For a decade, I thought mathematics was ugly. Long division was my least favorite. The teacher taught an acronym to remember the steps of long division. By the end of fourth grade, I was good at the rules.
I could fill a page with numbers and through these rules I could find the answer to a division problem. But to me, these rules had no history. I could not derive these rules; all I could do was work by them. Where did these rules come from? I guessed that they came from adults and so I took them to be true. Why does the multiplication of two negative numbers result in a positive number? Because my fourth grade teacher said so, and that was good enough. These rules were fleeting, memorized one minute, forgotten the next.
Mathematics was boring. I wrote numbers and symbols down mechanically and without thought. Plugging in was my least favorite. I spent ten years learning how to plug numbers into formulas and extract answers using my calculator. My mind remained inactive during this time. It required no creative thought; the answers could be found without searching.
Twenty five children per room, I practiced rules for ten years. It was robotic; I now own a calculator that can perform all of the rules I have taken ten years to learn. I think I was programmed to think of mathematics as number crunching using the “right” method. If the method was used correctly, then the answer would be correct. A child who excelled at school mathematics really only excelled at following the rules. “Formula sheets” hinder the creative child mathematician.
The adventure of finding something new is gone. The curiosity of mathematics is destroyed by formulas. There is no reason to dig and search for a rule when it is already given. There is no need to journey if we are already at our destination. During my eleventh year of practicing rules, I saw a glimmer of beautiful mathematics.
I had given up on searching for it because I didn’t believe such a thing existed. Instead, a mathematics artist painted it for me. In a rushed scrawl, he proved that the square root of two was irrational in three simple lines. That was the first time I fell in love with mathematics. It was a proof by contradiction, otherwise known as a counter proof. I had always “known” that the square root of two was irrational because it was a fact.
That was the first time I saw the underside and reasoning to mathematics. The way the numbers and symbols came together was beautiful. It was natural, and mathematics didn’t feel so adult made. The simplicity of the proof was what struck me most. Irrationality is such a complex concept that can be simply proved.
Although I had used the square root of two in previous math classes, I had never understood its complexity and importance. In just three lines, it was crystal clear. Not only was it clear that the square root of two is irrational, but it was also clear how little I knew about mathematics. I had never bothered to question the irrationality of the square root of two; I had just accepted it as a fact. By not questioning and figuring out mathematics by myself, I had missed out on so much. I had missed the definitions and meanings of the rules, symbols, and formulas I had always easily accepted.
A true understanding of the simplest and most basic concepts of mathematics is necessary to an understanding of mathematics. Rules, definitions, formulas, and symbols should all be questioned. One should not accept a mathematical concept as true until it is proved. Finally, I saw that mathematics is not a system of rules to follow. It is a complex abstraction of the mind that was governed by natural laws beyond the power of adults and teachers. Mathematics is not made up.
Mathematicians do not create rules and formulas; they find them. Mathematics is an inherent part of the world and mankind. The interrelatedness between these abstract constructions of the human mind and the physical world is astounding. The physical world can be described beautifully by mathematics, from markings on animal coatings to the patterns of honeycombs and shadows. Pi is the number of times the diameter can be wrapped around the circumference of a perfect circle. Surprisingly, it can be used to describe human population growth.
The rules of mathematics are inherent in nature; they lie in the ground we tread on. Mathematics was built into our planet, and so we must excavate the knowledge from nature. Mathematics is a way to represent nature in the human mind. But only through individual abstraction can we discover and understand more about nature and the mysterious system of mathematics. Though we must be given a shovel, we must individually dig up mathematics. Rules, numbers, figures, and formulas are not meaningful.
Calloused hands, sweat, and hard work bring meaning and life to mathematics. Each formula and rule should be individually derived through creativity and abstraction. Mathematics should only be arrived at when the brain has been fully stretched. We must spend days and years digging for mathematics. Each of us should learn our own method of shoveling meaningful mathematics from nature and our earth.
It takes a long time to understand how to properly excavate mathematics. Mathematics is the art form of finding the rules inherent in nature. Mathematics is not the memorization of rules and formulas, but the process and creativity. A good mathematician knows how to derive his formulas and rules because he has individually found them. He has tended to the simplest of equations with creativity and diligence, so he is more prepared to handle the complex.