Chronicle
December 20, 2013

Mathematics and Understanding

(Published in Aftenposten December 20, 2013, in Norwegian.)

There is currently a lot of talk about mathematics, standardized tests and Norwegian pupils' "mathematics skills". However, almost no one talks about what we actually mean when we say "mathematics", or what mathematical understanding and experience we want the students to achieve in school. Do we want the students to really understand what they are doing, or is it sufficient that they become good at calculations?

A Mathematician's Lament, Paul Lockhart, 2002
A Mathematician's Lament, Paul Lockhart, 2002

The American math teacher Paul Lockhart wrote in 2002 a pamphlet entitled "A Mathematician's Lament" (PDF), where he compares the teaching of mathematics in school with a music education without music, where you are focusing solely on music reading and notation and where you neither play instruments or listen to music.

He describes the American school as a school where mathematics is gone, where the only thing that remains is an empty shell, and where the mathematical understanding and experience no longer has a place. It is a harsh criticism, but I agree and think the same can be said about a lot of the mathematics in today's Norwegian school.

For what is really the essence of mathematics? Think about it before you continue reading. Imagine that you had to explain and define what mathematics is. What would you have said, what would you have focused on? Would you have begun to explain the four arithmetic operations (plus, minus, multiply and divide) and what numbers, algebra, fractions and functions are?

No. Mathematics is something deeper, more fundamental, colorful and beautiful than that. Many believe that mathematics mainly all about numbers, formulas and calculations. However, this is wrong. Mathematics is the art of thinking.

Mathematics is about making assumptions and reasoning towards the consequences of these assumptions. Think of it as making the rules of a game, like chess or connect-four. Once you have defined the rules, the game is given. If you make good rules, you get good games. This is also how it is in mathematics. All rules of the game are more or less arbitrary, but they are defined as they are for creating good games.

I often say to my students that they can assume anything, but they must also accept the consequences that follow. If they want to assume that 2+2=5, then that is perfectly fine. Mathematics is the art of thinking from assumptions.

Mathematics is about finding patterns and abstracting. Look around you and look for patterns and symmetries. It is almost impossible not to find them. When you recognize forms or see that two people are related, then you are a pattern detector. Mathematics is the art of finding patterns.

Mathematics is about finding good representations, and to be able to switch between them. If you're able to represent one and the same phenomenon in two different ways, then that enables you to see new connections. Why is there such a great focus on solving equations in school? It is not to make children good at applying calculation rules without thinking: it is because the equations represent something else. Mathematics is the art of finding good representations.

Above all mathematics is about using the imagination and being creative. If you imagine a circle, then this is something that exists only as an idea in your consciousness. And this circle is absolutely perfect, unlike circular looking objects in nature. Mathematics is the art of using the imagination in order to reason about such abstract ideas.

Doing math in this way is to be an explorer. To discover and prove mathematical truths becomes an art form. One asks questions like "What is really pi? How long is the diagonal of a square? Is infinity a number? Do infinite sets exist?" and spend time trying to understand, instead of using rules for calculation and putting numbers into formulas.

A great way to explore and discover mathematics is through programming. Why is not programming and computer science an integral part of school? I do not mean a word processing, spreadsheets, web pages and graphics, but proper programming, where you make your own programs and tell a computer exactly what to do.

In several ways, computer science and mathematics are two sides of the same coin: It is about making rules, finding patterns and representations, exploring consequences, abstracting, and solving problems. And doing this in a way that is dynamic and safe: When one makes a mistake, then one does not throw entire program away and start over; one looks for the mistake and modifies the program such that it becomes correct.

This debugging process is extremely constructive and useful and has great transfer value to other parts of life. By teaching children basic programming, they get a tool that enables them to explore mathematics themselves, experiment with models, define their own rules and simulate natural processes. A computer can thus be an extremely useful tool for achieving greater insight and understanding.

Many take pride in the fact that they never understood math. And when I ask children about what mathematics is, very many of them just answer, "boring". That does not surprise me. We have managed to make the most creative and colorful of all subjects into something that it is ok to dislike. If you think about it, it is not so strange. We teach a lot of rules, facts and formulas and expect that understanding in a magical way emerges through a lot of training and repetition of formulas.

It is an extreme optimism to believe that even more calculations training and exercise solving is what it takes to increase understanding of mathematics. The optimism lies in believing that understanding comes automatically. It does not. Being good at applying rules, in a precise and accurate manner, is not the same as understanding what you are doing.

We have taken away the discoveries, the creativity, the beauty, the experience, the colors and the essence. This has become a cultural problem that there is no simple solution to. However, we must give children and adults the opportunity to fall in love with mathematics, to look at it as something that is beautiful and harmonious, just like a fine piece of music.

We must get them to see the art and the experience in mathematics. We do not do this by even more calculations, more thoughtless application of formulas and more repetition. We do this by creating wonder, joy, fascination and curiosity, by opening up for exploration and by telling good stories.

by Roger Antonsen