A Theory Bite on Learning Through Mathematical Activity

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ACKNOWLEDGMENTS

I thank Andy diSessa, Amy Hackenberg, and Andy Norton for comments on earlier versions of this article, but I alone am responsible for its contents.

Notes

This definition of perturbation includes “inputs” that have origins internal to a system.

The coordination he established was that four threes is twelve and three fours is twelve.

Heuristically, one can think about Jason's gap as recognizing that he was yet to make a coordination that he had never made before.

Had Jason been asked to pull out two-thirds and three-fourths, he would have experienced a perturbation he could not eliminate and, hence, he would have had a problem.

For example, 1/12 can be used to measure both 1/3 and 1/4 in that 1/12 iterated three times is 1/4 and 1/12 iterated four times is 1/3.

The scare quotes are added to alert the reader that the circumstances are the researchers’ circumstances.

Given that Erin solved each problem correctly and straight away, perhaps it would be more appropriate to use “task” rather than “problem” given the authors’ definition of a problem.

There is a side issue concerning the innateness of reflection if one wants to build models using reflection. Concerning this issue, E. von Glasersfeld commented that, “I have often said that reflection is a feature of consciousness which remains an unexplained presupposition of the constructivist model. This has nothing to do with ‘innateness;’ it's a theoretical working hypothesis” (personal communication, June 23, 2009).

This phrase refers to the perceptual and motor aspects of a situation.

*“Theory bites” are short commentaries on articles that appear in the journal. They aim to highlight and characterize (a) the author's theoretical perspective, (b) alternative ways of thinking about relevant phenomena, and (c) the empirical and educational consequences of taking one or another position. The point of these non-oppositional, dialog-like contributions is to emphasize and contextualize the theoretical commitments behind the research reported in the journal and to advance the capacity of our community to theorize powerfully. A longer discussion of theory bites appears in the editor's note in Vol. 26, No. 4.