Commentary on Theories of Mathematics Education

196 B. Dahl
Pegg and Tall have shown that the theoretical frameworks referred to in their
model all share the same underlying local cycle of learning (2005, p. 474). It is
very interesting that they in this way bring together ‘different’ theories and show
that they in fact, share this similar cycle of concept development. They in fact build
a meta-theory on top of a number of ‘different’ theories. However, their strength
might also be their “weakness” since, being the devil’s advocate; what they show is
that rather similar theories are—rather similar. On the other hand, it does require an
analysis as deep as that of Pegg and Tall to in fact learn if a number of theories do
in fact share central features. But what about theories of cognitive development that
do not fit the action-object process? I will discuss this again in section ‘Merging the
Frameworks’.
Pegg and Tall also state that learning outcomes can be analysed in terms of the
SOLO UMR cycle and that this is a local framework. These UMR-levels furthermore
constitute the SOLO 2–4 levels in the five-step SOLO Taxonomy (Biggs and
Collis 1982, pp. 17–31; Biggs 2003, pp. 34–53; Biggs and Tang 2007, pp. 76–80).
SOLO 1 is the ‘pre-structural level’ where only scattered pieces of information have
been acquired and SOLO 5 is the ‘extended abstract level’ where we see further
qualitative improvements as the structure is generalized and the student becomes
capable of dealing with hypothetical information not given in advance. The SOLO
Taxonomy is among other things used for university teaching and converges on production
of new knowledge. Thus here it seems the model is used as a global model,
at least for the higher education sector. It was investigated by Brabrand and Dahl
(2009) where we for all science departments, except mathematics, were able to detect
progression in SOLO Taxonomy competencies from undergraduate (Bachelor)
to graduate (Master) studies at the faculties of science of two Danish universities
that had both changed their curricula and formulated intended learning outcomes
based on the SOLO Taxonomy. However, for mathematics, the SOLO Taxonomy
was not able to detect any SOLO-progression. It was concluded that for mathematics,
progression is more in the content which the SOLO Taxonomy with its focus
on verbs is not as able to describe. In any case, it can therefore be questioned if the
SOLO Taxonomy is indeed a global framework, at least for mathematics teaching
at the universities, Bachelor and Master’s programmes.
Another Framework of Cognitive Processes
In an attempt to move beyond various dichotomies in the psychology of learning
mathematics, I developed a model incorporating a number of different theories
focusing on cognitive processes. The theorists were Dubinsky (Asiala et al.
1996), Ernest (1991), Glasersfeld (1995), Hadamard (1945), Krutetskii (1976), Mason
(1985), Piaget (1970, 1971), Polya (1971), Sfard (1991), Skemp (1993), and
Vygotsky (1962, 1978). I chose theories that have reached a status of being “classics”.
The model goes across the theories and sorts the different theories’ statements
on six themes; in that sense “mixing” them. There are overlaps and some of the