Archive

Archive for the ‘math’ Category

I accept mathematical bribes

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

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

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

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

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

The front is a theorem:

wallet_front

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

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

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

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

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

I’ll give you all a moment…

OK almost ready?

OK cool here’s why this is a proof.

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

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

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

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

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

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

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

wallet_back_quarter

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

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

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

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

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

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

That means the overall generalized figure will have total area:

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

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

Notes:

  • this would be a fun thing to do with a kid.
  • there’s more math inside the wallet which I haven’t gotten to yet.
Categories: math, math education

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

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

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

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

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

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

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

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

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

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

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

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

Here’s the breakdown for “genius”:

genius-neg

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

 

Similar results occur with “brilliant”:

brilliant-neg

Now check out “bossy” and negative reviews:

bossy-neg

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

 

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

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

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

 

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

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

Is it being reported and understood well? Resounding no!

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

Male nerd privilege

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

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

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

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

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

So, I had two responses to her piece.

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

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

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

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

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

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

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

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

Notices of the AMS is killing it

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

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

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

Screen Shot 2014-10-01 at 8.57.17 AM

 

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

Screen Shot 2014-10-01 at 9.02.00 AM

 

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

Categories: math, modeling

The green-eyed/ blue-eyed puzzle/ conundrum

September 17, 2014 70 comments

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.

Set-up

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!”.

Puzzle

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.

Conundrum

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?

Saying

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.

Categories: math

Guest Post: Bring Back The Slide Rule!

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

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

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

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

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

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

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

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

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

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

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

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

I love math and I hate the Fields Medal

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.

spirograph

 

In high school I was incredibly lucky to attend HCSSiM and learn about the wonders of solving the Rubik’s cube with group theory.

rubiks-cube

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.

Categories: math
Follow

Get every new post delivered to your Inbox.

Join 2,924 other followers