Just catching up on some blog reading and came (via Sue) across Steven Strogatz writing about training to use inquiry-based learning in his class for the first time and feeling embarrassed when he couldn’t solve something as fast as his colleagues! This kind of narrative is so valuable. Our students need to know it’s not just them!
T was a third grader when all this went down.
At a previous session, I had asked T what she knew about multiplication, and she had told me, among other things, that is four sixes, and because that’s 24, she also knew that six fours would be 24. I asked why she said so and she didn’t know why. I asked her if she thought this would always be true for bigger numbers, or could it be possible that there were some big numbers like 30,001 and 5,775 for which 30,001 5775’s was different than 5,775 30,001’s. She wasn’t sure. I asked her if she thought it was a good question and she said she thought it was.
So this session I reminded her of this conversation. I forget the details of how we got going on it; I remember inviting her to wonder about the question and note that there is something surprising about the equality between four sixes and six fours. She could count up to 24 by 4’s and by 6’s and mostly you hit different numbers on the way up, so why do the answers match? And would it be true for any pair of numbers? But the place where I really remember the conversation is when we started to get into the nuts and bolts:
Me: Maybe to help study it we should try to visualize it. Can you draw a picture of four sixes?
T draws this –
Me: Cool! Okay I have a very interesting question for you. You know how many dots are here –
T: 24 –
Me: and you also told me that six 4’s is also 24, right?
Me: – so that means that there must be six 4’s in this picture! Can you find them?
T: I don’t understand.
Me [writing it down as well as saying it this time]: You drew a picture of four 6’s here, yes?
Me: And that’s 24 dots, yes?
Me: And you told me before that 24 is also six 4’s, yes?
Me: So it must be that right here in this picture there are six 4’s!
[It clicks.] T: Yes!
Me: See if you can find them!
At this point, I go wash my hands. An essential tool that has developed in my tutoring practice is to give the student the social space to feel not-watched while they work on something requiring a little creativity or mental looseness, or just anything where the student needs to relax and sink into the problem or question. The feeling of being watched, even by a benevolent helper adult, is inhibitive for generating thoughts. Trips to wash hands or to the bathroom are a great excuse, and I can come back and watch for a minute before I make a decision about whether to alert the student to my return. I also often just look out the window and pretend to be lost in thought. Anyway, on this particular occasion, when I come back, T has drawn this:
T: I found them, but it’s not… It doesn’t…
I am interrupting because I have to make sure you notice how rad she’s being. The child has a sense of mathematical aesthetics! The partition into six 4’s is uglifying a pretty picture; breaking up the symmetry it had before. It’s a kind of a truth, but she isn’t satisfied with it. She senses that there is a more elegant and more revealing truth out there.
This sense of taste is the device that allows the lesson to move forward without me doing the work for her. Her displeasure with this picture is like a wall we can pivot off of to get somewhere awesome. Watch:
Me: I totally know what you mean. It’s there but it doesn’t feel quite natural. The picture doesn’t really want to show the six 4’s.
Me: You know what though. You had a lot of choice in how you drew the four 6’s at the beginning. You chose to do it this way, with the two rows of three plus two rows of three and like that. Maybe you could make some other choice of how to draw four 6’s that would also show the six 4’s more clearly? What do you think? You wanna try to find something like that?
T: Yes! [She is totally in.]
At this point I go to the bathroom. I hang out in the hall for a bit when I get back because she seems to still be drawing. Finally,
Me: Did you find out anything?
T: I drew it a lot of different ways, but none of them show me the six 4’s…
She’s got six or seven pictures. One of them is this –
Me: Hey wait I think I can see it in this one! (T: Really??) But I can’t tell because I think you might be missing one but I’m not sure because I can’t see if they are all the same.
T immediately starts redrawing the picture, putting one x in each column, carefully lined up horizontally, and then a second x in each column. As she starts to put a third x in the first column, like this,
she gasps. Then she slides her eyes sideways to me, and with a mischievous smile, adds this to her previous picture:
The pieces just fell into place from there. Again I don’t remember the details, but I do remember I asked her what would happen with much bigger numbers – might 30,001 5,775’s and 5,775 30,001’s come out different? And she was able to say no, and why not. Commutativity of multiplication QED, snitches!
An obvious observation –
The last 6 or 7 years in public education policy seem to have been characterized by the following trends:
1) Tying teacher evaluation, hiring, firing, and teacher pay to student standardized test results.
2) Relatedly, using value-added measurements in making these decisions.
3) School closings and state takeovers.
4) Using VAM in making decisions about those too.
I.e. Stressing all the adults who work in schools the f*ck out.
5) Subcontracting to charter networks.
6) Direct funding cuts.
I.e. divesting from education as a public trust.
All in all, these trends, spearheaded by the US Dept. of Ed. under the leadership of Arne Duncan, but with numerous assists from other folks, representing both public and private interests (being in NYC, I’m lookin at you Mike Bloomberg), seem to me to have an obvious common theme:
Making public schools shittier places to work.
Recently, both the NYT and EdWeek have reported a national teacher shortage as enrollment in teacher training programs has dropped precipitously for several years in a row. Even TfA is having trouble recruiting.
Motherf*ckers, what did you think was gonna happen?
(Cynical voice at back of head: Ben, you are so effing naive. That’s what they wanted to happen. What better excuse to hire un-credentialed people to teach poor children? Me: No! I don’t believe it!)
Addendum (5/9/16): It came to my attention at some point that there was some debate last summer about the validity of NYT’s and EdWeek’s coverage. Is/was there really a national teacher shortage this fall, or were there certain districts with a shortage and others with a surplus? Michael Pershan had some tweets about this. So, let me just acknowledge this debate. This post was a quickly-fired-off response to seeing talk of a shortage in two major press outlets, after several years of running workshops with young teachers and feeling awe for their willingness to stay in the game even as working conditions have become shittier. If there is a shortage, I’m not surprised. If there’s not, then, let’s hope it stays that way. Go young people! That is all.
Dan Meyer’s most recent post is about how in order to motivate proof you need doubt.
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.