Over the years I’ve lost confidence in the idea that applications of mathematics make the subject any more interesting or easier to grasp. It is perhaps not a clever thing to say in public because of the obvious objections. But I’ve gradually swung over to this idea about a humanistic approach focusing the inherent interest and depth of the subject itself with its history and philosophy and weird language. And the applications of mathematics are naturally included under the banner of humanistic mathematics. Applications in science and technology and everyday life are clearly but aspects of human culture. A humanistic approach to mathematics thus naturally subsumes applications and there is no contradiction.
That is a good thing – it puts applications in its proper place – it is but one aspect of mathematics. Thinking about applications in a humanistic way may be an anti-dote against dry, dull, boring and silly standard applications of textbooks. So now I’ve said it.
A happy thing occurred a week ago in one of my calculus classes. Here is the background. I don’t remember if I’ve written on these pages about my experiments, going on for a couple of years now, having mathematics exams where I just require answers to the problems? Yes you read it correctly. I don’t care for the solutions to the problems. I only want to see the answers.
You may be so shocked that you cannot continue reading and perhaps I should explain my rational reasons for doing this. However that would distract me from what I want to say in this post, so bear with me, I will soon write about why I think Only-Answers-Exams is a very good thing in mathematics.
However, there is no denying that there is a point in the students being able to write up a solution to a problem. It’s just that a 4-hour exam is not the proper time to require writing up solutions to perhaps 8 problems. As a compromise, I have now included one problem where the focus is on writing up a solution. It should be an easy problem that almost everyone who has done the course can do so that the focus is indeed on writing up the solution.
Anyway, the calculus class was drawing towards its end and I was trying to come up with a good problem to require a solution to. The students are textile engineers, so I thought about such “applications” and finally came up with the example of calculating how long a thread can be rolled up on a bobbin. For every winding of the thread, the radius increases by the diameter of the thread, and the problem essentially boils down to summing an arithmetic series. I gave the students this problem as an exercise. For the exam I gave them the same problem but now requiring them to solve it by integrating the linear function for the radius as a function of the number of windings. The students became very engaged, they knew that some variant of this problem would appear on the exam, and that always leads to engagement, of course. We had an extra session where the students could ask question as a preparation for the exam. There was much discussion about various approaches to the thread winding problem.
I don’t think it was just the fact that the problem was an exam problem that lead to the engagement, these are ambitious and dedicated students. I believe that they found the problem intriguing in itself. It was simply an interesting problem that was engaging.
Could it be said that this is an application of calculus? Yes, probably so, but I prefer not to think of it as an “application”. I prefer to think about it as an “engaging problem”.
Let me try to generalize: we should not look for “applications” of mathematics. Since the real “reality” (so to speak) is so complicated, real “applications” become to difficult. Instead we (textbooks) come up with unrealistic simple “applications” such as: what is the volume of this bottle, the shape of which you get by rotating a function around the y-axis. That problem is silly and boring. In reality: just fill the bottle with water and then …
[I know, however, that the volume of wine bottles was an important problem in the early days of calculus, so there is some humanism in that too …]
Instead “engaging” problems occupy a middle ground between real applications and silly textbook “applications”. They are interesting in themselves, they are not too simple and they require quite some thinking, they can be solved in different ways, and the common sense solution (take the bobbin and wind up a thread, then rewind and measure the length) is not really feasible. The problem should be relevant for the students. And we don’t need lots and lots of such problems (and they are not easy to come up with). Perhaps three or four of them, designed with care, is sufficient for a calculus class.
What I’m trying to say is: thinking “applications” leads the thinking into the desert of boredom, thinking “engaging problems” leads the thinking into more fertile grounds. Thinking “applications” means thinking “lifting mathematics into reality”. Thinking “engaging problems” means “lifting reality into mathematics”.