Algebraic reasoning through exploration of the commutative principle

Developing early algebraic reasoning through exploration of the commutative principle

Student transition from arithmetical understandings to algebraic reasoning is recognized as an important but complex process. An essential element of the transition is the opportunity for students to make conjectures,

justify, and generalise mathematical ideas concerning number properties. Drawing on findings from a classroom-based study, this article outlines how the commutative principle provided an appropriate context for young students to learn to make conjectures and generalisations. Tasks, concrete material and specific pedagogical actions were important factors in students’ development of algebraic reasoning.

For those students who complete their schooling with inadequate algebraic understandings access to further education and employment opportunities is limited. An ongoing concern for these students internationally, has

resulted in increased research and curricula attention of the teaching and learning of algebraic reasoning. To address the problem one response has been to integrate teaching and learning of arithmetic and algebra as a unified curriculum strand in policy documents (e.g., Ministry of Education 2007, National Council of Teachers of Mathematics 2000). Within the unification of arithmetic and algebra, students’ intuitive knowledge of patterns and numerical reasoning are used to provide a foundation for transition to early algebraic thinking (Carpenter, Franke, & Levi 2003).

Importantly, this approach requires the provision of opportunities for students to make conjectures, justify, and generalise their mathematical reasoning about the properties of numbers. Carpenter and his colleagues explain that deep conceptual algebraic reasoning is reached when students engage in “generating [mathematical] ideas, deciding how to express them …justifying that they are true” (2003, 6). We know, however, from exploratory studies (e.g., Anthony & Walshaw 2002, Warren 2001) that currently many primary age students have limited classroom experiences in exploring the properties of numbers. These studies illustrated that more typically students experience arithmetic as a procedural process. This works as a cognitive obstacle for students when later they need to abstract the properties of numbers and operations. These studies also investigated student application of the commutative principle and illustrated that many students lack understanding of the operational laws. Both Anthony and Walshaw’s study of Year 4 and Year 8 students and Warren’s studies involving Year 3, Year 7 and Year 8 students demonstrated that many students failed to reach correct generalisations regarding commutativity. The students recognised the commutative nature of addition and multiplication; but also thought that subtraction and division were commutative. Anthony and Walshaw showed that although students offered some explanation of the commutative property none offered generalised statements nor were many students able to use materials to model conjectures related to arithmetic properties. These researchers concluded that very few students were able to draw upon learning experiences which bridged number and algebra.

Nevertheless studies (e.g., Blanton & Kaput 2003, Carpenter et al. 2003) which involved teaching experiments provided clear evidence that young children are capable of reasoning in generalised terms. These studies illustrated that they can learn to construct and justify generalisations about the fundamental structure and properties of numbers and arithmetic. Importantly, they demonstrated that when instruction is

targeted to build on students’ numerical reasoning they can successfully construct and test mathematical conjectures using appropriate generalisations and justifications.

The study sought to explore how student exploration of the commutative principle deepened their understanding of arithmetic properties whilst also supporting their construction of conjectures, justification and generalisations. Similar to the findings of Anthony & Walshaw (2002) and Warren (2001), many of these students initially failed to reach correct generalisations regarding commutativity. Extending the task beyond true and false number sentences and the introduction of equipment led to student modelling of conjectures and provision of concrete forms of explanatory justification. Importantly, teacher interventions were required to shift students to make generalised statements about the commutative principle. Results of this study support Carpenter and Levi’s (2000) contention that use of number sentences provides students with access to a notational system for expressing generalisations precisely. The symbolic representation of their conjectures coupled with the use of equipment and teacher press for generalisation led to more specific student generated generalisations.

Many of the students in the study deepened their understanding of arithmetic properties. However, the small proportion of students who continued to overgeneralise to include subtraction and division indicate the need for multiple opportunities over an extended period of time for students’ to develop deep understanding of operational laws.

Findings of the study affirm that the context of the commutative principle can provide students with effective opportunities to make and represent conjectures, justify and generalise. Appropriate tasks, concrete material and teacher intervention supported students to develop their understanding of the commutative principle. Opportunities to develop explanations with concrete material and use notation to represent conjectures led to students developing further generalisations.