Redish & Gupta, GIREP Conference Presentation (2009)

Making Meaning with Math in Physics

Edward F. Redish and Ayush Gupta

Contributed paper presented at GIREP2009, Leicester, UK, August 20, 2009.

Physics makes powerful use of mathematics, yet how this happens is often poorly understood. Professionals closely integrate their mathematical symbology with physical meaning, resulting in a powerful and productive knowledge structures. But because of the way the cognitive system builds expertise, instructors who are expert physicists may have difficulty in unpacking their well-integrated knowledge in order to understand the difficulties novice students have in learning their subject. Despite the fact that students may have previously been exposed to ideas in math classes, the addition of physical contexts can produce severe barriers to learning and sense-making. In order to better understand student difficulties and to unpack expert knowledge, we adopt and adapt ideas and methods from cognitive semantics, a sub-branch of linguistics devoted to understanding how meaning is associated with language. We illustrate this with examples spanning the physics curriculum.

Redish & Bing, GIREP Conference Poster (2009)

Using Math in Physics: Warrants and Epistemological Frames

Edward F. Redish and Thomas J. Bing

Prepared in conjunction with Symposium, “Mathematization in Physics Lessons: Problems and Perspectives”, R. Karam and G. Pospiech, organizers. GIREP meeting, Leicester, UK, 18. August, 2009.

Abstract: Mathematics is an essential component of university level science, but it is more complex than a straightforward application of rules and calculation. Using math in science critically involves the blending of ancillary information with the math in a way that both changes the way that equations are interpreted and provides metacognitive support for recovery from errors. We have made ethnographic observations of groups of students solving physics problems in classes ranging from introductory algebra based physics to graduate quantum mechanics. These lead us to conjecture that expert problem solving in physics requires the development of the complex skill of mixing different classes of warrants – the ability to blend physical, mathematical, and computational reasons for constructing and believing a result. In order to analyze student behavior along this dimension, we have created analytical tools including epistemic frames and games. These should provide a useful lens on the development of problem solving skills and permit an instructor to recognize the development of sophisticated problem solving behavior even when the student makes mathematical errors.

(List of references)

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.

Layman & Shama, Conference in Collegiate Mathematics Ed (1997)

The Role of Representations in Learning an Interdisciplinary Mathematics and Physics University Course

J. Layman & G. Shama, Presented at the Research Conference in Collegiate Mathematics Education, Central Michigan University (Sept 1997). (html version)

Abstract: The University of Maryland offers a physics course as part of the Maryland collaborative for teachers' preparation [MCTP] project. One of the course aims is to promote the learning of the concept of a function through the learning of physics. The students learn in small groups, through problem solving and with the aid of microcomputer based laboratories. Students are asked to examine and find connections between experiments, stories, graphs and algebraic representations. Analysis of observations of students' group work in the course revealed that experiments differed from stories in many characteristics as graphs differed from algebraic expressions. Similarities were found among the translation process from an experiment or a graph to a story or an algebraic representation. Similarities were also found between the opposing direction translations. These translations differed in many characteristics from translations between graphs to experiments and from translations between algebraic equations to stories. According to the above analysis the four situations -- experiment, story, graphical and algebraic representations -- can be presented has vertexes of a parallelogram. Each edge and diagonal of the parallelogram represents a bi-directional arrow of possible translations. Between parallel edges there are many similarities.

Elby, J of Mathematical Behavior (2000)

What students' learning of representations tells us about constructivism

A. Elby, Journal of Mathematical Behavior, 19, p 481-502 (1999). (html version)

Abstract: This paper pulls into the empirical realm a longstanding theoretical debate about the prior knowledge students bring to bear when learning scientific concepts and representations. Misconceptions constructivists view the prior knowledge as stable alternate conceptions that apply robustly across multiple contexts. By contrast, fine-grained constructivists believe that much of students' intuitive knowledge consists of unarticulated, loosely-connected knowledge elements, the activation of which depends sensitively on context. By focusing on students' intuitive knowledge about representations, and by fleshing out the two constructivist frameworks, I show that they lead to empirically different sets of predictions. Pilot studies demonstrate the feasibility of a full-fledged experimental program to decide which flavor of constructivist describes students more adequately.

Bao & Redish, Am J Phys (2002)

Understanding probabilistic interpretations of physical systems: A prerequisite to learning quantum physics

L. Bao & E. F. Redish, Am J Phys, 70(3), p 210-217 (2002). (html version)

Abstract: Probability plays a critical role in making sense of quantum physics, but most science and engineering undergraduates have very little experience with the topic. A probabilistic interpretation of a physical system, even at a classical level, is often completely new to them, and the relevant fundamental concepts such as the probability distribution and probability density are rarely understood. To address these difficulties and to help students build a model of how to think about probability in physical systems, we have developed a set of hands-on tutorial activities appropriate for use in a modern physics course for engineers. We discuss some student difficulties with probability concepts and an instructional approach that uses a random picture metaphor and digital video technology.

Steinberg, Wittmann & Redish, ICUPE AIP (1996)

Mathematical Tutorials in Introductory Physics

R. N. Steinberg, M. C. Wittmann & E. F. Redish, Sample class, presented at The International Conference on Undergraduate Physics Education (ICUPE), College Park, MD (July 31 - August 3, 1996) Proceedings to be published by the American Institute of Physics, E. Redish & J. Rigden (Eds.) (html version)

Abstract: Students in introductory calculus-based physics not only have difficulty understanding the fundamental physical concepts, they often have difficulty relating those concepts to the mathematics they have learned in math courses. This produces a barrier to their robust use of concepts in complex problem solving. As a part of the Activity-Based Physics project, we are carrying out research on these difficulties and are developing instructional materials in the tutorial framework developed at the University of Washington by Lillian C. McDermott and her collaborators. In this paper, we present a discussion of student difficulties and the development of a mathematical tutorial on the subject of pulses moving on strings.

Redish & Wilson, American Journal of Physics (1993)

Student Programming in the Introductory Physics Course: M.U.P.P.E.T. 

E. F. Redish & J. M. Wilson, American Journal of Physics, 61, 222 (1993). (html version)

Abstract: Since 1983, the Maryland University Project in Physics and Educational Technology (M.U.P.P.E.T.) has been investigating the implication of including student programming in an introductory physics course for physics majors. Many significant changes can result. One can rearrange some content to be more physically appropriate, include more realistic problems, and introduce some contemporary topics. We also find that one can begin training the student in professional research-related skills at an earlier stage than is traditional. We learned that the inclusion of carefully considered computer content requires an increased emphasis on qualitative and analytic thinking.