Philosophy of mathematics and its pedagogical implications.

Abstract

In this study, we have undertaken to analyse the extent to which 'mathematical language' is appropriate for pedagogic communication of the structure of mathematics, given that its symbolic - representational form and its verbal - propositional form do, in fact, interact in natural language. In chapter one we have indicated that the purpose of our analysis is to formulate criteria for understanding and qualifying 'mathematical language' as the means for communicating the structure of mathematics in education. To accomplish our mission we have performed a comparative analysis of the structures of mathematical and linguistic systems in chapter two. In this regard, we have found out that the structure of mathematics is both linguistically and cognitively ordered such that although mathematics and language have transactional and interactional purposes respectively, they are indeed functionally isomorphic. The isomorphism so established reveals that there is a linguistic foundation of mathematics made possible by harmonizing the principles of the three programmes in the foundations of mathematics. These programmes namely Intuitionism, Logicism and Formalism (ILF) are harmonized under 'The ILF Model' through structuralism. This has been done in chapter three. Chapter four traces the development of a mathematical system from ordinary human experience.

In Chapter five, for purposes of promoting mathematical pedagogy, we have utilized The Il.F Model, akin to a mathematical language theory in the identifaction of three forms of discourse in mathematical pedagogy, namely, problem exposition, problem representation and probem solution. Through these forms of discource, we have suggested an alternative methodological design for modelling levels of cognition in teacher aided learning of mathematics - a challenge to curricula developers. Curricula planners need to harmonize the relationship between techno-social-practical relevance of mathematics with communication of its structure which is at the centre of the classic problem of transfer. Finally we have developed an alternative approach to problems in terms of phases of intelligibility of problem solving procedure according to the principles of The Il.F Model from which teachers would benefit