In this paper, we present data from two iterative teaching experiments involving students' constructions of four basic counting problems. The teaching experiments were designed to leverage the generalizing activities of relating and extending to provide students with opportunities to reflect on initial combinatorial activity when constructing…

Chess experts remember meaningful knowledge in the form of networks or patterns. Applied to mathematics instruction, effective classroom approaches can use investigation to identify patterns or rules. Described are a class activity and a small-group activity to investigate addition of signed numbers and linear relationships. (MDH)