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Abstract Title: Probing understanding of the sophisticated use of "simple" mathematics in physics
Abstract: Introductory physics starts with mathematics that is familiar to most students yet is used in novel and sophisticated ways. University students taking introductory physics often succeed in executing mathematical procedures in context, but struggle with using mathematical concepts for sense making. Even students in calculus-based courses have difficulty with the basic algebraic reasoning that is a foundation for more advanced mathematical thinking.

An important component of physics is quantitative modeling, which requires conceptual understanding of not only the underlying physical phenomenon, but also of the mathematics used to describe it. The blend of physical, arithmetic, and algebraic reasoning is more challenging for students than many instructors realize. For example, the interdependence of physical quantities is at the heart of symbolic models in physics, yet many students have difficulties with co-variational reasoning even in the context of pure numbers in mathematics. The use of variables (instead of numbers) to represent quantities, which is also known to make reasoning more difficult in mathematics, can pose additional barriers to reasoning about physics concepts and relationships. Flexibility with the varied uses of negative quantities can be an indicator of mastery in algebra; however, in physics, negative quantities have additional nuances that are not always apparent to students. Similarly, differentials, which pose a significant challenge for learners of calculus, can present additional difficulties in physics as their application can differ depending on context.

This session targets student understanding of the application of mathematics concepts in physics contexts that may be regarded by experts as "just" basic math. It brings together researchers who have asked questions related to expert-like thinking and student understanding of conceptual quantitative ideas as well as exploring the space between already-learned mathematics and physics.
Abstract Type: Poster Symposium
Session Time: Parallel Sessions Cluster III
Room: Grand North

Author/Organizer Information

Primary Contact: Peter S. Shaffer
University of Washington
and Co-Presenter(s)
Moderator: P. Shaffer
Presenters:  S. Brahmia, A. Elby, A. Gupta, M. M. Hull, E. Kuo, T. Sikorski, E. Torigoe, J. Von Korff, G. White

Symposium Specific Information

Presentation 1 Title: Natures of Negativity in Introductory Physics
Presentation 1 Authors: Suzanne White Brahmia, Alexis Olsho, Trevor I. Smith, Andrew Boudreaux
Presentation 1 Abstract: Mathematical reasoning skills are an important desired outcome of many introductory physics courses, particularly the calculus-based courses. Positive and negative quantities are ubiquitous in physics, and the sign carries important and varied meanings. Unlike physics experts, novices struggle to understand the many roles signed numbers can play in physics contexts, and recent evidence shows that unresolved striggle carries over to subsequent physics courses. The mathematics education research literature documents the cognitive challenge of conceptualizing negative numbers as mathematical objects – both for experts, historically, and for novices as they learn. We contribute to the small but growing body of physics education research that focusses on student reasoning about signed quantities and the role of the negative sign in models. This paper contributes a framework for categorizing the various natures of the negative sign in physics contexts, modeled on the established natures of negativity in algebra from the mathematics education research community. It is our hope that such a framework can help facilitate innovation in methods and curricular activities that can catalyze a deeper mathematical conceptualization of signed quantities from the introductory courses and beyond.
Presentation 2 Title: Recognizing and Promoting Mathematical Cause-and-Effect Reasoning: An example of responsive teaching in a peer instruction-style lecture
Presentation 2 Authors: Andrew Elby, Ayush Gupta, Michael M. Hull, Eric Kuo
Presentation 2 Abstract: Physicists do not always calculate exact values to answer their questions.  At times, the blending of a qualitative understanding of physical systems with a quantitative understanding of relevant physics equations can lead to mathematical reasoning about causes and their effects (e.g., proportional reasoning, reasoning about dependencies).  While this type of mathematical sensemaking in physics is valuable, there is little work on how to help students engage in these practices spontaneously.  We propose that responsive teaching can support mathematical cause-and-effect reasoning by recognizing and building on students' existing ideas.  In this poster, we present an example from a peer instruction-style discussion in introductory mechanics where the instructor's responsive pedagogy made space in the classroom for students to produce sophisticated cause-and-effect reasoning.  We will examine the nature of the instructor's responsive moves and argue that this pedagogical approach supported mathematical sensemaking in two ways: (1) by providing opportunities to engage in this type of reasoning and (2) by supporting the epistemological stance that such reasoning is valuable in physics.
Presentation 3 Title: Student reasoning about whether a solution is "sensible"
Presentation 3 Authors: Gary D. White, Tiffany-Rose Sikorski
Presentation 3 Abstract: Practicing physicists value a variety of answer-checking behaviors such as reviewing units, limiting case analysis and numerical estimations, which we refer to as "the three usual ways" of answer-checking. Students, however, often do not adopt these behaviors even when the instruction includes explicit efforts to encourage it. In previous work we have documented settings in which all students demonstrate the ability to perform solution checks, finding that checking units is most readily adopted, limiting case analysis, less so, and numerical estimation, least of all. Here, undergraduates in two upper-level courses---electrodynamics and intermediate lab--- were asked to check whether a given solution is "sensible" and their responses were studied with an eye toward better understanding the "simple" mathematical and physical reasoning that was preferentially invoked. We report on stark differences between the two groups, largely related to previous instruction, we believe.
Presentation 4 Title: How Plug and Chug Interferes with Symbolic Problem Solving
Presentation 4 Authors: Eugene Torigoe, Andrew Meyertholen, Shuwang Zhang, and Alan He
Presentation 4 Abstract: Several studies indicate that physics students perform worse on symbolic problems, those with symbols and no numbers, than on numeric problems, those involving numbers. We studied the mathematical strategies used by students (N = 477) in a second semester introductory physics course to solve long answer, free response questions during a final exam in a second semester introductory physics course.  The students were randomly given symbolic or numeric versions of two problems. While there was one dominant strategy observed in the numeric problems, there were a variety of strategies used in the symbolic problems. We hypothesize that on the version with numbers, the students were guided by a plug-and-chug approach that focused on solving for the remaining non-numeric variable symbol. Without that structure in the symbolic version, they are much more likely to choose a random variable to isolate and solve.