Proofs
Sane people go crazy on you. They say "No, man, that's not the deal we made. I got to go. I got to go."
Not a math teacher...I’d like to hear from math thinkers on this one.
As my math teachers continue to wade deeper into the swimming pool of Peter Liljedahl, I’ve been asked to check out classes that use vertical surfaces and the activities that spring from them. (I continue to be a big fan and am interested in how this might be applied in other disciplines…my English teacher self wants to do annotations on boards, and my Social Studies self is mulling over what vertical surfaces might look like there.)
In this context, I was recently invited to a Geometry classroom, where the teacher was teaching the standard two-column proof. The kids’ goal was pretty typical stuff: to show that triangle ABC was congruent to triangle DEF using the SAS, SSS, ASA, or AAS postulates. I watched period 2 as trios of kids passed a whiteboard marker around and talked about sides and angles and definitions. I was into what I saw, with the teacher and paraeducator flitting about from board to board, making sure the trios understood one proof before moving on to the next thin-sliced problem. The teacher said that it would be tougher for her to do alone later that day, in 5th period, when the paraeducator had another commitment. My schedule was clear for 5th period, so I agreed to come back and teach rather than coach.
I hadn’t done a proof since 1984 (the literal 1984, not the metaphorical one). I am vaguely, faintly aware of some level of controversy over whether proofs should continue to be taught (or to be taught centrally) in Geometry classes. Maybe it’s not as hot a debate as I thought, as my Googling is not revealing anything big from NCTM or anywhere else. But that was in my mind as I watched these teenagers wrestle with triangles and postulates and logic.
Reader, my experience 5th period elicited a surprisingly strong response out of me…
I fell in love with proofs.
I didn’t come in expecting to have my world rocked. But I found that as I went from board to board, from trio to trio, as I sorted out who was cruising and who was struggling, that I became smitten with the thinking kids were compelled to do.
I was brought back to a memory at a car rental counter nearly 40 years ago. Sometime in my teens, I was taking a trip somewhere with my dad…in retrospect, this was most likely my college tour during spring break of my junior year, but I can’t swear by that. When we got off the plane and headed to get the car, the worker at the counter really struggled with whatever dot-matrix technology was in front of them. They couldn’t get the information we needed and eventually sputtered to a halt—a frustration stasis. They were completely stuck. Dad was annoyed.
Before things got too out of hand, the manager came in from the bullpen like Trevor Hoffman. After looking at the screen for a little bit, the manager scrunched up their brow, clicked a few things, and worked out of whatever cul-de-sac the original worker had brought us all down. Moments later, we had our car keys and were on our way to look at universities.
“That,” my Dad said as we walked through the parking lot, “is why it’s so important to teach math in high schools.”
I think I had trace elements of understanding of Dad’s logic that day. But I got it fully when I watched kids doing proofs.
The first set of kids I watched doing proofs had a buckshot approach to the problem. They listed about a bazillion different sets of congruent angles and congruent sides and then gave the reason of SAS (or SSS, or AAS, I don’t remember which) as to why the triangles were congruent. My response was to (1) say that they had proven that the triangles were congruent, and (2) to teach them the word “superfluous.” With that definition in mind, they fixed the problem on future proofs. Rather than giving me every piece of information and saying “See?” with a big smile, they gave me only the relevant information and thereby showed that their conclusion was backed up by facts.
Somewhere in there, and in the other eyeballing of kid thinking I was doing, I found that what we were doing was teaching kids to answer the question “how do we know a thing is true?” Yes, we are talking about congruent triangles, and because of that, kids will give their eye rolls and their “when are we ever going to have to use this” protests. But this pure, raw thought–this simple logic puzzle–I can sell as getting their brains stronger as squats get their bodies stronger. By eliminating all interpretation (no arguments about those pesky symbols in the poem or uncertain cause/effect relationships in social studies), we can get right to the heart of things. We know that triangle ABC is congruent to triangle DEF because angle A is congruent to angle E, AB is congruent to DE, and CA is congruent to FD. Full stop.
When a kid’s brain that can do that, and can work its way up, personal trainer style, to tougher and tougher tasks…that’s a good feeling, and one that I can sell is useful to the kids. And, at least that afternoon, I felt like proofs were uniquely suited to that sort of thinking. That’s the kind of thinking my dad was referring to in the rental car parking lot.
Since then, I have asked two people I respect a ton (one fantastic math teacher and one assistant principal who used to teach math) for their views on math, and both told me they can’t stand proofs. The teacher said she doesn’t like how the writing gets in the way of kids showing their understanding. The latter wrote this: “I hate them. The whole point of doing them is to practice logic and reasoning. And I think there are so many other ways we could do that.”
The writing issue is familiar and important. When I taught English, my colleagues and I had to be thoughtful that a kid’s struggles with writing didn’t prevent us from seeing their insightful reading. Similarly, I bet it’s plausible for a math teacher to see through the static of a kid’s struggle to write to see if they get the logical progressions at the heart of geometrical proofs.
As for the “so many other ways to do that…” I haven’t heard one yet. What are the other ways we can teach logic and reasoning, and why are they better than proofs? Because what I saw that day made my logical heart sing.
I come from a place of curiosity and not proselytization. Math teachers, math lovers, and math haters alike: talk to me. I’m listening. Why isn’t the proof the absolute best and purest way to teach thinking? Or am I right that it’s the gold standard of logical thinking?