Developing Model‐Based Reasoning

Abstract

Key elements of the structure and function of models in mathematics and science are identified. These elements are used as a basis for discussing the development of model‐based reasoning. A microgenetic study examines the beginnings of model‐based reasoning in a pair of fourth‐ and fifth‐grade children who solved several problems about chance and probability. Results are reported in the form of a cognitive model of children's problem‐solving performance. The cognitive model explains a transition in children's reasoning from tacit reliance on empirical regularity to a form of model‐based reasoning. Several factors fostering change in children's thinking are identified, including the role of notations, peer interaction, and teacher assistance. We suggest that model‐based reasoning is a slowly‐developing capability that emerges only with proper contextual and social support and that future study should be carried out in classrooms, where these forms of assistance can also be part of the object of study. Model‐based reasoning is a significant intellectual milestone because it bridges the worlds of personal, intuitive knowledge, on the one hand, and mathematical‐scientific theory, on the other. However, across disciplines, consensus is still forming about what model‐based reasoning comprises, and there is little knowledge about its ontogenetic origins or how it develops. We consider analogy as the core of modeling, because in model‐based reasoning a system in one domain is used to understand a system in another. To understand how models come to play a role in reasoning, it is important to initiate study of their origins. Accordingly, we report a microgenetic study examining the beginnings of model‐based reasoning in a pair of young children solving problems about chance and probability. In this study we are engaged in the enterprise of modeling the development of modeling. That is, we report our results in the form of a cognitive model of children's problem‐solving performance that explains a transition in reasoning from a tacit reliance on empirical regularity to a form of model‐based reasoning. It is important to note the two distinct meanings for the term model used in this article. The first describes how children come to understand and appropriate a system of reasoning exemplified in practices of modeling. The second describes a research tool, a model of human reasoning—specifically, how children in this study began to use models of probability to reason about uncertain events. In this report, we use the terms model or model‐based reasoning to refer to the former interpretation, whereas references to a cognitive model denote the simulation of children's thinking—in this case, implemented as a computer program. Before describing the empirical work, we first identify some key elements of the structure and function of models. Next, these elements of modeling are used as the basis for generating some conjectures about the development of model‐based reasoning. We describe a task that we used as a window to understanding progression in student reasoning toward reliance on models as tools for thought. We present our rationale for developing cognitive models of student performance and explain some choices concerning the implementation of the cognitive model reported here. Finally, we turn to the children's performance on chance and probability tasks and explain how that performance illuminates both what children do not understand about models and the kinds of relevant knowledge that they are acquiring.