### The World is an Ill-posed Problem

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I got into trouble the other day on one of my listserves. A
physics teacher asked if anyone had a collection of electrostatics questions
that might be appropriate for multiple-choice assessments in intro physics. I
had just completed a first pass on putting together a set of clicker questions
for intro physics, and many of them had been drawn from quiz and exam questions
that had worked well. So I posted a link to them on the listserve.[1]

I then got severely taken to task by one of the listserve
regulars because too many of my questions were ambiguous. The word he used was
“ill-posed”. I looked at some of the questions and realized that in
transitioning from exam or quiz question to a PowerPoint-displayed clicker
question, I had been forced to remove a lot of text, including some
specifications. Someone who wanted to use them as assessments would have to
reconstruct the more complete context. My bad. This means that I should not
only be posting our clicker questions, but our exam and quiz questions in their
original form in order for them to be useful. I’ll get right on it.

But I was interested in my reader’s reaction. I

*liked*my clicker questions and had used some of them extensively in class. Many were very effective for generating engaging and useful discussions. Some of the best discussions took place when some student said, “Hey! Wait a minute. What if …” – and they noted an unspecified ambiguity. I would have lost those discussions if my questions were all “well posed”. What was I trying to do?
An explanation relates to my perception of how we ought to
be teaching the mathematical content in our physics classes. This is
represented by my “four-box” diagram show below.

Note that this diagram is intended to represent the
philosophical imbedding of math into physics and NOT as a process that should
be followed and NOT as a model for how the brain interprets physics. The
processes of actually doing physics or thinking about math in physics are much
more intricate and contain lots of feedback loops.

The diagram IS intended as a way of thinking about the role of math
in physics – and of guiding us as instructors to be sure that we are including
some activities for each linking line. In this picture, the idea is as follows:
Math provides a set of symbolic structures that have been tested for
consistency (as much as possible –

*pace*Gödel) and that provide well-developed and tested procedures. We use these mathematical structures by first selecting a set of physical phenomena that we want to describe. We then create ways of making mappings of physical properties onto elements of a mathematical structure – a*mathematical model*of the physics system. From this, we inherit the processing and solution structures of the math, allowing us to develop relationship and solve complicated problems, well-beyond what we can manipulate or store in our heads at any single instant. This is extremely powerful and works well. It is a major part of our success in physics.
But the math is not the whole story by a long shot. The
other parts of the diagram are just as important. Choosing what bits of the
great blooming and buzzing world we are going to pay attention to and model
mathematically is definitely the hard part – and where the art is in doing
science. [2] Interpreting and evaluating whether the model works – and for what
– play a critical role in establishing and refining our mathematical model.

Now to get back to the point. “Ill-posed” is a mathematical
term. It means that the problem has not been stated in such a way that there is
an unambiguous and unique solution. But this lives firmly on the top side of
our diagram. Stating that a problem is ill-posed or well-posed means that
you are evaluating it as to how well framed it is as a mathematical problem – how
well it is situated in the upper-left box.

In my teaching, however, my focus is much less on whether my
students can turn the crank on a well-posed mathematical problem. I am much
more interested in whether they can “see the physics in the math” – model
physical systems mathematically and evaluate the math they generate in terms of
the physics. The physics is NOT identical to the math. The world is an ill-posed problem. A major part of what I want my students to do is consider what it

*means*for a problem to be ill or well posed. What has to be specified? What is the physical system like? What do I have to ignore to enable the construction of a mathematical problem that is well structured? Is it OK to ignore these things? Just for now? Always?
We often are so concerned with making sure that our students
can “handle the math” that we ignore – worse, suppress – the essential issue of
matching the physics to the math. In my experience, this appears to be one of
our habits that lead so many of our students to reject physics as “irrelevant
to real life.” [3] We might argue that it’s a part of our selection process for
physics majors. Physics majors have to be strong in the mathematical skills and
we can claim that they will learn modeling and evaluating approximations later
in their careers.

But I even object to this for physics majors. I did not
really learn that the math

*wasn’t*the physics until I was well into research. I would have been a stronger and more effective physicist earlier had I understood the difference between the math I was being taught and the physics that it was attempting to describe. In my advising of graduate students, I have seen many who had trouble with Jackson (Electricity & Magnetism) or Sakurai (Quantum Mechanics) because they focused too strongly on the math and failed to blend their physics knowledge with it. (For a more technical discussion of this, see my paper with Tom Bing in Phys Rev. [4])
What’s even worse is that most physics students are not
going on to be physicists. They are going to be engineers, biologists, and
doctors. Teaching only “well-posed” physics to these students tends to turn them
off. These students tend to care more about reality than about math and they want
to see the connection of the physics to the real world. You might say, “Well
that’s what well-posing does for them. It states what has to be specified in order
to use the math.” But until a student understands what’s going on it doesn’t look like
that to them. What they see is “You’re just giving me unrealistic [and, for them, by
implication uninteresting] problems.” They find learning to pose and evaluate
problems from a real-world physical situation (and then solving them) more
satisfying.

If we go back to the original argument – “this is for an
assessment, a quiz or exam, so you can’t give them ill-posed problems” – I
would concur, in a way. You don’t want your exam questions to be too ambiguous.
But if you only test them on mathematical manipulation, they will ignore the critical
other parts of my diagram.

Therefore, I explicitly try to include in my exams questions that test the students on the other parts of the diagram. I might assign a problem that starts with a real-world example, then propose a set of approximations, ask the student to solve it, but then ask, “of the proposed approximations, which would you want to put back first in order to get a more realistic result, if you had more time.”

Or I might give a multiple-choice question in which an experiment is proposed and ask the students, “From this experiment what can you conclude?” If the case is that the result being sought is known to the students, but the experiment is insufficient to show it, the correct answer may be not “the correct answer” (that is, what really happens), but “nothing can be concluded from

Therefore, I explicitly try to include in my exams questions that test the students on the other parts of the diagram. I might assign a problem that starts with a real-world example, then propose a set of approximations, ask the student to solve it, but then ask, “of the proposed approximations, which would you want to put back first in order to get a more realistic result, if you had more time.”

Or I might give a multiple-choice question in which an experiment is proposed and ask the students, “From this experiment what can you conclude?” If the case is that the result being sought is known to the students, but the experiment is insufficient to show it, the correct answer may be not “the correct answer” (that is, what really happens), but “nothing can be concluded from

*this*experiment.” Or, I may ask them to construct a problem or constrain a problem through order-of-magnitude estimations.
A major component of learning physics is learning to look at
the real world and generate a well-posed problem; not only to see what that
means but why you might want to do so and what are the implications of doing
it. In my current class, I often use ill-posed problems in the hope of
generating discussion and encourage my colleagues to consider moving in this
direction as well.

[1] These materials are being developed for a new physics
course for life-science majors. The goal of the course is NOT to introduce the
students to every physics concept they might ever see in their lives; rather,
it is designed to provide support for difficult concepts they encounter and use
is biology and chemistry, and to prepare them for upper division biology
classes. To see these materials, go to http://nexusphysics.umd.edu.
They are currently very much in a state of flux, so if you have suggestions or
find problems with them please let me know. (redish-at-umd.edu)

[2] E. Wigner, “On the
unreasonable effectiveness of math in the natural sciences”.

[3] E. F. Redish, J. M. Saul, & R. N. Steinberg, Student
expectations in introductory physics,

*Am. J. Phys.***66**, 212-224 (1998).
[4] T. Bing & E. F. Redish, Epistemic complexity and the
journeyman-expert transition,

*Phys. Rev. ST Phys. Educ. Res.*, Vol. 8 (Feb 2012), 010105. doi:10.1103/PhysRevSTPER.8.010105.
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