In this section, Wit Busza discusses the importance of explaining to students why we solve problems.
Why do we solve problems? We don’t often explain to students why we solve problems. But shouldn’t we? After all, when you really stop to think about it, the answer to this question is not at all trivial.
Experiencing One of the Biggest Mysteries of Science
The primary reason we solve problems is to experience one of the greatest mysteries of science. In the scientific method, what do we do? We try to take a physical situation and describe it in terms of mathematical equations. In other words, we have this belief that nature can be described in terms of mathematics. Then, once we derive these equations, we forget that they have anything to do with the real world. We take them as purely mathematical expressions. The amazing thing is that we can use mathematics to predict what will actually happen in the real physical situation. Take any problem, and you find you go through this process. You take the physical situation, you describe it in terms of equations, and you predict what will happen. If you repeat the process experimentally, you find it actually happens just as you predicted. If you really stop to think about this process, you realize it is absolutely amazing. Every time a professor explains a problem or a student does a problem, he or she is doing something truly remarkable.
Explaining why we solve problems—and the ways in which it is astounding that this process actually works—is a perfect opportunity to bring excitement into students’ learning experiences. It’s an opportunity for teachers to help students move away from problem solving as a formulaic experience toward experiencing what Eugene Wigner (1960) described as the “unreasonable effectiveness of mathematics in the natural sciences” and toward feeling what Albert Einstein meant when he noted that “the most incomprehensible thing about the world is that it is comprehensible.” Very few people appreciate the profundity of problem solving. And it’s one of the biggest mysteries there is in science! Now, I’m not saying I’ve made a big discovery here. I haven’t. I am just reminding you and emphasizing that we solve problems to experience first-hand this profound mystery and that we should explain to students this rationale for solving problems. We do them a disservice if we allow them to think that problem solving is isolated from the way nature behaves.
Once you accept the first reason for problem solving, then something very useful comes out of it. Suddenly you realize that you can predict what happens under situations you've never before encountered. If you’ve developed some technology or built something, you will want to know how it will behave in situations you haven’t yet experienced. So you set yourself a problem and solve it for a very practical purpose.
For students, problem solving is a way to learn the skills and techniques needed to make accurate predictions. Predicting what will happen in unknown situations also serves as a way to assess if students have understood the content of a lecture. I think it’s very important that we explain to students these reasons for solving problems.
Wigner, E. "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc.