ABSTRACT
This chapter deals with a few major ideas about mathematical concepts and the conceptualizing process. One must recognize that algebra deeply modifies the cognitive status of arithmetical concepts. Algebra is an important step in the learning of mathematics. Since Plato and Aristotle many questions have been raised about the empirical or non-empirical roots of mathematical knowledge, about intuition and formalism, about the nature of mathematical proofs, about the relationship of mathematics to logic, or about the possibility of proving the consistency of mathematics. There are several ways to approach the problem of the nature of mathematical concepts empirically. The most widely accepted is the historical approach. Mathematical concepts are involved both in schemes and in sentences. The concept of scheme is essential to any theory of cognition because it articulates into a unit both its behavioural and representational features: rules of action and operational invariants. Schemes are at the heart of cognition, and at the heart of the assimilation-accommodation process.
