Wednesday, April 6, 2016
On the college level, I think his point is mostly correct: Outside of the science and engineering disciplines, a lot of fields can and should modify their math requirements to be less abstract and more practical. It is still an argument for tracking to some extent--if you are a social science major who takes a lot of calculus and statistics you keep open the option of career paths that in which you're an early adopter of new tools, one who customizes them for the specific needs of your field. Every field has a few people like that. If you are a social science major who doesn't take those classes there are still plenty of things you can do in the workforce as a numerate person who understands a lot of things about people and societies, but you are closing certain doors. Still, I take his point: Most people just need to be numerate.
I still think that a broadly-educated person should attempt at least one real calculus or statistics class (as opposed to a "numeracy" class) if they are majoring in a subject that has highly quantitative sub-fields. But I see the point that most people will do fine in life if they don't take that class, or they try it but it doesn't work. I think most people are, on some level, fine with this sort of compromise, because college students are adults and choosing a major is all about turning away from most of the options in front of you to focus on a few closely-related possibilities.
But the other half of his argument is about high school, and here Americans tend to be much more ambivalent. For a number of reasons, I am OK with starting tracking in high school, partly because I think we unjustly devalue vocational education and partly because I see the consequences of pretending that we need everyone to be a STEM major. However, while I might personally be OK with it, this is a Big Issue. It's a significant departure from current practice, it involves educational issues that go well beyond math education (e.g. writing instruction for aspiring college students should be different from the writing instruction for people who plan to enter the workforce right after high school), and it hits on fundamental questions about democratic values. Emphasizing vocational education can do a lot for the middle class, but it will also lock a lot of people onto paths at a young age. There's a massive value judgment to be made there, and this book does not do justice to that issue. I can tell you that Hacker's argument would be a complete non-starter with most of my colleagues, and not just the mathematicians. Even the social scientists who would stand to benefit from greater emphasis on "numeracy" (which they might teach more effectively than most pure mathematicians) would have concerns about the equity issues in putting more people on non-STEM tracks.
Finally, while Hacker clearly respects math for its intellectual significance, I wish that he had shown that respect more openly, by taking the conversation one step beyond numeracy, at least at the college level. Let's grant that "college algebra" or a traditional statistics class should not be the gatekeeper courses for most non-STEM majors. Why should "numeracy" be the first and last encounter with math? Besides the fact that many people might benefit from a second course on statistics, what about math as a liberal arts discipline? The history of geometry encompasses agriculture, architecture, art (e.g. perspective drawing), and philosophy (via the Greeks' development of mathematics as a formal, logical, deductive discipline). Why not a general education course on that? Or a course on number theory (at an elementary level) and cryptography? I'm not talking about teaching humanities majors to do cryptography in any hard, formal, abstract sense, but why not have courses that try to give educated citizens a sense of what this tremendously important technology is really about, and why it ties into seemingly elementary ideas about arithmetic? How about formal logic courses taught one step down from what a math, philosophy, or computer science major might take, with the goal of getting the students to understand some aspect of the profound insights that Godel and Turing arrived at? I'd cheer for a school that required the non-STEM majors (and maybe even the STEM majors) to take a "numeracy" course plus one more math course from a menu that might include some of the ideas that I outlined here.
On pages 155-156, Hacker says that a country that is as unequal as ours cannot expect strong math performance across the board. Usually people assume that success in STEM classes will fix inequality, but Hacker dares to say that fixing inequality is what it would take to get more success in STEM. That is about as heretical as an Imam drinking bacon-flavored vodka. Then he goes on to question whether beating the rest of the world in math is even a valid goal. Saying that in the current climate of higher education in America is like pissing on the Virgin of Guadalupe in Mexico. Well done, sir.
Tracking is generally considered to be a horribly regressive idea, one for evil conservatives. It thus warms my heart that Hacker, having endorsed tracking in chapter 8, uses chapter 9 to endorse progressive "Discovery Mathematics" ideas. Not my cup of tea, but I appreciate when people break out of dichotomies.
Tuesday, April 5, 2016
Finally, I think Hacker makes one absolutely essential point: As much as we resist tracking, when we make a high school diploma contingent on algebra and geometry we are tracking some people to not attain a basic credential. That is a real cost that a lot of people pay. There might be fine arguments to insist on algebra anyway (personally, I think everyone should be required to try algebra), but we need to confront the bigger questions here. This really deserves a book of its own.
I get all of that, but he starts going too far in some of his assertions. Yes, I'm going to nitpick, but I'm nitpicking because he's tossing out a lot of things largely to fill space. His basic point could have been made, explored, defended, and extended in a 3-part series of articles in a high-profile venue, and it probably would have gotten more attention while including fewer irrelevant and inaccurate asides.
On page 112 he says that mathematicians have done such a bad job of articulating the real value of their subject that 2013 only 730 US citizens got PhDs in math. There's much to say about how academic disciplines have failed to articulate the true value and beauty of their disciplines to the captive audiences in intro classes, but I don't know that we need to complain about "too few" PhDs. Elsewhere in the chapter he was complaining that intro math is often taught by low-paid and low-status part-timers. Well, over-production of graduate degrees is one of the key economic factors underpinning that model. The applied mathematicians tend to get industrial jobs, but the pure mathematicians are produced in greater numbers than the academic job market can absorb into full-time positions.
He also argues that many math courses for non-mathematicians should be taught by non-mathematicians. I will grant that a course on numeracy or "basic statistics for voters" might be better taught by a social scientist than a pure mathematician. And certainly science and engineering departments do tend to have their own mathematical methods courses on topics specific to their disciplinary needs. However, I think it would be a mistake to not have any of this done by mathematicians. An engineering department doesn't know in advance which of their students will go on to careers where they are end-users of signal processing software (where they might not need to know lots of theoretical detail) and which will go on to careers where they develop signal processing algorithms (where they will most assuredly need lots of theory). A physics department doesn't know in advance which of their students will go on to careers where they mostly characterize materials in the lab (where they will certainly need a lot of quantitative reasoning but might not need lots of eigenfunction expansions), which will become a professional theoretician (a rare path, but one that needs theory, by definition), and which will be the applied physicist who develops the algorithms for more efficient software in that materials characterization lab (where they will need theory). So a certain amount of exposure to the thought processes of mathematicians is an important aspect of a broad education.
As is so often the case with 200 page books sold to a broad audience, Hacker has a reasonable basic point but is over-reaching to fill space without diluting his message. I'm coming to hate this genre of books.
Monday, April 4, 2016
In Chapter 6, I think Hacker is trying too hard. He starts by going after a common argument: That learning math is good for your overall intellect, and in some way unique to math, some way that other disciplines don't possess. I will agree with him that mathematical thinking is simply one way of thinking and not THE way of thinking. However, when he tries to claim that nobody makes a similar argument for humanities, he goes too far. Has he never read any of the numerous essays on how studying humanities makes you a broad-minded critical thinker, unlike those nasty, narrow STEM disciplines? There's a lot to be valued in humanities and social science, but let's not pretend that they don't get in on the "We're the REAL smart people" game.
On the other hand, I like his point about how mathematics, unlike some academic disciplines, can thrive even under despotic regimes. I don't consider that a BAD thing about math--I think it's great that there are some wonderful, beautiful areas of inquiry that even tyrants do not try to eradicate. But it also tells us something about how unthreatening many areas of STEM are for those in power. We might do well to ask ourselves why that is.