When we learn new knowledge, it takes up a large space in the brain—it literally occupies more room—as the brain works out what it means and where it connects with other ideas already learned. But as time goes on, the concepts we have learned are compressed into a smaller space. The ideas are still there so that when we need them, we can quickly and easily “pull” them from our brain and use them; they just take up less space. If I were to teach arithmetic to kindergarten students, the concepts would take up a large space in their brains. But if I asked adults to add 3 and 2, they would quickly do so, pulling the answer from their compressed knowledge of addition. William Thurston, a mathematician who won the Fields Medal, described compression in this way:
Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.
You may be thinking that few students describe math as a “real joy,” and part of the reason is that we can only compress concepts. So when students are engaging in mathematics conceptually—looking at ideas from different perspectives and using numbers flexibly—they are developing a conceptual understanding, creating concepts that can be compressed in the brain. When students believe that mathematics is about memorization, they are not developing a conceptual understanding or forming concepts that can then be compressed. Instead of compressed concepts in the brain, their math knowledge is more like a ladder of memorized methods that stack one on top of another, stretching, as it may seem to these learners, to the sky.
