Commentary on Theories of Mathematics Education

Knowing More Than We Can Tell
Nathalie Sinclair
Using a typical Piagetian conservation task, a child is asked whether the number
of checkers in two rows is different because the experimenter moved the checkers
in one row. He responds that they are different, “because you moved them”. At
the same time, he moves his pointing hand between the checkers in the row and
the checkers in the other row. The speech conveys one thing, that the boy has not
mastered conservation of number. The gesture conveys another, namely, a bodily
knowledge of one-to-one correspondence between numbers.
The question for the educator, and for the cognitive scientist, for that matter, is
what does the boy know? The significance of this episode, which was described by
the developmental psychologist Susan Goldin-Meadow (2003), lies in attention to
gesture as permitting a different interpretation of what the boy knows. Because such
a gesture may not have been noticed or recorded by the Piagetian researcher, the
conclusion would be that the boy does not know. An analysis of the boy’s verbal utterance
would lead to the same conclusion. In this case, gestures are functioning paralinguistically,
referring to the manner in which things are said, much like prosodic
features (tone, phrasing, rhythm, and so on). That things can be said without paralinguistic
features is obvious. However, that does not mean that the paralinguistic
is epiphenomenal to human expression and communication.
The term ‘paralinguistic’ is well defined and useful in the study of linguistics.
This paper aims to study an analogous, yet much less clearly defined, set of ways
of knowing that can only be defined in opposition to the literal, propositional, logical
and perhaps even linguistic ways of knowing that are sometimes referred to
as ‘strictly cognitive.’ The length and diversity of the following list of terms (and I
make no claims of exhaustiveness) I have in mind strikes me as overwhelming: tacit,
implicit, aesthetic, emotional, holistic, qualitative, creative, subconscious, intuitive,
personal, insightful, visual, instinctual, imaginative. They are united only in their
opposition to the linear, sequential, and detached ways of knowing that characterize
formal mathematics. Some scholars (see Shavinina and Ferrari 2004) refer to these
ways of knowing as being “extracognitive,” a term I find problematic if cognition
refers to what we know. Papert (1978) proposes the term “extralogical,” which he
views as ways of thinking that are not determined by mathematical logic. I propose
the term “propositional” to describe the ways of knowing to which I am staking
my opposition—this includes expressions made in language (spoken or written) or
N. Sinclair ()
Faculty of Education, Simon Fraser University, Burnaby, Canada
e-mail: nathsinc@sfu.ca
B. Sriraman, L. English (eds.), Theories of Mathematics Education,
Advances in Mathematics Education,
DOI 10.1007/978-3-642-00742-2_56, © Springer-Verlag Berlin Heidelberg 2010
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