The emerging theory of how the brain works gives us a new language to us for discussing how we teach, learn and communicate math.

Modules

Our minds have many functionalities.  They are implemented by what I called modules in Math and modules of the mind because I don’t understand very much about what cognitive scientists have learned about how these functionalities are carried out.  They talk about a particular neuron, a collection of neurons, electrical charges flowing back and forth, and so on, and it appears there is no complete agreement about these ideas.

The functions the modules implement are physical structures or activities in the brain.  At a certain level of abstraction we can ignore the mechanism.

Most modules carry out functionalities that are hidden from our consciousness.

• When we walk, the walking is carried out by a module that operates without our paying (much) attention to it.
• When we recognize someone, the identity of the person pops into our consciousness without us knowing how it got there.  Indeed, we cannot introspect to see how the process was carried out; it is completely hidden.

Reasoning, for example if you add 56 and 49 in your head, has part of the process visible to your introspection, but not all of it.  It uses modules such as the sum of 9 and 6 which feel like random access memory.  When you carry the addition out, you (or at least I) are conscious of the carry: you are aware of it and aware of adding it to 9 to get 10.

Good places to find detailed discussion of this hiddenness are references [2] and [4] below.

Chunking

Math ed people have talked for years about the technique of chunking in doing math.

• You see an algebraic expression, you worry about how it might be undefined, you gray out all of it except the denominator and inspect that, and so on.  (This should be the subject of a Mathematica demo.)
• You look at a diagram in the category of topological spaces.  Each object in the diagram stands for a whole, even uncountably infinite, space with lots of open and closed subsets and so on, but you think of it just as a little pinpoint in the diagram to discover facts about its relationship with other spaces.  You don’t look inside the space unless you have to to verify something.

Students have a hard time doing that.  When an experienced mathematician does this, they are very likely to chunk subconsciously; they don’t think, “Now I am chunking”.  Nevertheless, you can call it to their attention and they will be aware of the process.

There are modules that perform chunking whose operation you cannot be aware of even if you think about it.  Here are two examples.

Example 1. Consider these two sentences from [2], p. 137:

• “I splashed next to the bank.”
• “There was a run on the bank.”

When you read the first one you visualize a river bank.  When you read the second one you visualize a bank as an institution that handles money.  If these two sentences were separated by a couple of paragraphs, or even a few words, in a text you are likely not to notice that you have processed the same word in two different ways.  (When they are together as above it is kind of blatant.)

The point is the when you read each sentence your brain directly presents you with the proper image in each case (different ones as appropriate).  You cannot recover the process that did that (by introspection, anyway).

Example 2. I discussed the sentence below in the Handbook.  The sentence appears in references [3].

…Richard Darst and Gerald Taylor investigated the
differentiability of functions (which for our
purposes we will restrict to ) defined for
each by

In this sentence, the identical syntax appears twice; the first occurrence refers to the open interval from 0 to 1 and the second refers to the GCD of integers m and n.  When I first inserted it into the Handbook’s citation list, I did not notice that (I was using it for another phenomenon, although now I have forgotten what it was).  Later I noticed it.  My mind preprocessed the two occurrences of the syntax and threw up two different meanings without my noticing it.

Of course, “restricting to (0, 1)” doesn’t make sense if (0, 1) means the GCD of 0 and 1, and saying “(m, n) = 1” doesn’t make sense if (m, n) is an interval.  This preprocessing no doubted came to its two different conclusions based on such clues, but I claim that this preprocessing operated at a much deeper level of the brain than the preprocessing that results in your thinking (for example) of a topological space as a single unstructured object in a category.

This phenomenon could be called prechunking.  It is clearly a different phenomenon that zooming in on a denominator and then zooming out on the whole expression as I described in [1].

This century’s metaphor

In the nineteenth century we came up with a machine metaphor for how we think.  In the twentieth century the big metaphor was our brain is a computer.  This century’s metaphor is that of a bunch a processes in our brain and in our body all working simultaneously, mostly out of our awareness, to enable us to live our life, learn things, and just as important (as Davidson [4] points out) to unlearn things.  But don’t think we have Finally Discovered The Last Metaphor.

References

1. Zooming and chunking in abstractmath.org.
2. Mark Changizi, The vision revolution.  Benbella Books, 2009.
3. Mark Frantz, “Two functions whose powers make fractals”.  American Mathematical Monthly, v 105, pp 609–617 (1998).
4. Cathy N. Davidson, Now you see it.  Viking Penguin, 2011.  Chapters 1 and 2.
5. Math and modules of the mind (previous post).
6. Cognitive science in Wikipedia.
7. Charles Wells, The handbook of mathematical discourse, Infinity Publishing Company, 2003.