# Using Math in Physics

## Resources for helping students learn to use math in introductory physics

Edward F. (Joe) Redish

University of Maryland

Math in physics is different from math in math classes. In physics, we blend our physical knowledge with our knowledge of math. This changes the way we put meaning to math and even how we interpret equations.

Learning to think about physics with math instead of just calculating involves general scientific thinking skills often taken for granted (and rarely taught) in physics classes. If we want our students to learn to think with math we have to teach them how to do it.

In this series of papers, I give 7 tools that can help students develop these skills. Each paper includes an explanation and in-class or homework problems for integrating these skills into your instruction. The supplementary materials and modules in *the Living Physics Portal* provide additional activities and problems.

### Overview

*The Physics Teacher* **59** (2021) 314-318; https://doi.org/10.1119/5.0021129

In this paper, I present an overview of the series, explaining student difficulties with learning to think with math and why they may resist developing deeper understanding in favor of superficial memorization. I outline the tools described in this series and give links to the research supporting these ideas.

### 1. Dimensional analysis

*The Physics Teacher ***59** (2021) 397-400; https://doi.org/10.1119/5.0021244

Symbols in physics don’t stand for numbers but for measurements — physical quantities that may be represented by many different numbers, depending on the units chosen. Dimensional analysis provides the base for helping students make the conceptual shift.

- Supplementary materials — 1 in-class groupwork activity, 4 sample problems for a quiz, exam, and homework (Solutions at
*the Living Physics Portal*) - Additional materials at the
*Living Physics Portal*— 6 short student readings, 2 group learning activities, 45 homework problems with solutions and 38 problems appropriate for a quiz or exam from across the curriculum

### 2. Estimation

*The Physics Teacher,* **59** (2021) 525-528; https://doi.org/10.1119/5.0021823

Learning to estimate helps students build a sense of scale and see what’s important in a problem and what might reasonably be ignored.

- Supplementary materials — 2 in-class groupwork activities, 8 problems appropriate for homework, quizzes, or exams from across the curriculum (Solutions at
*the Living Physics Portal*) - Additional materials at the
*Living Physics Portal*— 7 short student readings, 2 group learning activities, 108 homework problems with solutions and 39 problems appropriate for an exam from across the curriculum

### 3. Anchor equations

*The Physics Teacher, ***59**** **(2021) 599-604; https://doi.org/10.1119/5.0023066

A few basic equations serve as anchors to code conceptual information and organize physics knowledge.

- Supplementary materials — links to 11 short “Reading the content in [equation]” for students at the ComPADRE website, 6 homework problems with from across the curriculum (Solutions at
*the Living Physics Portal*) - Additional materials at the
*Living Physics Portal*— 12 short “Reading the content in [equation]” for students, 87 homework problems with solutions from across the curriculum

### 4. Toy models

*The Physics Teacher,* 59 (2021) 683-688; https://doi.org/10.1119/5.0025840

In physics, we often begin analyzing a complex system using highly simplified or toy models. We tend to use our toy models without comment or motivation, and students then often infer that physics is irrelevant to the real world. Being explicit about the motivation and value of simple models (and discussing how to refine them to be more realistic) helps students see the value of modeling.

- Supplementary materials — 10 additional problems for a quiz, exam, or homework from across the curriculum. (Solutions at
*the Living Physics Portal*) - Additional materials at the
*Living Physics Portal*— 4 short student readings, 110 homework problems with solutions from across the curriculum

### 5. Functional dependence

*The Physics Teacher,*** 60 **(2022) 18-21; https://doi.org/10.1119/5.0040055

Understanding functional dependence — how physical variables change together (co-vary) — is rarely emphasized in introductory math classes. But it is a powerful way to see the implications of equations for qualitative as well as quantitative physical behavior.

- Supplementary materials — 11 additional problems for a quiz, exam, or homework from across the curriculum. (Solutions at
*the Living Physics Portal*) - Additional materials at the
*Living Physics Portal*— 2 short student readings, 1 group learning activity, 78 homework problems with solutions and 10 problems appropriate for a quiz or exam from across the curriculum

### 6. Reading the physics in a graph

*The Physics Teacher,** ***61** (2023) 651-65; https://doi.org/10.1119/5.0150860

Graphs can be a powerful tool in helping students learn to build the blend between physical concepts and mathematical symbology to develop their physical intuition and ability to think with math.

- Supplementary materials — answer keys to problems in text and 19 additional homework problems with answer keys
- Additional materials at the
*Living Physics Portal*— 9 short student readings, 6 group learning activities, 115 homework problems with solutions and 50 exam problems from across the curriculum

### 7. Telling the story

*The Physics Teacher*** 62**(2024) 5-11; https://doi.org/10.1119/5.0159037

To develop complex problem solving skills, students need to learn how to tell a coherent story: what the situation in the problem is, what the mechanism is, and what physics principles are appropriate to apply. This can be challenging for students who have been successful in science through memorizing answers and simply plugging numbers into equations.

- Supplementary materials — 14 solved homework problems
- Additional materials at the
*Living Physics Portal*— 3 short student readings, 242 solved homework problems, and some problems appropriate for a quiz or exam from across the curriculum