I realize that I’m poking at what has certainly become a central tenet of your work and thinking when I say this, but I take issue with the statement that a student needs to experience a wait-what moment to motivate proof. Of course that’s a motivation, but I can’t buy its necessity for a few reasons.

(1) Lots of proofs aren’t going to do anything in the neighborhood of satisfying that perplexity, and it’s tough to claim those proofs aren’t motivated. Non-constructive proofs are (by and large) also non-enlightening. I will certainly believe anything you prove to me by induction or contradiction, but I will also probably be about as perplexed as I was before our conversation.

(2) The moment of perplexity and the approach of provability might be so temporally distant as to be essentially unrelated. I learned very early on that the sum of two odd numbers is even. I’m sure it gave me a moment of wait-what. But by the time I could think about proving that fact, its perplexity had long since sublimated. In other words, the perplexity might very well precede the proof, but not in any meaningful way. But, when I become mature enough to think in terms of proof, I can certainly be motivated to shore up my “given” knowledge to my own satisfaction.

(3) If you look at my second point writ large, sometimes proof is motivated by the somebody taking a deep breath and saying, “Let’s measure twice and cut once, here. Can I be confident that these true things are true? They are? Awesome. Moving on…” They say it in papers and stuff, but that’s what they say. And that type of motivation is more external; it’s more about persuading other people that what I take to be true is a good thing to take to be true. It may never have perplexed me at all. I’m sure Gauss felt that way a lot.

Just some thoughts. Wait-what is a powerful thing, but I’m not sure it’s necessary.

Thanks for the plug.

That post seems to have resonated with a number of folks; I’m glad it spoke to you too.

I’ll be doing the hierarchy of hexagons in my NCTM New Orleans session in the spring. It will be a fun challenge to get a group of teachers to find and prove some new geometric truths. If you’re tired of those talks with prepared PowerPoints and talking points and want to watch a presenter squirm, come join us!

I really like how you make this proof “real.” Here’s my question, though: How would a English language learner access this problem. Can you use any of the new resources on the Understanding Language Web site to get at the language demands of these problems? http://ell.stanford.edu/teaching_resources/math. The “Language of Math” Task Templates are interesting — but I would love to hear your analysis on it.
Thanks.
I enjoy getting your blog in my inbox!

I guess my “wait-what” moment is “We are teaching geometric proofs?” Do we really think this is a transferable skill? It certainly isn’t a skill required by our adult lives (except for the minute percentage of people involved in academic math). I think the approach of starting from observations, asking questions, and gathering data to find answers is a great approach to many subjects and I applaud it (since it is essentially teaching the scientific method). However, at the end of the day is our real educational goal that 100% of our high school students need to learn the content knowledge of geometry. If it isn’t, why not teach problem solving divorced from academic math (ie would we accept geometry being replaced by a debate/philosophy class). It just seems like creating a more difficult problem for educators if we have to ask our students to learn how to problem solve, but we are stuck in a domain which is difficult for students to connect to and will do little to benefit them.

It’s obviously true! You just have to say why. Tell me anything more lifeless than that.

This is a really good post, and really thought-provoking. I want to raise two points here, both relating to the lifelessness of explaining stuff that you already know to be true:

1. One of my assumptions as a teacher is that if a lot of humans are choosing to to do something, it’s probably interesting. A lot of mathematicians choose to spend their time explaining stuff that they know to be true. So its probably interesting.

Whether we can help students see what’s cool about explaining known stuff is something worth talking about.

2. Let’s look at that “wait-what” structure very, very carefully. You do some experiment. You notice: the sums are all the same.

OK, pause.

At that very moment, do the kids believe that the sums are all the same? I’d say that they do.

Unpause, and now you’re asking “Why that should be true?”

There’s certainly a difference between this routine and the Given/Prove shtick, but it’s not the difference between explaining something known versus explaining something unknown. That moment that you’re giving your students is very much in line with “It’s obviously true. You just have to say why.”

I was just reading through the Park Math curriculum today and they do a lovely job setting up some of the ‘What do you notice? What do you wonder?’ type of reasoning. They present some radically different looking triangles and ask if you think that a circle can circumscribe them. The student is then asked to draw a variety of triangles to test out the theory at hand. I recognize that the text pushes the idea of circumscribing, but it feels like an interesting question. I hope that a willing group of students can be next convinced that it might be interesting to confirm what they think is true. Then, I’d whip out the GeoGebra and try to seal the deal.

Sense-making, arguing, and conjecturing are essential mathematical activities. However, it’s important to note that what’s distinctive about proof–as the term is used in the context of high school geometry courses–is not about argument or explanation. What makes “proof” different here is that the kinds of reasons we’re allowed to give is circumscribed in a formal way.

Now, the axiomatics of Euclidean plane geometry are thorny. It’s not a pleasant system to work in as a first-time proof writer. There are many axioms of many different kinds, and the statements that are taken as axioms and the statements that are derived as first propositions are easily interchangeable. Not to mention that this all comes at students fast and furious. No wonder that it’s hard to perceive the subtle structure that lies before them.

And then we have the audacity to ask them “why” things are true and won’t accept any of their perfectly reasonable but nonformal attempts at doing so–if they can muster anything to say at all. We mean for them to play a certain kind of unusual game, but then our verbal prompts make us appear to be asking for something perfectly natural and common sense: “why?”

Here is a direction I’ve taken to making the notion of a proof–embedded within a proof-system–graspable, with some success: http://ichoosemath.com/2011/11/22/introducing-proof-using-formal-systems/

In short: Proof here is not about finding truth or convincing. It’s about a certain kind of game we’re playing.

And a P.S.: I think it would be interesting and revealing to hear many math teachers’ responses to the question, “Why is the Pythagorean theorem true?”

@Colin Matheson
“Do we really think this is a transferable skill? It certainly isn’t a skill required by our adult lives”

I LOVED proofs in high school. The way I saw it, the task wasn’t about figuring out geometry. We already knew the geometry from experience, and most of it just makes visual sense. I loved the challenge of being given a set of rules and not taking them for granted, but actually picking out the ones that apply in the given situation, and connecting previously unconnected rules to each other. The transferable skill is in connecting discreet areas through logical arguments that link one accepted fact to another. For scienists, lawyers, anyone who wants to win an argument, geometric proofs teach something beyond geometric facts. Maybe this underlying outcome needs to be made more explicitly by teachers in order for students to look beyond the simple angle congruences that appear on the surface of the task.

Maybe that is your point, that we could teach this logical argument skill in some other way with less confusing content, but the superficial simplicity of geometric proofs is such a good vehicle for it. Lots of complex thinking goes into just 3 written lines of a logical, research-based student answer.

Thanks for replying, Dan. And I think that your response to my first point is dead on.

Your response to my second point makes me think that I didn’t communicate it clearly. I’m not saying that what counts as “obvious” is subjective. That’s obviously true. I was trying to argue that even in any of the proofs in your post the kids are still “given” a statement to prove.

This is the part where you’re probably thinking I’m nuts.

But let’s say you do these sorts of little experiments in class all the time. One day you ask kids to draw a rectangle and its diagonals, they do, and then you ask them what they notice. Sure enough, they notice that the diagonals seems to be the same length.

At this moment, I’d argue that the kids believe that the diagonals are the same length. They think it’s true. They’ve done these experiments before, and every time you’ve done them it results in some true thing that you have them prove.

Then you ask them “Why?”

You’ve just given the students a statement to prove. I’d argue that if kids find proving this more engaging than a text proof, then it’s not because the statement isn’t given. It is.

I cheated a bit by asking you to imagine a classroom where these little experiments have become a sort of learning cliche, where the kids know the drill. But I’d also argue — much more tentatively — that things aren’t much different even as a one-shot lesson. The kids still believe the proposition after the little experiment, and when you ask “Why?” you’re still giving them a proposition to prove. The proof still is an attempt to explain something that is — now — obviously true.

Despite this, it’s obvious to me that the proof is more interesting than if you hadn’t done the little experiment. But why?

@Michael Pershan: Regarding your concluding “why?”, I think it’s because an isolated empirical conclusion is hard to generalize, but a line of reasoning can be extended in interesting ways.

I don’t have a good Geometry example, but my favorite example is from physics. Since I know you work with periodic functions (and I’ll be stealing some of your lessons on that this year), this may interest you. A mass oscillating on a spring yields simple harmonic (i.e., sinusoidal) motion, and this can be derived from the fact that the spring force is directly proportional to the spring’s displacement from equilibrium. That’s cool, but not nearly as cool as how it can be extended.

Think about any object that’s in a stable equilibrium, which means when you displace it, it feels a force back toward its starting position. It could be a pendulum, a marble rolling back and forth in the bottom of a bowl, etc. When the displacement x is 0, the force F is 0, because it’s at equilibrium. When the displacement is x, the force is F(x), but we know from calculus that in the region around x = 0, F(x) is approximately linear. So it’s approximately proportional. So for small perturbations, ANY oscillating object is guided by the exact same equations as a mass on a spring. That one principle explodes to cover a massive range of phenomena.

You’d never get that if you only understood the mass on a spring empirically without deriving the connection to F being proportional to x. And once you’ve done the proof, you can see all sorts of surprising connections.

Just to add, I think these are the most important and meaty sentences in the post: “Questions that require proof are hard to create, hard to package in a textbook, and probably impossible to crowdsource. You’re trying to nail that point where the seemingly-true hasn’t yet turned into the obviously-true and that spot varies by the class and the student.”

I’m interested to hear more. Dan, you’ve been very successful in creating, packaging (in non-textbook fashion), and crowdsourcing other kinds of mathematical tasks. It makes me want to think about whether and how tasks involving proof are different.

A first attempt: proofwriting requires a certain kind of established culture. It’s much harder to make happen with a single task.

But really, I’m not sure.

Quoting Colin:
“I guess my “wait-what” moment is “We are teaching geometric proofs?” Do we really think this is a transferable skill? It certainly isn’t a skill required by our adult lives (except for the minute percentage of people involved in academic math)”

This is another piece that has been bothering me in teaching Geometry for the first time – what skills are transferable? Here’s what I come up with:

Creating rigorous proofs is transferable because you’re taking your conclusions, finding facts to back them up, and laying those out in a way that someone else will find hard to ignore. I think of a proof as saying “look, it has to be this way, the argument is final.” But I can’t figure out how to show students this, or how to build students up to this level, and I don’t know what is done in other (non-math) classes that I can build from.

For what it’s worth, I think Lockhart is seriously weighing in now with his recent book ‘Measurement’ (I’ve only read maybe 20% of it so far though).

I got the impression from ‘The Lament’ that his approach to “nailing that point where the seemingly-true hasn’t yet turned into the obviously-true” involved having a huge magazine of interesting problems to pose to students (or individuals) as needed. In ‘Measurement,’ Lockhart seems to be opening up his bag of tricks and he’s quickly becoming one of my biggest idols.

I’d also like that add that an appreciation for history seems to be a recurring theme here with Harel (via Dan):

“intellectual perturbations that are similar to those that resulted in the discovery of new knowledge”

With Serra (via Dan):

“The Great Pyramid was built in 3900 B.C. by rules based on practical experience: Euclid’s system did not appear until 3,600 years later. It is quite unfair to expect children to start studying geometry in the form that Euclid gave it. One cannot leap 3,600 years of human effort so lightly!”

with Dan:

“a perturbing moment experienced by so many athematicians before them”

And with Lockhart (sorry, no quotes… But his work I’ve seen is strewn with historical annecdote and reference).

I’ve been kicking myself for not having a stronger foundation in the history of math ever since I heard about the Bridges of Konningsburg this summer. I attended a session on math history just yesturday and learned this gem: the symbol e (maybe the value also, my notes were kinda furious) was first used by Euler in a paper called ‘Meditation on an experiment made recently on the firring of cannon.”

…If you involve cannon balls, is it even possible to setup a boring Act 1? (and now we need another two-column proof!)

@DavidTaub, you may find these posts by Ben Blum-Smith to be useful.

http://researchinpractice.wordpress.com/2010/05/07/pattern-breaking/

and

http://researchinpractice.wordpress.com/2010/07/12/pattern-breaking-ii/

One thing that occurs to me here is that we need to do a better job emphasizing figures that *don’t* satisfy the conditions of a theorem. That will give more “pop” to those that do.

For instance: “The diagonals of a rectangle are congruent.” If we show students some non-rectangles, and have them notice that the diagonals aren’t congruent, then it makes sense to wonder, “Okay — what is it about having *right angles* that makes the *diagonals* congruent? That is weird and sort of unexpected.”

Launch!

> showed them Euler’s clever proof that no solution was possible, they were genuinely interested in the proof.

This is important. In all other teaching, we present students with examples of the best: the best literature, the best art, the best physics experiments. We ought to do the same in math as well. Giving kids models of the sort of reasoning that actually goes into deductive argument can really engage them (and form them intellectually).

> There are as many fractions as whole numbers. Our intuition screams that this is crazy. Of course there are more fractions! Only a deductive proof can convince people that infinity is stranger than we give it credit.

There’s a lovely paper by Calkin and Wilf about this: http://www.math.upenn.edu/~wilf/website/recounting.pdf

I’ve often used it with enterprising calculus classes, as it’s short, surprising and, written in a readable style. It includes a couple of different patterns of reasoning to support one conclusion.

We generally spend five or six hours reading it together in class (as one might do with a tricky text in a literature class). There are vocabulary quizzes interspersed, and finally the students work in groups to construct presentations to make the argument accessible to seventh graders, which they then give.

I just read the Villiers paper that Rose linked to, and it’s really great. Thanks, Rose!