University of Maryland Physics Education Research Group


UMD PERG PhD Dissertations: Jonathan Tuminaro

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

Jonathan Tuminaro, Doctor of Philosophy, 2004

Dissertation directed by: Professor Edward F. Redish, Department of Physics


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.

Dissertation in PDF format.(154 pp., 1 MB)

Table of Contents

Chapter 1 Framing the Issue
Chapter 2 Review of previous research on studentsí use of mathematics
Chapter 3 Data and research methodology
Chapter 4 The cognitive building blocks students use to understand mathematics in physics: An introduction to resources
Chapter 5 Understanding the process of studentsí mathematics use in physics: An introduction to Epistemic Games and Frames
Chapter 6
A case study illustrating the use of mathematical resources, epistemic games, and frames in the analysis of studentsí mathematical thinking
Chapter 7
Understanding student mathematical errors in terms of resources, epistemic Games, and frames
Chapter 8
Summary and speculations for future research


Appendix A Homework Problems
Appendix B List of Epistemic Games


The Three Charge Problem

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Last modified 4 July 2005