One Mortal's Quest to Comprehend Himself and the World
When I was younger, I didn’t pay very much attention in my math classes, much like the other kids around me. The subject wasn’t necessarily boring, but it wasn’t interesting and it definitely wasn’t fun. I believed that math was something that I could perform adequately well at the time. However, I only studied enough to complete the homework assignments and pass the exams. I memorized the formulas and precisely executed all of the convoluted algorithms without questioning exactly what I was doing. Although I was a B student in my math classes, I didn’t completely understand the core concepts. I did not have a firm grasp of the absolute fundamentals. For example, you could’ve asked me what 2+2 equaled or what the Pythagorean Theorem was and I would’ve said four or a²+b²=c². If you were to then ask me to prove those claims or to explain why they were true, I would’ve been stumped. I wouldn’t have known where to begin. Why, exactly, did 2+2 equal four and not five? Why was the square of the hypotenuse equal to the sum of the squares of the other two sides?
It’s commonly said that math is cumulative. The things that you learn in the beginning will be useful for learning more advanced concepts in later courses. You start with basic axioms and definitions and then use them to construct basic theorems. You then progress to more advanced theorems and definitions using those basic theorems. You can’t simply learn a few concepts, take the exam, and then forget it all. That is a guaranteed ticket to confusion and failure. I believe this is one reason many school kids have trouble with mathematics. Teachers assume that they remember and understand key concepts, but they really don’t. They can’t understand A because they don’t understand B. They don’t understand B because they never mastered C. They never mastered C because they failed to grasp D, and so on. This problem gets worse as students advance to higher grades. Thus, whenever a student is learning math he or she must understand the fundamental ideas. Simply memorizing formulas and proofs is not constructive. Besides, there are way too many formulas to memorize.
It wasn’t until after college that I started to take an interest in mathematics. It all started when I read an article about imaginary numbers and how to think about them. When it was explained to me that multiplying a number by the imaginary number, i, ‘rotated’ the number by 90 degrees counterclockwise, something in my brain clicked. When I realized that addition and multiplication with complex numbers simultaneously rotated and scaled numbers, I felt as though I uncovered a deep fact about nature. The concept was so simple! Why didn’t they teach this in school? I immediately thought about the scaling and rotation of images and computer graphics. Perhaps you could do these operations without having to deal with sine and cosine functions? I quickly became obsessed with understanding the underlying relationships regarding mathematical constructs. However, I soon ran into a problem: there are just too many concepts to learn. How was I supposed to learn and remember them all?