Starting points: understanding children's pre-instructional intuitions about function tables

Functional thinking is an established route into algebra. However, the learning mechanisms that support the transition from arithmetic to functional thinking remain unclear. In the current study we explored children's pre-instructional intuitive reactions to functional thinking content, relying on a conceptual change perspective and using mixed methods. The sample included 20 grade 3 students and 24 grade 5 students. First, we assessed children's arithmetic skills and intuitive responses to generalisation tasks involving variation tables. The quantitative analysis showed that students' arithmetic skills correlated with functional aspects such as the following: identifying, and expressing function rules with words but not with the symbolic expression of function rules. The qualitative analysis revealed that students constructed framework theories that generated different intuitive conceptions of the algebraic ideas involved in noticing and expressing generalisation. Students' conflicts were concentrated in areas that determine key differences between arithmetic and algebra, such as generalisation, indeterminate quantities, and variable notation. We discuss how these results contribute to explaining the construction of algebra concepts.