### Why learn maths?

#### by Willie Wong

There is a quite well-known joke that goes something like this: a mathematician, a physicist, and an engineer are each handed a little red rubber ball, and are told to find out its volume. The mathematician takes a caliper and measures the diameter of the ball, and uses the formula . The physicist applies Archimedes’ principle: he submerges the ball in oil, and measures the displacement. The engineer goes back to his lab, walks over to the bookshelf, and opens this gigantic leather-bound volume to the Table of Little-red-rubber-balls.

To continue my earlier rant, it appears that I am not the only one to think that modern mathematics education is misguided. It took Underwood Dudley (warning: a subscription to the AMS Notices is required to access the PDF file) 8 pages to get to it, but in the end, he comes to the same conclusion as I did, albeit from a different starting point. The joke above is meant to illustrate Professor Dudley’s main argument, that *despite grandiose claims by the National Academy of Science or the National Research Council, a mathematics education does not provide a practical skill set that is necessary for most jobs*. (I will come back to the emphasis on the word “practical” in a bit.) With many examples, Dudley’s essay illustrates a common fallacy, that mathematics is important to learn because frequently in life (especially at work) one will encounter situations which calls for computations beyond basic arithmetic.

Perhaps I should make the distinction clear here: I am not claiming that mathematics education is useless. I am just observing that fact of life that most people, going about their everyday lives, will very infrequently, if ever, encounter a situation that requires the finer understanding of mathematics beyond the middle school level (US; elementary school for East Asia). And therefore we, as scientists/mathematicians/educators/parents, should not *oversell* the learning of mathematics as something crucial to one’s future, and bully the kids into studying it. Or, in other words, while I think that familiarity, nay, fluency, with mathematical concepts is a requirement for a well-educated man, I do not consider erudition to be a requirement to lead a productive life in society.

The mathematical curriculum (and by extension the physical sciences and history and all other of the more academic classes in American schools) should not invent reasons to convince the students that mathematics is used in all facets of life, and hence important. The knowledge of how to change a flat tire is likely to be much more practical for the average Joe over his lifetime, than the knowledge of how to solve the quadratic equation. (Most of what you need to know to succeed in life, you learn in kindergarten anyway.) That people come to question what goes into the general education based on potential on-the-job utility is completely misguided. A general education for the populace should *not* be equated with vocational training. A general education should train the students in the ability to reason, to think soundly, to approach problems logically. A flexible mind that is open to new ideas and is capable of solving problems is an asset applicable to any job. By narrow-mindedly restricting one’s attention to the immediate and direct applications of classroom subjects, one runs the risk of missing the grander picture in which the whole is more than just the sum of its parts.

In the version of the joke that I’ve heard, the engineer is the one who uses the calipers, while the mathematician only provides a proof that the answer exists and is unique. (Perhaps as per http://golem.ph.utexas.edu/category/2010/04/on_the_magnitude_of_spheres_su.html#more)

Jokes aside, a natural question from the above is to ask what mathematics education ought to be. I suggest that one of the reasons for mathematics education not providing a practical skill set is that it generally consists of rote learning of a set of rules. You’re successful in this game if you apply these rules to the (usually artificial) set problems, and hence pass the tests.

What is not taught to how to ‘think mathematically’ so as to be able to use mathematics in situations that you may encounter later on in life. For example, in purchasing a house you may be offered various finance packages, with differing interest rates, repayment plans, penalties and so on. How do you evaluate which is the best one for you? If you don’t know the necessary mathematics, how would you learn it? How would you check your answer? IMHO, the kinds of skills implied by the above questions (ability to identify/learn the relevant techniques, and apply them without an available ‘textbook answer’) would be a real ‘mathematics education’.

I’m not intending to imply that I know the answer for how this would be done in practice, nor am I blaming teachers in any way. I just think that changes could be made to help to build a ‘flexible mind that is open to new ideas and is capable of solving problems’, as you put it.