# Summary of Reference

**Author:**

Anthony, Glenda & Walshaw, Margaret

**Title:**

Swaps and Switches: Students’ understandings of Commutativity

**Bibliographic data:**

In* *B. Barton, K. Irwin, M. Pfannkuch, & M. Thomas (Eds.)(2002). *Mathematics Education in the South Pacific*: Proceedings of the 25th annual conference of the Mathematics Education Research Group of Australasia, (pp.91-99). Sydney: MERGA

**Summary:
**The development of algebraic reasoning is an important aspect of students’ mathematical development. The implicit use of algebraic reasoning is commonly characterised by reversing or changing the order in addition. The authors of this paper contend that this reasoning only becomes explicit when the student makes connections between the structure of algebra and the structural properties of the number system. This important distinction between implicit and explicit use of algebraic reasoning is about the student recognising that “preliminary ideas of algebra are merely the generalisation of arithmetic”.

They write that the most important algebraic ideas are to do with equivalence and the transformation of variables.

In reviewing the responses of a group of students participating in a range of NEMP tasks on addition, subtraction, and multiplication, the responses to the addition problem reflect a lack of awareness of the commutative law, a difference between year 4 and year 8 students in confidence and verbosity, and few generalised answers. Overall the authors conclude that student responses are mostly procedural and the extent of student understanding is unclear.

In the subtraction items in particular students frequently mis-represented the problem when asked to model it with cubes. The authors comment that “the frequency of these mis-representations suggest that modelling basic subtraction problems was not firmly established in the students’ experiences. The inability to directly model such elementary problems with concrete materials will invariably hinder the students’ propensity for visualisation and abstraction deemed necessary for algebraic thinking”. In the multiplication item they note that students rely on additive reasoning developed intuitively rather than multiplicative reasoning. They suggest that these responses reflect few opportunities to study multiplicative structures. This is problematic as it limits students’ conceptual development. They report that few students exhibited sophisticated multiplicative reasoning and commutative understanding.

The paper concludes that the building up of students’ structural understanding is limited through a focus in the curriculum on procedural learning. This has particular problems for the development of algebraic reasoning. The authors suggest that “forms of conjectorising and generalising about the properties of numbers need to be encouraged” and that “students need opportunities to make explicit their understanding of *why* number properties such as commutativity ‘hold good’”.