Proof is one of the most important concepts in mathematics. Proof separates math, where statements must be rigorously proven using deductive reasoning, from science, where statements of accepted truth are induced from observation of repeated trials. This is part of what makes Gauss claim that mathematics is the “queen of the sciences” and why mathematicians sometimes take a snooty attitude toward scientists, whose truth could be overturned at any moment by counter-evidence.
So how do we (and how should we) teach proofs? At the high school level, proof is usually only taught in Geometry, then ignored both before and after in the Algebra-Calculus sequence. And I’ve railed before at how proof is taught in geometry as a dulled-down, poorly-motivated, memorize-these-common-reasons-to-go-in-the-second-column-that-prove-things-which-are-already-obvious-anyway method, whose seeming pointlessness turns many students off to math even more than they had been before.
Irrational Cube ponders this question in a post on “Proof (1st day solo teaching)”:
Here lies the problem. Proofs in their final form are a series of steps from givens to a conclusion. However, that is not how people go about proving something. A proof requires thought of how to get from where you are to where you want to go. You feel around in the dark, think you’re getting somewhere but hit a dead end, give up, try and prove the opposite, give up again, start over, discuss it with friends, play with different ideas, sleep on it, draw a diagram, turn your diagram into a picture of a dragon eating the proof, give up again, wait a week, have a brilliant idea and then write down your proof. When we ignore all the intermediary steps then the question of “why are we substituting that” and “how do you know we’re supposed to do that” become unanswered. We are doing that because everything else that we didn’t try wouldn’t have worked. Of course, you can’t do all those things in a one hour period, so you skip to the end. In doing so, you deny the essence of proofs and equate them to unquestioningly following directions. No wonder students dislike proofs – they get the wrong idea about them.
What am I missing here. There has to be another way.
In my geometry class, I will be walking students through a proof that there are only five regular polyhedra (the Platonic solids) next week. I don’t expect them to come up with it on their own, but plan to do it as a whole-class activity, asking questions and having the students’ answers lead the proof along. I’m hoping to make sure the students want to know the answer before we delve in (there are infinitely many regular polygons, so many that we stopped naming them after a while and just call them things like 23-gon, so how/why should there only be five regular polyhedra?!?). I’m also excited because the proof connects back to our work with regular tilings of the plane, which in some sense are simply degenerate regular polyhedra, where the vertex angular sums didn’t allow the faces to fold up into a 3D shape.
On the whole, I try to encourage my students to notice patterns and make conjectures based on those patterns, though I need to do a better job of following through by allowing students to test those conjectures until they find counterexamples or have worked their way to a convincing argument for why their conjecture is true.
To get at a true answer to Irrational Cube’s question, though, we need not just to better motivate proofs and give students a few authentic experiences to prove things, but to overhaul our math education system. We need these experiences — noticing patterns, making conjectures, struggling with them mostly unaided by the teacher, researching appropriate related material, crafting a series of flawed-but-each-one-better-than-the-last arguments for why the conjecture could be true, creating a good logical argument and then tweaking it to be airtight — to be the essence of mathematics learning from a young age. There should be less emphasis on proof in geometry (so that there is room for other great geometric topics) but more emphasis on proof in algebra, so that proof is not just an isolated island in a sea of algebraic solving-for-x. A great take on how we might make these radical changes is given by Paul Lockhart in his A Mathematician’s Lament, worth the buy in book form, but a shorter essay of which is also available for free here.
But, since in the short term anyway, an overhaul of math ed in the USA is not realistic, what steps can we take (as math teachers) to help students understand and value the role of mathematical proof?