We also need to convey to students that mathematical truths are objective, persistent and timeless. They are not subject to changing authority, fads or fashion. A mathematical statement is either true or false; it’s something we all agree on. To paraphrase William Blake, mathematics “cleanses the doors of perception.”
What distinguishes us from cavemen is the level of abstraction we can reach. Abstraction enabled humans to move from barter to money, and from gold coins to plastic cards. These days, what’s left of “money” is often just an account record we read on a computer screen, and soon it could just be a line of code in a bitcoin ledger.
Today, abstraction is all around us — and math is the language of abstraction. In the words of the great mathematician Henri Poincare, mathematics is valuable because “in binding together elements long-known but heretofore scattered and appearing unrelated to one another, it suddenly brings order where there reigned apparent chaos.”
For the next generation to operate effectively, they must gain proficiency with abstraction, and that means mathematical knowledge plus conceptual thinking times logical reasoning — all things that a wider view of math would bring to the math classes at our schools.
I recently visited students in fourth, fifth and sixth grades at a school in New York to talk about the ideas of modern math, ideas they had never heard of before. They were young enough that no one had told them yet that math was impenetrable, that they wouldn’t get it. Their minds were still uncluttered with misconceptions and prejudice. They hadn’t yet been humiliated by poorly trained math teachers for making mistakes in front of their peers. Every question I asked them was met with a forest of hands.
I used a Rubik’s Cube to explain symmetry groups: Every rotation of the cube is a “symmetry,” and these combine into what mathematicians call a group. I saw students’ eyes light up when they realized that when they were solving the puzzle, they were simply discerning the structure of this group.