Session H42: Focus Session: Research in Mathematics Education and Mathematics in Physics Education

School mathematics is largely useless for learning physics. But it needn't be.

Physics educators often see mathematics as a toolbox for solving problems. This view fits naturally with the mathematics that most U.S. school teachers teach and that most students learn--meaningless rituals for getting answers. The two perspectives combine to make mathematics largely useless for learning physics. I will argue that a deep emphasis on ideas of quantity in school mathematics, and greater attention to the requirements of quantitative reasoning in physics, would benefit both math education and physics education greatly.   

Using Student Reasoning in Mathematics Instruction

Using student thinking and understanding as a basis for the development of mathematical ideas in the classroom is a challenging and often overwhelming task. In this session, I will report on two instructional tools, generative alternatives and record-of/tool-for mathematics and physics teachers can use to build on students' thinking and reasoning to develop mathematical concepts and processes. The instructional tools are rooted in the theory of Realistic Mathematics Education. Examples are drawn from a first course in undergraduate differential equations. The examples will illustrate ways in which a teacher can navigate the all-telling -- all-discovery continuum through the use of the generative alternative and record-of/tool-for tools.   

Investigating Student Understanding of Physics Concepts and the Underlying Calculus Concepts in Thermodynamics

In work on student understanding of concepts in advanced thermal physics, we are exploring student understanding of the mathematics required for productive reasoning about the physics. By analysis of student use of mathematics in responses to conceptual physics questions, as well as analogous math questions stripped of physical meaning, we find evidence that students often enter upper-level physics courses lacking the assumed prerequisite mathematics knowledge and/or the ability to apply it productively in a physics context. Our focus is in two main areas: interpretation of P-V diagrams, requiring an understanding of integration, and material properties and the Maxwell relations, involving partial differentiation. We have also assessed these mathematical concepts among students in multivariable calculus. Calculus results support the findings among physics students: some observed difficulties are not just with transfer of math knowledge to physics contexts, but seem to have origins in the understanding of the math concepts themselves.   

Student understanding of calculus within physics and mathematics classrooms

The earliest results in Physics Education Research demonstrated the challenges facing students in understanding the graphical interpretations of slope, derivative, and area under curves in the context of kinematics. As part of ongoing research on mathematical challenges that may underlie documented physics difficulties, we developed and administered a brief survey on single- and multivariable calculus concepts to students within physics and mathematics classrooms at both the introductory and advanced levels. Initial findings among students in multivariable calculus show that as many as one in five students encounter some type of difficulty when asked to rank the slopes at five different points along a single path. We will present further data on the extent to which students in a first semester calculus course and an introductory calculus-based physics course encounter similar challenges.   

Student Understanding of Basic Probability Concepts in an Upper-Division Thermal Physics Course

As part of ongoing research on student understanding in upper-division thermal physics, we developed a number of simple diagnostic questions designed to probe understanding of basic probability concepts. Preliminary results showed that many students had difficulty in distinguishing the concepts of microstate and macrostate, and in applying mathematical relationships for multiplicity of simple systems. We have tested a tutorial sequence designed to address some of the difficulties. We will summarize previous results, show post-test results from the target courses, and describe aspects of the tutorial sequence that are likely in need of modification.   

Exploring Student Difficulties with Multiplicity and Probability in Statistical Physics

We continue our research program on the teaching and learning of concepts in upper-division thermal physics at the University of Maine. Typical statistical physics textbooks introduce entropy (S) and multiplicity (w) [S = k ln(w)] with binary events such as flipping coins N times. Inherent confusion with probability and statistics, macrostates and microstates, and their varying dependence on N leads to student conceptual difficulties that persist after textbook-centered activities. We developed and implemented a guided-inquiry tutorial on the binomial distribution with student use of computational software to produce calculations of multiplicities, outcome probabilities, and graphs of their distributions as functions of N. This allows convenient exploration of statistics over more than seven orders of magnitude in N. Comparing student answers to pre- and post-tutorial questions, we find some, but not all of the intended learning results are achieved.