On the Construction of Learning Trajectories of Children: The Case of Commensurate Fractions

Abstract

Learning trajectories are presented of 2 fifth-grade children, Jason and Laura, who participated in the teaching experiment, Children's Construction of the Rational Numbers of Arithmetic. 5 teaching episodes were held with the 2 children, October 15 and November 1, 8, 15, and 22. During the fourth grade, the 2 children demonstrated distinctly different partitioning schemes-the equi-partitioning scheme (Jason) and the simultaneous partitioning scheme (Laura). At the outset of the children's fifth grade, it was hypothesized that the differences in the 2 schemes would be manifest in the children's production of fractions commensurate with a given fraction. During the October 15 teaching episode, Jason independently produced how much 3/4 of 1/4 of a stick was of the whole stick as a novelty, and it was inferred that he engaged in recursive partitioning operations. An analogous inference could not be made for Laura. The primary difference in the 2 children during the teaching episodes was Laura's dependency on Jason's independent explanations or actions to engage in the actions that were needed for her to be successful in explaining why a fraction such as 1/3 was commensurate to, say, 4/12.

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