Commentary on Theories of Mathematics Education

596 N. Sinclair
collections of signs that can be evaluated as true or false. The extrapropositional
invokes knowing that cannot be expressed by the knower, or of which the knower
might not even be aware; it refers to knowings that are not subject to being evaluated
or accepted as true or false, while still having underlying warrants.
The prefix ‘extra-’ (meaning “outside”) has the connotation of being optional or
superfluous, which may explain why the extracognitive and the extralogical have
often been studied in the context of special, elite, gifted, or expert mathematical
thinking—a point I return to later in this paper. I suggest that a better metaphor for
the set of ways of thinking I want to talk about is that of circle inversion: if the
propositional is what is in the circle itself, then the inversion with respect to this
circle—everything outside the circle—represents my area of interest. Indeed, the
prefix ‘para-’ as being ‘beside’ suggest that these ways of knowing are on equal
par, rather than epiphenomenal. Further, ‘para-’ also denotes the sense in which
parapropositional ways of knowing act beside the logical, as the episode above illustrates.
However, in addition to being a mouthful of a word, the prefix ‘para-’
continues the dichotomy and diminishes the value of what is termed ‘para-’: among
other things, a paralegal is not a legal, a paramedic is not a medic—but is characterized
and named in relation to what it is beside, but less than. Lastly, there is the
paranormal in relation to the normal, which I certainly wish to block. Therefore,
I have chosen the word ‘covert’ to describe these ways of knowing in contrast to the
logical, cognitive, propositional ways of knowing that are all overt.
My goal in this paper is threefold. First and foremost, I wish to offer some structure
for the somewhat amorphous area of research on covert ways of knowing by
examining how they are related in the specific context of mathematical thinking.
In so doing, I would like to remain attentive to how these covert ways of knowing
accompany, support, affirm and/or lead to public and formal mathematical knowing.
Second, I would like to examine the ways in which different constructs used to
study covert ways of knowing in mathematics education are related, to rejuvenate
some that have received relatively little attention, and to propose some that might be
more productive. Finally, I will consider methodological issues involved in trying
to study and understand phenomena that are intrinsically inarticulable or otherwise
unspecifiable.
Structuring Covert Ways of Knowing
That constructs such as ‘insight’ and ‘intuition’ are related is not surprising; we use
these terms almost interchangeably in everyday language, and there has not been
much of a need to distinguish them in the scholarly literature. However, the relationship
between constructs such as ‘aesthetic’ and ‘embodied’ are less evident, and
depend a great deal on one’s theoretical positioning of each. Further, both have very
deep and distinct histories of scholarly research, leading to vastly different research
programmes in mathematics education. It is these kinds of constructs that I will be
considering throughout the next three subsections; but instead of starting from their
contemporary positionings (usually based on theories and methodologies outside