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!
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.)
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:
Plus, I found something truly great, which is the ADL 3D repository. This is a repo of really good 3d models for simulation environments collected across government (seems like mostly DoD) projects and available to YOU for FREE. This is run by the Advanced DIstributed Learning labs, which is a DoD project that I wasn’t aware of before, but seems to be quietly making a serious run at the kind of computer-enabled education tech which I’m completely fascinated by and is being done with infinitely more PR heat (but I seriously doubt better actual results) by the MOOC community.
So I’ve got a game design, some models, and I’m following along happily with the Space Shooter tutorial series at Unity. Seems like a good day to me!
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:
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?
In August of 2007 I was lucky enough to give a talk on mathematics and quilting to the Cotton Patch quilters here in Athens. The talk was a lot of fun, and taught me quite a bit about quilts! You can read the slides at the link above.
In the summer of 2004, I ran a summer program on Mathematics and Visualization which enrolled four art students and four science students for a series of joint projects in science and art. We started with the project of building our own image filters in MATLAB. Here’s an example of a filter created by Seth Dowling based on reshaping the pixel data in an image using a space-filling curve:
Most of the student projects focused on the idea of tensegrity, in which different elements in tension and compression are balanced against one another to create intricate and sturdy shapes. The summer ended with a group art show at the main gallery in the Lamar Dodd School of Art at UGA.