Last Friday I spent far too long looking through my tutee’s notes searching for the “correct” way to name the property that allows one to subtract the same quantity from both sides of an equation. The ridiculousnes of spending n>0 seconds on this pursuit led to my infamous tweet of the other night:
Is it bad that I confessed to my tutee "this is why I hate properties" when talking about proving that -1*x=-x?—
Breedeen Murray (@btwnthenumbers) November 14, 2011
This is a slightly inaccurate depiction of what went down. In perfect honesty, I didn’t say that I hated properties; I said that I hated “these types of problems” i.e. algebraic properties style “proofs.” But that didn’t sound as snappy, so I changed it for the interwebs.
The thing is I love proofs. I was the totally nerdy freshman in high school who got jazzed about doing two-column geometry proofs. In hindsight, two-column proofs are the red-headed stepchild of real proofs, but, cut me some slack. I didn’t know any better at the time and it was my first real exposure to the idea of proving something rigorously.
You see, I am someone who loves being right. I can be the most bull-headed, stubborn pain in the you-know-what that you’ve ever met. In fact, I so dislike being wrong that I will do just about anything to avoid having to make a guess about something. Don’t believe me? Just ask Avery. He’ll vouch for me.
Proofs are like manna from heaven for someone who loves to be right ALL THE TIME. They let you give the intellectual “up yours” to any doubters that might be lurking around every single time you write QED, or draw that little black square at the end of the paragraph.
So I love — and I do mean LOVE — proofs.
But I HATE the way we teach proof to students. It makes me die a little on the inside when I see kids having to prove that -1x=-x. Or that multiplication distributes across subtraction, just like it does across addition. Gasp!
The problem with these “proofs” (and I use that word with reluctance) is what I am going to refer to as the Duh Factor. No one doubts that either of these ideas is true. Not even for a second. Everyone knows they are true because they’ve been using these ideas for years by this point in their mathematical careers. You don’t get that smug satisfaction of flipping someone the mathematical bird by proving that -1x=-x. It’s an obvious, accepted fact.
No one cares.
Proofs are the persuasive writing of math. Writing a letter to someone to try to convince them of all the reasons why two-year-olds shouldn’t be taking the LSAT’s isn’t going to impress your English teacher. Why then are we making students do the intellectual equivalent in math?
If we want to amaze students with the inherent beauty, the power, the sheer coolness of math, we need to stop giving them problems with the Duh Factor and start giving them concepts to grapple with that truly stretch their thinking in new ways. That blow their friggin’ minds. Because what makes proofs truly powerful is their ability to prove that something which seems at first unbelievable is actually, unequivocally true.