2013 Conference Proceedings - University of Nevada, Las Vegas

Figure 3. (A) Original drawing for 3 x 3. (B) Modified drawing for 4 x 4. Squares indicateadditions to original drawing.When asked to do the same thing for the problems 4 × 4, one teacher drew Figure 1(B), stating“I knew the answer to 3 × 3, and I knew how to count on. So I added one to every group [of 3],and I added a new group [of 4].” In this case, the parent genes were (a) her knowledge of 3 × 3(or of groups of dots in general) and (b) her counting on strategy. When asked why she used thisapproach, the teacher responded that “They [the problems] were close together, so it was easy todo.” In other words, the environment (problem) guided the notion of “fitness” of the idea. Thisnew approach was used later when this student took a geometric approach to this problem:Working in base-5, consider the sequence P 0 , P 1 , P 2 …. Find P 110 if:(a) P 0 = 0, and(b) P n+1 = P n + n + (n+1), for n ≥ 0This student calculated some terms by hand and recognized that P n are the square numbers. Shedemonstrated it as in Figure 2, which is very like Figure 1(B): in each case, one was added toeach (vertical) “group” in the original situation (P 3 ), and another new “group” (of size n+1) wasadded to get to the new situation (P 4 ). While not all genotypes will be used in multiple situations,the “transfer” of this genotype between situations indicates that this genotype is a thing in itself.It should be noted that it is often difficult to make these types of genotypic observationsbecause it needs a microgenetic (Kuhn, 1995) approach to observations. While this is possible inresearch and in classrooms, it requires a different stance on the part of the observer. It alsorequires a different approach to formative assessment, one in which the teacher looks not only atwhat problems can be solved, but also looks in detail at how they are solved.Multiplication in the LiteratureSteffe (1994) is an example of what happens when one focuses only on creating a taxonomyof the various aspects of learning think multiplicatively. The components identified (uniting,Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 193