## Nerding out: RSA on an iPython Notebook

Yesterday was a day filled with secrets and codes. In the morning, at The Platform, we had guest speaker Columbia history professor Matthew Connelly, who came and talked to us about his work with declassified documents. Two big and slightly depressing take-aways for me were the following:

- As records have become digitized, it has gotten easy for people to get rid of archival records in large quantities. Just press delete.
- As records have become digitized, it has become easy to trace the access of records, and in particular the leaks. Connelly explained that, to some extent, Obama’s harsh approach to leakers and whistleblowers might be explained as simply “letting the system work.” Yet another way that technology informs the way we approach human interactions.

After class we had section, in which we discussed the Computer Science classes some of the students are taking next semester (there’s a list here) and then I talked to them about prime numbers and the RSA crypto system.

I got really into it and wrote up an iPython Notebook which could be better but is pretty good, I think, and works out one example completely, encoding and decoding the message “hello”.

The underlying file is here but if you want to view it on the web just go here.

## The platonic solids

I managed to record this week’s Slate Money podcast early so I could drive up to HCSSiM for July 17th, and the Yellow Pig Day celebration. I missed the 17 talk but made it in time for yellow pig carols and cake.

This morning my buddy Aaron decided to let me talk to the kids in the last day of his workshop. First Amber is working out the formula for the Euler Characteristic of a planar graph with the kids and after that I’ll help them count the platonic solids using stereographic projection. If we have time we’ll talk about duals (update: we had time!).

Tonight at Prime Time I’ll play a game or two of Nim with them.

## Correlation does not imply equality

One of the reasons I enjoy my blog is that I get to try out an argument and then see if readers can 1) poke holes in my arguement, or 2) if they misunderstand my argument, or 3) if they misunderstand something tangential to my argument.

Today I’m going to write about an issue of the third kind. Yesterday I talked about how I’d like to see the VAM scores for teachers directly compared to other qualitative scores or other VAM scores so we could see how reliably they regenerate various definitions of “good teaching.”

The idea is this. Many mathematical models are meant to replace a human-made model that is deemed too expensive to work out at scale. Credit scores were like that; take the work out of the individual bankers’ hands and create a mathematical model that does the job consistently well. The VAM was originally intended as such – in-depth qualitative assessments of teachers is expensive, so let’s replace them with a much cheaper option.

So all I’m asking is, how good a replacement is the VAM? Does it generate the same scores as a trusted, in-depth qualitative assessment?

When I made the point yesterday that I haven’t seen anything like that, a few people mentioned studies that show *positive correlations* between the VAM scores and principal scores.

But here’s the key point: *positive correlation does not imply equality.*

Of course sometimes positive correlation is good enough, but sometimes it isn’t. It depends on the context. If you’re a trader that makes thousands of bets a day and your bets are positively correlated with the truth, you make good money.

But on the other side, if I told you that there’s a ride at a carnival that has a positive correlation with not killing children, that wouldn’t be good enough. You’d want the ride to be safe. It’s a higher standard.

I’m asking that we make sure we are using that second, higher standard when we score teachers, because their jobs are increasingly on the line, so it matters that we get things right. Instead we have a machine that nobody understand that is *positively correlated* with things we do understand. I claim that’s not sufficient.

Let me put it this way. Say your “true value” as a teacher is a number between 1 and 100, and the VAM gives you a noisy approximation of your value, which is 24% correlated with your true value. And say I plot your value against the approximation according to VAM, and I do that for a bunch of teachers, and it looks like this:

So maybe your “true value” as a teacher is 58 but the VAM gave you a zero. That would not just be frustrating to you, since it’s taken as an important part of your assessment. You might even lose your job. And you might get a score of zero many years in a row, even if your true score stays at 58. It’s increasingly unlikely, to be sure, but given enough teachers it is bound to happen to a handful of people, just by statistical reasoning, and if it happens to you, you will not think it’s unlikely at all.

In fact, if you’re a teacher, you should demand a scoring system that is consistently the same as a system you understand rather than positively correlated with one. If you’re working for a teachers’ union, feel free to contact me about this.

One last thing. I took the above graph from this post. These are actual VAM scores for the same teacher in the same year but for two different class in the same subject – think 7th grade math and 8th grade math. So neither score represented above is “ground truth” like I mentioned in my thought experiment. But that makes it even more clear that the VAM is an insufficient tool, because it is only 24% correlated *with itself*.

## Why Chetty’s Value-Added Model studies leave me unconvinced

Every now and then when I complain about the Value-Added Model (VAM), people send me links to recent papers written Raj Chetty, John Friedman, and Jonah Rockoff like this one entitled *Measuring the Impacts of Teachers II: Teacher Value-Added and Student Outcomes in Adulthood *or its predecessor *Measuring the Impacts of Teachers I: Evaluating Bias in **Teacher Value-Added Estimates*.

I think I’m supposed to come away impressed, but that’s not what happens. Let me explain.

Their data set for students scores start in 1989, well before the current value-added teaching climate began. That means teachers weren’t teaching to the test like they are now. Therefore saying that the current VAM works because an retrograded VAM worked in 1989 and the 1990’s is like saying I must like blueberry pie now because I used to like pumpkin pie. It’s comparing apples to oranges, or blueberries to pumpkins.

I’m surprised by the fact that the authors don’t seem to make any note of the difference in data quality between pre-VAM and current conditions. They should know all about feedback loops; any modeler should. And there’s nothing like telling teachers they might lose their job to create a mighty strong feedback loop. For that matter, just consider all the cheating scandals in the D.C. area where the stakes were the highest. Now that’s a feedback loop. And by the way, I’ve never said the VAM scores are totally meaningless, but just that they are not precise enough to hold individual teachers accountable. I don’t think Chetty et al address that question.

So we can’t trust old VAM data. But what about recent VAM data? Where’s the evidence that, in this climate of high-stakes testing, this model is anything but random?

If it were a good model, we’d presumably be seeing a comparison of current VAM scores and current other measures of teacher success and how they agree. But we aren’t seeing anything like that. Tell me if I’m wrong, I’ve been looking around and I haven’t seen such comparisons. And I’m sure they’ve been tried, it’s not rocket science to compare VAM scores with other scores.

The lack of such studies reminds me of how we never hear about scientific studies on the results of Weight Watchers. There’s a reason such studies never see the light of day, namely because whenever they do those studies, they decide they’re better off not revealing the results.

And if you’re thinking that it would be hard to know exactly how to rate a teacher’s teaching in a qualitative, trustworthy way, then yes, that’s the point! It’s actually not obvious how to do this, which is the real reason we should never trust a so-called “objective mathematical model” when we can’t even decide on a definition of success. We should have the conversation of what comprises good teaching, and we should involve the teachers in that, and stop relying on old data and mysterious college graduation results 10 years hence. What are current 6th grade teachers even supposed to *do* about studies like that?

Note I do think educators and education researchers should be talking about these questions. I just don’t think we should punish teachers arbitrarily to have that conversation. We should have a notion of best practices that slowly evolve as we figure out what works in the long-term.

So here’s what I’d love to see, and what would be convincing to me as a statistician. If we see all sorts of qualitative ways of measuring teachers, and see their VAM scores as well, and we could compare them, and make sure they agree with each other *and themselves* over time. In other words, at the very least we should demand an explanation of how some teachers get totally ridiculous and inconsistent scores from one year to the next and from one VAM to the next, even in the same year.

The way things are now, the scores aren’t sufficiently sound be used for tenure decisions. They are too noisy. And if you don’t believe me, consider that statisticians and some mathematicians agree.

We need some ground truth, people, and some common sense as well. Instead we’re seeing retired education professors pull statistics out of thin air, and it’s an all-out war of supposed mathematical objectivity against the civil servant.

## How Not To Be Wrong by Jordan Ellenberg

You guys are in for a treat. In fact I’m jealous of you.

I had a little secret about my survival in grad school, and that secret has a name, and that name is Jordan Ellenberg. We used to meet every Tuesday and Thursday to study schemes at the CallaLily Cafe a few blocks from the Science Center on Kirkland Street, and even though that sounds kind of dull, it was a blast. It was what kept me sane at Harvard.

You see, Jordan has an infectious positivity about him, which balances my rather intense suspicions, and moreover he’s hilariously funny. He’s really somewhere between a mathematician and a stand-up comedian, and to be honest I don’t know which one he’s better at, although he is a deeply talented mathematician.

The reason I’m telling you this is that he’s written a book, called How Not To Be Wrong, and available for purchase starting today, which is a delight to read and which will make you understand why I survived graduate school. In fact nobody will ever let me complain again once they’ve read this book, because it reads just like Jordan talks. In reading it, I felt like I was right back at CallaLily, singing Prince’s “Sexy MF” and watching Jordan flirt with the cashier lady again. Aaaah memories.

So what’s in the book? Well, he talks a lot about math, and about mathematicians, and the lottery, and in fact he has this long riff which starts out with lottery math, then goes to error-correcting codes and then to made-up languages and then to sphere packing and then arrives again at lotteries. And it’s brilliant and true and beautiful and also funny.

I have a theory about this book that you could essentially open it up to any page and begin to enjoy it, since it is thoroughly enjoyable and the math is cumulative but everywhere so well explained that it wouldn’t take long to follow along, and pretty soon you’d be giggling along with Jordan at every ridiculous footnote he’s inserted into his narrative.

In other words, every page is a standalone positive and ontological examination of the beauty and surprise of mathematical discovery. And so, if you are someone who shares with Jordan a love for mathematics, you will have a consistently great time with this book. In fact I’m imagining that you have an uncle or a mom who loves math or science, in which case this would be a seriously perfect gift to them, but of course you could also give that gift to yourself. I mean, this is a guy who can make nazi jokes funny, and he does.

Having said that, the magic of the book is that it’s not just a collection of wonderful mathy tidbits. Jordan also has a point about the act of scrutinizing something in a logical and mathematical fashion. That act itself is courageous and should be appreciated, and he explains why, and he tells us how much we’ve already benefited from people in the past who have had the bravery to do so. He appreciates them and we should too.

And yet, he also sends the important message that it’s not an elitist crew of the usual genius suspects, that in fact we can all do this in our own capacity. It’s a great message and, if it ends up allowing people to re-examine their need for certainty in an uncertain world, then Jordan will really end up doing good. Fingers crossed.

That’s not to say it’s a perfect book, and I wanted to argue with points on basically every other page, but mostly in a good, friendly, over-drinks kind of way, which is provocative but not annoying. One exception I might make came on page 256: no, Jordan, municipal bonds do not always get paid back, and no, stocks do not always go up, not even in expectation. In fact to the extent that both of those statements seem true to many people is the result of many cynical political acts and is damaging, mostly to people like retired civil servants. Don’t go there!

Another quibble: Jordan talks about how public policy makers make proclamations in the face of uncertainty, and he has a lot of sympathy and seems to think the should keep doing this. I’m on the other side on this one. Telling people to avoid certain foods and then changing stances seems more damaging than helpful and it happens constantly. And it’s often tied to industry and money, which also doesn’t impress.

Even so, even when I strongly disagree with Jordan, I always want to have the conversation. He forces that on the reader because he’s so darn positive and open-minded.

A few more goodies that I wanted to adore without giving too much away. Jordan does a great job with something he calls “The Great Square of Men” and Berkson’s Fallacy: it will explain to many many women why they are not finding the man they’re looking for. He also throws out a bone to nerds like me when he almost proves that every pig is yellow, and he absolutely kills it, stand-up comedian style, when comparing Ross Perot to a small dark pile of oats. Holy crap he was on a roll there.

So here’s one thing I’ve started doing since reading the book. When I give my 5-year-old son his dessert, it’s in the form of Hershey Drops, which are kind of like fat M&M’s. I give him 15 and I ask him to count them to make sure I got it right. Sometimes I give him 14 to make sure he’s paying attention. But that’s not the new part. The new part is something I stole from Jordan’s book.

The new part is that some days I ask him, “do you want me to give you 3 rows of 5 drops?” And I wait for him to figure out that’s enough and say “yes!” And the other days I ask him “do you want me to give you 5 rows of 3 drops?” and I again wait. And in either case I put the drops out in a rectangle.

And last night, for the first time, he explained to me in a slightly patronizing voice that it doesn’t matter which way I do it because it ends up being the same, because of the rectangle formation and how you look at it. And just to check I asked him which would be more, 10 rows of 7 drops or 7 rows of 10 drops, and he told me, “duh, it would be the same because it couldn’t be any different.”

And that, my friends, is how not to be wrong.

## Is math an art or a science?

I left academic math in 2007, but I still identify as a mathematician. That’s just how I think about the world, through a mathematician’s mindset, whatever that means.

Wait what *does* that mean? How do I characterize the mathematician’s mindset? I’ve struggled in the past to try, but a few days ago, a part of it got a little bit easier.

I was talking to my friend Matt Jones – an historian of science, actually – about the turf wars inside computer science surrounding functional versus object oriented programming. It seems like questions about which one is better or when is one more appropriate than the other have become so political that they are no longer inside the scientifically acceptable realm.

But that kind of reminded me of the turf war of the bayesian versus frequentist statisticians. Or the fresh water versus salt water economists. Or possibly the string theorists versus the non-string theorists in physics.

What’s going on in all of those fields, as best I can understand, is that different groups within the field have different assumptions about what the field may assume and what it’s trying to accomplish, and they fight over the validity of those sets of assumptions. The fights themselves, which are often emotional and brutal, expose the underlying assumptions in certain ways. Matt told me that historians often get at a fields assumptions through these wars.

Here’s the thing, though, math doesn’t have that. I’m not saying there are no turf wars at all in math, there certainly are, but they aren’t political in nature exactly. They are aesthetic.

In the context of mathematics, where nothing can be considered truly inappropriate as long as the assumptions are clear, it’s all about whether something is beautiful or important, not whether it is valid. Validity has no place in mathematics per se, which plays games with logical rules and constructs. I could go off an build a weird but internally logical universe on my own, and no mathematician would complain it’s invalid, they’d only complain it’s unimportant if it doesn’t tie back to their field and help them prove a theorem.

I claim that this turf war issue is a characterizing issue of the field of mathematics versus the other sciences, and makes it more of an art than a science.

To finish my argument I’d need to understand more about how artistic fields fight, and in particular that their internally hurled insults focus more on aesthetics than on validity, say in composition or painting. I can’t imagine it otherwise, but who knows. Readers, please chime in with evidence in either direction.

## Interview with a middle school math teacher on the Common Core

Today’s post is an email interview with Fawn Nguyen, who teaches math at Mesa Union Junior High in southern California. Fawn is on the leadership team for UCSB Mathematics Project that provides professional development for teachers in the Tri-County area. She is a co-founder of the Thousand Oaks Math Teachers’ Circle. In an effort to share and learn from other math teachers, Fawn blogs at Finding Ways to Nguyen Students Over. She also started VisualPatterns.org to help students develop algebraic thinking, and more recently, she shares her students’ daily math talks to promote number sense. When Fawn is not teaching or writing, she is reading posts on mathblogging.org as one of the editors. She sleeps occasionally and dreams of becoming an architect when all this is done.

Importantly for the below interview, Fawn is not being measured via a value-added model. My questions are italicized.

——

*I’ve been studying the rhetoric around the mathematics Common Core State Standard (CCSS). So far I’ve listened to Diane Ravitch stuff, I’ve interviewed Bill McCallum, the lead writer of the math CCSS, and I’ve also interviewed Kiri Soares, a New York City high school principal. They have very different views. Interestingly, McCallum distinguished three things: standards, curriculum, and testing. *

*What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they’re presented?*

I can’t speak for other teachers. I understand that the standards are not meant to be the curriculum, but the two are not mutually exclusive either. They can’t be. Standards inform the curriculum. This might be a terrible analogy, but I love food and cooking, so maybe the standards are the major ingredients, and the curriculum is the entrée that contains those ingredients. In the show Chopped on Food Network, the competing chefs must use all 4 ingredients to make a dish – and the prepared foods that end up on the plates differ widely in taste and presentation. We can’t blame the ingredients when the dish is blandly prepared any more than we can blame the standards when the curriculum is poorly written.

Similary, the standards inform testing. Test items for a certain grade level cover the standards of that grade level. I’m not against testing. I’m against bad tests and a lot of it. By bad, I mean multiple-choice items that require more memorization than actual problem solving. But I’m confident we can create good multiple-choice tests because realistically a portion of the test needs to be of this type due to costs.

The three – standards, curriculum, and testing – are not a “package deal” in the sense that the same people are not delivering them to us. But they go together, otherwise what is school mathematics? Funny thing is we have always had the three operating in schools, but somehow the Common Core State Standands (CCSS) seem to get the all the blame for the anxieties and costs connected to testing and curriculum development.

*As a teacher, what’s good and bad about the CCSS?*

I see a lot of good in the CCSS. This set of standards is not perfect, but it’s much better than our state standards. We can examine the standards and see for ourselves that the integrity of the standards holds up to their claims of being embedded with mathematical focus, rigor, and coherence.

Implementation of CCSS means that students and teachers can expect consistency in what is being in taught at each grade level across state boundaries. This is a nontrivial effort in addressing equity. This consistency also helps teachers collaborate nationwide, and professional development for teachers will improve and be more relevant and effective.

I can only hope that textbooks will be much better because of the inherent focus and coherence in CCSS. A kid can move from Maine to California and not have to see different state outlines on their textbooks as if he’d taken on a new kind of mathematics in his new school. I went to a textbook publishers fair recently at our district, and I remain optimistic that better products are already on their way.

We had every state create its own assessment, now we have two consortia, PARCC and Smarter Balanced. I’ve gone through the sample assessments from the latter, and they are far better than the old multiple-choice items of the CST. Kids will have to process the question at a deeper level to show understanding. This is a good thing.

What is potentially bad about the CCSS is the improper or lack of implementation. So, this boils down to the most important element of the Common Core equation – the teacher. There is no doubt that many teachers, myself included, need sustained professional development to do the job right. And I don’t mean just PD in making math more relevant and engaging, and in how many ways we can use technology, I mean more importantly, we need PD in content knowledge.

It is a perverse notion to think that anyone with a college education can teach elementary mathematics. Teaching mathematics requires *knowing* mathematics. To know a concept is to understand it backward and forward, inside and outside, to recognize it in different forms and structures, to put it into context, to ask questions about it that leads to more questions, to know the mathematics *beyond* this concept. That reminds me just recently a 6th grader said to me as we were working on our unit of dividing by a fraction. She said, “My elementary teacher lied to me! She said we always get a smaller number when we divide two numbers.”

Just because one can make tuna casserole does not make one a chef. (Sorry, I’m hungry.)

*What are the good and bad things for kids about testing?*

Testing is only good for kids when it helps them learn and become more successful – that the feedback from testing should inform the teacher of next moves. Testing has become such a dirty word because we over test our kids. I’m still in the classroom after 23 years, yet I don’t have the answers. I struggle with telling my kids that I value them and their learning, yet at the end of each quarter, the narrative sum of their learning is a letter grade.

Then, in the absence of helping kids learn, testing is bad.

*What are the good/bad things for the teachers with all these tests?*

Ideally, a good test that measures what it’s supposed to measure should help the teacher and his students. Testing must be done in moderation. Do we really need to test kids at the start of the school year? Don’t we have the results from a few months ago, right before they left for summer vacation? Every test takes time away from learning.

I’m not sure I understand why testing is bad for teachers aside from lost instructional minutes. Again, I can’t speak for other teachers. But I do sense heightened anxiety among some teachers because CCSS is new – and newness causes us to squirm in our seats and doubt our abilities. I don’t necessarily see this as a bad thing. I see it as an opportunity to learn content at a deeper conceptual level and to implement better teaching strategies.

If we look at anything long and hard enough, we are bound to find the good and the bad. I choose to focus on the positives because I can’t make the day any longer and I can’t have fewer than 4 hours of sleep a night. I want to spend my energies working with my administrators, my colleagues, my parents to bring the best I can bring into my classroom.

*Is there anything else you’d like to add?*

The best things about CCSS for me are not even the standards – they are the 8 Mathematical Practices. These are life-long habits that will serve students well, in all disciplines. They’re equivalent to the essential cooking techniques, like making roux and roasting garlic and braising kale and shucking oysters. Okay, maybe not that last one, but I just got back from New Orleans, and raw oysters are awesome.

I’m excited to continue to share and collaborate with my colleagues locally and online because we now have a common language! We teachers do this very hard work – day in and day out, late into the nights and into the weekends – because we love our kids and we love teaching. But we need to be mathematically competent first and foremost to teach mathematics. I want the focus to always be about the kids and their learning. We start with them; we end with them.

## Interview with a high school principal on the math Common Core

In my third effort to understand the Common Core State Standards (CC) for math, I interviewed an old college friend Kiri Soares, who is the principal and co-founder of the *Urban Assembly Institute of Math and Science for Young Women*. Here’s a transcript of the interview which took place earlier this month. My words are in italics below.

——

*How are high school math teachers in New York City currently evaluated?*

Teachers are now evaluated on 2 things:

- First,
*measures of teacher practice*, which are based on observations, in turn based on some rubric. Right now it’s the Danielson Rubric. This is a qualitative measure. In fact it is essentially an old method with a new name. - Second,
*measures of student learning*, that is supposed to be “objective”. Overall it is worth 40% of the teacher’s score but it is separated into two 20% parts, where teachers choose the methodology of one part and principals choose the other. Some stuff is chosen for principals by the city. Any time there is a state test we have to choose it. In terms of the teachers’ choices, there are two ways to get evaluated: goals or growth. Goals are based on a given kid, and the teachers can guess they will get a certain slightly lower score or higher score for whatever reason. Otherwise, it’s a growth-based score. Teachers can also choose from an array of assessments (state tests, performance tests, and third party exams). They can also choose the cohort (their own kids/ the grade/the school). The city also chose performance tasks in some instances.

*Can you give me a concrete example of what a teacher would choose as a goal?*

At the beginning of year you give diagnostic tests to students in your subject. Based on what a given kid scored in September, you extrapolate a guess for their performance in the June test. So if a kid has a disrupted homelife you might guess lower. Teacher’s goal setting is based on these teachers’ guesses.

*So in other words, this is really just a measurement of how well teachers guess?*

Well they are given a baseline and teachers set goals relative to that, but yes. And they are expected to make those guesses in November, possibly well before homelife is disrupted. It definitely makes things more complicated. And things are pretty complicated. Let me say a bit more.

The first three weeks of school are all testing. We test math, social studies, science, and English in every grade, and overall it depending on teacher/principal selections it can take up to 6 weeks, although not in a given subject. Foreign language and gym teachers also getting measured, by the way, based on those other tests. These early tests are diagnostic tests.

Moreover, they are new types of tests, which are called performance-based assessments, and they are based on writing samples with prompts. They are theoretically better quality because they go deeper, the aren’t just bubble standardized tests, but of course they had no pre-existing baseline (like the state tests) and thus had to be administered as diagnostic. Even so, we are still trying to predict growth based on them, which is confusing since we don’t know how to predict performance on new tests. Also don’t even know how we can consistently grade such essay-based tests- despite “norming protocols”, which is yet another source of uncertainty.

*How many weeks per year is there testing of students?*

The last half of June is gone, a week in January, and 2-3 weeks in the high school in the beginning per subject. That’s a minimum of 5 weeks per subject per year, out of a total of 40 weeks. So one eighth of teacher time is spent administering tests. But if you think about it, for the teachers, it’s even more. They have to grade these tests too.

*I’ve been studying the rhetoric around the CC. So far I’ve listened to Diane Ravitch stuff, and to Bill McCallum, the lead writer of the math CC. They have very different views. McCallum distinguished three things, which when they are separated like that, Ravitch doesn’t make sense. *

*Namely, he separates standards, curriculum, and testing. People complain about testing and say that CC standards make testing easier, and we already have too much testing, so CC is a bad thing. But McCallum makes this point: good standards also make good testing easier. *

*What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they’re presented?*

It’s much easier to think of those three things as vertices of a triangle. We cannot make them completely isolated, because they are interrelated.

So, we cannot make the CC good without curriculum and assessment, since there’s a feedback loop. Similarly, we cannot have aligned curriculum without good standards and assessment, and we cannot have good tests without good standards and curriculum. The standards have existed forever. The common core is an attempt to create a set of nationwide standards. For example, without a coherent national curriculum it might seem OK to teach creationism in place of evolution in some states. Should that be OK?

CC is attempting to address this, in our global economy, but it hasn’t even approached science for clear political reasons. Math and English are the least political subjects so they started with those. This is a long time coming, and people often think CC refers to everything but so far it’s really only 40% of a kid’s day. Social studies CC standards are actually out right now, but they are very new.

Next, the massive machine of curriculum starts getting into play, as does the testing. I have CC standards and the CC-aligned test, but not curriculum.

Next, you’re throwing into the picture teacher evaluation aligned to CC tests. Teachers are freaking out now – they’re thinking, my curriculum hasn’t been CC-aligned for many years, what do I do now? By the way, importantly, none of the high school curriculum in NY State is actually CC-aligned now. DOE recommendations for the middle school happened last year, and DOE people will probably recommend this year for high school, since they went into talks with publication houses last year to negotiate CC curriculum materials.

The real problem is this: we’ve created these new standards to make things more difficult and more challenging without recognizing where kids are in the present moment. If I’m a former 5th grader, and the old standards were expecting something from me that I got used to, and it wasn’t very much, and now I’m in 6th grade, and there are all these raised expectations, and there’s no gap attention.

Bottomline, everybody is freaking out – teachers, students, and parents.

Last year was the first CC-aligned ELA and math tests. Everybody failed. They rolled out the test before any CC curriculum.

*From the point of view of NYC teachers, this seems like a terrorizing regime, doesn’t it?*

Yes, because the CC roll-out is rigidly tied to the tests, which are in turn rigidly tied to evaluations of teachers. So the teachers are worried they are automatically going to get a “failure” on that vector.

Another way of saying this is that, if teacher evaluations were taken out of the mix, we’d have a very different roll-out environment. But as it is, teachers are hugely anxious about the possibility that their kids might fail both the city and state tests, and that would give the teacher an automatic “failure” no matter how good their teacher observations are.

So if I’m a special ed teacher of a bunch of kids reading at 4th and 5th grade level even through they’re in 7th grade, I’m particularly worried with the introduction of the new and unknown CC-aligned tests.

*So is that really what will happen? Will all these teachers get failing evaluation scores?*

That’s the big question mark. I doubt it there will be massive failure though. I think given that the scores were so clustered in the middle/low muddle last year, they are going to add a curve and not allow so many students to fail.

*So what you’re pointing out is that they can just redefine failure?*

Exactly. It doesn’t actually make sense to fail everyone. Probably 75% of the kids got 2’s or 1’s out of a 4 point scale. What does failure mean when everyone fails? It just means the test was too hard, or that what the kids were being taught was not relevant to the test.

*Let’s dig down to the the three topics. As far as you’ve heard from the teachers, what’s good and bad about CC?*

My teachers are used to the CC. We’ve rolled out standards-based grading three years ago, so our math and ELA teachers were well adjusted, and our other subject teachers were familiar. The biggest change is what used to be 9th grade math is now expected of the 8th grade. And the biggest complaint I’ve heard is that it’s too much stuff – nobody can teach all that. But that’s always been true about every set of standards.

*Did they get rid of anything?*

Not sure, because I don’t know what the elementary level CC standards did. There was lots of shuffling in the middle school, and lots of emphasis on algebra and algebraic thinking. Maybe they moved data and stats to earlier grades.

So I believe that my teachers in particular were more prepared. In other schools, where teachers weren’t explicitly being asked to align themselves to standards, it was a huge shock. For them, it used to be solely about Regents, and also Regents exams are very predictable and consistent, so it was pretty smooth sailing.

*Let’s move on to curriculum. You mentioned there is no CC-aligned curriculum in NY. I also heard NY state has recently come out against the CC, did you hear that?*

Well what I heard is that they previously said they this year’s 9th graders (class of 2017) would be held accountable but now the class of 2022 will be. So they’ve shifted accountability to the future.

*What does accountability mean in this context?*

It means graduation requirements. You need to pass 5 Regents exams to graduate, and right now there are two versions of some of those exams: one CC-aligned, one old-school. The question is who has to pass the CC-aligned versions to graduate. Now the current 9th grade could take either the CC-aligned or “regular” Regents in math.

I’m going to ask my 9th grade students to take both so we can gather information, even though it means giving them 3 extra hours of tests. Most of my kids pass 2 Regents in 9th grade, 2 in 10th, and 3 in 11th, and then they’re supposed to be done. They only take those Regents tests in senior year that they didn’t pass earlier.

*What are the good and bad things about testing?*

What’s bad is how much time is lost, as we’ve already said. And also, it’s incredibly stressful. You and I went to school and we had one big college test that was stressful, namely the SAT. In terms of us finishing high school, that was it. For these kids it’s test, test, test, test. I don’t think it’s actually improved the quality of college students across the country. 20 years ago NY was the only one that had extra tests except California achievement tests, which I guess we sometimes took as well.

*Another way to say it is that we did take some tests but it didn’t take 5 weeks.*

And it wasn’t high stakes for the teacher!

*Let’s go straight there: what are the good/bad things for the teachers with all these tests?*

Well it definitely makes the teachers more accountable. Even teachers think this: there is a cadre of protected teachers in the city, and the principals didn’t want to take the time to get rid of them, so they’d excess them out of the schools, and they would stay in the system.

Now with testing it has become much more the principal’s responsibility to get rid of bad teachers. The number of floating teachers is going down.

*How did they get rid of the floaters?*

A lot of different ways. They made them go into the schools, take interviews, they made their quality of life not great, and a lot if them left or retired or found jobs. Principals took up the mantle as well, and they started to do due diligence.

*Sounds like the incentive system for over-worked principals was wrong.*

Yes, although the reason it became easier for the principals is because now we have data. So if you’re coming in as ineffective and I also have attendance data and observation data, I can add my observational data (subjective albeit rubric based) and do something.

*If I may be more skeptical, it sounds like this data gathering was used as a weapon against teachers. There were probably lots of good teachers that have bad numbers attached to them that could get fired if someone wanted them to be fired.*

Correct, except those good teachers generally have principals who protect them.

*You could give everyone a bad number and then fire the people you want, right?*

Correct.

*Is that the goal?*

Under Bloomberg it was.

*Is there anything else you want to mention? *

I think testing needs to be dialed down but not disappear. Education is a bi-polar pendulum and it never stops in the middle. We’re on an extreme but let’s not get rid of everything. There is a place for testing.

Let’s get our CC standards, curriculum, and testing reasonable and college-aligned and let’s keep it reasonable. Let’s do it with standards across states and let’s make sure it makes sense.

Also, there are some new tests coming out, called PARCC assessments, that are adaptive tests aligned to the CC. They are supposed to replace Regents down the line and be national.

*Here’s what bothers me about that. It’s even harder to investigate the experience of the student with adaptive tests.*

I’m not sure there’s enough technology to actually do this anyway very soon. For example, we were given $10,000 for 500 student. That’s not going to go far unless it takes 2 weeks to administer the test. But we are investing in our technology this year. For example, I’m looking forward to buying textbooks and get my updates pushed instead of having to buy new books every year.

*Last question. They are redoing the SAT because rich kids are doing so much better. Are they just trying to get in on the test prep game? Because, here’s the thing, t**here’s no test that can’t be gamed that’s also easy to grade. It’s gotta depend on the letters and grades. We keep trying to shortcut that.*

Listen, this is what I tell the kids. What’s going to matter to you is the letter of recommendation, so don’t be an jerk to your fellow students or to the teachers. Next, are you going to be able to meet the minimum requirements? That’s what the SAT is good for. It defines a lower bound.

*Is it a good lower bound though?*

Well, I define the lower bound as 1000 in total. My kids can target that. It’s a reasonable low bar.

*To what extent do your students – mostly inner-city, black girls interested in math and science – suffer under the wholly gamed SAT system?*

It serves to give them a point of self-reference with the rest of the country. You have to understand, they, like most kids in the nation, don’t have a conception of themselves outside of their own experience. The SAT serves that purpose. My kids, like many others, have the dream of Ivy League minus the understanding of where they actually stand.

*So you’re saying their estimates of their chances are too high?*

Yes, oftentimes. They are the big fish in a well-defined pond. At the very least, The SAT helps give them perspective.

*Thanks so much for your time Kiri.*

## Billionaire money and academic freedom

If you haven’t seen this recent New York Times article by William Broad, entitled *Billionaires With Big Ideas Are Privatizing American Science, *then go take a look. It generalizes to all of scientific research my recent post entitled *Billionaire Money in Mathematics.*

My favorite part of Broad’s article is the caption of the video at the top, which sums it up nicely:

Funding the Future: As government financing of basic science research has plunged, private donors have filled the void, raising questions about the future of research for the public good.

In his article Broad makes a bunch of great points.

First, the fact that rich people generally ask for research into topics they care about (“personal setting of priorities”) to the detriment of basic research. They want flashy stuff, bang for their buck.

Second, academics interested in getting funding from these rich people have to learn to market themselves. From the article:

The availability of so much well-financed ambition has created a new kind of dating game. In what is becoming a common narrative, researchers like to describe how they begged the federal science establishment for funds, were brushed aside and turned instead to the welcoming arms of philanthropists. To help scientists bond quickly with potential benefactors, a cottage industry has emerged, offering workshops, personal coaching, role-playing exercises and the production of video appeals.

If you think about it, the two issues above are kind of wrapped up together. Flashy academic content goes hand in hand with flashy marketing. Let’s say goodbye to the true nerd who doesn’t button up their cardigan correctly. And I don’t know about you but I like those nerds. My mom is one of them.

This morning I thought of another way to express this issue, from the point of view of the individual scientist or mathematician, that might have profound resonance where the above just sounds annoying.

Namely, I believe that academic freedom itself is at stake. Let me explain.

I’m the last person who would defend our current tenure system. It’s awful for women, especially those who want kids, and it often breeds a kind of arrogant laziness post-tenure. Even so, there are good things about it, and one of them is academic freedom.

And although theoretically you can have academic freedom without tenure, it is certainly easier with it (example from this piece: “In Oklahoma, a number of state legislators attempted to have Anita Hill fired from her university position because of her testimony before the U.S. Senate. If not for tenure, professors could be attacked every time there’s a change in the wind.”).

But as we’ve seen recently, tenure-track positions are quickly declining in number, even as the number of teaching positions is growing. This is the academic analog of how we’ve seen job growth in the US but it’s majority shitty jobs. And as I’ve predicted already, this trend is surely going to continue as we scale education through MOOCs.

The dwindling tenured positions means there are increasing number of people trying to do research dependent upon outside grants and funding, and without the safety net of tenure. These people often lose their jobs when their funding flags, as we’ve recently seen at Columbia.

Now let’s put these two trends together. We’ve got fewer and fewer tenure jobs, which are precariously dependent on outside funding, and we’ve got rich people funding their own tastes and proclivities.

Where does academic freedom shake out in that picture? I’m going to say nowhere.

## Why is math research important?

As I’ve already described, I’m worried about the oncoming MOOC revolution and its effect on math research. To say it plainly, I think there will be major cuts in professional math jobs starting very soon, and I’ve even started to discourage young people from their plans to become math professors.

I’d like to start up a conversation – with the public, but starting in the mathematical community – about mathematics research funding and why it’s important.

I’d like to argue for math research as a public good which deserves to be publicly funded. But although I’m sure that we need to make that case, the more I think about it the less sure I am how to make that case. I’d like your help.

So remember, we’re making the case that continuing math research is a good idea for our society, and we should put up some money towards it, even though we have competing needs to fund other stuff too.

So it’s not enough to talk about how arithmetic helps people balance their checkbooks, say, since arithmetic is already widely known and not a topic of research.

And it’s also a different question from “Why should I study math?” which is a reasonable question from a student (with a very reasonable answer found for example here) but also not what I’m asking.

Just to be clear, let’s start our answers with “Continuing math research is important because…”.

Here’s what I got so far and also why I find the individual reasons less than compelling:

——

**1) Continuing math research is important because incredibly useful concepts like cryptography and calculus and image and signal processing have and continue to come from mathematics and are helping people solve real-world problems.**

This “math as tool” is absolutely true and probably the easiest way to go about making the case for math research. It’s a long-term project, we don’t know exactly what will come out next, or when, but if we follow the trend of “useful tools,” we trust that math will continue to produce for society.

After all, there’s a reason so many students take calculus and linear algebra for their majors. We could probably even put a dollar value on the knowledge they gain in such a class, which is more than one could probably say about classes in many other fields.

Perhaps we should go further – mathematics is omnipresent in the exact science. And although much of that math is basic stuff that’s been known for decades or centuries, there are probably many examples of techniques being used that would benefit from recent updates.

The problem I have with this answer is that no mathematician ever goes into math research because someday it might be useful for the real world. At least no mathematician I know. And although that wasn’t a requirement for my answers, it still strikes me as odd.

In other words, it’s an answer that, although utterly true, and one we should definitely use to make our case, will actually leave the math research community itself cold.

So where does that leave us? At least for me straight to the next reason:

**2) Continuing math research is important because it is beautiful. It is an art form, and more than that, an ancient and collaborative art form, performed by an entire community. Seen in this light it is one of the crowning achievements of our civilization.**

This answer allows us to compare math research directly with some other fields like philosophy or even writing or music, and we can feel like artisans, or at least craftspeople, and we can in some sense expect to be supported for the very reason they are, that our existence informs us on the most basic questions surrounding what it means to be human.

The problem I have with this is that, although it’s very true, and it’s what attracted me to math in the first place, it feels too elitist, in the following sense. If we mathematicians are performing a kind of art, like an enormous musical piece, then arguably it’s a musical piece that only we can hear.

Because let’s face it, most mathematics research – and I mean current math research, not stuff the Greeks did – is totally inaccessible to the average person. And so it’s kind of a stretch to be asking the public for support on something that they can’t appreciate directly.

**3) Continuing math research is important because it trains people to think abstractly and to have a skeptical mindset.**

I’ve said it before, and I’ll say it again: one of the most amazing things about mathematicians versus anyone else is that mathematicians – and other kinds of scientists – are trained to admit they’re wrong. This is just so freaking rare in the real world.

And I don’t mean they change their arguments slightly to acknowledge inconvenient truths. I mean that mathematicians, properly trained, are psyched to hear a mistake pointed out in their argument because it signifies progress. There’s no shame in being wrong – it’s an inevitable part of the process of learning.

I really love this answer but I’ll admit that there may be other ways to achieve this kind of abstract and principled mindset without having a fleet of thousands of math researchers. It’s perhaps too indirect as an answer.

——

So that’s what I’ve got. Please chime in if I’ve missed something, or if you have more to add to one of these.

## Interview on Math-Frolic with Shecky Riemann

*Crossposted from mathtango.*

I’ve been reading Cathy O’Neil’s “* Mathbabe*” blog off-and-on pretty much since its inception, but either I’ve changed or her blog has, because for the last several months almost every entry seems like a gem to me.

**Cathy is somewhat outside-the-box of the typical math bloggers I follow… a blogger with a tad more ‘attitude’ and range of issues. She is a Harvard (PhD) graduate (also Berkeley and MIT) and a data scientist, who left the finance industry when disillusioned.**

Political candidates often talk of having a “fire in the belly,” and that’s also the sense I’ve had of Cathy’s blog for awhile now. So I was very happy to learn more about the life of the blogosphere’s mathbabe, and think you will as well:

************************************

**1) To start, could you tell readers a little about your diverse background and how you came to be a sort of math “freelancer” and blogger… including when did your interest in mathematics originally arise, and when did you know you wished to pursue it professionally?**

I started liking math when I was 4 or 5. I remember thinking about which numbers could be divided into two equal parts and which couldn’t, and I also remember understanding about primes versus composites, and for that matter g.c.d., when I played with spirographs and taking note of different kinds of periodicities and when things overlap. Of course I didn’t have words for any of this at that point.

Later on in elementary school I got really into base 2 arithmetics in 3rd grade, and I was fascinated by the representation of the number 1 by 0.9999… in 7th grade. I was actually planning on becoming a pianist until I went to a math camp after 9th grade (HCSSiM), and ever since then I’ve known. In fact it was in that summer, when I turned 15, that I decided to become a math professor.

Long story short I spent the next 20 years achieving that goal, and then when I got there I realized it wasn’t the right speed for me. I went into finance in the Spring of 2007 and was there throughout the crisis. It opened my eyes to a lot of things that I’d been ignoring about the real world, and when I left finance in 2011 I decided to start a blog to expose some of the stuff I’d seen, and to explain it as well. I joined Occupy when it started and I’ve been an activist since then.

**[Because so many carry the stereotyped image of a mathematician as someone standing at a blackboard writing inscrutable, abstract symbols, I think Cathy's "activism" has been one of the most appealing aspects of her blog!]**

**2) You’re involved in quite a number of important activities/issues… what would you list as your most ardent (math-related) goals, for say the next year, and then also longer-term? **

My short- or medium- term goal is to write a book called “**Weapons of Math Destruction**” which I recently sold to Random House. It’s for a general audience but I’ve been giving a kind of mathematical version of it to various math departments. The idea is that the modeling we’re seeing proliferate in all kinds of industries has a dark side and could be quite destructive. We need to stop blindly assuming that because it has a mathematical aspect to it that it should be considered objective or benign.

**[...Love the title of the book.]**

Longer term I want to promote the concept of open models, where the public has meaningful access to any models that are being used on them that are high impact and high stakes. So credit scoring models or Value-Added Teacher models are good examples of that kind of thing. I think it’s a crime that these models are opaque and yet have so much power over people’s lives. It’s like having secret laws.

**3) Related to the above, you’ve been especially outspoken about various financial/banking issues and the “Occupy Wall Street” movement… I have to believe that there are both very rewarding and very frustrating/exasperating aspects to tackling those issues… care to comment? **

I’d definitely say more rewarding than frustrating. Of course things don’t change overnight, especially when it comes to the public’s perception and understanding of complex issues. But I’ve seen a lot of change in the past 7 years around finance, and I expect to see more skepticism around the kind of modeling I worry about, especially in light of the NSA surveillance programs that people are up in arms over.

**4) Your blog covers a wider diversity of topics than most “math” blogs. Sometimes your blogposts seem to be a combination of educating the public while also simultaneously, venting! (indeed your subheading hints at such)… how might you describe your feelings/attitude/mood when writing typical posts? And what are your favorite (math-related) subjects to write about or study?**

Honestly blogging has crept into my daily schedule like a cup of coffee in the morning. It would be really hard for me to stop doing it. One way of thinking about it is that I’m naturally a person who gets kind of worked up about how people just don’t think about a subject X the right way, and if I don’t blog about those vents then they get stuck in my system and I can’t move past them. So maybe a better way of saying it is that getting my daily blog on is kind of like having an awesome poop. But then again maybe that’s too gross. Sorry if that’s too gross.

**[Let's just say that I may never think about composing blog posts in quite the same way again! ;-) ] **

**5) Is “Mathbabe” blog principally “a labor of love” or is it more than that for you (some sort of means to an end)? i.e., You’re writing a book and you do speaking engagements, along with other activities… is the blog a mechanism to help promote/sustain those other endeavors, or do you view it as just a recreational side activity? **

I’ve been really happy with a decision to never let mathbabe be anything except fun for me. There’s no money involved at all, ever, and there never will be. Nobody pays me for anything, nobody gets paid for anything. I do it because I learn more quickly that way, and it forces me to organize my half-thoughts in a way that people can understand. And although the thinking and learning and discussions have made a bunch of things possible, I never had those goals until they just came to me.

At the same time I wouldn’t call it a side activity either. It’s more of a central activity in my life that has no other purpose than being itself.

**6) Go ahead and tell us about the book you have in the works and its timetable…**

It’s fun to write! I can’t believe people are willing to let me interview them! It won’t be out for a couple of years. At first I thought that was way too long but now I’m glad I have the time to do the research.

**7) How do you select the topic you post about on any given day? And are there certain blogposts you’ve done that stand out as personal favorites or ones that were the most fun to work on? From the other side, which posts seem to have been most popular or attention-getting with readers?**

I send myself emails with ideas. Then I wake up in the morning and look at my notes and decide which issue is exciting me or infuriating me the most.

I have different audiences that get excited about different things. The math education community is fun, they have a LOT to say on comments. People seem to like Aunt Pythia but nobody comments — I think it’s a guilty pleasure.

**[Yes, I was skeptical of Aunt Pythia when you announced it (seemed a bit of a stretch), but it too is a fun read... though I most enjoy the passionate posts about issues tangential to mathematics.]**

I guess it’s fair to say that people like it when I combine venting with strong political views and argumentation. My most-viewed post ever was when I complained about Nate Silver’s book.

**8) What are some of the math-related books you’ve most enjoyed reading and/or ones you would particularly recommend to lay folks? **

I don’t read very many math books to be honest. I’ve always enjoyed talking math with people more than reading about it.

But I have been reading a lot of mathish books in preparation for my writing. For example, I really enjoyed “**How to Lie with Statistics**” which I read recently and blogged about.

Most of the time I kind of hate books written about modeling, to be honest, because usually they are written by people who are big data cheerleaders. I guess the best counterexamples of that would be “**The Filter Bubble**,” by Eli Pariser which is great and is a kind of prequel to my book, and “**Super Sad True Love Story**” by Gary Shteyngart which is a dystopian sci-fi novel that isn’t actually technical but has amazing prescience with respect to the kind of modeling and surveillance — and for that matter political unrest — that I think about all the time.

**9) Anything else you’d want to say to a captive audience of math-lovers, that you haven’t covered above?**

Math is awesome!

**[INDEED!]**

************************************

Thanks so much, Cathy, for filling in a bit about yourself here. Good luck in all your endeavors!

Cathy tweets, BTW, at @mathbabedotorg and she did this fascinating interview for PBS’s **“Frontline”** in 2012 (largely on the financial crisis):

http://www.pbs.org/wgbh/pages/frontline/oral-history/financial-crisis/cathy-oneil/

(I *highly* recommend this!)

## Guest Post: Beauty, even in the teaching of mathematics

*This is a guest post by Manya Raman-Sundström.*

**Mathematical Beauty**

If you talk to a mathematician about what she or he does, pretty soon it will surface that one reason for working those long hours on those difficult problems has to do with beauty.

Whatever we mean by that term, whether it is the way things hang together, or the sheer simplicity of a result found in a jungle of complexity, beauty – or aesthetics more generally—is often cited as one of the main rewards for the work, and in some cases the main motivating factor for doing this work. Indeed, the fact that a proof of known theorem can be published just because it is more elegant is one evidence of this fact.

Mathematics is beautiful. Any mathematician will tell you that. Then why is it that when we teach mathematics we tend not to bring out the beauty? We would consider it odd to teach music via scales and theory without ever giving children a chance to listen to a symphony. So why do we teach mathematics in bits and pieces without exposing students to the real thing, the full aesthetic experience?

Of course there are marvelous teachers out there who do manage to bring out the beauty and excitement and maybe even the depth of mathematics, but aesthetics is not something we tend to value at a curricular level. The new Common Core Standards that most US states have adopted as their curricular blueprint do not mention beauty as a goal. Neither do the curriculum guidelines of most countries, western or eastern (one exception is Korea).

Mathematics teaching is about achievement, not about aesthetic appreciation, a fact that test-makers are probably grateful for – can you imagine the makeover needed for the SAT if we started to try to measure aesthetic appreciation?

**Why Does Beauty Matter?**

First, it should be a bit troubling that our mathematics classrooms do not mirror practice. How can young people make wise decisions about whether they should continue to study mathematics if they have never really seen mathematics?

Second, to overlook the aesthetic components of mathematical thought might be to preventing our children from developing their intellectual capacities.

In the 1970s Seymour Papert , a well-known mathematician and educator, claimed that scientific thought consisted of three components: cognitive, affective, and aesthetic (for some discussion on aesthetics, see here).

At the time, research in education was almost entirely cognitive. In the last couple decades, the role of affect in thinking has become better understood, and now appears visibly in national curriculum documents. Enjoying mathematics, it turns out, is important for learning it. However, aesthetics is still largely overlooked.

Recently Nathalie Sinclair, of Simon Frasier University, has shown that children can develop aesthetic appreciation, even at a young age, somewhat analogously to mathematicians. But this kind of research is very far, currently, from making an impact on teaching on a broad scale.

Once one starts to take seriously the aesthetic nature of mathematics, one quickly meets some very tough (but quite interesting!) questions. What do we mean by beauty? How do we characterise it? Is beauty subjective, or objective (or neither? or both?) Is beauty something that can be taught, or does it just come to be experienced over time?

These questions, despite their allure, have not been fully explored. Several mathematicians (Hardy, Poincare, Rota) have speculated, but there is no definite answer even on the question of what characterizes beauty.

**Example**

To see why these questions might be of interest to anyone but hard-core philosophers, let’s look at an example. Consider the famous question, answered supposedly by Gauss, of the sum of the first n integers. Think about your favorite proof of this. Probably the proof that did NOT come to your mind first was a proof by induction:

Prove that S(n) = 1 + 2 + 3 … + n = n (n+1) /2

S(k + 1) = S(k) + (k + 1)

= k(k + 1)/2 + 2(k + 1)/2

= k(k + 1)/2 + 2(k + 1)/2

= (k + 1)(k + 2)/2.

Now compare this proof to another well known one. I will give the picture and leave the details to you:

Does one of these strike you as nicer, or more explanatory, or perhaps even more beautiful than the other? My guess is that you will find the second one more appealing once you see that it is two sequences put together, giving an area of n (n+1), so S(n) = n (n+1)/2.

Note: another nice proof of this theorem, of course, is the one where S(n) is written both forwards and backwards and added. That proof also involves a visual component, as well as an algebraic one. See here for this and a few other proofs.

**Beauty vs. Explanation**

How often do we, as teachers, stop and think about the aesthetic merits of a proof? What is it, exactly, that makes the explanatory proof more attractive? In what way does the presentation of the proof make the key ideas accessible, and does this accessibility affect our sense of understanding, and what underpins the feeling that one has found exactly the right proof or exactly the right picture or exactly the right argument?

Beauty and explanation, while not obvious related (see here), might at least be bed-fellows. It may be the case that what lies at the bottom of explanation — a feeling of understanding, or a sense that one can ”see” what is going on — is also related to the aesthetic rewards we get when we find a particularly good solution.

Perhaps our minds are drawn to what is easiest to grasp, which brings us back to central questions of teaching and learning: how do we best present mathematics in a way that makes it understandable, clear, and perhaps even beautiful? These questions might all be related.

**Workshop on Math Beauty**

This March 10-12, 2014 in Umeå, Sweden, a group will gather to discuss this topic. Specifically, we will look at the question of whether mathematical beauty has anything to do with mathematical explanation. And if so, whether the two might have anything to do with visualization.

If this discussion peaks your interest at all, you are welcome to check out my blog on math beauty. There you will find a link to the workshop, with a fantastic lineup of philosophers, mathematicians, and mathematics educators who will come together to try to make some progress on these hard questions.

Thanks to Cathy, the always fabulous mathbabe, for letting me take up her space to share the news of this workshop (and perhaps get someone out there excited about this research area). Perhaps she, or you if you have read this far, would be willing to share your own favorite examples of beautiful mathematics. Some examples have already been collected here, please add yours.

## Upcoming talks

A few months ago I gave a talk entitled “Start Your Own Netflix” talk that was part of the MAA Distinguished Lecture Series, the slides for which are available here and a short video version here.

Today I’m planning to modify that talk so I can give a longer and more technical version of it on Friday morning at the Department of Mathematical Science of Worcester Polytechnic Institute, where I’ve been invited to speak by Suzy Weekes.

In about a month I’m going to Berkeley for a week to give a so-called MSRI-Evans talk on Monday, February 24th, at 4pm, thanks to the kind invitation of Lauren Williams. I still haven’t decided whether to give a “The World Is Going To Hell” talk, which would be kind of the technical version of my book (and which I gave at Harvard’s IQSS recently), or whether I should give yet another version of the Netflix talk, which is cool and technical but not as doomsday. If you’re planning to attend please voice your opinion!

Finally, I’m hoping to join in a meeting of some manifestation of the Noetherian Ring while I’m at Berkeley. This is a women in math group that was started when I was an undergrad there, back in the middle ages, in something like 1992. It’s where I gave my first and second math talks and there was always free pizza. It really was a great example of how to create a supportive environment for collaborative math.

## If it’s hocus pocus then it’s not math

A few days ago there was a kerfuffle over this “numberphile” video, which was blogged about in Slate here by Phil Plait in his “Bad Astronomy” column, with a followup post here with an apology and a great quote from my friend Jordan Ellenberg.

The original video is hideous and should never have gotten attention in the first place. I say that not because the subject couldn’t have been done well – it could have, for sure – but because it was done so poorly that it ends up being destructive to the public’s most basic understanding of math and in particular positive versus negative numbers. My least favorite line from the crappy video:

I was trying to come up with an intuitive reason for this I and I just couldn’t. You have to do the mathematical hocus pocus to see it.

What??

Anything that is hocus pocus* isn’t actually math*. And people who don’t understand that shouldn’t be making math videos for public consumption, especially ones that have MSRI’s logo on them and get written up in Slate. Yuck!

I’m not going to just vent about the cultural context, though, I’m going to mention what the actual mathematical object of study was in this video. Namely, it’s an argument that “prove” that we have the following identity:

Wait, how can that be? Isn’t the left hand side positive and the right hand side negative?!

This mathematical argument is familiar to me – in fact it is very much along the lines of stuff we sometimes cover at the math summer program HCSSiM I teach at sometimes (see my notes from 2012 here). But in the case of HCSSiM, we do it quite differently. Specifically, we use it as a demonstration of *flawed mathematical thinking*. Then we take note and make sure we’re more careful in the future.

If you watch the video, you will see the flaw almost immediately. Namely, it starts with the question of what the value is of the infinite sum

But here’s the thing, that *doesn’t actually have a value*. That is, it doesn’t have a value until you assign it a value, which you can do but then you ~~might want to~~ *absolutely positively must* explain how you’ve done so. Instead of that explanation, the guy in the video just acts like it’s obvious and uses that “fact,” along with a bunch of super careless moving around of terms in infinite sums, to infer the above outrageous identity.

To be clear, sometimes infinite sums do have pretty intuitive and reasonable values (even though you should be careful to acknowledge that they too are assigned rather than “true”). For example, any geometric series where each successive term gets smaller has an actual “converging sum”. The most canonical example of this is the following:

What’s nice about this sum is that it is naively plausible. Our intuition from elementary school is corroborated when we think about eating half a cake, then another quarter, and then half of what’s left, and so on, and it makes sense to us that, if we did that forever (or if we did that increasingly quickly) we’d end up eating the whole cake.

This concept has a name, and it’s *convergence*, and it jibes with our sense of what would happen “if we kept doing stuff forever (again at possibly increasing speed).” The amounts we’ve measured on the way to forever are called *partial sums*, and we make sure they converge to the answer. In the example above the partial sums are and so on, and they definitely converge to 1.

There’s a mathematical way of defining convergence of series like this that the geometric series follows but that the series does not. Namely, you guess the answer, and to make sure you’ve got the right one, you make sure that *all of the partial sums are very very close* to that answer if you go far enough, for *any* definition of “very very close.”

So if you want it to get within 0.00001, there’s a number N so that, after the Nth partial sum, all partial sums are within 0.00001 of the answer. And so on.

Notice that if you take the partial sums of the series you get the sequence which doesn’t get closer and closer to anything. That’s another way of saying that there is no naively plausible value for this infinite sum.

As for the first infinite sum we came across, the that *does* have a naively plausible value, which we call “infinity.” Totally cool and satisfying to your intuition that you worked so hard to achieve in high school.

But here’s the thing. Mathematicians are pretty clever, so they haven’t stopped there, and they’ve assigned a value to the infinite sum in spite of these pesky intuition issues, namely , and in a weird mathematical universe of their construction, which is wildly useful in some contexts, that value is internally consistent with other crazy-ass things. One of those other crazy-ass things is the original identity

[Note: what would be really cool is if a mathematician made a video explaining the crazy-ass universe and why it's useful and in what contexts. This might be hard and it's not my expertise but I for one would love to watch that video.]

That doesn’t mean the identity is “true” in any intuitively plausible sense of the word. It means that mathematicians are scrappy.

Now here’s my last point, and it’s the only place I disagree somewhat (I think) with Jordan in his tweets. Namely, I really do think that the intuitive definition is qualitatively different from what I’ve termed the “crazy-ass” definition. Maybe not in a context where you’re talking to other mathematicians, and everyone is sufficiently sophisticated to know what’s going on, but definitely in the context of explaining math to the public where you can rely on number sense and (hopefully!) a strong intuition that positive numbers can’t suddenly become negative numbers.

Specifically, if you can’t make any sense of it, intuitive or otherwise, and if you have to ascribe it to “mathematical hocus pocus,” then you’re definitely doing something wrong. Please stop.

## The coming Calculus MOOC Revolution and the end of math research

I don’t usually like to sound like a doomsayer but today I’m going to make an exception. I’m going to describe an effect that I believe will be present, even if it’s not as strong as I am suggesting it might be. There are three points to my post today.

**1) Math research is a byproduct of calculus teaching**

I’ve said it before, calculus (and pre-calculus, and linear algebra) might be a thorn in many math teachers’ side, and boring to teach over and over again, but it’s the bread and butter of math departments. I’ve heard statistics that 85% of students who take any class in math at a given college take only calculus.

Math research is essentially funded through these teaching jobs. This is less true for the super elite institutions which might have their own army of calculus adjuncts and have separate sources of funding both from NSF-like entities and private entities, but if you take the group of people I just saw at JMM you have a bunch of people who essentially depend on their take-home salary to do research, and their take-home salary depends on lots of students at their school taking service courses.

I wish I had a graph comparing the number of student enrolled in calculus each year versus the number of papers published in math journals each year. That would be a great graphic to have, and I think it would make my point.

**2) Calculus MOOCs and other web tools are going to start replacing calculus teaching very soon and at a large scale**

It’s already happening at Penn through Coursera. Word on the street is it is about to happen at MIT through EdX.

If this isn’t feasible right now it will be soon. Right now the average calculus class might be better than the best MOOC, especially if you consider asking questions and getting a human response. But as the calculus version of math overflow springs into existence with a record of every question and every answer provided, it will become less and less important to have a Ph.D. mathematician present.

Which isn’t to say we won’t need a person at all – we might well need someone. But chances are they won’t be tenured, and chances are they could be overseas in a call center.

This is not really a bad thing in theory, at least for the students, as long as they actually learn the stuff (as compared to now). Once the appropriate tools have been written and deployed and populated, the students may be better off and happier. They will very likely be more adept at finding correct answers for their calculus questions online, which may be a way of evaluating success (although not mine).

It’s called progress, and machines have been doing it for more than a hundred years, replacing skilled craftspeople. It hurts at first but then the world adjusts. And after all, lots of people complain now about teaching boring classes, and they will get relief. But then again many of them will have to find other jobs.

Colleges might take a hit from parents about how expensive they are and how they’re just getting the kids to learn via computer. And maybe they will actually lower tuition, but my guess is they’ll come up with something else they are offering that makes up for it which will have nothing to do with the math department.

**3) Math researchers will be severely reduced if nothing is done**

Let’s put those two things together, and what we see is that math research, which we’ve basically been getting for free all this time, as a byproduct of calculus, will be severely curtailed. Not at the small elite institutions that don’t mind paying for it, but at the rest of the country. That’s a lot of research. In terms of scale, my guess is that the average faculty will be reduced by more than 50%, and some faculties will be closed altogether.

Why isn’t anything being done? Why do mathematicians seem so asleep at this wheel? Why aren’t they making the case that math research is vital to a long-term functioning society?

My theory is that mathematicians haven’t been promoting their work for the simple reason that they haven’t had to, because they had this cash cow called calculus which many of them aren’t even aware of as a great thing (because close up it’s often a pain).

It’s possible that mathematicians don’t even know how to promote math to the general public, at least right now. But I’m thinking that’s going to change. We’re going to think about it pretty hard and learn how to promote math research very soon, or else we’re going back to 1850 levels of math research, where everyone knew each other and stuff was done by letter.

**How worried am I about this? **

For my friends with tenure, not so worried, except if their entire department is at risk. But for my younger friends who are interested in going to grad school now, I’m not writing them letters of recommendation before having this talk, because they’ll be looking around for tenured positions in about 10 years, and that’s the time scale at which I think math departments will be shrinking instead of expanding.

In terms of math PR, I’m also pretty worried, but not hopeless. I think one can really make the case that basic math research should be supported and expanded, but it’s going to take a lot of things going right and a lot of people willing to put time and organizing skills into the effort for it to work. And hopefully it will be a community effort and not controlled by a few billionaires.

## Billionaire money in mathematics

During the recent JMM AMS panel I was on, where the topic was the Public Face of Math, the issue came up repeatedly that we mathematicians might want to find a billionaire who could solve all our PR problems (although we didn’t quite seem to agree on what these PR problems are).

Indeed billionaire money seemed to represent a panacea even though it originated with a slightly facetious suggestion of a super PAC for mathematics from Congressman Jerry McNerney. The idea was taken quite seriously and repeated by at least 3 audience members.

I think this happened for a few reasons. First, mathematicians are mostly apolitical and don’t think of politics or PR as part of their job. They also don’t think they’re good at that stuff, and they are happy for someone else to do it. Who else but a rich guy interested in that stuff and who “has people” who are good at it.

Second, Jim Simons has been doing good stuff for math lately and people trust him. I totally get that, and I don’t entirely disagree, although invitation-only conferences in the Virgin Islands is not my idea of easy and transparent access to ideas that many mathematicians strive for. I hear his Quanta Magazine is awesome.

Here’s the thing. We lose something when we consistently take money from rich people, which has nothing to with any specific rich person who might have great ideas and great intentions.

The first thing we lose is power, and specifically control over our own image. That might seem like a fair deal now, since at least someone is working on it, but it’s not obvious that it would always be.

It means, for example, that one person has a huge amount of influence about, say, how the math community deals with the NSA. As we know this is an recent and ongoing discussion, but it came up pretty suddenly, as issues do, and it might be weird to all of a sudden need to know what some rich guy thinks of a specific issue.

Another example of why taking money from a few super rich people might not be a great idea requires the idea of a funding feedback loop, which well articulated by Benjamin Soskis and Felix Salmon with respect to the public parks in New York City.

The basic idea is that, as public funding dries up for something like public parks (or from the NSF) and as a community gets desperate for basic operating funds, money from rich individuals seems like a godsend. But over time two things happen.

First, the public funding never ever comes back. Because, after all, why should it? It looks like everything is well-funded. And the individuals who are part of that community are not agitating for the return of that funding since they have jobs.

Second, it’s not clear that the new money will be distributed in a good governance type of way. It might be distributed based on where rich people live, in the case of parks, or what their preferred mathematical subjects are, in the case of math. And the community has no recourse on those decisions, because the entire system depends on the generosity of someone who could change his mind at any moment.

And I’m not saying NSF doesn’t have weird rubriks for which fields (and which people!) get funded as well, but at least we can have a public discussion about that and make noise. And the decisions are made by different groups of mathematicians every year.

My suggestion is that we should think about representing ourselves in this PR campaign, if we have one to wage, and we should focus efforts on things that would improve NSF funding instead of getting us addicted to private funding. And it should be a community conversation where everyone participates who cares enough.

What are the chances that will ever happen? In terms of whether typical mathematicians will ever be willing to become politically active, my vote is on “yes” and “very soon,” and the reason I say this is that I believe mathematics research is being hugely (if quietly) threatened by the oncoming Calculus MOOC Revolution, which I plan to write about very soon.

## JMM

It occurs to me, as I prepare to join my panel this afternoon on Public Facing Math, that I’ve been to more Joint Math Meetings in the 7 years since I left academic math (3) than I did in the 17 years I was actually in math (2). I include my undergraduate years in that count because when I was a junior in college I went to Vancouver for the JMM and I met Cora Sadosky, which was probably my favorite conference ever.

Anyhoo I’m on my way to one of the highlights of any JMM, the HCSSiM breakfast, where we hang out with students and teachers from summers long ago and where I do my best to convince the director Kelly and myself that I should come back next summer to teach again. Then after that I spend 4 months at home convincing my family that it’s a great plan. Woohoo!

Besides the above plan, I plan to meet people in the hallways and gossip. That’s all I have ever accomplished here. I hope it is the official mission of the conference, but I’m not sure.

## Two thoughts on math research papers

Today I’d like to mention two ideas I’ve been having recently on how to make being a research mathematician (even) more fun.

**1) Mathematicians should consider holding public discussions about papers**

First, math nerds, did you know that in statistics they have formal discussions about papers? It’s been a long-standing tradition by the Royal Statistical Society, whose motto is “Advancing the science and application of statistics, and promoting use and awareness for public benefit,” to choose papers by some criterion and then hold regular public discussions about those papers by a few experts who are not the author, about the paper. Then the author responds to their points and the whole conversation is published for posterity.

I think this is a cool idea for math papers too. One thing that kind of depressed me about math is how rarely you’d find people reading the same papers unless you specifically got a group of people together to do so, which was a lot of work. This way the work is done mostly by other people and more importantly the payoff is much better for them since everyone gets a view into the discussion.

Note I’m sidestepping who would organize this whole thing, and how the papers would be chosen exactly, but I’d expect it would improve the overall feeling that I had of being isolated in a tiny math community, especially if the conversations were meant to be penetrable.

**2) There should be a good clustering method for papers around topics**

This second idea may already be happening, but I’m going to say it anyway, and it could easily be a thesis for someone in CS.

Namely, the idea of using NLP and other such techniques to cluster math papers by topic. Right now the most obvious way to find a “nearby” paper is to look at the graph of papers by direct reference, but you’re probably missing out on lots of stuff that way. I think a different and possibly more interesting way would be to use the text in the title, abstract, and introduction to find papers with similar subjects.

This might be especially useful when you want to know the answer to a question like, “has anyone proved that such-and-such?” and you can do a text search for the statement of that theorem.

The good news here is that mathematicians are in love with terminology, and give weird names to things that make NLP techniques very happy. My favorite recent example which I hear Johan muttering under his breath from time to time is *Flabby Sheaves*. There’s no way that’s not a distinctive phrase.

The bad news is that such techniques won’t help at all in finding different fields who have come across the same idea but have different names for the relevant objects. But that’s OK, because it means there’s still lots of work for mathematicians.

By the way, back to the question of whether this has already been done. My buddy Max Lieblich has a website called MarXiv which is a wrapper over the math ArXiv and has a “similar” button. I have no idea what that button actually does though. In any case I totally dig the design of the similar button, and what I propose is just to have something like that work with NLP.

## AMS Panel on The Public Face of Mathematics

A week from today I’ll be at the Joint Math Meetings in Baltimore to join a panel discussion on the Public Face of Mathematics. I’ll steal the blurb for my panel from this page:

The Public Face of Mathematics, Friday, 2:30 p.m.–4:00 p.m.Moderated by Arthur Benjamin, Harvey Mudd College. Panelists Keith Devlin, Stanford University; Jerry McNerney, U. S. Congress; Cathy O’Neil, Johnson Research Labs; Tom Siegfried, Freelance Journalist; and Steve Strogatz, Cornell University, will share ideas and lead discussion about how the mathematics community can mobilize more members to become proactive in representing mathematics to the general public and to key audiences of leaders in discussions of public policy. Sponsored by the Committee on Science Policy and the Committee on Education.

One thing I’ve already noticed that might make me different from some of the other panelists is that I don’t spend too much time explaining math to the general public, although my notes on teaching at HCSSiM might arguably be the exception. And sometimes I explain how to do modeling, but that’s not stuff I learned as a mathematician.

Mostly what I do, at least from my perspective, is comment on the culture of mathematics (for women, for example) or talk about how unfortunate it is that the public’s trust in mathematics and mathematician is being perverted into a political campaign about the (supposed) objectivity of mathematical modeling by people like Bill Gates. I specialize in calling out the misapplication of mathematical imprimatur.

Anyhoo, two questions for my readers:

- Are you going to JMM too and wanna hang with me? Please know I’m only there during the day Friday, it’s a short visit. But please contact me!
- The moderator, Art Benjamin, is asking us panelists for questions that he should ask the panel next week. Please comment below with your suggestions, and thanks!

## On being a mom and a mathematician: interview by Lillian Pierce

*This is a guest post by Lillian Pierce, who is currently a faculty member of the Hausdorff Center for Mathematics in Bonn, and will next year join the faculty at Duke University.*

I’m a mathematician. I also happen to be a mother. I turned in my Ph.D. thesis one week before the due date of my first child, and defended it five weeks after she was born. Two and a half years into my postdoc years, I had my second child.

Now after a few years of practice, I can pretty much handle daily life as a young academic and a parent, at least most of the time, but it still seems like a startlingly strenuous existence compared to what I remember of life as just a young academic, not a parent.

Last year I was asked by the Association for Women in Mathematics to write a piece for the AWM Newsletter about my impressions of being a young mother and getting a mathematical career off the ground at the same time. I suggested that instead I interview a lot of other mathematical mothers, because it’s risky to present just one view as “the way” to tackle mathematics and motherhood.

Besides, what I really wanted to know was: how is everyone else doing this? I wanted to pick up some pointers.

I met Mathbabe about ten years ago when I was a visiting prospective graduate student and she was a postdoc. She made a deep impression on me at the time, and I am very happy that I now have the chance to interview her for the series Mathematics+Motherhood, and to now share with you our conversation.

LP: Tell me about your current work.

CO: I am a data scientist working at a small start-up. We’re trying to combine consulting engagements with a new vision for data science training and education and possibly some companies to spin off. In the meantime, we’re trying not to be creepy.

LP: That sounds like a good goal. And tell me a bit about your family.

CO: I have three kids. I got pregnant with my first son, who’s 13 now, soon after my PhD. Then I had a second child 2 years later, also while I was a postdoc. I also have a 4 year old, whom I had when I was working in finance.

LP: Did you have any notions or worries in advance about how the growth of your family would intersect with the growth of your career?

CO: I absolutely did worry about it, and I was right to worry about it, but I did not hesitate about whether to have children because it was just not a question to me about how I wanted my life to proceed. And I did not want to wait until I was tenured because I didn’t want to risk being infertile, which is a real risk. So for me it was not an option not to do it as a woman, forget as a mathematician.

LP: What was it like as a postdoc with two very young children?

CO: On the one hand I was hopeful about it, and on the other hand I was incredibly disappointed about it. The hopeful part was that the chair of my department was incredibly open to negotiating a maternity leave for postdocs, and it really was the best maternity policy that I knew about: a semester off of teaching for each baby and in total an extra year of the postdoc, since I had 2 babies. So I ended up with four years of postdoc, which was really quite generous on the one hand, but on the other hand it really didn’t matter at all. Not “not at all”—it mattered somewhat but it simply wasn’t enough to feel like I was actually competing with my contemporaries who didn’t have children. That’s on the one hand completely obvious and natural and it makes sense, because when you have small children you need to pay attention to them because they need you—and at the same time it was incredibly frustrating.

LP: It’s interesting because it’s not that you were saying “I won’t be able to compete with my contemporaries over the course of my life,” but more “I can’t compete right now.”

CO: Exactly, “I can’t compete right now” with postdocs without children. I realize—and this is not a new idea—that mathematics as a culture frontloads entirely into those 3 or 4 years after you get your PhD. Ultimately it’s not my fault, it’s not women’s fault, it’s the fault of the academic system.

LP: What metrics could departments use to be thinking more about future potential?

CO: I actually think it’s hard. It’s not just for women that it should change. It’s for the actual culture of mathematics. Essentially, the system is too rigid. And it’s not only women who get lost. The same thing that winnows the pool down right after getting a PhD—it’s a whittling process, to get rid of people, get rid of people, get rid of people until you only have the elite left—that process is incredibly punishing to women, but it’s also incredibly punishing to everybody. And moreover because of the way you get tenure and then stay in your field for the rest of your life, my feeling is that mathematics actually suffers. The reason I say this is because I work in industry now, which is a very different system, and people can reinvent themselves in a way that simply does not happen in mathematics.

LP: Do you think industry, in terms of the young career phase, gets it closer to “right” than academia currently does?

CO: Much closer to right. It’s a brutal place, don’t get me wrong, it’s brutal. I’m not saying it’s a perfect system by any stretch of the imagination. But the truth is in industry you can have a 3 year stint somewhere that is a mistake. Forget having kids, you can have a 3 year stint that was just a mistake for you. You can say “I had a bad boss and I left that place and I got a new job” and people will say “Ok.” They don’t care. One thing that I like about it is the ability to reinvent yourself. And I don’t think you see that in math. In math, your progress is charted by your publication record at a granular level. And if you’re up for tenure and there’s a 3 year gap where you didn’t publish, even if in the other years you published a lot, you still have to explain that gap. It’s like a moral responsibility to keep publishing all the time.

LP: How are you measured in industry?

CO: In industry it’s the question “what have you done for me,” and “what have you done for me lately.” It’s a shorter-term question, and there are good elements to that. One of the good elements is that as a woman you can have a baby or a couple babies and then you can pick up the slack, work your ass off, and you can be more productive after something happens. If someone gets sick, people lower their expectations for that person for some amount of time until they recover, and then expectations are higher. Mathematics by contrast has frontloaded all of the stress, especially for the elite institutions, into the 3 or 4 years to get the tenure track offer and then the next 6 years to get tenure. And then all the stress is gone. I understand why people with tenure like that. But ultimately I don’t think mathematics gets done better because of it. And certainly when the question arises “why don’t women stay in math,” I can answer that very easily: because it’s not a very good place for women, at least if they want kids.

LP: You mention on your blog that your mother is an unapologetic nerd and computer scientist; the conclusion you drew from that was that it was natural for you not to doubt that your contributions to nerd-dom and science and knowledge would be welcomed. How do you think this experience of having a mother like that inoculated you?

CO: One of the great gifts that my mother gave me as a Mother Nerd was the gift of privacy—in the sense that I did not scrutinize myself. First of all she was role-modeling something for me, so if I had any expectations it would be to be like my mom. But second of all she wasn’t asking me to think about that. I think that was one of the rarest things I had, the most unusual aspect of my upbringing as a girl. Very few of the girls that I know are not scrutinized. My mother was too busy to pay attention to my music or my art or my math. And I was left alone to decide what I wanted to do—it wasn’t about what I was good at or what other people thought of my progress. It was all about answering the question, what did I want to do. Privacy for me is having elbow space to self-define.

LP: Do you think it’s harder for parents to give that space to girls than to boys?

CO: Yes I do, I absolutely do. It’s harder and for some reason it’s not even thought about. My mother also gave me the gift of not feeling at all guilty about putting me into daycare. And that’s one of my strongest lessons, is that I don’t feel at all guilty about sending my kids to daycare. In fact I recently had the daycare providers for my 4-year-old all over for dinner, and I was telling them in all honesty that sometimes I wish I could be there too, that I could just stay there all day, because it’s just a wonderful place to be. I’m jealous of my kids. And that’s the best of all worlds. Instead of saying “oh my kid is in daycare all day, I feel bad about that,” it’s “my kid gets to go to daycare.”

LP: Where did this ability not to scrutinize come from? Where did your mother get this?

CO: I don’t know. My mother has never given me advice, she just doesn’t give advice. And when I ask her to, she says “you know more about your life than I do.”

LP: How do you deal with scrutiny now?

CO: It’s transformed as I’ve gotten older. I’ve gotten a thicker skin, partly from working in finance. I’ve gotten to the point now where I can appreciate good feedback and ignore negative feedback. And that’s a really nice place to be. But it started out, I believe, because I was raised in an environment where I wasn’t scrutinized. And I had that space to self-define.

LP: The idea of pushing back against scrutiny to clear space for self-definition is inspiring for adults as well.

CO: Women in math, especially with kids, give yourself a break. You’re under an immense amount of pressure, of scrutiny. You should think of it as being on the front lines, you’re a warrior! And if you’re exhausted, there’s a reason for it. Please go read Radhika Nagpal’s Scientific American blog post (“The Awesomest 7-Year Postdoc Ever”) for tips on how to deal with the pressure. She’s awesome. And the last thing I want to say is that I never stopped loving math. Cardinal Rule Number 1: Before all else, don’t become bitter. Cardinal Rule Number 2: Remember that math is beautiful.