Course Goals and The Essential Questions: “What is Physics?” and “What is Math?”

The idea began with my IPLS II course which has a clear set of essential questions: “What is an Electron?” and “What is Light?” After careful consideration and discussion with colleagues, I settled on the essential questions of “What is Physics?” and “What is Math?” Fundamentally, I feel that no student should leave an introductory physics class without being able to answer these questions, and, as is visible in the poster, students do not generally have expert-like opinions on the nature of physics. Most answers to this question on the first day of class center on kinematic quantities with “The study of motion” being a common theme.

First Days

We begin the discussion of these essential questions in the first days of class. On the very first day, students are asked these same essential questions. They first think individually and then collaborate with the folks near them (the formal teams described in the Team-Based Learning section are not formed at this point) to come up with a common answer which they put on the board. The results are often of the form listed below:

What is Physics? “The Study of Movement + Energy”

What is Math? “The Study of Numbers.”

Typical student responses to the essential questions of the course on the first day of class.

After this discussion, I provide my own answer to these questions:

The goal of physics is to uncover the fundamental concepts and rules which govern all processes in nature.

•Physis is Greek for nature!
•This includes those that underly biology.

We then want to be able to apply and use those rules to learn new things or gain new insights into how the world works!

My answer to “What is Physics?”

We will be thinking about math a bit differently than you are probably used to thinking about it in a math course.
We will use math as a language (one of many) to describe physics concepts.

My answer to “What is Math?”

I then use these answers to justify the structure of the course, “The prep is the place students learn the “rules” and class is where we learn to apply them. Similarly, we will take a concepts-first approach and translate our concepts in to mathematics.”

How this manifests in the preparation

As discussed in the section on Team-Based Learning, preparation is a key element in our class. Our class’s fundamental questions suggest that the physics part of the preparation should be purely conceptual: with nothing the students would recognize as math. However, for the purposes of inclusion, there is math review on topics such as trigonometry, systems of equations, etc. However, this math review is distinct in both form and location: the math review is generally contained in appendices.

This conceptual physics focus includes exposure to fundamental concepts such as conservation of energy and specific ideas such as kinetic energy. For example, in the introduction of kinetic energy, students are expected to learn that kinetic energy depends on mass and speed, with speed being more important. Moreover, they are expected to realize that kinetic energy must be positive. In terms of conservation of energy, they are expected to know that it must be conserved and should be able to do problems of the type of different balls rolling down different shaped hills of the same height and recognize that the kinetic energy at the bottom should be the same. Thus, our book must be specifically tailored and unique.

How this manifests in class

Writing Equations

The first thing we do when we encounter a concept in class is write the relevant equation. As already stated, students are expected to know some conceptual information before hand and then be able to translate them. Following guidance from ELL education, students write it individually first and then work collaboratively to come up with an equation or proportionality. Frequently, most groups get the correct result even for concepts that are not at all covered in high-school physics such as entropy.

Solving Problems

Once students have mastered the fundamentals of the concepts in class, we then move on to their deeper understanding and application during the class sessions. As a coherent theme guiding the structure of class time, I use a custom problem solving method, described below, which puts the concepts first as well. The work of Paul and Webb somewhat dramatically shows that such an approach is more equitable.

Step 0 – DON’T PANIC: You Don’t Need to Know All the Steps, You Only Need to Know What to Do Next.

This is actually an effort at inclusion. Many life-science students enter physics nervous about the subject. This anxiety is increased when they are told that they will be asked to solve problems in contexts they have not encountered before (part of my answer to “What is physics?” above). This step serves an important role in addressing that anxiety straight on.

The second aspect, that “you only need to know what to do next” is a direct challenge to many of the problem solving strategies out there which seem to imply that students will know all the steps upon looking at a new situation. Of course, if they know all the steps, it is not a problem but an exercise as is pointed out in Heller and Heller. Thus, we should NOT expect them to know all the steps. However, students, particularly life-science students, seem to expect that they should know all the steps. Thus, addressing this conception head-on is critical.

Step 1 – Tell a Story In Terms of the Physical Principles

Again, this is somewhat following the paradigm of Heller and Heller with an additional aspect of facilitating students’ transition from novice to expert by encouraging them to look past what Chi et al. identify as “surface features” and on to deeper considerations of physical principles such as energy an Newton’s Laws.

In fact, for the more computationally complex units centered on energy and forces, I completely divide the unit, again following the suggestions of Webb and Paul. I make a list of all the problems I want to explore. We then approach them all conceptually first: telling the story of each situation. Students are told to keep this work. Only after this process of “telling stories” for all problems, do we loop back around and apply computational techniques. This approach not only has the benefit of potentially being more equitable, but also addresses the common student statement that “I understand the concepts, but I just cannot do the problems.” Asking students to write stories first makes it clear to them that, in fact, it is their conceptual understanding which is incomplete, not their mathematical skill.

These stories can take multiple forms including diagrams and words. In both cases, I tell students “Math is not your first language. Do physics in your first language first!”

Step 2 – Translate the Story into Mathematics via a Principle

In this step, students begin their mathematical application. The motivation for this step is three-fold:

1. Describing the process as translation helps reinforce the math-as-language paradigm which is central to this course.
2. The structure is clearly concepts first: students must take their conceptual story, represented in figures and text.
3. The goal is to explicitly facilitate their transition from novice to expert by requiring that the story must begin with a physical principle (conservation of energy, Newton’s Laws) and the types of energy (kinetic energy K, spring forces), as opposed to definitions associated with surface features (kx for springs etc.).

Step 3 – Definitions

Now students can apply their definitions of specific concepts: kx for springs etc. Again the goal is to get students to think in terms of forces, energies, etc. before they get “focused on formulas.”

Step 4 – Algebra

This framing makes the algebra, “something the students know” and makes it possible to go through a bit more quickly, not showing every step. I believe in holding students accountable for the prerequisites for the course. As STEM majors, I feel that it is important for students to develop a certain facility with fundamental algebra. Moreover, my experience shows they can do this when asked. Of course, there are concerns with equity and inclusion here, but the preparation helps mitigate such differences in prior experience by making the exact level of mathematical familiarity explicit through homework. Particularly challenging algebraic steps can even be put in the homework and then quizzed on the day on which they will be used.

Step 5 – Numbers

Almost all physics instructors I have met have expressed their struggles with students’ desires to put numbers in their equations early in the process. Of course, if the numbers make too early an appearance, not only is the opportunity to practice symbolic manipulation and interpretation lost, but also the ability to find one’s mistakes. This second point seems to help “sell” the idea of late numbers to students: they are willing to concede that they are likely to make mistakes and the ability to find them is an appealing prospect.

Step 6 – Interpretation

If math is a language, and physics is where students are starting to put vocabulary on that language, then they must be able to not just write but also read their results. Limiting cases etc. can be discussed here.

How this manifests on exams

On exams, students are exposed to a new physics concept not covered in class. Typically I use a video to demonstrate the fundamental principles. Students must then select (not write in full) the correct matching mathematical description.

Bibliography

• D. J. Webb and C. A. Paul, Attributing Equity Gaps to Course Structure in Introductory Physics, Phys. Rev. Phys. Educ. Res. 19, 020126 (2023).
• Heller, Kenneth and Heller, Patricia, Cooperative Problem Solving in Physics A User’s Manual: Why? What? How?, in (n.d.).
• M. T. H. Chi, P. J. Feltovich, and R. Glaser, Categorization and Representation of Physics Problems by Experts and Novices*, Cognitive Science 5, 121 (1981).