The Microevolution of Mathematical Representations in Children's Activity

In this article, I discuss children's design of mathematical representations on paper, asking how material displays are constructed and transformed in activity. I show that (a) the design of displays during problem solving shapes one's mathematical activity and sense making in crucial ways, and (b) knowledge of mathematical representations is not simply recalled and applied to problem solving but also emerges (whether constructed anew or not) out of one's interactions with the social and material settings of activity. A detailed characterization of student-designed tables of values to solve problems about linear functions is also presented. The role of notational systems in mathematical activity is often assumed to be two-fold: (a) supporting cognitive processing, and (b) mediating communication (Kaput, 1987). Fey (1990) added that, from an auxiliary role, representations can become the object of mathematics itself and yield the study of "unanticipated patterns in concrete situations" (p. 73). Drawing on Skemp's (1979) work, Pimm (1987) detailed the role of notational systems in mathematical activity by listing the following uses that symbols can be put to: "communicating," "recording and retrieving knowledge," "helping to show structure [among ideas]," "allowing routine manipulation to be made automatic," and "making reflection possible" (p. 138). In classical mathematics education, the analysis of symbol use tends to oppose cognitive processing (labeled as internal) to the actual manipulation of inscriptions (labeled as external), giving rise to mapping models of the kind proposed