The ‘subject knowledge’ of teachers of mathematics, especially but not only those in primary schools, has been a high-profile issue for more than a decade. This new-found interest in teacher knowledge extends to policy makers and policy documents too, an exemplary case being the formulation of the Standards for Initial Teacher Training . Those commentators with more than the most superficial appreciation of the issue recognize that ‘subject knowledge’ for teaching has a pedagogical dimension (sometimes quaintly referred to as ‘putting it across’) as well as the more commonplace kind of knowledge acquired in studying mathematics at school or at university.
If, as is widely suggested, there is a ‘problem’ of some kind with teachers’ mathematics subject knowledge, the nature of this knowledge has to be understood with a sophistication adequate to talk about it in conceptually useful ways, yet with sufficient simplicity to avoid being overwhelmed by it. A recent instance of lack of conceptual clarity in this respect can be seen in remit 4 for the recent Williams review, which asked “What conceptual and subject knowledge of mathematics should be expected of primary school teachers ...?”. We acknowledge that this was penned by a politician, or a civil servant, but nevertheless we could ask why 'conceptual' and 'subject' are juxtaposed in this way. Is subject knowledge never conceptual?
Shulman’s (1986) classic taxonomy of teacher knowledge famously introduced the notion of pedagogical content knowledge (PCK) as an essential professional adjunct to subject-matter knowledge (SMK). The conceptual distinction between PCK and more generic pedagogic notions is that PCK is specific to the subject-matter being taught. Knowledge of commonplace mathematical misconceptions would be an example of PCK, in the case of the subject ‘mathematics’, and a demonstration that PCK goes beyond what the educated citizen would be expected to know. In a nutshell, SMK is “the amount and organization of the knowledge per se in the mind of the teacher” whereas PCK consists of “the ways of representing the subject which makes it comprehensible to others ....[it] also includes an understanding of what makes the learning of specific topics easy or difficult...” (Shulman 1986, 9).
Recent research by Deborah Ball and her colleagues at the University of Michigan unravels and clarifies these concepts. Shulman’s SMK is separated into ‘common content knowledge’ (CCK) and ‘specialized content knowledge’ (SCK), while his ‘pedagogical content knowledge’ is divided into ‘knowledge of content and students’ and ‘knowledge of content and teaching’ (Ball, Thames and Phelps, submitted). In our view, the distinction between CCK and SCK within the category SMK is not very clear. A different distinction, due to Joseph Schwab (1978) but also explicit in Shulman’s mental map (Shulman and Grossman 1988) is that between substantive and syntactic subject-matter knowledge. Substantive knowledge encompasses the key facts, concepts, principles, structures and explanatory frameworks in a discipline, whereas syntactic knowledge concerns the rules of evidence and warrants of truth within that discipline, the nature of enquiry in the field, and how new knowledge is introduced and accepted in that community – in short, how to find out. This distinction comes close to that between content (substantive) and process (syntactic) knowledge, although syntactic knowledge seems to entail greater epistemological awareness than process knowledge. Schwab’s choice of the word ‘syntactic’ is perhaps unfortunate: ‘syntax’ suggests formal structure only, whereas the heuristics of enquiry are at the heart of the intended meaning. Nevertheless, we stick with the word. The substantive/syntactic distinction is important, not least because research with trainee primary teachers (Goulding et al 2002) suggests that syntactic knowledge cannot be adequately addressed or learned within initial teacher education.
Our proposal therefore is that a minimal-but-adequate conceptual taxonomy for policy and practice in mathematics teacher education is the three-fold framework comprising (i) substantive subject-matter knowledge (ii) syntactic subject-matter knowledge (iii) pedagogical content knowledge. These are exemplified in the following sections, with data from our own studies and from Alan Bishop’s writing.
We suggest that the three-fold framework comprising (i) substantive subject-matter knowledge (ii) syntactic subject-matter knowledge (iii) pedagogical content knowledge, though minimal, is adequate for discussion of mathematical knowledge in teaching. We used data from primary classrooms to illustrate each of the three aspects of the framework. We suggest that it can be applied more generally to the analysis of mathematics teaching, and also that it is sufficiently focused to identify how very different kinds of teacher knowledge are made visible in teaching. We would be interested to know, in the first instance, whether other teacher educators share our view concerning the adequacy of this framework, and whether they find it useful – as we do - in their own work in mathematics teacher education.
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