I was poking around one of my favorite textbooks for higher math, and I came across a problem that I thought was obvious.

I felt, in myself, a weird combination of reactions:

- "Of course this is true!"
- "I have no idea how to prove this."

What a funny pair of things to think!

I thought about the problem for a bit, and I realized how I might prove this pretty obvious fact. And

*that*felt satisfying also, like I had cracked a puzzle or made a connection.
I see a strong parallel between my reaction and what students of geometry proof often feel. They'll see a problem and it just seems

*impossible*to prove because it's so. damn. obvious.
As Dan puts it:

To motivate a proof, students need to experience that “Wait. What?!” moment of perplexity, the moment where the seemingly-true has revealed itself, a perturbing moment experienced by so many mathematicians before them.

But maybe that's not the whole story. There are at least two ways to mess with the "too obvious to prove" dilemma. The first is to wear away at our certainty in the to-be-proved, to introduce doubt and confusion. The second might be to ensure that students

*have*reasons to give.
Maybe there's room for proving obvious things, just as there might be room for working on easy problems. Our job, partially, is to make sure that students have reasons to give in their arguments.

In sum, I think there might be two big things that teachers of proof can do to help students enjoy and understand it:

- Create genuine moments of confusion that lead to natural, informal mathematical arguments and debates that become increasingly formal in the course of time.
- Make explicit for students the reasons (e.g. "vertical angles are congruent") and patterns of argument (e.g. area-equivalence arguments) that one can use in mathematical contexts, and help them make connections between these and their informal tendencies.

Hypothesis: One won't work without the other.

Well, when I read question 4.1 I was a bit puzzled. My brain said "What does it mean to say that two integers are congruent?" . Then I thought "They mean congruent mod m to each other". They should have written it, as otherwise the statement is rubbish. Do I get full marks for this?

ReplyDeleteThe other problem is more general. How far back are they expected to go ? Euclid 's definition of parallel, indirectly put, is that if two lines cross a third line and the sum of the adjacent angles (correct term ??) is not equal to two right angles then the first two lines will meet, so in the example they are using two things about the lines in the picture which are not axioms.

I quote you : "Our job, partially, is to make sure that students have reasons to give in their arguments."

ReplyDeleteThis should be done with all sorts of things, not just proofs. An account of an algebraic argument, say for solving a pair of simultaneous equations, should really have a connecting description stating or explaining the reasons for at least some of the steps.