Critical Thinking v Problem Solving II - Problem Solving from Start to Finish

In order to explore problem solving and critical thinking a little more, I would like to talk about problem solving. Mainly, I'll talk about a 1963 paper by Slagle, who programmed the first symbolic integrator using punch cards. And a very clever algorithm. It was able to solve 52 of 54 integrals on what sounds like a Calculus II final at MIT* using a ladder of proximate goals reaching up to an ultimate goal, the solution of the integral. It needed, to mimic some aspects of human problem solving by implementing two sets of rules: those that are always good and those that are heuristics and sometimes break down. But it does get to the answer. This isn't so surprising in itself, since commercial products, e.g., Mathematica, have been doing so for decades, but the fact that it could be done with a 1959 mainframe computer and with so few rules is an astounding fact. I will use this algorithm as the backbone for how to think about problem solving.

The general way that people solve math problems is by a searching method.** You start with what you're given and then you search the space with known rules and inspiration and try to move forward, one step at a time. From time to time you get stuck, and then you take a few steps back and try another likely path. One of the differences between an expert and a novice is how they react to a setback: a novice usually chooses one way of doing a problem, then never steps back.*** An expert tries several tactics, until they complete the task. Matthew Shoenfeld's work quantified this behavior, and private discussions make it clear that many people consider this searching, a la Polya,(*4) almost a moral imperative of reasoning.

Slagle's system worked on a method of proximate goals. That is, you start with a main goal, where you'd like to get to, and then as you work towards it, you sometimes identify a subgoal. When you do, you keep track of that subgoal. Sometimes as you work, you can reach a point where several directions are possible. In these cases you keep all of the goals, and place them in your tree. You don't try to work all of the goals together. Instead, you use various methods to assign the goals priority. Some goals can be reached through automatic processes, processes that are guaranteed to move you toward the main goal. You do these first. Others are heuristics, and you look to those only when there is no automatic option. Furthermore, for heuristics, you need to judge how costly each one is in the given situation: how hard it is compared to the goal. And you try the least costly first, as long as its line remains the least costly. When you find a way to reach the main goal, you have solved the problem.

This should remind you of the method that people use to solve problems in the real world, as observed on Betamax by Shoenfeld: you search the space with known rules, trying to move forward, and when you get stuck, you take a few steps back and try a likelier path.

There were three types of methods used by the program to solve the integrations problem. The first was a short integration table, a list of "Standard Forms." If you came to a standard form, then you had effectively solved the problem. In effect, the rest of the edifice is built to put a non-standard form in the image of a standard form. The second set of methods were those that always improved the situation. Whenever you came to one of these you tried it, and then checked it against the standard forms. These had no deviations and didn't really require any interesting tracking. You tried one after another until there were no obvious steps to try. Finally, there were the heuristics. These are rules that sometimes improve the situation, but sometimes do not (sometimes, trying them makes matters worse). When you came to this point, the program would try all of the applicable variants, and then assess the character of each try to judge which is the best way to proceed. And as I said, it worked in 52 of the 54 cases, and the other two could not be solved because the IBM 7090 didn't have enough memory for more entries in its integral table.

This goal-directed reasoning is what I'll use as a paradigm for problem-solving. I often characterize it as a tangram in my classes: physics gives you a set of tiles with which to form the shape of the solution. The tiles are limited, but the forms are infinite.(*5) Arranging the tiles into the correct form is called thinking.

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* At MIT, it's part of 18.01 Single Variable Calculus. 'most everywhere else, it's Calculus II - Methods of Integration -- with some idiosyncratic identifier cataloged by the inane, parochial system used by universities (a college catalog is a definitive, smack-down, irrefutable argument against expert judgement, although listening to the faculty senate is even better). I first learned of this algorithm from a very good lecture by Patrick Winston from 6.034, Reasoning: Goal Trees and Problem Solving. A short description of the algorithm can be found in Winston's Artificial Intelligence textbook.

** See Mathematical Problem Solving, Matthew Shoenfeld. I would like to show some diagrams from his work, but at this time I don't have a way to use images.

*** A college freshman thinks that problems should take 2-3 minutes to solve, and literally thinks that a problem that takes more than 10 minutes is impossible.

(*4) See, e.g., How to Solve It, George Polya.

(*5) Well, not really. Firstly, of course, there is room for disagreement about how many tiles there are, and what constitutes a different tile. I am sure there are many fewer than the 60-odd list of things the engineering college says it wants the students to learn (about one per 25 minutes of instruction), since "solve a quadratic equation," "Newton's Third Law," and "Kepler's Laws" are all very different types of things. Also, I think that there aren't technically an infinite number of solutions. I am 95% sure there are only seven one-dimensional kinematics problems for one process on one object (and 100% certain for uniformly accelerated motion). Well, I guess, then, if I allow an infinite number of processes and an infinite number of objects, I could end up with an infinite number of kinematics problems. Probably. And these can be sutured onto more complicated dynamics problems, which unlike kinematics, are really physics.

How Long Does It Take to Do an Infinite Number of Things?

Infinite processes shred our intuition to Hell like nothing other than probabilities and relativity. This was the infernal currency of Zeno of Elea, who proposed devilish paradoxes in order to prove the conjecture that movement is impossible. Zeno would construct a simple scenario that would show that everyday aspects of life, like chasing a tortoise or shooting an arrow. We really only have accounts from other authors, basically Aristotle, who disagree with him. Some of these arguments intertwine ideas about space and time, the arrow argues that an object is motionless at any instant, and both the dichotomy and Achilles and the Tortoise make arguments about the impossibility of doing an infinite number of things. I will eventually move on to an example to calculate the total time that it takes for a ball to bounce an infinite amount of time, but I will first describe the dichotomy using Philocetes' Arrow as a story (rather than Aristotle's bare-boned description from the Physics [1]).

Philocetes looses an arrow from the Bow of Heracles at the Trojan prince, Paris. In order for the arrow to strike the Son of Priam, first, it must fly half the distance between the heroes. But, it is clear that in order for an arrow to travel half way to the midpoint before it can get to the midpoint. And it has to travel half way to the point before that, ad infinitum. Therefore, in order to move from one spot to another, no matter how close, you have to move an infinite number of times in a finite amount of time to get anywhere, so the arrow never flies and Trojan War never ends.

This is paradox because we do move, but at least the first time your hear the argument, you don't have a good reason why it is wrong. The arrow must move, but logically it cannot. And this is applicable to every kind of motion. Before you can eat your Wheaties in the morning, you have to get out of bed and get into the kitchen. But, you have to move through an infinite number of small separations to get there.

And you can't do an infinite number of things before breakfast.

Or can you?

Most people think that the invention of Calculus resolved Zeno's paradoxes. This is certainly true in the case of the dichotomy: Leibniz used and even dirtier trick with infinity than Zeno did. Leibniz built calculus out of the idea of an infinitesimal to align with his cosmological ideas. An infinitesimal is a chunk of the universe that is smaller than the smallest division, basically the reciprocal of infinity. The integral calculus would define the distance that the arrow must travel as a sum of all of the infinitesimal chunks of space between Philocretes and Paris. The nature of infinitesimals is that they are smaller than the smallest fraction, there are an infinite number of them between each rational number. The infinity of the infinitesimals is that of the real numbers (the continuum) and the infinity of the dichotomy is that of the rational numbers (countable). So, if you can construct a theory of motion that adds up all the infinitesimal points, it will automatically encompass the infinity of halves used by Zeno.

This solve the dichotomy, but it does so indirectly with an end around.

By subsuming the motion of the dichotomy into a single, continuous process that can be analyzed separately, we show that motion can exist. We solve the riddle by changing the problem, though. However, this leaves Zeno's premise unchallenged: an infinite number of processes take an infinite amount of time. What I'd like to do here is take on the premise that an infinite number of discrete, sequential processes needs to take an infinite amount of time. You could do the same thing with the dichotomy, as well, but since that is an arbitrary partition of a single, continuous process, which I feel is a little different.

The specific question I ask is: how long does it take a rubber ball to stop bouncing? The physics here is quite simple. It can be done with kinematics using the simplest of deflection theories: the coefficient of restitution. The model uses the simple rule that the velocity of the rebound of an object is proportional to its original speed, and that proportionality (the coefficient of restitution) remains the same after each bounce. The duration of the air time of the ball is given by uniformly accelerated motion. The sum of a sequence of such bounces will lead to an infinite series with a known sum (thanks again, calculus), and this sum will be finite.

Where can this go wrong? Well, it's not in the assumption of uniform acceleration. Yes, it's not quite true, but it's pretty accurate at low velocities for short times, which a rubber ball acts in. If we complicate the problem by adding in air resistance, that will give us a slightly more accurate estimate at the cost of an annoying integral (no thanks, calculus). This accuracy will give us a time that is strictly smaller than the uniform acceleration version by giving us a factor similar to the coefficient of restitution itself. The significant assumption that would break this analysis, if it were relevant, would be that the time of the bounce itself will be the same each time the ball hits the ground if the bounce is modeled on an elastic restoring force, which is probably the best model available. Even though this will be small, at some point it will be larger than the air time per bounce, and since it remains the same, adding an infinite number of them would create an infinite time for the bounce.

But, to answer the basic question, can an infinite number of processes be completed in a finite amount of time, eliminating the time of the bounce is justifiable.

So what happens in this case? Well, from basic kinematics, we find that the time of an individual flight is proportional to the initial speed of that bounce.* Since the initial speed of each process is the coefficient of restitution is just the initial speed of the previous process, the duration of the subsequent process is scaled down by the sane proportionality,

t_n = r t_n-1 = rⁿ t₀

which means that flight is scaled down by a power of the coefficient of restitution.
When these are summed, we find an infinite series in powers of the coefficient of restitution that has a known sum: the inverse of one less the coefficient [ 1/(1-r) ]. So, the total time the ball bounces is finite if r < 1 (which is must be unless it is gaining energy from the environment somehow).

So, an infinite number of bounces takes a finite amount of time

t = 1/(1-r) t₀.

This is a reasonable answer because if r = 1 the bouncing goes on forever and if r = 0 it stops after the first flight. This should be the same result you'd find if you were to sum the time to travel each segment of the arrow's path, but here we have distinctive processes represented by the flights between bounces. Our hero Leibniz has defeated Zeno of Elea's Satanic dichotomy.

So the Trojan War terminates, and you can do an infinite number of things before breakfast.



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* The proportionality constant is 2/g.

[1] All of my Aristotle is missing. Most of what I know about this comes from Sainsbury's Paradoxes, >although I was using the Stanford Encyclopedia of Philosophy.

Critical Thinking vs. Problem Solving I - How Are They Different?

I've spent a lot of time thinking about critical thinking and problem solving. Nominally, my course is required to include a critical thinking component by the university so that it order to satisfies a distribution requirement. Over the past five years, I have never really been able to discern the difference between critical thinking and problem solving, a more common term for what we "teach" in physics. Some papers explicitly say that problem solving is a kind of critical thinking, [1] others say that it's a separate skill. [2] I don't really know the proportion. I originally just went for a rather strict problem solving format from the University of Minnesota* that I feel shows me how a student is thinking about problems,and based on Force Concept Inventory scores, I implemented it well (gain around 35-45%).

The students hated it, but my job isn't to be adored.

However, one spring day in 2020, I was informed that my course was to be reviewed for its critical thinking component.** When reviewing a course, all I have to do is to submit a product for each student that shows their critical thinking skills. This product can be a test question, a homework problem, a paper. I have a section on the tests where students individually provide an explanation of how to solve a problem, and I felt that this would do. It is a little stilted. What do you want to do? "Find the velocity." How are you going to do it? "Use conservation of energy." How do you represent that? "1/2 m v^2 = 1/2 m u^2 + mgh." It shows exactly what the student is thinking and how they utilize the data. I felt it would be a good way to show the development of the students' problem solving capabilities over the semester.

But I was given a rubric, and the rubric told me that I was wrong.

These are some highlights from the rubric:

Explanation of Issues. Problem is stated clearly and described comprehensively, including all relevant information.

Evidence. Viewpoints of experts are questioned thoroughly.

Context. Thoroughly analyzes assumptions and carefully evaluates the relevance of contexts when presenting a position.

Student's Position. Position is imaginative and other's views are synthesized within it.

Conclusions. Conclusions are logical and reflect the evidence and perspectives in priority order.

Try that with conservation of energy.

For some of this, expanding the selection from just he planning phase of the problem solving process would probably do. For others, it seems irrelevant. In fact, some of the categories seem to be completely irrelevant to the course ("Influence of Context and Assumptions" is the full title). But, looking at the rubric for the curriculum component, I feel at minimum it requires a term paper, and probably a thesis. The school implicitly takes the side that problem solving is not a part of critical thinking.

Critical thinking, as described by the rubric, is really separate from problem solving.

However, I still think that there should be some overlap. I think in come coming posts, I'll talk about what I think problem solving and critical thinking are, possibly in several posts each, and then I'll talk about some specific problem-solving tools for first year physics students.

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* That, I think, they don't use it any more.

** And just after I wrote this, they told me that this was the evaluation of the engineering students' "written communication."

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[1] Willingham, D., "Critical Thinking: Why Is It So Hard to Teach?" American Educator (2007).

[2] Pasquinelli, E., M. Farina, A. bedel, and R. Casati, "Naturalizing Critical THinking: Consequences for Education, Blueprint for Future Research in Cognitive Science." Mind, brain, and Education 15, 168 (2021).