Practical Mathematics

Many students say that they want to learn practical, useful mathematics, and that is what I want to teach as well. Let's consider some aspects of doing mathematics in the real world:

• Real world problems don't come labelled with the solution method.
• Real world problems demand combining methods which you may have learned in disparate contexts.
• Real world problems may require learning methods which you have never met before.
• The real world doesn't have an answer book.
• You really have to solve the problem in the real world. It's your job. Or someone is depending on you. Or you are really, really interested in the engineering, or science, or business situation, and you really, really want to understand what's going on.
• Real world problems occur in a social context: you have to solve the problem together with a team, or you have to solve the problem and explain the solution to someone.

What are some consequences for the way you have to work in the real world?

• You have to really get involved with the problem and understand what it is. You have to use the problem to figure out what methods may be relevant. You have to be very flexible and open minded and ready to try all sorts of approaches, including symbolic, numerical, and graphical methods.
• Since you really have to solve the problem, you have to be self-reliant, determined, and persistent.
• Since there is no answer book, you aren't finished when have your "solution." You have to look critically at your solution, and test it for reasonableness in any way that you can: does it agree with common sense, is it internally consistent, does it have a reasonable order of magnitude, did the units come out right, does it make reasonable predictions.
• You have to be able to explain your work coherently, in an appropriate mixture of real English sentences and mathematical notation.

How does this compare with the ususal cycle of work in a school mathematics course?

• In school mathematics, the work is organized by techniques: a book section contains a technique, and the exercises which are given are suited precisely to be solved by that technique. The problems do come labelled with the relevant technique, and you generally do not have to combine disparate methods.
• You have an answer book.
• Most students get used to skipping the harder work and waiting for an explanation, since this turns out to be the most "efficient" way to get the work finished. (That is, you don't really care about the problem or the solution, and you don't feel that you have to solve it.)
• Generally, students are not required to explain their work coherently, and they don't try to do so.

In short, the usual pattern of work in school mathematics is pretty much a disaster for learning to do practical mathematics.

What can we do to make this course more useful for learning to do real mathematics? The organization of the course is fixed, and it is organized in the traditional way, by techniques. If it were otherwise, you might feel pretty uncomfortable, since you already have 12-13 years of experience with school mathematics and no experience at all with real mathematics.

I have the following suggestions:

• Take advantage of every opportunity to work as in the real world: Get involved with the problem and be determined to figure it out yourself. Getting stuck is an opportunity, an opportunity to persist and work something out for yourself.
• Breeze through the more straightforward computational work, but then lavish your time on the less clear cut, open ended exercises. Note that "theoretical" exercises are probably the most practical work you are doing, and the "practical" exercises the least practical! That's because the theoretical exercises give you the opportunity to practice the intellectual habits which are actually useful in the real world.
• Throw away the answer book. (OK, don't throw it away, but avoid it as long as possible.) Test your answers yourself for reasonableness in any way you can. Compare your answers with those of your colleagues and resolve the differences.
• Make a real effort to explain and present your work well. A chaotic presentation allows a vague and chaotic understanding. A logical and clear presentation demands a logical and clear understanding.