What we’re debating when we debate “misconceptions”

Is ‘misconceptions’ a bad word? I’ve had the conversation about misconceptions a number of times, most recently when I wrote this post. Here is a bit from the conclusion:

We see misconceptions in children because it really is true that there’s stuff that they don’t yet know. Noticing this doesn’t have to be an act of violence — in fact, I don’t think that it usually is. Usually it’s like me playing with my son and noticing there’s stuff he doesn’t yet know how to do, even as my mind is blown because oh my god my son is into puzzles! When did our baby turn into a kid?

Is it good pedagogy to ask people who don’t already see their pedagogy as abusive to forswear from using words that they use all the time? Isn’t this exactly the sort of “intellectual violence” that we’re being urged to refrain from? Shouldn’t we start with the way people actually see the world, rather than asking them to use language that is not their own?

That excerpt did not convince anybody at all, but my goal here isn’t to convince. Really all I want to do is bring up something I learned about the constraints of this argument.

There are a couple people I’ve met who have flirted with the idea of cutting out all evaluative language from discussions of teaching, but it’s largely an unsustainable position. You can’t cut out value from teaching, and the thought that you can is a bad mistake. Even if you don’t talk of “misconception” you’re still in need of language to describe thinking that isn’t yet what it could be. Maybe there are no misconceptions, but there is thinking that is e.g. inflexible, procedural, memorized, additive-but-not-yet-multiplicative, trick-reliant, stage one, whatever it is you want to say.

Plus, the math education community very clearly want to be able to understand problematic language and ideas for what they are. We want to be able to call ideas or patterns of thought racist, sexist, colonialist, etc. That’s very different than the “all thinking is just thinking” position.

And so the discussion is only ever about what is particularly harmful (or not) about the term “misconception” and its popular usage. Though people frequently talk about the issues with evaluative language in general when discussing misconceptions, that argument just confuses things. We need to be able to talk about thinking in terms of what it could, even should ideally be.

So there are really just two questions that are relevant for this discussion. Is the term “misconception” particularly harmful, compared to other evaluative language? And even if the term is intrinsically fine, is it used in particularly harmful ways?

I’ve shared my answers, but I’d make the case that those are the right questions.

Quick direct instruction and interesting practice at math camp

Today is my first day of summer break, sort of. I spent the last six weeks working for a wonderful math camp where the teaching is so much fun.

My class is the closest that students come to school content during camp. It’s a fractions class that students get placed into based on an assessment. If the camp thinks that students could use more time working on fractions — that’s who I teach.

That said, the course content is tricky because I don’t want to simply repeat what they’ve seen in school. That would be boring for a lot of kids, and I’m aiming to approach familiar ideas in unfamiliar ways. I’m trying to work on skills, but from interesting perspectives.

Here’s a one-two-three sequence from my fractions course that I think worked particularly well.

First, I ask students to think about visuals. This was a focus of the previous lesson, but I want to make sure every students has it at the front of their minds.

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Source for image is fractiontalks.com

I’m trying to give everyone a chance to figure out what fraction a piece is by multiplying. (“There are four pieces, this is divided into fourths, that would make sixteen in total.”)

I’m teaching this both because it’s a useful bit of visual fluency, but also because I want to use this as part of my direct instruction.

Next is the direct instruction. I’m trying to teach students a mental shortcut: if you’re dividing a fraction by an integer (e.g. 1/2 divided by 10 is 1/20) you can multiply the denominator by the integer because that’s simply making the pieces 10 times smaller. I use visuals to explain this.

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This is a classroom poster with a version of my explanation. Below it is the next mental strategy I teach in the course. 

I immediately give students a few chances to try out this new technique on some mental math problems. (Below is my little cheat sheet — this is what I ask students, but I don’t give them this paper.)

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That’s the basics. But how are we going to practice it further? And how are we going to keep it interesting, and make sure students start using this technique in other contexts?

I then move to the third activity in this lesson, some mobile problems (designed by me on the EDC site). They’ve been carefully designed to give us a chance to use that mental shortcut we’ve just studied.

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A lot of the lessons in my fractions course seem to follow something like this pattern: reminder, quick explanation, interesting practice.

What exactly is it that worked about this? I think this pattern of quick direct instruction followed by interesting practice is a useful one. Of course not every topic is amenable to quick direct instruction (some skills need to be taught in larger chunks) but some are. And after some quick “are we on the same page” questions, it was nice to follow it up with interesting practice. And what made it interesting? I think that it looked different than the direct instruction, but there was still the chance to use it frequently.

This is a way of engineering challenging classroom experiences around stuff that you want to just explicitly teach. I think a lot of people think of these things as incompatible, but they clearly aren’t. At the same time, for a lot of my groups during the year I am trying to make things more accessible — I’m not trying to make it more challenging.

Or maybe I should be? Maybe this pattern of instruction would work just fine in my school-year work. One issue during the year is that I’m much more cautious about whether the practice is actually going to help with the skill. There is a risk to practicing in a different context than instruction. It’s always possible that kids won’t make the connections, that it will be either too hard or students won’t actually practice the thing you thought they would.

So, I’m not sure whether this is something I’ve learned about teaching camp or teaching school or teaching math. Time will tell, I guess.

Q&A on Humanizing Mathematics

Do you like students and mathematics? 

Yes, definitely.

Do you want students to know that their teacher cares for them? Is curious and interested about their passions? 

Yes!

Do you want to help your students understand what is beautiful and vibrant about mathematics as a discipline?

Yes. Of course I balance that with all sorts of other competing desires (they and their parents have goals too) but, in general, yes.

Do you like the phrase “humanizing mathematics”? 

No, I do not.

How about “math as a human endeavor,” with emphasis on “human”?

No, I don’t like that either.

That seems ridiculous. Do you disagree that math is something done by humans

Of course I don’t disagree — who else could be doing the math?

No no, that’s not the point at all. The point is that the things that humans do, we also do in math. Humans play, mathematicians play. Humans love beautiful things — well, so do mathematicians. Truth, Justice, Love: human values, and mathematical values too. 

Hey, did you read Francis Su’s Mathematics for Human Flourishing?

Yeah, I totally did. But I still don’t like this way of talking about math or math teaching.

Are you just being annoying? Are you trolling?

I don’t think so?

So what’s your deal?

Is work human? Is understanding human? Is thinking human? I’m just confused as to what it means for students to think that math is non-human, or not done by humans (with emphasis).

Aren’t they doing mathematics in their classrooms? Aren’t they human?

But the point is students don’t think of mathematics as something they can create. Do students see it as something they can love? Can they seem themselves doing it outside of school? Do they see it as something that was just done by the INVENTORS OF MATHEMATICIANS in some distant past, or could they see themselves and people who look like them doing it?

Is that what it means to be human?

Come on now, of course it is!

No, really. Is the implication here that you’re only human if you are creating mathematics, not if you’re learning someone else’s mathematics?

Is this philosophy? I hate philosophy.

Don’t worry then, this is not philosophy.

Good. Humans are creative and enjoy creating things. I agree that you’re not somehow being not-human if you aren’t being creative, but being creative is to be fully human.

I disagree, and I think that’s a disturbing idea.

What?!

Really! Tell me this: is it a good thing to tell students that if they don’t end up in a creative line of work they somehow aren’t being fully human? That if someone is working as a home aide, an Uber driver, a warehouse worker, that they aren’t fully human?

I don’t think it’s awful to say that those lines of work are less creative and therefore less meaningful. Therefore less of an expression of one’s humanity. We should hope to prepare every student for creative, meaningful and (therefore) more human work.

I’m just not comfortable with it.

Another thing: are we sure that our mathematical values are really universal? I once wrote a piece about how in Ancient Greece there were two vibrant mathematical cultures: one that is all about play, love, the abstract, etc., and the other about algorithms, application, practical knowledge.

When we tell our students that true human flourishing in mathematics is all about the playful, beautiful, loving side of mathematics, do we alienate some students who (legitimately, it seems to me) are interested in using math for the sake of other things? I think we’re taking a narrow slice of the mathematical world and making a claim for universality when we slap it with the “human” label.

Wait, are you sure this is not philosophy?

I promise.

So you don’t like the phrase. Don’t use it — why are you making such a big deal about this?

First, I apologize if this sounds like a big deal. I don’t think it’s a big deal.

But I think this matters. Talk of “math as a human endeavor” is relatively new (to me) but the message behind it is not new. For decades, progressive math educators have been agitating for students to do a wider range of mathematical activities, and to thereby see themselves as creators (discoverers) of mathematics. When you step behind the new way of putting it, how different is this message from the message of: inquiry, discovery, creativity, doing math, math as a verb, and so on?

I don’t think it’s very different at all, this call feels familiar.

And so why not call it using the more familiar language? If it’s a call for doing certain things in class because they’re important, let’s talk about that. It’s clearer.

You’re missing the point, which is this new language of “human endeavor” is a chance to unify a bunch of different activities under a single value: humanity. Yeah, these activities and ideas have frequently gone together in the past, but this is a new way to unify them under a single header.

But what is that value, exactly? The valuing of humanity? What does that mean?

Sigh, we’ve been through this. It’s the idea that we want students to know they can be fully human in math class…

But that’s the thing! We’re taking this controversial package of views about teaching and saying, look, this isn’t radical. It’s just being human. You don’t disagree with being human, do you?

And of course I don’t! I love my students and I want them to be able to be served well by mathematics and school. I don’t know how to convince you of this through words — I really do care, a lot.

It may or may not be a good idea to teach for a growth mindset, to use certain routines, to give kids a chance to explain themselves, to give kids chances to act like mathematicians, to talk about different mathematicians, to share new research, and so on, and so on, and I really do many of these things. But my vision of humanity is big enough to realize that this is not what it takes for something to be a human endeavor.

You’re getting pretty worked up about something you don’t think is a big deal.

I’m sorry, it’s really a bad habit.

Are you going to get in trouble for saying this?

I really hope not.

Well, good luck to you!

Thanks!

How does this end?

I don’t know.

I mean the Q&A.

I know what you meant. I don’t know.

Should it keep going? This is getting weird and cutesy.

Alright, you hang up first.

No, you.

OK, we’ll do it together. One, two,..

Are you still there?

You didn’t hang up!

This is getting silly.

OK, I’ll just stop. Three!

This post is part of the Virtual Conference on Humanizing Mathematics.

My political views

This is Pew’s political typology quiz. Online talk of politics is often just hinted at, I thought it would be fun to get explicit. I took the quiz, and below are my responses.

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Top response, with low confidence. I really have no deep understanding of how national debt works but I feel the government should do more to help needy Americans. I don’t know where the money should come from. Everything has unintended consequences.

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Can I say both? No? I think about the subway system here in NYC. I read a ton of articles about how much of a mess the MTA is, but at the end of the day it’s a pretty remarkable system. And public schooling is part of the government, and thinking of that definitely makes me think that “both” is the right response. I guess the second, with low confidence.

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Is there really no way to skip questions on this quiz? I don’t know anything about this. I chose the top response just to shake things up.

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Top response, high confidence. Put it like this: slavery is the main reason why many black people can’t get ahead these days.

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Top response, medium confidence. I think it really depends on the industry and the kind of regulation, though. I’m glad there’s government regulation of over drugs but I think there must be a better way to do it and might be doing more harm than good at the moment.

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Top response, high confidence.

Brief aside: What does it mean to be ‘accepted by society’? To me that means you should never receive hate or discrimination for being who you are.

My sense is that for other people ‘accepted by society’ means ‘it should become invisible to society, not even abnormal.’ That’s something that I don’t hope for, and mostly this is because I’m a religious Jew who wears a yarmulke in public and I carry my differences with me wherever I go. I don’t think being or feeling different is a bad thing, and I don’t think it should be a bad thing. In fact! In fact the idea that everybody deserves to feel not different is an oppressive majoritarian idea. Moving on.

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Most? Most corporations make a fair and reasonable amount of profit. I think there are some industries that should be taxed more significantly because they do not serve the public interest. I’m mostly thinking of certain parts of the financial industry.

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No clue. It depends on the laws and regulations, I guess. You can’t pass, so I’m going to go with the less-liberal option just to keep things interesting. Certainly it’s sometimes true that stricter environmental laws don’t really help anybody.

By the way, part of my response here is that none of the environmental laws will actually be strong enough to impact climate change. If presented with a government program that could convincingly impact climate change I would absolutely hurt the economy and sacrifice jobs.

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I chose the top one with high confidence, because I’m a good old-fashioned coastal elite and nearly all economists are pro-immigration in the US.

But even if I did think that immigrants were a burden on our country (I guess I’m willing to believe that) I would still be opposed to using cruel methods to expel people from the country. Those two things don’t have to go together.

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I really don’t know what to do with this choice. The first is just mean and wrong. Poor people don’t have it easy even if they get benefits. I also don’t know if the government is the main reason why poor people have hard lives. Also being poor doesn’t mean you have a “hard life.”

I went with the second because I took this to be asking “should poor people get more or less government money on the margins” and I think more.

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I vote the second option, with medium confidence. (Though: what even is fair, man?)

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In a TV interview Malcolm X once said the following:

If you stick a knife in my back nine inches and pull it out six inches, there’s no progress. If you pull it all the way out that’s not progress. Progress is healing the wound that the blow made. And they haven’t even pulled the knife out much less heal the wound. They won’t even admit the knife is there.

I think there’s probably no better year to be a black person in the United States than 2019, and also that’s not saying much. I’m pretty optimistic that things will continue to get better for black people in our country but part of that calculation is taking into consideration continued outrage about police beatings and killings.

Second option, high confidence.

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Oh god, I don’t know. The first, I guess.

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The second option, high confidence. Though it really does depend on what “success” means.

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I think this is another “both” situation for me.

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I really have very few opinions about foreign policy. and all of them are weakly held.

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I like voting and I live in NYC, so I’m a registered Democrat. I don’t strongly identify with the party.

Apparently this all makes me an Opportunity Democrat? That’s what Pew tells me, at least.

 

A weird idea: would you like to go off twitter and turn on blogs with me for a few months?

I miss blogs, and while I’m peace with being very much on Twitter given the state of things, I know that I prefer the quieter, slower version of the internet where I have more control over what I see.

Here’s an idea that I’ve wanted to try for some time: do you want to spend several months blogging with me? The only rule is that you’d have to sign off social media entirely.

Here is what I think it would take for this to really work:

  • auto-posting to your social media feed would be OK
  • comments would have to be open to anyone in the group
  • we would need a group of bloggers with similar(ish) interests
  • we would need a group of at least 10, ideally closer to 20
  • agreeing on the length of the experiment
  • you’re allowed (but not required) to have an anonymous blog

I doubt very much if there are many of us out there who would be eager to turn off Twitter and replace it with blogs. And I very much don’t have the energy to be an agitator for this at the moment, but I think about it frequently.

The opening chapter of the novel RED PLENTY is all about mathematical abstraction

RED PLENTY by Francis Spufford was so good. A great deal of the novel is about the frustrated attempt by Soviet economists and mathematicians to reform the Russian economy.

The book opens on Leonid Vitalevich, about to discover linear programming:

Today he had a request from the Plywood Trust of Leningrad. “Would the comrade professor, etc. etc. grateful for any insight, etc. etc., assurance of cordial greetings, etc. etc.’ It was a work-assignment problem. The Plywood Trust produced umpteen different types of plywood using umpteen different machines, and they wanted to know how to direct their limited stock of raw materials to the different machines so as to get the best use out of it. Leonid Vitalevich had never been to the plywood factory, but he could picture it. It would be like all the other enterprises which had sprung up around the city over the last few years, multiplying like mushrooms after rain, putting chimnies at the end of streets, filing the air with smuts and the river with eddies of chemical dye…

To be honest, he couldn’t quite see what the machines were doing. He had only a vague idea of how plywood was actually manufactured. It somehow involved glue and sawdust, that was all he knew. It didn’t matter: for his purposes , he only needed to think of the machines as abstract propositions, each one effectively an equation in solid form, and immediately he read the letter he understood that the Plywood Trust, in its mathematical innocence, had sent him a classic example of a system of equations that was impossible to solve. There was a reason why factories around the world, capitalist or socialist, didn’t have a handy formula for these situations. It wasn’t just an oversight, something people hadn’t got around to yet. The quick way to deal with the Plywood Trust’s enquiry would have been to write a polite note explaining that the management had just requested the mathematical equivalent of a flying carpet or a genie in a bottle.

But he hadn’t written that note. Instead, casually at first, and then with sudden excitement, with the certainly that the hard light of genesis was shining in his head, brief, inexplicable, not to be resisted or questioned while it lasted, he had started to think. He had thought about ways to distinguish between better answers and worse answers to questions which had no right answer. He had seen a method which could do what the detective work of conventional algebra could not, in situations like the one the Plywood Trust described, and would trick impossibility into disclosing useful knowledge. The method depending on measuring each machine’s output of one plywood in terms of all the other plywoods it could have made. But again, he had no sense of plywood as a scractchy concrete stuff. That had faded into nothing, leaving only the pure pattern of the situation, of all situations in which you had to choose one action over another action. Time passed. The genesis light blinked off. It seemed to be night outside his office window. The grey blur of the winter daylight had vanished. The family would be worrying about him, starting to wonder if he had vanished too. He should go home. But he groped for his pen and began to write, fixing in extended, patient form – as patient as he could manage – what’d come to him first unseparated into stages, still fused into one intricate understanding, as if all its necessary component pieces were faces and angles of one complex polyhedron he’d been permitted to gaze at, while the light lasted; the amazing, ungentle light. He got down the basics, surprised to find as he drove the blue ink onward how rough and incomplete they seemed to be, spelt out, and what a lot of work remained.

It’s the optimism generated by ideas like these that are the true subject of the book, which is the story of the rise and fall of this optimism. The book points out that in a society governed by engineers it was mathematicians and abstract theoreticians that were the main sources of cultural idealism. (In contrast to a place like the US, he says, where lawyers rule the land and writers and artists are the main source of social idealism.)

If you know what happened to the Soviet economy you know the end of this story. The entire book presents itself as a kind of mathematical tragedy, the destruction of the idea of utopian abundance in a planned economy.

An extremely brief summary of what I’ve learned about math anxiety and timed tests over the past few days

  • There are many studies that find math anxiety impacts how well a kid does in math. This includes performance on timed tests.
  • There are pretty much no studies that attempt to find evidence that timed tests contribute to math anxiety. (See this thread for the full conversation.)
  • There are a handful of studies (two that I saw) that surveyed teacher candidates and basically asked them what makes them anxious about math. Along with some other things like word problems, timed tests are implicated.

  • A few people made the argument that “timed tests cause math anxiety” is an untestable hypothesis because it’s unethical. A few researchers chimed in: not untestable in practice, with caveats. (Researchers love caveats.)

  • A few people wanted to know where the evidence was that timed tests don’t cause math anxiety. But that would necessarily involve the same sorts of studies that don’t yet exist; the studies that find a causal connection are precisely the ones that would be useful for showing there is no connection. Anyway, I wasn’t saying either of those things. I don’t have an opinion about the relationship between timed tests and math anxiety.
  • Well, OK, I have a few opinions.
  • A lot of people told me stories about the stress caused by timed tests. I hear you! Research isn’t the only thing that matters. We should keep telling our stories — about our children, experiences as students, what we’ve learned as teachers. True, it would be wise to hold off on the biggest and strongest proclamations (“WE KNOW THAT TIMED TESTS CAUSE MATH ANXIETY”) but just because something hasn’t been validated by research doesn’t mean that it’s not true.
  • But I’m suspicious of much of what YouCubed produces precisely because they present everything as a research result, an absolute law of Brain Science. When you look closer, the research results aren’t there — which isn’t to say that I, Michael Pershan, know that timed tests don’t contribute to math anxiety. Just that there’s a difference between what one thinks and what the research says.

Three Trivial Curiosities

Nobody knows for sure where the term ‘Indian Summer’ comes from.

The word ‘hippocampus’ refers to three different things:

The hyphen distinguishes between the mythological creature and the fish.

In Agatha Christie’s Murder At the Vicarage the term ‘shemozzle’ appears in the following dialogue:

“I expect you’d find her in the studio in the garden — sitting to Lawrence Redding.”
“There’s been quite a shemozzle about him,” said Lettice. “With father, you know. Father’s dreadful.”

‘Shemozzle’ is borrowed from the Yiddish ‘shlimozel,’ usually meaning ‘misfortune.’

Straightedge and Compass

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John Donne’s A Valediction Forbidding Mourning ends with two lovers compared to the arms of a geometric compass over several stanzas:

Our two souls therefore, which are one, 

   Though I must go, endure not yet 

A breach, but an expansion, 

   Like gold to airy thinness beat. 




If they be two, they are two so 

   As stiff twin compasses are two; 

Thy soul, the fixed foot, makes no show 

   To move, but doth, if the other do. 




And though it in the center sit, 

   Yet when the other far doth roam, 

It leans and hearkens after it, 

   And grows erect, as that comes home. 




Such wilt thou be to me, who must, 

   Like th' other foot, obliquely run; 

Thy firmness makes my circle just, 

   And makes me end where I begun.

I came across this in Stephanie Burt’s book Don’t Read Poetry. She writes:

Each lover “leans and hearkens” after the other, as if Donne and his intimate friend, lover, or wife heard each other across the sea. The balanced eight-syllable lines, with their alternating rhymes, depend on each other too. Their closure seems “just” both mathematically and morally; in their mutual response, one or both of the lovers stands up, or becomes “erect” (yes it’s a penis joke).
If you yourself have ever felt unique or confused or confusing to others, especially in matters of the heart; if you have ever felt that your connection to somebody else–whether or not it is romantic, or exclusive, or recognized by the law–requires some explanation of deserves a passionate defense; if you have friends in a stubborn long-distance relationship; if you have been in any such situation, you might see Donne’s elaborate, challenging metaphors not as barriers to sincerity but as ways to achieve it, ways that take advantage of the tools–metaphor, indirection, complex syntax, rhythm–that we can find in poems. You might even, at least if you are looking for them, see in Donne’s great love poems, this one among them, defenses of what we now call queer relationships, relationships not sanctioned by custom or law, relationships most people in your own society can’t quite understand.

That image at the top, by the way, is a set of compasses held by the British Library from Donne’s time, the 16th century.

Encore!

Another school year: done.

I love the dismantling that happens after the kids leave. For months this place has been made just so, and it takes just moments for the entire accumulation process to be set in reverse. Posters are first in the line of fire, but sooner or later everything is headed either to a drawer or the trash. The recycling bins were stuffed with homework this afternoon.

I’ve never been involved with anything theatrical (ok fine once) but I imagine it’s similar to what it feels like to strike the set the day after everything closes. Everything of significance has got to go. It’s all trash now, but just a few hours before the whole thing was whatever it is you call trash’s opposite. This stuff was indisposable by mutual assent. Now: nah.

And what that made me think of was the artifice of this whole enterprise. School is such a weird performance of the strangest kind of pretend — they call me ‘mister’! People sometimes point to the artificiality of schools as a critique: this isn’t what learning looks like in its natural state. And of course that’s exactly right, it’s not. It’s all fake. And that guy up on the stage — did you know he’s not really a wizard?

Teaching is weird, it’s fake. Teaching is not medicine. Medicine is someone is sick and you can help them, the most natural thing in the world. Driving a cab is not weird. People want to get to x, they pay you to take them there. Natural. Getting paid to cook people food is a reasonable transaction, it is not a weird. Lawyering, on the other hand: super weird. And as long as we’re on the topic: basketball player, musician, researcher, middle manager, writer, actor, teacher, weirdos all.

So it’s summer and the show is over and all but, come autumn, let’s do it again! I’ve got a school I love working at and I’m ready to play pretend for another year. Year nine is over.