I had a teacher in college named Shlomo. He's a fantastic teacher; his lectures are rife with information, you learn a ton, and he imparts a lot of passion and excitement into the material. Shlomo told a story one time about going to an art museum, just for kicks. He wasn't an art fan, but wanted to see what the fuss was about. He remembers seeing a painting and not being very impressed with it. He went back to the same museum later, and there was a tour taking place, so he tagged along. The tour guide stopped at this same painting and espoused all of the things he loved and respected about this painting. Shlomo left with the same love of the painting as his tour guide, and he learned something too: Sometimes people don't appreciate something until they learn why it should be appreciated.
A little later (this might have happened in the same lecture, in fact) Shlomo is working through a proof of something. I don't know what the proof was on, but it was in a geometry class. He looks at an equation and says "Maybe you don't like this equation. You don't like it? You change it. Where else can this happen in life but in mathematics? You want something, you get it. Only in mathematics."
I'd never thought of this. Truly I didn't really stop to think about it until much, much later. His lectures were dense, and I usually left feeling a bit punch-drunk as my brain tried to wrap itself around the new keys to the universe it had discovered. One day later I was solving an equation with a bunch of fractions, and instinctively I got rid of the fractions. No one likes working with fractions.
It hit me at that moment. I didn't want the fractions, so I got rid of the fractions. I was able to get what I wanted immediately. This doesn't happen in life! Only in mathematics!!
The above equation has 4 different fractions with no common denominators. I have to teach my students how to solve these types of equations, and when I show the first one their faces go white. I say "What, you don't like fractions? Fine. I don't either. Let's get rid of them." If we multiply everything by 30 (the lowest common denominator of all 4 fractions), we get the equation 20x + 24 = 15x - 25. No fractions. Hallelujah.
I pause here and yell at my class. We didn't like the fractions so we got rid of them. You have that kind of power! You have that freedom! If you don't like something in math, you can change it! You're given rules to follow and you follow the rules, but do whatever you want. Where else in life does this happen? Where else can you want something and get it right away? If you want to get stronger, you have to work hard. If you want to get richer or prettier, you have to be extremely lucky or you have to work hard. If you want to get taller, well you can't. But in math, when you want to get rid of the fractions, you get rid of the fractions. I love that about mathematics. I love many things about mathematics, but that's a big one.
My hope is this is a turning point for my students, where they go from what is their worst-nightmare-of-a-problem, to a realization that math is potentially wonderful. I try to keep sharing this point, and don't mind it when they laugh at my nerdiness. Just like how Shlomo inherited the love of a painting from someone else, Shlomo had to show me this appreciation for math, and then I inherited it. I hope if I can show this appreciation to my students, they can inherit it as well.
Well, I've thought more about this, of course - this relationship of rules and freedom in mathematics. As long as you follow the rules you can do what you want. My parents have often said that kids want boundaries. They want to have clear rules of what they can and cannot do. They want to have these rules enforced consistently and fairly. If they have freedom within those rules they will be more or less happy and content. This is just like mathematics. There are extremely clear rules that are very, very consistently enforced, but if you follow those rules you can do as you please.
So then I think about society. Ideally, society would be full of easy-to-understand, all-encompassing, non-loop-holey rules that are enforced but are fair. This doesn't happen, and that leads to all kinds of problems, but wouldn't it be great of society was more like mathematics in this sense?
(Heck, isn't this kind of like how God gives us rules to follow, but freedoms within those rules?)
You are thinking "But Grant, what about art? What about freedom of expression? What about diversity? What if some rules work for some people and not for others?" My reply? Shut up hippie. Seriously, though, that's all true. We won't ever have that idyllic society where a fair and just government makes and enforces laws that are for the equal protection and benefit of all. It won't happen. That sucks.
But I can have that idyllic world in my small, misunderstood world of mathematics. There's freedom in all of those restrictions and rules that students don't understand. I just need to get them to appreciate it.
A little later (this might have happened in the same lecture, in fact) Shlomo is working through a proof of something. I don't know what the proof was on, but it was in a geometry class. He looks at an equation and says "Maybe you don't like this equation. You don't like it? You change it. Where else can this happen in life but in mathematics? You want something, you get it. Only in mathematics."
I'd never thought of this. Truly I didn't really stop to think about it until much, much later. His lectures were dense, and I usually left feeling a bit punch-drunk as my brain tried to wrap itself around the new keys to the universe it had discovered. One day later I was solving an equation with a bunch of fractions, and instinctively I got rid of the fractions. No one likes working with fractions.
It hit me at that moment. I didn't want the fractions, so I got rid of the fractions. I was able to get what I wanted immediately. This doesn't happen in life! Only in mathematics!!
Let me explain a bit more, but it involves actual math so feel free to skip down to more good stuff.
I pause here and yell at my class. We didn't like the fractions so we got rid of them. You have that kind of power! You have that freedom! If you don't like something in math, you can change it! You're given rules to follow and you follow the rules, but do whatever you want. Where else in life does this happen? Where else can you want something and get it right away? If you want to get stronger, you have to work hard. If you want to get richer or prettier, you have to be extremely lucky or you have to work hard. If you want to get taller, well you can't. But in math, when you want to get rid of the fractions, you get rid of the fractions. I love that about mathematics. I love many things about mathematics, but that's a big one.
My hope is this is a turning point for my students, where they go from what is their worst-nightmare-of-a-problem, to a realization that math is potentially wonderful. I try to keep sharing this point, and don't mind it when they laugh at my nerdiness. Just like how Shlomo inherited the love of a painting from someone else, Shlomo had to show me this appreciation for math, and then I inherited it. I hope if I can show this appreciation to my students, they can inherit it as well.
Well, I've thought more about this, of course - this relationship of rules and freedom in mathematics. As long as you follow the rules you can do what you want. My parents have often said that kids want boundaries. They want to have clear rules of what they can and cannot do. They want to have these rules enforced consistently and fairly. If they have freedom within those rules they will be more or less happy and content. This is just like mathematics. There are extremely clear rules that are very, very consistently enforced, but if you follow those rules you can do as you please.
So then I think about society. Ideally, society would be full of easy-to-understand, all-encompassing, non-loop-holey rules that are enforced but are fair. This doesn't happen, and that leads to all kinds of problems, but wouldn't it be great of society was more like mathematics in this sense?
(Heck, isn't this kind of like how God gives us rules to follow, but freedoms within those rules?)
You are thinking "But Grant, what about art? What about freedom of expression? What about diversity? What if some rules work for some people and not for others?" My reply? Shut up hippie. Seriously, though, that's all true. We won't ever have that idyllic society where a fair and just government makes and enforces laws that are for the equal protection and benefit of all. It won't happen. That sucks.
But I can have that idyllic world in my small, misunderstood world of mathematics. There's freedom in all of those restrictions and rules that students don't understand. I just need to get them to appreciate it.
No comments:
Post a Comment