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:

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.

Q.E.D.

I’m pretty much on the same page with you here. As you know, I’d like to teach those algebra properties by choosing some other operations and investigating whether the properties actually hold or not, and then appreciating what the consequences might be of those results. That’s some substantial fraction of why I chose the problem I did for Friday’s session.

On the other hand, “Because what makes proofs truly powerful is their ability to prove that something which seems at first unbelievable is actually, unequivocally true.” … I’m not so sure I agree with that. I usually say we prove things for two main reasons: (1) to understand why something happens (as opposed to merely being convinced that it does) and (2) to show connections between different topics. The example I give is the proof using parallels that the angles of a triangle add to 180 degrees: it shows that there’s a connection between angle sums and parallels, and tells you something about what’s going to happen in noneuclidean geometry, that when you start messing with the parallel postulate you’re also going to be messing with angle sums.

To get back to agreeing with you, though, I like to start my Euclidean geometry classes with a bit of circle geometry, and the first thing I have the students prove is that an angle inscribed in a semicircle is a right angle. That’s a bit surprising and non-obvious, at least enough to be interesting, and it’s also useful in proving lots of other things.

Breedeen,

Stumbled across your little spot on the web. Found some great thoughts but I wanted to respond to this article. I feel like you are missing part of the power of proofs. You seem to see them independently of one another (as I interpreted from this entry). Like each one is its own isolated puzzle. If when going through Euclid’s Elements we ignored all the “Duh Factor” proofs, much of book 1 would be eliminated and we would be left picking and choosing constructions and proofs worthy of our attention. The beauty is in its entirety. Sure its very obvious that an equilateral triangle can be constructed or that alternate interior angles are congruent. But these proofs are the foundation for the “blow their friggin minds” proofs coming later in Euclid’s grand textbook. Book 13 loses its power when a teacher says “assume you have already done all the duh factor proofs, those propositions say this, that and this and now lets do a real humdinger!” If a teacher wallows in the “duh factor” proofs, I agree but if its part of the grand structure they are building, then more power to them.

Thanks,

Brett Edwards

Hi Brett, Thanks for reading and for commenting.

I appreciate what you’re saying about how the whole structure that mathematicians can build from a few axioms is one of the most powerful aspects of proof. I totally agree with you on this. I was arguing that the fact that we

students to proof via these “duh factor” proofs is a practice that I think is far less than ideal.introduceOnce students realize that proof is powerful enough to do amazing things, then they have some motivation to explore what they can prove from as few axioms as possible. I have a high level of skepticism that students find this process inherently compelling.

Agreed. Thanks for the response. I enjoyed viewing your prezi. Well done.