Reported is the everyday teaching and curriculum planning activities of the Madison Project, operated by teachers and mathematicians. Two kinds of informal explanatory experiences are provided in order to involve students in the discovery of significant mathematical concepts--experiences where children do something and experiences where a "seminar" of children discuss something, under the leadership of a teacher. The Project has developed a set of seven criteria for selecting appropriate classroom "informal exploratory experiences:" (1) The child should have adequate previous "readiness", (2) informal exploratory experiences should be related directly to fundamental ideas, (3) the student must have an active role, (4) concepts must be learned in context, (5) interesting patterns must be developed in every task, (6) the experiences should be appropriate to the age of the child, and (7) the sequence of "informal exploratory experiences" must seem to be worthwhile. A list of some of the objectives of the Madison Project is indicated under two headings--Mathematical objectives and general objectives. (RP)