From Concepts to Problem-Solving

Connecting concepts to problem-solving Stephen Kanim Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003

Traditional quantitative problems of concepts. In contrast, novices tend to the type commonly found at the end of classify problems based on the chapters in physics textbooks are surface features of the problem. assigned to students most introductory  Experts use multiple physics courses. Many students use a representations to clarify a problem. formula-driven approach to solve these Mathematical solution typically problems that does not rely on follows a restatement of the problem understanding underlying physics based on general principles. concepts and that does little to Novices often start with an equation, encourage the problem-solving skills substituting given values and then employed by experts. In another paper looking for additional equations to presented at this conference, we gave an give other values. Many students see example from electric circuits to the concepts as offering at best some illustrate the use of “bridging exercises” guidelines for selection of equations. as part of students’ homework to encourage students to solve problems by Results reported from the Maryland starting with developed physics concepts Physics Expectations Survey of students and models.1 In this paper, we describe enrolled in introductory physics courses our attempts to use the same approach in are also pertinent to understanding the context of electrostatics. student approaches to problem solving.3 After completing an introductory Previous Research mechanics course, only about 50% of the students who completed an Much of what we know about student introductory mechanics course at a large problem solving in physics is a result of public university agreed with the ‘expert/novice’ studies, in which the statement “When I solve most exam or problem solving strategies of experts homework problems, I explicitly think (typically university physics faculty about the concepts that underlie the members) are compared to those of problem.” Many students do not seem novices (typically students in to make a strong connection between introductory physics courses).2 the concepts and the problems that are Researchers have found important part of introductory physics courses. differences: Effect of conceptual development on  Experts arrange their problem solving -- electric circuits knowledge in a hierarchical fashion with more general ideas related to Does an increased emphasis on the progressively more specific pieces of development of concepts in the information. Novices have less introductory course promote more knowledge and their knowledge expert-like problem solving? In the tends to be sparsely connected and context of electric circuits, we found poorly organized. that students often did not apply learned concepts to quantitative problems. We  Experts tend to classify physics designed homework worksheets that problems based on underlying included "bridging exercises" intended to form a more explicit link between the rod, and then integrating to find the concepts developed in tutorials and end- total electric field. of chapter problems. After completing P the worksheets, more students were able to solve quantitative problems related to y current in electric circuits.1 Bridging exercises for other topics L Encouraged by our experience with A thin nonconducting rod of length L electric circuits, we wrote worksheets carries a total charge q, spread uniformly for other topics in the calculus-based along it. Show that E at point P on the electricity and magnetism course. These perpendicular bisector in the figure is worksheets also attempted to guide given by q E = 1 2 2 1/2 students in their consideration of the 2peoy (L + 4y ) concepts underlying more traditional problems. For many of these topics, Figure 1. Electric field problem. however, comparatively little research The initial steps of the homework into specific student difficulties existed worksheet are shown in Figure 2. These upon which to base the worksheets. We questions were intended to allow tried to anticipate student’s difficulties students to reason qualitatively about with the concepts and with the factors that determine the electric field. mathematical steps required to solve commonly assigned quantitative Criticize the following statement: problems, and to encourage solution of "From Coulomb's law, we would expect a these problems without resorting to force on a positive charge qt placed at formula-driven approaches. point P to have a magnitude 1 q q 1 q q F = 1 2 = 1 t . We offered optional problem-solving 2 2 4peo r 4peo y sessions for students in sections in So the electric field at pointP is the force which the worksheets were first divided by the test charge: assigned. Students worked through the F 1 q E = = ." assignments in small groups, with q 2 t 4peo y instructors available to answer Would you expect the electric field at point questions. Our observations of these P to be larger orsmaller than the value that small groups led us to realize that for would be obtained using the method above? many topics the worksheets were poorly Figure 2. Initial questions for matched to students' needs. electrostatics worksheet. As an example, we discuss our We expected that these questions experience with a worksheet intended to would be relatively easy for students guide students through a standard (especially compared to the more problem in electrostatics shown in mathematical steps that would follow). Figure 1. The steps required to solve Most of the students in the optional this problem include breaking the rod up problem-solving session, however, were into small elements of length dx, finding unable to answer these questions even the electric field due to one of these after significant guidance. It became charge elements, calculating the clear that we did not have an adequate component of this field perpendicular to understanding of student’s conceptual difficulties in electrostatics to allow us questions in a manner that is consistent to design effective “bridging” with a belief that electric fields can be worksheets. We decided that if we blocked by the presence of nearby hoped to improve students’ ability to conductors or insulators. Some students solve quantitative problems in reason that insulators act to insulate a electrostatics in a more expert-like charge from the effects of an electric manner we would first need to field. Other students give similar understand more about the nature of the reasoning about the effect of conductors. underlying conceptual difficulties. Some of the difficulties revealed by Conceptual difficulties in electrostatics our investigation were difficulties with fundamental underlying concepts. For We found that for electrostatics, example, many students had difficulty many difficulties exhibited by students differentiating between charge and were difficulties with the addition of charge density. We asked students the vectors.4 Many students treated vectors questions shown in Figure 3 after the as scalar quantities when they were completion of instruction in asked to add them. Other students were electrostatics up to Gauss’ law. able to show graphically what the sum of vectors would be, but then gave algebraic answers that were inconsistent A plastic block of length with their drawings but were consistent w, height h, and thickness with treating the vectors as scalars. Still t contains a net positive h other students added only the charge Qo distributed component of a vector that was collinear uniformly throughout its w with the other vector in a vector sum, volume. t neglecting the effect of perpendicular A. What is the volume charge components. Additional difficulties density of the plastic block? with the relationship between vectors and their components were elicited by The block is now broken into two pieces. some of the questions that we asked as The pieces are labeled part of this study. Many of these A and B as shown. h difficulties with vector addition were B. Rank the volume also elicited in topics other than charge densities of the 2w electrostatics. For example, after original block (r ), o 3 w noticing that some students claimed that piece A (r ), and piece 3 vectors of equal magnitude representing A t B (rB). Explain your electrostatic forces or fields completely reasoning. cancelled one another (in cases where they were pointing at an acute angle to Figure 3. Charged block question. one another), we asked a similar About 70% were able to find the question in the context of momentum in charge density from the given a mechanics class. This same difficulty information. However, only about 55% was exhibited by about the same recognized that the charge densities of proportion of students. the pieces would be the same. About We also discovered difficulties that 20% answered that piece B would have were specific to electrostatics. For the largest charge density. Often example, many students answer student's explanations suggested that they were thinking in terms of the equation for charge density, but were many topics, student's exhibited holding the charge constant: Similar unanticipated difficulties, often with difficulties with expressions involving fundamental underlying concepts. If we more than 2 variables have been noted hope to improve students' ability to in different contexts, for example the solve problems in a more expert-like ideal gas law. manner, we need to understand the About 20% answered that the original conceptual issues associated with the piece would have the largest charge problems we are posing, and we need to densities. These responses often have developed effective methods for included reasoning that suggested a addressing these conceptual issues. difficulty differentiating between charge Otherwise, student difficulties with and charge density. problem solving will often reflect conceptual errors with the underlying Student difficulties with the material. interpretation of information about charge density were also noted when Acknowledgements students were asked to solve problems I would like to thank the many involving distributed charges. For students who participated in the studies example, on examination questions described in this paper. The assistance where students were expected to use of all members of the Physics Education Gauss’ law, only about 30% were able Group at the University of Washington to find the enclosed charge. Many of is gratefully acknowledged, and in the incorrect responses included a particular the substantial contributions substitution of the given charge density made by Peter Shaffer and Lillian information into equations that required McDermott. a charge. References Summary 1S. Kanim, "Connecting concepts Where common conceptual about current to quantitative circuit difficulties for a certain topic in physics problems," in these conference are well understood, it is possible to proceedings. design curricula that addresses many of these difficulties. We have found that in 2J. Larkin, J. McDermott, D. Simon, and these cases an explicit link between the H. Simon, “Expert and novice developed concepts and associated performance in solving physics quantitative problems (“bridging problems,” Science 208(20), 1335-1342 exercises”) increased the number of (1980). students who were able to solve 3E. Redish, J. Saul, and R. Steinberg, quantitative problems. Moreover, the “Student expectations in introductory approach taken by students in these physics,” American Journal of Physics cases was often less formula-driven and 66(3), 212-224 (1998). was more likely to include explicit reference to developed concepts. 4S. Kanim, “An investigation of student difficulties in qualitative and In electricity and magnetism, quantitative problem solving: Examples however, we found that often we did not from electric circuits and electrostatics,” have a sufficient understanding of Ph.D. dissertation, Department of student’s conceptual difficulties to Physics, University of Washington design effective bridging exercises. For (1999).