I’ve loved math since I can remember. When I was 5 I played with spirographs and learned about periodicity, which made me understand prime numbers as colorful patterns on a page. I always thought 5-fold symmetry was the most beautiful.
Then I got to college at UC Berkeley and in my second semester was privileged to learn algebra (and later, Galois Theory!) from Ken Ribet, who became my very good friend. He brought me to have dinner with all sorts of amazing mathematicians, like Serge Lang and J.P. Serre and Barry Mazur and John Tate and of course his Berkeley colleagues Hendrik Lenstra and Robert Coleman and many others. Many of the main characters behind the story of solving Fermat’s Last Theorem were people I had met at dinner parties at Ken’s house, including of course Ken himself. Math was discussed in between slices of Cheese Board Pizza and fresh salad mixes from the Berkeley Bowl.
How lucky was I?!?
And I knew it, at least partially. Really the best thing about these generous and wonderful people was how joyful they were about the serious business of doing math. It was a pleasure to them, and it made them smile and even appear wistful if I’d mention my difficulties with tensor products, say.
They were incredibly inviting to me, and honestly I was spoiled. I had been invited into this society because I loved math and because I was devoting myself to it, and that was enough for them. Math is, after all, not an individual act, it is a community effort, and progress is to be celebrated and adored. And it wasn’t just any community, it was a really really nice group of guys who loved what they did for a living and wanted other cool and smart people to join.
I mention all this because I want to clarify how fucking cool it can be to be a mathematician, and what kind of group involvement and effort it can feel like, even though many of the final touches on the proofs are made inside closed offices. Being part of such a community, where math is so revered and celebrated, it is its own reward to be able to prove a theorem and tell your friends about it.
Hey, guess what? This is true too! We always suspected it but now we can use it! How cool is that?
Now that I’ve explained how much I love math (and I still love math very much), let me explain why I hate the Fields Medal. Namely, because that group effort is utterly lost and is replaced with a synthetic and false myth of the individual genius working in isolation.
Here’s the thing, and I can say this now pretty confidently, journalism has rules about writing stories that don’t really work for math. When journalists are told to “put a face on the story,” they end up with all face and no story.
How else is a journalist going to write about progress in some esoteric field? The mathematics itself is naturally not within arms reach: mathematics is by nature deep and uses multiple layers of metaphor and notation which even trained mathematicians grapple with, never mind a new result on the very far edge of what is known. So it makes sense that the story becomes about the mathematician himself or herself.
It’s not just journalists, though. Certain mathematicians do their best to represent research mathematics, and sometimes it’s awesome, sometimes it kind of works, and sometimes it ends up being laughably or even embarrassingly simplistic. That’s the thing about math, it’s deep. It’s hard to boil down to a nut graf.
So here’s the thing, the Fields Medal is easy to understand (“it’s the Nobel Prize for math!”) but it’s incredibly and dangerously misleading. It gives the impression that we have these superstars who “have it” and then we have a bunch of wandering nerds who “don’t really have it.” That stereotype is a bad advertisement for mathematics and for mathematicians, who are actually much more generous and community-spirited than that.
Plus, now that I’m in full rant mode, can I just mention that the 40-year-old age limit for the award is just terrible and obviously works against certain people, especially women or men who take parenting seriously. I am not even going to explain that because it’s so freaking clear, and as a 42-year-old woman myself, may I say I’m just getting started. And yes, the fact that a woman has won the Fields Medal is a good things, but it’s a silver lining on an otherwise big old rain cloud which I do my best to personally blow away.
And, lest I seem somehow mean to the Fields Medal winners, of course they are great mathematicians! Yes, yes they are! They’re all great, and there are many great mathematicians who never get awards, and doing great math and making progress is its own reward, and those mathematicians who do great work tend to be the ones who already have lots of resources and don’t need more, but I’m not saying they shouldn’t be celebrated, because they’re awesome, no question about it.
Here’s what I’d like to see: serious outward-facing science journalism centered around, or at least instructive towards, the incredible collaborative effort that is modern mathematics.
Everyone I know who codes uses stackoverflow.com for absolutely everything.
Just yesterday I met a cool coding chick who was learning python and pandas (of course!) with the assistance of stackoverflow. It is exactly what you need to get stuff working, and it’s better than having a friend to ask, even a highly knowledgable friend, because your friend might be busy or might not know the answer, or even if your friend knew the answer her answer isn’t cut-and-paste-able.
If you are someone who has never used stackoverflow for help, then let me explain how it works. Say you want to know how to load a JSON file into python but you don’t want to write a script for that because you’re pretty sure someone already has. You just search for “import json into python” and you get results with vote counts:
Also, every math nerd I know uses and contributes to mathoverflow.net. It’s not just for math facts and questions, either, there are interesting discussions going on there all the time. Here’s an example of a comment in response to understanding the philosophy behind the claimed proof of the ABC Conjecture:
OK well hold on tight because now there’s a new online forum, but not about coding and not about math. It’s about all the other STEM subjects, which since we’ve removed math might need to be called STE subjects, which is not catchy.
So far only statistics is open, but other stuff is coming very soon. Specifically it covers, or soon will cover, the following fields:
- Cognitive Sciences
- Computer Sciences
- Earth and Planetary Sciences
- Science & Math Education
- History of Science and Mathematics
- Applied Mathematics, and
I’m super excited for this site, it has serious potential to make peoples’ lives better. I wish it had a category for Data Sciences, and for Data Journalism, because I’d probably be more involved in those categories than most of the above, but then again most data science-y questions could be inserted into one of the above. I’ll try to be patient on this one.
Here’s a screen shot of an existing Stats question on the site:
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”.
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.
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.
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.
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.
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.