Epistemology, normativity and mathematics education

Theories of mathematics education can aim for several goals: to account for the nature of mathematical content and knowledge, to describe and explain the teaching and learning of mathematics, and to make sense of performance and thinking within mathematics. Four prominent contemporary theories---von Glasersfeld's radical constructivism, Lakoff and Nunez's cognitive science, Ernest's social constructivism and Davis's complexity science---all claim to naturalize mathematics and mathematics education. Each in its turn fails in its ambitions, but in instructive ways.;The theories' epistemological claims are tested against the Kripke-Wittgenstein paradox. Each fails to account of the possibility of mathematical understanding because it cannot provide the normative content of rule following. Positive responses to the paradox are provided through appeals to the work of Charles Taylor and Thomas Nagel.;In addition, none of the theories gives a defensible account of mathematical intersubjectivity. Habermas's theory of communicative action is shown to provide sufficient resources to give a rich account of intersubjectivity of mathematics education.;Building on Taylor's, Nagel's and Habermas's theories, suggestions for mathematics education research are given. I argue that it is necessary for educational purposes to assume the content and practice of mathematics as part of the initially unquestioned background against which mathematical thinking and learning can take place and be understood by assessors and researchers. Further, I argue that in order to do mathematics, one must treat mathematical objects as something that transcend the first-person perspective. Finally, I argue that a communicative model of understanding holds promise in educational assessment and research, because it may provide means of inferring student understanding either through direct discourse or via models that reconstruct internalized dialogue. One understands a mathematical concept, I claim, insofar as one can provide reasons that would be compelling to any relevantly situated interlocutor, at least in principle.;These four theories are analyzed in terms of their programmatic content. Each theorist has a declared political intent, namely, to provide an alternative to perceived elitism in mathematics education. The political claims of each theory are compared against the theory's epistemological aspirations, and in each case an insuperable gap is shown.