The World is an Ill-posed Problem

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 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.