Students' conceptual understanding of derivatives in freshmen calculus. (c2015)

Abstract

The freshmen calculus curriculum has witnessed many changes in the past few years due to the development of computer technology and dynamical mathematical software. This study aims to examine the students’ conceptual understanding of derivative in a calculus I course. Fifty- two students participated in this study consisting of 27 males and 25 females. All students attended a calculus I course at a private Lebanese University. Two groups, each of 26 students, are considered: a control group (taught by instructor X) learning derivatives using Book 1 that emphasizes the symbolic approach of the concept, and an experimental group (taught by instructor Y) learning derivatives using Book 2, which emphasizes the multiple- representations approach of the concept. In the experimental group, cooperative learning, technology (Autograph), and a series of activities incorporating the APOS (action- process, object- schema) levels were integrated in the teaching and learning of derivative. Data were collected using qualitative and quantitative methods such as the content analysis of two calculus books (Book 1 and Book 2), observations, and questionnaires on derivatives administered to all students before and after the implementation of the unit on derivatives. Moreover, data was collected using a test, consisting of five conceptual-understanding based problems on derivatives, and clinical interviews conducted with twelve students. Results show that students in the experimental group have better conceptual understanding of the derivative concept than those in the control group. Many students in the experimental group seem to have an object conception and almost a comprehensive understanding of the derivative particularly concerning the slope of a tangent line at a point, the instantaneous rate of change, and the relation between a function and its derivative. However, many students in the control group had deficiencies in their understanding, showing an action/ process conceptions of the derivative. Findings in this study are parallel to the findings of several studies. In addition, the quantitative analysis of both the questionnaires and the tests reveal significant statistical differences in the mean scores between the two groups in favor of the experimental group. Finally, the observations reveal that many students in the experimental group were more interested and motivated to learn mathematics. On the other hand, some experimental students resisted the approach used and found it difficult and demanding. This study highlights the need for all stakeholders to work collaboratively to integrate technology in teaching calculus and to consider changes in the calculus curriculum and the books adopted to encourage the use of multiple representations.