Gender and racial achievement gaps in math

I spent the morning watching this one hour lecture by David Kung, who has been studying the gender and racial achievement gaps in mathematics. Interesting stuff, with historical perspective – math has a sad history – and a call for the end to passive lecturing and much more:

Watch it if you have time. You can skip to 7:20 to start.

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Greek debt and German banks

Are you fascinated by the “debt as moral weight” arguments you see being tossed around and viciously debated over in Germany and Greece nowadays? It seems like the moral debate has superseded the economic reality of the situation. Even the IMF has declared the current Greek deal untenable, but that hasn’t seemed to interfere with the actual negotiations.

What gives? Many point to history to explain this. Besides the whole Nazi thing, or maybe exactly because of it, the Greeks keep reminding the Germans that they (and others) forgave half of existing German debt after World War II, with the1953 London Debt Agreement. The Germans have responded vehemently that such ancient history is irrelevant, and that the Greeks are a bunch of lazy olive-eating tax avoiders. It’s a dirty fight, and getting dirtier every week.

I maintain we don’t have to examine the history of 60 years ago to understand at least some of the moral anxiety. Instead we should look a mere 7 years ago, at the enormous German bailout of their own banks, which had invested quite recklessly in all sorts of the most risky financial instruments and, most relevantly, Greek bonds.

Start with the basic facts. German and French banks invested very heavily in Greek bonds, partly because they were allowed by European Basel “risk regulation” laws to set the risk of those Greek bonds at zero, and partly because they were just investing in anything and everything with a relatively high yield. Since Greek bonds were at a higher yield than other government bonds that maybe deserved the “zero risk” designation more, they naturally bought an asston of those.

[Side note: whenever there’s a market with a spectrum of products, the ones with the biggest yield for a given risk profile will be snatched up the fastest, because people want to maximize profits. We’ve seen that this almost always is a bad thing and creates bubbles very quickly. But it’s also the reason people are constantly inventing new products that hide risk. In this case they didn’t need to “invent” anything, because it was a political decision to designate Greek bonds at zero risk.]

There are two ways to look at this story from a morality standpoint. One is that, no matter who owns this debt now, the Greek government is on the hook for borrowing it and needs to figure out how to pay it back. From this point of view it was a mistake of the Greeks to issue too much debt and to spend it unwisely, while not cracking down on tax avoiders.

The other way to look at it is that, German banks should have known better to buy this debt in the first place. After all, it’s a free market, and nobody forces you to buy things, and after all if there really were no risk at all on it there would also be no yield (beyond inflation). But the very reason Greek bonds had yield was because the market was differentiating it from German bonds. From this point of view it was a mistake of the German bankers.

Either way, when the Germans bailed out their banks, they took what was a bank problem and made it into a taxpayer problem.

Have I oversimplified? I’ll also admit that, after that whole bailout went down, a series of “Greek bailouts,” all of which were clearly insufficient, made the European governments even more involved, and the Greeks owed way more on paper to the European taxpayers, which layered on the debts while destroying the Greek economy. But most of those bailouts were simply loans which were used to pay back the original loans. Put another way, the Greeks might not have needed bailing out if the original Greek bonds had been refused by risk-averse bankers in the first place.

This is not to suggest that there was perfect planning going on by the previous Greek governments. But I do think that, if we’re looking for who deserves blame in this story, we might want to circle back to the German bankers who couldn’t resist subprime mortgages and Greek bonds back in the early 2000’s.

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The 17-armed spiral within a spiral

Last Friday I visited my high school math camp, HCSSiM, where I became a nerd. I also taught there multiple times over the years, and in 2012 I blogged my lectures.

Why the visit? You see, we loyal alums of HCSSiM have a tradition of going back every July 17th to celebrate “Yellow Pig day,” which consists of a talk where founder and director (David) Kelly talks extensively about fun facts regarding the number 17, which happens before dinner, and then after dinner we sing “yellow pig carols” and eat an enormous amount of cake in the shape of a yellow pig. You can learn more about this ridiculous and hilarious tradition here.

Anyhoo, this year we (I went with other nerds) missed the 17 talk because of traffic in Connecticut but we made it for the dinner and carols. Luckily at dinner I had the chance to talk to Kelly, and I asked him if there were any new 17 facts this year. He told me there was one, and it was slightly mysterious. This post is an attempt to explain it a bit.

The mathematical set-up is explained here. Namely, we start with something called the Ulam Spiral, which is simply a way to label the boxes of an infinite two-dimensional grid with the natural numbers. You start at some place and then spiral outwards from there. Here’s a picture:

The center of the Ulam Spiral

The center of the Ulam Spiral

OK, so the first thing to say is that, when you label the plane like this, primes tend to cluster along lines. I think this is what Ulam thought was cool about his spiral:

Primes are black. This is the spiral with 200 layers.

Primes are black. This is the spiral with 200 layers.

Now comes the observation. You need to know what a triangular number is first, though. Namely, it’s a number that corresponds to counting up how many dots you need to form a triangle. We say the nth triangular number corresponds to a triangle with n rows. Here are the first few:

Screen Shot 2015-07-22 at 4.50.32 AM

You can also draw these triangles so that consecutive ones fit together to form squares.

When you highlight the triangular numbers in the Ulam Spiral, instead of the primes, then you get something that looks weird:

Green dots are triangular.

Green dots are triangular numbers within the Ulam Spiral.

OK so if you count those spiral arms, you’ll see there are 17 of them. But does that last forever? And if so, why?

Well, the answer is going to be yes. And here’s a rough proof. Rough because it uses asymptotic limits, so technically I will not show that the above picture extends perfectly, but rather that it eventually does look like a spiral with 17 arms.

A famous story about Gauss tells us that the formula for the nth triangular number is

T_k = k \cdot (k+1) /2.

Also, by construction of the Ulam Spiral, the bottom right corner of each “spiral layer” is an odd square, and that if we call that number n^2, there will be 4 \cdot n + 4 boxes on the very next layer, corresponding to the 4 sides of the next layer plus the 4 corners of the next layer.

Now imagine that there’s a triangular number right on that bottom right corner. That would mean that for some k,

k \cdot (k+1) /2 = n^2, or in other words that

k^2 + k = 2 \cdot n^2.

This is when things get asymptotic. Imagine that n is very very large. That would mean that k is too (everything here is a positive integer), and in particular that the k^2 term would dwarf the k term above. In other words, we could approximate:

k = \sqrt{2} \cdot n.

My next question is, how many triangular numbers would lie on the next layer of the spiral? Well, as we said above there are 4 \cdot n + 4 spots in the next layer, which we will approximate by 4 \cdot n, and the triangular number coming after T_k is T_{k+1}, which is k+1 bigger than T_k, corresponding to adding one layer to a triangle with k rows. We will approximate k+1 by \sqrt{2} \cdot n, again ignoring small terms.

For that matter, the next few triangular numbers after T_k come regularly, about k spots after the first. Therefore there are about 4 \cdot n / (\sqrt{2} \cdot n) triangular numbers in the next row of the Ulam spiral. That comes out to 2 \sqrt{2}, which is about 2.83.

So far we’ve figured out that, when n is huge, then after meeting the kth triangular number on the nth row, we will see two more, and get most of the way to a third, by going one more row.

Now let’s do that 6 more times. After traveling 6 rows past a triangular number, we will meet about 12 \sqrt{2} more triangular numbers. But

12 \sqrt{2} = 16.9705627485...,

which is very close to 17. So after traveling the Ulam Spiral for 6 rows, we will just about hit 17 triangular numbers, which will be more or less evenly spaced from each other.

What this means is that we should expect to see a spiral with 17 arms, but that when the picture is enlarged to include a very large number of rows, we will see the spiral shifting very slightly to the other direction.

By the way, I didn’t figure this out immediately. First I had a most delightful time understanding when, exactly, square numbers and triangular numbers coincide. In other words, I wanted to understand when there is a X and an X so that:

X \cdot (X+1) /2= Y^2, or

X^2 + X= 2 \cdot Y^2.

I might write this up in another post, but play around with it for a while if you get bored on the subway.

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Protest against gender and racial inequality tomorrow morning! #OccupySummerSchool

The Occupy Summer School students are organizing a clever demonstration tomorrow morning to protest racial and gender inequality. From the men/women pay gap to how the police arrest black and brown people for minor nuisance crimes, the girls from the UAI have figured out thoughtful ways of raising consciousness while having fun.

The plan is to have two tables. At the first table, there will be a “bake sale” where cupcakes will be “sold” for $1 to men but 77 cents to women, to protest unequal pay. They will be handed over using a plate which details many other kinds of gender inequalities. In actuality, anybody who shows up to the protest can get a free cupcake.

At the other table, we’ll be handing out brownies with toothpick flags, which are toothpicks with facts about racial inequalities taped to them. For example one might read, “While people of color make up about ​30 percent​of the United States’ population, they account for ​60 percent​of those imprisoned.”

Everyone loves free food, of course, but given that black women like Sandra Bland get killed in this country for minor traffic infractions, there’s a deeply serious side to it as well.

If you have time, please join us. The event will take place on Cadman Plaza near Tillary, in downtown Brooklyn, tomorrow morning from 9-11am. The girls will appreciate your visit.

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Star Trek uniforms for everyone

When I was a young idealistic mother, pregnant with my first kid, I had this crazy idea that I’d dress my kids in gender neutral clothing, like they have on Star Trek. In fact my actual goal was to get them Star Trek uniforms, but I knew that might be slightly difficult. It was 1999 and we were all worried about Y2K.

Little did I realize, until after the kiddo was born, how difficult it would be to get anything remotely gender neutral. Especially because I was rarely willing to spend lots of money on clothes I knew would be immediately outgrown, I ended up shopping at places like Toys R Us and similar, and man oh man are those clothes gendered. There’s a pink section and a royal blue and red section. Nothing in between, and no overlap.

Well, things have changed in the past 16 years, and nowadays there are clothing companies deliberately creating kids clothing that doesn’t have the awful princess/superhero dichotomy embedded into every garment. According to this Bloomberg article, there are now pink and purple clothes for boys and dinosaur, pirate, and science clothes for girls. Svaha, for example, sets itself up as a place that makes “clothes that empower your children.” Here’s an example of a girls’ shirt:


There’s also a boys’ shirt, also pink, with flowers and test tubes. That would have been great for my first son, whose favorite color was, as he described it at the time, “light red.”

A couple of things. First, these shirts are $25. That’s approximately 4 times more than these shirts that are standard issue “boy” clothes. Partly that’s just because it’s not a concept that’s really taken off, so we don’t have huge factories in Bangladesh churning out these shirts at ridiculous rates. But even so, it means that, like organic food, open-ended gender categorical clothing is firmly within the realm of the well-off parent.

Second, I don’t think it’s all that reasonable to say a shirt “empowers” a kid. Most times, when a kid is defined externally, through a shirt or a social convention or an adult’s comment, or even another kid’s comment, it’s an exercise in limiting that kid, not expanding him or her. Kids assume they can do anything until we tell them otherwise. When you say to a young girl, “You can be a scientist too, you know!” she thinks, “I never thought I couldn’t. Wait, why should I think I couldn’t?”. It’s not until they’re teenagers that they get this stuff, and can have a critical mindset about it.

In other words, I’m going back to Star Trek uniforms for everyone. The great thing about them is how utterly vapid they are of style or message. If you had to pin a message on to them, it would be an awesome (but distant) future.

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Occupy Summer School in the Metro!

Yesterday, as I was accompanying Adam Reich to the Occupy Summer School on the downtown 2 train, he pointed over my shoulder at someone reading a Metro, because the girls were on the front page:


We also were on the second page:

Screen Shot 2015-07-15 at 7.03.35 AM

After I got to the UAI, the high school where we run Occupy Summer School, I found the online version of the Metro story as well, which is also exciting.

Since Occupy Summer School (OSS) is half over, I think it’s a good time to update you on what’s been happening.

  • Last Monday we introduced ourselves, met the students, talked a little bit about Occupy, agreeable disagreement, and had a discussion about what they wanted to focus on using “stack.” Among the issues they came up with: inequality, Black Lives Matter, taxing the rich, how teenagers are unfairly targeted, and gender issues.
  • Tuesday Ale and Mo, a high school activist, came and talked to the girls about activism. We discussed how organizing actually works, what were the props for events, like stickers, flyers, signs and banners, how to get the word out among networks via text or twitter or other social media, and so on. We ended the day by quickly planning a protest against overly lengthy standardized tests.
  • Wednesday Tamir’s friends from Local 79 came and talked about unions and union organizing. The girls didn’t know much about unions, and were interested to learn how power can be created through numbers. 
  • Thursday Gerald provoked a fantastic discussion on #BlackLivesMatter and related topics. This was the first time where the girls really took over the discussion and the grownups in the room were merely listening and every now and then joining the discussion.
  • Friday Marni dazzled the girls with her approach to creative protests. She brought her own entourage, which ended up being how we got into the Metro. The day ended by planning a bake sale where women would be charged 78 cents and men one dollar for the same brownies, to illustrate the difference in pay.
  • This Monday I spoke to the girls about “why high school is free but college is expensive,” and then about debt more generally. We ended the day by planning a protest around a $100 ticket one of the girls had gotten for “doubling up” in the subway with her cousin, who didn’t have a student Metro card. The demand was to be unlimited metro rides for high school students on all days, not just school days.
  • Yesterday Adam came and talked to them about sociology, what is power (power is the opposite of dependence), and his work helping orgainze workers at Walmart. By the end of it he had two volunteers who wanted to join the cause.

I can’t wait for the rest of OSS! I’ll write another update at the end of next week when it’s over.

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Puerto Rico’s debt situation

As you already know, Puerto Rico is in a debt crisis. It’s unsustainable – take a look at some of the numbers – and people are suffering. There’s very high prices, few jobs, and on top of that there’s a terrible drought as well. I’m trying this week to learn what some of the details are of this situation, which is incredibly complicated because Puerto Rico is a U.S. commonwealth, not a state, and has historically been ignored by the political system.

Here’s what I know. The bond markets for Puerto Rico have historically been attractive to investors because the bonds are “triple exempt,” which basically means no taxes are applicable to them. This made it too easy for Puerto Rico to borrow money and has put it in a hole, very analogous to the Greek situation. And now we have to decide how much the people should suffer for the results of the bond markets.

Yesterday I reblogged a post by Marc Joffe, who argued that the U.S. should extend Chapter 9 to Puerto Rico. Hypothetically this would allow Puerto Rico to declare bankruptcy and restructure its debts in some reasonable way. However, as Kristi Culpepper explained in this Medium piece (hat tip Tom Adams), it actually wouldn’t give Puerto Rico the relief that it needs, first of all because it would redefine Puerto Rico as a “state” but states are not eligible to declare bankruptcy, and secondly because the corporate bonds issued by Puerto Rico’s public corporations have a special status that also prevents them from restructuring.

Culpepper also notes in her piece that people who cry foul at the concept of restructuring debt after it has been issued can rest assured that there is precedent for it. Personally, I don’t even understand that complaint; surely everyone realizes that any debt might go into default, and it hardly matters exactly what that procedure looks like.

Culpepper recommends something else entirely, namely a federal financial control board. The idea is that there’s also precedent for this, in the 1990’s in Washington D.C.. However, it would essentially mean handing over control over its finances to the board. Culpepper notes that this could even happen without consent. I think the Puerto Rican people may have something to say about this. She also suggests we could provide liquidity for Puerto Rico if we wanted, although it might look something like a bailout.

The biggest problem is that, even now, no politician seems to really care about Puerto Rico, except to fight against it becoming a state.

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