Curriculum Design and Development for Studens, Teachers and Researchers

The major part of the chapter describes our manner of understanding the phenomenological analysis of mathematical concepts (or mathematical structures) that Freudenthal proposed in his book Didactical Phenomenology of Mathematical Structures (Freudenthal, 1983). For this purpose we outline the essential characteristics of a conception of the nature of mathematics that is compatible with our way of understanding Freudenthal’s phenomenology and that also includes the idea of the generation of concepts from proofs, which is characteristic of the work of Lakatos. We also discuss Freudenthal’s distinction between mental objects and concepts, and the consequences for curriculum development, which derive from the opposition that Freudenthal proposed between the constitution of mental objects and the acquisition of concepts. In this discussion, we use our semiotic viewpoint as a basis for interpreting the distinction established by Freudenthal, using as an example some considerations for a LTM for studying the uses of natural numbers. In the context of these considerations, we present the distinction between three types of sign —icons, indices and symbols— which Peirce himself used to describe algebraic expressions as iconic, while the letters in them are indices, and signs such as those of operation or equality are symbols.