Lesh R., Doerr H. M., Carmona G. & Hjalmarson M. (2003) Beyond constructivism. Mathematical Thinking and Learning 5(2–3): 211–233.

Lesh R., Doerr H. M., Carmona G. & Hjalmarson M.
(

2003)

Beyond constructivism.
Mathematical Thinking and Learning 5(2–3): 211–233.
In a recent book titled Beyond Constructivism: A Models & Modeling Perspective on Mathematics Problem Solving, Learning & Teaching (Lesh & Doerr, 2003a), the concluding chapter describes a number of specific ways that a models and modeling perspective moves significantly beyond the implications that can be drawn from constructivist theories in the context of issues that are priorities to address for teachers, curriculum developers, or program designers. In that chapter (Lesh & Doerr, 2003b), the following topics were treated as cross-cutting themes: (a) the nature of reality, (b) the nature of mathematical knowledge, (c) the nature of the development of children’s knowledge, (d) the mechanisms that drive that development, (e) the relationship of context and generalizability, (f) problem solving, and (g) teachers’ knowledge and the kinds of teaching and learning situations that contribute to the development of children’s knowledge. In this article, we organize our comments directly around the preceding topics and describe how a models and modeling perspective provides alternative ways of thinking about mathematics teaching and learning that enable teachers, researchers and others to produce useful and sharable conceptual tools that have powerful implications in the context of decision-making issues that are of priority to practitioners.

Steffe L. P. (2004) On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning 6(2): 129–162. https://cepa.info/2113

Steffe L. P.
(

2004)

On the construction of learning trajectories of children: The case of commensurate fractions.
Mathematical Thinking and Learning 6(2): 129–162.
Fulltext at https://cepa.info/2113
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.