Friday, February 14, 2014

Why Do We Learn Math?

I have been thinking a lot about how math and its role in higher education lately.  Before getting into my thoughts, let me quote (again) from a classic article by Gale and Shapley  [link here], which makes clear that math isn't numbers or calculus even, but a way of thinking:
Most mathematicians at one time or another have probably found themselves in the position of trying to refute the notion that they are people with "a head for figures," or that they "know a lot of formulas." At such times it may be convenient to have an illustration at hand to show that mathematics need not be concerned with figures, either numerical or geometrical. For this purpose we recommend the statement and proof of our Theorem 1. The argument is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical, while people without mathematical training will probably find difficulty in following the argument., though not because of unfamiliarity with the subject matter.
The insight that Gale and Shapley put forth -- that mathematical training helps with forming and following structured logical arguments -- is a really important reminder for what role math should play in high school, college, and graduate education.  If our goal in education is to enhance our students' abilities to think critically and effectively analyze problems, mathematical training is indispensable.

This reason for taking math seriously is quite different than the usual motivation to push for better math and science education.  Usually, people (politicians, commentators, etc.) argue that math and science are the most practical of the subjects, and because we're going to be using these tools every day, we had better study them and master them.  That's how we end up to be engineers and scientists, and those are good, high paying jobs.

It is true that math is practical for these professions that use math, but math is so much more than that.  Mathematical training helps structure logic, logic facilitates problem solving, and any high-skilled job is going to require a lot of problem solving.  We should take math seriously because math is a language that is particularly well suited to posing, understanding, and solving problems.

On a closing note, the tricky thing about languages is that when you become fluent in them, you do not notice how much you use it.  This is especially true with mathematical training.  You may not use calculus every day, but the very fact that you learned calculus changes how you think and process information.  You're better at solving problems because you took calculus, and thus, have a richer experience in solving problems.  This is true whether or not you need to use calculus to solve the problem.  

2 comments:

  1. I wonder how much studying math actually changes the way the average person thinks. It's an empirical question, someone should do a study.

    There is already plenty of work that measures how much "material" people actually retain from math classes (or any classes) in general. The consensus seems to be: depressingly little. But do certain classes change them in other ways? Do they think differently afterwards, and do they have different beliefs, even if they can't remember the reasoning that led them to these beliefs?

    Suppose the vast majority of students forget the fine details of how to solve any of the problems from their economics classes. They might still remember that it did once make sense to them, that there is a sensible way to think through certain problems, and even though they don't remember what it is anymore, they may trust economists more than they used to.

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    Replies
    1. Thanks for the comment, Xan!

      I agree that it is an empirical question, but it is an important one. There is an important distinction between remembering subject matter as a tool ("fine details" as you put it) versus being influenced to think differently by the subject matter. Education should be about thinking differently, rather than mastering fine details and tools.

      Even so, remembering the fine details is neither a necessary nor sufficient condition for "being influenced" to think differently (e.g., I don't remember much of my measure theory, but I'm still glad I have taken it for math sophistication purposes).

      It is for this reason that, when I teach, I focus on the big picture, the process, the reasoning, and the problem solving, rather than the formulas, the tasks and the tools. In my view, the problem solving parts of the educational process are more durable (and more fungible across tasks), and thus, can have much greater impact on careers and lives than really mastering how to compute a standard deviation, a derivative, maximizing a utility function, or finding where D and S intersect!

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