### Archive

Archive for the ‘math’ Category

## Gender And The Harvard Math Department

This is a guest post by Meena Boppana, a junior at Harvard and former president of the Harvard Undergraduate Math Association (HUMA). Meena is passionate about addressing the gender gap in math and has co-lead initiatives including the Harvard math survey and the founding of the Harvard student group Gender Inclusivity in Math (GIIM).

I arrived at Harvard in 2012 head-over-heels in love with math. Encouraged to think mathematically since I was four years old by my feminist mathematician dad, I had even given a TEDx talk in high school declaring my love for the subject. I was certainly qualified and excited enough to be a math major.

Which is why, three years later, I think about how it is that virtually all my female friends with insanely strong math backgrounds (e.g. math competition stars) decided not to major in math (I chose computer science). This year, there were no female students in Math 55a, the most intense freshman math class, and only two female students graduating with a primary concentration in math. There are also a total of zero tenured women faculty in Harvard math.

So, I decided to do some statistical sleuthing and co-directed a survey of Harvard undergraduates in math. I was inspired by the work of Nancy Hopkins and other pioneering female scientists at MIT, who quantified gender inequities at the Institute – even measuring the square footage of their offices – and sparked real change. We got a 1/3 response rate among all math concentrators at Harvard, with 150 people in total (including related STEM concentrations) filling it out.

The main finding of our survey analysis is that the dearth of women in Harvard math is far more than a “pipeline issue” stemming from high school. So, the tale that women are coming in to Harvard knowing less math and consequently not majoring in math is missing much of the picture. Women are dropping out of math during their years at Harvard, with female math majors writing theses and continuing on to graduate school at far lower rates than their male math major counterparts.

And it’s a cultural issue. Our survey indicated that many women would like to be involved in the math department and aren’t, most women feel uncomfortable as a result of the gender gap, and women feel uncomfortable in math department common spaces.

The simple act of talking about the gender gap has opened the floodgates to great conversations. I had always assumed that because no one was talking about the gender gap, no one cared. But after organizing a panel on gender in the math department which drew 150 people with a roughly equal gender split and students and faculty alike, I realized that my classmates of all genders feel more disempowered than apathetic.

The situation is bad, but certainly not hopeless. Together with a male freshman math major, I am founding a Harvard student group called Gender Inclusivity in Math (GIIM). The club has the two-fold goal of increasing community among women in math, including dinners, retreats, and a women speaker series, and also addressing the gender gap in the math department, continuing the trend of surveys and gender in math discussions. The inclusion of male allies is central to our club mission, and the support from male allies at the student and faculty level that we have received makes me optimistic about the will for change.

Ultimately, it is my continued love for math which has driven me to take action. Mathematics is too beautiful and important to lose 50 percent (or much more when considering racial and class-based inequities) of the potential population of math lovers.

## Nick Kristof is not Smarter than an 8th Grader

This is a post by Eugene Stern, originally posted on his blog sensemadehere.wordpress.com.

About a week ago, Nick Kristof published this op-ed in the New York Times. Entitled Are You Smarter than an 8th Grader, the piece discusses American kids’ underperformance in math compared with students from other countries, as measured by standardized test results. Kristof goes over several questions from the 2011 TIMSS (Trends in International Mathematics and Science Study) test administered to 8th graders, and highlights how American students did worse than students from Iran, Indonesia, Ghana, Palestine, Turkey, and Armenia, as well as traditional high performers like Singapore. “We all know Johnny can’t read,” says Kristof, in that finger-wagging way perfected by the current cohort of New York Times op-ed columnists; “it appears that Johnny is even worse at counting.”

The trouble with this narrative is that it’s utterly, demonstrably false.

My friend Jordan Ellenberg pointed me to this blog post, which highlights the problem. In spite of Kristof’s alarmism, it turns out that American eighth graders actually did quite well on the 2011 TIMSS. You can see the complete results here. Out of 42 countries tested, the US placed 9th. If you look at the scores by country, you’ll see a large gap between the top 5 (Korea, Singapore, Taiwan, Hong Kong, and Japan) and everyone else. After that gap comes Russia, in 6th place, then another gap, then a group of 9 closely bunched countries: Israel, Finland, the US, England, Hungary, Australia, Slovenia, Lithuania, and Italy. Those made up, more or less, the top third of all the countries that took the test. Our performance isn’t mind-blowing, but it’s not terrible either. So what the hell is Kristof talking about?

You’ll find the answer here, in a list of 88 publicly released questions from the test (not all questions were published, but this appears to be a representative sample). For each question, a performance breakdown by country is given. When I went through the questions, I found that the US placed in the top third (top 14 out of 42 countries) on 45 of them, the middle third on 39, and the bottom third on 4. This seems typical of the kind of variance usually seen on standardized tests. US kids did particularly well on statistics, data interpretation, and estimation, which have all gotten more emphasis in the math curriculum lately. For example, 80% of US eighth graders answered this question correctly:

Which of these is the best estimate of (7.21 × 3.86) / 10.09?

(A) (7 × 3) / 10   (B) (7 × 4) / 10   (C) (7 × 3) / 11   (D) (7 × 4) / 11

More American kids knew that the correct answer was (B) than Russians, Finns, Japanese, English, or Israelis. Nice job, kids! And let’s give your teachers some credit too!

But Kristof isn’t willing to do either. He has a narrative of American underperformance in mind, and if the overall test results don’t fit his story, he’ll just go and find some results that do! Thus, the examples in his column. Kristof literally went and picked the two questions out of 88 on which the US did the worst, and highlighted those in the column. (He gives a third example too, a question in which the US was in the middle of the pack, but the pack did poorly, so the US’s absolute score looks bad.) And, presto! — instead of a story about kids learning stuff and doing decently on a test, we have yet another hysterical screed about Americans “struggling to compete with citizens of other countries.”

Kristof gives no suggestions for what we can actually do better, by the way. But he does offer this helpful advice:

Numeracy isn’t a sign of geekiness, but a basic requirement for intelligent discussions of public policy. Without it, politicians routinely get away with using statistics, as Mark Twain supposedly observed, the way a drunk uses a lamppost: for support rather than illumination.

So do op-ed columnists, apparently.

## How many NYC are arbitrarily punished by the VAM? About 578 per year.

There’s been an important update in the thought experiment I started yesterday. Namely, a reader (revuluri) has provided me with a link to show how many teachers are considered “ineffective,” which was my shorthand for scoring either third or fourth in the four categories.

According to page 5 of this document, that percentage was 16% in 2011-2012, 17% in 2012-2013, and 16% in 2013-2014. We’ll take this to mean that the true cutoff is about 16.3%. Using my formula from yesterday, that means that after 4 years, about

$1- (0.837^4 + 4 \cdot 0.837^3 \cdot 0.163) = 0.127,$

or 12.7% of teachers going up for tenure in the new system will be arbitrarily denied tenure based only on their VAM score.

How many people is that in a given year? Well, this document explains that in 2000, 9,000 teachers were hired and in 2008, 6,000 teachers were hired. I’ll assume my best guess for “teachers hired” in a given year is something between those two numbers, but I’ll also assume it’s closer to the latter since it is more recent information. Say 7,000 new teachers per year.

Of course, not all of them go up for tenure. There’s attrition. Say 35% of those teachers leave before the tenure decision is made (also guessing from this document). That leave us with about 4,550 teachers going up for tenure each year, and 12.7% of them is 578 people.

So, according to my crude estimates, about 578 people will be denied tenure simply based on this random number generator we call VAM. And as my reader said, this says nothing about the hard-to-measure damage done to all the good teachers trying to teach their kids but having to deal with this standardized testing nonsense. It’s a wonder anyone is willing to work here.

Please comment if you have updated numbers for anything here.

Categories: education, math

## I accept mathematical bribes

Last Friday I traveled to American University and gave an evening talk, where I met Jeffrey Hakim, a mathematician and designer who openly bribed me.

Don’t worry, it’s not that insidious. He just showed me his nerdy math wallet and said I could have one too if I blogged about it. I obviously said yes. Here’s my new wallet:

It’s made of the same kind of flexible plastic they use on the outside of buildings. Or something. I expect it will last for many years.

You might notice there is writing and pictures on my new wallet! They are mathematical, which is why I don’t feel bad about accepting this bribe: it’s all in the name of education and fun with mathematics. Let me explain the front and back of the wallet.

The front is a theorem:

Here’s the thing, I’ve proven this. I have even assigned it to my students in the past to prove. We always use induction. This kind of identity is kind of made for induction, no? Don’t you think?

Well Jeffrey Hakim had an even better idea. His proof of Nicomachus’s Theorem is represented as a picture on the back of the wallet:

It took me a couple of minutes to see why this is a proof.

Here’s what I’d like you all to do: go think about why this is a proof of the above identity. Come back if you can’t figure it out, but if you can, just go ahead and pat yourself on your back and don’t bother reading the rest of this blogpost because it’s just going to explain the proof.

I’ll give you all a moment…

OK cool here’s why this is a proof.

First, convince yourself that this “pattern,” of building a frame of square boxes around the above square, can be continued. In other words, it’s a square of 4 1×1 boxes, framed by 2×2 boxes, framed by 3×3 boxes, and so on. It could go on forever this way, because if you focus on one side of the outside of the third layer, there are 4 3×3 boxes, so length $4 \cdot 3$, and we need it to also be the length inside the 4th frame, which has 3 boxes of length 4. Since $4\cdot 3 = 3\cdot 4$, we’re good. And that generalizes when it’s the $n$th layer, of course, since the outside of the $n$th layer will have $n+1$ boxes, each of length $n,$ making the inside of the $n+1$st have $n$ boxes, each of length $n+1$.

OK, now here’s the actual trick. What is the area of this box?

I claim there are two ways to measure the area, and one of the ways will give you the left hand side of Nicomachus’s Theorem but the other way will give you the right hand side of Nicomachus’s Theorem.

To be honest, it’s just one bit more complicated than that. Namely, the first way gives you something that’s 4 times bigger than the left hand side of Nicomachus’s Theorem and the second way gives you something 4 times bigger than the right hand side of Nicomachus’s Theorem.

Why don’t you go think about this for a few minutes, because the clue might be all you need to figure it out.

Or, perhaps you just want me to go ahead and explain it. I’ll do that! That’s why I got the wallet!

OK, now imagine isolating the top right quarter of the above figure. Like this:

That’s a square, obviously, so its area is the square of the length of any side. But if you go along the bottom, the length is obviously $1 + 2 + 3 + 4,$ which means the area is the square of that, $(1 + 2 + 3 + 4)^2.$

And since we know we can generalize the original figure to go up to $n$ instead of just 4, one quarter of the figure will have area $(1 + 2 + 3 + 4 + \dots + n)^2,$ which is to say the entire figure will have area $4(1 + 2 + 3 + 4 + \dots + n)^2.$

That’s 4 times the right-hand side of the theorem, so we’re halfway done!

Next, we will compute the area of the original figure a different way, namely by simply adding up and counting all the differently colored squares that make it up. Assume that we continue changing colors every time we get a new layer.

So, there are 4 1×1 squares, and there are 8 2×2 squares, and there are 12 3×3 squares, and there are 16 4×4 squares. In the generalized figure, there would be $4n$ $n\times n$ squares.

So if you look at the area of the generalized figure which is all one color, say the $n$th color, it will be of the form $4\cdot n \cdot n^2 = 4 \cdot n^3.$

That means the overall generalized figure will have total area:

$4 \cdot 1^3 + 4 \cdot 2^3 + 4 \cdot 3^3 + \dots + 4 \cdot n^3 = 4 \cdot (1^3 + 2^3 + 3^3 + \dots + n^3).$

Since that’s just 4 times the left-hand side of the theorem, we’re done.

Notes:

• this would be a fun thing to do with a kid.
• there’s more math inside the wallet which I haven’t gotten to yet.
• After staring at the picture for another minutes, I just realized the total area of the whole (generalized) thing is obviously $(n\cdot (n+1))^2,$ which is to say that either the left-hand side or right-hand side of the original identity is one fourth of that. Cool!
Categories: math, math education

## Guest post: Be more careful with the vagina stats in teaching

This is a guest post by Courtney  an assistant professor of mathematics at Hamilton College. You can see her teaching evaluations on ratemyprofessor.com. She would like you to note that she’s been tagged as “hilarious.” Twice.

Lately, my social media has been blowing up with stories about gender bias in higher ed, especially course evaluations.   As a 30-something, female math professor, I’m personally invested in this kind of issue.  So I’m gratified when I read about well-designed studies that highlight the “vagina tax” in teaching (I didn’t coin this phrase, but I wish I had).

These kinds of studies bring the conversation about bias to the table in a way that academics can understand. We can geek out on experimental design, the fact that the research is peer-reviewed and therefore passes some basic legitimacy tests.

Indeed, the conversation finally moves out of the realm of folklore, where we have “known” for some time that students expect women to be nurturing in addition to managing the class, while men just need to keep class on track.

Let me reiterate: as a young woman in academia, I want deans and chairs and presidents to take these observed phenomena seriously when evaluating their professors. I want to talk to my colleagues and my students about these issues. Eventually, I’d like to “fix” them, or at least game them to my advantage. (Just kidding.  I’d rather fix them.)

However, let me speak as a mathematician for a minute here: bad interpretations of data don’t advance the cause. There’s beautiful link-bait out there that justifies its conclusions on the flimsy “hey, look at this chart” understanding of big data. Benjamin M. Schmidt created a really beautiful tool to visualize data he scraped from the website ratemyprofessor.com through a process that he sketches on his blog. The best criticisms and caveats come from Schmidt himself.

What I want to examine is the response to the tool, both in the media and among my colleagues.  USAToday, HuffPo, and other sites have linked to it, citing it as yet more evidence to support the folklore: students see men as “geniuses” and women as “bossy.” It looks like they found some screenshots (or took a few) and decided to interpret them as provocatively as possible. After playing with the tool for a few minutes, which wasn’t even hard enough to qualify as sleuthing, I came to a very different conclusion.

If you look at the ratings for “genius”  and then break them down further to look at positive and negative reviews separately, it occurs predominantly in negative reviews. I found a few specific reviews, and they read, “you have to be a genius to pass” or along those lines.

[Don’t take my word for it — search google for:

rate my professors “you have to be a genius”‘

and you’ll see how students use the word “genius” in reviews of professors. The first page of hits is pretty much all men.]

Here’s the breakdown for “genius”:

So yes, the data shows that students are using the word “genius” in more evaluations of men than women. But there’s not a lot to conclude from this; we can’t tell from the data if the student is praising the professor or damning him. All we can see that it’s a word that occurs in negative reviews more often than positive ones. From the data, we don’t even know if it refers to the professor or not.

Similar results occur with “brilliant”:

Now check out “bossy” and negative reviews:

Okay, wow, look at how far to the right those orange dots are… and now look at the x-axis.  We’re talking about fewer than 5 uses per million words of text.  Not exactly significant compared to some of the other searches you can do.

I thought that the phrase “terrible teacher” was more illuminating, because it’s more likely in reference to the subject of the review, and we’ve got some meaningful occurrences:

And yes, there is a gender imbalance, but it’s not as great as I had feared. I’m more worried about the disciplinary break down, actually. Check out math — we have the worst teachers, but we spread it out across genders, with men ranking 187 uses of “terrible teacher” per million words; women score 192. Compare to psychology, where profs receive a score of 110.  Ouch.

Who’s doing this reporting, and why aren’t we reading these reports more critically?  Journalists, get your shit together and report data responsibly.  Academics, be a little more skeptical of stories that simply post screenshots of a chart coupled with inciting prose from conclusions drawn, badly, from hastily scanned data.

Is this tool useless? No. Is it fun to futz around with? Yes.

Is it being reported and understood well? Resounding no!

I think even our students would agree with me: that’s just f*cked up.

## Male nerd privilege

I recently read this essay by Laurie Penny (hat tip Jordan Ellenberg) about male nerd privilege. Her essay stemmed from comment 171 of Scott Aaronson’s blogpost about whether MIT professor Walter Lewin, who was found to be harassing women, should also have had his OpenCourseWare physics course taken down. Aaronson says no.

Personally, I think it should, because if I’m a woman who was harassed by that dude, I don’t want to see physics represented by my harasser up on MIT’s website; it would not make me feel welcome to the MIT community. Physics is a social community activity, after all, just like mathematics, and it is important to feel safe doing physics in that community. Plus the courses will be available on YouTube and other places, it’s not like the physics represented in the course has been lost to humanity.

Anyhoo, I did really want to talk about white male nerd privilege. Penny makes a bunch of good points in her essay, but I think she misses a big opportunity as well.

Quick summary. Aaronson talks about how he spent his youth and formative years terrified, since he was a shy nerd boy. Penny talks about how she did too, but then on top of it had to deal with structural sexism. Good point, and entirely true in my experience. Her best line:

At the same time, I want you to understand that that very real suffering does not cancel out male privilege, or make it somehow alright. Privilege doesn’t mean you don’t suffer, which, I know, totally blows.

So, I had two responses to her piece.

First was, she was complaining about her childhood, but she wasn’t even fat! I mean, GAWD. She was complaining about being too skinny, of all things. Plus it’s not clear whether or not she came from an abusive home. So I’ve got like, at least two complaints up on her. She thinks she’s had it bad?!

My point being, we can’t actually win when we count up all the ways we were miserable. Because the truth is, most people were actually miserable in their childhood, or soon after it, or at some time. And by comparing that stuff we just get stuck in a cycle of feeling competitively sorry for ourselves and pointing fingers. We need to sympathize, not only with our former selves, but with other people.

And although she does end the essay with the idea that we have to transcend all of our personal bruises and wrongs, and call each other human, and forget our resentments, it doesn’t seem like she’s giving us a path towards that.

Because, and here’s my second point, she doesn’t do the big thing of naming all of her privileges. Like, that nerds get good jobs. And that white people get loads of resources and attention and benefit of the doubt just for being white. At the end of the day, we are privileged to be sitting around talking about privilege. We are not worried about dying of hunger or exposure.

When Aaronson complained that naming male privilege is shaming, I’m prone to agree, at least if it’s done like this. What I’d propose is to figure out a way to talk about these structural problems in an aspirational way. How can we help make things fairer? How can we move this problem to the next level? Scott, you’re wicked smart, want to be on a taskforce with me?

Would it help if we gave it another name? Basic human rights, perhaps? Because that’s what we’re talking about, at the end of the day. The right to be free, to not get shot by the police, the right to hold a good job and care for your family, stuff like that.

Of course, there are plenty of people who are unwilling to move to the next level because they don’t acknowledge the structural racism, sexism, and other stuff at all. They don’t see the current situation as problematic. But on the other hand, there are loads of people who do, and Aaronson is clearly one of them.

As for problems for women in STEM, we’ve already studied this and we all know that both men and women are sexist, so it’s obviously not a blame game here. Instead, it’s a real cultural conundrum which we would like to approach thoughtfully and we’d like to make progress on as a team.

## Notices of the AMS is killing it

I am somewhat surprised to hear myself say this, but this month’s Notices of the AMS is killing it. Generally speaking I think of it as rather narrowly focused but things seem to be expanding and picking up. Scanning the list of editors, they do seem to have quite a few people that want to address wider public issues that touch and are touched by mathematicians.

First, there’s an article about how the h-rank of an author is basically just the square root of the number of citations for that author. It’s called Critique of Hirsch’s Citation Index: A Combinatorial Fermi Problem and it’s written by Alexander Yong. Doesn’t surprised me too much, but there you go, people often fall in love with new fancy metrics that turn out to be simple transformations of old discarded metrics.

Second, and even more interesting to me, there’s an article that explains the mathematical vapidness of a widely cited social science paper. It’s called Does Diversity Trump Ability? An Example of the Misuse of Mathematics in the Social Sciences and it’s written by Abby Thompson. My favorite part of paper:

Oh, and here’s another excellent take-down of a part of that paper:

Let me just take this moment to say, right on, Notices of the AMS! And of course, right on Alexander Yong and Abby Thompson!

Categories: math, modeling