Empirical generalisation as an inadequate cognitive scaffold to theoretical generalisation of a more complex concept

The impact of prior learning on new learning is highlighted by the case of Dean, a Year 8 student who developed his own method to find the sum of the interior angles of a polygon without knowing why his method worked. Enriched transcripts and visual displays of the cognitive, social (Dreyfus, Hershkowitz, & Schwarz, 2001) and affective elements (Williams, 2002) of Dean's interrupted abstraction process informed the identification of factors that inhibited Dean's constructing process. It was found Dean possessed an empirical, not theoretical, generalization (Davydov, 1990) about sums of interior angles of triangles that was an inadequate cognitive artifact for constructing the new more complex theoretical generalization. The study suggests use of tasks designed with the opportunity develop assumed knowledge in conjunction with new concepts. This case forms part of a broader study of factors that promote or inhibit the process of student-initiated and student-directed abstraction of mathematical concepts, without mathematical input from an external source (like the teacher, text-book, or students external to the group) during the abstraction process. Dean's case illustrates an interesting phenomena; his inability to utilize the procedure specified and demonstrated by th