Subtlety and complexity of mathematics teachers’ disciplinary knowledge
Brent Davis, University of Calgary
What sort of mathematical competencies must a teacher possess in order to teach the subject
well? Variations of this question have been of longstanding interest in the mathematics education
research community, but they have proven difficult to investigate. Indeed, as theories and tools
have grown more sophisticated, answers to the question have grown more complex. A current
consensus is that teachers’ knowledge of formal mathematics remains “inert” unless accompanied
by a rich pedagogical content knowledge. Unfortunately there is a lack of consensus around the
nature of that knowledge. Two perspectives prevail, neither yet associated with a research base
that enables strong claims about practice. The majority of current investigations focus on explicit
knowledge that might be assessed directly through observation, interview, or written test, with a
parallel research emphasis on the formal contents of teacher education programs. A second
school of thought, and the one developed in this lecture, is that the most important competencies
tend to be tacit. Among other elements, teachers’ tacit knowledge comprises many of the
instantiations – that is, analogies, metaphors, images, exemplars, and applications – invoked to
introduce and elaborate concepts.
I frame the discussion with an elaboration of Ma’s construct of “profound understanding
of fundamental mathematics,” arguing that it might be more productive to think in terms of
“profound understanding of emergent mathematics.” That is, situating the discussion in
complexity science, I explore the utility of construing mathematics and learning as ever-evolving
organic systems. The presentation will be illustrated with reference to ongoing collaborative
research with teachers that is focused on the excavation, interrogation, and elaboration of their
usually tacit mathematics knowledge, looking in particular at how teachers might be encouraged
to develop dispositions to investigate and extend their own mathematics as they help learners
make sense of unfamiliar situations in ways that are mathematically sufficient but neither overly
rigid nor overwhelmingly complex. There are strong indications in this work that explicit
interrogations of tacit knowledge can have immediate, significant, and sustained impacts on
teachers’ knowledge of mathematics, perspectives on learning, and classroom practices, as well
as student engagements, understandings, and attitudes.