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.

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!

I just finished uploading a large collection of my tight knots and links to Thingiverse, where they can be downloaded for 3d printing or sent to Shapeways or another service. The ever-awesome Laura Taalman did a bunch of these over the summer, and I was so inspired that ever since I’ve wanted to put the whole collection online somewhere.

Hopefully, someone will actually print these! Here’s a sort of random selection of some of the models available for your printing pleasure…

So I’ve been working for a few days on understanding LuxRender, which if you’ve never seen it is a quite impressive global illumination renderer. I’ve been trying to piece together a Mathematica interface which will allow me to take quick snaps of tubes, knots, random polygons and the like and been completely befuddled.

Aside from the usual confusion about coordinate systems and the like, the real thing that threw me about physically realistic rendering was the idea of tone mapping. The problem is that images that you actually see in the real world have very high dynamic range: there are regions in the image where lots and lots of light is coming into your eye, and others where very little is. That causes a real problem for a renderer, which comes up with brightness values which are way larger in range than anything which you can display in a reasonable file format (also EXR seems to be a graphics format designed for exactly this sort of thing). I’ve figured out the settings enough to produce the image below right from the data below left:

which I call “pink popcorn explosion”. That seems like a pretty good day’s work to me! Of course, the image right suffers from the usual problems of bad photorealistic rendering: too few lights, shadows, are too harsh, color choices are kind of random, and there’s no sense of scale anywhere in the image. Plus the camera angle or the data seems to have changed between renders. Still, as a first attempt: seems like a good day’s work to me!

I just finished making a small contribution to this image (hopefully to appear on the cover of the Proceedings of the National Academy of Sciences) built by Tammy Cantarella and Aaron Abrams.

The structure illustrates some of Aaron’s research on Dehn functions. Aaron’s work is way deeper than this example, but the example is still pretty neat. It shows a space where a curve can enclose *exponentially* much area compared to its length (as opposed to curves in the plane, which can only enclose area proportional to the square of their length). The rectangles on the complex above are each considered to have the same area. At each branch point, the lines split off along the three leaves of the tree in cyclically repeating order, so the number of lines on each leaf is one-third of the number on the “parent” leaf. This means that a curve which goes far out from the central spine can enclose a lot of rectangles near the spine with a very short length.

Tammy made this picture by actually building the tree from paper and carefully photographing it. Then she edited the image in Photoshop and added the horizontal and vertical lines. Aaron provided the math, and I provided a little design advice and mostly translated between the two of them.

Kudos to Tammy and Aaron– this looks awesome!

Most people know that you can always solve a maze (eventually) by turning left. Here’s a more visual solution to the maze problem which I developed for a elementary school class at Waseca Montessori School in Athens, Georgia. Instead of thinking about the maze, think about the walls of the maze: if there’s a path through, that path has to divide the walls into two disconnected pieces. If you color in the walls differently, the solution to the maze becomes obvious!

Of course, you can make an even harder maze, with several different solutions, by adding some more connected pieces to the maze (with more colors, of course). Here’s one with 5 components instead of two.

I made a bunch of mazes with Mathematica for the students to try, which you can print out as PDF files: Level 2 Maze, Level 2 Maze, Level 3 Maze, Level 4 Maze, Level 5 Maze, Level 6 Maze, Level 13 Maze, Level 16 Maze.

You can play with the demonstration by downloading Coloring Mazes (CDF) which is a file in the Wolfram Computable Document Format, which is really pretty neat! This is now also available as part of the Wolfram Demonstrations Project.

Erik Demaine, Marty Demaine, and Anna Lubiw proved the following amazing thing a few years ago:

- You can fold a piece of paper so that ANY shape with straight edges can be cut out of the paper with a single cut.

see Erik’s page on Fold-And-Cut problems to read more about the history of this fact. (Did you know that Betsy Ross cut out the five-pointed stars on the first American flag this way? I didn’t.) During (most of) the fall, I wrote some code with CGAL and Mathematica to generate fold-and-cut creases.

The resulting pictures made fun origami puzzles, so I’m posting them here for your amusement. Each of these images can be folded (along the creases shown) so that you can cut out the shaded letter with a single straight cut. You won’t need any ‘more’ creases, but you might not use all of the creases shown. You only have to crease the dotted lines, not the outline of the letter or shape. Can you do it?

I suggest that you print out these PDF versions (U G A) and try it! Be sure to make the creases before you start folding.

I learned the really interesting theorem from Joe Fu that a flat polyhedral surface always has the property that the spherical polygon enclosed by all the surface normals at a vertex always encloses area zero. As a favor to Joe, I made some neat Mathematica movies showing the folding process taking a flat sheet with three folds at 120 degree angles to a square pyramid that you get by folding a pair of faces into the center of the pyramid. The resulting animations are here and in a top view here. The Mathematica notebook that I used to create them ishere; it contains some code which computes the folded shape in terms of the first n-3 bending angles (the last three are determined by the intrinsic geometry of the surface).