Archive for the ‘math education’ Category

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. I expect it will last for many years.

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.

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 almost ready?

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 nth layer, of course, since the outside of the nth layer will have n+1 boxes, each of length n, making the inside of the n+1st 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 nth 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.


  • 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: A Discussion Of PARCC Testing

This is a guest post by Eugene Stern, who writes a blog at Sense Made Here, and Kristin Wald, who writes a blog at This Unique* Weblog. Crossposted on their blogs as well.

Today’s post is a discussion of education reform, standardized testing, and PARCC with my friend Kristin Wald, who has been extremely kind to this blog. Kristin taught high school English in the NYC public schools for many years. Today her kids and mine go to school together in Montclair. She has her own blog that gets orders of magnitude more readers than I do.

ES: PARCC testing is beginning in New Jersey this month. There’s been lots of anxiety and confusion in Montclair and elsewhere as parents debate whether to have their kids take the test or opt out. How do you think about it, both as a teacher and as a parent?

KW: My simple answer is that my kids will sit for PARCC. However, and this is where is gets grainy, that doesn’t mean I consider myself a cheerleader for the exam or for the Common Core curriculum in general.

In fact, my initial reaction, a few years ago, was to distance my children from both the Common Core and PARCC. So much so that I wrote to my child’s principal and teacher requesting that no practice tests be administered to him. At that point I had only peripherally heard about the issues and was extending my distaste for No Child Left Behind and, later, Race to the Top. However, despite reading about and discussing the myriad issues, I still believe in change from within and trying the system out to see kinks and wrinkles up-close rather than condemning it full force.


ES: Why did you dislike NCLB and Race to the Top? What was your experience with them as a teacher?

KW: Back when I taught in NYC, there was wiggle room if students and schools didn’t meet standards. Part of my survival as a teacher was to shut my door and do what I wanted. By the time I left the classroom in 2007 we were being asked to post the standards codes for the New York State Regents Exams around our rooms, similar to posting Common Core standards all around. That made no sense to me. Who was this supposed to be for? Not the students – if they’re gazing around the room they’re not looking at CC RL.9-10 next to an essay hanging on a bulletin board. I also found NCLB naïve in its “every child can learn it all” attitude. I mean, yes, sure, any child can learn. But kids aren’t starting out at the same place or with the same support. And anyone who has experience with children who have not had the proper support up through 11th grade knows they’re not going to do well, or even half-way to well, just because they have a kickass teacher that year.

Regarding my initial aversion to Common Core, especially as a high school English Language Arts teacher, the minimal appearance of fiction and poetry was disheartening. We’d already seen the slant in the NYS Regents Exam since the late 90’s.

However, a couple of years ago, a friend asked me to explain the reason The Bluest Eye, with its abuse and rape scenes, was included in Common Core selections, so I took a closer look. Basically, a right-wing blogger had excerpted lines and scenes from the novel to paint it as “smut” and child pornography, thus condemning the entire Common Core curriculum. My response to my friend ended up as “In Defense of The Bluest Eye.”

That’s when I started looking more closely at the Common Core curriculum. Learning about some of the challenges facing public schools around the country, I had to admit that having a required curriculum didn’t seem like a terrible idea. In fact, in a few cases, the Common Core felt less confining than what they’d had before. And you know, even in NYC, there were English departments that rarely taught women or minority writers. Without a strong leader in a department, there’s such a thing as too much autonomy. Just like a unit in a class, a school and a department should have a focus, a balance.

But your expertise is Mathematics, Eugene. What are your thoughts on the Common Core from that perspective?

ES: They’re a mix. There are aspects of the reforms that I agree with, aspects that I strongly disagree with, and then a bunch of stuff in between.

The main thing I agree with is that learning math should be centered on learning concepts rather than procedures. You should still learn procedures, but with a conceptual underpinning, so you understand what you’re doing. That’s not a new idea: it’s been in the air, and frustrating some parents, for 50 years or more. In the 1960’s, they called it New Math.

Back then, the reforms didn’t go so well because the concepts they were trying to teach were too abstract – too much set theory, in a nutshell, at least in the younger grades. So then there was a retrenchment, back to learning procedures. But these things seem to go in cycles, and now we’re trying to teach concepts better again. This time more flexibly, less abstractly, with more examples. At least that’s the hope, and I share that hope.

I also agree with your point about needing some common standards defining what gets taught at each grade level. You don’t want to be super-prescriptive, but you need to ensure some kind of consistency between schools. Otherwise, what happens when a kid switches schools? Math, especially, is such a cumulative subject that you really need to have some big picture consistency in how you teach it.


ES: What I disagree with is the increased emphasis on standardized testing, especially the raised stakes of those tests. I want to see better, more consistent standards and curriculum, but I think that can and should happen without putting this very heavy and punitive assessment mechanism on top of it.

KW: Yes, claiming to want to assess ability (which is a good thing), but then connecting the results to a teacher’s effectiveness in that moment is insincere evaluation. And using a standardized test not created by the teacher with material not covered in class as a hard percentage of a teacher’s evaluation makes little sense. I understand that much of the exam is testing critical thinking, ability to reason and use logic, and so on. It’s not about specific content, and that’s fine. (I really do think that’s fine!) Linking teacher evaluations to it is not.

Students cannot be taught to think critically in six months. As you mentioned about the spiraling back to concepts, those skills need to be revisited again and again in different contexts. And I agree, tests needn’t be the main driver for raising standards and developing curriculum. But they can give a good read on overall strengths and weaknesses. And if PARCC is supposed to be about assessing student strengths and weaknesses, it should be informing adjustments in curriculum.

On a smaller scale, strong teachers and staffs are supposed to work as a team to influence the entire school and district with adjusted curriculum as well. With a wide reach like the Common Core, a worrying issue is that different parts of the USA will have varying needs to meet. Making adjustments for all based on such a wide collection of assessments is counterintuitive. Local districts (and the principals and teachers in them) need to have leeway with applying them to best suit their own students.

Even so, I do like some things about data driven curricula. Teachers and school administrators are some of the most empathetic and caring people there are, but they are still human, and biases exist. Teachers, guidance counselors, administrators can’t help but be affected by personal sympathies and peeves. Having a consistent assessment of skills can be very helpful for those students who sometimes fall through the cracks. Basically, standards: yes. Linking scores to teacher evaluation: no.

ES: Yes, I just don’t get the conventional wisdom that we can only tell that the reforms are working, at both the individual and group level, through standardized test results. It gives us some information, but it’s still just a proxy. A highly imperfect proxy at that, and we need to have lots of others.

I also really like your point that, as you’re rolling out national standards, you need some local assessment to help you see how those national standards are meeting local needs. It’s a safeguard against getting too cookie-cutter.

I think it’s incredibly important that, as you and I talk, we can separate changes we like from changes we don’t. One reason there’s so much noise and confusion now is that everything – standards, curriculum, testing – gets lumped together under “Common Core.” It becomes this giant kitchen sink that’s very hard to talk about in a rational way. Testing especially should be separated out because it’s fundamentally an issue of process, whereas standards and curriculum are really about content.

You take a guy like Cuomo in New York. He’s trying to increase the reliance on standardized tests in teacher evaluations, so that value added models based on test scores count for half of a teacher’s total evaluation. And he says stuff like this: “Everyone will tell you, nationwide, the key to education reform is a teacher evaluation system.” That’s from his State of the State address in January. He doesn’t care about making the content better at all. “Everyone” will tell you! I know for a fact that the people spending all their time figuring out at what grade level kids should start to learn about fractions aren’t going tell you that!

I couldn’t disagree with that guy more, but I’m not going to argue with him based on whether or not I like the problems my kids are getting in math class. I’m going to point out examples, which he should be well aware of by now, of how badly the models work. That’s a totally different discussion, about what we can model accurately and fairly and what we can’t.

So let’s have that discussion. Starting point: if you want to use test scores to evaluate teachers, you need a model because – I think everyone agrees on this – how kids do on a test depends on much more than how good their teacher was. There’s the talent of the kid, what preparation they got outside their teacher’s classroom, whether they got a good night’s sleep the night before, and a good breakfast, and lots of other things. As well as natural randomness: maybe the reading comprehension section was about DNA, and the kid just read a book about DNA last month. So you need a model to break out the impact of the teacher. And the models we have today, even the most state-of-the-art ones, can give you useful aggregate information, but they just don’t work at that level of detail. I’m saying this as a math person, and the American Statistical Association agrees. I’ve written about this here and here and here and here.

Having student test results impact teacher evaluations is my biggest objection to PARCC, by far.

KW: Yep. Can I just cut and paste what you’ve said? However, for me, another distasteful aspect is how technology is tangled up in the PARCC exam.


ES: Let me tell you the saddest thing I’ve heard all week. There’s a guy named Dan Meyer, who writes very interesting things about math education, both in his blog and on Twitter. He put out a tweet about a bunch of kids coming into a classroom and collectively groaning when they saw laptops on every desk. And the reason was that they just instinctively assumed they were either about to take a test or do test prep.

That feels like such a collective failure to me. Look, I work in technology, and I’m still optimistic that it’s going to have a positive impact on math education. You can use computers to do experiments, visualize relationships, reinforce concepts by having kids code them up, you name it. The new standards emphasize data analysis and statistics much more than any earlier standards did, and I think that’s a great thing. But using computers primarily as a testing tool is an enormous missed opportunity. It’s like, here’s the most amazing tool human beings have ever invented, and we’re going to use it primarily as a paperweight. And we’re going to waste class time teaching kids exactly how to use it as a paperweight. That’s just so dispiriting.

KW: That’s something that hardly occurred to me. My main objection to hosting the PARCC exam on computers – and giving preparation homework and assignments that MUST be done on a computer – is the unfairness inherent in accessibility. It’s one more way to widen the achievement gap that we are supposed to be minimizing. I wrote about it from one perspective here.

I’m sure there are some students who test better on a computer, but the playing field has to be evenly designed and aggressively offered. Otherwise, a major part of what the PARCC is testing is how accurately and quickly children use a keyboard. And in the aggregate, the group that will have scores negatively impacted will be children with less access to the technology used on the PARCC. That’s not an assessment we need to test to know. When I took the practice tests, I found some questions quite clear, but others were difficult not for content but in maneuvering to create a fraction or other concept. Part of that can be solved through practice and comfort with the technology, but then we return to what we’re actually testing.

ES: Those are both great points. The last thing you want to do is force kids to write math on a computer, because it’s really hard! Math has lots of specialized notation that’s much easier to write with pencil and paper, and learning how to write math and use that notation is a big part of learning the subject. It’s not easy, and you don’t want to put artificial obstacles in kids’ way. I want kids thinking about fractions and exponents and what they mean, and how to write them in a mathematical expression, but not worrying about how to put a numerator above a denominator or do a superscript or make a font smaller on a computer. Plus, why in the world would you limit what kids can express on a test to what they can input on a keyboard? A test is a proxy already, and this limits what it can capture even more.

I believe in using technology in education, but we’ve got the order totally backwards. Don’t introduce the computer as a device to administer tests, introduce it as a tool to help in the classroom. Use it for demos and experiments and illustrating concepts.

As far as access and fairness go, I think that’s another argument for using the computer as a teaching tool rather than a testing tool. If a school is using computers in class, then at least everyone has access in the classroom setting, which is a start. Now you might branch out from there to assignments that require a computer. But if that’s done right, and those assignments grow in an organic way out of what’s happening in the classroom, and they have clear learning value, then the school and the community are also morally obligated to make sure that everyone has access. If you don’t have a computer at home, and you need to do computer-based homework, then we have to get you computer access, after school hours, or at the library, or what have you. And that might actually level the playing field a bit. Whereas now, many computer exercises feel like they’re primarily there to get kids used to the testing medium. There isn’t the same moral imperative to give everybody access to that.

I really want to hear more about your experience with the PARCC practice tests, though. I’ve seen many social media threads about unclear questions, both in a testing context and more generally with the Common Core. It sounds like you didn’t think it was so bad?

KW: Well, “not so bad” in that I am a 45 year old who was really trying to take the practice exam honestly, but didn’t feel stressed about the results. However, I found the questions with fractions confusing in execution on the computer (I almost gave up), and some of the questions really had to be read more than once. Now, granted, I haven’t been exposed to the language and technique of the exam. That matters a lot. In the SAT, for example, if you don’t know the testing language and format it will adversely affect your performance. This is similar to any format of an exam or task, even putting together an IKEA nightstand.

There are mainly two approaches to preparation, and out of fear of failing, some school districts are doing hardcore test preparation – much like SAT preparation classes – to the detriment of content and skill-based learning. Others are not altering their classroom approaches radically; in fact, some teachers and parents have told me they hardly notice a difference. My unscientific observations point to a separation between the two that is lined in Socio-Economic Status. If districts feel like they are on the edge or have a lot to lose (autonomy, funding, jobs), if makes sense that they would be reactionary in dealing with the PARCC exam. Ironically, schools that treat the PARCC like a high-stakes test are the ones losing the most.

Opting Out

KW: Despite my misgivings, I’m not in favor of “opting out” of the test. I understand the frustration that has prompted the push some districts are experiencing, but there have been some compromises in New Jersey. I was glad to see that the NJ Assembly voted to put off using the PARCC results for student placement and teacher evaluations for three years. And I was relieved, though not thrilled, that the percentage of PARCC results to be used in teacher evaluations was lowered to 10% (and now put off). I still think it should not be a part of teacher evaluations, but 10% is an improvement.

Rather than refusing the exam, I’d prefer to see the PARCC in action and compare honest data to school and teacher-generated assessments in order to improve the assessment overall. I believe an objective state or national model is worth having; relying only on teacher-based assessment has consistency and subjective problems in many areas. And that goes double for areas with deeply disadvantaged students.

ES: Yes, NJ seems to be stepping back from the brink as far as model-driven teacher evaluation goes. I think I feel the same way you do, but if I lived in NY, where Cuomo is trying to bump up the weight of value added models in evaluations to 50%, I might very well be opting out.

Let me illustrate the contrast – NY vs. NJ, more test prep vs. less — with an example. My family is good friends with a family that lived in NYC for many years, and just moved to Montclair a couple months ago. Their older kid is in third grade, which is the grade level where all this testing starts. In their NYC gifted and talented public school, the test was this big, stressful thing, and it was giving the kid all kinds of test anxiety. So the mom was planning to opt out. But when they got to Montclair, the kid’s teacher was much more low key, and telling the kids not to worry. And once it became lower stakes, the kid wanted to take the test! The mom was still ambivalent, but she decided that here was an opportunity for her kid to get used to tests without anxiety, and that was the most important factor for her.

I’m trying to make two points here. One: whether or not you opt out depends on lots of factors, and people’s situations and priorities can be very different. We need to respect that, regardless of which way people end up going. Two: shame on us, as grown ups, for polluting our kids’ education with our anxieties! We need to stop that, and that extends both to the education policies we put in place and how we collectively debate those policies. I guess what I’m saying is: less noise, folks, please.

KW: Does this very long blog post count as noise, Eugene? I wonder how this will be assessed? There are so many other issues – private profits from public education, teacher autonomy in high performing schools, a lack of educational supplies and family support, and so on. But we have to start somewhere with civil and productive discourse, right? So, thank you for having the conversation.

ES: Kristin, I won’t try to predict anyone else’s assessment, but I will keep mine low stakes and say this has been a pleasure!

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

This is a guest post by Courtney Gibbons, an assistant professor of mathematics at Hamilton College. You can see her teaching evaluations on 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 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.

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.

Aunt Pythia’s advice

Aunt Pythia has something in the works for you dear people, but it’s not quite ready yet, and you’ll have to wait another week. Rest assured, it will be worth it. And apologies to subscribers who received an errant test post this week.

In the meantime, Aunt Pythia is going to write a quick column today from a Montreal hotel room after an amazing workshop yesterday which she will comment on later in the week.

So quick, get some tea and some flannel-lined flannel, because damn it’s wintery outside, all snowy and shit. Aunt Pythia’s about to spew her usual unreasonable nonsense!

This week in Montreal. From

From earlier this week in Montreal. 

LET’S DO THIS PEOPLES!!! And please, even if you’ve got nothing interesting to say for yourself, feel free to make something up or get inspired by Google auto complete and then go ahead and:

ask Aunt Pythia your question at the bottom of the page!

By the way, if you don’t know what the hell Aunt Pythia is talking about, go here for past advice columns and here for an explanation of the name Pythia.


Dear Aunt Pythia,

This may not really be an “Aunt Pythia” question. But could either you or Mathbabe comment on this article on sexism in academic science?

I can imagine many ways they could be misrepresenting the statistics, but I don’t know which.

No Bias, Really?

Dear No Bias,

I was also struck by the inflammatory tone and questionable conclusions of this article. But you know, controversy sells.

So, here are a couple of lines I’ll pull out. First:

Our country desperately needs more talented people in these fields; recruiting more women could address this issue. But the unwelcoming image of the sexist academy isn’t helping. Fortunately, as we have found in a thorough analysis of recent data on women in the academic workplace, it isn’t accurate, either.

And second:

Many of the common, negative depictions of the plight of academic women are based on experiences of older women and data from before the 2000s, and often before the 1990s. That’s not to say that mistreatment doesn’t still occur — but when it does, it is largely anecdotal, or else overgeneralized from small studies.

I guess right off the bat I’d ask, how are you collecting data? The data I have personally about sexist treatment at the hands of my colleagues hasn’t, to my knowledge, been put in any database. The sexist treatment I’ve witnessed for pretty much all of my female mathematics colleagues has, equally, never been installed in a database to my knowledge. So yeah, not convinced these people know what they are talking about. It’s famously hard to prove something doesn’t exist, especially when you don’t have a search algorithm.

One possibility for the data they seem to have: they interviewed people after the fact, perhaps decades after the fact. If that’s the case, then you’d expect more and better data on older women, and that’s what we are currently seeing. There is a lag on this data collection, in other words. That’s not the same as “it doesn’t exist.” A common mistake researchers make. They take the data as “objective truth” and forget that it’s a human process to collect it (or not collect it!). Think police shootings.

The article then goes on to talk about how the data for women in math and other science fields isn’t so bad in terms of retention, promotion, and other issues. For there I’d say, the women have already gone through a mighty selection process, so in general you’d expect them to be smarter than their colleagues, so in general their promotion rates should be higher, but they aren’t. So that’s also a sign of sexism.

I mean, whatever. That’s not actually what I claim is true, so much as another interpretation of this data. My overall point is that, they have some data, and they are making strong and somewhat outrageous claims which I can dismiss without much work.

I hope that helps!

Aunt Pythia


Dear Aunt Pythia,

In his November “Launchings” column, David Bressoud has presents some interesting data on differences between male and female college calculus students. As much as I’ve appreciated all of Bressoud’s careful explorations of mathematics education, I find I’m a bit irritated by his title, “MAA Calculus Study: Women Are Different,” because it appears to take the male experience as the norm.

Perhaps I was already annoyed because of a NYTimes op-ed, “Academic Science Isn’t Sexist”, in which Wendy Williams and Steven Ceci claim that “[w]e are not your father’s academy anymore,” and that the underrepresentation of women in math-intensive fields is “rooted in women’s earlier educational choices, and in women’s occupational and lifestyle preferences.” Here, too, the message seems to be “don’t worry about changing the academy — women are different from the norm, which is (naturally) that which works for men.”

My question for you, Aunt Pythia, is this: am I overreacting here?

I received my PhD in mathematics in 1984, and I’ve seen significant change for the better in the academy since then. Child care at AMS meetings? A crowd in the women’s rest room at same? Unthinkable when I started. But if women are still disproportionately “choosing” to go into other fields, might we look a little more closely at the environments in which girls and women are making their educational and “lifestyle” choices?

I welcome your thoughts. If you’re eager for more data analysis, I’d also love to hear your take on the paper by Williams, Ceci, and their colleagues.

Still One of the Underrepresented After All These Years


Without even reading that article, I can say without hesitation that yes, it’s a ridiculous title, and it’s infuriating and YOU ARE NOT OVERREACTING. To be clear, that is bold-faced, italicized, and all caps. I mean it.

The word “different” forces us to compare something to a baseline, and given that there is no baseline even mentioned, we are forced to guess at it, and that imposes the “man as default” mindset. Fuck that. I mean, if the title had been, “There are differences between male and female calculus students,” I would not have been annoyed, because even though “male” comes first, I’m not a stickler. I just want to acknowledge that if we mention one category, we mention the other as well.

To illustrate this a bit more, we don’t entitle a blog post “Whites are different” and leave it at that, because we’d be like, different from whom? From blacks? From Asians? From Asian-Americans? See how that works? You need to say different from some assumed baseline, and the assumed baseline has to be a cultural norm. And right now it’s white male. Which is arguable one reason that calculus students act differently when they are men (har!).

As for the other article, I already shit on that in the previous answer but I’m happy to do it once again. It’s bullshit, and I’m disappointed that the Times published it.

As for the article, I don’t have time now but I’ll take a look, thanks!

Aunt Pythia


Dear Aunt Pythia,

I am twenty years old, near the halfway point in my senior year of a mathematics BS at a large, well-regarded public university in the Northeast. I’ve been aiming my energies at graduate school, and I am now looking at PhD program applications. Most apps ask for two or three letters of recommendation from a faculty member who is familiar with your work. This poses a very big problem, because all of my professors hate me.

Okay, maybe it’s not quite like that. But I’ve had a really lousy time in the math department at LWRPUN. My fellow students are dispassionate, unresponsive, and unfriendly. My professors are dry, uncommitted to their students, and the ones who aren’t mathematically incompetent are lousy teachers. On top of all this, a crippling bureaucracy has prevented me countless times from taking classes I’m interested in (few as they are in this catalog), substituting instead ANOTHER REQUIRED SEMESTER OF ANALYSIS.

So I haven’t made any personal connections of the sort that might benefit me in the form of a letter of rec. My work hasn’t even been that good; my depression and anxiety (in general as well as re all this) have increasingly prevented me from completing even easy homework assignments. Nobody here has seen my best mathematical work, and for that matter, nobody anywhere else has either*.

And for four years, everyone I’ve come to with this gathering creeping progressively life-eating concern has given me the same old BS about You should really put yourself out there! and It’s just so important to go to your professor’s office hours! without considering maybe — I’ve tried, I really have.

What can I do, Aunt Pythia? I’m really passionate about mathematics, but I’m worried I won’t be able to pursue my studies without these magic papers.

Reports Embargoed by Crummy Lecturers, Earnestly Seeking Solace

*I thankfully have a professor from an outside experience willing to write about my teaching credentials, but that one letter is surely not sufficient to show my potential as a graduate student and researcher.


I am afraid I will have to call bullshit on you, RECLESS. Plus your sign-off doesn’t actually spell anything.

Here’s the thing, there are no mathematically incompetent lecturers at large, well-regarded public universities. There are, in fact, mathematically very competent people who can’t get jobs at such places. Such is the pyramid-shaped job market of mathematics. So whereas I believe you when you say your lecturers have been uninspired, and uncommitted to their students, the fact that you added “mathematically incompetent” just makes me not believe you at all, in anything.

Here’s what I think is happening. You think you’re really into math, but you’ve never really understood your classes, nor have you understood that you’ve never understood your classes, because your self-image is that you’re already a mathematician, and that people have just not acknowledged your brilliance.

But that’s not how math actually works. Math is a social endeavor, where you have to communicate your ideas well enough for others to understand them, or else you aren’t doing math.

I’m not saying you haven’t had bad luck with teachers. It’s a real possibility. But there’s something else going on as well, and I don’t think you can honestly expect to go to the next level without sorting stuff out. In other words, even if you don’t love the teacher, if you loved the subject, got into it, and did the proofs, you’d still be getting adequate grades to ask for letters. The thing about writing letters, as a math prof, is that you don’t have to like the student personally to write a good letter, you just need to admire their skills. But since you can’t do that either, you won’t get good letters, and moreover I don’t think you’d deserve good letters. And therefore I don’t think you should go to grad school.

Suggestion: look carefully at your own behavior, figure out what it is you are doing that isn’t working. Maybe think of what you love about math, or about your own image of being a mathematician, and see if there’s something you really know you’re good at, and other people know it to, and develop that.

Good luck,

Aunt Pythia


Dearest Aunt Pythia,

I have a sex question for you! Kind of. You have to get through the boring back story first…I’m a 19 year old female physics major. I’m quiet, rather mousy, and awkward. A lot of the time I feel like I have more to prove than the boys do, because I’m a girl, and because of the aforementioned shyness.

People seem to automatically assume I’m unintelligent. I think I’m just as intelligent as the boys in my program, but I don’t come off that way! Point is, I want to be this cool, strong, independent, successful, respectable girl who doesn’t take shit from misogynistic people who assume I’m inferior.

However, I feel extremely guilty about my sexual preferences. I’m pretty submissive. I’d like power exchange in my relationships…hair pulling, bondage, spanking, being bossed around, the whole bit. I like to be dominated by men. Older men. Smart older men. Hopefully I’ve successfully conveyed my dilemma. I want to be respected by the men (and women, and others) I’m surrounded by in my academic life, but taken control of as a girlfriend.

Why does what I despise happening to me in an academic setting please me so much in a romantic/sexual one? Agh, I feel like such a bad girl! (and not in the arousing way…)

Much Love,

Dear Conflicted,

This is such a relief – finally, a sex question! – and it’s honestly one of the best questions I’ve ever gotten, ever, in Aunt Pythia or elsewhere. I’m so glad I can answer this for you.

It is absolutely not in conflict to want something in a sexual context that is abhorrent to you in normal life. It is in fact a well-known pattern! You shouldn’t feel at all weird about it! Lots – LOTS – of the submissives I’ve met are, in their day jobs, the boss, literally. They have companies and are extremely fancy and in control. And then they love to be bossed around and spanked. Seriously. If anything, my feeling is that your sexual proclivities point to being alpha in real life, but maybe I’m going overboard.

So yeah, no problem here. You are killing it. And in 3 or 4 years I want you to write back and explain to me how you’ve found an amazing lover who gives you what you want in the bedroom and worships your physics prowess outside it. There will, in fact, be people lining up for this role.

And those people in your program? Do your best to ignore them. Men are just impossibly arrogant at that age, but time will humble them somewhat even as your confidence will rise as you learn more. I’m not saying it ever evens out entirely but it does improve.

Also: find other women (and super cool men) to study with. Surround yourself with supportive people. Take note of obnoxious people and avoid them. Trade up with friends whenever possible.

Love always,

Aunt Pythia


Well, you’ve wasted yet another Saturday morning with Aunt Pythia! I hope you’re satisfied! Please if you could, ask me a question. And don’t forget to make an amazing sign-off, they make me very very happy.

Click here for a form or just do it now:


Educational feedback loops in China and the U.S.

Today I want to discuss a recent review in New York Review of Books, on a new book entitled Who’s Afraid of the Big Bad Dragon? Why China Has the Best (and Worst) Education System in the World by Yong Zhao (hat tip Alex). The review was written by Diane Ravitch, an outspoken critic of No Child Left Behind, Race To The Top, and the Common Core.

You should read the review, it’s well written and convincing, at least to me. I’ve been studying these issues and devoted a large chunk of my book to the feedback loops described as they’ve played out in this country. Here are the steps I see, which are largely reflected in Ravitch’s review:

  1. Politicians get outraged about a growing “achievement gap” (whereby richer or whiter students get better test scores than poorer or browner students) and/or a “lack of international competitiveness” (whereby students in countries like China get higher international standardized test scores than U.S. students).
  2. The current president decides to “get tough on education,” which translates into new technology and way more standardized tests.
  3. The underlying message is that teachers and students and possibly parents are lazy and need to be “held accountable” to improve test scores. The even deeper assumption is that test scores are the way to measure quality of learning.
  4. Once there’s lots of attention being given to test scores, lots of things start happening in response (the “feedback loop”).
  5. For example, widespread cheating by students and teachers and principals, especially when teachers and principals get paid based on test performance.
  6. Also, well-off students get more and better test prep, so the achievement gap gets wider.
  7. Even just the test scores themselves lead to segregation by class: parents who can afford it move to towns with “better schools,” measured by test scores.
  8. International competitiveness doesn’t improve. But we’ve actually never been highly ranked since we started measuring this.

What Zhao’s book adds to this is how much worse it all is in China. Especially the cheating. My favorite excerpt from the book:

Teachers guess possible [test] items, companies sell answers and wireless cheating devices to students, and students engage in all sorts of elaborate cheating. In 2013, a riot broke out because a group of students in Hubei Province were stopped from executing the cheating scheme their parents purchased to ease their college entrance exam.

Ravitch adds after that that ‘an angry mob of two thousand people smashed cars and chanted, “We want fairness. There is no fairness if you do not let us cheat.”’

To be sure, the stakes in China are way higher. Test scores are incredibly important and allow people to have certain careers. But according to Zhao, this selection process, which is quite old, has stifled creativity in the Chinese educational system (so, in other words, test scores are the wrong way to measure learning, in part because of the feedback loop). He blames the obsession with test scores on the fact that no Chinese native has received a Nobel Prize since 1949, for example: the winners of that selection process are not naturally creative.

Furthermore, Zhao claims, the Chinese educational system stifles individuality and forces conformity. It is an authoritarian tool.

In that light, I guess we should be proud that our international scores are lower than China’s; maybe it is evidence that we’re doing something right.

I know that, as a parent, I am sensitive to these issues. I want my kids to have discipline in some ways, but I don’t want them to learn to submit themselves to an arbitrary system for no good reason. I like the fact that they question why they should do things like go to bed on time, and exercise regularly, and keep their rooms cleanish, and I encourage their questions, even while I know I’m kind of ruining their chances at happily working in a giant corporation and being a conformist drone.

This parenting style of mine, which I believe is pretty widespread, seems reasonable to me because, at least in my experience, I’ve gotten further by being smart and clever than by being exactly what other people have wanted me to be. And I’m glad I live in a society that rewards quirkiness and individuality.

Student evaluations: very noisy data

I’ve been sent this recent New York Times article by a few people (thanks!). It’s called Grading Teachers, With Data From Class, and it’s about how standardized tests are showing themselves to be inadequate to evaluate teachers, so a Silicon Valley-backed education startup called Panorama is stepping into the mix with a data collection process focused on student evaluations.

Putting aside for now how much this is a play for collecting information about the students themselves, I have a few words to say about the signal which one gets from student evaluations. It’s noisy.

So, for example, I was a calculus teacher at Barnard, teaching students from all over the Columbia University community (so, not just women). I taught the same class two semesters in a row: first in Fall, then in Spring.

Here’s something I noticed. The students in the Fall were young (mostly first semester frosh), eager, smart, and hard-working. They loved me and gave me high marks on all categories, except of course for the few students who just hated math, who would typically give themselves away by saying “I hate math and this class is no different.”

The students in the Spring were older, less eager, probably just as smart, but less hard-working. They didn’t like me or the class. In particular, they didn’t like how I expected them to work hard and challenge themselves. The evaluations came back consistently less excited, with many more people who hated math.

I figured out that many of the students had avoided this class and were taking it for a requirement, didn’t want to be there, and it showed. And the result was that, although my teaching didn’t change remarkably between the two semesters, my evaluations changed considerably.

Was there some way I could have gotten better evaluations from that second group? Absolutely. I could have made the class easier. That class wanted calculus to be cookie-cutter, and didn’t particularly care about the underlying concepts and didn’t want to challenge themselves. The first class, by contrast, had loved those things.

My conclusion is that, once we add “get good student evaluations” to the mix of requirements for our country’s teachers, we are asking for them to conform to their students’ wishes, which aren’t always good. Many of the students in this country don’t like doing homework (in fact most!). Only some of them like to be challenged to think outside their comfort zone. We think teachers should do those things, but by asking them to get good student evaluations we might be preventing them from doing those things. A bad feedback loop would result.

I’m not saying teachers shouldn’t look at student evaluations; far from it, I always did and I found them useful and illuminating, but the data was very noisy. I’d love to see teachers be allowed to see these evaluations without there being punitive consequences.

Guest Post: Bring Back The Slide Rule!

This is a guest post by Gary Cornell, a mathematician, writer, publisher, and recent founder of StemForums.

I was was having a wonderful ramen lunch with the mathbabe and, as is all too common when two broad minded Ph.D.’s in math get together, we started talking about the horrible state math education is in for both advanced high school students and undergraduates.

One amusing thing we discovered pretty quickly is that we had independently come up with the same (radical) solution to at least part of the problem: throw out the traditional sequence which goes through first and second year calculus and replace it with a unified probability, statistics, calculus course where the calculus component was only for the smoothest of functions and moreover the applications of calculus are only to statistics and probability. Not only is everything much more practical and easier to motivate in such a course, students would hopefully learn a skill that is essential nowadays: how to separate out statistically good information from the large amount of statistical crap that is out there.

Of course, the downside is that the (interesting) subtleties that come from the proofs, the study of non-smooth functions and for that matter all the other stuff interesting to prospective physicists like DiffEQ’s would have to be reserved for different courses. (We also were in agreement that Gonick’s beyond wonderful“Cartoon Guide To Statistics” should be required reading for all the students in these courses, but I digress…)

The real point of this blog post is based on what happened next: but first you have to know I’m more or less one generation older than the mathbabe. This meant I was both able and willing to preface my next point with the words: “You know when I was young, in one way students were much better off because…” Now it is well known that using this phrase to preface a discussion often poisons the discussion but occasionally, as I hope in this case, some practices from days gone by ago can if brought back, help solve some of today’s educational problems.

By the way, and apropos of nothing, there is a cure for people prone to too frequent use of this phrase: go quickly to YouTube and repeatedly make them watch Monty Python’s Four Yorkshireman until cured:

Anyway, the point I made was that I am a member of the last generation of students who had to use slide rules. Another good reference is: here. Both these references are great and I recommend them. (The latter being more technical.) For those who have never heard of them, in a nutshell, a slide rule is an analog device that uses logarithms under the hood to do (sufficiently accurate in most cases) approximate multiplication, division, roots etc.

The key point is that using a slide rule requires the user to keep track of the “order of magnitude” of the answers— because slide rules only give you four or so significant digits. This meant students of my generation when taking science and math courses were continuously exposed to order of magnitude calculations and you just couldn’t escape from having to make order of magnitude calculations all the time—students nowadays, not so much. Calculators have made skill at doing order of magnitude calculations (or Fermi calculations as they are often lovingly called) an add-on rather than a base line skill and that is a really bad thing. (Actually my belief that bringing back slide rules would be a good thing goes back a ways: when that when I was a Program Director at the NSF in the 90’s, I actually tried to get someone to submit a proposal which would have been called “On the use of a hand held analog device to improve science and math education!” Didn’t have much luck.)

Anyway, if you want to try a slide rule out, alas, good vintage slide rules have become collectible and so expensive— because baby boomers like me are buying the ones we couldn’t afford when we were in high school – but the nice thing is there are lots of sites like this one which show you how to make your own.

Finally, while I don’t think they will ever be as much fun as using a slide rule, you could still allow calculators in classrooms.

Why? Because it would be trivial to have a mode in the TI calculator or the Casio calculator that all high school students seem to use, called “significant digits only.” With the right kind of problems this mode would require students to do order of magnitude calculations because they would never be able to enter trailing or leading zeroes and we could easily stick them with problems having a lot of them!

But calculators really bug me in classrooms and, so I can’t resist pointing out one last flaw in their omnipresence: it makes students believe in the possibility of ridiculously high precision results in the real world. After all, nothing they are likely to encounter in their work (and certainly not in their lives) will ever need (or even have) 14 digits of accuracy and, more to the point, when you see a high precision result in the real world, it is likely to be totally bogus when examined under the hood.


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