ABSTRACT

Many researchers have stressed the embodied nature of mathematical understanding. Here we explore how embodied knowledge may evolve as students learn a basic calculus concept: the rate of change. We examined undergraduate students with different levels of calculus knowledge working in pairs to model the rate of change in an everyday phenomenon. Our findings revealed substantial differences between advanced and introductory students in how they represented rate of change with their speech and gestures. In particular, the advanced students’ embodied representations showed alignment with the formal symbolic process of integration and reflected more advanced ways to coordinate the relation between multiple changing variables. For example, advanced students often made gestures that represented a “disk” when explaining how the rate at which water rose in a bottle was related to the change in height. Thus, both introductory and advanced students demonstrated embodied knowledge through speech and gestures, but the advanced students constructed different concrete representations that reflected an advancement in their embodied knowledge. Our findings are relevant to calculus education and to the study of gestures in mathematics learning.

摘要

许多研究人员强调了数学理解的具体本质。在这里，我们探讨了当学生学习基本的微积分概念时，具体知识如何演变：变化率。我们检查了具有不同微积分知识水平的本科生，他们结对工作，以模拟日常现象的变化率。我们的研究结果揭示了高级学生和入门学生在如何用语言和手势表示变化率方面存在很大差异。特别是，高年级学生的具身表征显示出与整合的形式符号过程一致，并反映了协调多个变化变量之间关系的更先进的方式。例如，高年级学生在解释瓶中水的上升速度与高度变化之间的关系时，经常会做出代表“圆盘”的手势。因此，入门学生和高级学生都通过言语和手势展示了具身知识，但高级学生构建了不同的具体表征，反映了他们具身知识的进步。我们的研究结果与微积分教育和数学学习中的手势研究有关。