PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 9, 020108 (2013) Understanding student use of differentials in physics integration problems Dehui Hu and N. Sanjay Rebello* Department of Physics, 116 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506-2601, USA (Received 22 January 2013; published 26 July 2013) This study focuses on students’ use of the mathematical concept of differentials in physics problem solving. For instance, in electrostatics, students need to set up an integral to find the electric field due to a charged bar, an activity that involves the application of mathematical differentials (e.g., dr, dq). In this paper we aim to explore students’ reasoning about the differential concept in physics problems. We conducted group teaching or learning interviews with 13 engineering students enrolled in a second- semester calculus-based physics course. We amalgamated two frameworks—the resources framework and the conceptual metaphor framework—to analyze students’ reasoning about differential concept. Categorizing the mathematical resources involved in students’ mathematical thinking in physics provides us deeper insights into how students use mathematics in physics. Identifying the conceptual metaphors in students’ discourse illustrates the role of concrete experiential notions in students’ construction of mathematical reasoning. These two frameworks serve different purposes, and we illustrate how they can be pieced together to provide a better understanding of students’ mathematical thinking in physics. DOI: 10.1103/PhysRevSTPER.9.020108 PACS numbers: 01.40.Àd mathematical symbols [3]. For instance, mathematicians I. INTRODUCTION often use x; y asR variables and the integrals are often written Mathematical integration is widely used in many intro- in the form fðxÞdx, whereas in physics the variables ductory and upper-division physics courses. Developing could be r, l, and t, each representing a different meaning expertise to use integration in physics is both necessary associated with the corresponding physical system. The and important for students to fully understand physics difference of using symbols in math and physics causes concepts as well as transfer their learning to new situations. many difficulties for students as they use mathematical Studies have reported that students are typically good at notations [4]. Differential terms are context dependent, evaluating a predetermined mathematical integral, but often representing an infinitesimal amount or an infinitesi- often struggle with setting up integrals in physics problems mal change of a physical quantity. Compared with expert [1,2]. Setting up integrals in a physics problem can be physicists, novice students have not developed a sophisti- divided into several steps: setting up the expression for cated understanding of mathematical concepts and they an infinitesimal quantity (e.g., dE, dB), accumulating the have less experience in physics problem solving. In this infinitesimal quantity, determining the variable of integra- study, we investigate students’ reasoning about differen- tion, and turning the integral into a form that can be tials in physics problems requiring integration. We inves- evaluated mathematically [2]. Hence, using integration in tigate student reasoning of this topic through the lens of physics requires students to not only understand the con- resources [5] and conceptual metaphors [6]. cept of an integral as representing a Riemann sum but also There are three main purposes of this paper. First, we to correctly use a differential concept to express relations focus on the mathematical resources that students activate between physical quantities. For example, to construct the with their application of the differential concept in a phys- dE ¼ kdq=r2 expression from the fundamental Coulomb’s ics context. The analysis of mathematical resources pro- E ¼ kq=r2 law , a student needs to develop the understand- vides us with deeper insights into how students apply their dq dE ing of the differential terms (i.e., , ) in the context of mathematical learning to physics problem solving. Second, an electric field. In this paper, we mainly focus on students’ we identify the conceptual metaphors involved in students’ reasoning of the differential terms in physics equations as use of mathematical differentials and prove the existence they solve physics integration problems. of metaphors in students’ higher-level mathematical think- Physicists construct notations based on a specific ing (i.e., calculus). First studied in cognitive linguistics, physical situation instead of following prescriptive conceptual metaphors are a kind of intuitive knowledge that describes mappings between the source domains, usu- *[email protected] ally from our interactions with the physical world, and abstract concepts in target domains such as mathematics. Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- Identifying the conceptual metaphors that appear in student bution of this work must maintain attribution to the author(s) and discourse illustrates how concrete experiential notions the published article’s title, journal citation, and DOI. might affect students’ construction of scientific reasoning. 1554-9178=13=9(2)=020108(14) 020108-1 Published by the American Physical Society DEHUI HU AND N. SANJAY REBELLO PHYS. REV. ST PHYS. EDUC. RES. 9, 020108 (2013) It offers an alternative and perhaps fundamental explana- ‘‘amount of x’’ or ‘‘x increment’’; in other words, it was tion of student thinking. Third, we amalgamate the two more or less the same as x. It is clear that students did not frameworks—resources and conceptual metaphors— understand the symbols of differentiation or the approach which have previously typically been used separately. We to differentiation. Orton pointed out that ‘‘some students illustrate how each framework serves different purposes are introduced to differentiation as a rule to be applied and how they could be fused together to understand student without much attempt to reveal the reasons for and justi- reasoning. fications of the procedure.’’ In another interview about the In the next section, we give a brief review of related integration concept, Orton found that very few students literature on students’ understanding of a differential con- understood integration as the limit of a Riemann sum [1]. cept in the context of physics integration problems. In Artigue et al. [9] administered questionnaires to both Sec. III, we discuss the conceptual framework that we physics and mathematics students to investigate their con- chose for this study. In Sec. IV, we describe our research ceptions about differentials (e.g., dx, dl). In particular, she setting which includes the student population and method- discussed two extreme tendencies in students’ responses. ology used to collect and analyze our data. In Sec. V,we First, the differential element lost all meaning except that it present the results of our study. In Sec. VI, we discuss the indicated the variable of integration which may have possible implications for instruction. In Sec. VII, we dis- excluded other meanings. Second, the differential element cuss the main findings and the limitations of this study, as had material content, such as ‘‘dl is a small length’’ or ‘‘a well as future work. little bit of wire.’’ Artigue argued that the terms (e.g., small and little bit) were associated with ‘‘looseness in reason- II. RELEVANT LITERATURE ing’’ indicating students’ lack of a rigorous understanding of the mathematical concept, even though this reasoning is In this section we review previous studies in two areas convenient and effective in physics. which are closely related to our work. First, we briefly discuss several prior studies about students’ understanding Several other studies [10,11] in math education also of the differential concept in mathematics education. Since showed that students do not have a deep conceptual under- all of the students in our calculus-based physics classes standing of calculus concepts. Students learn the concepts have previously taken calculus, previous research in as procedures and their understanding is often bereft of mathematics education provides us with the necessary contextual meaning. background to understand their prior mathematics knowl- edge and what we could possibly observe in their mathe- B. Student application of calculus in physics matical thinking in physics contexts. It also allows us to see In physics education research, several studies indicated what mathematicians value in students’ understanding of students’ lack of ability to apply the integral concept the differential concept and reflect on which of these successfully in physics contexts [2,12,13]. In mathematics, attributes physicists might perceive as valuable in students’ students often learn how to follow an algorithm to evaluate thinking. Next, we provide an overview of the literature on an integral and operate symbols without referring to any students’ difficulties with setting up integrals from studies concrete physical situations. Some of their conceptual in physics education research. Our work is motivated and difficulties with physics integration problems could be built upon these previous studies. attributed to their insufficient understanding of mathemat- ics concepts, such as their understanding of an integral as a A. Student understanding of calculus concepts sum and the area-integral relation