On Problem Solving

Posted Thu Sep 20, 2001 in

Since my hydraulic design class last Wednesday, I’ve been thinking about student life and problem solving. One of my students asked for an example of a problem assigned to them as homework. (Which was a reasonable request. ;) What was so notable to me was her comment, which will always stick with me: “Monkey see—monkey do.”

When I meet with my advisees each semester, one of the things I try to do for them is to give them some insight into problem solving. I remember clearly when, as an undergraduate student at Missouri-Rolla (although it’s now > 20 years ago) there was a period during which I discovered the difference between “pattern-matching” solutions and development of a solution from real knowledge. As I grew to understand more about how I personally learn, the latter became my standard approach. Furthermore, when I found myself falling back into the pattern-matching approach, I knew that my understanding was incomplete.

That leads me to the following essay. It is probably more an attempt to clarify my thinking as much as anything else.

How to Solve It

There’s a great little book by George Polya called How to solve it. I’ve looked at this book several times and it could easily form the basis for an introduction of students to the problem-solving approach. Anyway, I’m getting ahead of myself.

Pattern Matching

The classic approach that most students use is what I’ve termed the pattern-matching approach. It’s advantage is simple: the learner (student) simply finds a solved problem that is of the same type (or class) of problem, then mimics the original solver’s steps to the solution. The learner believes that, having duplicated the solution, they have “learned” how to solve the problem. By having a large enough database of problems memorized, the learner (student) is prepared to approach an in-class exercise (my euphemism for an examination).

I know I’ve used this approach on numerous occasions and expect to use again. Pattern-matching is an appropriate tool when first learning a new technique. However, the difficulty with this approach is that maintenance of the problem database is required. After a few semesters, that’s a lot of stuff to remember! In fact, what happens is that the learner (student) forgets some or all of their problem database. Therefore, the individual “forgets” what they’ve learned. This explains the excuse that I so often hear: “It’s been three semesters since I had fluids; I don’t remember that.” (To which I sometimes respond, “It’s been 20 years since I had fluids, but I still remember.” With feisty students, which I appreciate, I sometimes draw the response, “But, you’re the teacher!”)

So here we are with the pattern-matching technique. Is there a better way? Sure there is.

Synthesis

I discovered sometime during my sophomore year that I was learning general principles from which I could construct problem solutions without having to look at other’s solutions to similar problems! That is, I was generalizing (or synthesizing) how systems work and how to apply general principles to solution of engineering problems. I distinctly remember working my way through second-semester calculus (the integral calculus) and how difficult it was because there was so much rote memorization. In order to solve problems, I simply had to memorize the formulae.

At about the same time I was taking an engineering class (either fundamental soil mechanics or fluids or something like that). I recall how much easier a time I was having with that class, not because I knew so much about the material but because I understood the fundamental principles and could formulate a solution to problems I was presented with. I recall the striking difference between the two approaches.

I’ve asked myself why this should be many, many times. I think that it’s because the focus is not on the specific problem but on the process. Therefore, there is less material to remember. (I’ve always hated rote memorization!)

This leads me to how engineers work. (I see another essay on engineering coming on—I’ll write it another day. ;) I’ve seen plenty of engineers, when faced with a design problem, head over to the plan file to see what’s been done before. (Yes, I’ve BTDT but no T-shirt!) Certainly we need standard details for components and assemblies that we use as standard designs. However, many problems deserve or require a new (unique) design. Therefore, looking in the plan drawer is going to provide no help in developing a design for such a problem.

That leads me to the punch line: Real Engineers learn how systems work. They combine knowledge of materials, physics, chemistry, and mathematics into a special blend of ability that enables them to produce systems useful to others. Because our systems are complex, there is in general no pattern-matching solution. Young engineers who move beyond the pattern-matching approach while in engineering school have an easier transition into professional life than those who don’t.

So, the take-home message is to learn to synthesize general principles from your studies. Use pattern-matching as a first step but strive to move beyond it to application of general principles (from your personal knowledge-bank) as quickly as possible.