Introduction: How hard is the "hard core" of a scientific program? / Massimo Piattelli-Palmarini -- pt. 1. The debate: 1. Opening the debate: The psychogenesis of knowledge and its epistemological significance / Jean Piaget -- On cognitive structures and their development: a reply to Piaget / Noam Chomsky -- 2. About the fixed nucleus and its innateness: Introductory remarks / Jean Piaget -- Cognitive strategies in problem solving / Guy Cellerier -- Some clarifications on innatism and constructivism / Guy Cellerier (...) -- 3. Artificial intelligence and general development mechanisms: The role of artificial intelligence in psychology / Seymour Papert -- 4. Initial states and steady states: The linguistic approach / Noam Chomsky -- 5. Cognitive schemes and their possible relations to language acquisition: Language and knowledge in a constructivist framework / Bäé́ Thom -- Appendix C: Localist hypothesis and theory of catastrophes: note on the debate / Jean Petitot. (shrink)

In this paper we look at some of the ingredients and processes involved in the understanding of mathematics. We analyze elements of mathematical knowledge, organize them in a coherent way and take note of certain classes of items that share noteworthy roles in understanding. We thus build a conceptual framework in which to talk about mathematical knowledge. We then use this representation to describe the acquisition of understanding. We also report on classroom experience with these ideas.

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