The Real Satisfaction of Mathematics

“In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others.”

I return often to this quote from Bill Thurston. Here’s what I’m getting out of it today: the mathematical researcher is just looking for something new to share with others, a distinctive piece of the mathematical universe that they can teach the world. The search is for something new, but only because new ideas are something worth sharing.

Mathematical teachers, professionally speaking, are chasing the same goal via the opposite strategy. We don’t bother seek new mathematics, which after all is only one way among many to find something worth sharing. But what we do aim to share, we aim to share more widely than any researcher could.

I wonder whether there’s an alternate history of mathematics out there. Rather than focusing on those who were mathematically innovative, it would be a history of learners and sharers of mathematics par excellence. Would that be the same history? I don’t think it would.

Why should mindset and purpose interventions work equally well?

Screenshot 2018-06-06 at 10.08.41 PM

This figure is from a 2015 paper, Mind-Set Interventions Are a Scalable Treatment for Academic Underachievement, and it comes out of the PERTS group, which generally does great work (as far as I as an outsider can tell).

There’s something fascinating about this study. I think, very quietly, their work represents a conceptual shift in research on mindset. The move is away from mindset and toward interventions as the main object of study.

I have Carol Dweck’s Mindset book, and it’s pretty clear that mindset is a uniquely powerful force in our motivational psychology. There really are two types of people: people who view intelligence as fixed or malleable, and this is a major factor in your motivation and subsequent success in a variety of arenas.

But check out this 2015 paper and check out that figure — there are two interventions that they tested, and only one of them has anything to do with mindset. First, the typical implicit theory of intelligence intervention:

Growth-mind-set interventions convey that intelligence can grow when students work hard on challenging tasks—and thus that struggle is an opportunity for growth, not a sign that a student is incapable of learning.

But then there’s the sense of purpose intervention which has nothing to do with the malleability of anything:

Sense-of-purpose interventions encourage students to reflect on how working hard and learning in school can help them accomplish meaningful goals beyond the self, such as contributing to their community or being examples for other people.

The theory that supports this intervention is entirely unrelated to growth mindset theory. It takes no position on whether someone thinks of human attributes as essentially fixed or malleable. If you thought that growth mindset was a hugely impactful factor that governs motivation, there’s no reason at all why you’d think a sense of purpose intervention works.

(There’s a reeaaaal cool move when the authors call both of these “academic mindset interventions” in that paper.)

And the results of this study found that both of these types of interventions worked about as well as each other. And their benefits didn’t seem to combine, which is also interesting, because why wouldn’t they, if they’re separate motivational concerns?

One possibility: people tend to be demotivated because of theory of intelligence or because of absence of purpose, but not by both. Another possibility is that demotivated people tend to be equally motivated by either intervention.

(I imagine there’s a lot of ways to sort this out with the data they’ve already collected. Which intervention works better for students assessed as having a fixed mindset?)

The second possibility — that both interventions work equally well for at-risk students — would represent a really interesting possibility, which is that the theory behind the mindset intervention doesn’t matter a ton. What if all this under the hood theory doesn’t matter a great deal? What if motivational interventions and their design is the thing worth studying, and the basic theory underlying them doesn’t matter a great deal?

If it’s true, this would make a great deal of sense to me. Dweck’s mindset theory would have not predicted that you could get the same results with an intervention like sense of purpose that uses an entirely different mechanism. (People who underwent the purpose intervention didn’t have changed beliefs about intelligence — they checked.) Mindset was supposed to be the big thing. The fact that it’s being considered as part of a menu of motivational interventions along with purpose seems significant. We’ve already moved most of the way away from seeing it as a uniquely powerful theory for explaining motivation.

And maybe the authors are saying as much in their paper. After all, it seems that now a mindset researcher doesn’t study “mindsets” at all but “mindset interventions,” which is a totally different thing.

I eagerly await something that will help clarify things. Speaking of, does anybody have a copy of this preprint? I wish I’d held on to it before it was taken down. (Update: oh, I think this is it. If so seems like sense of purpose interventions weren’t in play.)

Some questions about the problem of teachers leaving the classroom

Is it actually a problem for kids? Would schools be more effective places if more teachers on the margins of leaving were to stay in the classroom? How do we know? Is there a correlation between ambitious and teaching skill? What is the correlation?

Do master teacher programs improve learning for a district?

How much of the stress in education about people leaving the classroom could be explained by how uniquely meaningful working with children is? After all, going into management involves a change at work across professions. (Sales managers don’t go on sales calls; you leave the regular police work to get a desk job; you still do rounds occasionally but mostly you don’t see patients, etc.) How much of the problem is that there is a huge emotional gap between teaching and higher-paying work that keeps teachers in the classroom, marginally?

Would people be more effective at their administrative jobs if they were partly in the classroom? Would they be more influential?

William Carlos Williams was a doctor by day, poet by night. No one suggests that there should be more doctor/poet jobs. How do we decide what sorts of jobs their ought to be?

I find this so confusing. What questions do you have? Comments are open.

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.

Would you like to read together at The Old Reader?


Do you miss Google Reader?

Way back when Google Reader closed up shop, my roommate and I decided to migrate ourselves over to The Old Reader. Per the title, it’s a near-clone of Google’s late RSS aggregation tool.

Here are things I love about it:

  • It’s simple.
  • Unlike the rest of the internet, it’s not built around images. It’s built around text.
  • Sharing and commenting is exactly as it was with Google Reader.

My (now former) roommate and I have been continuously using The Old Reader to share and discuss stuff we see on the internet for the past however many years. What I’ve lost, however, is the posse of friends with other interests whose enthusiasms I no longer have access to. They aren’t on The Old Reader; they’re on Facebook, Twitter, elsewhere.

Twitter is, like, I hope I don’t say the wrong thing. The Old Reader is, like, let’s think about this together and we both might be wrong. Very much in the spirit of “rough draft thinking.”

Would you like to join? If so, sign up for The Old Reader (free, but I pay for unlimited subscriptions), and we’ll follow each other and read together.

The Power of Words


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


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.


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.


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.


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.

The intellectual work that teachers can do but researchers probably can’t

[I’ve written versions of this post many times before. Here here here. Don’t read those, this version is probably better.]

Tomorrow night, I’m going to teach teachers about teaching. I think a legitimate question is, on what grounds am I claiming to know anything about teaching at all?

To be sure, I am pretty confident that I know something very important about my own teaching — in my school, with my students, in my courses, given my personality, etc. I observe my own classrooms (imperfectly) every day. The cumulative evidence of all that observation makes me pretty (not fully) sure that I’ve figured something out.

But the tricky thing about teaching is that this stuff often doesn’t translate to other situations. Just because something works in my classroom (according to me) doesn’t mean that it’ll work in vastly different contexts. To get really specific about this for a second, I teach students who are among the wealthiest children in America. This reality impacts my school in a bajillion ways. Who says that my dumb ideas about feedback will mean anything to the other teachers in my department. Teachers in other schools, and especially high-poverty schools? Forget about it.

(To be fair to myself for a second: I haven’t only taught in my school.)

The point is that there are obvious reasons to doubt that the things I think I know are really truths of teaching. This is even true if we move past the particular practices that I advocate and get behind the thinking and values that support those practices. I think I have a useful way of thinking about teaching, or I think I’ve identified some value that is important for the student experience. Who says that this is anything but my own thinking?

This is the natural place that research on teaching enters the conversation. Whatever you want to say about research, it’s not about my classroom. In general, it’s about forming generalizations in a way that improves upon (e.g.) my ability to make stuff up about my teaching.*

There are lots of interesting edge cases to consider, but I think the generalization about generalizations stands. Researchers might write cases grounded in particulars or engage in a teaching experiment, but the point of those is to contribute to the formation of generalizations that are broadly useful. 

This is getting pretty abstract so let me just get to the point: could researchers ever respect the generalizations that teachers make about teaching as knowledge that stands on par with their own?

The usual way of talking about teacher/researcher parity is to say that researchers excel at making generalizations, while teachers contribute crucial local knowledge. And it’s totally true that teachers do have local knowledge.

But does this really create parity between teachers and researchers? The whole point of broadly useful knowledge about teaching is that it goes beyond local knowledge — it makes a generalization. If what teachers can contribute is local knowledge, then I think we’re just saying that teachers are at best a source of data to the researcher. The teacher inputs local knowledge, the researcher generates broadly useful generalizations.

It’s true that there’s no reason to inherently value general vs. local knowledge, so in a certain sense there can be parity between teachers/researchers. But at the end of the day, what’s broadly useful are generalizations, and teacher knowledge can’t really compete with what researchers contribute.

Or…can we?

I want to speculate a bit about some different ways of sorting out the relationship between what researchers and teachers can contribute. To start, I want to ignore local knowledge for a second and talk about how teachers contribute to generalizations, i.e. researcher turf.

I have a few ideas here, and they’re very rough, so bear with me.

First, researchers are institutionalized while teachers are necessarily amateurs at producing generalizations. The relationship between teachers/researchers can then be folded into the general relationship between amateurs and experts. And, of course, we need experts. But the ecosystem isn’t healthy if it’s entirely populated by experts.

Amateurs play a lot of important roles, even when it comes to forming generalizations that are broadly useful. Here are a few that I’ve read about (too lazy to cite right now):

  • Amateurs can disrespect the boundaries of fields or sub-professions and put together ideas that from an institutional perspective are incongruous
  • Amateurism is in general lower stakes/lower payout than being an expert. If I’m an amateur and my ideas are wrong or useless, my career isn’t on the line. So there’s a way in which amateurs can attend to riskier ideas, or work on lines of thought that are perceived to be less rich in reward or are in general undervalued.
  • Amateurs play an important role in teaching and spreading expert generalizations, but in doing so amateurs often simplify or otherwise improve the results of experts in significant ways.

But this way of framing things — teachers as amateurs, researchers as experts — doesn’t really leave room for teachers to ever get institutional respect from experts as generators of generalizations about teaching, and two further points on that:

  • this is probably true
  • this is much more exciting to me than institutional respect

It’s not good for my $$$, but I am really quite fine accepting the role of an amateur in all this. It’s exciting to try to smash fields together and to not be beholden to conventional wisdom in the field. I can chase ideas about teaching, throw them out there for others and see what resonates for others. We make up the rules as we go. It’s fun!

That’s the spirit in which I’m going to teach this class tomorrow night. I’ve got this stuff I’ve figured out about teaching. I don’t want to make myself sound like a tin-hat Alex Jones-type, but I do think that what I’ve learned about teaching goes against a certain conventional, institutional, expert way of thinking. And it is the result of mashing up a bunch of things — trial-and-error in the classroom, reading research, experimenting by giving presentations to teachers. And if it’s not broadly useful as a generalization about teaching? Hey, that’s OK too. There’s very little at stake here.

So, I’m not an expert, and neither are you: maybe these ideas are useful to you? Let’s find out. That’s the way I approach this stuff right now, as an amateur.

What is retrieval practice when you’re learning math?

I’ve never really carefully read the retrieval practice literature, but I think it gets confusing when people talk about retrieval practice when talking about math skills, as opposed to mathematical facts.

Here is the description from @poojaagarwal‘s website, committed to promoting retrieval practice among practitioners:

Retrieval practice is a strategy in which calling information to mind subsequently enhances and boosts learning. Deliberately recalling information forces us to pull our knowledge “out” and examine what we know. For instance, I might have thought that I knew who the fourth U.S. President was, but I can’t be sure unless I try to come up with the answer myself (it was James Madison).

But how does this apply to math skills? Can trying a problem (i.e. practicing the skill) ever count as retrieval practice? Does it make sense to use the metaphor of ‘calling information to mind’ to describe what’s going with skills practice?

I think not. But I also am finding retrieval practice useful in my lesson planning. There is a great deal of knowledge that is useful for students to know when they’re learning something new. This sort of knowledge is the sort of thing that I’d like my students to know (i.e. retrieve from memory), more than I’d like them to derive.

Often, at the beginning of class, the first thing I ask my students to do is to remember some facts that they may (or may not yet) know from memory. Some constraints:

  • I don’t ask students to solve a problem and call it retrieval practice — that’s skills practice, not retrieval practice, and tickles other parts of the mind.
  • I only ask students questions that I think they could remember, even if it might be difficult to recall these things. Ideally, these would be things that either students could derive if they can’t recall them.
  • Because stuff from the last few days of class can often get forgotten really quickly, I often use these prompts to strengthen the memory of what we’ve recently done. (The prompt “Summarize what we did yesterday” is surprisingly difficult!)

Here are some prompts I’ve recently used with students:

“Draw a pair of ramps that are pretty close to being of equal steepness.”

“Write an equation of a quadratic, describe what it would look like.”

“What happens when you use the tan button on the calculator? Give some examples.”

“Write several pairs of decimals, and write the number that is between them.”

The truest ‘retrieval practice’ of these is the one about the tan button. Next in line is the one about the equation of the quadratic, since I’m prompting kids to remember what the features of the graph are (though it’s also skills practice). What made me think about these as retrieval practice is that they were all calling back on the previous day’s class.

Here are some purer examples of retrieval practice prompts in math:

“What’s the Pythagorean Theorem?”

(If a specific procedure is supposed to be known for converting a decimal into a fraction:) “How do you convert a decimal into a fraction?”


As I’m messing around in graph theory, I’m noticing that there are a lot of things that would be useful to remember — particular proofs that could serve as paradigms, constraints (in the form of inequalities) on possible planar or non-planar graphs, theorems, specific graphs that are useful examples, etc. If I had a teacher of graph theory, I’d want that teacher to prompt me to remember these things so that I could have more of them available as resources when I’m trying to learn something new or do some creative proving or problem solving.

(I should probably bust out some flash cards at some point…)

As an aside, I think that retrieval practice is sometimes mixed up with spaced practice, but I think these are different things. Spaced practice might be a better fit for what people are describing when they talk about intentionally building time-separated practice of skills into their courses and assignments. I think this requires a different sort of finesse than retrieval practice, though, as the problem with spaced practice is making sure students have something productive to do if they’ve actually forgotten the material.