Commentary on Theories of Mathematics Education

Preface to Part VI
The Fundamental Cycle of Concept Construction
Underlying Various Theoretical Frameworks
by John Pegg and David Tall
Stephen J. Hegedus
I wish to preface Pegg & Tall’s paper with a word of caution for the reader—be
ready for a thorough, detailed and rich adventure in comparing multiple theoretical
frameworks drawing on concept formation and more generally, epistemology. I
would recommend that graduate students read this paper with a mind to diving into
each theory described simultaneously to find detailed examples and lengthier descriptions
of the ideas contrasted in this paper. John and David do a thorough job at
comparing various theories that address local and global issues in cognitive growth
and formation of rich mathematical concepts. They move beyond simply comparing
various cognitive development theories to offering issues concerning the learning
of mathematics and empirical studies that could arise from this meta-framework
synthesis. Indeed, they report on data from various longitudinal studies and smallerscale
studies, upon which I believe future research could be established.
The paper addresses the need to contrast local and global issues in utilizing one or
more theories of cognitive growth. Local broadly means the focus on processes and
concepts whereas the global positions are local mathematical cognitive behaviors
in a broader and more longitudinal development of mathematical knowledge for an
individual. Here the important contrast is an epistemological one again in defining
the very nature of knowledge formation.
Primarily, the authors contrast the global stage of development in the theories of
Piaget, van Hiele and Bruner with the SOLO (Structure of the Observed Learning
Outcome) model by Kevin Collis (who the paper is dedicated to) and John Biggs.
One important feature of the SOLO model is how it appreciates how stages of a
child’s intellectual development are nested in prior ones instead of replacing earlier
ones and in doing so increasingly more sophisticated thinking can evolve. A second
important feature is the focus on students’ responses rather than which stage of development
they are situated within. The SOLO model offers a more social theory
for interpreting the structures of responses of multiple individuals in a variety of
S.J. Hegedus ()
School of Education, Public Policy and Civic Engagement, University of Massachusetts
Dartmouth, Dartmouth, USA
e-mail: shegedus@umassd.edu
B. Sriraman, L. English (eds.), Theories of Mathematics Education,
Advances in Mathematics Education,
DOI 10.1007/978-3-642-00742-2_18, © Springer-Verlag Berlin Heidelberg 2010
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