Mathematical Reasoning

In recent times, the field of mathematical reasoning has been the focus of increased research from a number of quarters. Cognitive psychologists, mathematics educators, mathematicians, philosophers, and linguists are demonstrating the important role of mathematical reasoning in human learning and development. This special issue highlights some of these different perspectives on mathematical reasoning. Nunez and Lakoff address a new domain of study, namely, the cognitive science of mathematics, which they define as "the study of mathematical ideas, from the perspective of research on our largely unconscious everyday conceptual systems as they are embodied in the human brain" (p.85). The authors illustrate their approach by presenting a comprehensive cognitive analysis of continuity, one of the most important ideas in 20th Century mathematics. A focus on children's abilities to reason by analogy appears in a paper by Stephen White, Patricia Alexander, and Martha Daugherty, and in a paper by Lyn English. Such reasoning is one of the most important mechanisms underlying human thought, and is particularly important in the learning of mathematics, where learners must see connections and relationships between mathematical ideas and apply this understanding to the solution of new problems. Reasoning with physical or visual representations of mathematical structures and concepts is a significant component of mathematical learning. Reasoning with conceptual representations forms the focus of Karen Fuson's paper. In presenting a model of conceptual-support nets, Fuson addresses issues of matches and mismatches among: conceptual supports, the conceptual and procedural accessibility of various multi-digit algorithms, learners' solution methods, their level of language development, and their mathematical achievement.