Today I want to share a puzzle that my friend Aaron Abrams told me a few days ago. I’m sure some of you have heard it before, but it’s confusing me, so I’m asking for your help.
Here’s the setup. There’s an island of people, all of whom have either blue eyes or green eyes. By social convention they never discuss eye color, because there’s a tragic rule that states that, if you ever figure out your eye color, you have to leave the island within 24 hours. Oh, and there are no mirrors.
OK, get it? So think of the island as pretty small, maybe 100 people, so you know everyone else’s eye color but not your own.
Here’s what happens next. Some castaway arrives by swimming onto the island, stays for a few days and hangs out with the folks there eating island food and having island parties, and then after building himself a boat he prepares to leave. Not being trained in the social customs of the island, on the day he leaves he says, “hey, it’s good to see some people with green eyes here!”.
So the puzzle is, what happens next?
Here’s what’s obvious. If you are a person who only sees blue eyes, you know by his statement that you must have green eyes. So you have to leave the next day.
But actually he said “some people.” So even if you only see one other person with green eyes, then you have to leave, with that other green-eyed person, after one day.
With me so far?
But hey, what if you see two other people with green eyes? Well, you might think you’re safe, and you’d wait to see them leave together the next day. But what if they don’t leave after one day? That must mean that you also have green eyes. Then all three of you have to leave, after two days. Get it?
Then you work by induction. If you see N other people with green eyes, they should all leave after N-1 days, or else you have green eyes too and all (N+1) of you leave after N days.
OK, so here’s the conundrum. The guy who started this whole mess really didn’t do much. He just stated what was obvious to everyone already on the island, namely that some people had green eyes. I mean, yes, if there were really only 2 people with green eyes, then he clearly added real information, because both those people had thought only 1 person had green eyes.
But just for the fun of it, let’s assume there were 17 people with green eyes. Then they guy really didn’t add information. And yet, 16 days after the guy left, so do all the green-eyed islanders. So really the guy just started a count-down more than anything.
So, is that it? Is that what happened? Or was the original set-up inconsistent? Is it not an equilibrium at all? Or is it an unstable equilibrium?
In any case, Aaron and his friend Jamie have developed a saying, it’s a green-eyed/ blue-eyed thing, which means it’s an apparently information-free fact which changes everything. I think I’ll use that.
I was was having a wonderful ramen lunch with the mathbabe and, as is all too common when two broad minded Ph.D.’s in math get together, we started talking about the horrible state math education is in for both advanced high school students and undergraduates.
One amusing thing we discovered pretty quickly is that we had independently come up with the same (radical) solution to at least part of the problem: throw out the traditional sequence which goes through first and second year calculus and replace it with a unified probability, statistics, calculus course where the calculus component was only for the smoothest of functions and moreover the applications of calculus are only to statistics and probability. Not only is everything much more practical and easier to motivate in such a course, students would hopefully learn a skill that is essential nowadays: how to separate out statistically good information from the large amount of statistical crap that is out there.
Of course, the downside is that the (interesting) subtleties that come from the proofs, the study of non-smooth functions and for that matter all the other stuff interesting to prospective physicists like DiffEQ’s would have to be reserved for different courses. (We also were in agreement that Gonick’s beyond wonderful“Cartoon Guide To Statistics” should be required reading for all the students in these courses, but I digress…)
The real point of this blog post is based on what happened next: but first you have to know I’m more or less one generation older than the mathbabe. This meant I was both able and willing to preface my next point with the words: “You know when I was young, in one way students were much better off because…” Now it is well known that using this phrase to preface a discussion often poisons the discussion but occasionally, as I hope in this case, some practices from days gone by ago can if brought back, help solve some of today’s educational problems.
By the way, and apropos of nothing, there is a cure for people prone to too frequent use of this phrase: go quickly to YouTube and repeatedly make them watch Monty Python’s Four Yorkshireman until cured:
Anyway, the point I made was that I am a member of the last generation of students who had to use slide rules. Another good reference is: here. Both these references are great and I recommend them. (The latter being more technical.) For those who have never heard of them, in a nutshell, a slide rule is an analog device that uses logarithms under the hood to do (sufficiently accurate in most cases) approximate multiplication, division, roots etc.
The key point is that using a slide rule requires the user to keep track of the “order of magnitude” of the answers— because slide rules only give you four or so significant digits. This meant students of my generation when taking science and math courses were continuously exposed to order of magnitude calculations and you just couldn’t escape from having to make order of magnitude calculations all the time—students nowadays, not so much. Calculators have made skill at doing order of magnitude calculations (or Fermi calculations as they are often lovingly called) an add-on rather than a base line skill and that is a really bad thing. (Actually my belief that bringing back slide rules would be a good thing goes back a ways: when that when I was a Program Director at the NSF in the 90’s, I actually tried to get someone to submit a proposal which would have been called “On the use of a hand held analog device to improve science and math education!” Didn’t have much luck.)
Anyway, if you want to try a slide rule out, alas, good vintage slide rules have become collectible and so expensive— because baby boomers like me are buying the ones we couldn’t afford when we were in high school – but the nice thing is there are lots of sites like this one which show you how to make your own.
Finally, while I don’t think they will ever be as much fun as using a slide rule, you could still allow calculators in classrooms.
Why? Because it would be trivial to have a mode in the TI calculator or the Casio calculator that all high school students seem to use, called “significant digits only.” With the right kind of problems this mode would require students to do order of magnitude calculations because they would never be able to enter trailing or leading zeroes and we could easily stick them with problems having a lot of them!
But calculators really bug me in classrooms and, so I can’t resist pointing out one last flaw in their omnipresence: it makes students believe in the possibility of ridiculously high precision results in the real world. After all, nothing they are likely to encounter in their work (and certainly not in their lives) will ever need (or even have) 14 digits of accuracy and, more to the point, when you see a high precision result in the real world, it is likely to be totally bogus when examined under the hood.
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.