# What we talk about when we talk
about physics problem solving

Noa Perlmutter  Program in Cognitive Science, The University of Chicago  Zosia
Krusberg  Department of Physics, The University of Chicago

## Introduction

I am a second-year cognitive science major, and as a student who has completed
my physical science distributive requirements, I will likely never again come
across Gauss’s law. So why do I feel that the time and effort I devoted to
solving Gauss’s law problems was worth it? Partly, I inherently enjoy the
learning process and the new perspective on the physical world I have acquired
by understanding electromagnetism. But I was also inspired by the ways in
which physics problems train the mind in effective problem-solving strategies.
(Of course I was—I am a cognitive science major!) Two themes emerged as I
reflected on this realization. First, physics problems serve as useful toy
models for more complex problems outside of physics, training us in broadly
transferable problem-solving skills. Second, the physics problem-solving
process invites us to reflect on our unique cognitive and affective processes.
These themes are interconnected and complimentary. An improved metacognitive
understanding of our minds facilitates solving progressively more complex
problems, and the act of solving increasingly difficult problems provides
further insight into our minds. In what follows, Professor Zosia Krusberg and
I consider nine general lessons offered by the physics problem-solving
process.

## Nine Lessons

  1. 1.

Physics problems often initially strike us as impossible to solve, leading to
a great deal of stress. We struggle to make sense of the wording of the
problem, to visualize the system, to discern relevant and irrelevant
information, and to identify applicable physical principles. However, after
solving countless physics problems that initially appeared overwhelming, we
begin to develop a sense of confidence in our problem-solving abilities, both
within and beyond the physics classroom. Indeed, when we come across real-
world problems that initially feel overwhelming, physics problem solving can
teach us not to let first impressions of a problem daunt us. After all, in the
context of problem solving, perhaps one of the least useful states is being
overwhelmed: it turns off our sense of focus and curiosity, disabling two of
our main skills for solving problems in the world.111A wonderful example of
what can be accomplished when someone is not overwhelmed by a problem is the
story of the American mathematician George Dantzig. As a graduate student,
Dantzig once arrived late to a statistics lecture, and assumed that the two
problems written on the board were homework problems. When he handed in
solutions to the problems a few days later, his professor informed him that he
had just solved two previously unsolved statistics problems.

  2. 2.

Once we have convinced ourselves that a problem is solvable, we need to decide
how to begin our solution. In physics, there are a number of effective steps
to get the problem-solving process started, including drawing a simple visual
representation of the system, labeling relevant physical variables, and
identifying the objective of the problem. These methods work because
successful problem solving begins with a clear picture of the problem. Such
strategies can be effective outside the physics classroom as well. For
instance, we often describe complex, real-world problems quite broadly, like
“the problem of climate change” or “the problem of Covid-19,” but in order to
begin to address such immense problems, it is critical that we understand what
specific aspects of these problems that we are trying to confront. In that
way, we can focus our awareness and attention and begin to take action.

  3. 3.

Physics problems also teach us that the problem-solving process can be
simplified by the productive use of assumptions [1, 2]. In physics,
acknowledging assumptions included either implicitly or explicitly in the
problem statement, as well as assumptions invoked to simplify and constrain
the problem, is an essential problem-solving strategy. I developed the habit
of stating my assumptions explicitly early in the problem-solving process,
noting whether I could safely assume that dissipative forces could be
neglected, whether there were no other charges close enough to affect the
system, and so on, which made the process of determining what physical
principles to invoke significantly easier. Our brains regularly make conscious
and unconscious assumptions in order to simplify our experience in the world.
However, when faced with the task of solving real-world problems, it seems
critical that we should be aware of and curious about our assumptions, to be
willing to challenge them, and to be open to letting them go.

  4. 4.

While physics problem solving makes extensive use of assumptions and
simplifications, the use of orders of approximation allows us to identify the
most important factors contributing to a solution and provides us with a tool
to determine whether we may have oversimplified the problem. For instance, if
we want to determine the trajectory of a basketball launched toward a hoop, to
find a first-order solution we may assume that the ball is a point particle in
a vacuum. Then, to determine second-order corrections we may account for
effects such as air resistance and the rotation of the ball. Then, and only if
we are concerned with obtaining an answer to an extreme degree of precision,
we may account for relativistic effects. So, the use of orders of
approximation can simplify complex problems dramatically by offering a
strategy to identify the most important factors contributing to a solution.
Since it provides us with a way to calculate the relative contributions of
various terms to a solution, this technique also allows us to determine
whether we have oversimplified the problem—if we initially made an assumption
that neglected a factor whose contribution to the final answer is an order of
magnitude larger than the factors we chose to take into consideration,
something has clearly gone awry!

  5. 5.

Once we have made sense of a problem, invoked assumptions, and verified that
we have accounted for the most important factors contributing to our solution,
we need to identify the fundamental physical principles at play. One of the
difficulties we might encounter in this process is that different physics
problems may appear quite similar based on their superficial features, yet may
be solved using very different physics—and vice versa. For instance, in one
version of a collision problem, mechanical energy may be conserved. In
another, non-conservative forces may need to be considered. And in yet
another, relativistic effects may be non-negligible. In each case, a different
physical law would be invoked to solve the problem. Clearly, a critical aspect
of the problem-solving process is to differentiate between fundamental
principles and surface features of problems [3, 4, 5]. The skill of
identifying fundamental principles when we are stuck on some surface features
of a problem seems equally critical in the world outside of physics problem
solving. Patients with the sniffles may all appear similar at first glance,
however, in order to determine the proper treatment, it is critical that we
establish whether they are suffering from allergies, a bacterial infection, or
the flu.

  6. 6.

To move forward in the problem-solving process, we make use of
heuristics—cognitive tools and techniques built up in the course of our
experience with problem solving [6, 7]. In physics, useful heuristics include
such strategies as drawing new diagrams, comparing the problem to a similar
one that we have solved in the past, and explaining the problem out loud (a
technique I found particularly helpful in my own problem-solving process).
Other heuristics prompt the unconscious mind to generate insight into the
problem, such as brainstorming related concepts, taking a break from the
problem, even sleeping. From both a cognitive science as well as a practical
perspective, it is interesting that in the context of problem solving, relying
entirely on the conscious mind through single-minded focus is sometimes not
sufficient to arrive at a solution, and insights emerging from our unconscious
mind can play an important role in the problem-solving process [8, 9]. The
critical point is that our problem-solving toolbox consists of both domain-
specific and domain-general heuristics that have proven to be helpful for us
in our unique experience as problem solvers.

  7. 7.

Although our focus thus far has been on our inner, individual problem-solving
experience, there is no doubt that the diverse perspectives of other
individuals enrich the problem-solving process [10, 11, 12, 13, 14]. The study
groups that emerged in my physics classes frequently highlighted the limits of
my understanding and the value of other people’s cognitive processes in
solving the physics problems we had been assigned. When the groups functioned
well, there was something magical about how we helped one another—the part of
the problem I was stuck on was the part someone else understood and vice
versa. These groups also exposed me to some of the inherent challenges of
collaboration, such as keeping an open mind to other perspectives, making
space for all individuals to be heard, and handling disagreements
respectfully, all of which exist at grander scales in the world beyond the
physics classroom. Indeed, the sciences are highly collaborative fields of
study and practice. In research groups around the world, individuals come
together every day to present lines of reasoning, listen critically, solicit
and provide feedback, and use that feedback to deepen understanding. When we
encounter obstacles, we turn to one another, since modern scientific problems
demand combinations of skills and knowledge that no single individual
possesses.

  8. 8.

Once we have completed a problem, our solutions offer an excellent opportunity
for reflection and introspection on our experience of the problem-solving
process [15, 16]. At the conclusion of a problem, we can look back on the path
from problem statement to solution, making note of successful and unsuccessful
strategies. We can also reflect on moments where we were stuck and experienced
a sudden moment of insight—evidence of the work of the unconscious mind—making
note of what preceded the insight. Was it a particular way we played with the
problem? Was it how someone else described it? Was it a process in the
unconscious mind that just needed time to work? This is also a great time to
determine whether our solutions make intuitive and mathematical sense, which
effectively constitutes a check of all the procedural steps leading to the
solution—a nonsensical answer would be a red flag that, somewhere along the
way, we made an error. In addition to performing these cognitive checks, we
can also reflect on our affect during the problem-solving process [17]. We
might note the feelings that emerged when we first encountered the problem,
when problem-solving decisions brought us to a dead end, and when we had
moments of insight and understanding. This can be a critical process to
prepare ourselves for future problem-solving endeavors—after all, it is easier
to accept feelings of standstill and frustration when we have a collection of
experiences in which such feelings eventually lead to feelings of joy and
satisfaction, which in turn can result in feelings of curiosity and excitement
when encountering new problems.

  9. 9.

Along the same lines, mistakes offer insight into our cognitive patterns and
invite us to construct new and more effective ones [18]. By deliberately
reflecting on feedback on graded assignments, we can assess whether we have a
tendency to make certain types of errors: misinterpreting problem statements,
excluding essential details on diagrams, carrying out incorrect mathematical
operations, and so on. When we discover such cognitive patterns, we can make
internal or external reminders to warn us when we are about to repeat them,
making us more effective problem solvers. Even mistakes that, at first glance,
appear completely nonsensical sometimes can offer insight into how our minds
are trying to integrate new concepts and problem-solving strategies [19].
Unsurprisingly, simply stating, “That was silly, I’ll never do that again”—as
many of us do when confronted with evidence of our errors—is generally
completely unhelpful and all but ensures that we will repeat them.222For an
interesting study on what happens when students don’t reflect on their errors,
see Andrew Mason and Chandralekha Singh. Do advanced physics students learn
from their mistakes without explicit intervention? American Journal of
Physics, 78(7):760, 2010. If, on the other hand, we take the errors seriously,
we are offered a wonderful opportunity to examine and modify the dynamic
processes in our minds.

## Conclusion

We have made the case that training in physics problem solving prepares us for
the more challenging problems that we will come across in our lives. To begin,
physics problem solving teaches us to approach problems with confidence, and
to obtain a clear understanding of a problem and its context (Points 1, 2, 3,
and 4). It is rare to find problems in other domains that allow us to so
sharply hone our holistic understanding of a situation. Also, most problems
that we encounter in our physics classes are at an unusual level of difficulty
in that they are challenging enough to require sophisticated and broadly
transferable problem-solving techniques, yet simple enough to be completely
solvable (Points 5, 6, and 7). Furthermore, clear and prompt feedback on our
physics problem solutions offers us the opportunity to reflect both on the
state of our problem-solving skills and on the cognitive and affective
processes involved in our unique experience solving problems (Points 8 and 9).

In conclusion, we would like to encourage physics teachers at all levels to
make the case to their students that physics problem solving is a complex yet
reasonably well-understood process and to provide explicit, formal instruction
in problem solving. We would also like to invite students to look at their
introductory physics courses as being about more than Newton’s laws, Maxwell’s
equations, and the laws of thermodynamics, and instead as opportunities to
develop effective problem-solving skills that may be of use far beyond the
physics classroom, as well as invitations to explore the beautiful
complexities of their minds.

## Acknowledgements

We would like to extend our gratitude to Elam Coalson, Kacy Gordon, and Ella
Herz for their extensive and insightful feedback on early drafts of this
paper.

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