**Project No.**

**PI Name**

**Institution**

### Abstract 1

### Collaborative Research: Research on learning and teaching at the physics-mathematics interface

**Presentation Type**

**Team**

**Need**

Our work expands the research base in upper-division physics learning, focusing on a course (math methdods) that is commonly offered in physics departments but that has received little prior attention from researchers. Our work will function as one of the few in-depth and rigorous studies of knowledge transfer across disciplinary boundaries by expanding our understanding of how studentsﾒ mathematical knowledge changes while learning physics. Undergraduate physics majors are motivated, reasonably articulate in describing their thought processes, and relatively sophisticated in their understanding, but not yet experts; thus their location on the boundary of expertise makes them particularly suitable for studying the development of learners from novices to experts with a lens focused on interdisciplinary knowledge transfer. It is here that this research has the potential to extend far beyond the boundaries of the study itself: it expands upon the study of transfer that Wagner first began with the publication of Transfer in Pieces.

**Goals**

The goal of this project is to perform a coordinated program of research and curriculum development centered on the evolution of studentsﾒ mathematical understanding in the context of learning physics, particularly while solving challenging problems posed in upper-division physics courses. One key area of focus for the project will be sophomore- and junior-level physics courses in mathematical methods, a portion of the curriculum that is largely untouched by education research. We will perform fundamental research on student learning of key content areas covered in these courses, including the underlying mathematical ideas as well as the physics contexts in which these ideas are typically deployed.

The work brings together colleagues from the physics education research (PER) and research in undergraduate mathematics education (RUME) communities as well as the researchersﾒ individual methodologies and analytical lenses. It blends efforts at identifying and addressing student conceptual difficulties in physics with efforts toward understanding knowledge transfer between mathematics and physics.

There are three intellectual threads through the work: research on student understanding of physics and mathematics concepts; research on knowledge transfer, specifically from mathematics to physics; and development and assessment of instructional resources. Results in these three threads will, respectively, inform the undergraduate teaching of mathematics and physics, expand our understanding of interdisciplinary knowledge transfer, and produce instructional resources usable by other practitioners to improve student learning of physics, mathematics, and the use of mathematics in physics.

With attention to key topics and project threads, we ask the following research questions:

ﾕ What is the nature of studentsﾒ knowledge and resources for understanding these targeted mathematics concepts in upper-division physics courses?

? What conceptual challenges are particular to specific physical contexts and which of these are related to difficulties in applying mathematical concepts in new situations?

ﾕ How does studentsﾒ understanding of these core mathematical ideas evolve and transfer as they apply them in increasingly sophisticated physical contexts?

? In particular, how does the application of these mathematical ideas within physics contexts affect studentsﾒ understanding of both the mathematics they are using and the physics they are learning?

ﾕ How does the coevolution of studentsﾒ mathematics and physics understanding inform researchersﾒ knowledge about interdisciplinary transfer?

ﾕ How can emerging answers to these questions be used to inform the development of instructional resources to improve the learning of both mathematics and physics?

**Approach**

The project has practice-oriented and transfer-oriented threads. The practice-oriented work involves primary research on student reasoning as well as the development of instructional interventions. The transfer-oriented work focuses on models of transfer as applied to student use of mathematics in physics. Our primary research methods are analysis of written student responses and individual student interviews.

**Outcomes**

For this poster we would present a results from student thinking and reasoning on two or three of the following:

- Student reasoning with coordinate systems and unit vectors

- Quantitative reasoning with units and limiting cases

- The gap between procedures and the construction of physical integrals

- Expanding the concept of vector to include functions and series

**Unexpected Challenges**

Collaboration across institutions and disciplines has been our greatest challenge.

**Citations**

This is a new project, so the only published work thus far is in conference proceedings:

Quantitative reasoning skills in math methods, M. E. Loverude, accepted for publication, Physics Education Research Conference Proceedings 2015.

BEYOND PROCEDURES: QUANTITATIVE REASONING IN UPPER-DIVISION MATH METHODS IN PHYSICS, M. E. Loverude, short report accepted for Research in Undergraduate Mathematics Education Proceedings 2016.