Abstract
Wittgenstein's work on rule-following and language games is well-known within philosophical circles. His work on the foundations of mathematics is also celebrated and contested with equal measure. This paper seeks to apply some of Wittgenstein's core philosophical writings on rule-following to justify a very particular type of mathematical pedagogy for mathematics and its teaching. It will be argued that since mathematics is fundamentally rule-like in nature, that Wittgenstein's work on rule-following directly applies. Other key elements of Wittgenstein's philosophy will also be central to this argument. For example, the notion of training, and of language-games, as well as what it means to engage in a meaningful way in these language-games. It will be argued that a rule in itself cannot guide, just like a signpost can point in a direction, but it cannot ensure that people will follow it correctly. So too in mathematics, it will be shown that the rules of mathematics can be interpreted in infinitely many ways, so that teaching mathematics as rule-like only would lead to inherent problems. As such, the case for some other guiding faculty will be made, and this, it will be shown, is the core justification for teacher-centred pedagogy in mathematics.
