What It Comes Down To Monday, Jun 13 2016 

I was just reading A. K. Whitney’s piece in The Atlantic about the Hacker-Tanton debate. She gets to the heart of the matter.

Actually, not just the heart of the matter of the Hacker-Tanton debate, but, like, The Heart of The Matter in math education.

Is math for everybody?

I have come to feel like I can hear this question somewhere in the background of almost every debate about math education and math education policy that I encounter.

Almost everyone will say “yes.” But do they mean it? Or more precisely, what do they mean?

Is ‘rithmetic for everybody but that abstract stuff is just for eggheads? Is being put through the paces of the corpus of school math for everybody but enjoying it is just for dorks or smartypants? Is having to take tests for everybody but math as a tool to exercise agency is just for white and Asian men?

Or is all of it for everybody?

The History of Calculus / Honor Your Dissatisfaction Saturday, May 21 2016 

I was just rereading an email exchange with a friend (actually the O of this post), and found that I had summarized the history of calculus from the 17th to 20th centuries, up through and including Abraham Robinson’s invention of nonstandard analysis, in the form of a short play! I’m sharing it with you.

Mainly this is for fun, but it’s also part of my ongoing campaign promoting the value of honoring your dissatisfaction. The dialectic between honoring our impulse to invent ideas to understand the world better and honoring our dissatisfaction with these ideas is where mathematics comes from.

Here’s the play!

The History of Calculus, in 4 Extremely Short Acts

Featuring a lot of oversimplification and a certain amount of harmless cursing

Act I

Late 17th century

Leibniz, Newton: Look everybody, we can calculate instantaeous speed!

Everybody: How??

Leibniz: well, you consider the distance traveled during an infinitesimal interval of time, and you divide distance/time.

Everybody: Leibniz, what do you mean, “infinitesimal”? Like, a millisecond?

Leibniz: No, way smaller than that.

Everybody: A nanosecond?

Leibniz: Nah, dude, you’re missing the point. Smaller than any finite amount.

Everybody: So, zero time?

Leibniz: No, bigger than that.

Some people: Oh, cool! Look we can use this idea to accurately calculate planetary motion and stuff!

Other people: WTF are you talking about Leibniz? That makes no effing sense.

Act II

18th century

Bernoullis, Euler, Lagrange, Laplace, and everybody else: Whee, look at everything we can calculate with Newton and Leibniz’s crazy infinitesimals! This is awesome!

Bishop George Berkeley: But nobody answered the question of WTF they are even talking about. “What are these [infinitesimals]? May we not call them the ghosts of departed quantities?”

Lagrange: Hold on, let me try to rebuild this theory from scratch, I will make no mention of spooky infinitesimals, and will do the whole thing using the algebra of power series.

Everybody: Cool, good luck with that.

Act III

19th century

Cauchy: Lagrange, homie, it’s not gonna work. e^{-1/x^2} doesn’t match its power series at zero.

Lagrange: Sh*t.

Everybody: I think we don’t actually understand this as well as we thought we did.

Ghost of departed Bishop Berkeley: OMG I HAVE BEEN TRYING TO TELL YOU THIS.

Cauchy: How about we forget the whole “infinitesimal” thing and just say that the average speeds are approaching a certain limit to whatever desired degree of accuracy. As long as we can identify the limit and prove that it gets as close as we want it to, we can call that limit the “instantaneous speed” without ever trying to divide some spooky infinitesimals by each other.

Everybody: Awesome.

Weierstrass: I have an even better idea. Let’s formalize Cauchy’s thinking into some tight symbols and quantifiers. “Let us say that the limit of a function f(x) at c is a number L if for every \varepsilon > 0 there exists a \delta > 0 such that whenever 0 <|x-c|<\delta, it follows that |f(x)-L|<\varepsilon…”

All the mathematicians: AWESOME. Down with spooky infinitesimals! Calculus can be built soundly on the firm footing of “for any \varepsilon>0 there exists a \delta>0 such that…” and you never have to talk about any spooky sh*t!

All the mathematicians, in private: … but thinking about infinitesimals sure streamlines some of these calculations…

[Meanwhile all the physicists and engineers miss this whole episode and continue blithely using infinitesimals.]

Act IV

20th century

Scene i

Mathematicians: Infinitesimals are satanic voodoo!

Physicists and engineers: What are you talking about, what about CALCULUS?

Mathematicians: Whatever dude, don’t you know about Weierstrass and \varepsilon and \delta?

Physicists and engineers: Um, no, and I don’t care either! What’s the point when everything already works fine?

Mathematicians, in public: No, dude, there are all these tricky convergence issues and you will F*CK UP EVERYTHING IF YOU’RE NOT CAREFUL!

Mathematicians, in private: … but those infinitesimals are indispensible as a heuristic guide…

Scene ii

Abraham Robinson: Um, whatever happened to infinitesimals?

Mathematicians: I mean we rejected them as satanic voodoo because nobody was ever able to tell us WTF THEY ARE.

Robinson: I have a proposal. How about we consider them to be [fancy-*ss definition based on formal logic and other fancy sh*t]. Would you say that constitutes an answer to “wtf they are?”

Mathematicians: … why, yes!

Some mathematicians: omg awesome I can now RESPECTABLY use infinitesimals in calculations, I don’t have to hide anymore!

Other mathematicians: Whatever, I have no need to do the work to master this fancy sh*t. It doesn’t do anything good ole’ Weierstrass \varepsilon and \delta couldn’t do.

Physicists and engineers: wow, you guys are way over-concerned with the little stuff. Literally.

End

(Long-time readers of this blog will recognize the bit of dialogue with Leibniz from something I shared long ago.)

The point is that the whole episode is driven by uncertainty about what is even being discussed. The early developers of calculus shared the conviction that there was something there when they talked about “infinitesimals”, but none of them (not even Euler) gave a definition that was satisfying to everybody at the time (let alone to a modern audience). But this encounter, between the intuition that there’s something there and the insistence of the world to honor its dissatisfaction until a really satisfying account was given, was a generative encounter, resulting in several hundred years’ worth of powerful math progress.

So. Honor your dissatisfaction.

Pershan’s Essay on Cognitive Load Theory Monday, May 9 2016 

Just a note to point you to Michael Pershan’s motherf*cking gorgeous essay on the history of cognitive load theory, centered on its trailblazer, John Sweller.

Read it now.

I’m serious.

I tend to think of Sweller as, like, “that *sshole who thinks he can prove that it’s bad for learning if you think hard.”

On the other hand, any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees, so you know that he’s talking about a real thing.

Michael has just tied it together for me, tracing how Sweller’s point of view was born and evolved, what imperatives it comes from, other researchers who take cognitive load theory in related and different directions, where their imperatives come from, and how Sweller’s relationship to these other directions has evolved as well. I have more empathy for him now, a better sense of his stance, and a better sense of why I see things so differently.

Probably the biggest surprise for me was seeing the connection between Sweller’s point of view on learning, and the imperatives he is beholden to as a scientist. I get so annoyed at the limited scope of his theory of learning, but apparently he defends this choice of scope on the grounds that it supports the scientific rigor of the work. I understand why he sees it that way.

The remaining confusion I have is why the Sweller of Michael’s account, ultimately so clear on the limited scope of his work (“not a theory of everything”) and the methodological reasons for this limited scope, nonetheless seems to feel so empowered to use it to talk about what is happening in schools and colleges. (See this for an example.) Relatedly, I’m having trouble reconciling this careful scientific-methodology-motivated scope limitation with Sweller’s stated goal (as quoted by Michael) to support the creation of new instructional techniques. The problem I’m having is this:

Is his real interest in supporting the work of the classroom or isn’t it?

If it is, well, then this squares both with the fact that he says it is, and that he’s so willing to jump into debates about instructional design as it is implemented in real classrooms. But it doesn’t square with rigorously limiting the scope of his theory, entirely avoiding conversations about obviously-relevant factors like motivation and productive difficulty, which he says he’s doing for reasons of scientific rigor, as in this quote:

Here is a brief history of germane cognitive load. The concept was introduced into CLT to indicate that we can devise instructional procedures that increase cognitive load by increasing what students learn. The problem was that the research literature immediately filled up with articles introducing new instructional procedures that worked and so were claimed to be due to germane cognitive load. That meant that all experimental results could be explained by CLT rendering the theory unfalsifiable. The simple solution that I use now is to never explain a result as being due to factors unrelated to working memory.

On the other hand, if his interest is purely in science, in mapping The Truth about the small part of the learning picture he’s chosen to focus on, then why does he claim he’s doing it all for the sake of instruction, and why does he feel he has something to say about the way instructional paradigms are playing out inside live classrooms?

Michael, help me out?

Reality Check: Does Anybody Care What Andrew Hacker Thinks? Monday, Apr 4 2016 

So I’m just trying to figure out who, if anybody, cares what Andrew Hacker thinks about math education.

This is an earnest question.

At least since 2012, when he had an opinion piece in the NYT, he has been going on about how we should stop requiring “advanced” math, from algebra up, in schools. He was in the NYT twice again recently and has a new book out about it.

Now, of course both math educators and mathematicians are going to “care” in the sense that it annoys us. We are spending all this time trying to figure out how to improve students’ appreciation for and understanding of algebra etc., and out comes this dude talking about “scrap that whole project.” I remember there being at least a few responses in the MTBoS back in 2012, although Dan Meyer’s and Patrick Honner’s are the only ones I remember specifically. (Dan had some more fun with it a few years later.) And I was moved to write this reading mathematician Evelyn Lamb’s piece in Slate responding to Hacker’s book. (Dan’s responses succinctly summarize Hacker’s lack of imagination. Mr. Honner sees algebra in what Hacker wants to replace algebra with. And if you want to get more into the details, go read Lamb’s piece, it’s great.)

But annoying math teachers and mathematicians is definitely not the same thing as being remotely relevant. I mean, he is suggesting to do away with required algebra precisely at the point in history when, between the Common Core, the increasing quantity and stakes of standardized testing, and the incessant press handwringing about international competitiveness,[1] it seems to me that math, including advanced math, is more centrally ensconced in the curriculum than it’s ever been. (In my lifetime anyway. Possible historical exception of the immediate post-Sputnik era.) Is anybody taking him remotely seriously?

Yes, he has coverage in the New York Times and the Chronicle of Higher Ed. This doesn’t answer the question. He’s being intentionally provocative and succeeding in getting a rise. Is anybody taking his proposals seriously?

I eagerly await your thoughts.

*****

Postscript: Let me indulge myself to go ahead and give you my take, just for the record. Disclaimer that I haven’t read his book. I’m going on the 3 NYT pieces.

At the level of fundamental goals, he is upset about the fact that so many Americans have traumatic experiences with their math education, meanwhile graduating without basic numeracy needed for citizenship, and he wants to do something about it. I’m not mad at this, nor could I be.

I’m also sort of delighted that being a public intellectual counts for enough that this 86-year-old Queens College political scientist can mouth off in the NYT whenever he wants. I hope that when I’m 86, the NYT still exists and I can mouth off in it whenever I want.

But on to the merits themselves. I think he sees a real problem but I, like Dan, think he lacks imagination about both (a) what math education could be, and (b) what math is for – he doesn’t get it as a domain of human inquiry, or an intellectual inheritance, just as a tool, so he applies a utilitarian standard to it I’m sure he’d never apply to history, or art, say. But also, (c) I think he misdiagnoses the problem if he thinks removing required algebra (and up) will solve it. Algebra isn’t the first point in the curriculum where massive numbers of American children jump ship emotionally. This is already happening with fractions, and may begin much earlier. Hacker would never propose to take fractions, or even more fundamental stuff, out of the curriculum, since it’s obviously (even to him) part of the “citizen mathematics” he champions.[2] Doing a good job teaching math is a problem to be solved, not avoided. Finally, (d) I think he would probably puke if he really thought through the antidemocratic implications of a general public without advanced technical literacy while all the contemporary centers of power – finance, info tech, biotech, etc. – are technocratic and growing more so. He sometimes argues that you don’t need to know algebra to learn to code. Depends on what you are coding I suppose, but in any case this is beside the point. Sergey Brin does know algebra. Jamie Dimon does too.

Update 4/5: Sam Shah sends this graphic of number of people on feedly who “saved” or “favorited” Hacker’s Feb. 27 piece (which is about what he wants to replace algebra with):

benbs

Update 6/28: Patrick Honner wrote something relevant on the Math for America blog back in May – When It Comes to Math Teaching, Let’s Listen to Math Teachers.

[1] I’m only lumping these three things (CCSSM, high-stakes testing, and international-comparison handwringing) together from the point of view that all three seem to me to be moves in the direction of consolidating the consensus on the centrality of math in contemporary American education. I do not have them confused with each other and I don’t feel the same way about each of the three at all. For exmaple, I basically dig the CCSSM but (as any regular reader of this blog knows) I do not at all dig high-stakes testing.

[2]Patrick Honner’s post points out that Hacker’s notion of “citizen mathematics” almost surely involves algebra as well…

Lessons from Bowen and Darryl Thursday, Jan 28 2016 

At the JMM this year, I had the pleasure of attending a minicourse on “Designing and Implementing a Problem-Based Mathematics Course” taught by Bowen Kerins and Darryl Yong, the masterminds behind the legendary PCMI teachers’ program Developing Mathematics course, with a significant assist from Mary Pilgrim of Colorado State University.

I’ve been wanting to get a live taste of Bowen and Darryl’s work since at least 2010, when Jesse Johnson, Sam Shah, and Kate Nowak all came back from PCMI saying things like “that was the best math learning experience I’ve ever had,” and I started to have a look at those gorgeous problem sets. It was clear to me that they had done a lot of deep thinking about many of the central concerns of my own teaching. How to empower learners to get somewhere powerful and prespecified without cognitive theft. How to construct a learning experience that encourages learners to savor, to delectate. That simultaneously attends lovingly to the most and least empowered students in the room. &c.

I want to record here some new ideas I learned from Bowen and Darryl’s workshop. This is not exhaustive but I wanted to record them both for my own benefit and in the hopes that they’ll be useful to others. In the interest of keeping it short, I won’t talk about things I already knew about (such as their Important Stuff / Interesting Stuff / Tough Stuff distinction) even though they are awesome, and I’ll keep my own thoughts to a minimum. Here’s what I’ve got for you today:

1) The biggest takeaway for me was how exceedingly careful they are with people talking to the whole room. First of all, in classes that are 2 hours a day, full group discussions are always 10 minutes or less. Secondly, when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They never ask for hands, and they never cold-call. This means they already know more or less what the students are going to say. Thirdly, they have a distinction between students who try to burn through the work (“speed demons”) and students who work slowly enough to receive the gifts each question has to offer (“katamari,” because they pick things up as they roll along) – and the students who are asked to present an idea to the class are only katamari! Fourthly, a group discussion is only ever about a problem that everybody has already had a chance to think about – and even then, never about a problem for which everybody has come to the same conclusion the same way. Fifthly, in terms of selecting which ideas to have students present to the class, they concentrate on ideas that are nonstandard, or particularly visual, or both (rather than standard and/or algebraic).

This is for a number of reasons. First of all, the PCMI Developing Mathematics course has something like 70 participants. So part of it is the logistics of teaching such a large course. You lose control of the direction of ideas in the class very quickly if you let people start talking and don’t already know what they’re going to say. (Bowen: “you let them start just saying what’s on their mind, you die.”) But there are several other reasons as well, stemming (as I understood it anyway) from two fundamental questions: (a) for the people in the room who are listening, what purpose is being served / how well are their time and attention being used? and (b) what will the effect of listening to [whoever is addressing the room] be on participants’ sense of inclusion vs. exclusion, empowerment vs. disempowerment? Bowen and Darryl want somebody listening to a presentation to be able to engage it fluently (so it has to be about something they’ve already thought about) and to get something worthwhile out of it (so it can’t be about a problem everybody did the same way). And they want everybody listening to feel part of it, invited in, not excluded – which means that you can’t give anybody an opportunity to be too high-powered in front of everybody. (Bowen: “The students who want to share their super-powerful ideas need a place in the course to do that. We’ve found it’s best to have them do that individually, to you, when no one else can hear.”)

2) Closely related. Bowen talked at great length about the danger of people hearing somebody else say something they don’t understand or haven’t heard of and thinking, “I guess I can’t fully participate because I don’t know that idea or can’t follow that person.” It was clear that every aspect of the class was designed with this in mind. The control they exercise over what gets said to the whole room is one aspect of this. Another is the norm-setting they do. (Have a look at page 1 of this problem set for a sense of these norms.) Another is the way they structure the groups. (Never have a group that’s predominantly speed-demons with one or two katamari. If you have more speed-demons than katamari, you need some groups to be 100% speed demon.)

While this concern resonates with me (and I’m sure everybody who’s ever taught, esp. a highly heterogeneous group), I had not named it before, and I think I want to follow Bowen and Darryl’s lead in incorporating it more essentially into planning. In the past, I think my inclination has been to intervene after the fact when somebody says something that I think will make other people feel shut out of the knowledge. (“So-and-so is talking about such-and-such but you don’t need to know what they’re talking about in order to think about this.”) But then I’m only addressing the most obvious / loud instances of this dynamic, and even then, only once much of the damage has already been done. The point is that the damage is usually exceedingly quiet – only in the mind of somebody disempowering him or herself. You can’t count on yourself to spot this, you have to plan prophylactically.

3) Designing the problem sets specifically with groupwork in mind, Bowen and Darryl look for problems that encourage productive collaboration. For example, problems that are arduous to do by yourself but interesting to collaborate on. Or, problems that literally require collaboration in order to complete (such as the classic one of having students attempt to create fake coin-flip data, then generate real data, trade, and try to guess other students’ real vs. fake data).

4) And maybe my single favorite idea from the presentation was this: “If a student has a cool idea that you would like to have them present, consider instead incorporating that idea into the next day’s problem set.” I asked for an example, and Bowen mentioned the classic about summing the numbers from 1 to n. Many students solved the problem using the Gauss trick, but some students solved the problem with a more visual approach. Bowen and Darryl wanted everybody to see this and to have an opportunity to connect it to their own solution, but rather than have anybody present, they put a problem on the next day’s problem set asking for the area of a staircase diagram, using some of the same numbers that had been asked about the day before in the more traditional 1 + … + n form.

I hope some of these ideas are useful to you. I’d love to muse on how I might make use of them but I’m making myself stop. Discussion more than welcome in the comments though.

Steven Strogatz talking about feeling dumb for not solving something fast! Saturday, Aug 22 2015 

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!

Some work I’m proud of Friday, Aug 14 2015 

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 4\times 6 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 –

Tessa 4 6's cropped

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?

T: yes.

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?

T: Yes.

Me: And that’s 24 dots, yes?

T: Yes.

Me: And you told me before that 24 is also six 4’s, yes?

T: 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:

Tess 4 6's recircled cropped

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.

T: Yeah.

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 –

4x6 cropped

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,

neat one cropped

she gasps. Then she slides her eyes sideways to me, and with a mischievous smile, adds this to her previous picture:

final cropped

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!

And, a teacher shortage… Tuesday, Aug 11 2015 

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.

Also:

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.

Uhm sayin Saturday, Aug 1 2015 

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:

  1. It motivates proof
  2. It lowers the threshold for participation in the proof act
  3. It allows students to familiarize themselves with the vocabulary of proof and the act of proving
  4. 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.

“You have made us the enemy. This is personal.” Thursday, Apr 9 2015 

Just caught this from the Washington Post blog 2 months ago. Word.

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