Today in Geometry Class I started it off with a quick round of “do you want to see what we did in Calculus today?” Before I erase the board I try to point out to them any instance where we are using algebra or geometric formulas to solve something. Today we used the area of triangle formulas to find integrals of linear functions.
After that we looked at a pattern in blocks that I learned at UNLV in a math methods course that I’d used in Calculus. The pattern starts with one block, then 1 +3 in a pyramid shape. You keep adding rows so you get 1 + 3 + 5, then1 + 3 + 5 +7. If you actually draw this out you’ll see that you can always rearrange the blocks into a perfect square. To find the sum of the first n odd numbers you just square n! One of my students who tends to struggle with algebra was the first one to figure this out. I’m always surprised what happens when I put non-traditional problems up on the board. She had a smug and satisfied look on her face when she solved the Calculus problem.
(Nerdy side note:The reason we were doing it is Calculus is that you can relate this series to left Riemann sums to approximate the area under the curve 2x+1. Then we see that the area has to do with x squared and the anti-derivative.)
One kid exclaimed, “we are doing real math now” because he liked the problem so much. I said, well, I try not to teach fake math too often. Then of course they wanted to know what fake math was and I told them fake math was “memorizing formulas you don’t understand because your teacher told you to.” Someone said “every single test I ever take in history is fake history” and I was pleased that they saw the connection between memorizing random historical facts and memorizing random formulas. Of course, some students thrive on memorization because it is lower level thinking that they know they can do but I’m trying to show them just how much fun math can be.
Next a student asked “do you ever have Calculus students do Geometry problems?” I thought for a moment and pulled out “The World’s Hardest Easy Geometry Problem” The kids were able to understand the problem immediately. It was a triangle with some angles listed and an x in one part of the triangle. It seems like it is going to be possible because they’ve been able to solve every other problem like this. This problem is incredibly hard however to the point that unless I sat down with it for at least 2 or 3 hours I don’t think I’d even approach a solution. I know one student solved it last semester but it took him and his family all weekend. The solution requires drawing a bunch of additional lines and use of SAS congruence postulate. The students were hooked. They barely wanted to correct answers on their homework. I decided not to even try to do what I’d planned for the day (easier geometric proofs) because I figured they were excited about math and practicing the same type of reasoning that was on the homework anyways. One student who has forgotten most of Geometry A and never seemed interested in class being hard asked “Can we do problems like this for homework?” Another said “I’d rather solve one really hard problem than a bunch of easy ones.” Problems are not the solution to all of problems in math education but kids respond to them. Even if they are as pointless as finding x in a triangle that has no real world connections. It seems that the quality of the thinking required to solve the problem matters just as much as the potential utility of whatever is being practiced. I’m excited to get into the book Thinking Mathematically at the conclusion of our unit on geometric proof because it has lots more problems like this.
At the end of the class one student asked “what types of proofs will be on the test?” and I literally had to say, flip over the triangle problem because I want you to listen to the answer to this question. I’ve never had to ask kids to flip over a math problem before! Then a few kids were late to the next class because they said “we are so close…” and agreed that as a group they would continue to work on the problem in the next class.
So I ask myself: Are they learning just as much as they might from textbook geometric proofs? Are they learning more? How do they transfer these experiences? How does it affect their dispositions towards math? What happens when only one of them can solve it because it really is the world’s hardest easy geometry problem? Do I warn them that it will be really hard to solve so they don’t feel frustrated or do I encourage them to keep them working on it? While they explain their reasoning to me they are able to use reasons and theorems and creativity even if they were not able to make these leaps in vastly easier proofs. Why does use of a hard problem encourage the ability to think logically? What is the value in having them write down their answers to this?
All I know is that I need to not be lazy about finding problems like this! They make class interesting. I hope that kids are looking forward to math in a way they haven’t before.