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.

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.

No, I don’t think the job market should decide whether or not we teach math.

I understand where the confusion is coming from, but I don’t think school should just reflect job market needs.

I also don’t think that science is a more meaningful context for math.

I think pure math can be meaningful, and it’s easier to apply the math to new situations if it’s mastered in a more abstract, contextless form. (Of course, that abstraction needs to be meaningful to students.)

What I do believe is that high school math goes beyond the mathematics that is meaningful and broadly useful for everybody to learn. With the exception of something like exponential functions (which in the US appears in Algebra 1), I think we’re requiring too much. And there are two — well, three — reasons that you’d require more math than everybody needs.

First, because the content you’re requiring is awesome and deep and wonderful.

Second, because it keeps open career pathways for more students who would not elect to take the courses if they were optional.

While math is wonderful, I don’t think high school students are having awesome, deep, and wonderful experiences in their coursework. I don’t actually think this is intrinsic to the content of algebra — I’m a high school math teacher, after all! I think it’s more that we’re requiring too much learning to happen and all previous learning is prerequisite; if you’re behind by 9th Grade, it usually gets worse.

So let’s cut the math! Let’s change the curriculum! Yes, but then will students still have every opportunity open at the end of high school?

Really, the idea that you can keep every option open to every student the way we currently school our children is a myth. But it’s one that is widely held and makes it hard to make any change in the curriculum — learning about rearranging rational expressions is a universal right! You hear these things.

That was when I said to myself, OK, let’s keep those options open for kids. Instead of requiring math, let’s require the subjects that we are supposedly preparing them for careers in. Let them teach the math that you need for those subjects as units in those courses, or even as the first half of the year if that’s what’s best. True, science teachers these days don’t teach math very well…but they also aren’t charged with it. I’m sure they could figure it out as well as high school math teachers can.

Why would this be a better experience for students? Well, maybe we could pare down the curriculum while simultaneously keeping those pathways open. And while math is beautiful and wonderful, so is science. And while students certainly deserve the opportunity to experience the kind of thinking that makes mathematics unique…come on, they’ve already got 8 or 9 years of school to study pure math. At some point you have to stop requiring it!

All these speculative ideas are tough to judge because they’re all fantasy. Which fantasy is more realistic? I don’t know. Maybe it’s better to fantasize that we’ll replace Algebra 2 with Graph Theory, or that we’ll drastically cut the algebra requirements while beefing up support systems so that every student can success in the algebra sequence. Maybe we should require coding or statistics courses.

(Honestly, requiring statistics or coding as math classes might be a fantastic compromise in our current system, a little bit less fantastical. For me the key question is how dependent success in a course is on success in previous coursework. Students, especially in high school, would benefit from more courses that represent something like a fresh start. Otherwise the failures just compound.)

But, probably, none of this will happen, and the reason is because the real role that math is playing in high school isn’t about the importance of the subject as a humanity or its value on the job market — it’s a third reason: sorting students and signalling their academic potential to universities. And that should be a major concern for mathematicians and math educators. While I do love math and would love to share more of what I love with students, if cutting math meant I could lower the stakes for math students in high school in a significant way, I would do it with no hesitation.

Not that the plan I articulating in a few vague sentences is anything like a solution. Other people have other ideas. But I hope we can get on the same page about what the problems are.

YouCubed, Reviewed

This exponents activity is neither original nor at all an interesting version of the idea. It’s no better than what most teachers would make on their own, if they wanted to teach exponent rules inductively.

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Better versions of this are readily available in practically any textbook, but Illustrative Math has a totally free and online unit on exponents that does this activity better. It’s less tedious and repetitive and it asks questions to push students towards generalizations, rather than asking kids to churn out rows and notice the structure at the very end (“discovery”).

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Yes, it’s at a Grade 8 level, but this lesson is pretty much there too. And if you can wait a few months, you’ll have the high school version available too.

“Equity” is dead, long live equity

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By the time organizations — even organizations whose work I really like — start using the language of equity to advertise their work, it’s a sign that we’ve overtaxed the latest bit of edu lingo. “Equity” is at that point in the edu fad life cycle; it’s beginning to mean just about anything.

I don’t know if there’s anything to do about this. I think this is less about education and more about the corporate world — business lingo isn’t much better than edu lingo. People want to signal that they get it, without getting too bogged down in what exactly “getting it” entails.

The thing I try to remind myself is to be specific and to use familiar, boring words whenever possible. In place of stuffing meaning into abstract terms, I try to put it into sentences. And instead of “equity” I try to talk about the particulars: unsafe classrooms, hot schools, bad water, inexperienced teachers, and so on. This is my personal resistance to the educational world’s endless desire for catchy language, as I think it’s really all we’ve got.

Some of my assumptions for communicating about teaching

These all might be wrong, but I think some of them are worth exposing. Maybe you’ll help me see how I’m wrong?

1. When have something I want to say about teaching or learning, there is a temptation to coin a new word that identifies a new concept. I try to avoid this temptation.

Suppose, for example, that I get up at a conference and say “math should be sticky.” There are some risks. First, there’s the risk someone will spend a lot of time puzzling over what I mean by “sticky,” remember the phrase, and have no idea what I meant by it in context. (This happens often — people remember memorable tags but struggle to articulate what they mean.) Probably then I’ll start hearing people say that I believe that you should teach in such-and-such a way because it’s “sticky” when that’s not what I meant. There’s also a risk that my word will have connotations that I didn’t expect. (Oh, you think “sticky” is gross and bad? Oops.)

So as a rule — a writing rule, a speaking rule — I try very hard to only use words that I think everybody pretty much uses in the same way.

This is not easy, because (I associate this thought with Ilana Horn) the meanings people assign to seemingly clear words like “discover” in teaching varies a great deal. I might say “worksheet” and you might imagine “evil packet that kids work on in silence and struggle” and I imagine “a bunch of problems on a page that hit the sweet spot for kids, who are asking questions and talking together about math.”

So it’s not easy, but I do try. It helps to keep an eye out for words (like “worksheet”) that could be misunderstood, and to replace those with context and sentences that make it clearer what’s happening and what I’m imagining.

2. I try to avoid advocating for practices unconditionally. What I mean is that I never say “we should do this in class more!” without suggesting when it might be useful to do that in class. I’m thinking about this right now with worked examples. I think example-based learning is great and cool and fun, but I would never give a talk (I think) calling for greater use of examples in teaching. Instead, though, I would give a talk describing situations that especially call for worked-examples and teaching people how examples can be useful in that context. (Here are two: “examples as feedback” and “examples as models for really complex thinking.”)

Likewise, I try never to talk in general about teaching, or about teaching math in general. I try to stay conditional.

***

These two things, I think, make communication about teaching easier. As a consequence, I think it ensures that nobody thinks that I mean something I don’t mean, and nobody thinks that I have solutions to many of their teaching problems, or a message that would revolutionize math teaching.

And, as a further result of that, what I have to say is less broadly meaningful, polarizing and also less popular. That’s the tradeoff, I think. Clarity for popularity.

Addendum: I have nobody in particular in mind with this post, but it was inspired by a lot of the tweets I saw from the NCTM conference. I’ll say that the “unconditional” thing was inspired by advocacy for a lot of the thinking prompts that don’t call for precise answers — numberless word problems, goal-free problems, estimation problems, notice/wonder, etc.

These are all incredibly useful, but (I think) far more useful when a topic is new to a student. So I think the general direction is that these more open prompts are great ways in, but you sort of want to call for more and more precision in your prompts as the learning progresses.

I was once talking to a friend who felt burned by Estimation180. Why, I asked. Well, she was trying to use it every day to improve her students’ number sense, but it hadn’t worked. She was disillusioned.

I’m not disillusioned. I know that Estimation180 tasks are useful in some situations and less useful in others. I have some thoughts about where and when they’re useful in my teaching. I try to stick to talking about that when I’m talking about teaching and estimation.

The Power of Words

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[highly speculative, probably wrong, maybe patronizing, etc.]

I.

Does it matter whether we describe kids as being “gifted” or as “ready for more”?

Is it just harmless to describe things as crazy, or lame?

Should we refrain from using terms like misconception, mistake, or error when describing a student’s thinking of mathematical work?

Here is a perspective — I don’t think it’s my own — from which the answers to these questions are, yes, it matters, no, it’s not harmless, and of course we should refrain from those terms.

The reason is not so much because of how those words will impact others. Because the words are routine and normal, they wouldn’t really draw attention from other people, unless those people have already been woken up to the true implications of those words. The language isn’t harmless, but it doesn’t necessarily harm anybody.

The first source of harm is what the words you choose do to you. The words you use are sort of like an infection, from this perspective. They dive deep into your psyche and start unconsciously impacting the way you see things. They form the seeds of unconscious bias, silently impacting the way you see the world. If you’re always talking about mistakes and errors that a kid makes, that’s going to shape the grooves of your mind, making it easier to fall into noticing the negatives. Ditto with ableist language. The Sapir-Whorf hypothesis is true, and serious.

The second source of harm is about what you’re not doing when you don’t guard your language, which is you don’t correct your biases. The point is that word choice is a discipline, a moral discipline. Focusing intently on the words you use is an active way of cultivating a thoughtful, less-biased personality.

How? Say that you’re looking at a piece of student work, and you’re trying to describe it. You see something that from a certain perspective is mathematically inaccurate. You reach for a word. You think, hmm, should I call this a ‘mistake’? And you have a moment where you reflect and say, hey, I should not focus on what this kid does not know, I should think about what they can do. Repeat that reflection on word choice a dozen times a day, hundreds of times a month. Reflecting on word choice can change a person.

II.

Spring is here, and it’s math education conference week in the US. And, as usual, I’m feeling a contradictory slush of mixed feelings about sitting it out — relief that I don’t have to go and jealousy towards those who can.

Whether at conferences or following along at home, the main thing I feel is loneliness and alienation from the profession at large. This isn’t even necessarily a criticism of the math education profession. I just feel out of sync with things, that’s all. Things that lots and lots of other people find inspiring or useful, I just don’t find useful.

The flip side is true also, by the way. There are lots of things that I find meaningful and inspiring about teaching that generally don’t get much airtime in these conferences. Meaning, I’m not some sort of humorless scold, incapable of inspiration or fun. I guess there really isn’t any way to prove this and I am doth protesting too much but I think it’s true. Friends, family: a little support here??

I’m not exactly sure how to describe the things at conferences that feel orthogonal to the way I think. I think there are three big things that don’t fit with me naturally:

  1. expressing laws of teaching (e.g. do not teach by telling, ban worksheets and timed tests)
  2. inflating the stakes (e.g. mathematical violence)
  3. managing word choice (e.g. misconception, mistake)

Some time, over the next week, I’m going to see a slide from a conference presentation declaring “Let’s ban talk of misconceptions — it’s mathematical violence!” and I am going to be incredibly triggered and have to remind myself that I’m the one with the problem, not the speaker.

III.

Are words powerful? Yes and no, I guess is what I’d say.

I had been ready to dismiss the Sapir-Whorf hypothesis (that language determines thought), but Douglas Hofstadter gives me pause:

I myself was once most disdainful of this hypothesis, but over time came to realize how deeply human thought — even my own! — is channeled by habit and thus, in the last accounting, by the repertoire of mental chunks (i.e., perceptual attractors) that are available to the thinker. I now think that it is high time for the Sapir-Whorf hypothesis to be reinstated, at least in its milder forms.

(To a lesser extent George Lakoff gives me pause.)

I remain skeptical that self-consciously changing the language someone else uses could impact their thinking. Like, suppose that you actually could actually ban or mandate language. (Not legally, but as a social matter we definitely can do this.) If you really think that language choice unconsciously impacts your own thinking or the thinking of others, then (modulo some unsavory side-effects) you should probably think that a ban or a mandate would reduce harmful biases. People will use better words, the words will have a better impact on their psyche, and as a result their biases will be improved.

But psychological biases are strong, and reinforced by the strong forces of society and culture. Can they really be reduced by the unconscious, subtle grooving of word choice? And without a reflective, conscious effort to change one’s habits? I remain skeptical.

But maybe word choice can improve oneself. If it does, I think it would be in the way I outlined above, as a reflective, moral practice that allows the speaker to frequently reflect on their thoughts and actions.

In other words, I can see how word choice can function as a moral practice that resembles a lot of the religious practices I uphold. Does eating matzah on Passover make you a better person? Of course not, but it could give you a chance to. And it also serves an important social function, centering a community around a shared practice.

IV.

When I was in high school, I remember there was a poster outside my classroom. It exhorted us to guard our tongues and refrain from lashon hara, evil speech, gossip, mean remarks, etc.

There were people I knew who were exceptionally cautious to follow these Jewish laws of speech. These laws were only articulated fairly recently in the Jewish legal game, in 1873, by Rabbi Israel Meir Kagan, but they were entirely widespread when I was in yeshiva. These friends of mine lived in a state of mild panic, in fear they would accidentally utter a bit of unnecessary gossip.

(Complicating things, Kagan also says that accidentally listening to forbidden speech is forbidden, so those who kept these laws carefully had to always be on guard against someone saying something that they shouldn’t hear. To be really safe, you really end up just having fewer conversations.)

There are all sorts of exceptions and guidelines for the rules, but what Kagan changed, for a sliver of the population, was the basic assumption: speech isn’t neutral, and what you express might be forbidden.

Even in high school, this didn’t fit with me. I mean, I certainly felt guilty about how I spoke. But I was a kid that liked to make people laugh. It’s my go-to social coping strategy, and has been for three decades or so. As a student, I tried very hard to stay away from meanness, but, well, let’s just say that Kagan didn’t have a chapter on jokes. The entire enterprise was perilously close to forbidden.

I do believe that I should try not to harm people with my words, and that’s as far as I go with Kagan. I’m not scrupulous about this, just as I’m not scrupulous with my language in general. I try to be scrupulous about other things, and I do think that they “matter” more for me.

But who am I to begrudge anyone their discipline, the thing that stops them in their tracks, helps them to think about themselves and their bad habits? Everyone needs a discipline, and guarding one’s language seems pretty good to me.

What can “I’m not good at this” mean?

Here’s a quick note from the field.

Class today was largely about some fraction arithmetic, and one student was having trouble with it. In class, a couple times, this student said “I’m just not good at this” or “This doesn’t make sense” and “I’m bad at calculating stuff.”

But, all through class, the student stuck with it — I mean that they tried problems on their own and asked questions. And there was definite progress. The student was becoming able to handle this type of question on their own, and starting to make sense of things.

At the end of class I told this student that I thought they were getting the hang of this. I saw a sort of relief pass over this student. Then the student told me:

  • They had a hard time with this topic
  • But this student sort of blames previous teachers
  • This student recently signed up for extra math that is more like our class’
  • The math of our class is the “only math they’re good at”

So what did this student mean when they said “I’m not good at this” etc. in class? In retrospect, it was a statement of fear and anxiety precisely because it was the exception to the rule, at least the rule in our class. This student feels generally competent in our class, so much so that they’ve signed up for similar, optional math next year, and was feeling nervous that this good thing was in danger.

This doesn’t contradict the standard story that people tell about kids saying that they’re “bad at math.” (Especially because this student’s statements weren’t that they were bad at math, in general. It was more specifically about arithmetic and algebra.) But I do think that it illustrates one way that these statements can be an attempt to express something subtler than what they’re usually credited for.

Three things seem important to me about this case:

  1. The student clearly didn’t think that “being bad at arithmetic” was a fixed quality, as this student was simultaneously asking for help. This goes against the “fixed mindset” interpretation, I think.
  2. The student clearly doesn’t, in general, think that they’re bad at all math. These expressions of frustration can be about anxiety that a student’s good thing is under threat. So while it’s never a good thing to hear a kid say that they’re bad at something, it can be a local, specific issue rather than a global one about their status in math class.
  3. The student didn’t think that it was socially OK to be bad at arithmetic or bad at math. This wasn’t a student being proud of being bad at math, and it wasn’t a sign of anything but their own frustration.

I’d be curious to hear other people’s stories about students who make “I’m bad at…” statement in class, to learn more about the different contexts in which kids say this. I’m sure, at the end, we’ll find that they always come at moments of frustration. But I suspect that if a kid is saying that they’re bad at math in class, there’s more to that story, and maybe it’s a sign that (paradoxically) some fundamental things are working for that student in class.

Thinking about Francis Su’s “Mathematics for Human Flourishing”

Francis Su:

What I hope to convince you of today is that the practice of mathematics cultivates virtues that help people flourish.  These virtues serve you well no matter what profession you choose.  And the movement towards virtue happens through basic human desires.

I want to talk about five desires we all have.  The first of these is play.

Yeah, so the first of these fundamental human desires is play, and the whole list looks like this:

  1. Play
  2. Beauty
  3. Truth
  4. Justice
  5. Love

The way Su sees things, if you use math to chase these desires then you end up cultivating virtues. Here are some of those virtues:

  • Hopefulness
  • Perseverance
  • Joy
  • Transcendence
  • Rigorous Thinking
  • Humility
  • Circumspection

(Justice and Love are desires that don’t end in virtues for Su, it seems, as he doesn’t name any virtues in those sections of his talk. Instead, pursuing love is supposed to enable your other pursuits because it results in seeking meaningful human connections: “Love is the greatest human desire.  And to love and be loved is a supreme mark of human flourishing.  For it serves the other desires—play, truth, beauty, and justice—and it is served by them.”)

(So, then what’s the role of Justice? Maybe for Su it’s sort of like a necessary consequence of valuing all these other things. To truly desire all these other things is to desire them for everyone, and that necessarily is the pursuit of justice? I’m just making stuff up here, I’m not sure what Su thinks.)

Anyway, basically none of Su’s talk resonates with me. I don’t mean that I think he’s wrong.* I mean it doesn’t resonate — it doesn’t hit my frequency, make me hum.

* Well, I guess I don’t have any confidence that the study of math itself can impart any of these virtues. I don’t know if he’s claiming that they will, though I think he is.

OK, fine, but then what do I think?

One of my favorite games is “long response to short thing OR short response to long thing.” Let’s play “short response to long thing”:

Michael Pershan’s Version of Math for Human Flourishing:

Five Fundamental Desires:

  1. Understanding
  2. Belonging
  3. Growing
  4. Teaching
  5. I’m not sure but I came up with four

Virtues that one MIGHT develop in math, but I make no commitments about how frequently or reliably even an idealized version of math education could promote these in our students, since virtues are BIG things and math is just ONE thing:

  • Humility
  • uhhh this is also hard
  • I’m totally stuck

Maybe not “virtues,” but ways in which I think math makes my life richer:

  • Math helps me know that, to really understand something or someone, I need to give it my full attention.
  • It gives me an arena in which to grow.
  • It gives me questions to share with others.
  • Is there a name for that thing where you’re walking down the street and you can’t not see parallel lines or tessellations or similar triangles? The math curse? I love that.
  • I love theory. I love that thing where you take a perfectly conventional idea and flesh it out, completely. I recently read that topology came, in part, in response to Cantor’s proof that the line and the plane were equinumerous. But clearly there is a difference between the two — how can we capture that? “How can we capture that?” is one of my favorite questions to ask.
  • I love that math gives me things to help other people think about.

Responding to Criticism from @blaw0013

I wasn’t sure whether to respond to this or not. I want to be the sort of person that gives people stuff to think about, and (just like in the classroom) there’s a point where you have to step back and give people a chance to speak.

But: “deny joy of and access to maths for many”? It’s an interesting criticism, one that I have a lot of thoughts about.

I don’t see micro-skills as denying joy and access to students. And I think it’s partly about seeing joy in maths as something that happens in the abstract versus something that happens in the context of school.

If you think “abstractly” about what joy in math involves, your mind would probably start thinking about the sort of math that is joyous and exciting, the very coolest stuff that math has to offer. You would think of noticing surprising patterns, of unusual theorems, the endorphin release of cracking a puzzle.

Francis Su is the current leading expositor of this side of math, the beautiful, joyous, elegant side:

Pursuing mathematics in this way cultivates the virtues of transcendence and joy.  By joy, I refer to the wonder or awe or delight in the beauty of the created order.  By transcendence, I mean the ability to embrace mystery of it all.  There’s a transcendent joy in experiencing the beauty of mathematics.

If you think abstractly, and ignore the context that students of math actually encounter math in, then you’d look at something like “micro-skills” as just the opposite of all this. And yet I think if you look at the reality of students’ lives (instead of a radical proposal for what students’ lives should be) then I think you can see where joy comes into the picture.

Yesterday I gave students a no-grades quiz in algebra. A student who, I had been told at the start of the year, frequently struggles in math, has been having a lot of success lately. She knew exactly how to handle both of the systems of equations that were on this short quiz, but she got stumped at one of the resulting equations:

-1.7x = 4.3x + 3.6

I didn’t know what to say when she got stuck, exactly, but I was fairly confident that this was an example of a micro-skill that she was missing.

She and I agreed that she’d like me to write a little example on the side of her page, so I wrote this:

-2x = 5x + 7

[I drew some arrows going down from each side labeled “+2x.”]

0 = 7x + 7

My student read the example and then exclaimed (in a way I can only describe as “joyous”), Oh wait, you can make 0 there?!

You can! It’s very cool, and to the mind of a child learning algebra it’s surprising, elegant, beautiful, joyous. This is what I’m talking about — not treating the moments when kids get stuck as “forgettings” or “bugs” in some universal algorithm, and instead thinking of them as opportunities for students to prove mini-theorems, try mini-strategies, learn mini-skills.

And to treat these as moments lacking joy is also to ignore the major impediment to joy in a classroom: feelings of incompetence, worries about status, anxieties about math.

I’m no psychic and my students’ story isn’t mine to tell, but she showed all appearances of being happy and relieved when she understood how to go about solving this problem. How could she experience this, given that she was dealing with the drudgery of a micro-skill? Well, part of it is that (it’s easy to forget) things that are drudgery to teachers are often rich, problematic (in a good way) terrain for students.

But part of it is that these are children in school, surrounded by other children in school. Joy can’t be separated from that social context. Students can’t experience joy if they don’t feel competent, and conversely there is joy in competence. I see this every day.

If, like me, you care both about helping kids experience joy in math and joy from competence in math (hard to separate) then you need to find opportunities in your teaching to do both. The above is how I’m currently thinking, and I’d be interested to read Brian’s take on all this — maybe he and I can find a way to write up a case that illustrates the different choices we’d make in a situation like this. I love the idea of collaborations to resolve differences.