ABSTRACT

The purpose of this study is to investigate how students interpret expressions from calculus statements in the graphical register. To this end, I conducted 150-minute clinical interviews with 13 undergraduate mathematics students who had completed at least one calculus course. In the interviews, students evaluated six calculus statements for various real-valued functions depicted in graphs in the Cartesian plane. From my analysis of these interviews, I found four distinct interpretations of expressions in the graphical register that students used in this study while evaluating the statements using the graphs. I describe the characteristics of these four interpretations, which I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. For some students, the use of these interpretations supported their graphical reasoning and correct evaluations of the statements. For other students, the use of some interpretations rather than others presented obstacles to their graphical understanding of the expressions in the statement. For instance, seven of the students never used a magnitude interpretation (interpreting an expression as a distance in the graph), even when working with difference expressions. I discuss implications of these findings for teaching with graphs across levels and directions for future research.

摘要

本研究的目的是调查学生如何解释图形寄存器中微积分语句中的表达式。为此，我对至少完成了一门微积分课程的 13 名本科数学学生进行了 150 分钟的临床访谈。在访谈中，学生评估了笛卡尔平面中图形中描述的各种实值函数的六个微积分语句。从我对这些访谈的分析中，我发现学生在本研究中使用图形评估语句时使用的图形寄存器中有四种不同的表达解释。我描述了这四种解释的特征，我称之为 (1) 名义的，(2) 序数的，(3) 基数的，和 (4) 量级的。对于一些学生来说，这些解释的使用支持了他们的图形推理和对陈述的正确评估。对于其他学生来说，使用某些解释而不是其他解释会阻碍他们对陈述中表达的图形理解。例如，七名学生从未使用过幅度解释（将表达式解释为图中的距离），即使在使用差异表达式时也是如此。我讨论了这些发现对未来研究的跨层次和方向的图形教学的影响。七名学生从未使用过幅度解释（将表达式解释为图中的距离），即使在使用差异表达式时也是如此。我讨论了这些发现对未来研究的跨层次和方向的图形教学的影响。七名学生从未使用过幅度解释（将表达式解释为图中的距离），即使在使用差异表达式时也是如此。我讨论了这些发现对未来研究的跨层次和方向的图形教学的影响。