# Representation in the Math Major

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.)

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