I flipped around this Illustrative Math lesson — it went pretty well!

Every year, this lesson has given my students trouble:

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Here were the problems I had with the lesson:

  • My students didn’t have strategies for making sense of why the faster bug would go on the bottom. Kids would quickly misidentify which line was which bug, and we’d have to back up and talk about that for a while before doing anything else.
  • So they didn’t really get a chance to engage with the math. And I wasn’t sure what the math exactly was, beyond this little tricky graph that puts time on the y-axis.
  • Also the formatting was tricky, because the ladybug/ant race happens on one page and the graph on the other. You don’t necessarily need cognitive science to tell you that swapping between two very separated images makes learning hard, but it does.

Each of the Illustrative Mathematics lessons has a Summary at the end of the lesson. It’s good, but meant as a reference — it’s not really designed for classroom use.

So here was my redesign idea:

  • Turn the summary into sample student work i.e. a worked example
  • Pair that with analysis prompts and a follow-up question i.e. an example-problem pair
  • Redesign the actual materials so that the graph and the race are next to each other

Here’s what I did:

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Rewriting 8.3.1Rewriting Ladybug and Ant

We used the original “warm up” from the materials. Then shifted into the example-based materials I created. Then the redesigned activity from the lesson itself.

It went well! Here’s how I know that it went well: most kids got to the extension questions, and the students were able to focus their on those.

That seems to me the basic tradeoff. If you leave ideas a bit more implicit, then kids will spend more time uncovering them. That can be good, mathematical thinking, of course. The other choice is to make things more explicit at the outset. Then, maybe you have a better shot of diving into what would otherwise be “advanced” “challenge” problems.

I usually make that second choice, and part of why is because I think good mathematical thinking can happen with the example-based materials I shared. After that warm-up (where I asked kids to notice as many details about the graphs as possible and didn’t really push the “wonder” question) I covered up the “student solution” and showed my students the “problem” I had created. Then, I uncovered the student work. There was a pause — followed by “oh!” and “ah that makes sense.” There’s a little mathematical thrill you get every time you figure something out — a few kids got that when I revealed the work.

Then I asked kids to talk about it with partner, and then to solve a similar problem with neighbors. I listened in on conversations and was able to figure out if kids were understanding the example or not. A few times I inserted myself into conversations to help. And then I led a discussion about the example where kids shared the following ideas:

  • That the ratio between the heights of points on each line that are directly atop each other stays equal.
  • That you could also compare points that are at the same horizontal.
  • That if the axes were swapped the top line would represent the faster racer, because then the top point covers more distance in the same time.

And the kids most eager to share were not the ones who usually solve problems with the most confidence (and therefore least likely to share if all of this had come through problem solving).

And then we did the activity that had given my kids trouble each year of the past, and they were able to be struggle productively i.e. they had the “compare points on the same vertical/horizontal” strategy. And they got to extension problems.

I’m going to keep looking for chances to do make this same trade.

How do you make yourself nicer?

I’m not, like, some expert on philosophy. But I have a degree in it! I took a bunch of ethics classes in college, ethics is great. I love ethics. You can quote me on that.

Here’s a question we never studied and nobody ever asked in an ethics class: how do you become a better person?

We talked (often endlessly) about what it means to be a better person, sure. How do you know if you’re a good person or not, what is a good action, is there really such a thing as good/bad. (Philosophy is all about those italics.)

How a person actually goes about the actual process of going from an OK person to a better person along any dimension is, apparently, not a question of philosophy at all. I can’t name a single philosopher I read who says much about it at all. Maybe that’s a limitation of my reading, maybe not, I don’t know, but doesn’t that say something?

I am also a religious person and was raised religious, but I’m only interested in the secular version of this question. Of the religious texts I’ve studied there are a few that tackle the question of moral self-improvement, but it’s still pretty vague. The most explicit Jewish writing about becoming more ethical is from the mussar movement and they are…intense and weird? Though sometimes insightful?

So let’s say that you woke up one morning and realized you were a jerk. You’d been denying it for years, but it’s time to get honest. You’re mean to people, and for no reason at all. You get angry — not “murder” angry but not-nice angry, and with some frequency. You have relationships, but it’s increasingly clear that these are people who tolerate and recognize your flaws. Because you’re not all bad — but you’re mean, and a bit of a jerk.

You realize this, and make a commitment to do something about it. It’s time, gosh darn it! (You’re working on your language, too.)

What exactly would you do about it?

[Oh, by the way, please nobody mention The Good Place to me. It’s good, I like it.]

Here are the only plausible things I can think of that might help you be nicer.

  1. Keep a diary about your efforts. But I have no idea how that would help. Maybe it would slow down your quick thinking so your slow brain could take control…is that how people get better? Their rationality just lashes out at the emotions, until emotions grow docile and tame? Look, maybe.
  2. Maybe you need a role model? I’ve sometimes felt myself be nicer to others after reading a biography about a really good person. For a few days that person lives in my head, asking questions like “What would Abraham Lincoln do?” or “Can you imagine what Gandhi would think of that?” The problem is that after a few days I go back to normal. But maybe there is a way to keep their presence around? If you wanted to be nicer, maybe spend some time each morning thinking about a really really nice person? I don’t know, I’m making this up.
  3. Put up sticky notes everywhere? But everything eventually loses its force and fades into the background.
  4. Study ethical texts with regularity? Not because they’ll help you manage complicated moral issues (though maybe they will) but because it’s a way to keep things fresh and interesting while still giving you a regular reminder that you’re aiming to be better?

I have no idea. I don’t think I’m a jerk, but I’d like to be a better father/husband/son/friend/teacher/person, sure. I don’t know how much I care — probably not enough. Most of us don’t care, though we probably should, as Eric Schwitzgebel argues.

I’ve looked around, there seems to be practically no empirical work on any of these questions. Moral psychology is the name of the field to look in, but they steer clear of the developmental question. I assume there’s a good reason, it veers too closely to “in this study I started a cult”, or it’s unstudiable for some other reason.

But this seems like a shame for our jerk, and for all of us too. Don’t we want to study the actual process of moral improvement? For are we not all jerks, wishing just a bit that we were a little better?

High, Holy Days: A Playlist

A lot of you have been asking where my Elul/Rosh HaShana/Aseret Yemei Teshuva/Yom Kippur playlist is. “Is it ready yet?” people ask. “You promised.”

Well, it’s not quite done. I’m still tinkering with it. But it’s as ready as it’s ever going to be. Here it is, on Spotify.

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You want me to what? Explain it? That defeats the whole point of a playlist. It would be reductive to go song by song and explain its presence and purpose. I mean, seriously.

Still, there is what to say.

We open as the month of Elul does, with the arrival of the Infanta heralded by the shofar. The Queen is in the field, Elul is in the sky, and the question is what you’re going to do about it. Mad Men, indeed.

It’s time to start asking the big questions. Turn off your mind, relax — but not too much. You need to rethink things, to pay attention. It is not dying, but it’s not not dying either. Because there are certain things to keep in mind when you hear the shofar. Everybody here is a cloud. Don’t forget. If I’m alive, next year.

Sinnerman, Troubleman, Man, it doesn’t matter what you’re called. It matters what you are.

Here’s the deal about “Who By Fire?”: I don’t like any of the versions on Spotify. This is one of those times when a song has a single correct version, and it’s the version with the saxophone.

The whole thing doesn’t work with that Mediterranean guitar intro/accompaniment nearly as well. I think it’s just a fundamental difference between guitar and sax. Sax is a horn, guitar is a string instrument. Your guitar can do a lot of things (e.g. it can weep) but it’s not powered by breath and it never will be. To put it another way, guitar is your siddur but sax is your shofar.

This is the version that should be on the playlist. If it were a mixtape I’d have ripped it, etc.

After this you get to go down to the river for three songs. This is tashlich, but it can also be the mikveh before Yom Kippur if you want to keep things moving roughly chronologically. (I originally tried to make this thing match the chronology more closely. It was a mess.)

From “Get By”:

This morning, I woke up
Feeling brand new and I jumped up
Feeling my highs, and my lows
In my soul, and my goals
Just to stop smokin’, and stop drinkin’
And I’ve been thinkin’ – I’ve got my reasons
Just to get by

And you get one last wordless prayer. Then, the whole thing ends, and we’re on to a new year. Will there be feasting and dancing in Jerusalem this year?

But pat yourself on the back for a second — you have made it to a new year, this year.

May you be written in the book of life. Ketivah v’chatimah tovah.

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.

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.

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.