ABSTRACT

Despite the increasing amount of research on students’ quantitative reasoning at the secondary level, research on students’ quantitative reasoning at the undergraduate level is scarce. The present study used task-based interviews to examine 16 high-performing undergraduate calculus students’ quantitative reasoning in the context of solving three related rates of change problems-two geometric and one non-geometric. A qualitative analysis of verbal responses and work written by the students when solving the three problems revealed that 15 students created and used diagrams to support their quantitative reasoning in the geometric problems, and that the creation of these diagrams helped the students to solve the two problems. In addition, seven students exhibited lack of facility with the product or quotient rule of differentiation. We argue that the use of diagrams in calculus instruction at the undergraduate level could play an important role in supporting students’ quantitative reasoning in topics that require students to make sense of relationships between quantities such as related rates of change problems. Furthermore, we argue that helping students develop greater facility with rules of differentiation such as the product or quotient rule could increase students’ success in many calculus application problems, including related rates of change problems. Directions for future research are included.

摘要

尽管中学阶段对学生定量推理的研究越来越多，但在本科阶段对学生定量推理的研究却很少。本研究使用基于任务的访谈法，在解决三种相关的变化率（两个几何问题和一个非几何问题）的背景下，检查了16位高性能本科生的微积分学生的定量推理。定性分析学生在解决这三个问题时的口头反应和工作成果表明，有15名学生创建并使用了图表来支持他们在几何问题中的定量推理，并且这些图表的创建帮助学生解决了这两个问题。另外，七名学生表现出缺乏产品或商的差异化规则的设施。我们认为，在本科阶段的微积分教学中使用图表可能会在支持学生定量推理方面发挥重要作用，这些主题要求学生理解数量之间的关系，例如相关的变化率问题。此外，我们认为，通过差异规则（例如乘积或商数规则）帮助学生发展更大的设施，可以提高学生在许多微积分应用问题（包括相关的变化率问题）中的成功率。包括未来研究的方向。我们认为，通过差异规则（例如乘积或商数规则）帮助学生发展更大的设施，可以提高学生在许多微积分应用问题（包括相关的变化率问题）中的成功率。包括未来研究的方向。我们认为，通过差异规则（例如乘积或商数规则）帮助学生发展更大的设施，可以提高学生在许多微积分应用问题（包括相关的变化率问题）中的成功率。包括未来研究的方向。