COLUMBUS PANEL Annie Selden and John Selden UME Trends, Vol. 2, No. 5, December 1990, 1, 3 & 7 “Research on learning affects us all.” This is what MAA President Lida Barrett said when she introduced the CTUM panel on research in learning undergraduate mathematics at the Summer Meeting in Columbus. James Kaput began the panel of three by giving an overview of different kinds research: (1) surveys giving descriptive information on large populations, e.g., NAEP and international comparison studies; (2) the “agricultural-botany” type of controlled experiments, which dominated educational research until the late 1970's, where one studied the effect of some small change in what one did--with statistical significance being a main result; (3) historical or philosophical research looking at longer term events; (4) formative or summative research evaluating the outcome of a change in curriculum materials; (5) case studies or clinical research involving careful studies of individuals in complex learning situations with the aim of providing detailed models of cognitive processes. The latter, which Kaput sees at the most promising, aims at descriptions of what’s going on in the minds of individuals as they learn mathematics. An important outcome of this research is the identification of students' primitive knowledge structures about limits, continuity, etc. The misconceptions research of the 1970’s and 1980’s was in this vein. Since then, however, there's been a shift away from treating students' knowledge structures as “enemies, towards figuring out how to build on them. Another outcome is the description of mature knowledge structures. Not only do experts have a lot more knowledge than beginners, their knowledge is also highly structured or “chunked.” Experts use a lot of heuristics, many of which are not addressed by the curriculum, but are considered major components of mathematical maturity. Researchers want to know how to build such mature knowledge structures and about obstacles to building them. Research into student thinking and learning challenges some fundamental assumptions about the way we do things. For example, most people think of calculus as a course, whereas, in reality, it is a “web of ideas,” which grows slowly in one's head. Kaput thinks technology might provide early access to this “world of big ideas” without reference to algebra. He demonstrated his current research project, Math Cars, in which students drive simulated vehicles, and at the same time, generate position and velocity graphs. His underlying idea is to map the phenomenologically rich world of experience (visual and audio) onto the structured world of mathematics. Next Joan Ferrini-Mundy indicated some theoretical perspectives and research findings that might help teachers of undergraduate mathematics. This work appears where mathematicians do not normally see it--sometimes in mathematics education research journals, but often in conference proceedings of groups like PME (the International Group for the Psychology of Mathematics Education) or in working papers. Much current