Doodling the Axioms of Set Theory

Our son is almost three, and he’s starting to really like to draw. He’s also getting to the age where, if he doesn’t have something to do, he’ll tear our apartment to shreds, so most Sunday afternoons we dump a lunchbox of crayons and whatever paper we have on the floor. It’s arts and crafts time, Yosef!

Now, don’t get me wrong, I like watching my kid draw as much as the next parent. (“It’s a fish? No, a dog? Oh, wow, that’s terrific.”) But, what can I say, I get a bit bored.

More to the point, drawing with crayons is so much fun. Arts and crafts time is great. So I draw along with him. And very often I find myself trying to doodle some math.

Lately I’ve been studying a book called Classic Set Theory. It’s been really working for me. It has great exercises, clear organization, oodles of historical context. It’s subtitled “For Guided Independent Study,” and it’s making me wonder why every math textbook isn’t for guided independent study too.

Set theory and logic was my way back into math. I was a philosophy major and had all sorts of worries about whether I could handle a college math class. Then I took a philosophy class about Frege, Russell and Wittgenstein, figures who stand at the creation of mathematical logic and set theory.

In that class, I was finally starting to understand how I had to study this stuff — line by line, ask myself lots of questions, don’t leave anything fuzzy. (Which is more of less how I know to study Talmud.) And I was realizing that if I put in this sort of effort, and if I was honest with myself about what I got and what I didn’t, I could understand some (if I do say so) ridiculously abstract stuff at at least a basic level.

“At some level,” because there was still a lot of stuff that I didn’t get. Since college, set theory and logic has been some of the math that I’ve read the most on. But I still haven’t felt like I really understood set theory, as I hadn’t been able to make much headway in any set theory text.

This is part of why Classic Set Theory is so much fun for me to read. Finally, I feel like I’m starting to get this stuff. Finally!

Here’s the question I found myself asking while drawing with my son today: what could the axioms of set theory* look like if you drew them?

The axioms of Zermelo-Fraenkel set theory with the Axiom of Choice, I mean. I’d love to understand some of the other set theories better. During that class I took in college we studied type theory as Russell’s attempt to patch up the contradiction he discovered in Frege’s system, but I think I only ever had a surface-level understanding of what this meant.

There are nine axioms in all. I know that explanations work better when the objects to be explained are doing stuff, so I tried to make the axioms as active as possible in the drawings. In practice, this means I interpreted the axioms as little machines, rather than as rules or laws. (Axioms have always been presented to me as rules, and until the last couple of days that’s always how I’d always thought of them.)

The toughest one to visually represent was Replacement. The ones I’m most worried about having misrepresented are the Axiom of Infinity and of Foundation. Honestly, all of them are probably flawed in some way. There might be mathematical errors or misinterpretations — as long as you’re nice about it, would you point those out to me?

But I’m not sharing these as resources or to make some point about teaching. I’m sharing these out of celebration, and a bit of relief, that I might be finally figuring out how to learn some math on my own.

IMG_0645
Extensionality, Empty Set, and the Axiom of Pairs
IMG_0646
Axiom of Separation
IMG_0647
Power Set Axiom
IMG_0648
Union Axiom
IMG_0651
Axiom of Replacement
IMG_0649
Axiom of Infinity
IMG_0650
Axiom of Foundation
IMG_0652
Axiom of Choice
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Study an example, see the world

I’ve been a math teacher in New York City since 2010, a few months after I graduated from college. It’s the only job I’ve ever had, besides for little things over the summer when I was a teen. (In order: babysitter, camp counselor, Pepsi vendor at a baseball stadium, tutor. All kind of relevant to teaching, come to think of it.)

Though I teach math, math didn’t feel easy for me as a student. It was never where I shined. An exception was geometry, with its heavy emphasis on proof. Proof felt natural for me in a way that algebra didn’t.

When I began teaching, I realized that for many students the situation is reversed — it’s proof that feels unnatural and cumbersome. Writing a proof involves combining statements in ways that seemed to mystify many students. This was especially true early in my career.

After a few years of hitting my head against the wall, I started to understand what made this such a difficult skill to teach. Proof is the closest that mathematics comes to writing, and writing itself is impossible without reading. How can a student who has never read an essay possibly write one? I concluded that my students needed to read more proofs.

It took me a few more years to understand how exactly to pull this off in class. My big frustration was that my students wouldn’t devote enough attention to the proof examples I shared. I would distribute a completed proof and ask the class to read it with care. Very often, it seemed that they missed the whole point of the proof. They couldn’t read it carefully yet — they didn’t know how.

Now, things go better when I share proofs in class. One big difference is I have a much better understanding of all the subtle conceptual understandings that go into a proof, many which were invisible to me at first. (In teaching, it can be trouble when a topic comes naturally to you.) There are many aspects of a proof that I need to help them uncover.

Besides for a better understanding of the subtleties of proof, I’ve learned to structure my activities in sturdier ways. I’ve learned to design these activities so that they have three parts:

  • The proof example
  • Comprehension questions about the example
  • Proof-writing practice, with the example as a model

I didn’t come to this structure on my own, by the way. I came to it through reading about Cognitive Load Theory (where these are sometimes called “example-problem pairs”) and especially from seeing it in some especially well-designed curricular materials:

Screenshot 2017-11-30 at 6.49.21 PM.png

Also:

Screenshot 2017-11-30 at 6.50.45 PM.png

 

(In fact, I didn’t really understand how to make my own example activities until I saw many models in these curricular materials. I needed examples, myself.)

So, for instance, I created this proof example for my students this year:

DM8ANAWWsAACjSC.jpgLooking back, the example isn’t perfect. It ended up being a bit visually crowded, and it might have been better to eliminate some of the letter-abbreviations. In class, I actually covered up each stage of the proof to focus their attention on each part.

In any event, this activity shows a lot of what I’ve learned about teaching proof. I knew I wanted to make explicit the complicated two-stage structure of some congruence arguments, so I worked hard to create a pretty clear example for my students. I then called on students to answer a trio of analysis questions about the proof — there’s a lot to notice, and students don’t yet know how to notice the underlying structure of this kind of proof all on their own. Finally, I ask students to use what they’ve noticed on a related pair of problems, so that students see that there’s something here that’s generalizable to many different kinds of diagrams.

Even when my proof activities aren’t structured so rigidly, I try to include variety and a chance to practice. Here is a simpler activity, but I still call for students to do a bit of proof-completion in the second prompt:

DM8ANArXkAItunW.jpgSometimes when I talk to other teachers about examples, they tell me they’re worried that kids will just try to unthinkingly copy the model. I do know what they mean, but it’s not what I see with my kids. I think that part of the reason is that I reserve example-analysis for when I worry that the math is going to be difficult, even overwhelming for many students. There is certainly a way to misuse these activities, and perhaps if I used these sorts of tasks on less complex material I would see unthinking imitation.

One of my jobs is to help students see things that they can’t yet see — things like the logical structure of a good mathematical argument, or the way just a tiny bit of information about a shape can guarantee a whole lot more. When things don’t come naturally to my students, what I’m learning to do is to design an activity that opens up a little window into the mathematics so they can learn to see new things.

Something I Wrote on NCTM’s Membership Forum about Equity and NCTM Policy

I also think there should be an opportunity to review NCTM’s policy commitments and their relation to equity at the national conference. It’s not at all clear to me that the policies that NCTM gets behind lobbying for in DC help to reduce inequality. This should be a matter for member discussion.

My understanding — just from things that I’ve read — is that much of NCTM’s lobbying happens as part of the STEM Ed Coalition [http://www.stemedcoalition.org/about-us/our-leadership/]. Here are aspects of the STEM Ed Coalition platform that trouble me, from the perspective of equity:

  • They call for expanding accountability measures and testing to include science, but testing regimes are frequently used to support inequity in practice.
  • They call for private money to flow into education to support STEM education, though in practice private money has been used to support inequity.
  • The foundation for their STEM platform is the belief that STEM jobs are necessary for national security and economic reasons. This kind of nationalistic justification is often used to support inequities in education — after all, it’s in the national interest to have our very best students, and only our very best, in STEM.

If NCTM wants to put equity at the center, then NCTM policy needs to be revisited at the conferences.

Teaching, in General

I. 

If you give a quiz covering lots of different topics, you’re going to get a lot of different mistakes. Which leaves you with a dilemma: how do you address those mistakes?

Yesterday’s quiz in geometry was a review quiz, so the topics were from all over the place:

  • angles in isosceles triangles
  • inscribed angles in a circle
  • area of triangles, parallelograms and trapezoids
  • congruence proofs

As expected, kids distributed their not-quite-there work fairly evenly across these topics. (OK so that’s not true, there were a lot of issues with the congruence proofs. There always are and always will be. Sigh.)

Here were two bad options for returning the quiz:

  • Try to address all the issues with individual comments. First, it’s a game of whack-a-mole that is guaranteed to drive me insane. Second, what should I do? Try to leave perfect hints? Say nothing, and let kids figure out on their own what they did wrong? Show them the correct way to answer the question, and thereby eliminate anything for the kids to actually think about when I return the quizzes?
  • Pick just one thing to focus on. Reteach that one thing in a careful way, then return the quizzes and ask kids to revise.

The second of the two options is great when there the mistakes are in the same galaxy. (I wrote about this in a post, Feedbackless Feedback.) But, I’m realizing now, this isn’t a terrific move when the mistakes are distributed across many topics. Because on what basis should I pick something to focus on reteaching? Any choice would be equally bad.

II. 

While reviewing the class’ quizzes, I found myself falling into written comments, at least until I figured out what else to do with the quizzes.

I used to write long, wordy comments that were essentially hints on the margins of the page. (“Great start! Have you tried multiplying both sides of the equation by 3?”) I came to dislike those sort of comments, as they just focus focus focus attention all on THIS problem. But I don’t particularly care about whether a students gets this problem correct; I care about the generalization.

What I’ve fallen into is, whenever possible, writing a quick example that’s related (but not identical) to the trouble-problem (the problem-problem) on the page. I do this below on the second question:

image-3.jpg

Then, I ask kids to revise the original on the basis of the example (or anything else they realized).

After writing a few of these example-comments, I realized I was taking a lot of time doing this, and repeating myself somewhat. I also realized that I don’t know if I could repeat this on every page for the congruence proofs, as the problem itself was reasonably complex:

image-2.jpg
Hard to read. I use the highlighter to flag errors, but this student highlighted the triangle himself.

I wasn’t sure what to do. Then, I remembered something I had read from Dylan Wiliam — I think it’s in Embedded Formative Assessment. His idea there was that you can give all the class’ comments to everyone, and then kids have to decide which comments apply to them.

I thought, OK, I can work with this. So I quickly (quickly!) made a page of examples, one for every mistake I saw on the quiz:

ums-copier@saintannsny.org_20171115_095927 (1)-page-001.jpg

My routine in class went like this:

  • Hand out the examples for revision.
  • Hand back the quizzes with comments.
  • Search for an example that’s relevant to your mistake.
  • Call for revision on the basis of the examples. Work with friends, neighbors. Of course, I’m available to help.
  • Then, try the extension task.

This was my first time trying this, but I thought it went well. Solid engagement, really good questions, no unproductively stuck students.

When you do something good in teaching, you never really know if it’ll work again, but I’ve got a good feeling about this one. It feels like a lot of what has already worked for me, but in a better order.

III.

Harry Fletcher-Wood is very nice and has a lot of interesting thoughts about feedback. As such, Harry and I very nicely disagree about a pretty interesting question about feedback: how can you teach people how to give better feeedback?

The usual caveats apply: I am not a teacher teacher, but Harry is involved in teacher education, and I have no idea if I’m right on this.

In any event, Harry recently published a really cool post where he tried to synthesize a lot of the research on feedback into a decision tree:

Feedback-decision-tree-1.6

Now, this is awesome as a synthesis. But just because something is a good description of feedback doesn’t mean that it’s useful prescriptive advice. My favorite example of this comes from Pólya’s strategies for mathematical problem solving. Alan Schoenfeld has a nice way of putting it in Learning to Think Mathematically — the strategies have descriptive, but not prescriptive validity:

In short, the critique of the strategies listed in How to Solve It and its successors is that the characterizations of them were descriptive rather than prescriptive. That is, the characterizations allowed one to recognize the strategies when they were being used. However, Pólya’s characterizations did not provide the amount of detail that would enable people who were not already familiar with the strategies to be able to implement them.

In other words, just because a heuristic is a good description of practice doesn’t mean that it is an effective pedagogical tool. And that’s precisely my concern with Harry’s decision tree.

Feedback is a high-level concept that describes a TON of what happens in teaching. And any guidelines for how to give feedback effectively are also going to be high-level in a way that reminds me of Pólya’s moves like “find a simpler problem” or “draw a picture.”

And just as Pólya’s moves struggle because they aim to guide problem solving in geometry, algebra, topology, etc., all areas of math, Harry’s decision tree seems to me an attempt to guide feedback in all areas of teaching — math, history, medical school, etc.

Of course, Harry doesn’t intend for this to be the only thing guiding students, but neither did Pólya. My question is whether these generalizations themselves are helpful, beyond whatever ways that teacher educators can make them concrete and specific for teachers.

But what’s the alternative?

I don’t know yet. I can say a few things now that I couldn’t a few years ago:

  • I think domain-specific — math-specific, history-specific — generalizations will be more useful than domain-general ones.
  • I think that the generalizations can productively come in the form of instructional routines.

And, with this post and the other one, I now have two generalizations I can make about giving feedback in math class.

First: if there’s a problem that a lot of students have trouble with, consider a reteaching/revising cycle like the one in this image:

Screenshot 2017-11-15 at 1.24.03 PM

Second: if mistakes are sprinkled across too many topics, consider something like the revision routine I described in this post.

IV.

My bet is that a lot of knowledge about teaching looks like this. It’s not that there isn’t knowledge about teaching that accrues, but that we look for ways to scale things out of their contexts. Then we call those things myths and talk about how we have to kill ’em.

In general, generalizations about teaching are hard to come by. But nobody teaches in general. All teaching is intensely particular. These kids. These schools. This idea.

Some people are skeptical of the possibility of making generalizations about teaching, and the vast majority of people are cheery about making sky-high generalizations that cross every context. There’s a middle position that I want to find. There’s a sweet spot for knowledge about teaching, though I don’t know if we’ve all found it yet.

Addendum: On Discovery and Inquiry

I.

I appreciated some of the disagreement that got aired as a response to my last piece, on discovery. In particular, some told me that guided-inquiry or discovery really is more memorable than other forms of instruction.

Either because the stuff you learn from discovery is more meaningful (and hence more memorable):

Screenshot 2017-11-07 at 9.31.30 AM

…or because discovery involves surprise, and surprises are more memorable and lead to stronger learning:

Screenshot 2017-11-07 at 9.29.10 AM

I’m not satisfied with either of these arguments.

The thing about discovery activities is that the new idea — by definition of discovery — comes at the end of the activity. That means that kids are spending most of the activity thinking about stuff besides the new, often difficult, idea. It takes time to understand new ideas — to make them meaningful, to “own” them — and most of the time in a discovery activity is spent thinking about other stuff.

That’s certainly the case for the triangle angle activity that I critiqued in my post. While working on the activity, a student’s attention is drawn to many mathematical things — the angles, protractors, adding angles — and only very little of the time is spent thinking about what exactly a triangle’s angles sum to. (This is especially true if the idea is truly new to a student — they’ll only be thinking about the sum once they discover it, towards the end of the activity.)

It’s also true in the trapezoid/triangle area task that I shared. There was a ton of excitement precisely because my class hadn’t discovered the relationship between bases and area yet. That was where the joy was coming from — that also means that they were thinking about the discovered relationship for comparatively little of the time spent on the activity.

As I argued in the original post, that’s OK for me. It was fun and beautiful, and kids should have a chance to articulate slippery patterns and feel the pleasure of discovery. That’s part of math that I enjoy sharing with kids.

Anyway, that’s my response to the idea that discovery is more memorable because it’s more meaningful. Ideas are meaningful when you have time to get used to them, and that’s precisely what gets lost in a discovery activity.

As far as the idea that guided inquiry is surprising, and surprising stuff is more effective: why can’t you structure an explanation to elicit prior knowledge and surprise students? Aren’t explanations sometimes surprising? I think they can be.

Of course, how to craft effective explanations — that surprise and really engage students — is not easy, but it doesn’t get any easier if we don’t talk and write about it. That was part of my argument in Beyond “Beyond Explaining.”

II.

This is all theory, though. What happened in class today, after the weekend, after the memorable discussion on Friday?

I ask them to find the area of a trapezoid and…it’s like Friday never happened.

Wait what?

Hold on what do you mean the same as a triangle?

Could we go over this again?

The only kid who remembered how to find the area of the trapezoid — and I promise this is true, and not just me making up details to annoy advocates of discovery — was the kid who had connected Friday’s lesson to a formula that she once knew.

No guys, it’s the sum of the bases times half the height.

This is sort of surprising and disappointing. Friday’s class was so good! And nearly everybody was involved in the inquiry/discovery/discussion. It felt wonderful and it was fun.

That class, for me, was discovery that’s about as good as it usually gets. And yet it failed to stick over the weekend.

And yet this isn’t that surprising. The kids didn’t get a chance to practice the idea on Friday because we spend the class time uncovering some super-cool math. Kids need practice to remember ideas, and discovery takes a long time. This is just how it goes.

But if it’s not surprising, it’s also not disappointing. It was a lot of fun, and everybody was involved. It’s not what my class is like every day, and it would probably frustrate kids if it were.

So, at least this time, anecdote matches argument. And since we started practicing finding the area of trapezoids today,  it’s getting a lot more meaningful for my kids.

A typically wishy-washy take on discovery in math class

I think of myself as a non-discovery math sort of teacher, but every once in a while I find myself asking kids to discover stuff. I recently did this in my geometry classes, with a dot-paper area activity.

I really like dot paper. On dot paper you can make precise statements about area that typically generalize nicely to non-dot scenarios. The same basic relationships apply, it’s just easier to see them in a dot world.

My idea was to share this picture with students, and ask them to find the area of each shape. (This is the version that I marked up under the camera in class. None of the purple or blue ink was in the original.)

Picture 1.JPG

All of these shapes have the same height — 4 — but they otherwise differ. The first is a rectangle that has area 8. Next was a parallelogram: also area 8. Then a trapezoid. Many students came up with the idea of drawing a line to split it into a triangle and a rectangle — together they have area 8. This didn’t occur to every kid, though, so there was a good deal of neighbor-asking and chat to get the hang of those trapezoids.

Spoilers: every shape here has area 8! There’s something else they share too: the sum of their top and bottom bases is always 4. (The weirdest case is the triangle, that has a bottom of base of 4 and [arguably] a top base of length 0!)

So that was what I wanted my kids to come to notice and articulate. And I wanted it to be fun, and feel like they were discovering something new.

Not because I think that learning is more effective when kids discover something new, or that they’re working on their discovering skills or anything. Just because I think it’s fun for them to uncover patterns. It’s a cool part of math, and I’m trying hard to share more of the cool parts of math with my students, along with working on their skills and knowledge etc.

Here then is my take on discovery in math class:

  • It can be fun to discover cool stuff about math.
  • It takes longer for kids to understand something via discovering, and doesn’t really confer a learning advantage.
  • But if the activity is relatively brief and I can spare the time, why not? I want my students to think math is fun and cool.

This is my unprincipled take. I like discovery for fun and color in class, and I don’t feel the need to aim for 100% efficiency in every second of my teaching.*

Whether you feel such a need — or need to feel such a need — probably varies a lot depending on your school, administration, students, etc. My guess, though, is that the vast majority of teachers don’t feel this need, and probably are correct in this feeling.

Lots of discovery activities are uninteresting to me. Though I absolutely love the practice exercises in Discovering Geometry, the discovery activities largely leave me cold. Here’s an example of the sort of thing I’ve fallen out of love with:

dg discovery.png

My issue with this activity isn’t that it’s discovery. It’s that it’s not such a cool pattern (most kids have seen it before taking a geometry class), so discovering it isn’t as surprising or fun. The activity takes a while — do it once, check, do it twice — and all that is probably to protect against the risk of incorrect measurement, which is another tedious aspect of this discovery activity. And, at the end of all this, what cool math have you uncovered? Relatively little — just a sum. You don’t see any new relationships or geometric structure that guarantees that the triangle will have such a sum.

Why bother with all this? I’ve decided that this sort of discovery activity isn’t much use for me. But that’s not a principled objection against discovery — it’s just that I don’t think this type of activity is worth it.

Speaking of “types of activity,” I think it’s fair to categorize this angle sum activity as discovering something easy to articulate. Check an example, check an example, check an example, woah it’s always the same. You don’t uncover the geometric relationship in this activity. I think that’s part of what makes it not much fun and sort of tedious.

I think discovering something hard to articulate (I’m not quite sure what to name this) tends to be more fun, more cool.

To illustrate this, here’s the conversation that went along with my “area 8” activity in class.

I began the conversation with a prompt to my students: what do all these shapes share? how do these shapes differ?

I called on James first.

James: They all have heights and bases of 4.

I said that I didn’t entirely get that, and asked if anyone also saw that. Robin came up to the board to point. She also subtly refined Jame’s claim.

Robin: A lot of them have a common theme that either the height is 4 or the base of 4.

Then Liam chimed in to make it precise and accurate:

Liam: The height is always 4. The bases are different though.

Luiz: Yeah, the bases are either 2 or 4…or no they vary. Sometimes it’s 1, 2, sometimes it’s none.

Then Sara chimes in. She started articulating a generalization — she was WORKING HARD to try to articulate some sort of generalization. Her first one wasn’t entirely accurate though. I loved how she put it as a question.

Sara: Wait, does that mean that any shape that has a height of 4 has an area of 8?

The class and I (and Sara) agreed: this just could not be true. (Luiz says: well you could have a base of 1000.) Jess tried to get clearer about what was special about all these shapes.

Jess: No this is just because they are all parallelograms and…wait the third one is sort of confusing. What is that?

[Insert a minute of discussion about which of these are trapezoids and which are parallelograms.]

After this, I decided that we wouldn’t be able to restart the conversation unless I summarized and took some notes. So I wrote some notes on the board.

This time, though, I decided to take a heavier hand to draw attention to something really cool. The kids hadn’t noticed it yet, and I wanted to make sure that they did.

I wrote the numbers that you saw in purple ink in my image. I wrote the top and bottom bases.

Sara: So couldn’t you say that it’s base 1 plus base 2, times the height and that’s the area. Like for shapes in general.

Very close! There are some gasps and agreements.

Samantha: So wait does that work for everything?

Sara: Yeah that’s what I’m asking.

Me: That’s a good question. I’m trying to find some dot paper.

Joe: Wait so does it?

That question just hangs there for a second. And here’s a choice I could make. I could act coy, refuse to answer, and insist that the thinking here come from the students.

But then you get this weird dynamic in class where kids never know if they’re getting a straight answer from the teacher or not. I don’t like that dynamic. I like it when kids ask questions about math, and I like that they can get a straight answer out of me. And would they spend more time thinking about this cool relationship if I answered that question, or if I refused to?

“The answer is yes, sort of.”

GASPS.

Sara: Does it have to have parallel lines? Does it have to be a trapezoid…wait does a parallelogram count as a trapezoid.

Good thinking, Sara!

While there’s thinking kind of just toppling out of Sara’s head, I’m searching for a blank piece of dot paper, because there’s something that I realize might help. I don’t want them to get too used to the area 8 case — that might lead to a false generalization, since Sara keeps on saying that it’s double the sum of the bases. (That’s true when the height is 4, but not when the height is something else.)

So I draw this:

Picture 2.JPG

We clocked in at about 10 minutes there. No question that this was not as effective as a worked example or something else more carefully designed for learning. But I wasn’t aiming for efficacy. I was aiming for those half-articulations, those gasps, that enthusiasm. And as long as I don’t come to worship those gasps and chase them exclusively, class will be a bit closer to being fun, cool.

Addendum: this follow-up post.

 

 

YouCubed is Sloppy About Research

I.

There’s a lot of sloppy talk of science that gets tossed around education. Every teacher knows this — or least, I hope they do — and I try not to get too worked up about it. I also try not to get worked up by people who wear backpacks on crowded subways. Not that it’s OK, but lots of people do it (the backpack thing) and picking any one person to bark at hardly seems like it would help.

The other thing is that people can be really passionate about sloppy science. This happens all the time, it’s nothing special about education. But passion makes it hard to talk critically about the research without it seeming like you’re attacking everything else that the person is passionate about.

I think a lot of the time it’s because we assume that the attack on the research isn’t really about the research, it’s about everything else it supports. It’s like, Why are you bothering to poke holes in [research that supports X]? You’d only do that if you were really against [X]. 

Which leads to an interesting question. Is it possible, at all, to avoid this trap? Is it possible to critique sloppy use of research without being heard as if you’re trashing a person, their organization, and everything they stand for?

II.

What if you ask a lot of rhetorical questions — does that help?

Anyway, let’s talk about YouCubed. I think there’s something that — if we slow down, turn off passion, turn on curiosity — we can agree is a serious mistake. Here’s a popular quote from their popular page, Mistakes Grow Your Brain:

When I have told teachers that mistakes cause your brain to spark and grow, they have said, “Surely this only happens if students correct their mistake and go on to solve the problem correctly.” But this is not the case. In fact, Moser’s study shows us that we don’t even have to be aware we have made a mistake for brain sparks to occur.

When teachers ask me how this can be possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged and the challenge results in growth.

Indeed — this does sound really, really surprising! So, applied to math, it sounds like if you solve an equation incorrectly you learn from that experience (brain sparks?) in a way that you wouldn’t if you had solved it correctly. The mistake you made causes struggle — even if it’s not a struggle that you’re aware of. You might not feel as if you’re struggling, but your brain is on account of the mistake.

The citation of Moser is very clear, so, ok, let’s go upstream and check out Moser. Though a lot of research is behind paywalls, a lot of it isn’t and a quick Google Scholar search gives us a copy of the paper, here.

The study was an fMRI (EEG, see below. -MP) study. Here’s my understanding of the paper. The researchers wanted to know, on a neurological level, what’s different about how people with a growth mindset or a fixed mindset react to mistakes. So they gave everybody a test, to figure out if they had a fixed or a growth mindset. Then they hooked subjects up to fMRI EEG machines. While in the machines, participants worked on a task that yields a lot of mistakes. Researchers recorded their neural activity and then analyzed it, to see if there was a meaningful difference between the fixed and growth mindset groups, after making errors.

Spoilers: they found a difference.

Also: the whole paper is premised on people being aware of the errors that they made. This is not a subtle point, buried in analysis — the paper mentions this like nine times, including towards the end where they write, “one reason why a growth mind-set leads to an increased likelihood of learning from mistakes is enhanced on-line error awareness.”

This is so clearly different than what the YouCubed site says that I’m starting to doubt myself. The paper seems to be entirely about what happens when you realize you’ve made a mistake. Yet it’s cited as supporting the notion that you learn (“brain grows”) from mistakes, even if you aren’t aware of them.

Is there something that I’m missing here?

III. 

This doesn’t seem to me like an isolated issue with YouCubed.

A while ago Yana Weinstein and I made a document together trying to collect errors in YouCubed materials, explain why, and suggest improvements. I don’t have much else to say about that, except that if you’re interested you might check it out here.

IV.

What strikes me about YouCubed is that the errors just seem so unnecessary. The message is a familiar one, and I’m OK with a lot of it: don’t obsess over speed, think about mindset, don’t be afraid of mistakes. But there’s this sloppy science that gets duct taped on to the message. What purpose does that serve?

There’s also the question of why so few people in the math education community talk about this. I mean, it’s not like we lack the critical capacity. Every so often I see people in math education whip out their skeptical tools to tear apart a piece of research. Why not with YouCubed?

I don’t want to be cynical, but I want to be truthful. The first reason, I think, is because the message of YouCubed (besides the science) is widely admired. A lot of teachers love it, and nobody wants to ruin a fun time.

But I don’t think we have to worry about that. We can talk about the science of YouCubed in a way that doesn’t entangle the rest of the YouCubed message.

The second reason is, I think, that YouCubed and Jo Boaler’s popularity makes it difficult for the most visible people in math education to seem critical — no one wants to turn on one of their own. Especially since Boaler has often been subject to unreasonable attacks in the past — nobody wants to be unfair, cruel or sexist to her.

I’m actually very sympathetic to that. But this is also why it’s important for people who aren’t part of the research or PD world to have platforms to discuss ideas. We don’t have the reputation or the connections to lose, and so we can take a closer look and ask, wait, does that really make sense?

Corrections:

Actual neuroscientist Daniel Ansari points out that I don’t know the difference between an EEG and an fMRI, which is true.

 

NCTM Journals: Be Interesting, Not Useful

My department subscribes to Math Horizons, a journal “intended primarily for undergraduates interested in mathematics.” I really like it. I recently found an old issue around school, and was reminded how much fun it can be. Here are opening lines, pulled from articles in the issue:

  • “The year 2014 is an especially good time to tell this tale of disguise, distance, disagreements, and diagonals.”
  • “What made you decide to be a math major?”
  • “Being in charge of a math club can be exhausting.”
  • “Time to end it all, Ellen thought.”
  • “More than 65 years ago, William Fitch Cheney Jr. conceived one of the greatest mathematical card tricks.”
  • “What’s your favorite number?”
  • “I grew up around decks of cards.”

Following these openers, one can read interviews with mathematicians, longer pieces about the history of math, book reviews, mathematical exposition, and even fiction.

After rereading Math Horizons, I went searching around my apartment for an issue of an NCTM journal. I’ve subscribed to each of the three journals since first becoming a member, always hoping that the other journal would interest me more. I finally found the latest issue of Teaching Children Mathematics smooshed in with a pile of other magazines.

Here are first sentences pulled from the September issue of TCM:

  • “When you think of ‘modeling’ in the mathematics classroom, what comes to mind? With the inclusion of Model with mathematics as one of the Standards for Mathematical Practice (SMP), the Common Core (CCSSI 2010) puts forth a vision of modeling in the mathematics classroom that moves beyond using concrete materials or other visual representations to give meaning to mathematics.”
  • “We recently conducted a randomized controlled trial that showed a significant impact of teachers’ lesson study, supported by mathematical resources, on both teachers’ and students’ understanding of fractions (Gersten et al. 2014; Lewis and Perry 2017).”

These are long. At risk of losing my own readers, I’ll include one last, even longer opening line:

  • “I am always in pursuit of resources that will add to my knowledge as described in Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014), which posits how crucial it is for math educators to continue to “recognize that their own learning is never finished and continually seek to improve and enhance their mathematical knowledge for teaching, their knowledge of mathematical pedagogy, and their knowledge of students and learners of mathematics. (p. 99)”

In response to my frequent kvetching about the journals, an NCTM board member emailed me. He asked, “What would you like to see in the journal?”

Fair enough! I would like NCTM to publish interesting articles.

Nobody sets out to publish boring articles, of course. But I have reason to think that “is this interesting?” is not being asked nearly enough at the NCTM journals right now.

For instance: I recently completed a twenty-two question survey about the NCTM journals. Four of the questions asked me about what I found useful. What sort of articles do I find the most useful? The least? Which departments are useful or not to me?

To be fair, one question asked, “Would you be interested in reading articles about…? (check all that apply).” That makes a four-parts usefulness to one-part interestingness ratio, which sounds about right for what NCTM is currently putting out. Invert the ratio, and I don’t think the above quotes make the cut any longer.

The other thing about interest vs. usefulness is something Henri Picciotto calls “the seemingly obligatory genuflection at NCTM’s sacred texts, most recently Principles to Action.” He means the way so many of the pieces published include the line “…as demanded by the Standards for Mathematical Practice,” or “…as detailed in Principles to Actions.” And, in fact, all three editorial teams officially require articles to show consistency with Principles to Actions:

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It’s simply hard to tell a good, interesting story about teaching while also projecting your adherence to a set of teaching standards. As a writer, you start losing options. One of the sturdiest formats I’ve found for writing about teaching is narrating learning. You develop some question, and then you take the reader along in your attempt to answer it. It is immeasurably harder to do this if in the very first sentence you announce that we already know how best to teach.

Each of those juicy opening lines from Math Horizons helps generate space to tell a story — about a trick, about a career, about a number. In turn, each of the NCTM openers eliminates space that might otherwise be occupied by a story.

An NCTM journal that aimed to be mostly interesting — four-parts to one, let’s say — could therefore change in these three ways:

  1. Publish crisp, engaging writing that tries to capture attention.
  2. Discourage writers from trying to adhere to standards; publish writing that disagrees with NCTM policy and teaching documents.
  3. Seek articles from the range of reader interests: math, math history, classroom dilemmas, policy debates, interviews, and so on, and so on. Even research, but for heaven’s sake keep it interesting!

This won’t be an easy change to make. I know it will be difficult to find writers willing to veer from what NCTM has published in the past. A word of advice on the editorial process, then: don’t seek submissions, seek writers. Find people that you’d like to write, and then ask them to pitch ideas. When one strikes an editor’s eye as especially interesting, help the writer develop it. Ask for snippets, early thoughts, rough drafts, and help craft the pieces into something that you expect to capture reader interest.

And all of this is worth it, because courting interest is a matter of respect. A piece that doesn’t attempt to capture attention (like a textbook) projects the opposite message: you really ought to read this. And, after all, isn’t that the main message of NCTM to teachers? That you really ought to teach like this, because we have the standards, the experts, the research and the know-how to train and educate you. Sure, this may be a slog to read, but aren’t you a professional? And you’ll read what you need to for your professional development.

Of course, like the speaker who comes in with gimmicks and cheap jokes, writing can miss the mark the other way. Bad writing can suggest a lack of seriousness.

But when done well, engaging writing can project trust and respect to the reader. We know you’re busy and discerning, it says, and that you have the intelligence to decide how to think and what to think about. You and us both. But, how about this?

So, stop trying to be so useful, NCTM! Relax, and try to be interesting instead.

My NCTM Benefits

I.

I’ve been getting those emails again — “30 Days Left of Your NCTM Benefits.”

If you talk to someone who works with NCTM, one of the first things they’ll tell you is the organization is committed to making big changes. Depending on the person, they’ll also tell you it’s because the organization is scared, staring straight into a crisis. Membership and income is dropping, and nothing they’ve tried has tempered this trend.

I’m committed to NCTM. But if the question is do you personally enjoy the benefits of membership?, my answer is “no.”

I mostly don’t enjoy the conferences, and it’s not easy for me to justify missing school for them. I mostly don’t enjoy the journals — I mean just that I don’t usually want to read them. And seeing as conference discounts and a journal are the major personal benefits that NCTM offers members, there’s just not much for me, personally speaking.

But what if the question isn’t about personal benefits, but of benefits to the field of math education? Volunteers often talk about the role of NCTM in making policy when explaining why they support NCTM: Sure, you can take or leave the personal benefits. But there is no other organization that has a voice for math education on the state or national level. When NCTM talks, people listen. Don’t you want to support that?

A lot of people are passionate on this point. In fact, I was once talking to a former employee of NCTM, and I suggested that I’d rather NCTM didn’t pay lobbyists to try to influence policy. We had been having a fairly radical talk: what if the organization eliminated conferences, changed the journals, restructured the volunteer board, etc. When I mentioned cutting lobbying, though, this person showed real emotion. This was unthinkable, she said. Every professional organization lobbies in Washington, and NCTM is a professional organization.  

Maybe that’s right. All I can say is that I’ve never been asked by NCTM what my policy views are, so I don’t know how they can claim to represent them. A survey would be a nice start, but hardly enough. Maybe there should be debates at the national conference or something, or a right to vote on policy positions if you pay your dues. People sometimes worry that the policy work of NCTM is invisible to membership. I bet a lot more teachers would know about the policy arm if they could influence it.

It’s not hard to come up with a suggestions like these, and it’s not hard to come up with many others like it, that would increase the role of teachers in the organizations. Why haven’t these ideas been taken up yet? Why is NCTM structured the way it is?

II.

NCTM is an organization that wants math teachers to pay for the right to defer to experts. The experts are math education professors, consultants, coaches, administrators, and a few teachers on their way to becoming one of the above.  

Here’s an excellent point that Henri Picciotto made to me: teachers are hardly the only members of NCTM. “The organization is not uniquely or even primarily a teachers’ organization,” he said. This sounds exactly right, and you can see it in everything that NCTM does.

Let’s get concrete. I love writing and reading, and I’ve thought about writing pieces for the NCTM teacher journals. I’ve been turned off, though, sinceI learned that my submission would be judged, in part, by its “consistency with the mathematics teaching practices as described in Principles to Actions.” (link)

In other words, you might not know what great teaching looks like, but NCTM does. If you have any great examples showing how to put NCTM’s vision into practice, by all means, submit an article. But if you’re not ready to sign on to the NCTM vision of teaching, well, best to take your writing elsewhere. And, hey, why shouldn’t you adhere to the NCTM vision? It simply describes elements of good teaching we all agree on. After all, it was written and packaged by the experts.

The “Principles to Actions” clause (and similar requirements across the journals and at conferences) is a shame. First, it narrows the bandwidth of ideas that we’re allowed to talk about in math education. Second, it makes for a duller reading experience; to me the journals have a sort of corporate feel to them. But the most significant thing of all, I think, is that it greatly reduces the creative work that teachers are encouraged to do, and teachers want to do creative work.

“Putting research into practice” can mean a lot of things, but most often it signifies that we all already agree on the best ways to teach. What’s left is to convince colleagues, boards, parents, kids, etc. And when you start with the sorts of vision-documents that NCTM has produced, you end up with very little left for teachers to do.

So across the board — for policy, journals, conferences, PD, publications, resources — NCTM’s pitch to teachers is: don’t you want us be your experts? And the question is, does that pitch still resonate with teachers if membership is dropping?

III.

NCTM seems to get that the internet changed things, but I think they’re wrong about why. It’s not just that there are free alternatives to NCTM publications, or that people now expect digital copies of stuff. The bigger problem for them is the web has allowed teachers to find alternatives to the institutional trust that NCTM currently seeks to trade on. Now, you can choose your experts.

So the current relationship — where teachers are asked to pay NCTM because they trust the experts — is no longer tenable for the organization. This leaves NCTM with really one alternative, which is to focus on what math teachers want, whatever that happens to be.

Is NCTM heading towards this? I don’t know. I do know that, for the first time in its history, NCTM now lists “Building Member Value” as a organizational goal. They are now institutionally committed to the following:

“NCTM fosters communities that engage members to improve the teaching and learning of mathematics.”

Which is great! I think that NCTM will do a better job earning members when it aims to serve the needs of those members, rather than asking us to pay for the right to be influenced. The direction of the organization needs to be reversed; math education professionals will need to trust teachers.

For the journals, this could mean publishing stuff that isn’t already 100%-certified nutritious. And it would mean, I think, that writers would have to start making the case for their vision of teaching without merely citing NCTM consensus documents for authority. But really, NCTM would have a mandate to publish whatever there are readers for in math education.

As far as the conferences go, I’m confused by the role that math plays in these math education conferences. The last NCTM conference I attended was Nashville, two years ago. I remember (and wrote about) being surprised why there wasn’t more learning and doing math for teachers at these conferences. What if it turned out that NCTM members wanted more chances to learn math with other teachers? What if we’ve heard enough about formative assessment?

But all of this is just fleshing out the details. NCTM won’t regain its membership by focusing on PD, making statements, or publishing new guidelines. It needs to stop trying to fix math education, and start serving its teachers.

Mathematicians Ask for Help

Lately I’ve been struggling to finish a piece about growth mindset research, a topic that I can’t seem to leave alone. I always come back to it, for reasons that aren’t entirely clear to me.

Summer is ending, and teachers are putting their classrooms back together. A lot of classrooms — if social media is to be believed — have bulletin boards that look like this:

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There were no bulletin boards like this at the high school I attended. I don’t remember if there were any bulletin boards at all — there must have been, but only for announcements and intramural schedules. No teacher would have dreamed of decorating their classrooms in this way. We wouldn’t have taken it seriously; it probably would have been destroyed the second the teacher turned their back to us.

****

I only had two math teachers in all of high school. Rabbi Weiss covered 9th, 11th and 12th Grade math for the honors track. He hated geometry, so he found someone else to cover that. Rabbi Weiss also taught me Talmud/Halacha in 10th Grade, so he was my teacher for all of high school.

Yeshivas, in my experience, are incredibly competitive places. (All-male yeshivas, I mean.) Who would make it to the top class? Who would be offered advanced placement in the Israeli schools? Who would win Torah Bowl? (Yep! Torah Bowl.)

Rabbi Weiss gave long, difficult exams in both math and Talmud. There were two competitions on every exam: who would finish first? who would finish last? Because Rabbi Weiss gave you as much time to finish these monster tests as you needed, and you could look up any sources you wanted (for Talmud — you were on your own for math). You could win for speed or you could win on endurance.

I won the speed competition on the first Talmud exam in 10th Grade, but that was the last time I won that. For every other test that year I was one of the last to finish. I’m not sure what changed.

(Come to think of it, grades were totally a part of our competition too. Getting a perfect score on one of those exams was another thing we fought for.)

There’s more to say about all of this: about Rabbi Weiss’ pedagogy, how badly I miss the summer Talmud classes that met in his basement, his sense of humor, and how even though all of us were highly competitive we were also best of friends, studying together and nudging each other along.

I have to say a bit more about Torah Bowl. I was made captain of a team, and how we made it to the championship. I wasn’t the fastest and never had scary-good memory for trivia, but I also drafted well and our crew was formidable. I could tell you about the legal question — from Bava Kamma if you care — we were asked in sudden death, to crown a winner. It was about damages: is such-and-such more like starting a fire, or digging a treacherous pit? And while I don’t remember the answer, I remember that I raised my hand and answered wrong, losing the contest.

****

I signed up for Multivariable Calculus in my first semester at college. I had just come from studying in a yeshiva in Israel for a year. I had a great time, but there was no question: I returned from Israel with a bad case of angst and melodrama. I was obsessed with questions of self-worth, all of which had been highlighted by the constant talk of “who’s a genius?” that permeated that world. This was the state of mind I was in when I started college.

Here’s what I wanted to do: I wanted to show up in tough classes and kick some ass, because otherwise what are you worth? You can only contribute that unique something if you have that unique something.

I wasn’t really prepared for Multivariable Calculus. Rabbi Weiss taught strictly through note-giving and homework-reviewing. It wasn’t terrible for us — along with a bunch of my classmates, I aced the AP exams — but it left me with relatively shallow reserves to draw on in my first college class.

Most importantly, though, I saw Multivariable Calculus as a referendum on me. I didn’t ask questions in class. It was only near the end of the semester that, sheepishly, I arrived at my teacher’s office for help. I profusely apologized for, like, a whole minute before my teacher (whose English wasn’t great) made it clear through intense eye-rolling that I was being ridiculous. Of course he was right — I was ridiculous.

In the end of the year, my classmates managed to get this guy a teaching award. I walked around campus rolling my eyes — haha, my turn now! — because they were giving an award to this guy? The guy who frequently stopped class to ask for English translations of mathematical terms? The guy who, I felt, had given me nothing, no life-vest, no rope, no help?

The big, big thing I was missing was that all the non-grumps in the class liked him precisely because he would ask questions. In doing so, he made everyone else feel as if they could ask too. That was the whole thing.

****

A few years ago I taught a 9th Grader who came with a warning: his teacher last year had been able to get nothing out of him. He shuts down, I was told, and this was absolutely confirmed by what I saw in class during the first few weeks.

At the start of my career, I would have diagnosed him with a struggle-allergy. He wasn’t willing to dig in; he was used to things being handed to him in math; he didn’t know that struggle is normal, a sure sign that learning is happening.

I don’t want to dismiss all of this, but I’ve found a different strategy more helpful. It’s simpler too — which is good, because I don’t do well with complex. I need simplicity in my teaching, as much as possible.

Here’s what I did for my 9th Grader: I told the entire class, “I want you all to ask me questions. Lots of questions. When you’re feeling stuck: ask me for help.”

And, then, when my 9th Grader didn’t ask me questions I walked over to him: “I really want you to ask me some questions if you’re stuck.”

When that didn’t work (“I’m doing fine Mr. P”) I went back to him and I said: “You’re going to start having an easier time with these problems when you start asking me some questions.”

And, finally, when he asked me a question, I answered it as best I could and said, “This was great — please keep asking questions.”

At risk of driving home the point a bit too strongly: I really, really wanted him to ask me questions.

When a student is working on their own — tinkering away, seemingly content — it might not be that they’ve embraced struggle. It might be that they’re embarrassed to ask for help. Kids sometimes end up thinking that you’re supposed to deal with problems on your own, and that in fact dealing with issues on your own is a sign of intelligence and academic worth. It’s certainly what I thought, sitting in the back of Rabbi Weiss’ class or in my professor’s office hours.

It’s the thing I look for, most of all, in evaluating how a student is doing. If they’re asking questions, they expect to learn. If they aren’t, it could very well be that they’ve given up, or are considering it.

I’m not great at classroom culture — kids like me OK, I think — but this is one thing I know that I do. It’s one thing, nothing complicated, but I beg kids to ask me questions. It’s how you grow.

****

On and off for the past seven years, I’ve been trying to learn more math. Not just to solve problems, but to learn a new discipline of math, or to relearn my college material in more depth.

Each summer I sign myself up for a new mathematical project; each year I fail. What I’ve realized, though, is that I can’t do this on my own. I need to ask for help. This summer has been my most exciting summer for learning math, and it’s entirely because I’ve realized that I just need help to learn new stuff.

(Shout out to Anna, Ben, Ben, David, Evelyn and anybody else who has helped me out with math over the past few months! Thank you.)

A lot of teachers — myself included — find it helpful at times to talk about the nature of mathematical work with students. So: mathematicians prove things; mathematicians struggle; mathematicians make mistakes; etc.

The thing is, though, that mathematicians do a lot of things. We get to pick and choose which aspects of mathematical culture we want to promote with kids. Mathematicians prove things, sure, but they also invent discriminatory algorithms. (Put that on a poster!) So we make choices.

It’s a choice to emphasize struggle, mistake-making and individual effort in our classes. What we’re trying to do is emphasize that one’s success in class is in one’s control. And that’s often true, but I don’t think that it mostly happens by trying harder on problems, which is what our growth mindset messages seem to emphasize.

Mathematicians struggle, it’s true, but mathematicians also ask for help. And when it comes to helping kids who have given up, I don’t find it helpful to emphasize the normality of struggle and frustration in math. (We’re all frustrated! might not be the most compelling sales-pitch on behalf of our subject to these students.)

I do find it helpful to beg kids to do this one thing: ask, ask, ask. It’s how you get someone to help you learn. Ask!

****

Barry Mazur (another of my math teachers) helped prove Fermat’s Last Theorem:

KEN RIBET: I saw Barry Mazur on the campus, and I said, “Let’s go for a cup of coffee.” And we sat down for cappuccinos at this cafe, and I looked at Barry and I said, “You know, I’m trying to generalize what I’ve done so that we can prove the full strength of Serre’s epsilon conjecture.” And Barry looked at me and said, “But you’ve done it already. All you have to do is add on some extra gamma zero of m structure and run through your argument, and it still works, and that gives everything you need.” And this had never occurred to me, as simple as it sounds. I looked at Barry, I looked at my cappucino, I looked back at Barry, and I said, “My God. You’re absolutely right.”

Mathematicians ask questions. Sometimes these questions are fun and playful, but other times the questions are more straightforward: can you help me understand this?

I wish that a teacher had told me — no, begged me — to ask questions that weren’t aimed at impressing anybody. Maybe then I could have been better equipped for math in college, and I wouldn’t have had to run away from it after that first taste. There’s so much more that I could have learned during those years if I had been more comfortable seeking clarity from those who had it.

So, put it on a poster: When you feel stuck, ask for help.