The End.

I’ve done this before and I’m doing it again. This blog is done, over, finished. No more posts.

The reasoning is idiosyncratic, so I won’t bother getting into it. Basically, this part of my writing life feels like it has run its course. I’m eager to try writing in different ways and in different places, and that’s easier for me if I formally close up this shop.

Truth be told, in 2019 it hardly even matters. If you read my writing, mostly you’re finding it through social media anyway. What difference does it make if you find it here or somewhere else, really?

If you want to keep up with what I’m doing, you can always check out my homepage which I periodically update, and I’m still on Twitter.

Alright — it’s been fun! Let’s try something new.


Excerpts from my outbox: Where are all the good readings about feedback?

Question: Do you have any recommendations for good (i.e. accessible, useful, reasonable) readings on feedback?

Answer: First I’ll get the self-recommendations out of the way.

I did a blog series about feedback a few years ago. It’s definitely idiosyncratic, but it might be more readable than some of the academic stuff that I’m about to recommend. Also, there are some pretty detailed case studies as “chapters” in this series, so that might be helpful:

Series On Feedback and Revision

I really do have a lot of writing scattered around about feedback, and I can’t help but recommend more of my own stuff.

In terms of readings…well, I think honestly most stuff is not very good. If you end up reading some of my writing on this you’ll find out why I think that is. In short, I just don’t believe in feedback as a teaching concept. Conceptually it’s far too broad a category. Feedback can only (I think) be successfully studied in lab conditions that artificially restrict interactions between students and learning materials — so we end up hearing that feedback is very important, but really there’s very little useful to say about how or why. In teaching, feedback is constant but not always productive for teaching.

Dylan Wiliam is the standard recommendation, though I feel his notion of “formative assessment” isn’t much better than “feedback” in terms of covering a huge amount of teaching territory. (It does emphasize that student work should be used to impact teaching, but remains an incredibly broad category.) That said, I was definitely influenced by his piece in the NCTM research handbook, which he reworked later into a series of books — this is his Embedded Formative Assessment. I think the most useful things in this book are some of the ideas of specific formative assessment techniques. It’s still pretty jargony. I don’t love it, but it’s probably the best thing out there.

Valerie Shute wrote a long literature review. That’s on the researchy side of things. Again, I think it suffers from all the challenges of trying to organize a vast and unwieldy teaching concept, but I do find it useful as evidence that the research literature is a mess.

Both Wiliam and Shute cite Kluger & DeNisi’s meta-analysis and reviews of the feedback literature, because they did a great job underscoring how much of a mess the literature was/is. They’re the ones that found that roughly 1/3 of studies found that feedback had a negative impact on learning.

Another canonical citation is Ruth Butler’s experiment with gradeless comments — though honestly in this day and age we should be wary of psych studies with eye-popping results that haven’t been carefully replicated, and I don’t know if this has been. That’s just my editorial. Raymond wrote up the paper in a very nice way.

If I’m thin on readings that I haven’t written, that’s largely because I grew very unhappy with the dominant ways of talking about feedback. Once you take the lesson from Wiliam that kids don’t learn from feedback — they learn from thinking about feedback, or using feedback to do something else — then there’s no reason to think that grades or comments are any more “feedback” than your next classroom activity.

The thing that does matter is (a) figuring out what kids know and (b) crafting activities that are responsive to what kids know. If feedback is good for anything, it would have to be good for that. And that really has nothing to do with grades or comments on pages.

But in some ways, it’s hard to put the genie back in the bottle. People already think of feedback as comments/grades. So I don’t really know what to do with most of the existing writing about feedback, because I don’t know if I really want to talk about feedback as it’s usually defined at all.


The Chinese Remainder Theorem by Example

I learned some math today. What happened was during lunch I was trying problems on Alcumus and I came across a problem I absolutely hated. It went like this:

The marching band has more than 100 members but fewer than 200 members. When they line up in rows of 4 there is one extra person; when they line up in rows of 5 there are two extra people; and when they line up in rows of 7 there are three extra people.

How many members are in the marching band?

I sighed loudly to no one in particular and started working it out. I wrote down a lot of numbers that fit the “two extra people when there are rows of 5” constraint (which I know as “2 mod 5”). I wrote 102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 107, 117, 127, 137, 147, 157, 167, 187, and — wait for it — 197. And then I started crossing numbers off the list that didn’t work for the “rows of 7” constraint (the first ten numbers) and then searched one by one to find the last one.

I showed it to my friend who I was sitting next to, he said that there has got to be a better way to do this.

Anyway, I find the solution and punch it into the website. I got it right (woohoo) and start trying to make sense of the solution, which is a truly awful Wall of Text. See?:

Screen Shot 2019-01-23 at 1.17.05 PM.png

OK, but it mentions the Chinese Remainder Theorem, which I know as “that thing from group theory that I didn’t understand.”

I wanted to learn about the Chinese Remainder Theorem so I went to wikipedia, which is useless for this sort of thing, and then I said to my friend (who was still sitting nearby): I’ve got to find an example.

So I search, looking for something I can make sense of without already understanding it, and then I find this. It’s not great but it’s thankfully mostly-free of symbols, so I invest some effort into making sense of it. I work this through, step-by-step on paper.


My friend and I keep talking about it, and I think I figured out how it worked. And then I thought that I’d write the sort of examples that I wish I had been able to find in my initial search. So, while I’m no fan of Search Engine Optimization, I want to help this post make its way into Google Image search so I’m going to (Chinese Remainder Theorem Example) try my best (Chinese Remainder Theorem Example) to (Chinese Remainder Theorem Example) help Google (Chinese Remainder Theorem Example) find it.

This is not a complete explanation — there are definitely some gaps, and I’m sure it could be improved — but hopefully it’ll give you a start.

So here’s the explanation, with examples in the style of my favorite example resource, which is also the style of example I mostly use in my teaching.

This all begins with linear congruences. For example, there are lots of numbers that are 2 mod 5 (i.e. they have a remainder of 2 when you divide them by 5). 12 is congruent to 2 mod 5. So is 22. So is 1002, 10000002, etc., etc., there are a lot such numbers.

There are likewise numbers that have a remainder of 1 when you divide them by 3. Like 4, which is congruent to 1 mod 3. So is 34, 667, 333333333334, etc., etc.

So are there numbers that are both? And how do you find them? The Chinese Remainder Theorem says that there is a process that works for finding numbers like these. Here is an example of that process in action:

chinese remainder theorem examples - google docs-page-001 (2)

There’s probably no way to understand this without working through each step of the example — sorry! — but part of what I think is cool here is that this is a constructive process. What you’re doing is building this number, and the number is pieced together from parts that are designed to work. The reason this has to work comes in two parts:

  1. Ingredient A: Anything times 7 is 0 mod 7. Anything 5 times anything is 0 mod 5. Anything times 13 is 0 mod 13. 156 x 45 x 23 x 8 is 0 mod 45.
  2. Ingredient B: You can multiply numbers and get back to 1. So if you start with 3 mod 4, you can get back to 1 by doing 3 x 3 mod 4. If you start with 2 mod 5, you can get back to 1 by doing 2 x 3 mod 5. Can you always do this? Good question, it had better work for us here though.

These are the two ingredients that we use in that example. We make a number that is a sum of a multiple of 7 and a multiple of 3, which I imagine in this case to be little self-destruct signals waiting to be activated. When you want the number to be 2 mod 3, the multiple of 3 part is activated and it blows up, counting for 0, leaving just the multiple of 7 that you multiply by the thing that sets it back to 1 (thanks Ingredient B) and is then scaled by 2. Tada: 2 mod 3.

And you can do it again to get 1 mod 7. This time the mod 7 part explodes, counting for 0. You’ve multiplied 3 by the number that makes it 1, mod 7, and that’s perfect. You get 1 mod 7.

There are subtleties here. To test yourself, you might want to see if you can spot the mistake in this example:

chinese remainder theorem examples-page-003

One other little twist is that so far, we’ve just discussed how to find any number that fits both conditions. But what if you’re looking for the smallest positive number that fits the bill? In short, just count backwards, though at first you may be surprised by what you have to count backwards by…

Chinese Remainder Theorem Examples-page-002 - Edited.jpg

A beautiful thing is that this process works exactly the same way when there are three conditions instead of two. That is, it almost exactly works the same way. Because we now want to plant even more little self-destruct signals in each part of the number. We want to rig things so that there are three terms, two of which self destruct every time we consider one of those congruence conditions.

Here we go:

chinese remainder theorem examples-page-006

And that’s that! The procedure just keeps going for more and more terms.

One thing I haven’t thought a great deal about yet is where to go from here, mathematically. What would be fun ways to extend or apply this idea? What are some good problems to try next?

I don’t know, I just started understanding this a few hours ago. But here are a few ideas:

  • If you can do it, do it backwards. (I learned this from Kate Nowak.) But it’s not so much fun to start with a number and find some congruency conditions. Like, yes, divide 46 by 5 and you get remainder 1 and divide it by 7 you get 4, big whoop. But what’s kind of cool is that each of these uses of the Chinese Remainder Theorem end up partitioning the number. So can you know how to partition the number without going all the way through? Does each partition work for some initial conditions?
  • The process always works for producing a number that fits the congruency conditions, but it doesn’t always produce the smallest. What’s up with that? How much bigger do they end up? Is there a way to know how many times bigger your number will be than the smallest possible integer?
  • Does any of this relate to a procedure for solving systems of linear equations? Could you solve this graphically? Is there a nice graphical way of representing any of this?

I’ll keep thinking!

Excerpts from my outbox, on Cognitive Load Theory

Thanks for pointing me to that paper. I wasn’t sure if this was worth blogging about, and I don’t even know if this is something that needs clarifying — maybe you already have heard me out on this — but I never really thought I was advocating for CLT to incorporate motivational factors into its theory.

My point [in my essay] was less a practical and more an academic one. That is, I thought that the question as to whether include motivation as part of CLT or not was essentially a choice. That choice has consequences, and there are tradeoffs either way: a more complex theory would encompass more variables but at the risk of becoming less useful. This is a tension I saw CLT grappling with at various points in its development. I didn’t see this as a criticism of CLT, but more of as representing a tension that exists throughout educational research.

So from my point of view, it was never a criticism of Sweller to point out that there had been times when he’d considered incorporating motivation into CLT (along with van Merrienboer).

And the point was never to suggest that CLT should be discredited as less scientific or more subjective on account of its history. Rather, the overall point is that theory-making involves making choices. Those choices can be reasonable or not. To an extent theories are determined by the evidence, but theories are always underdetermined by the evidence.

I think that this is a good thing for people to know about how science works, and I think it’s a useful lens through which to see some debates between more anthropologically inclined educational researchers and those more grounded in psychology. That is, different educational fields have different views on how best to grapple with complexity. Whether those different fields are equally successful in generating insights is an entirely different (but relevant!) question.

Moreover, I am personally agnostic as to whether CLT would be better off with or without motivational variables included as part of their models. I think of that as essentially a question about the viability of competing research programs, and I don’t know whether CLT research would be more or less informative with motivational factors included. I see this paper as making the case one way, but others have made the case the other way. I think the proof is in the pudding, and based on some of his papers from the 1990s, I’d think that Sweller would say the same.

The main regret I have about my initial writing about CLT is that I downplayed the role of worked examples. At the time I hadn’t made worked examples part of my teaching, but since then I certainly have, and it’s greatly improved my teaching. At this point I’ve given talks about constructed example-problem pairs and using worked examples as feedback, and I’ve written about examples a number of different times. I think it’s an incredibly rich aspect of teaching and useful in so many different teaching contexts.

I don’t necessarily regret downplaying worked examples in the essay itself, but certainly in my initial blogging after the essay itself I wish I’d known what I now know.

A very simple game that kids can play to practice multiplication with their flashcards


I have no idea what to name this game. Here are several ideas for names, and the ideas will probably give you a sense of what the game is about: Card Capture; Guess the Back; Collect Points for Knowing Stuff; Actually I Am Terrible At Naming Things, You Name It Whatever You’d Like.

Here’s how to play.

Step One: Take out a bunch of cards from your practice deck that you want to practice. Turn them to the “result” side. Shuffle them up.

Step Two: Deal out a bunch of cards. 4 works well. So does 6. Place them with the products facing up.

Step Three: Take turns. If it’s your turn, you get to pick a card and guess what is on the other side of the card. If you are correct, you get that many points. Meaning, if you guess “6 x 5” and that’s on the other side of your card, congrats, you get 30 points.

Step Four: Etc., that’s the whole game.

The only subtlety is what to do when there are multiple possibilities for what’s on the other side. I think a good way to play this could be that the other person (the non-picker, whose turn it is not) could also issue a guess as to what’s on the other side, and if the picker is wrong maybe they could steal the card? That seems a bit complicated though and what I like about this is how simple it is.

This game only makes sense if you have flashcard decks for every kid in your class, but then again why wouldn’t you? They’re great.

A rant about mathematical whimsy and p-adic numbers


I’m not saying that I’m Mr. Fun or anything, but I like a good time. Well, a good quiet time, but my point is that I’m not exactly a dark and brooding personality — most of the time. I like happiness; I’m certainly not anti-joy. And yet there are some topics that, whenever they come up, make me sound like the biggest, baddest grump on the planet.

For whatever reason, mathematically speaking, it’s whimsy. I hate whimsy. It drives me up the wall.

Maybe you know what I mean. Mathematical whim is when the ultimate justification for some mathematical pursuit is a version of ‘because mathematicians — we’re just some wiiiiild and cra-zay guys!’ 

Mathematical whim is when you invent a new number because you can. It’s when you extend something beyond the point of reason, because why not? It’s when you sort of suggest that once you enter Math Club you’re powerful and in charge and nothing, not even reason itself, can keep you from playing this meaningless, arbitrary game with yourself…

…and there I go again, turning into Oscar the Grouch, but for real this whole thing irrationally bugs me.

The first time mathematical whim really bugged me was when I came to dislike the way math teachers typically introduce imaginary numbers. The common pedagogical move is to point to previously unsolvable equations and suggest that we invent a solution. So, up until now we haven’t had a solution to x^2 = -100? What. if. we. just. made. one. up.

“Oh my god, math teacher, can you actually do that? What is this? I didn’t realize doing math was so cool that you could just do whatever you want whenever you want?”

Five reasons why I dislike this exercise in mathematical whimsy:

  • It’s historically false. We recognized the value of imaginary numbers when they were useful, when treating them as numbers and therefore as things you could add/subtract/multiply/divide was useful because doing that arithmetic helped you find real solutions to polynomial equations.
  • It’s pedagogically false. It gives students no appreciation for why imaginary numbers are at all useful or interesting.
  • It’s sociologically false. Mathematicians don’t play the role in society that this teaching suggests that they do. Mathematicians don’t get NSF funding because they’re the red-nosed court jesters of science. Mathematicians play the role they do in society because, along with the rest of paid science, the nation thinks that math is crucial to the economy and to national defense.
  • It’s psychologically false. Most people pursue things for a reason.
  • It’s personally false. I like things that make sense. I don’t like putting in a lot of work to understand something that we just made up because, why not? It’s not a way of thinking about math that at all connects with who I am and what I value.

In the case of imaginary numbers, this made me upset enough that I spent a lot of time trying to put together materials that expressed a different introductory vision of what these things are. And I think that non-Euclidean geometries are similarly misunderstood and mispresented to students.

And the whole thing makes me feel grumpy, and like I’m no fun at all.


This thought recently came up as I’ve been studying p-adic number theory with a colleague, because this is an area of math that it is very easy to present whimsically.

Here’s a whimsical presentation: Hey, you know how we normally find the distance between two points? You know, directly, like this:

screenshot 2019-01-09 at 8.00.48 pm

Well, what if distance worked differently? What if there are other, alternate ways of measuring the distance between two points? Maybe, like this, so that the distance between A and B is 7:

screenshot 2019-01-09 at 8.02.01 pm

And so that sets us out on a quest to clarify what it really means to be a measure of distance, and to search for alternate ways to satisfy those conditions.

Via wikipedia, here is what those conditions might be:

Screenshot 2019-01-09 at 8.27.37 PM.png

In other words, distance should not be negative, your distance is only 0 to yourself, for distance the order of your points doesn’t matter, and the “direct” route can’t be longer than the “indirect” route that takes a stop at some other point along the way.

Huzzah! We can now explore alternate measures of distance. Whim, engaged.

But…wait. Both the taxicab metric and the conventional (“Euclidean”) metric are defined using the absolute value function.

The taxicab metric measures the distance between two points (x_1, y_1) and (x_2, y_2) as D_t = |x_1 - x_2| + |y_1 - y_2|.

The conventional way of measuring distance pythagorizes the taxicab terms: D = \sqrt{|x_1 - x_2|^2 + |y_1 - y_2|^2}.


So, let’s do it again: let’s figure out the core qualities of what makes something an “absolute value” and then try to find weirdo, alien-planet absolute values that fit the axioms but differ from our own normy absolute value.

Here we go again, from wikipedia, here’s what it means to be an absolute value:

screenshot 2019-01-09 at 8.38.49 pm

And now we tell our students — or, in this case, me, since I’m the student here — that this is what p-adic numbers are. They are the answer to this particular call of whim, a response to the desire to explore alternate worlds and possibilities.

P-adic distance is the distance you get when you’re using these alternate absolute values. The numbers that you create, when using these alternate ways of measuring distance, are analogous to the Real numbers (Real numbers are created with conventional aka boring absolute value), but they are awesomer: they are the p-adic numbers (the p-adic completion of the rationals).



Except, what?!

On what basis can you abstract the properties of what it means to be an absolute value from the one paradigmatic case of the absolute value? Who is to say that the properties that are essential to being an absolute value aren’t just every single one of the properties of the conventional absolute value?

The whole whimsical direction doesn’t make sense.

Now what is true is that there are some VERY cool things that you can do with this wider perspective on absolute values. They really do operate like a whole family of related functions. And there is a terribly stunning theorem that says that there are only two kinds of absolute values: the familiar one, and the p-adic ones.

So all of that is cool, but it still left me unhappy. Where does this idea of what it means to be an absolute value come from? Why would anyone care about this?

And what’s especially frustrating is that I’m just not there yet. I’m in the middle of learning about all these things. I went back to some of the papers introducing p-adic numbers towards the end of the 19th century, and I wasn’t able to connect the dots between their concerns with algebraic numbers and what I’ve been reading in my text. And there’s no reason to think that I will be able to understand any of it until I persist a bit further in my learning.

Which leads me to a troubling thought: what if mathematical whimsy is a useful lie? What if it’s a shot of instant-motivation that’s necessary to get students over that initial hump? What if it’s the sort of thing that makes itself useless in time, a ladder that a successful student will throw away once they’ve reached a higher point of vantage?

Nah, forget that. And forget whimsy too. Someone should be able to tell me why we’d bother creating these alternate p-adic absolute values in a way that makes sense.

So, you want to be a teacher who does other things in education too.

It is frequently mentioned that people in education are not interested in asking classroom teachers to do things in education. Sometimes this is stupid. Sometimes it makes sense.

Here’s the question: suppose you want to try to do it. You want to do valuable work that reaches beyond your classroom, and you want to get paid doing it. What kind of opportunities can you reasonably expect to find, as a classroom teacher?

I feel like this is the sort of thing I would’ve liked reading about when I was starting out. I will share what I have seen.

OK first let’s point out that the question is in a sense ridiculous, because how on Earth can you find the energy to have a second job besides for the classroom work? And I think there’s an obvious thing that people don’t always talk about but let’s say it: some teaching jobs are MUCH more stressful than others. It’s much harder to have a side gig in education when work life is emotionally taxing.

The first school that I taught at was a draining place to be at. Terrible boss, pressure to cover material for state tests, unhappy colleagues, a few lovely people who simply didn’t teach kids stuff, kids who destroyed the classroom computers, walls and windows were trashed, there was once a bottle of pee in the hallway that created a minor kerfuffle, I once caught a teacher trying to steal a laptop, I got a Michael Scott-style hazing on my way out from my principal, I was there three years, it was weird.

During those years I did manage to start writing, speaking, and trying to find myself in the larger world of education. I can’t imagine doing that plus working there plus dealing with little kids at home, which is my life now.

The school that I’m at now is an extremely lovely place. Any classroom work puts you in contact with stressful situations, that’s just the name of the game. But I feel very happy at the place I’m at now (I’ve been there six years) and that has allowed me to handle parenting plus whatever stuff I’ve been able to do on the side.

OK, but what are the actual things that you might be able to do in education from a classroom stance, and even get paid for them? Here are some “yes”s and some “no”s, in the order that I think of them.

Part-time Curriculum Work – Yes. I know a lot of people who have done this in one form or another, and I have done this. Sometimes there are local organizations that want to hire teachers. A lot of the very-online teachers (myself included) have been able to leverage their online profile into work creating lessons or activities or lesson guides.

I wouldn’t know how to actively pursue this work, though. Like I don’t know where you’d apply to get a job writing curriculum. The opportunities I’ve known of seem spotty — never know when it’ll open up — and I’ve only known teachers who have connected with paid curriculum work through their various networks. I’m sure there are other ways to get connected to this kind of work, but I don’t know how.

Research – Mostly no, you can’t be a researcher who gets paid to do research as a classroom teacher. In fact, no, you can’t be a researcher who gets anything published as “research” as a classroom teacher. There are a few exceptions, but they prove the rule. Magdalene Lampert was a researcher already when she was a teacher-researcher. There are PhDs who continue to publish here and there after they return to the classroom. But if you want to be a researcher (especially one who gets paid in any form) there is only one path towards that and it goes through grad school.

Now…I have gotten paid here and there for things that are kinda research. So I once got paid to read a lot of papers and summarize them. (That was a lot of fun. Someone should pay me to do that again.) And another time I got paid to design something — that only worked because the research team I was working with just wanted me to design something and write about it. And I’ve done little paid research-assistant things here and there.

Every opportunity that I’ve had here has been 100% a consequence of my being a very-online math teacher. People knew and liked what I had shared online, and they thought I could do a good job — I am sure in some cases they liked the idea that they could pay me less than they could pay someone with a real background.

PD – Yes. You can do this part-time, and you can also transition into doing this full-time. This is an extremely well-worn pathway that I’ve seen classroom teachers take. One way of getting started here seems to be getting a coaching position in your school or district. That will get you in a lot of PD-offering situations, it seems to me. Or you can become a technology specialist for your school or district. TOSAs seem to often move in this direction.

The other thing is to try to leverage conference appearances into some sort of PD gig. That seems tougher to me at the national level unless you make a big impression in your talks. Maybe if you’re speaking locally, someone will ask you if you’ll run PD for their teachers?

I once taught a mini-course at Math for America, but that was weird because I got involved more through my blogging than any speaking. (I’d do it again, though!)

Keynoting/Paid Speaking Gigs – I know of one or two classroom teachers that get paid speaking gigs. I certainly never have done anything like this.

The tricky thing is that to get a paid keynote or speaking gig, you need to have an independent profile that would make someone aware that your perspective is unique and interesting. That immediately raises the question as to how you’d pull this off, how do you become so visible?

You can try becoming highly-visible online, though I think it’s tougher now if you’re starting out than it was in 2005-2010, that’s just my impression.

You can become a highly-regarded teacher, win a lot of impressive awards, basically just impress lots of people.

Other than that, I don’t know.

Teacher education – I have no idea how, but I know a couple of people who have become adjuncts and taught teacher education classes. This is apparently a thing that is possible to aspire to as a classroom teacher, but I couldn’t tell you how you get such a gig.

Writing books – There are some teachers who have done this, yes. Though I have trouble thinking of many of them off the top of my head. Book-writing in education is more or less like PD, since the books are usually textual PD.

If it’s textual PD that you’d like to write, teaching is a hard way to gather material for a book. It just is. It’s easier to generate material for a PD book when you’re regularly doing PD, or in charge of figuring out what your message is for a group of teachers. If your goal is to write this kind of book, I think the best thing to do is to find chances to do PD with teachers, or try to do some teacher training.

Other than that, there’s no law that says teachers can’t write general interest books. But that’s taking us beyond the world of education and beyond the purview of this post.

Writing articles – I know a couple people who freelance while teaching, and you can do this too. It’s just that writing is hard, is all.

One thing that doesn’t work, I don’t think, is trying to use your classroom perspective as an advantage. Unless you have a kvetchy op-ed about kids these days, I don’t know who will publish what you’re saying purely on the strength of your classroom vantage. If you want to write something about education, you need to borrow some other vantage, or find a way to artfully supplement your own perspective. Reading is good — history, research, math, these are all good.

Tutor – YES, everyone tutors. Well, I don’t any more because I stopped enjoying it and took on extra responsibilities at school, but this must be the #1 thing you’re qualified to do as someone who teaches a lot of people at once — teach one person at a time also.

Social Media Influencer – Yes, ew.

Sell your lessons online – Yes, clearly, but mostly ew.

That’s all I can think of. Have I forgotten anything?

History of Math: “The Two cultures of mathematics in ancient Greece”


The article is “The two cultures of mathematics in ancient Greece,” and it’s by Markus Asper, a contemporary historian. Let’s read it, as it highlights some tensions that are worth poking at.

Quick: first thing that comes to your mind when I say “ancient Greek math”!

What did you come up with? Was it,

  • Euclid?
  • proof?
  • geometry?
  • Pythagoras?
  • irrational numbers?

If you thought of any of these words, congratulations! You’ve located the theoretical branch of Greek math, which is precisely one of the two cultures that Asper points to in this piece.

In fact, his whole point is that it wasn’t just Euclid, Pythagoras, proof, etc.:

The words of, for example, Euclid or Archimedes appear to be of timeless brilliance, their assumptions, methods, and proofs, even after Hilbert, of almost eternal elegance…Recently, however, a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.

The rest of that iceberg wasn’t theoretical math. Everyone else was doing practical math, and much of this article reviews the careful historical clues we have as to what practical math looked like in ancient Greece.

The fact is that we do need clues, because the theoretical camp were the elites. They wrote the books, they dominated our picture of mathematics today. And the theoretical mathematicians that we know and love today had no love for practical mathematics. The mathematics of Euclid was an elite activity that was at a comfortable distance from anything that might be mistaken as professionally useful. Practical mathematics — algorithmic, procedural, professional, useful — was seen as common and not proper.

Nowadays you can enter any educational conference and hear a thousand sessions with something like the following message: “Don’t teach people to just follow procedures! Teach them to prove, to generalize, don’t just tell them how to do it — help them understand it!”

Euclid would have likely agreed, but Euclid wasn’t the only mathematician hanging around Athens. There was a whole collective of lower-class mathematics engaged in practical math, and they were all about following procedures.


Who were these practical mathematicians?

For one, they were pebble calculators. In ancient Greece, pebbles were used to perform difficult calculations (see Netz) and there were likely professional pebble calculators, but we don’t know much about them — like most ancient professionals, they rarely made it into literature. There are hints here and there, and also archaeological evidence of counters, but it’s not much.

(Netz makes the very cool point that coins as a technology are essentially a riff on the counter. A coin has the manipulative properties of counters, but they’re also made of precious heavy metals that are imbued with  intrinsic value.)

Beyond numerical calculators in ancient Greece, there were also geometric measurers and calculators — experts in surveying area and volume.

If you have read anything about ancient Mesopotamian mathematics, what the Greeks did will feel familiar. As it should, because their methods of measuring and calculating area and volume were almost certainly an inheritance from their Near East neighbors.

In other words, ancient Greek math included things like this:

Concerning stones and things needed to build a house, you will measure the volume according to the rules of the geometer as follows: the stone has 5 feet everywhere. Make 5 x 5! It is 25. That is the area of the surface. Make this 5 times concerning the height. It is 125. The stone will have so many feet and is called a cube.

These texts have several features. First, they sound like someone is trying to teach you something — that’s because they are! Second, they sound mired in specifics, and it might be tough to learn general procedures all on your own. That’s also right. This was valuable knowledge, so it was protected. You’d need a teacher to help, an apprentice, to help you really get the hang of things. Asper explains:

Since this kind of knowledge was of economic importance, it was probably not popular or widespread but rather guarded, perhaps by guild-like social structures. Performing as a practical mathematician in one of these arts was a specialized profession. For some of these people, a Greek name has survived: there was a professional group called harpedonaptai (‘rope stretchers’), obviously surveyors operating with ropes for measuring purposes.

I don’t mean to blur distinctions between the past and present, but I do think it’s worth taking a moment to think of all the professions today that might be called practitioners of practical mathematics. We’d be looking for people who aren’t hunters of knowledge, proof, or generalization — we’re looking for people who are calculating, measuring, counting.

Computers have changed things, obviously, but they haven’t done so entirely. I’d say actuaries and accountants probably ought to make the cut. Probably not cashiers, in this day and age. Actual surveyors, definitely. Computer programmers? Arguably.

How is math used in other professions, especially those that don’t require a college degree? I don’t really know. Probably not much more than measuring a cup of this, 5 ml of that, counting 9:30 plus 3 hours. I’m just guessing here — I’d like to know more about practical math used in common professions.


And what about theoretical mathematics? It’s what you’d think. I’ll quote some juicy bits:

Perhaps it would be adequate to think of theoretical mathematics as some form of game rather than something pertaining to a professional occupation, which it as become today, and which practical mathematics has always been. The persons who played this game were certainly at home in the upper circles of Athenian society (evidence collected by Netz 1999a, 279f), similarly to Plato and his followers who eagerly absorbed theoretical mathematics. From the majority’s perspective, comedians could already make fun of mathematicians in 414 BC. They must have felt like an elitist little group among Athenians. For them, theoretical mathematics was probably a status practice, perhaps enforced by the fact that the most common status practice, that is, politics, became quite dangerous for the old upper class at the end of the fifth century. Mathematics was, as philosophy was to become, a status-conscious way to keep one’s head down.

Here is Aristophanes making fun of Meton, an astronomer/geometer:

PISTHETAERUS In the name of the gods, who are you?

METON Who am I? Meton, known throughout Greece and at Colonus.

PISTHETAERUS What are these things?

METON Tools for measuring the air. In truth, the spaces in the air have precisely the form of a furnace. With this bent ruler I draw a line from top to bottom; from one of its points I describe a circle with the compass. Do you understand?

PISTHETAERUS Not the very least.

METON With the straight ruler I set to work to inscribe a square within this circle; in its centre will be the market-place, into which all the straight streets will lead, converging to this centre like a star, which, although only orbicular, sends forth its rays in a straight line from all sides.

PISTHETAERUS I want to give you a proof of my friendship. Use your legs.

METON Why, what have I to fear?

PISTHETAERUS ‘Tis the same here as in Sparta. Strangers are driven away, and blows rain down as thick as hail.

METON Is there sedition in your city?

PISTHETAERUS No, certainly not.

METON What’s wrong then?

PISTHETAERUS We are agreed to sweep all quacks and impostors far from our borders.

METON Then I’m off.

What makes theoretical mathematics distinctive — the proofs, the generality, the passive-aggressive distancing from anything practical — to Asper is a tell:

I suggest that many of the odd features of the theorists, such as expressly refusing to mention any practical applications or any useful effect, worked intentionally as distinctive markers, meant to distinguish the precious game of distinction from sordid occupations that were carried out by people for hire. Plato himself defines, quite polemically, the difference between practical calculations and theoretical mathematics…A late and, almost certainly inauthentic anecdote illustrates my point nicely:

“Someone who had taken up geometry with Euclid, asked after he had understood the first theorem: ‘What is my profit now that I have learned that?’ And Euclid called for his servant and said: ‘Give him a triobolon, since he must always make a profit out of what he learns.”

Theoretical mathematics, the playground of the elite.


Here is the final tally of the distinctive features of practical and theoretical mathematics in ancient Greece:

Greek practical math:

  • was derivative of older traditions that, ultimately, originated in the ancient Near East;
  • solved ‘real-life’ problems;
  • communicated actual procedures in order to convey general methods;
  • used written texts (if at all) as secondary means of knowledge storage and instruction;
  • employed ‘social’ technologies of trust, that is a rhetoric based on institutional authority; for example, the guild’s pristine tradition, the specialist status of its practitioners, and the knowledge’s commonly accepted usefulness;
  • worked within a stable and highly traditional social–that is, institutional–framework;

Greek theoretical mathematics:

  • emerged in sixth- to fifth-century Greece, at least partly from a practical background
  • was a theoreticians’ game with artistic implications, pointedly removed from ‘real life’;
  • communicated general theorems concerning ideal geometrical entities;
  • depending on writing and produced autonomous texts;
  • employed epistemological technologies of trust based on evidence and logic;
  • was not institutionalized, at least no during its formative stages.

So with all this, what exactly are we doing in schools today? What are students learning?

It’s a weird mix, which I suppose reflects the weirdo development that led us here. If you were in a schoolhouse in the 19th century in the US, my sense is that you were mostly studying calculation with a practical eye. You’d learn some basics and then solve problems with an eye towards applications in business or the market. Usually, that was that.

If you were wealthy, you went to secondary school or even university. And there you were certainly studying theoretical mathematics — even ancient Greek theoretical mathematics. But then in the 20th century, we greatly expanded secondary school enrollment. So now there’s all this theoretical math that everyone is studying.

Though the way we instruct all these kids is more in line with the way practical math was taught (“used written texts as secondary means of knowledge storage and instruction”). And now even the practical stuff is not practical. And to bridge the gap between high school and elementary school, we have moved more of the theoretical work into ever earlier grades. Not that this is a bad thing, mind you.

Overall, the distinctions of ancient Greek mathematics don’t map cleanly on to school math today, I don’t think.


I had two purposes in this blog post. The first was to just share and summarize a really interesting article. Go ahead, use it however you’d like.

The second purpose was to help me put my finger on a tension that I keep on seeing.

If I got up in front of a bunch of people in math education in 2019 and told them, “too many people only think of a few famous men when they think of Greek math,” there would be agreeable nodding.

If I then continued and said, “in fact, people in every culture and class were practicing mathematics in ancient Greece in ways that history, with its focus on elites, tends to ignore,” I’d get some some sympathetic social media shares.

If I went further and said, “we really need to respect the many different cultures of mathematics, and not pretend that math always looks the same in every culture,” that’s something I could get some applause with, if I could borrow some keynoter charisma, I think.

And all of those things are true. But if I then said, “many mathematical cultures don’t even care at all about why something is true, and they also don’t care about proofs or making theoretical generalizations — they just want to know how to solve the problem,” well well well right this way Mr. Pershan, yes very interesting alright have a nice day we’ll call you.

And I think that tension is generally present when we correctly expand our notions of who does math and what math looks like. Looking at the ancient Greek example makes me think that we really ought to respect practical mathematics — which by definition is mathematics that is not concerned with the “why.”

And yet there is so often disdain among some teachers for “mindless” calculation or “thoughtless” problem-solving. That seems unfair to me.

There are any number of reasons why we might want to teach students the “why” or proof or generalizations — the theoretical stuff that is woven into our teaching, and is perhaps the profession’s dominant value. But I think it’s wise to ask ourselves if we can respect our students’ own mathematics — even if it’s practical and disinterested in proof or justification, and even as we then make moves to help them see a bit more theoretically. Because theoretical mathematics was born of elitism.

The instinct to respect, the instinct to expand

Meet Ana, a child who weaves baskets. From a young age, Ana had watched her family weave, and slowly she learned to make baskets herself. Eventually, she was taught how to make baskets in several different styles. There are distinct patterns to learn, and she has learned them to the point where she can quickly and efficiently make a variety of designs.

If you ask Ana to explain how to make baskets, she could explain to you how to do it, step by step. First you raise this strand, then put it under, and so on and so on.

If you ask her to explain why this works? Well, that’s a very abstract question. Her knowledge is essentially practical, and it’s worth as much respect as some other mathematician’s theoretical knowledge. There are, after all, multiple ways of knowing, each worthy of respect.

But later, Ana goes to school. Her teacher announces on the first day of class that it’s not enough to solve a problem — you have to be able to explain why you’re doing something. In fact, what he really cares about is the why — that’s what mathematics is about, anyway. I mean, anybody can compute stuff. A computer can compute stuff! If a machine can do it, heck, a machine should do it.

But can’t a machine weave a basket? Does it count as math if you can’t explain why? Does practical math count?

There are two ideas out there in math education-land, and I wonder how they work together:

Idea #1: A lot of kids just get taught how to do stuff without understanding it. That’s not really mathematics.

Idea #2: A lot of activities — like basket weaving, hair braiding, making change — are mathematics, even if they don’t look like school math.

So the only thing valuable about school mathematics is understanding stuff, but outside of school the standards seem different.

Maybe those two ideas are really attempts at expressing two educational instincts. There’s the instinct to respect what a person already knows, and there’s the instinct to expand their horizons. We’re trying to find ways of putting those essentially social instincts into language, but it’s awkward

I feel the tension between those two instincts whenever I’m listening and talking to a student. (Not just with students, though.) People want to be understood — teachers need to understand. But teachers want to help expand what people know, and people want to learn. Those instincts are also desires.

So as instincts or emotions, I understand what’s going on. But as ideas, I’m not exactly sure how to put the pieces together. Why shouldn’t you treat pure computation as beautiful, practical mathematics?

(This post was partly inspired by Two Cultures of Greek Mathematics, a really cool article that I might blog about, especially if somebody wants me to.)

Checking Education Week: How Wealthy are Calculus-Taking Kids?


Last spring, Education Week published an article about calculus. It said:

More than half of students who take calculus in high school come from families with a household income above $100,000 a year, according to a study this month in the Journal for Research in Mathematics Education. By contrast, only 15 percent of middle-income students and 7 percent of those in the poorest 25 percent of families take the course.

But this is a mistake, because the linked-to article says absolutely nothing about this. The JRME piece was a big survey of college freshman in calculus classes. The professors handed out surveys asking about their background and previous courses, and then at the end of the semester attached grades to the surveys and sent them back to the researchers. Then the researchers looked for connections between high school background and the final semester grade.

Here’s the thing though: the survey didn’t even ask about income. And as far as I can tell (correct me if I’m wrong!) the article mentions absolutely nothing resembling the EdWeek stat.

That’s fine. People make mistakes. People should correct mistakes when they find them, and it’s sad that the corrections never get as much attention as the original. Hopefully EdWeek will fix the link, or the writer will issue a correction, and then redesign their website because it’s quite clunky and I can never find my password.


Yes yes yes, mistakes were made, but that’s boring. Interesting: what’s the actual percentage of high school calculus students whose parents make such and such amount?

I couldn’t find a source for the Education Week quote, but I did find a lovely table that answer the exact opposite of our question. It doesn’t tell us what percentage of calculus students are rich, it tells us what percentage of rich students take calculus.

It’s from a report from the NSF that is built on NCES’ High School Longtitudinal Study of 2009, aka not some rando. And no surprises here:

Students in the highest SES quintile were more likely to take advanced mathematics courses than their peers in the middle and lowest SES quintiles …For example, the percentage of students in the highest SES quintile taking calculus or higher was four times higher than the percentage of students in the lowest SES quintile (37% versus 9%) and two times higher than the percentage of students in the middle SES quintiles (37% versus 16%).

Here’s the full table:

Screenshot 2018-12-17 at 7.49.52 PM.png

Just a quick note that the denominator here isn’t high school students, it’s high school completers. So this is the wealthiest 20% of people who finish high school. Out of that 20%, 36.7% take calculus (or a higher level class).

It took me a while of staring at these numbers to make sense of them, as there are a bunch of units involved. Here is the diagram that helped me visualize what’s going on:

Screenshot 2018-12-17 at 7.45.38 PM

But if we want to know what percentage of calculus takers are wealthy, we need to switch units from HS completers to calculus takers. Meaning, this is our new “whole”:

Screenshot 2018-12-17 at 8.02.03 PM

The blue region in the above diagram is 36 out of 36 + 9 + (16 x 3), or 38.7% of people who take (at least) calculus in high school. Repeating those calculations (correct me if I’m wrong!) gives me this:

Wealthiest 20%: 38.7% of calculus (or higher) takers

Middle 60%: 51% of calculus (or higher) takers

Poorest 20%: 9% of calculus (or higher) takers

But this isn’t just the wealthiest 20% of US households — this is the wealthiest 20% of US households of high school completers. So to really answer the question that Education Week raises, we have to connect this with some numbers about income and high school completion…

But I think this is actually OK, because the sample was nationally representative and by 2016 when this table was created, the vast majority of the study group had completed high school.

Screenshot 2018-12-17 at 8.39.18 PM.png

It’s 96% completers of any kind of high school credential, and I’m sure that the last 4% is skewed towards poorer students, but look it’s 96% and we’re not trying to land on the moon here.

(There’s info about the socioeconomic breakdown of high school dropouts for this cohort in the 2012 follow-up, but I couldn’t find a similar table from the 2016 follow-up.)

I’m not sure how to translate socioeconomic status into parental income — I’m pretty sure you can only do this heuristically — but using this table from BLS, what the heck, I’ll give it a shot.

Household income is $108,040 or higher: 38.7% of calculus (or higher) takers

Household income is between $19,868 and $108,040: 51% of calculus (or higher) takers

Household income is less than $19,868: 9% of calculus (or higher) takers

I hope I’ve failed to convince you that this analysis is correct, as I am clearly playing around with things that I don’t totally know how to handle.

But my calculations lead me to believe that it’s not super-duper plausible that 50% of students taking calculus live in households that make more than $100,000 a year. More like 40%ish.