The Role of Teachers' Knowledge of Functions in Their Teaching: A Conceptual Approach with Illustrations from Two Cases

We investigate whether and how personal mathematical knowledge at an advanced level impacts on teaching at a lower school level. We study this in the context of functions because understanding them permeates secondary and advanced mathematics. Textbook treatment of these can be patchy, implying a need for knowledgeable teachers to rectify weaknesses. A small observational study of two mathematically well-qualified teachers, teaching at a range of levels, leads to a deeper understanding of how their personal mathematical knowledge is manifested in their teaching. Finally, we present a theory of pedagogy combining dual development, intellectual necessity, and repeated experience of reasoning, for which these teachers' practices provide illustrations.