Sunday, July 29, 2012

Algebra

There's been discussion about this recent article by Andrew Hacker, Is Algebra Necessary?, in which he argues that we should do away with the standard algebra curriculum and replace it with things like "citizen mathematics." It is, of course, pure nonsense; the problem is not that we teach algebra but that (1) we do not teach it soon enough and (2) we teach it by formula and drill in endless problem sets, and while practice of this sort is necessary, when unalleviated it is a good way to bore people to death. We probably could also add that (3) we do not teach it very coherently, since if you pick up a typical algebra book, you'll find it goes from one thing to another without much connection between topics (part, although not all, of the reason for this is that its intimate connection with geometry often goes completely untaught, and so students don't grasp, as Hacker himself doesn't grasp, that these are not arbitrary symbol-strings to be manipulated but descriptions of shapes, curves, relations, proportions, topological transformations, and so forth). But this is a matter of flawed approach. And we could easily do it; elementary school students in fact do problems that are simple algebra problems already. ("What do you have to add to two if you want to get four?" "What times itself gets you nine?") Likewise, any "citizen statistics" course will by its very nature be backdoor algebra, despite Hacker's insistence that it is not.

I was somewhat amused by Hacker's comment that "it's not easy to see why potential poets and philosophers face a lofty mathematics bar." In reality I think this merely shows that Hacker doesn't really have much of a grasp on what philosophers do, since philosophers in general usually need more mathematics than they actually get. Certainly in this day and age it doesn't hurt to be able to follow at least the very basics of abstract algebra, which is a higher order of abstraction than the elementary algebra you'd likely get in high school.

What actually irks me about the article is that it makes the standard anti-liberal-arts move, in which 'course that makes people better as citizens' is read as 'course that makes people more useful', when the distinctive characteristic of a citizen is not useful but free. What we need is to teach mathematics as a liberal art -- i.e., as the making of intellectual instruments that can be applied to many different tasks -- and drilling should be like drills in sports, practice for challenging problems rather than the main content of the course. Cantor famously said that the essence of mathematics is freedom; and algebra is precisely the gateway to the parts of mathematics where this becomes breathtakingly clear. People need to know that world is there. The problem really is that algebra is not taught with any sense of this intrinsic direction toward freedom at the heart of algebra, its ability to carry the mind past the limitations of mere narrow-viewed calculation.

2 comments:

  1. Here's what an article that I think makes another good counterargument:

    http://blogs.scientificamerican.com/observations/2012/07/30/abandoning-algebra-is-not-the-answer/

    I think several other points should be brought up:

    1)  When I think back to my childhood, my guess is that mathematics is the subject that I probably spent the most time doing homework on.  In college, I would say that trend continued as I was certainly spending a lot more time on homework as a math major than my business school roommates.  While there are differing degrees of mathematical aptitude in every student, I think the bottomline is that mastering mathematics as a student probably isn't going to happen unless you are putting the work in.  The fact that it has a high failure rate I think likely has to do with its higher homework requirements more than anything.

    2) Perhaps he misidentified the cause of academic troubles.  He cites inability to pass math classes as something that affects retention rates and provides the solution of simply replacing those math classes that are creating difficulty for students.  But it seems to me that by looking at some of the other classes that CUNY report cites students having trouble with, even some of the classes outside the math department (Economics, Accounting, Chemistry, etc.) also have a fairly strong mathematical component and several of these contain algebra.  Perhaps the solution here isn't less math, but rather more.

    3)  Hacker makes the point that you don't need it for your job unless you are in a select few professions.  However, even that point I would question.  I currently work as a Landman, doing title research for oil and gas companies to figure out who has the mineral rights we need to lease in order for the exploration companies to drill the lands their geologists inform them will be profitable.  While this seems like it wouldn't be a very math intensive job (you spend most of your day going to the court house to read through deeds and other legal documents), math and even (*gasp*) algebra still have a way of creeping in.  For example, creating formulas in an Excel spreadsheet where you use another column in your spreadsheet as a variable isn't complex, but it is an applied version of algebra.  And that's without even pointing out that I think the mathematic classes I have taken in subjects like algebra has assisted in the development of my abstract reasoning skills.

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  2. That was a good argument.

    It occurs to me that another problem with going with a pure 'citizen mathematics' approach is that you have to presuppose what mathematics 'citizens' would find most useful. But precisely the usefulness of something like algebra is its general flexibility -- you don't have to assume, you just teach the general principles, and thus make it possible for people to focus later on whatever they might need.

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