Here is an interesting article in the Scientific American on math education.
It seems to be based on a study, which claims that an application centered approach can double the interest of the students in the topic and improve the results.
This is nothing new. In fact, I read articles from around 1900 claiming just the same: Public math education should be for applications in physics and other fields, not for mathematicians. If you go through this blog or any other source devoted to math education, you find the same discussion over and over. I don’t know how often I have spent the lunch break with just that topic.
At 1900, math strived to become an axiomatic and logically complete system of reasoning. Even though that idea was shattered by Gödel later, math got more and more abstract, algebraic and guided by abstract functional analysis. Not even the strange consequences of the axiom of choice prevented mathematicians from using it for basic theorems. „It is a heaven that nobody can take away from us“, Hilbert said. So, it is no surprise that applications of mathematics and students of these fields felt uncomfortable in math classes.
However, times have changed long ago. At least, since computers are available, the more abstract aspects of mathematics didn’t make sense any longer for applied sciences. I need to explain this to prevent being misunderstood.
First, let me start to outline a more modern approach to math for applied sciences.
Let us take the topic of differential equations as an example. In the old school, the students learned some tricks to solve simple cases, involving great skill in calculus and learning the right „ansatz“. The existence and uniqueness of the solution was the next target, followed by theorems on stability. I can say, with confidence, that this approach fails on most students of applied sciences. They simply don’t have the skills, and they have no time to acquire them in their packed first semester schedule.
But I know that many profs now take a more relaxed approach for this audience. We can now use the computers in the classroom to use the Euler method to solve and visualize the results and the effects of the step size easily. These numerical methods have the further advantage that they are discrete and can use differences instead of differentials. Looking to a phenomenon in time deltas is always the first step in physics. Differentials are a higher step in abstraction which should be done after the first step. Then, the benefits and the limits of calculus will be apparent to the student.
Only after studying and seeing simple examples, more abstract questions like uniqueness or stability can be discussed. And we have now the tools to demonstrate these issues on the computer in real time.
The same can be done in probability theory. The computer can be easily used to do Monte-Carlo simulations. This not only verifies theoretical results, but also allows getting new ones which cannot be derived with algebraic methods. Moreover, it implants a solid meaning of the term „probability“, which so often is abused and misunderstood leading to self-contradicting cases.
Another change that computers force on us is that skills in computation are no longer essential to solve mathematical problems. This starts in primary classes, where calculators replace the external drill of written computations. Later, we should start to ask ourselves if extensive skill and practice in tricks of calculus is really needed any longer. Low-level computations in the head and an understanding of the sizes of numbers (which most people lack) looks more important to me. And I’d say we should rather concentrate on understanding the logic behind problem-solving and forget the drills. This brings me to my last topic.
I should end this blog entry with a warning: Logical reasoning won’t become easier with computers. From my teaching, I know that the students struggle most with the logic of a scientific idea. Neither can they verbalize the idea in precise form, nor can they apply it. Asking a precise definition of continuity or an eigenvalue is one of the most challenging things you can do in a verbal examination. Asking the students to explain what their computations mean and do, logically, is a good way to see their skills.
Logical reasoning needs to be learned by training and repetition. I cannot stress enough, how important it is for the teacher to not expect the students to immediately understand the meaning of a mathematical statement. Looking back, a teacher should remember that he did not understand most things immediately himself. Everybody struggled with logic at some point. Never forget that!
So, while it is an „old hat“ to talk about a more applied mathematical education for applied sciences, we still need to exploit what the computers can do for math education. And we still face the problem of logic reasoning, which doesn’t teach itself and needs a lot of attention.