**Project No.**

**PI Name**

**Institution**

### Abstract 1

### Paradigms in Physics: Representations of Partial Derivatives

**Presentation Type**

**Team**

**Need**

A common feature across STEM disciplines is that we are interested in studying change, whether we are studying how changing a design parameter a?ects a device, how changes in temperature a?ect a measurement, or how pressure changes when we adiabatically compress a gas. Indeed, the nature of scienti?c measurement is to control all parameters of an experiment except for the one parameter being studiedﾗwhich itself is being changed. Mathematically, we express the concept of changing one parameter while ?xing others by using partial derivatives. However, how we use partial derivatives, and how we talk about partial derivatives varies dramatically across STEM disciplines. We have found that many studentsﾗeven those with a strong mathematics backgroundﾗ?nd partial derivatives particularly di?cult. This raises the question of how we can best prepare students to use partial derivatives in their ?elds.

**Goals**

Our current grant continues the joint work of two very successful projects: The Paradigms in Physics Project, begun in 1997, and the Vector Calculus Bridge Project, begun in 2001. Written materials produced by these projects include more than 250 group activities and class notes for 20 separate courses.

The major goals of this phase of the project are to:

ﾕ Explore how experts use and represent change;

ﾕ Move students toward a robust understanding of the quantification of change;

ﾕ Develop and test curricular materials for middle-division math and physics courses;

ﾕ Establish students' initial and ongoing levels of understanding as they progress through these materials;

ﾕ Make these materials freely available online.

**Approach**

A theoretical appreciation of representations is helpful in understanding how students interpret, use, and move between di?erent representations. We draw on the perspective of distributed cognition, which provides an account for the role of external entities (including tools, other people, and representations) in cognition. As a part of this this project, we will study the representations used to work with partial derivatives across the STEM disciplines, and will use these results to analyze and construct learning trajectories for students, as they progress from novice to expert throughout their university career.

**Outcomes**

Michelle Zandieh's 2000 framework for describing student concept images of the derivative has been extended to partial derivatives. In addition, the notion of 'thick' derivative has been introduced in order to address the idea of numerical or experimental data that is approximate but 'good enough'. Our extended framework has appeared in refereed papers for both the mathematics and physics communities and been presented at several conferences and colloquia.

We are continuing to design and classroom test our learning trajectories and classroom materials.

We have begun an active collaboration with researchers from the ﾓRaising Calculus to the Surfaceﾔ project (NSF 1246094).

**Broader Impacts**

This project will directly impact mathematics and physics education at the middle-division undergraduate level by providing classroom-tested curricular materials and associated instructor resources to the education community through existing, proven online resources (an activities wiki and textbook). Mathematics materials will support learning trajectories in multiple STEM disciplines, not just mathematics and physics. The addition of the new materials will make the existing resources easier to adopt by providing more complete coverage, in line with most common course structures.

**Unexpected Challenges**

None

**Citations**

Early publications describing the Paradigms project as a whole:

Corinne A. Manogue and Kenneth S. Krane, Paradigms in Physics: Restructuring the Upper Level, Physics Today 56, 53ﾖ58 (2003).

Corinne A. Manogue, Philip J. Siemens, Janet Tate, and Kerry Browne (Department of Physics) & Margaret L. Niess and Adam J. Wolfer (Department of Science and Mathematics Education), Paradigms in Physics: A New Upper-Division Curriculum, American Journal of Physics 69, 978ﾖ990 (2001).

Publications which reference the current grant:

David Roundy, Tevian Dray, Corinne A. Manogue, Joseph F. Wagner, Eric Weber, An Extended Theoretical Framework for the Concept of the Derivative, RUME18 proceedings, eds. Tim Fukawa-Connelly et al., MAA, pp. 838ﾖ843.

David Roundy, Eric Weber, Tevian Dray, Rabindra R. Bajracharya, Allison Dorko, Emily M. Smith, and Corinne A. Manogue, Experts' understanding of partial derivatives using the Partial Derivative Machine, Physical Review Special Topics ﾖ Physics Education Research 11, 020126 (2015).

David Roundy, Mary Bridget Kustusch, and Corinne A. Manogue, Name the experiment! Interpreting thermodynamic derivatives as thought experiments, American Journal of Physics, 82(1), pp. 39-46 (2014).

Mary Bridget Kustusch, David Roundy, Tevian Dray, and Corinne A. Manogue, Partial derivative games in thermodynamics: A cognitive task analysis, Physical Review Special Topics - Physics Education Research, 10, 010101 (2014).

Justyna P. Zwolak and Corinne A. Manogue, Assessing student reasoning in upper-division electricity and magnetism at Oregon State University, Physical Review Special Topics ﾖ Physics Education Research 11, 020125 (2015).

Justyna P. Zwolak and Corinne A. Manogue, Revealing Differences Between Curricula Using the Colorado Upper-Division Electrostatics Diagnostic, 2014 Physics Education Research Conference, eds. P.V. Engelhardt, A.D. Churukian, and D.L. Jones, AAPT, pp. 295ﾖ298.

**Project Page**

**Project Document**