Analysing and stimulating flexibility in mathematics education
Center for Instructional Psychology and Technology
Katholieke Universiteit Leuven
In his foreword to the book The development of arithmetic concepts and skills. Constructing adaptive expertise edited by Baroody and Dowker (2003), the late Hatano (2003, p. xi) argues that one of the most important issues is how students can be taught curricular subjects so that they develop adaptive expertise. He describes adaptive expertise as "the ability to apply meaningfully learned procedures flexibly and creatively" and opposes it to routine expertise, i.e. "simply being able to complete school mathematics exercises quickly and accurately without (much) understanding".
Although the constructs of adaptive and routine expertise were introduced by Hatano already more than two decades ago (Hatano, 1982) and although terms like adaptivity and flexibility have been used with increasing frequency by researchers and practitioners in the field of (mathematics) education for a long time, few attempts have been made to rigorously and systematically study adaptive expertise as a competence, and its acquisition and cultivation.
In this plenary lecture, I will, first, present a conception of flexibility
(or: adaptivity) as the efficient adaptation of one’s strategies, models
or representations to task, subject, and context variables, and argue
that any definition of this notion that neglects one of these variables,
is incomplete and therefore somehow problematic. Then I will report
emerging empirical support – mostly coming from for our own research
in the domain of mathematics education the claim that experts demonstrate
more flexibility in their choices of strategies, models and representations
than non-experts and that this directly and substantially accounts for
their better performance in their field of expertise. Finally, I will
address some educational issues dealing with the when, for whom, and
how to teach for such flexibility.