Another addition this semester was a “problem solving process.” While most physics textbooks include problems solving processes, I have a fundamental disagreement with the philosophical underpinnings implied by these published sequences. Many of these processes implicitly suggest that students should be able to look at a problem and see all the steps before beginning work; that they should be able to “outline a solution” before even beginning the math. In my experience, this is not how physicists solve problems. Frankly, a situation is not really a problem if you know all the steps upon setting out. I want students to learn to sit with the discomfort of *not *knowing all of the steps at the outset and to develop the confidence needed to figure out problems as they go.

This implicit suggestion that one should immediately see all the steps need to solve a problem is, I feel, one source of many sources of frustration that many students have with physics. In many other introductory courses, students often can look at an exam problem and know the solution or, at the very least, see all the steps they need to complete to arrive at that solution. A good example of this second type is the titration calculation in chemistry as visible in OpenStax Chemistry 2e where there is a flowchart for the series of steps for this type of problem. This expectation is, I suspect, the origin of some of the common questions/comments in physics courses such as, “What are the series of steps for this type of problem?”, or “Is there a type of problem for this situation where…?” or, very commonly, “I feel that the exam problems should basically be the same as those we did in class (with different numbers of course).” All these questions/comments belie a “I just need to be able to memorize the process for each type of problem and I will be good” mindset. Of course, as any physics instructor will tell you, this goal of “memorizing the process for each type of problem” is fundamentally anathema to physics. Physics is about starting with basic principles that are widely applicable and using them to describe a variety of phenomena.

## Some example textbook strategies

The popular textbook *Sears and Zemansky’s University Physics with Modern Physics *(13^{th} edition) by Young and Freedman, which I used as an undergraduate at University of Arizona, employs an example of such a problem solving method which implies an ability to know all of the steps from the get go. The method, called *I SEE –* *Identify, Setup, Execute, and Evaluate*, is shown below in an image from page 3. Another example, also shown below from page 35, is from Cutnell and Johnson’s* Physics *(9th ed) which I have used as an instructor.

Both these strategies embody a few features which I find problematic.

- The first steps Young and Freedman’s
*Setup*as well as Cutnell and Johnon’s*Reasoning*are rather simplistic, formulaic, and go all the way to the end of the problem. - Both are very equation focused: Young and Freedman instructs students to “choose the equations that you’ll use.” While Cutnell and Johnson do a similar process in
*Modeling the Problem*. Both mention the process of listing all knowns and unknowns. Cutnell and Johnson do this very explicitly in a table, while Young and Freedman describe a similar process in their identify step. - Neither encourages students to strongly think about the reality these equations (the focus of these frameworks) are trying to model.
- Both are presented as fool proof.

The issue with the first point has already been described: it implies that students should be able to know all the steps to solving a problem upon first glance. This implication arises from the position early in the process and from the fact that all the steps to the end are depicted. The often meandering or cyclical nature of true problem solving is absent. Moreover, there rather straightforward and simplistic language used neglects the fact that this stage is often the source of most struggle for students.

The second point reinforces a common conception of physics with which many students enter introductory courses: that physics is simply a set of equations that need to be strung together. Tuminaro and Redish call this the “plug-and-chug epistemic game” (Tuminaro and Redish, 2007). In this problem-solving view, the goal is to list all the knowns and unknowns and then find an equation that connects them with little regard for the principles that underlie these equations. Such a mindset leads to a mistake that all physics instructors have seen: inserting a value into an equation about an entirely different concept. For example, a student may input a momentum into an equation about pressure as both are represented by the variable *p*.

Another problem arising from the equation focus represented by the second point is the neglect of the value that other representations can bring to the problem-solving process. Most physicists will, as a matter of course, draw some type of diagram for almost every problem. Novice physics students, on the other hand, will often not draw a diagram unless explicitly required; if one is required, students will often draw it and then not use it for the rest of the problem (Chi et al, 1981).

The third issue, a failure to explicitly encourage students to invoke reality, can lead to other student pitfalls. I feel that many students, particularly non-majors, approach introductory physics with a mindset that “nature does not actually work like this.” I have had a student say that direct quote to me in an office hour. By separating reality from the problem-solving process as Cutnell and Johnson do, or relegating that consideration to an “afterthought” of a problem solving process as Young and Freedman do by including it as only a part of the *Evaluate *step further reinforces this mindset.

Beyond conflicting with many of the goals for my courses, such a mindset can also lead to missteps in solving traditional problems. Students will often miss key information for solving a problem because of the mental disconnect between the equations in front of them and reality. A classic example in Physics 132 is in problems regarding light in a vacuum. Many students will make mistakes or get confused because they forget to use the fact that all light in vacuo travels at *c* = 3×10^{8} m/s. If you ask students directly, the speed of light, they will almost without fail get it correct. However, due to their mental disconnect between the equations and conceptual knowledge, they do not bring that conceptual knowledge to the problem-solving process.

The final issue, of “foolproofness” is also an issue. If a situation is truly a problem, then there will, by definition, be some meandering and wrong turns in the process of solving it. Presenting a seemingly foolproof process without acknowledging the messy reality of true problem solving can lead to discouragement, “I followed all the steps, but I still couldn’t get it to work. I clearly am just no good at physics!”

## My solution – list of problem-solving tips

My solution, first codified in Physics 132 during the Spring 2022 semester, is therefore less a series of steps and more a set of points to keep in mind. The numbers are not sequential, but a listing of priorities. I would write these on a board at the front of the classroom as shown below (with the capitalization) and refer to it repeatedly while helping students.

- DON’T PANIC! (All caps intentional!).
- You SHOULDN’T know all the steps at once. You only need to figure out what to do NEXT.
- It is OKAY to make a wrong turn. 😊
- Start with a PRINCIPLE.
- Remember, you are describing REALITY.
- Start simple and add complexity as you need it.
- Do one-thing-at-a-time.

These steps avoid several of the flaws previously mentioned. They acknowledge that problem solving is just that: using tools to learn something you did not know before. They also affirm that making mistakes is okay. They also encourage a more conceptual approach – there is no mention of equations anywhere. These steps work just fine for a graphical or word-based analysis as well as for mathematical ones removing the distinction between “mathematical” and “conceptual” problems that many students have.

Anecdotally, these steps have been a benefit. Below are a few comments from my end-of-semester evaluation.

- “That it is okay if I don’t understand a problem or set of problems at first and ways to go about solving them piece-by-piece with less panic. Starting with principles (what we fundamentally know is true), worry about one thing at a time until the next thing, remembering that an answer/solution will be consistent with reality.”
- “I learned that I should start with principles and visualize what is happening in a particular problem and not just jump into it with equations and writing out knowns vs. unknowns .”
- “One skill we learned in class was how to go about solving problems. They are as follows: don’t panic, start with a principle, and know what to do next. I think this was an essential skill to learn because I feel as though we all have the misconception that when we see a problem we should immediately know how to get to the answer, but this is not a good or efficient way to think about it. The key skill of knowing what to do next was essential in another class too, Organic chemistry, specifically in the synthesis problems. In the synthesis problems we are given an initial product and a final product and we have to solve for steps that it took to get to the final product. It is almost impossible to know exactly how to get to the final molecule from just looking at the initial one, which is why you just have to know which reaction will get the product to the next step, which will get it closer to the final product. Knowing what to do next is definitely key in solving these challenging problems.”

These comments are clear reflections of what I want students to learn!

## References

*4.5 Quantitative Chemical Analysis – Chemistry 2e | OpenStax*, https://openstax.org/books/chemistry-2e/pages/4-5-quantitative-chemical-analysis.

H. D. Young, R. A. Freedman, A. L. Ford, and F. W. Sears, *Sears and Zemansky’s University Physics: With Modern Physics*, 13th edition (Pearson, Boston, 2012).

J. D. Cutnell, K. W. Johnson, D. Young, and S. Stadler, *Cutnell & Johnson Physics*, 9. ed., internat. student version (Wiley, Hoboken, NJ, 2012).

J. Tuminaro and E. F. Redish, *Elements of a Cognitive Model of Physics Problem Solving: Epistemic Games*, Phys. Rev. ST Phys. Educ. Res. **3**, 020101 (2007).

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).