Progression towards understanding functions: What does spatial generalization contribute?

We focus on a ‘typical’ task in which students have to give a functional generalisation in algebraic form of a growing sequence of spatial structures. We analyse the contribution of this task to a coherent knowledge of functions. Despite a plethora of research about misconceptions and the teaching of functions, little is known about the overall growth of students’ understanding of functions throughout schooling. We aim to map the development of students’ understanding of concepts which contribute to understanding functions in two different curriculum systems: the UK and Israel. The research uses a survey
instrument that was developed in collaboration with a group of teachers and the task for this paper is one of six that span several routes to understanding functions. Our data appears to contradict some other studies as well as to suggest conjectures about how students' willingness to use covariational reasoning depending to some extent on task features.