I’m teaching Geometric Proof. And it’s hard. Some kids understand what I’m talking about when I mention making a list of reasons that prove something. They totally understand when I say, don’t use what you are trying to prove as one of your reasons. Don’t jump to conclusions about something that would be true if the conclusion was true even though you know the conclusion is true and that that fact will also be true. Today we were proving that if you draw an altitude in an isosceles triangle it divides it into two congruent triangles. If you can imagine that you’ll see that the altitude is the leg of two right triangles and that to use the Hypotenuse Leg Theorem you need to technically say that the two triangles have congruent legs because they overlap.
Technically, this is called the reflexive property. My students want to know why they need to include this step because it is so obvious. I know that “real mathematicians” wouldn’t always include this step because it is too obvious. Calculus books wouldn’t include that step if they were trying to write a proof of a theorem involving shapes. It’s obvious. It’s boring. I understand that it’s there but how in the heck do I explain why it’s important to the students. I know that details of proofs should only be skipped when you understand them but when they barely understand the structure of a proof, crap like the reflexive property is so much harder than actually thinking logically through the problem. In another proof that vertical angles are congruent first you notice that the angles form two linear pairs, then you notice that linear pairs are supplementary and then you notice that supplementary angles by definition are 180 degrees. A bit of substitution and subtraction property of equality later you’ve got the theorem. Vertical angles are congruent. YUCK! This is not proof. This is some sick bastardization of proof.
Must proofs be dry and boring and obvious to be easy enough for a high school student? Of course the answer is no, but must we continue doing these banal geometric proofs with the stupid reflexive property. Does it get us at increased clarity of thinking? Where does pickyness about reflexive properties get us. One of my students has a geometry tutor and returned to class with a precisely written proof, reflexive property and all. I was a little taken aback. How did you already know how to do this I asked? We haven’t even learned this formal structure yet. Her tutor had helped her was the obvious answer though I didn’t want to admit to her that I entirely doubted that she could produce such work in a weekend. She said “My tutor told me to ask you if we need all these details. She said some teachers require them and some don’t.” Later in class a student, kind of annoyed by the reflexive property asked me if I was going to require such detail on obvious facts. “It’s up to me. Real mathematicians don’t include details like this in proofs because they are obvious. They know they are there and could include them but don’t. The number of details that is included depends on the audience.”
She wasn’t satisfied with that. “So will you require them on tests in this class?” And once again I’m faced with this question of details. I know that how I respond to this general line of questioning is going to influence how people think about proof. If I say we must always include the reflexive property kids are going to kind of resent the redundancy. I don’t like writing things like that either. If I say that we don’t need it I do lose some mathematical rigor. So I tried to go back to first principals. What am i really trying to do. Logical thinking. Deductive reasoning. Writing clear arguments. I hedged the question “If you know it’s supposed to be there, please put it but I’m not going to attach huge points to it.” While checking proofs that day I told kids who’d put “same line” instead of reflexive property that their thinking was logical and good. I think I’m grading these proofs on logic more than conventional details but I’m struggling to assess what really matters when they ask. Frankly, I want to just get to thinking mathematically and ditch all these boring Geometric proofs as soon as I can. Boring is such the wrong word too. There are some beautiful and wonderful geometric proofs that seem way to hard for my kids at the moment but which might be great later. I have no problem with geometry as a field, but I don’t want to be marking down points for did you include “definition of supplementary” as the reason for “angles add to 180″ as opposed to skipping straight to the needed result. Though, honestly, if I look at my math career I was perfectly content to prove that two evens added are always even and that there are infinitely many primes in 10 different ways. The actual utility of the item proved does not always matter to me. I don’t always care if I’m already convinced something is true with inductive reasoning. There is something satisfying about understanding infinite decent in Fermat’s Last Theorem or why the harmonic series diverges and the sum of the square number is connected to pi.
After being a little on the spot confused about what I really wanted from the kids in terms of proof I said “Do you want to do a proof that isn’t obvious and boring?” I wonder if these are actually more efficient at helping them understand the point of proof because they are not struggling to identify what they know for sure and what they also know because they already know the answer and everything is so obvious but can’t technically use in the proof. Also. Two Columns. I read somewhere that these were invented as a scaffold for high schoolers who were not developmentally ready for deductive logic. How do I feel about them? In some ways I like the idea of a scaffold. There must be some way to help my kids who have a blank page staring at them get something accomplished with the proof. One of my students(one who also said she loves notes and structure) said to me today that she doesn’t understand where proofs come from. All of these ideas come out of no where and I don’t know which ones to pick. I started talking about music and how I have no idea how people write songs. How do they decide which note comes next? How can they pick from the millions of combinations. There is no formula for that. And I admit that I feel defeated when sitting with a guitar and asked to write a song. I can however make some random guesses, see how they sound and honestly as much as I might resist I probably could learn some general rules of song writing and be able to write a passable song. One that my mom might enjoy at least.
We returned to the triangle and altitude problem. She started with the given. I said “well, they use the word altitude, so even if I’m not sure it’s going to be helpful I look up the definition and write that down.” Then I notice that there are triangles and we are trying to prove they are congruent so I read through all the congruence triangle theorems until I find one that matches. I wonder if this is actually what I do and if my thinking applies. I know when faced with a ridiculous proof on a take home final I read through the book until I found something similar and started with that. It felt almost like cheating because I realized that I was throwing excessive time at the proof as opposed to excessive cleverness. What do I actually think about in Geometric proofs? I can see all the steps from beginning to end on these so reflecting on the minute snapshots of my thinking and laying them out for others to see is hard. And I’m not even sure if my thinking is the model. I have not done many proofs on the board for them because I don’t want them to think that proofs are something to copy or to memorize. I wonder if that is a mistake and if more direct instruction would help. I learned proof in college with lots of direct instruction. Tomorrow I think I might try the “even + even = even proof” or perhaps the harmonic series diverging. Of course divergence is a really hard topic because kids think it must diverge because you are adding up infinitely many terms. At least these proofs seem more interesting. My co-teacher was going to look for accessible proof for the kids but I might need to do some looking myself to struggle to find something interesting and possible.
One final note on proof before I wrap up this longish post. After we went over a proof in class I asked the kids “honestly, and I promise that I don’t care what you say, what is your opinion of these proofs?” I’d already expressed a bit of negativity about the reflexive property and was curious what they thought(though of course they’d been influenced.) One said “I can see how this will help us eventually in life.” Another pointed out that it seems like it would make more sense to prove things that we not obvious. I’m sure someone mentioned the picky little detail steps being annoying. And I know some are still wondering where all of this comes from. I now understand why some people say “some kids get proof and some don’t.” With equally small amounts of instruction half the class comes back with proofs and the other half comes back confused with white spaces on their paper. I’m already planning differentiation.