Specialised Content Knowledge: The Convention for Naming Arrays and Describing Equal Groups' Problems

Specialised Content Knowledge (SCK) is defined by Ball, Hoover-Thames, and Phelps (2008) as mathematical knowledge essential for effective teaching. It is knowledge of mathematics that is beyond knowledge which would be required outside of teaching; for instance, the capacity to determine what misconception(s) may lie behind an error in calculation. Such knowledge should be core business of teachers of mathematics, and any perceived shortfall in SCK viewed as problematic. The research reported on here is part of a large study about multiplicative thinking involving approximately two thousand children between nine and twelve years of age and their teachers. Data were generated from semi-structured interviews and a written diagnostic assessment quiz. As part of that large project, forty-four Australian and New Zealand primary and middle school teachers were asked to respond to student work related to multiplicative thinking, particularly to concepts of numbers of equal groups and commutativity. Participants' responses reflected confusion about a pivotal piece of SCK, the convention for naming arrays. As well, questionable assumptions about the children's work samples were made. Given that there is not unanimous agreement amongst mathematics educators about naming conventions, these observations may not be surprising. Due to the sample size, broad generalisations cannot be made, but the results suggest that many teachers may have limited SCK with regards to the important mathematical area of Multiplicative Thinking (MT).