Dan Meyer’s most recent post is about how in order to motivate proof you need doubt.
This is something I was repeatedly and inchoately hollering about five years ago.
As usual I’m grateful for Dan’s cultivated ability to land the point cleanly and actionably. Looking at my writing from 5 years ago – it’s some of my best stuff! totally follow those links! – but it’s long and heady, and not easy to extract the action plan. So, thanks Dan, for giving this point (which I really care about) wings.
I have one thing to add to Dan’s post! Nothing I haven’t said before but let’s see if I can make it pithy so it can fly too.
Dan writes that an approach to proof that cultivates doubt has several advantages:
- It motivates proof
- It lowers the threshold for participation in the proof act
- It allows students to familiarize themselves with the vocabulary of proof and the act of proving
- It makes proving easier
I think it makes proving not only easier but way, way easier, and I have something to say about how.
Legitimate uncertainty and the internal compass for rigor
Anybody who has ever tried to teach proof knows that the work of novice provers on problems of the form “prove X” is often spectacularly, shockingly illogical. The intermediate steps don’t follow from the givens, don’t imply the desired conclusion, and don’t relate to each other.
I believe this happens for an extremely simple reason. And it’s not that the kids are dumb.
It happens because the students’ work is unrelated to their own sense of the truth! You told them to prove X given Y. To them, X and Y look about equally true. Especially since the problem setup literally informed them that both are true. Everything else in sight looks about equally true too.
There is no gradient of confidence anywhere. Thus they have no purchase on the geography of the truth. They are in a flat, featureless wilderness where all the directions look the same, and they have no compass. So they wander in haphazard zigzags! What the eff else can they do??
The situation is utterly different if there is any legitimate uncertainty in the room. Legitimate uncertainty is an amazing, magical, powerful force in a math classroom. When you don’t know and really want to know, directions of inquiry automatically get magnetized for you along gradients of confidence. You naturally take stock of what you know and use it to probe what you don’t know.
I call this the internal compass for rigor.
Everybody’s got one. The thing that distinguishes experienced provers is that we have spent a lot of time sensitizing ours and using it to guide us around the landscape of the truth, to the point where we can even feel it giving us a validity readout on logical arguments relating to things we already believe more or less completely. (This is why “prove X” is a productive type of exercise for a strong college math major or a graduate student, and why mathematicians agree that the twin prime conjecture hasn’t been proven yet even though everybody believes it.)
But novice provers don’t know how to feel that subtle tug yet. If you say “prove X” you are settling the truth question for them, and thereby severing their access to their internal compass for rigor.
Fortunately, the internal compass is capable of a much more powerful pull, and that’s when it’s actually giving you a readout on what to believe. Everybody can and does feel this pull. As soon as there’s something you don’t know and want to know, you feel it.
This means that often it’s enough merely to generate some legitimate mathematical uncertainty in the students, and some curiosity about it, and then just watch and wait. With maybe a couple judicious and well-thought-out hints at the ready if needed. But if the students resolve this legitimate uncertainty for themselves, well, then, they have probably more or less proven something. All you have to do is interview them about why they believe what they’ve concluded and you will hear something that sounds very much like a proof.
5 thoughts on “Uhm sayin”
I was thinking about the pointlessness of proving that vertical angles are equal, and then it occurred to me that this would be a real test of the euclidean axioms – do they really do the job of defining geometry.. So a proof is a test of the axioms.
Here’s another thing.
1. prove, or see a proof of, the irrationality of root(2)
2. What about root(3) ?
3. What about root(2) + root(3) ? This is decidedly not obvious !
It surely does not have to be only geometry.
Yeah, proof-as-test-of-axioms (or definitions) is a really beautiful, generative task. The thing to keep in mind is that it requires students really to already understand proof. So it’s not a good introduction to proof. But once they can prove, it becomes interesting again in exactly the way you describe.
there was a guy in my 9th-grade geometry class… true story…
that illustrated your “no gradient of confidence” situation perfectly.
it was his first exposure to “two-column proofs”
(presumably of the why-prove-anything-so-obvious
variety, but i don’t remember this much detail
and i promised a true story).
and (i paraphrase wildly) he wanted to know
—in the first line, we copy out the “Given”
part in the first column (as a statement)
and write “given” in the second (as a “reason”).
—so why can’t we just copy the “Prove” part
into the first column and write “prove” as
and he wasn’t kidding. he wanted to pass and was willing
to put up with being confused while he found out how.
i worked through quite a bit of the course with him that
semester and he *did* pass. this’ll have been my first
serious bite of the math-teacher bug. i don’t know if
he ever did get the *philosophy*… but he could crank
out some decent two-column proofs if they were
enough like the ones we practiced. thanks, “dave”!
(another great post, ben.)
That is a gorgeous illustration!