Formalism and Intuition: more problems in teaching math

topOne of the big issues that interests Occupy Math is the teaching of mathematics. In earlier blogs we have looked at teachers being blamed for things that are not under their control, problems with teachers being chained to high stakes standardized tests, math teaching strategies that implement fads without understanding them, teaching topics in silos, and the problem of thinking of math as a form of ritual magic. Parts of this last topic are examined in finer detail in today’s post, where we look at the difference between formal math and understanding math.

Math is a formal subject and mathematicians try to choose exactly the most useful formalism. Many of Occupy Math’s readers may have encountered functions. Formally a function is:

A set of ordered pairs with the first coordinate of the pair drawn from the domain of the function and the second drawn from the range so that no two ordered pairs have the same first coordinate.

Occupy Math’s editor found this definition to be gibberish! Without examples and definitions of “ordered pair”, “domain”, and “range”, it probably is gibberish, but please stay tuned. At one point one of Occupy Math’s professors from his undergraduate days asked a class what a function was and a student repeated the definition verbatim. My old professor then asked for an example of a function and the student looked at her blankly. The student had memorized the formalism while gaining no intuition from it at all. It did not bother the student that the math, as stated, was nonsensical.

In fact, we have many tools, like the vertical line test, for helping students gain intuition about functions. We have tools for taking things that are not functions, like circles, and either breaking them into multiple functions (the top half and the bottom half) or changing the point of view to make then functions. A circle in standard x-y coordinates is not a function. A circle in polar coordinates is a function — but of different variables. See Occupy Math’s earlier post on making flowers for an application of polar coordinates.

The intuition gap

Occupy Math has argued that, aside from physical sciences and mathematics majors, that calculus is a poor choice of a first university math course. One of the core reasons for this is that the intuition of calculus is pretty hard to get across: infinitesimals and limits are among the most difficult things to understand, but there are many intellectual sand-traps in calculus.

This problem is made worse by the fact that too many math teachers have the formalism of math in hand, but not the intuition. They teach the students how to pass a test, but not how to understand the math, what it means, and its implications for the real world and their lives. This makes math education far less useful to the students — and much harder. Occupy Math took electromagnetic physics after his fourth semester of calculus and differential equations. This meant that he could remember two surface integrals (don’t ask!) instead of twenty-odd formulas for the magnetic field around some arrangement of conductors. Understanding the formalism of the math would not have been enough — Occupy Math needed the intuition that explained how calculus lets you compute the shape of magnetic fields around conductors.

I tried to be nice!

At a conference in Brazil, Occupy Math spent some time with a colleague who teaches engineers. He was complaining that his students had the mathematical formula for the Fourier transform, but no intuition at all. For a cool intuitive video on this topic, check out Three Blue One Brown. Briefly, if you have a function of time (something that tells you a measured quantity at each point in time), this transform gives it to you as a sum of sine and cosine functions. This lets you see, transparently, what the periodic elements of the function are. Temperature, for example, will have a noticeable 24 hour and 365 day periodic component. Basically Fourier transforms let you see periodic elements in data or formulas with ease.

Occupy Math’s colleague asked his students “If we delay the start of acquisition of data, what will the effect on the Fourier transform will be?” The answer was, “The magnitude of the transform will not change, but the phase will shift.” With sine and cosine functions, the magnitude is how high they are and the phase tells you where their waves cross zero. My colleague thought this was a gift, an easy problem that could be answered without needing to do a bunch of calculations. A majority of the students did not answer the question, because their intuition, their understanding of the meaning of the Fourier transform, was missing!

What can you do?

A definition, by itself, is pretty dry. If you want to give students the chance of developing their intuition about the concept you are defining, then you should give several examples of things that fit the definition — and they should be different enough from one another to illustrate just how far the definition stretches. To make a literary analogy, a definition supported by a cast of good examples is a well-rounded character; the definition by itself is a cardboard cutout that conveys little knowledge.

Occupy Math has asked, on several occasions, what he thought were low work gift questions that turned out to be horribly hard because they relied on the students understanding the math rather than having the correct rituals set up and ready to go. Here are a few steps you can take:

  1. Teach the intuition, not just the formalism, when you are the teacher.
  2. Care about the intuition, the meaning of the math, when you are the student.
  3. Give examples that illustrate the intuition.
  4. Give homework and practice problems that permit students to develop and practice using their intuition.

Occupy Math would be happy to try to do posts about the intuition involved in specific areas of math. Some of the most popular posts in the blog have been initiated by reader requests — you are better than Occupy Math at figuring out what interests you! Requests can be made to and your identity will be held in confidence or please comment on this post!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


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