Exploring early secondary students' algebraic generalisation in geometric context

Exploring multiple representations of a relationship between two variables has been shown to help students learn conceptually about functions in algebra. This paper discusses an investigation of 102 Year 7 students, prior to their study of algebra, which considered their making sense of a linear functional relationship in two geometric contexts - through generalisation of a 2-dimensional figural growing pattern and through representation of its variables on a Cartesian plane. This provided the opportunity to consider how algebraic and geometric understandings might reinforce each other in different representations of the same function. Students used a variety of ways to describe the structure of the figural pattern and 40 were able to use this to create an explicit rule, with over 20 already using a symbolic expression or full equation. Two thirds of the students created an incorrect graphical representation of the relationship and the other third drew either a column or line graph. Implications for middle-school algebra learning are discussed.