# Mathematization

## What is "mathematizing"?

Mathematizing is a part of problem solving where students convert a physics problem into a form on which they can "do math". Learning how to mathematize is an important part of learning physics. It's vital in homework during university studies, and it's also necessary for reaching conclusions in all types of work undertaken by scientists and engineers.

The Mathematization project studies how upper-division physics students use mathematics across multiple courses and in several modalities. Our research takes place primarily in Classical Mechanics, Electromagnetic Fields, and Quantum Mechanics, three of the core upper-division physics theory classes. We conduct network analyses of their homework solutions to see which ideas are connected within problems, and what kinds of problems elicit which ideas. We analyze their in-class small-group problem solving to see how their epistemic framing affects (and is affected by) their peers, the instructor, and the kinds of problems they work on. We track how their ideas about "looking ahead" in numerical solutions vary by physics context, and we examine how they coordinate multiple representations to generate new ideas.

## Goals

Only a few limited studies have been undertaken to attempt to understand how students mathematize and the difficulties that they encounter while doing it. Thus, the results of this research will provide new insights into the problem solving process that must be developed by scientists and engineers.

This research will result in new models of student development of important STEM problem solving capabilities. The novel focus will provide a unique way to investigate the problem solving process. Our data and results will help physics education researchers and practitioners generalize and create models of the process and thus create new teaching strategies. These new strategies will enable students to learn important components of the problem solving process better. Thus, a broader impact of this work will be improved STEM education which will serve the students beyond their undergraduate years. The research methods which are developed during the project will be applicable to similar studies in other STEM disciplines.

## Design

Our research questions are large:

- What malleable and moderating factors influence students' mathematization?
- How does students' problem mathematization evolve over the students' undergraduate career?

Our prior research shows that students struggle with parsing a word problem and representing it in a form that is solvable using algebra, calculus, or calculational devices: they lack adequate mathematizing skills. Building on these studies, we zoom in from our previous research on problem solving to problem mathematization. We identify the following mathematizing strands that are important across multiple sub-disciplines in physics and that require further research:

- setting up integrals,
- setting up differential equations,
- using approximations, and
- choosing appropriate coordinate systems.

This study looks at mathematizing across the full range of the undergraduate physics curriculum from the introductory calculus-based course to senior-level efforts. We use both cross-sectional studies (all the students in Mechanics, for example), and longitudinal ones.

To delve into these questions, we engage in three kinds of data collection:

- natural,
*in situ*observations of classroom activities; - contrived research settings like think aloud interviews and focus group interviews;
- written artifacts of students' work, like exams and homework.

We triangulate among these three kinds of data to build a strong picture of how students' mathematizing changes as they move up the four-year physics curriculum.

## Personnel

PI: Eleanor Sayre

Co-PIs: Dean Zollman

Postdocs: Ulas Ustun, Deepa Chari, Hai Nguyen

Graduate students: Nandana Liyanage, Bahar Modir, Dina Zohrabi-Alaee, Tra Huynh

Undergrads: John D Thompson

This project is funded by the NSF IUSE program.