Math is More Than Calculation

Before attempting to start talking about what math is and why I am so interested in it, I want to clarify what it is not. What I’d call math is probably is not what you learned in school. I think a better label for what most people think of when they hear “math” is calculation. What is taught in school, or at least the way a lot of people approach what is taught in school, is memorized methods for arriving at results. You memorize your times tables, you memorize how to add fractions, you memorize the quadratic formula. Note the repeated use of the word “memorize.” You learn that certain techniques are right, and try to replicate them. And that’s the end of it. No questions asked. When we do calculation, we do essentially the same thing a computer does. We receive instructions and execute those instructions.

So if calculation is not mathematics, what is? I don’t really take issue with calculation; memorizing certain techniques is quite useful. There is a place for that in math, just like learning basic vocabulary has an important place in English class. But in English, you don’t stop there. The problem with math classes, or the way people approach them, is that most people stop at calculation. I think the best way to express what I mean is to share how I first fell in love with math.

I was in second grade. We had already learned our times tables, and we were beginning to learn problems like 32 times 28 using the following method: \begin{aligned} & 32 \\ \underline{\textnormal{x}} & \underline{28} \\ 2 & 56 \\ +\underline{6} & \underline{40} \\ 8 & 96\end{aligned}

In case you didn’t learn this method, the general idea is that you multiply 8 by 32, and make that result a new row, you then place one zero in the next row, and then place 2 times 32 to the left of the zero. The last step is to add together the two rows.

I had already been curious about multiplication when I learned how to do something like 32 times 8, but got even more curious when I learned this way to multiply numbers like 32 and 28. It caught my imagination. I wanted to know why we did things this way. I thought about it, and it took a while but I eventually came up with some ideas. I knew that multiplication was about grouping things together, that is, the sentence “2 times 4 equals 8” can be rephrased as “2 groups of 4 make a total of 8”. Now, say we have 28 groups of 32 people. This is really the same thing as 20 groups plus 8 more groups, and I realized this is why you had the two rows, one for the 8 and one for the 20. I also knew that 20 = 2 x 10, and so 32 x 20 = 32 x 2 x 10. So, the 0 at the right side of the second row is there because of the 10, and the other work is just 32 x 2. So I started to see why we were doing things this way.

Around this time, I had an idea that made me even more curious. I thought that, perhaps, the same thing will work for bigger numbers. If I wanted to do 789 x 123, for example, couldn’t I make add together three rows, one for 100, one for 20, and one for 3? And since the row for 10 had one zero, I should put two 0’s on the row for 100. In the notation from earlier, I thought that maybe this would work: \begin{aligned} &789 \\ \underline{\textnormal{x}} & \underline{123} \\ 2 & 367 \\ 15 & 780 \\ +\underline{78} & \underline{900} \\ 97 & 047 \end{aligned}

We hadn’t learned this yet, so I went to my teacher after school, very excited, and asked her if that was correct. She told me it was right! I was really happy and proud, but it got better. I could see now that the size of the number didn’t matter, I could use the same trick! This teacher graciously helped me learn more math after school when she could, and let me use her whiteboards to multiply huge numbers together for fun. What I had noticed I later learned had a name – the distributive law. We normally write this as $A \times (B + C) = A \times B + A \times C$

What I had noticed in multiplying numbers like 32 and 28 is that the “standard method” we were being taught was taking advantage of this rule in the following way: $32 \times 28 = 32 \times (20 + 8) = 32 \times 20 + 32 \times 8$

The additional step I noticed is that you can use the same rule for larger numbers like this: $789 \times 123 = 789 \times (100 + 20 + 3) = 789 \times 100 + 789 \times 20 + 789 \times 3$

Challenge for the Reader: Don’t just take my word for it! Convince yourself that the distributive property still work when more than two numbers are being added together.

This may sound weird to some, but I would probably rank this moment in the top five or ten most joyful moments of my life, and spurred a strange excitement with doing long and arduous multiplication problems. It was never calculating in and of itself that I enjoyed, though there is a sort of childish glee looking at incomprehensibly large numbers written on a whiteboard. The reason this moment had such an impact on me is because this wasn’t just something I’d been taught, this was something I discovered on my own. I understood where it came from, I knew how to explain it, and most importantly, I found it on my own.

This, I would say, is what math is really about. It is about understanding not just how things are, but why they are that way. It is about finding patterns, asking questions, and discovering why we see the patterns we do. Even more than this, math is an art form that transcends time and language. In the words of G.H. Hardy in his classic essay A Mathematician’s Apology, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

The difference between a calculation and true mathematics is like the difference between a road sign and Shakespeare. Both are written word, but the first is dull and the second is artistry. A simple calculation on a receipt is dull, but the writings of the great mathematicians of the past are works of art. To quote Hardy again, “there is no permanent place in the world for ugly mathematics.” We hardly glance at road signs, but we still read Shakespeare hundreds of years after his death. So it is with great mathematics.

So what does real mathematics look like? What does it mean to show why a pattern is true? Why do I and so many others find mathematics so incredibly fascinating? These will be topics for posts soon to come. But if you learn anything from this post, calculation and mathematics are not the same thing!