This is the summary of a presentation given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

What do mathematicians do? What does it mean to do mathematics? Can undergraduates, high school students, and secondary school mathematics teachers do real mathematics? We addressed these questions by offering students in an introductory course for non-mathematics majors, in a seminar for mathematics majors (many of whom plan to be school mathematics teachers), and in a graduate program for high school teachers of mathematics and science an opportunity to work on a variety of open questions and unusual closed questions in number theory and combinatorics and some interesting questions in the mathematics of social choice. In this talk, we present some of the questions and problems students were asked to investigate, we describe the results of student investigations, and we discuss ways in which the secondary teachers changed their teaching as a result of having done some interesting mathematics.

Emelie Kenney, Siena College, Loudonville, NY 12211
James Matthews, Siena College, Loudonville, NY 12211
Lester Rubenfeld, Rensselaer Polytechnic Institute, Troy, NY 12180.