner," noting its implications for research on teaching and raising questions about how practitioner wisdom might best be tapped. Two methods for studying teachers (process tracing, and description and reflection) are cited, and their advantages and disadvantages outlined. It is stated that information-processing research indicates teachers must simplify the teaching world to make it manageable. Researchers must conduct supplemental studies, using different methodologies, to determine how and why teachers choose certain modes of simplification rather than others. It is suggested that researchers can develop at least some teaching algorithms, which teachers would transfer into viable heuristics. Successful implementation of research requires effective dissemination of findings to teachers. A number of questions for future study are also presented. (Author/MP)