Devin's Construction of a Multiplicative Double Counting Scheme: Dual Anticipation of Start and Stop

In this case study with Devin (pseudonym), which was part of a larger, constructivist teaching experiment with students identified as having learning difficulties in mathematics, we examine how a fourth grader constructed a dual anticipation involved in monitoring when to start and when to stop the simultaneous count of composite units (numbers larger than 1) in multiplicative tasks. We postulate that such a dual anticipation underlies the first, Multiplicative Double Counting (mDC) scheme (Tzur et al., 2013) that marks children's conceptual progress from additive to multiplicative reasoning. Data from three teaching episodes with Devin focus on his anticipation of the start/stop features of his double counting activity. We discuss theoretical implications of these findings in terms of similarity between the dual anticipation in additive and multiplicative reasoning, and practical implications in terms of task design and sequencing. [For the complete proceedings, see ED583608.]