The field of mathematics education has been fashioned by a diversity of theoretical and philosophical perspectives. The purpose of this study is to add to this field an analysis of the philosophical position of critical realism. To achieve this objective, the study addresses the following questions: what does critical realism have to offer mathematics education? How may critical realism underlabour for this discipline? In addressing these questions, the study provides an overview of the basic theories and the possible weak points (...) of and arguments against critical realism and realism in general. It then draws upon the notion of a dialectical phenomenology to provide a sequence of Achilles' heel critiques of some of the perspectives that constitute broadly the theoretical landscape of mathematics education research. This critique proceeds with an analysis of didactically oriented empiricism, hermeneutics, pragmatism, postmodernism, traditionally recognized forms of constructivism, traditionally recognized theories of activity, and ethnomathematics. The last section summarizes the implications of philosophical underlabouring for a more beneficial science of mathematics education. (shrink)

Kitcher and Aspray distinguish a mainstream tradition in the philosophy of mathematics concerned with foundationalist epistemology, and a ‘maverick’ or naturalistic tradition, originating with Lakatos. My claim is that if the consequences of Lakatos's contribution are fully worked out, no less than a radical reconceptualization of the philosophy of mathematics is necessitated, including history, methodology and a fallibilist epistemology as central to the field. In the paper an interpretation of Lakatos's philosophy of mathematics is offered, followed by some critical discussion, (...) and an extension to a social constructivist position (which might well have been unacceptable to Lakatos). (shrink)

Journal of Philosophy of Education, EarlyView.

This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61students in six small sections of a “bridge" course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate the top-level logical structure of a proof. For simplified informal (...) calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or evaluate their validity. (shrink)

This paper claims that the awareness of crucial philosophical questions and controversies, which have arisen during the historical evolution of fundamental concepts, ideas and processes in mathematics, should be an essential component of the professional knowledge of student teachers who intend to teach children mathematics.

The purpose of this paper is to critically analyse and discuss the views of constructivism, on the teaching and learning of mathematics. I provide a background to the learning of mathematics as constructing and reconstructing knowledge in the form of new conceptual networks; the nature, role and possibilities of constructivism as a learning theoretical framework in Mathematics Education. I look at the major criticisms and conclude that it passes the test of a learning theoretical framework but there is still a (...) gap between theory and mathematics classroom practice. (shrink)