Tuminaro & Redish, PER Conference Proceedings (2003)

Understanding Students' Poor Performance on Mathematical Problem Solving in Physics

J. Tuminaro & E. F. Redish, Proceedings of the Physics Education Research Conference, Madison, WI (Aug 6-7, 2003).

Abstract: Many introductory, algebra-based physics students perform poorly on mathematical problem solving tasks in physics. There are at least two possible, distinct reasons for this poor performance: (1) Students lack the mathematical skills needed to solve problems in physics, or (2) students do not know how to apply the mathematical skills they have to particular problem situations in physics. Many physics faculty assume that the lack of mathematical skills is the problem. We present evidence suggesting that the major source of students’ errors is their failure to apply the mathematical knowledge they have or to interpret that knowledge in a physical context. Additionally, we present an instructional strategy that can help students employ the mathematical knowledge they already possess.

Tuminaro, PhD Dissertation (2004)

A cognitive framework for analyzing and describing introductory students' use and understanding of mathematics in physics

J. Tuminaro, Ph.D. Dissertation, E. F. Redish (advisor), (2004). (html TOC and abstract)

Abstract: Many introductory, algebra-based physics students perform poorly on mathematical problem solving tasks in physics. There are at least two possible, distinct reasons for this poor performance: (1) students simply lack the mathematical skills needed to solve problems in physics, or (2) students do not know how to apply the mathematical skills they have to particular problem situations in physics. While many students do lack the requisite mathematical skills, a major finding from this work is that the majority of students possess the requisite mathematical skills, yet fail to use or interpret them in the context of physics.

In this thesis I propose a theoretical framework to analyze and describe studentsí mathematical thinking in physics. In particular, I attempt to answer two questions. What are the cognitive tools involved in formal mathematical thinking in physics? And, why do students make the kinds of mistakes they do when using mathematics in physics? According to the proposed theoretical framework there are three major theoretical constructs: mathematical resources, which are the knowledge elements that are activated in mathematical thinking and problem solving; epistemic games, which are patterns of activities that use particular kinds of knowledge to create new knowledge or solve a problem; and frames, which are structures of expectations that determine how individuals interpret situations or events.

The empirical basis for this study comes from videotaped sessions of college students solving homework problems. The students are enrolled in an algebra-based introductory physics course. The videotapes were transcribed and analyzed using the aforementioned theoretical framework.

Two important results from this work are: (1) the construction of a theoretical framework that offers researchers a vocabulary (ontological classification of cognitive structures) and grammar (relationship between the cognitive structures) for understanding the nature and origin of mathematical use in the context physics, and (2) a detailed understanding, in terms of the proposed theoretical framework, of the errors that students make when using mathematics in the context of physics.

Tuminaro & Redish, Phys Rev STPER (2007)

Elements of a Cognitive Model of Physics Problem Solving: Epistemic Games

J. Tuminaro & E. F. Redish, Phys Rev ST PER, 3, 020101 (2007). 

Abstract: Although much is known about the differences between expert and novice problem solvers, knowledge of those differences typically does not provide enough detail to help instructors understand why some students seem to learn physics while solving problems and others do not. A critical issue is how students access the knowledge they have in the context of solving a particular problem. In this paper, we discuss our observations of students solving physics problems in authentic situations in an algebra-based physics class at the University of Maryland. We find that when these students are working together and interacting effectively, they often use a limited set of locally coherent resources for blocks of time of a few minutes or more. This coherence appears to provide the student with guidance as to what knowledge and procedures to access and what to ignore. Often, this leads to the students failing to apply relevant knowledge they later show they possess. In this paper, we outline a theoretical phenomenology for describing these local coherences and identify six organizational structures that we refer to as epistemic games. The hypothesis that students tend to function within the narrow confines of a fairly limited set of games provides a good description of our observations. We demonstrate how students use these games in two case studies and discuss the implications for instruc-tion.

Redish, Scherr & Tuminaro, The Physics Teacher (2006)

Reverse Engineering the Solution of a "Simple" Physics Problem: Why learning physics is harder than it looks

E. F. Redish, R. E. Scherr & J. Tuminaro, published in a slightly abbreviated version in The Physics Teacher, 44, p 293 (May 2006).

Abstract:  Problem solving is the heart and soul of most college physics and many high school physics courses. The “big idea” is that physics tells you more about a physical situation than you thought you knew — and you can quantify it if you use fundamental physical principles expressed in mathematical form. Often, the results of your problem solving can lead you to understand and rethink your intuitions about the physical world in new and more productive ways. As a result, physics is a great place (some of us would claim the best place) to learn how to use mathematics effectively in science.

As physics teachers, we often stress the importance of problem solving in learning physics. Unfortunately, many of our students appear to find problem solving very difficult. Sometimes they generate ridiculous answers and seem satisfied with them. Sometimes they can do the calculations but not interpret the implications of the results. Sometimes, despite apparent success in problem solving, they seem to have a poor understanding of the physics that went into the problems.1 We give them explicit instructions on how to solve problems (“draw a picture,” “find the right equation,” …) but it doesn’t seem to help.

We might respond that they need to take more math prerequisite classes, but in the algebra-based physics class at the University of Maryland, almost all of the students have taken calculus and earned an A or a B. Many of them have been successful in classes such as organic chemistry, cellular biology, and genetics. Why do they have so much trouble with the math in an introductory physics class?

As part of a research project to study learning in algebra-based physics,2 the Physics Education Research Group at the University of Maryland videotaped students working together on physics problems. Analyzing these tapes gives us new insights into the problems they have in using math in the context of physics. One problem is that they have inappropriate expectations as to how to solve problems in physics (some of it learned, perhaps, in math classes). This is discussed elsewhere.3 A second problem seems to lie with the instructors. As instructors, we may have misconceptions about how people think and learn, and this has important implications about how we interpret what our students are doing.

In this paper, we want to consider one example of students working on a physics problem that showed us in a dramatic fashion that we had failed to understand the work the students needed to do in order to solve an apparently “simple” problem in electrostatics. Our critical misunderstanding was failing to realize the level of complexity that we had built into our own “obvious" knowledge about physics.