Cognitive Modeling of Analogy Events in Physics Problem Solving From Examples

Cognitive Modeling of Analogy Events in Physics Problem Solving From Examples Matthew Klenk (m-klenk@northwestern.edu) Qualitative Reasoning Group, Northwestern University 2133 Sheridan Road, Evanston, IL 60208 USA Ken Forbus (forbus@northwestern.edu) Qualitative Reasoning Group, Northwestern University 2133 Sheridan Road, Evanston, IL 60208 USA Abstract The Companions Architecture Understanding how analogy is used in problem solving is an important problem in cognitive science. This paper describes a model of using worked solutions to solve new problems, in terms of structure-mapping processes in the Companions cognitive architecture. The Educational Testing Service independently evaluated the flexibility of the system by using AP Physics problems that were systematically varied to test different types of transfer. We also show that the model provides an explanation for many of the analogy events in VanLehn’s (1998) analysis of the use of analogy by students solving physics problems. Keywords: Analogy; Problem Solving The Companions architecture is exploring the hypothesis that structure-mapping operations (Gentner 1983; Forbus & Gentner 1997) are important building blocks for modeling reasoning and learning. This hypothesis suggests that within domain analogies, where new situations are understood in terms of prior understood examples, provide an important source of breadth and robustness of human common sense reasoning. Forbus & Hinrichs (2006) provides an overview of the Companions architecture; for this paper, the key processes to understand are analogical matching and retrieval. We summarize each in turn. The Structure-Mapping Engine (SME) models the structure-mapping process of comparison (Falkenhainer, Forbus, & Gentner 1989). Structure-mapping postulates that this process operates over two structured representations (the base and target), and produces one or more mappings, Introduction Cognitive science research has shown that analogy plays important roles in problem solving and learning (Gentner & Gentner, 1983; Holyoak 1985; Ross 1987; Novick 1988). One role is facilitating problem solving by using worked solutions. For example, VanLehn and Jones (1993a) observed that students used analogical reasoning in solving physics problems even when the underlying first principles knowledge was already known and had been successfully used before. This paper describes how the Companions cognitive architecture (Forbus & Hinrichs 2006) models example use in solving AP Physics problems. The AP Physics exam is taken by high school students in order to receive college credit. This is an interesting domain because students find such problems difficult and there is a wealth of prior cognitive science solving research on physics problem. Figure 1 shows four examples generated for this work by the Educational Testing Service (ETS), the company which administers the AP Physics exam. We start by briefly reviewing the Companions architecture, focusing on the key analogical processes used. Then, we describe how a Companion uses these processes to solve physics problems. Next, the results of an external evaluation demonstrating the system’s ability to successfully transfer across different problem variations are summarized. This is followed by an analysis of the 90 problems from this evaluation in terms of analogy events (as defined by VanLehn (1998)), showing a qualitative match with the patterns of analogy events found in protocols. Finally, we discuss other related work and future plans. 1. A ball is released from rest from the top of a 200 m tall building on Earth and falls to the ground. If air resistance is negligible, which of the following is most nearly equal to the distance the ball falls during the first 4 s after it is released? (a) 20m; (b) 40m; (c) 80m; (d) 160m. 2. An astronaut on a planet with no atmosphere throws a ball upward from near ground level with an initial speed of 4.0 m/s. If the ball rises to a maximum height of 5.0 m, what is the acceleration due to gravity on this planet? (a) 0.8m/s 2 ; (b) 1.2m/s 2 ; (c) 1.6m/s 2 ; (d) 20m/s 2 ; 3. A box of mass 8kg is at rest on the floor when it is pulled vertically upward by a cord attached to the object. If the tension in the cord is 104N, which of the following describes the motion, if any, of the box? (a) It does not move; (b) It moves upward with constant velocity; (c) It moves upward with increasing velocity but constant acceleration; (d) It moves upward with increasing velocity and increasing acceleration. 4. A block of mass M is released from rest at the top of an inclined plane, which has length L and makes an angle q with the horizontal. Although there is friction between the block and the plane, the block slides with increasing speed. If the block has speed v when it reaches the bottom of the plane, what is the magnitude of the frictional force on the block as it slides? (a) f = Mgsin(q); (b) f = Mgcos(q); (c) f = MgLsin(q)- ½Mv 2 ;(d) f = [MgLsin(q)- ½Mv 2 ]/2. Figure 1: Example AP Physics problems