Written by Guest Blogger, Gene McKenna

Introduction by Steve Hare:
One of the great benefits of being on Twitter has been the opportunity to meet and get to know Gene McKenna. His love of mathematics and desire to improve math education are nothing short of inspirational. Want proof? Check out the T-Shirt Puzzle website he created in his spare time for the pure love of sharing math concepts and calling attention to their beauty. The following is an article about mathematics as an art form that Gene shared on Twitter a while back, and that I just loved; he's graciously agreed to let us post it here.

Inspired by an exchange with the amazing @mpershan

My son and I play a game I call Hit the Tonic. It’s not a drinking game with gin - it’s a game on the piano. He plays the first 7 notes of a scale, then guards the piano trying to prevent me from relieving the tension by playing the final note of the scale - the “tonic”. I’m like Roger Rabbit who has to complete “Shave and a Hair Cut” or he gets physically ill.

Returning to the tonic is one of the simplest progressions in music composition. More advanced music composition has hundreds of other progressions through two or more chords used to tell a story or express certain feelings. A composer reaches into a rich musical toolbox to lay these out in a specific order, expressed a certain way, in a style which may impose constraints or accepted norms. No one doubts this is art.

Enter algebra

Math is often criticized as nothing but memorizing procedures and math facts. Usually the word “rote” is thrown in to indicate disdain. And yet, “solving for x” is much more like composition. It is art.

What is the procedure for solving 2x - 15 = 30 ? The “procedure” is easily stated:

Do the same things to both sides of the equation until x is isolated and known.

But those are not clear, and unambiguous instructions. We have multiple ways of getting to the result and, just like art, we can have discussions about the relative beauty and elegance of each. On the left we add 15 to both sides, then divide both sides by 2. On the right we divide both sides by 2 and then add 7.5 to both sides. They arrive at the same result but one is more elegantly done and more beautifully expressed.

If math is really all about memorization and procedures, we would expect to have an exact procedure to follow for each and every algebraic equation. But we don’t. Why?

Because it’s an art

There absolutely are procedures with exact and very specific steps in math. Getting to the point of algebra requires memorizing: 400 math facts (See FactFreaks). It requires memorizing procedures like multi-column addition and long division. By algebra, about 450 facts and/or rules have to be memorized ideally to a level of automaticity (aka “muscle memory”).

Theoretically one can do algebra without memorizing these things. Calculators are available. Similarly one can compose music without being able to read it, or knowing scales or any theory. It’s just very, very difficult without the foundation of the simpler concepts.

In all disciplines, basic skills to be memorized are also to be understood. There are reasons why certain martial arts forms are important and why some combinations of forms go together. Learning the “how” and the “why” go together and are equally important. Neither is ever learned perfectly, but getting better at “how” gives a deeper understanding of “why”. Knowing more about “why” makes remembering “how” easier and makes knowing “when” to use a technique possible.

Wax on, Daniel-son

Once a student is equipped with a good understanding and has become automatic with the tools of their craft, they have the tools to make art. This is as true in math as music or dance. Famously, Ralph Maccio’s character Daniel learns, in the Karate Kid, that we may not see the point of some of the basic skills right away, but once we can do them automatically, putting them together in novel sequences on-the-fly can be both a beautiful expression and a life-saving skill.

Solving an equation for x is like a composer who “feels it in their bones” and therefore “just knows” what to do to move a piece along to the next step, and the next and the next.

None dare say dance or martial arts is only memorizing positions or forms. But neither do we rail against instructors for requiring their mastery.

All Kids Can Love Math

Critics of requiring mastery of the skills which make mathematical art possible - memorization and procedural fluency - repeatedly fail to understand how art is made or how mathematical art springs forth from those skills. Many who claim to “want all to love math”, now want to delay art-making until later grades. What could be more soul crushing? “Sorry Johnny, practice your scales for another year. We’ll make music next year.”

But in this day and age, why does anyone need to memorize anything? Dance moves, jazz chords, martial arts forms … surely these are all available on youtube? But imagine a dance performance, a sparring match or a concert where the artist had to pause every 3 moves and go watch a video. Pretty horrific, right? But it isn’t just for the viewer’s pleasure that memorization is needed.

In all fields we more easily learn the advanced concepts and combinations when the basics are deeply ingrained and effortlessly recalled. (see Cognitive Load Theory) Music, dance, art and martial arts instructors require it. Increasingly math teachers do not.

Perhaps if we renamed Algebra as Mathematical Art we’d better communicate the necessity and value of all the work students do prior to arriving at algebra. And perhaps we could dispense with the notion that calculators and YouTube mean basic skills are no longer necessary.

All students can be mathematical artists and can love math. But only if we understand it for what it is.