A calculus student’s understanding of graphical approach to the derivative through quantitative reasoning

Aytug Ozaltun-Celik

doi:10.31129/lumat.9.1.1663

Abstract

The concept of derivative is used in many areas including applied problems and requiring mathematical modelling in different disciplines. One of the most important approaches for teaching the derivative is to support students in visualizing the concept. Also, it is necessary to shift researchers and teachers’ focuses to students’ dynamic mental actions while learning derivative in order to conduct effective teaching process. With this necessity, I focused on the perspective of quantitative reasoning related to the graphical approach to the derivative. This study aims to reveal a calculus student’s mental actions related to the graphical approach to the derivative. The data were collected from a first-year calculus student engaged in the task requiring graphical interpretation of the derivative. Results showed that the student’s understanding of the slope shaped her inferences about the tangent line because the quantity of ratio is prior knowledge for learning the instantaneous rate of change. Besides, as the student had the idea of correspondence related to the concept of function, she had difficulties in interpreting the global view of the derivate. This result suggests that having global view of the derivative requires a strong understanding of function and rate.

Highlights

The derivative of a function is a fundamental concept for the basis of calculus (García et al, 2011) and is used in many areas including requiring mathematical modeling of several situations in different disciplines such as engineering, physics, economics, etc
Thompson (1990) has emphasized the idea that a rate is conceived of as constituting a functional relationship may be a foundation for the derivative in calculus because it is consistent with conceptions of a single-variable derivative evaluated at a point as being the slope of a tangent line
I illustrate a calculus student’s mental actions based on the perspective of quantitative reasoning while engaging in the task including the graphical approach to the concept of derivative

Summary

Introduction

The derivative of a function is a fundamental concept for the basis of calculus (García et al, 2011) and is used in many areas including requiring mathematical modeling of several situations in different disciplines such as engineering, physics, economics, etc This concept was historically constructed as a way to represent rate of change which explains how one quantity changes in relation to another quantity (Weber et al, 2012). It would require that students be able to meaningfully connect the ideas of slope (instantaneous rate of change) in both algebraic ratio and geometric ratio for perceiving as an internal concept (Nagle et al, 2019) Based on these ideas, I focused on the graphical approach to the concept of derivative and the quantities related to the graph. This study proposes a task implementation for supporting students’ mental actions regarding instantaneous rate of change, on the other hand, presents reveal a calculus student’s mental actions related to the graphical approach to the concept of derivative

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