A rant about mathematical whimsy and p-adic numbers

I.

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.

II.

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}.

WHIMSY TIME: WHAT IF THE ABSOLUTE VALUE WAS ENTIRELY DIFFERENT???

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).

Tada!

III.

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.

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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”

I.

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.

II.

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.

III.

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.

IV.

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.

V. 

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?

I.

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.

II.

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
PDF: https://nces.ed.gov/pubs2018/2018139.pdf

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.

Here’s what I was trying to say about diversity, Jews, and education

I hate the feeling of realizing that I’ve been unclear, but it happens. Here’s what I was trying to say about diversity, Jews, and education in this post and the comments.

Q: Why should educational spaces worry about being diverse?

A: I think because Black students currently are not served (even abused) by the school system. I think things would be better for these students if there were more Black educators in the system.

Q: Why? 

A: Because, one way or another, a lot of the problem is a systematic lack of empathy for Black students from non-Black people in education.

Q: Wait, but this is just about Black educators. I asked about diversity…

A: I know.

Q: So what about everyone else. If we care about diversity, we should care about other groups being present too.

A: Which other groups?

Q: All other groups.

A: Well I don’t think that we need that kind of diversity. I don’t think we need to work hard to include every group.

Q: But surely we do!

A: I don’t think so. For example, I’m (more or less) an Orthodox Jew, and I definitely don’t think that we need to work hard to increase the presence of Orthodox Jews in educational spaces.

Q: But isn’t that entirely different? It’s not like Orthodox Jews are systematically excluded from conferences or keynotes or panels?

A: Actually, I think that we are. It’s very hard to travel to conferences because of the religious restrictions. These conferences are not designed with Orthodox Jews in mind, and as a result it’s just hard to be at them. That’s one thing that explains why Orthodox Jews aren’t at these things.

Likewise, a lot of Orthodox Jewish teachers prefer to avoid getting involved with public schools, because of the difficulty of managing the holiday calendar and kosher food.

So I think that there is a sense of exclusion, but that this doesn’t matter.

Q: You’re saying that a group can be excluded, but that it shouldn’t bother us?

A: That’s right. Just because you’re excluding a group doesn’t, by itself, matter very much. Nobody should be worried about whether they are excluding Orthodox Jews, redheads, people who don’t like hot weather, etc. Just because a group is excluded doesn’t mean that it’s a problem.

It’s only a problem when it hurts people. As in the case of the exclusion of Black educators, and the way that ends up harming Black children. That’s why it’s important to be inclusive concerning Black educators.

Q: So you don’t want conferences to give you kosher food or whatever?

A: I mean I would gladly accept it, but I don’t think that anyone should worry about this. There is no moral issue with not being maximally inclusive to Orthodox Jews. I mean if you can do it, terrific, why not. But it just doesn’t rise to the level of an issue that anyone should worry about.

Q: Why do you keep talking about Orthodox Jews? Who cares?

A: I want to make talk of “diversity” somewhat problematic. Exclusion of a group is not necessarily something that should trouble us — that’s what I’m arguing. But I don’t want to pick on someone else’s group, so I focused on my own identity.

Diversity claims that all groups should be included, and that it’s a problem when any group is excluded. But, thinking aloud about my own case, I think that’s not true.

So instead I’m saying that it’s a problem only when there are bad results from the exclusion and lack of “diversity.” I believe this is the case for the exclusion of Black educators.

Q: Only Black educators?

A: No, but that’s all I’m going to talk about here.

Q: Why do you care about all this?

A: Because I think “diversity” is a problematic concept. I’ve never heard a fully satisfying answer as to why it’s important for a group to be diverse, in the sense of being varied in general. (This is something that Marian has done a tremendous job pushing at.)

Diversity tries to talk about difference in general, but I think diversity doesn’t get off the ground unless it’s specific. There is a specific case for why we should aim to increase the number of Black educators in education. For some other groups — such as Orthodox Jews –there is no such case, even though they are excluded from various educational spaces.

I think “diversity” as a concept lets us off the hook of talking about these specifics. But I think it’s important to talk about the specifics.

Q: I feel like you could have said that without the confusing stuff about Orthodox Jews.

A: I think you’re right. It’s just how my mind was working and I couldn’t snap out of it.

Q: But isn’t this still different? Orthodox Jews aren’t really targeted by society in the way others groups are.

A: That’s why I brought up the story about some guy on the subway yelling at me yesterday. I get this kind of abuse just from walking around NYC with my yarmulke on. It happens once or twice a year, but it’s enough to keep me on edge a lot of the time. It’s the sort of thing you don’t usually have to worry about if you’re just a person sitting reading your book on the subway.

I’m not saying it’s the same sort of abuse — anti-Semitism is just totally different in its nature than racism — but I think it’s fair to say that we are targeted, just for who we are and how we appear.

And that was the contrast I was trying to set up. You have two groups. Both experience prejudice, both are de facto excluded from many educational spaces, but in one case it matters and in the other case (my own) I strongly believe that it doesn’t. This flies in the face of the rhetoric surrounding diversity — that we need to be inclusive to all groups at all times. I want to make the case that we need to talk about specific groups and the specific costs of excluding them if we want diversity to make any sense at all.

And I suggest this is why there are so few good explanations of why diversity matters.

When and Why Diversity Matters

Whenever I think about diversity — about whether workplaces, conferences, panels, etc. need to be diverse — I think about Orthodox Jews.

Given who I am and where I’m from, it’s natural for me to think this way. The “suburban shtetl” is a cliche but it describes something real. The (Modern) Orthodox world I grew up in really is a culture apart from the rest of the world. Sometimes, when I want to scandalize my students, I let slip a detail or two about what my high school was like. Like that hours went from 7:30 AM (prayer) to 8:30 PM (when evening Talmud study wrapped up), or that we ate three meals a day at school, or that we didn’t even start studying “secular” studies until 3 PM. (Pity my teachers!)

And despite a few boring intra-Jewish technicalities (we’re members of a decidedly non-Orthodox synagogue) it is still my life now. You can tell this as soon as you see me, because I wear my yarmulke everywhere I go.

So when I think about diversity, I use my own situation to help think things through.

Wherever I go in education, I’m usually the only Orthodox Jewish person around. (Assuming that I count as Orthodox, it’s complicated, etc., almost no one cares about this hedge so I’ll stop making it.) This is particularly true when I’ve traveled places for conferences or other edu gatherings, as it’s ridiculously annoying to travel while keeping traditional Jewish dietary practices i.e. kosher. The details truly are boring so suffice it to say that restaurant eating is, in general, out of the question and it’s always a scramble to find food when in e.g. Oklahoma or downtown Denver.

There is a lot of other stuff too that can conflict with conferences. For example the Jewish holidays or fast days, or anything that ever happens on Saturday, etc., etc.

And wherever I go, I also experience a lot of…I’m not exactly sure what to call it. Heckling. Shouting things at me. Stopping me on the street. On the subway today some guy was having a bad day and he walked past me and shouted, “Even the Jewish guy isn’t paying attention! What’s wrong with you? Does God like that?”

Right, right, back to diversity. So the thing is that there are very few Orthodox Jews who are part of these educational conferences or gatherings. And whenever I think about the value of diversity for an educational space I ask myself, is there a diversity problem with Orthodox Jews? Is there an imperative to make sure more Orthodox Jews are part of these spaces? Should more Orthodox Jews be in positions of power and influence? Is it a problem that more Orthodox Jews aren’t serving on panels or keynoting?

I think the answer is absolutely not. No. There is no diversity imperative that is relevant for Orthodox Jews in education.

The thing is that this immediately would contradict a lot of the reasons people give for valuing diversity. So if you say that you value diversity because you want to make sure your conference is open to everybody, then you have to really start worrying about Orthodox Jews. Or if you say that you want to learn from diverse people or diverse cultures, because different people have different perspectives…well, Orthodox Jews have a different perspective. Nothing special about Orthodox Jews as a group here, obviously, but that’s the point.

And another thing: it’s fair to say that Orthodox Jews have been systematically kept from attending these conferences, in the sense that (a) they often happen on Saturdays and (b) in places where it’s hard to get kosher food. So if the reason to value diversity is to undo systematic exclusion, Orthodox Jews should be part of the discussion.

But, again, I don’t really think that’s right.

As a (more-or-less) Orthodox Jew, I encounter a world that was not made for me, or people like me. There are professional opportunities or conferences that I have to pass on — I am excluded from them. And yet I also think that, basically, this holds no moral weight when it comes to diversity.

That’s because there isn’t anything valuable about diversity, in general. There’s always the question of who and why. So let’s get concrete and very specific about this for a second.

There are very many Black children in schools. There are not very many Black teachers in classrooms, compared to the number of Black people in the country. There are good reasons to want to increase the number of Black teachers a Black student encounters. Besides this, Black educators have been less likely to rise to influential positions within education. This is because of plain-old prejudice, geographic segregation, higher levels of poverty, more things too, and further this is a bad situation, because better decisions would be made if the system had marginally more sympathy for Black students.

Contrast this with the case of Orthodox Jews: there are very few Orthodox Jews in the public school system, further they are on the whole not educationally mistreated, there is no particular imperative for increasing the number of Orthodox Jewish educators and there isn’t any reason to think things would be better if more Orthodox Jews had power in education.

My point is this: exclusion is not always a big deal.

I mean, sure, if you can easily not exclude people, go for it. But it’s really not a big deal. Everyone — every conference — excludes people. If your conference is in California, it excludes poor people who live far away. If it’s on a Thursday, it excludes people who hate Thursdays. If it’s anytime or anywhere outside of a major Jewish area on a weekday, it excludes Orthodox Jews. Look, it’s tough. What can you do?

Exclusion only becomes a problem if…it’s a problem. Diversity is only an imperative if there is some imperative reason why you need to include a group of people. There is no such imperative for my own group, but there is for Black educators.

(There is also likely an imperative for the inclusion of other groups too. My point is that it needs to be considered on a group by group basis, that’s the only way the whole thing holds together.)

The whole language of diversity seems to me an elaborate way to beat around the bush. It’s popular precisely because it allows us to talk about race without talking about race, but the whole thing only holds together if we think about racism in frank terms. Whenever I think about my own group, the value of diversity in education just slips away.

Here’s how it is, I think: Orthodox Jews have very little power in education, and that seems about right. Black people should have more power in education. That’s why diversity matters.

Interesting Paragraphs

Having come this far it should be noted that although the quadrivium was officially part of the curriculum on medieval universities it was on the whole rather neglected. When taught the subjects were only taught at a very elementary level, arithmetic based on the primer of Boethius, itself an adaption of Nicomachus, geometry from Euclid but often only Book One and even that only partially, music again based on Boethius and astronomy on the very elementary Sphere of Sacrobosco. Often the mathematics courses were not taught during the normal classes but only on holidays, when there were no normal lectures. At most universities the quadrivium disciplines were not part of the final exams and often a student who had missed a course could get the qualification simple by paying the course fees. Mathematics only became a real part of the of the university curriculum in the sixteenth century through the efforts of Philip Melanchthon for the protestant universities and somewhat later Christoph Clavius for the Catholic ones. England had to wait until the seventeenth century before there were chairs for mathematics at Oxford and Cambridge.

From this, by Thony Christie. I didn’t know about this.

It reminds me of something else that I think about all the time, which is the article from 1920 announcing the creation of NCTM:

About ten years ago an organization known as the National Council of Teachers of English was formed. Today it has a membership of 5,000….

During the same period high school mathematics courses have been assailed on every hand. So-called educational reformers have tinkered with the courses, and they, not knowing the subject and its values, in many cases have thrown out mathematics altogether or made it entirely elective. The individual teachers and local organizations have made a fine defense to be sure, but there could be no concerted action. Finally, the American Mathematical Association of america came to the rescue and appointed a committee to study the situation and to make recommendations. Already two valuable reports have been issued and others are in preparation. The pity of it is that this work, wholly in the realm of the secondary schools, should have to be done by an organization of college teachers. True they have generously called in high school teachers to help, but the fact is that it remained for the college people to initiate the work. They could do it because they possessed a live, vigorous organization.

To help remedy the existing situation the National Council of Teachers of mathematics was organized last spring at the N.E.A meeting held in Cleveland.

Math has a far less secure foothold in the curriculum than you might think. Historically speaking, at least.

Mathematicians Behaving Badly

I write this post in a state of mild panic. New York Public Library has made a margin call on When Genius Failed: The Rise and Fall of Long-Term Capital Management, and I still have a few pages left in the epilogue. Holding on to this asset is costing me, I’m losing equity, etc., etc.

Having now rushed to finish the book over the past few days, I can’t recommend it highly enough.

The story, in short, is this: Long-Term Capital Management was a hedge fund created by “geniuses.” These included two university mathematicians who could make legitimate claims to being key creators of contemporary mathematical finance. (They won Nobel Prizes for this while working for the fund.) It included key disciples of these mathematicians. There were PhDs and Ivy League degrees up the wazoo.

They quickly made a bajillion dollars. They made this money by developing mathematical models that could help them identify pairs of items that were mispriced, relative to each other. They then bet that those prices would converge. They bet this over and over gain in many, many different situations. They always won.

Then, they didn’t. They very quickly lost it all and had to be bailed out by the NY Fed and a consortium of banks.

This was happening in the ’90s. The people involved with LTCM were (if I understand correctly) early pioneers of using quantitative models to capitalize on the mathematical theories of modern finance. I know that this previous sentence is pretty vague, but my point is that this is a story about mathematics and mathematicians.

While reading about these mathematicians, I recognized an arrogance that I’ve experienced in mathematical cultures waaaay too often:

Hilibrand finally offered to dispatch Scholes [Nobel Prize winner — MP] to give the bank a lesson on option pricing, but Pflug was too smart to go head-to-head with the guy who had invented the formula. “You can overintellectualize these Greek letters,” Pflug reflected, referring to the alphas, betas, and gammas in the option trader’s argot. “One Greek word that ought to be in there is hubris.”

What hubris did Pflug divine? The partners were not arrogant in their mannerisms or even in their speech; it was more deep-seated. It was the arrogance of people who had been to Harvard and MIT — of people who really believed that they were more intelligent than others. “Do you know why we make so much money?” Greg Hawkins once asked an old friend from Salomon. “It’s because we’re smarter.” Once the Hawk even tried to lecture a colleague’s wife about molecular biology, her longtime specialty. “You’re full of shit,” she finally replied.

In particular, it seems to me this is partly the arrogance of believing that math is truly about everything. I see this in books that claim math gives you the tools for being logical, in general, or in somehow not being wrong — again, in general.

I also see it in the belief that math can give students general-purpose reasoning skills, ready to be deployed willy-nilly in any context, whatsoever.

It’s something that comes up in the belief that models can be usefully deployed on any dataset, which is one way that LCTM dug its own grave:

Characteristically, Meriwether encouraged the firm to explore new territory. Even at Salomon, the troops had always sought to extend their turf…In retrospect, such moves had been baby steps, not bold new departures. But the partners’ experience — to them, at least — seemed to belie the adage that it is dangerous to try to transport success to unfamiliar ground. Trusting their models, they simply rebooted their computers in virgin terrain.

This is an attitude I even saw in something like Cathy O’Neill’s Weapons of Math Destruction. I don’t have the book in front of me (so apologies if I get this a bit wrong) but there are various moments when she suggests that the same algorithms that are currently deployed oppressively could simply be turned towards more socially useful causes.

I don’t think that, on reflection, O’Neill would deny that deep knowledge of particular context matters, but a reader of her book could be forgiven for thinking that it’s just a matter of where you aim your weapon.

(Hannah Fry’s Hello World does a better job thinking aloud about how deep knowledge of context is necessary for making mathematical models work. So, for example, medical experts might use a trained AI to help identify likely patches of cancerous cells. But the human is the crucial bit here; AI is the tool.)

I’m rambling now, so let’s wrap this up. When Genius Failed: I recommend it. And in general I recommend looking at finance as a major site in the contemporary mathematical landscape. If we want to understand what math is in the world these days, we have to look at finance.

Smoothing the path from simpler to more complex systems of equations

We played with Venn Diagram puzzles, and now my students had a better sense of what it means to be a solution to a system. So, onwards! But I noticed that my 8th Grade curriculum jumped pretty quickly from questions like this

Screen Shot 2018-12-03 at 11.34.53 AM

…to questions like this.

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To bridge that gap, I paired an estimation question and a worked example activity.

The purpose of the estimation question was just to make sure everyone was ready for the example. I wanted to remind kids that each of these equations would produce a line, and I wanted to remind them that their existing techniques wouldn’t work to provide a precise solution.

Screen Shot 2018-12-03 at 11.40.20 AM.png

I was hoping that there would be a nice example activity as part of the Algebra by Example collection. Sadly, nah, it jumps into too much, too quickly for my group. So I was forced to make my own (doc here):

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I find it really necessary to add a “Ready for more?”-type question to these, as some kids really need time to study the example carefully. A good reminder that it all works together — a good extension for fast-finishing students is an accommodation for my slower-finishing students.

I ran the example activity using the routine that I described here.

What makes this bloggable, for me, is that it shows how various things can work together. A worked example, an estimation question, connecting graphical and algebraic representations — there are people who advocate for any one of these, but they only really work for me when carefully aligned.

  • The graph/equation piece is an important bit of non-procedural knowledge.
  • The estimation bit makes sure students understand what a type of problem is asking.
  • The example helps students focus on a useful procedure, and the prompts to explanation make sure they understand it.
  • The “ready for more?” challenges students while giving me more time to help those who need it.

Today was a day when every piece mattered for me.