Test of Understanding Graphs in Calculus: Test of Students’ Interpretation of Calculus Graphs

Abstract

Studies show that students, within the context of mathematics and science, have difficulties understanding the concepts of the derivative as the slope and the concept of the antiderivative as the area under the curve. In this article, we present the Test of Understanding Graphs in Calculus (TUG-C), an assessment tool that will help to evaluate students' understanding of these two concepts by a graphical representation. Data from 144 students of introductory courses of physics and mathematics at a university was collected and analyzed. To evaluate the reliability and discriminatory power of this test, we used statistical techniques for individual items and the test as a whole, and proved that the test's results are satisfactory within the standard requirements. We present the design process in this paper and the test in the appendix. We discuss the findings of our research, students' understanding of the relations between these two concepts, using this new multiple-choice test. Finally, we outline specific recommendations. The analysis and recommendations can be used by mathematics or science education researchers, and by teachers that teach these concepts.

Testing students ability to use derivatives, integrals, and vectors in a purely mathematical context and in a physical context

“…A decade later, Beichner [47] designed the test of understanding graphs in kinematics (TUG-K), a multiplechoice test based on six dimensions corresponding to different categories of typical students' mistakes that he identified: graph as picture errors, confusion between slope and height, variable confusion, non-origin slope errors, area ignorance, and confusion among area, slope, and height. Recently, Zavala et al [48] have proposed a modified version of the TUG-K improving the parallelism between the different dimensions of the test, and Dominguez et al [49] have developed the test of understanding graphs in calculus (TUG-C), the counterpart of the TUG-K in a purely mathematical context.…”

Application of the Complementarities of Two Theories, APOS and OSA, for the Analysis of the University Students’ Understanding on the Graph of the Function and its Derivative

“…Some researchers (Amit, & Vinner, 1990;Asiala, Cotrill, Dubinsky, & Schwingendorf, 1997;Dominguez, Barniol, & Zavala, 2017) discuss specific difficulties that students have with the graphical representation of the derivative. These authors explain some difficulties students have with basic calculus concepts.…”

Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative

“…Third, we designed a written test comprised of 4 questions to compare the performance of the experimental group with the performance of another equivalent student group (control group). The test was designed by consulting relevant resources in mathematics education, in particular the teaching and learning of derivative (e.g., Asiala et al, 1997;Cooley et al, 2007;Dominguez et al, 2017).…”

“…Previous studies reported although students' procedural knowledge about differentiation is usually adequate, most of them have little intuitive or conceptual knowledge of the derivative (Asiala, Cottrill, Dubinsky, & Schwingendorf, 1997;Clark et al, 1997;Dominguez, Barniol, & Zavala, 2017;Ferrini-Mundy & Graham, 1994;Orton, 1983;Tall, 1993;Thompson 1994;Uygur & Özdaş, 2005). It seems that several students perform poorly because they are unable to adequately handle information given in symbolic form which represents abstract entities, for example functions, or/and they lack adequate schema or frameworks, which helps to organize and link different objects (Maharaj, 2005).…”