I was thrilled yesterday to see this very cool video posted by Henry Segerman, showing some of the tight prime knot models that I did with Eric Rawdon, Michael Piatek, and Ted Ashton. Thank you, Henry! (I don’t actually have printouts of the models as nice as the ones in the video, since the shapes are actually quite tricky to print without extensive support material!) But this is a really great explanation of what that project was all about.
It’s worth saying that the problem of finding an explicit parametrization for the trefoil knot (or any other) is still open, though Pieranski and Przybyl have amazing data now which gives some great clues to what the structure has to look like.
The Journal of Physics runs an occasional series of articles explaining particularly interesting papers which appeared in various series of the Journal. I liked today’s post especially because it deals with problems that I’ve spent a lot of time on– the shape of the tightest overhand (or trefoil) knot, and the shape of the tight ‘simple clasp’ formed by two ropes.
The picture at left is from an amazing series of very high-resolution simulations performed by Piotr Pieranski and Sylwek Przybyl. They reveal some of the amazing tiny and intricate details of the structure.
Maybe someday we’ll understand the whole picture and have a proof. In the meantime, it’s still kind of amazing to think that nobody really knows the shape of a tight knot.
I’ve always been fascinated with perpetual motion machines, since I was a small child. I even built one when I was 6 or 7– the idea was that you’d have a magnet on the end of a rotating arm like the minute hand of a clock and you’d put a bunch of other magnets arranged in a circle around the perimeter, say at every number on the clock. My theory was that if you started the minute hand with a push, it would be attracted to the nearest number’s magnet (true) and then momentum would carry it right past that magnet (also true) until it came close enough to the next magnet to be pulled in by that magnet, and so forth… clearly, the arm would spin forever.
If only I’d understood symmetry, I would have realized before I actually built it that you wouldn’t get any net energy from passing a magnet on the clock face. The arm just oscillates back and forth until friction slows it to hover over the magnet on the face. This was painfully obvious once you tried it.
My next idea was to replace each magnet with an electromagnet and turn the electromagnets off once the hand passed. This would break the symmetry and allow you to keep the arm spinning (as long you added power to the electromagnets). I was briefly convinced that this was a genius invention until my father the electrical engineer gently broke the news to me that this was basically the operating principle for electric motors.
I got thinking about all this today when I came upon this great webpage about overbalanced wheels, written by Donald Simanek at Lock Haven University. He poses the following interesting calculus question:
Suppose you have a circular wheel with three chambers and 3-fold symmetry. Suppose that when the chambers are empty, the center of mass of the wheel is at the center of the circle. Now suppose you add the same volume of liquid to each chamber.
Is the wheel balanced, or will it move?
Simanek links to a nice argument by Alex Pannis, which says
That’s an easy one… The center of gravity will always be lower than the center of the wheel. So, it would require an external force to move it [continually] in either direction.
So far, so good. If we fill each chamber completely, the center of mass is still at the origin (after all, it was at the origin when the chambers had constant density 0). And then if we take some liquid out of each chamber we’ve removed mass from the top (but not the bottom), so the center of mass has to move down.
Since the center of gravity will always be lower [than the rotation axis], independent of the position of the wheel (and the shape of the volume occupied by the liquid), the wheel will stay in equilibrium, no matter what its initial position.
This part seems less clear to me. After all, the center of mass could have moved “down and right” or “down and left” when we took liquid out of the top of each chamber– maybe as we rotate the wheel, the center of mass describes some loop in the plane (always staying below the height of the center of the circle, of course). So the wheel doesn’t have to be in equilibrium just because the center of mass is below the origin– you’d have to convince me that the center of mass is always in the same place.
So how about it? Any clever answers? You should, of course, be able to generalize to n-fold symmetry, though n-even is probably too easy. 🙂
So, a little bit of backstory: about 5 years ago, I became very interested in triangle centers for a while, because I thought they were going to solve my problems with the square-peg theorem (they didn’t). Triangle centers are various special points that you can define for a triangle in the plane: say, the center of the inscribed circle, or of the circumscribed circle, or the point where the medians meet. These points are generally all the same for the equilateral triangle, but for an arbitrary triangle they can be quite different. And there are tons of beautiful relations between different centers.
This picture, from Wikipedia, shows five different centers for an example triangle: notice that four of them are on a straight line. I wanted to run some experiments with them, so I found the coordinates at the comprehensive Encyclopedia of Triangle Centers page. Since the formulas alone weren’t particularly useful to me, I spent a day or two writing a program to auto-translate them into Perl code that I could execute.
I had found out what I needed to know, and the effort I’d put into it wasn’t particularly novel. But I thought, well, maybe I can save someone else some time. And I packaged the thing up as the Perl Encyclopedia of Triangle Centers, put it on my webpage, and promptly forgot all about it. I think I tracked that there had been a few hundred downloads at one point, but I never got any feedback about it, and I figured that it was probably just one of those many things you do in science which are cool at the time, but don’t particularly go anywhere.
But the other day, I got a pingback for the page, which led me to discover this post, and find out that for at least since 2011, the Perl ETC has been a tiny little part of GeoGebra. Which means that thousands and thousands of people are running, every day, a program with a little bit of my code in it! Which is a pretty good thing to find out on a random Tuesday in the summer. 🙂
The moral of the story, if there is one, is that there are a lot of things that mathematicians produce which can be usefully contributed to the world which are more “research-byproducts” than “research”. So we probably ought to at least think: could this be useful to someone else? Is there an easy, efficient way to package it for distribution? And if so, I encourage you to do it. You never know when something might really catch on!
So I just got back from the TPSE Math (Transforming Post-Secondary Education (in Math)) meeting in Austin this weekend. First off, it was in fact really cool, with a ton of interesting people with lots of interesting things to say about the future of math teaching in college. The experience got me in the mood to write about teaching again, and today’s topic is… FERPA. The Family Educational Rights and Privacy Act of 1974.
Now if you’re not a professor, you probably think of FERPA as that thing which you periodically waive your rights under in order to get a letter of recommendation, or maybe as that thing which keeps your parents from finding out that you failed Chemistry (again). But it’s a lot more than that, and some of the effects of it are downright damaging for your education.
Suppose you were in the hospital, and on your third or fourth day in, after receiving numerous treatments, tests, and prescriptions, your doctor rotated off shift and the new doctor came in the door. Only, instead of your chart, the doctor was holding a sheet of paper with your name, your email address, and your university ID number. “So”, the doctor says, “tell me why you’re here in the radiology department again?”. Your blood runs cold.
You see, when I teach a new class, that’s all I get from the University about you. Do I know your major? No. Do I know when you took the prerequisites for this course? No. Do I know if you took the prerequisites for this course or tested out of them with AP credit? No. Can I see your placement exam to see whether you might benefit from a quick review of, say, trigonometry, before we start using it in the calculus course? No. If your preparation is really good, can I find out where you went to high school and ask what they are doing right? No. If your preparation is lacking something, can I find out where you went to high school and complain that they are doing something wrong? Heck, no. After you leave my course, does anyone tell me how you did in calculus II so I can see if I did a good job? How about how you did in your physics major? Or once you graduate? (I think you’re starting the guess the answers here.)
Although we live in a world where Google knows that I like high-end coffee shops, dark chocolate, and MakerBots and Amazon knows that I’ve read all the Parker novels, to your college professors, you’re a blank slate. Every piece of information about you that we might possibly use to help us improve your education is between hard and impossible for us to get. We can’t even get your picture in order to greet you by name when you show up for our classes.
And forget improving our teaching by comparing the results to other classes, or other institutions. Amazon runs hundreds of A/B tests each year to see which version of the front page attracts more clicks and more sales. But though you fill out hundreds of online homework questions in my course alone, I can barely get that data out of the system to assign you a grade, and there’s no way I can compare it to results from other sections or other colleges.
Now does FERPA as written forbid me from getting this data? Actually, it does not. The law states that student records may be accessed “for any legitimate educational purpose, including studies for the purpose of improving instruction”. You might think that customizing a course to fit the students or improving teaching at the University would qualify as a “legitimate educational purpose”. However, the University legal department can have different opinions about which purposes are “educational” or “legitimate”.
The problem is that it’s institutionally very difficult to actually get any particular record. This is considered “a security risk”. Now, I understand that risk management is an important function in any large organization, and there’s an old Jewish proverb which says that “you should build a fence around the Torah”, meaning that you should avoid anything that’s even close to violating Jewish law. I get it. But this is why large organizations are so often very very slow to innovate. After all, the UGA legal team will not be rewarded if we teach better. But they will certainly be punished if we’re sued. The problem is that teaching is the core mission of the institution, and once the various arms of an institution start to lose sight of the fact we are all here to accomplish a mission, the mission starts to suffer. And that’s where we are today.
So, folks (and if anyone in the legislature is listening) we could do a lot better teaching in higher education and we could do it a lot cheaper if we had a regulatory environment a little more like Facebook in 2014 and a little less like Delta Airlines in 1964. Nobody is going to die if a freak security breach reveals everyone’s Calculus grade in the year 2007 to a botnet run by teenage Estonian hackers.
So can’t we make the precautions match the risks? Can’t we try to unleash innovation and reduce costs with a little deregulation of our industry? Just this once?
I just read a very interesting answer on Quora about whether we have too much math and science education in the U.S. school system. You can read the whole thing at the link, but it made a pretty good discussion about why I’m shifting my calculus class to do some much larger, integrative labs (with writing) as well as the standard “put the n in front of the x and replace it with an n-1” sort of problem solving in calculus.
Here are some excerpts from the Quora post, written by a student, Monika Kothari:
I’m a social science student, but I’ve take more math than I’ll ever need or use, through calculus III. I earned decent grades in those classes by working my butt off. But those skills have never proven useful to me, and I forgot most of them within a year. Most high school students graduate having passed Algebra II, at least in my school district where it’s a requirement to complete four years of math. But most adults don’t know how to apply that algebra to basic life problems. Why?
Take a look at how math is taught, how it’s presented in textbooks, how standardized tests analyze math skills. The emphasis is completely misplaced. When are most people ever going to encounter a problem in life that explicitly tells you to “solve for x”? When are most people ever going to have to construct a geometry proof?
Here’s the thing: they’re not. Problems pop up all the time in every day life, problems that may even require some simple algebra. But people don’t know how to apply the skills they already have to solve those problems. They don’t know what the numbers represent, which formulas to use, which variables are which. They can’t even set up the problem. This is because, instead of teaching students the logic behind math, its uses and flexibility, its applications to everyday problems, etc. math education in the United States tends to emphasize rote memorization and the theoretical at the cost of useful lessons that could applied across fields. (Maybe more word problems would help solve this issue, but that’s something textbook writers need to address as well.)
So here’s my takeaway from this. There are lots of places, especially in social science where some more sophisticated statistical analysis could unlock a lot of useful information, where calculus could be really useful.
But I can guarantee you that outside of the context of a multiple-choice test, any amount of technical skill in calculus is definitely not going to be useful if you don’t know how to find the calculus problem when confronted with a real-world situation. This is why my class spends so much time wrestling with the problem of getting a robot to throw a ball into a cup. It’s not because I think they are all going into robotics or physics (though some are!), it’s because this is a good example of a real-world problem where the really interesting part is defining the math problem. And that process uses all of the skills that we’d really like to see in our students: logical thinking, analysis, problem-solving, and communication.
I was recently invited to a meeting on “Transforming Post-Secondary Education in Mathematics” in Austin. One of the sessions is about “Opening Pathways”, especially for “underrepresented groups” in the math majors. This got me thinking about the term “underrepresented groups”. Who is underrepresented? What is the desirable level of representation and how should we calculate it? Are we achieving these goals, or falling short? And if we’re falling short for particular groups, is the solution at the post-secondary level or the K-12 level?
This blog post is a first attempt to think through some of these issues– though comments are disabled, I’d love to hear feedback by email (my name at gmail).
First, one baseline goal for the profession of college math teachers is to convert mathematically prepared students into math majors. If we’re doing that well, the demographics of mathematically prepared students should look a lot like the demographics of math majors. Of course, you can ask which students are “mathematically prepared” (and you can very much hope to convert some unprepared students into math majors, too, but this is presumably harder).
One reasonable definition of “mathematically prepared” might be to score higher than some cutoff on the math portion of the SAT. You can get this data from the College Board, which I did for the class of college bound seniors of 2013. You can get statistics on the demographics of every single graduating math major at an accredited U.S. institution (!) from the NSF Science and Engineering Indicators project. I used the 2011 data, which was the most recent available at the time of this post (2014). I realize that comparing the demographics of prepared students entering in 2014 to majors graduating in 2011 is bogus, but I wanted to use the most recent data I could. (And this is a blog post, not a journal paper!) At first, I compared the results to the overall composition of the U.S. population from the Census Bureau site, but I decided after some helpful discussions with colleagues that this isn’t really the right comparison. Instead, I used the college board’s demographics for “College Bound Seniors” to see the broadest population that the math major might be reasonably be expected to recruit from.
Here are the results by gender, using various cutoffs for SAT Math score to represent “mathematically prepared” incoming students.
You can see that the results are pretty interesting.
First, if we take the definition of “mathematically prepared” to be “SAT > 600”, the distribution of male and female math majors (second to last column on the right) is almost exactly the same as the distribution of prepared students. This would seem to indicate that all the things we’re doing at the college level to encourage women to major in math have basically succeeded at encouraging well prepared women to major in math. That is, this is the population we’re reaching well.
Notice also that if we look at lower levels of preparation, or at the proportion relative to the overall population of college-bound seniors, women are definitely underrepresented.
Parenthetically, if you choose SAT cutoffs between 650 and 750, women are overrepresented, and if you selected math majors only from students with SAT math 800, women are significantly underrepresented again. In fact, women get about 60% of the perfect math scores on the SAT. However, I think these are basically statistical quirks, since I don’t think anyone would argue that we should or do recruit math majors from students with math SAT scores > 600. In the first version of this post, I included these in the graph, and took them more seriously, but I’ve rethought that.
I think the conclusion you can draw is that if we want women to be more proportionally represented in math, we should look at our efforts in recruiting women whose K-12 preparation is reasonable but not exceptional (that is, the 400-600 range). It may not be realistic for us to hope to get a large number of additional math majors from the < 400 group without significant remediation efforts.
A colleague of mine pointed out that even this might not a fair split, since stereotype threat might artificially depress Math SAT scores among women. I regenerated the graph above, assuming that the scores for women were 50 points lower than their true ability, and settling on Math SAT > 450 (F) and > 500 (M). You get this:
which is to say, we’re under-recruiting women into the major relative to both stereotype-threat-corrected preparation and the population of entering students. It’s good to get a handle on the size of the effect, which is at least not huge: college bound seniors in 2013 were 53% female, while graduating math majors in 2011 were 43% female.
The demographic split by ethnicity/race is maybe even more interesting:
So there are a lot of ways to view this data. One is to note the plausibly sort-of-okness of the profession’s efforts to recruit mathematically prepared black students into the major. At a cutoff > 500, we seem to be getting a proportionate fraction of black students (and perhaps even some less prepared students) to successfully complete the major. This is at least progress, and perhaps it ought to be more broadly celebrated. Obviously, all the things we’re doing to encourage black participation in the major are having some effect and we should keep going with them.
By contrast, it seems that we are still failing at recruiting latino students into the math major regardless of preparation. Maybe we should consider more targeted strategies for this population? (This effect persists at higher cutoffs, but I no longer think it means much for those values of n.)
Asian students are more complicated. If we recruited from students with moderate preparation (say, SAT >= 500), we’re recruiting at a roughly proportional rate. If you believe that we recruit majors from a pool with (on average) better preparation, you’d see that asian students are also significantly underrepresented in the math major. (This is really dramatic for silly cutoffs like Math SAT = 800, where a strong plurality of the perfect scores go to asian students.)
If you make a correction for stereotype threat as above, you get:
which makes the results for black students look somewhat more serious, and keeps the results for latino students at roughly the same level.
I think that this gives a couple of reasonable ways of looking at the scale of differing representation effects in the math major. What it doesn’t do is explain them or decide whether they can or should be changed. As another colleague pointed out, it may be the case that students from some groups are disproportionately attracted to fields which are perceived as more highly paid (e.g. business, medicine) or more practical (e.g. engineering). Or it may be the case that we’re simply failing at running the math major in a welcoming and inclusive way.
Notes and caveats:
1. Of course, this is only attempting to be a first draft, and may well duplicate somebody’s published work along these lines. Does anyone know a reference I should be citing?
2. Note that I’m implicitly rejecting the strategy “steal mathematically prepared students from other majors” in an effort to increase the number of majors relative to the population. Clearly, this would benefit us, but I’m not at all sure that it’s a strategy that is overall beneficial for society. However, I do recognize that this strategy exists.
3. The data on cutoffs comes from the College Board percentiles.
4. Much like Piketty, I sort of feel like posting all my data is posting a giant “kick me” sign on the internet. However, much like Piketty, I also feel like the truth is important, and I’d be happy to hear about it and correct this blog post accordingly if I simply typed in the numbers wrong. So here’s my Mathematica worksheet if you want to play with the numbers yourself, or simply check my work.
5. There’s a tendency to conflate the proposition “efforts to address inequalities are working” with the proposition “efforts to address inequalities are pointless” or the proposition “there are no inequalities to address in the first place”. I would read the data above to support the proposition that our efforts are (for some groups, on balance) probably working reasonably, not either of the others. (If you really care, I can support that, but that seems like another post entirely.)
6. You could argue that this whole thing is bogus because the Math SAT doesn’t measure anything important to defining the pool of potential math majors (either because you don’t believe in standardized tests in general or the SAT in particular, or because you think something like high-school grades are a better measurement), or because Math SAT scores tend to underrepresent the preparation of certain groups due to stereotype threat or other factors. I attempted to address that (at least) with the “corrected” graphs above. I’m sympathetic to the objection that other data might be better, but those data are much harder to get.
8. You could argue that the whole thing is bogus because the pool of potential math majors is (at least) the entire population of college-bound seniors and (maybe) the entire U.S. or world population by definition. I am also sympathetic to that point of view for philosophical reasons. But in terms of measuring efforts on the part of university math faculty in particular, which is what I’m trying to do here, I think it’s reasonable to stipulate that we shouldn’t be expected to make math majors out of people who don’t go to college in the first place (even if they should or could). Further, it’s going to be very difficult to make math majors out of students who arrive at college with very serious underpreparation in math. (This is not to say that we shouldn’t try to do it. But we should probably have a conversation about how much we should spend on remediation versus maintaining our current programs.)
Every year that I teach the numerical analysis sequence, we end the year with a project on image compression. This year, the students used a technique called Principal Component Analysis (PCA) to sort though large datasets of images, looking for a common structure in the image data. Once they discovered the structure, they could use it to compress the images by only storing some of the pixel data and using what they knew about images to reconstruct the rest. The technique doesn’t work as well as it can on images which are already compressed with JPEG, since that method loses some detail already.
But the results on uncompressed data from a RAW file? Pure genius. Kristen Bach of Treehouse and beautyeveryday and Karen Gerow of Double Helix STEAM Academy donated some of their very excellent photography for the students to try their work on.
After a semi-rigorous set of A/B comparisons between different compressions of various images, the class decided that the best results were due to Fred Hohman (in the 50% compression category, meaning that Fred uses half of the image to predict the other half), Irma Stevens (in the 90% compression category, meaning that Irma used 10% of the image data to predict the rest) and Ke Ma (in the 99% compression category, meaning that Ke used only 1% of the image data to predict the rest).
Here are their results!
Since I’ve been the beneficiary of a lot of life-changingly good teaching throughout my life, I thought I’d post a little honor roll of teachers that I really appreciate this week:
For the last few days, I’ve been working through an unexpected problem with lovely, dependable Anki, which is a flashcard program that I use to memorize everything from my students’ names (it’s genius for this) to tricky integral and derivative formulas.
It turns out that all previous versions of Anki (so far as I can tell) worked with whatever TeX distribution you install on your Mac, as long as
This is not true with Anki 2.022 (presumably it will continue to fail for higher versions of Anki), which won’t find latex at all unless you’ve installed MacTeX (which installs in /usr/texbin) or BasicTeX (which I guess installs either in the same place, or someplace else that Anki can find it).
If, like me, you have a TeXLive distribution, you’re in trouble until you discover the “Edit_LaTeX_and_dvipng_calls” add-on for Anki (tip of the hat to one Soren Bjornstad for this one– you, sir, have saved my bacon royally with this one). The trick is to open the add-on, which gives you a piece of Python code containing the line
newLatex = [“latex”, “-interaction=nonstopmode”]
which you need to edit to insert the complete path to latex, as in
newLatex = [“/usr/local/texlive/2013/bin/x86_64-darwin/latex”, “-interaction=nonstopmode”]
If you don’t know the path, you can get it by opening terminal and typing “which latex”. This should show you the entire path on your machine.
Repeat the process with the line
newDviPng = [“dvipng”, “-D”, “200”, “-T”, “tight”]
which, again, needs the entire path to dvipng, as in
newDviPng = [“/usr/local/texlive/2013/bin/x86_64-darwin/dvipng”, “-D”, “200”, “-T”, “tight”]
As before, you can get the path (if you don’t know it) by typing “which dvipng” in terminal.
It’s likely that if you have TeXShop (or TeXWorks) or something similar, then you have either TeTeX or TeXLive as the underlying TeX distribution, and you’ll have to do this. Or, I guess, install MacTeX on top of your existing distribution.
Happy memorizing! And many thanks to Damien Elmes of Anki, who helped me work through all this.