Personal essays
Playing by the rules
Obvious
I recently posted that maybe we shouldn’t be teaching Geometry in school and I got a ton of pushback.
When I took and later taught High School Geometry I loved a lot of the ideas but the justification was always that it taught students proof and reasoning.
I didn’t think so.
I was told that it taught students to build up a foundation of little things and then build bigger things on top of those.
Mostly, I found it to be petty and trivial where the things the students were actually asked to do was to take small and obvious steps based on what they were just shown.
We did “Statement-Reason” proofs where we drew a vertical line and put the steps of our argument on the left side of the line and the reasons justifying each step on the right side of the line.
For the most part I thought the reasons were “Obvious”.
I went to a talk by the mathematician Peter Hilton where he told the story of his son having the same experience. His son was getting bad grades in high school geometry because he didn’t think obvious steps needed to be justified.
Hilton told his son to go visit the teacher in his office and knock on the door. If the teacher said “come in” then Hilton would take his son’s side, if the teacher said “open the door and come in” Hilton would take the teacher’s side.
It was Hilton’s proof of what the teacher really thought about obvious steps.
Just my imagination
I followed a link in one of my news feeds to an article on the BBC about Imaginary Numbers.
It began, “Just because imaginary numbers don’t exist, it doesn’t mean they are completely useless.”
What does that mean that imaginary numbers don’t exist?
What does it mean for a number to exist?
Of course I’d say that. I’m part of the conspiracy. My site is underwritten by Big Math.
In a sense, all of math is imaginary. Requiring that it match the world around us is not as important to me as that we agree on some basic rules and we see what follows.
Often what follows leads us to better understand the world around us - but I’m content if that understanding is no more concrete than the feeling we get about the world around us after listening to music we like.
“But Daniel,” you say, “doesn’t geometry help us understand the world around us?”
The real world
When I type a location into Apple Maps I’m given a distance.
For example, I type in CWRU (a local university) and Maps tells me it’s 2.1 miles away.
So I tap on it to get directions and my favorite route is 3.0 miles (the shortest is 2.9 miles).
The real world doesn’t care that the shortest distance between two points is a straight line. In fact, on the surface of a round earth, I’m not sure that most people know what a straight line is.
I reverse the order to find out the routes from CWRU to my house and my favorite 3.0 mile route is now 3.4 miles because some of the roads are one way.
So the distance from here to there is not the same as the distance from there to here.
Suddenly I’m Dr. Seuss.
Geometry
If Geometry is taught right, I’d love to see it remain.
To me, Geometry - all math - is about playing a game.
We agree on some rules and we see what the results are.
When the rules change, the game changes.
Check out the original rules of basketball. It doesn’t resemble today’s rules. There were no free throws after being fouled, dribbling was not part of the game, and the number of players on a team was not specified.
When I was growing up, the NBA had a shot clock but college basketball didn’t and so you’d see incredibly low scoring games where teams that were ahead would run down the clock by passing the ball around the perimeter. I worked with a guy whose instructions to his team was they were not to shoot until a minute had elapsed since they took possession.
When the NBA introduced the three point shot nearly fifty years ago it changed the game.
When you change the rules, you change the game.
To me - that’s the big idea of Geometry. It’s like a board game where first we sit down to understand the rules and then we see what we can build on these rules and finally we see what we can build on these things we’ve built on these rules.
Then - well then, we ask what happens if we change the rules.
My latest video series is a quick exploration of geometry - playing games and changing rules. What are some simple things that follow from a basic set of rules in geometry and what happens if we change one of them.
I’d love to see students leave geometry class with the feeling that it’s fun. I’d love students to get as involved in their fellow students presenting a proof as they do watching their team play basketball.
Essay from Dim Sum Thinking Newsletter 180. Read the rest of the Newsletter or subscribe