2013 Conference Proceedings - University of Nevada, Las Vegas

student learning calls for different approaches to instruction.The first point is related to Vygotsky’s notion of zone of proximal development (ZPD)(Vygotsky, 1978). Although Vygotsky did not put it in terms of genotype vs. phenotype, he didnote that, although two students may have equal “mental age”, their future performance coulddiffer greatly; this is part of his argument for the existence of a ZPD. Hence, it is important thatteachers assess something more than just phenotypes: Mere performance on problems is notnecessarily a good indicator of future learning.Second, planning for instruction is neither merely about careful presentation (a la thetransmission model of teaching) nor merely about setting up an environment to explore. Whilethe latter is closer to the mark, “setting up an environment” could be done solely by looking atphenotypes. Care must be taken, however, that the environment will present opportunity forchanges in genotype, and this happens only when there are appropriate opportunities forgenotype changes and the combination of existing genes.The usual curriculum design process in schools - which does not take into account the actualunderstandings of the students, but rather posits a series of problems to be mastered - is not likelyto consider the evolution of student genotypes. Successful mathematics curricula, such as thosedescribed in Fosnot & Dolk (2001) or Kamii (2000), while recognizing an overall structure oftopics, also explicitly take into account student reasoning, and do so in ways that respond tostudent genotypes as well as phenotypes. Even though a full examination of curriculumdevelopment is beyond the scope of this paper, it is worth noting that a “well-constructed”sequence of problems is one which takes into account a genotypic view of student understanding.Multiplication: A Classroom ExampleCertified childhood teachers in a graduate course focusing on the understanding of basicmathematical operations were working with alphabitia (Bassarear, 2011) which essentially asksteachers to rediscover and explore basic operations in base-5. While studying multiplication,there were several opportunities to observe these teachers developing new (genotype) ideas as anoffspring of prior ideas. During this process, artifacts from these teachers were collected andsemi-structured interviews were performed with the teachers to gain insight into their work.One (rather non-standard) demonstration of the combination of existing ideas occurred earlyon; the teacher’s work is illustrated in Figure 1. The teachers were asked to justify that 3 × 3 wasequal to 14, in base-5. Almost all the teachers drew a picture that was similar to Figure 1(A).Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 192