A weird idea: would you like to go off twitter and turn on blogs with me for a few months?

I miss blogs, and while I’m peace with being very much on Twitter given the state of things, I know that I prefer the quieter, slower version of the internet where I have more control over what I see.

Here’s an idea that I’ve wanted to try for some time: do you want to spend several months blogging with me? The only rule is that you’d have to sign off social media entirely.

Here is what I think it would take for this to really work:

  • auto-posting to your social media feed would be OK
  • comments would have to be open to anyone in the group
  • we would need a group of bloggers with similar(ish) interests
  • we would need a group of at least 10, ideally closer to 20
  • agreeing on the length of the experiment
  • you’re allowed (but not required) to have an anonymous blog

I doubt very much if there are many of us out there who would be eager to turn off Twitter and replace it with blogs. And I very much don’t have the energy to be an agitator for this at the moment, but I think about it frequently.

The opening chapter of the novel RED PLENTY is all about mathematical abstraction

RED PLENTY by Francis Spufford was so good. A great deal of the novel is about the frustrated attempt by Soviet economists and mathematicians to reform the Russian economy.

The book opens on Leonid Vitalevich, about to discover linear programming:

Today he had a request from the Plywood Trust of Leningrad. “Would the comrade professor, etc. etc. grateful for any insight, etc. etc., assurance of cordial greetings, etc. etc.’ It was a work-assignment problem. The Plywood Trust produced umpteen different types of plywood using umpteen different machines, and they wanted to know how to direct their limited stock of raw materials to the different machines so as to get the best use out of it. Leonid Vitalevich had never been to the plywood factory, but he could picture it. It would be like all the other enterprises which had sprung up around the city over the last few years, multiplying like mushrooms after rain, putting chimnies at the end of streets, filing the air with smuts and the river with eddies of chemical dye…

To be honest, he couldn’t quite see what the machines were doing. He had only a vague idea of how plywood was actually manufactured. It somehow involved glue and sawdust, that was all he knew. It didn’t matter: for his purposes , he only needed to think of the machines as abstract propositions, each one effectively an equation in solid form, and immediately he read the letter he understood that the Plywood Trust, in its mathematical innocence, had sent him a classic example of a system of equations that was impossible to solve. There was a reason why factories around the world, capitalist or socialist, didn’t have a handy formula for these situations. It wasn’t just an oversight, something people hadn’t got around to yet. The quick way to deal with the Plywood Trust’s enquiry would have been to write a polite note explaining that the management had just requested the mathematical equivalent of a flying carpet or a genie in a bottle.

But he hadn’t written that note. Instead, casually at first, and then with sudden excitement, with the certainly that the hard light of genesis was shining in his head, brief, inexplicable, not to be resisted or questioned while it lasted, he had started to think. He had thought about ways to distinguish between better answers and worse answers to questions which had no right answer. He had seen a method which could do what the detective work of conventional algebra could not, in situations like the one the Plywood Trust described, and would trick impossibility into disclosing useful knowledge. The method depending on measuring each machine’s output of one plywood in terms of all the other plywoods it could have made. But again, he had no sense of plywood as a scractchy concrete stuff. That had faded into nothing, leaving only the pure pattern of the situation, of all situations in which you had to choose one action over another action. Time passed. The genesis light blinked off. It seemed to be night outside his office window. The grey blur of the winter daylight had vanished. The family would be worrying about him, starting to wonder if he had vanished too. He should go home. But he groped for his pen and began to write, fixing in extended, patient form – as patient as he could manage – what’d come to him first unseparated into stages, still fused into one intricate understanding, as if all its necessary component pieces were faces and angles of one complex polyhedron he’d been permitted to gaze at, while the light lasted; the amazing, ungentle light. He got down the basics, surprised to find as he drove the blue ink onward how rough and incomplete they seemed to be, spelt out, and what a lot of work remained.

It’s the optimism generated by ideas like these that are the true subject of the book, which is the story of the rise and fall of this optimism. The book points out that in a society governed by engineers it was mathematicians and abstract theoreticians that were the main sources of cultural idealism. (In contrast to a place like the US, he says, where lawyers rule the land and writers and artists are the main source of social idealism.)

If you know what happened to the Soviet economy you know the end of this story. The entire book presents itself as a kind of mathematical tragedy, the destruction of the idea of utopian abundance in a planned economy.

An extremely brief summary of what I’ve learned about math anxiety and timed tests over the past few days

  • There are many studies that find math anxiety impacts how well a kid does in math. This includes performance on timed tests.
  • There are pretty much no studies that attempt to find evidence that timed tests contribute to math anxiety. (See this thread for the full conversation.)
  • There are a handful of studies (two that I saw) that surveyed teacher candidates and basically asked them what makes them anxious about math. Along with some other things like word problems, timed tests are implicated.

  • A few people made the argument that “timed tests cause math anxiety” is an untestable hypothesis because it’s unethical. A few researchers chimed in: not untestable in practice, with caveats. (Researchers love caveats.)

  • A few people wanted to know where the evidence was that timed tests don’t cause math anxiety. But that would necessarily involve the same sorts of studies that don’t yet exist; the studies that find a causal connection are precisely the ones that would be useful for showing there is no connection. Anyway, I wasn’t saying either of those things. I don’t have an opinion about the relationship between timed tests and math anxiety.
  • Well, OK, I have a few opinions.
  • A lot of people told me stories about the stress caused by timed tests. I hear you! Research isn’t the only thing that matters. We should keep telling our stories — about our children, experiences as students, what we’ve learned as teachers. True, it would be wise to hold off on the biggest and strongest proclamations (“WE KNOW THAT TIMED TESTS CAUSE MATH ANXIETY”) but just because something hasn’t been validated by research doesn’t mean that it’s not true.
  • But I’m suspicious of much of what YouCubed produces precisely because they present everything as a research result, an absolute law of Brain Science. When you look closer, the research results aren’t there — which isn’t to say that I, Michael Pershan, know that timed tests don’t contribute to math anxiety. Just that there’s a difference between what one thinks and what the research says.

Three Trivial Curiosities

Nobody knows for sure where the term ‘Indian Summer’ comes from.

The word ‘hippocampus’ refers to three different things:

The hyphen distinguishes between the mythological creature and the fish.

In Agatha Christie’s Murder At the Vicarage the term ‘shemozzle’ appears in the following dialogue:

“I expect you’d find her in the studio in the garden — sitting to Lawrence Redding.”
“There’s been quite a shemozzle about him,” said Lettice. “With father, you know. Father’s dreadful.”

‘Shemozzle’ is borrowed from the Yiddish ‘shlimozel,’ usually meaning ‘misfortune.’

Straightedge and Compass

16th-century-compasses-BM-1344603001.jpg

John Donne’s A Valediction Forbidding Mourning ends with two lovers compared to the arms of a geometric compass over several stanzas:

Our two souls therefore, which are one, 

   Though I must go, endure not yet 

A breach, but an expansion, 

   Like gold to airy thinness beat. 




If they be two, they are two so 

   As stiff twin compasses are two; 

Thy soul, the fixed foot, makes no show 

   To move, but doth, if the other do. 




And though it in the center sit, 

   Yet when the other far doth roam, 

It leans and hearkens after it, 

   And grows erect, as that comes home. 




Such wilt thou be to me, who must, 

   Like th' other foot, obliquely run; 

Thy firmness makes my circle just, 

   And makes me end where I begun.

I came across this in Stephanie Burt’s book Don’t Read Poetry. She writes:

Each lover “leans and hearkens” after the other, as if Donne and his intimate friend, lover, or wife heard each other across the sea. The balanced eight-syllable lines, with their alternating rhymes, depend on each other too. Their closure seems “just” both mathematically and morally; in their mutual response, one or both of the lovers stands up, or becomes “erect” (yes it’s a penis joke).
If you yourself have ever felt unique or confused or confusing to others, especially in matters of the heart; if you have ever felt that your connection to somebody else–whether or not it is romantic, or exclusive, or recognized by the law–requires some explanation of deserves a passionate defense; if you have friends in a stubborn long-distance relationship; if you have been in any such situation, you might see Donne’s elaborate, challenging metaphors not as barriers to sincerity but as ways to achieve it, ways that take advantage of the tools–metaphor, indirection, complex syntax, rhythm–that we can find in poems. You might even, at least if you are looking for them, see in Donne’s great love poems, this one among them, defenses of what we now call queer relationships, relationships not sanctioned by custom or law, relationships most people in your own society can’t quite understand.

That image at the top, by the way, is a set of compasses held by the British Library from Donne’s time, the 16th century.

Encore!

Another school year: done.

I love the dismantling that happens after the kids leave. For months this place has been made just so, and it takes just moments for the entire accumulation process to be set in reverse. Posters are first in the line of fire, but sooner or later everything is headed either to a drawer or the trash. The recycling bins were stuffed with homework this afternoon.

I’ve never been involved with anything theatrical (ok fine once) but I imagine it’s similar to what it feels like to strike the set the day after everything closes. Everything of significance has got to go. It’s all trash now, but just a few hours before the whole thing was whatever it is you call trash’s opposite. This stuff was indisposable by mutual assent. Now: nah.

And what that made me think of was the artifice of this whole enterprise. School is such a weird performance of the strangest kind of pretend — they call me ‘mister’! People sometimes point to the artificiality of schools as a critique: this isn’t what learning looks like in its natural state. And of course that’s exactly right, it’s not. It’s all fake. And that guy up on the stage — did you know he’s not really a wizard?

Teaching is weird, it’s fake. Teaching is not medicine. Medicine is someone is sick and you can help them, the most natural thing in the world. Driving a cab is not weird. People want to get to x, they pay you to take them there. Natural. Getting paid to cook people food is a reasonable transaction, it is not a weird. Lawyering, on the other hand: super weird. And as long as we’re on the topic: basketball player, musician, researcher, middle manager, writer, actor, teacher, weirdos all.

So it’s summer and the show is over and all but, come autumn, let’s do it again! I’ve got a school I love working at and I’m ready to play pretend for another year. Year nine is over.

Don’t ask “does it work?”

From Larry Cuban:

Do Core Knowledge Programs Work?

As for many school reforms over the past century, answering the “effectiveness” question–does it work?–is no easy task. The first major issue is answering the question of whether Core Knowledge was fully implemented in classrooms. If not completely implemented, then judging outcomes become suspect. Many of the early studies of Core Knowledge in schools were mixed, some showing higher test scores and some showing no positive effects (see herehere, here, and here). The Core Knowledge Foundation has a list of studies that they assert show positive outcomes. What is so often missing from research on reforms such as Core Knowledge are descriptions of the contextual conditions in which the reform is located and researchers saying clearly: under what conditions does this program prove effective? That is too often missing including the research on Core Knowledge schools.

The primary job of education research shouldn’t be to figure out what works, or to put it another way we shouldn’t expect the have a yes/no answer to that question. How does it work? When does it work? When it didn’t work, why didn’t it work? When it’s not used with fidelity, why wasn’t it used with fidelity? Education is not served well by the way research on program efficacy seems to frequently be done.

Mathematics that makes itself

Can something be true, just because you say it?

One example might be a promise. If you promise somebody that you’ll feed their cats…well, all of the sudden there is a promise there. The act of promising creates a promise. All of the sudden, there it is. It makes itself.

Anyway, maybe mathematics can sometimes pull off a trick like that. In 2003, MacKenzie and Millo argued that this is precisely what happened in financial markets with the Black-Scholes formula, a highly successful mathematical model used to find “correct” prices for a stock option:

Option pricing theory—a “crown jewel” of neoclassical economics—succeeded empirically not because it discovered preexisting price patterns but because markets changed in ways that made its assumptions more accurate and because the theory was used in arbitrage.

In other words, the use of the formula itself made the formula more reliable. It was a self-fulfilling mathematical model, a piece of mathematics that reshaped the world to conform to its assumptions. Wow.

(I found this interesting blog post that dives a bit deeper into the logic of a self-fulfilling equilibrium.)

If this feels eerie, it’s only because we’re forgetting how strange and self-referential the notion of predicting the markets really is: markets are hard to predict because they are predictions. This is a way that finance and economics is fundamentally unlike the natural sciences. In finance there is always the possibility that the scientist will influence the subject.

Black, Scholes, and Merton’s model did not describe an already existing world: when first formulated, its assumptions were quite unrealistic, and empirical prices differed systematically from the model. Gradually, though, the financial markets changed in a way that fitted the model. In part, this was the result of technological improvements to price dissemination and transaction processing. In part, it was the general liberalizing effect of free market economics. In part, however, it was the effect of option pricing theory itself. Pricing models came to shape the very way participants thought and talked about options, in particular via the key, entirely model‐dependent, notion of “implied volatility.” The use of the BSM model in arbitrage—particularly in “spreading”—had the effect of reducing discrepancies between empirical prices and the model, especially in the econometrically crucial matter of the flat‐line relationship between implied volatility and strike price.

To be clear, Ed Throp used option pricing to make a killing before the markets were influenced by Black-Scholes. So it’s not like the formula created its own reality entirely. The claim can only be one of degrees — that the model became more reliable, that the markets grew more like what the model predicted. I am unable to evaluate the evidence on its own and haven’t dived deeper into any of this literature but, huh, it makes you think doesn’t it?

It reminds me of Ben Blum-Smith’s excellent post about voting theory, where he suggests that mathematicians have at times gotten lost in their models and believed in them too strongly, more because of their mathematical properties than for any of their use in application. But what if — only at times, and only by degrees — your mathematical model could be its own fulfillment by changing the world to more closely accord to its predictions? Wouldn’t that be something.

On Benoit Mandelbrot’s “New Methods in Statistical Economics” [Part 1]

I’ve been reading Mandelbrot’s 1963 paper “New Methods in Statistical Economics.” It’s one of his many papers from the 1960s where he argues that people should pay more attention to non-Normal distributions (see my post). This is my attempt to explain the first two sections of his piece.

(My main purpose is to clarify things for myself here, so if you have questions or can explain issues with my exposition I would VERY VERY appreciate hearing from you!)

I come to the paper with two interests. First, Mandelbrot loudly argued that the price fluctuations of many commodities and securities is best thought of as nonGaussian. This is a paper where he makes that argument. That’s cool because finance is cool and important and interesting.

But the other reason is because just a few years after this, Mandelbrot became Mr. Fractal and published “How Long is the Coast of Britain?” This was the piece that popularized the notion of fractal dimension (something Mandelbrot calls a Trojan Horse for the way it represented a safe, neutral topic that served as a vector for his dimensional ideas).

But Mandelbrot also said that parts of this paper — which really seems to have nothing to do with fractals — represent the seeds of his geometric ideas. So, for example, he says this in the appendix of the reprinted edition of this 1963 paper:

The many footnotes in the original, except one, were easily integrated in the text. But Footnote 4 did not fit, and it cried out to be emphasized, because it was an early allusion to the theme of self-similarity that came to dominate my life and led to fractals. This footnote 4 read as follows:

“The various criteria of invariance used by physicists are somewhat different in principle from those I propose in economics. For example, the principle of relativity was not introduced to explain a complicated empirical relation, such as scaling. I am indebted to Harrison White for suggesting that I should stress the nuances between my methods and those of physics.”

So what does that have to do with fractals? It’s a subtle thing! Let’s dive in to the paper to see where he’s coming from. I’ll quote (sometimes at length) and then comment below the text.

Mandelbrot:

The approach I use to study the scaling distribution arose from physics. It occurred to me that, before attempting to explain an empirical regularity, it would be a good idea to make sure that this empirical identity is “robust” enough to be actually observed. In other words, one must first examine carefully the conditions under which empirical observation is actually practiced. The scholar observes in order to describe but the entrepreneur observes in order to act. Both know that most economic quantities can hardly ever be observed directly and are usually altered by manipulations.

Here is a thought experiment involving biology, not physics, but I think it illustrates the point. Suppose that you have a theory about trees: that the distribution of tree heights follows a nice bell curve. 

You are a lazy scientist that doesn’t want to measure anything, let’s say. Anyway, the world is large and you didn’t get funding so you can’t travel. Fortunately, a lot of other people have already done measuring! You discover this by googling.

But suddenly you run into a problem. Sure, some people have directly measured the heights of trees. But other people measured the total height of forests. That stinks! Sure, you can divide the forests by the number of trees to get an average tree height to put into your tree data. But that’s going to be messy.

So it’s a challenge to deal with data coming from many different sources. And if the data needed to explore your theory only could come from measuring every tree (and if you really can’t do that) then you probably should work on a different problem.

In most practical problems, very little can be done about this difficulty, and one must be content with whatever approximation of the desired data is available. But the analytical formulas that express economic relationships cannot generally be expected to remain unaffected when the data are distorted by the transformations to which we shall turn momentarily. As a result, a relationship will be discovered more rapidly, and established with greater precision, if it “happens” to be invariant with respect to certain observational transformations. A relationship that is noninvariant will be discovered later and remain less firmly established. Three transformations are fundamental to varying extents.

So in natural science, OK, it’s a problem. But in economic problems Mandelbrot is saying it is a MAJOR problem.

Suppose you think that stock prices tend to move along randomly, with the changes plucked from a Log-normal distribution, which looks like this:

LogNormal(median=3,stddev=2).png

OK, so you start looking for data. And some of your data comes from daily prices, some from weekly prices, other from yearly price variations. But there’s a problem: there’s no simple way to describe the relationship between the daily and the weekly data. You might want to simply add up a bunch of the daily data to compare it with the weekly data (or to take the weekly data and divide it by 5).

Well, you can’t. The sum of a bunch of log-normal distributions is not another log-normal distribution. So if your theory is true and the log-normal distribution is what guides the stock market’s random price changes, things are weird. If I understand correctly (not at all sure that I do) then there are two reasons why this is weird. First, it means that your attempt to compare different sources of data is likely to be a mess, as there is no easy way to compare and combine the different sources. Second…shouldn’t the daily and weekly prices show the same distribution? Wouldn’t it be weird if daily and weekly prices were governed by different distributions?

There is something exceptionally nice about the idea that small small variation is reproduced at higher scaled. People come in all shapes and sizes; the deepest levels of physical reality are governed by molecules randomly humming and bumping into each other. The idea that these scales are connected — that what we see is the cumulative result of the way the smallest things are — is highly attractive in both nature and economics. I think that this is a potential connection to the idea of fractals and self-similarity.

So, what are the ways that data might need to be stable when it’s transformed? Mandelbrot names three:

  1. Stable after being aggregated
  2. Stable after being mixed
  3. Stable when you only pay attention to the extremes

The first source of stability is the most important. Here’s an example Mandelbrot gives:

The distributions of aggregate incomes are better known than the distributions of each kind of income taken separately.

So if we are interested in the total distribution of income, we might be looking at a collection of different categories of income. Honestly, I can only guess what these categories might be. People sometimes talk about three forms of income (active, portfolio, passive) so maybe he means that? Or maybe he means that you have income distributions for each of the 50 US states, and you want to aggregate that into a national income distribution?

Anyway:

There is actually nothing new in my emphasis on invariance under aggregations. It is indeed well known that the sum of two independent Gaussian variables is itself Gaussian, which helps use Gaussian “error terms” in linear models. However, the common belief that only the Gaussian is invariant under aggregation is correct only if random variables with infinite population moments are excluded, which I shall not do (see Section V). Moreover, the Gaussian distribution is not invariant under our next two observational transformations.

This is indeed a very nice thing about the Gaussian distribution! You add them together, you get another one. Lots of little measurement mistakes add up to a Gaussian distribution of final measurements. (Lots of little differences at the cellular level lead to big differences in people.)

Mandelbrot is saying, you’d want this to remain true in your economic models. It would make the problems tractable to research; the prices and things really ought to add in this way too. And the vast majority of distributions (those “analytic equations” cited above) do NOT have this property. But Mandelbrot is going to argue in favor of an especially favorite family of distributions that do have this additive property — along with the invariance under the other two kinds of transformations.

These are the distributions that are called “Stable Distributions,” and include the Pareto distributions and the Gaussian distributions as members.

Levy_distributionPDF.png

More on the math of transforming various statistical distributions in Part 2.

Two classroom things

#1

Ask anybody: I’m not some sort of god teacher in the classroom. But the other day a teacher complimented me on my classroom management, in particular my ability to hold attention without using my voice. And I was like, yeah! That’s something I do very much on purpose.

I forget if I came up with this on my own or (more likely) picked it up from something I read, but I try very very hard not to interrupt the flow of the class with a redirection to a kid. None of that “I’ll wait” or “settle down” or “I need all eyes up here.” Not if I can help it, at least.

The thought is that the content of the class shalt not be interrupted, and I undermine this premise when I use my voice to interrupt. Plus, it’s just confusing to kids who are following along to hear this other thing, and then to dive back in. PLUS, it directs everyone’s attention to some kid and their shenanigans. PLUS PLUS, it creates a little pocket of energy in the room for a few moments and that can get filled with nonsense or accidental attention-grabbing things.

So I use looks, pauses, taps on the shoulder, etc. I know (I think) a lot of other teachers do this stuff, but I do it too and maybe it’s worth sharing.

#2

Very frequently there are kids that aren’t feeling confident and you can sort of feel them withdraw from the classroom. Very frequently there is an implicit contract these kids have with their teachers: I will be silent and confused, but you will not mind because I will not cause you an ounce of trouble.

Forget that! Cause me trouble.

So there’s this thing that sometimes happens in my classrooms. It’s a bit awkward, but I think it works out OK, most of the time. Which is that sometimes I’ll see a kid who is settling into that withdrawal, and so I’ll cold call on them.

Putting a kid on the spot in front of the class — especially one who is having a rough time in class — is a tricky proposition, and my purpose is not to embarrass a kid for not engaging. It’s just that they’ve got this contract, and they’re trying to figure out if I’m on board or not. But I’m not — I’m going to invite you in to whatever our class is doing, and I’m going to keep on doing that.

“Sarah, do you agree that 40 divided by 4 is 10? Why does that make sense, do you think?”

I’d be curious to hear the recording of that conversation. In that moment I felt myself trying very hard to balance two contradictory stances: a jokey, non-threatening, supportive teacher who is very persistently insisting that you share your thinking.

I also felt myself very much caught up in another contradiction, which was not signaling to Sarah (or the rest of the class) that I was “dumbing things down” for her, while also trying to find a mathematically interesting way for her to engage. This was also a high-wire act, even if it only lasted a few seconds. In these moments one of my moves is to just produce more and more on my own, but to be insistent that the student take on why.

“So from skip-counting we know that ten 4s makes 40. Why would that help us solve 40 divided by 4?”

Sarah ended up saying something nice, which was that 4 x 10 made 40 and that this gave you the answer to the division problem. (Did I mention yet that this is in my 3rd Grade class? I should’ve.)

Having survived this ordeal, I saw her withdrawing again, not looking at the board, not hearing other people, the sorts of things that raise all of my alarms as a teacher.

So I did something unexpected, which is for the next problem I cold-called on Sarah again. And we did that same thing, the same conversation. DEFINITELY a gamble — this is a risky teaching move. Looking at her face, I started getting very nervous…I thought tears were a distinct possibility. But what could I do? I had to keep going. So I turned back to that last problem.

“You had said 4 x 10 makes 40 to help with the last problem. So how can we think about 48 divided by 4?” 

She made it through that conversation. Thank god for her friend, Kya, who gave her a big high-five after that conversation wrapped up. Sarah put on a big smile.

This was a big risk, but at least this time I feel that it paid off. Sarah is pretty good at skip-counting, and she was able to use that to solve some problems on her own without any prompting from me. I was able to give her a high-five also, to compliment her on thinking to do that all on her own.

And though certainly it could have gone bad, the whole point of this to me was that, no, I do not accept the terms of that contract.