Sunday, July 29, 2012

Is Mathematics Only for the Mathematically Inclined?

A colleague just sent me a link to an opinion piece in the New York Times Sunday Review with the somewhat provocative title "Is Algebra Necessary?" The author (Andrew Hacker, an emeritus professor of political science), while acknowledging the profound role of mathematics in society, questions whether we should require algebra (or, as I read the piece, anything beyond basic arithmetic and some basic level of statistical numeracy) of all school children. He quotes a professor of educational psychology (John P. Smith III) from Michigan State University asserting that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.” (For what it's worth, MSU has a nationally renowned College of Education ... but I have a hard time trusting anyone who looks that much like a young Geraldo Rivera.)

I don't want to get bogged down in the details of mathematics education in the U.S., and I certainly agree that the curriculum content (as well as the delivery) needs to be updated periodically. The mathematics of computing was not on anyone's radar when I was an urchin, and it's extremely important today. I'll also stipulate that not every adult in the U.S. (or elsewhere for that matter) needs to be proficient in algebra or trigonometry, let alone calculus. That said, my immediate reaction was that I thought I saw two flaws, one perhaps minor but the other crucial, in the author's argument.

I'll address the minor flaw first. Professor Hacker says (near the middle of the second page):
Of course, people should learn basic numerical skills: decimals, ratios and estimating, sharpened by a good grounding in arithmetic.
By "estimating" I assume he means computing (or intuiting) an approximate value (order of magnitude, interval, ...) for some uncertain quantity (variable). That implies the ability to determine a formula for the variable in terms of other parameters of the problem. For example, if I have a recipe (for a meal, or for a medication I must administer) that calls for a water-based solution with 40% active ingredient by volume, and I need five fluid ounces of the solution, about how much active ingredient do I need? That sounds to me like solving a simple linear equation, which I believe falls into the category of "algebra". (As a sidebar, in the event that this is a medication to be administered to me, and a nurse or pharmacist is doing the calculation, I would really appreciate it if they could do better than just an "estimate".) What if I know that I have five fluid ounces of active ingredient (which must all be used) and require a 40% solution? Can I estimate the amount of water required without knowing some basic algebra?

The major flaw with Professor Hacker's argument is that if we reduce K-12 mathematics education to the bare minimum, as he suggests, it will shut the doors to many college majors before students reach college. When I was in school, separate math classes did not begin until junior high school, so through grade 6 I received the same mathematics instruction everyone else did ... which in Professor Hacker's universe would be thin gruel indeed. I assume the system still operates approximately that way.  The mathematically inclined public school student (which described me at that age) will hopefully be able to go beyond the basic curriculum ... somehow ... at least if his or her parents can afford private tutoring. The generic college-bound student, however, will matriculate with the ability to figure out the tip on a restaurant check, and not much else.

I'm reminded of an incident from graduate school. Back then, registration was done manually in a large gym called "The Pit", and our math department used graduate students to relieve the secretaries of some of the duty (doling out punch cards to students seeking classes). One summer, I was working The Pit with a fellow doctoral student, "Mean Dave" Green. (Don't ask about the nickname, just trust me: he earned it.) An undergrad came up and requested Algebra I. Dave looked at him and asked "Didn't you take that from me fall quarter?" The kid indeed had (and failed the class). Dave asked why he hadn't repeated it immediately, in winter quarter, while the memory was fresh.  The kid had - and failed again. As he did spring quarter. So here he was asking for it for the fourth time. I innocently inquired what his major was. His answer: electrical engineering. The requirements for EE then were: algebra I and II; calculus I through IV; linear algebra; and ordinary differential equations.

Now picture that kid's children (or, um, grandchildren? sigh) reaching college today with Professor Hacker's suggested mathematics curriculum under their belts. Electrical engineering? Not going to happen; the math prerequisites alone would add two years to their degree, charitably assuming that a university was willing and able to staff that many remedial math classes, and charitably assuming that someone 18 years old with a carefully honed disdain or fear (or both) of mathematics is still able to learn math. Other engineering majors? Not likely. (If, somehow, Professor Hacker's position prevails and universities continue to pump out civil engineers nonetheless, remind me to stay off bridges and out of tunnels.)

Computer science uses enough mathematics that I suspect it would be tough to handle. Some business majors might still be open (personnel management?), but the more lucrative ones (finance and accounting in particular) might be off the table for those who had not gotten private tutoring. Physical sciences? So long.

Before I added "emeritus" to my title, I sometimes found myself answering questions from students (or parents) about how much math to take during freshman and sophomore years of college. My response was typically some variation of "as much as you can stand". Most college students do not commit to a major until their junior year. At that point, if you have racked up more math than your major requires, you can usually count the rest as elective credits; but if you are short of math, it can freeze you out of some majors. I've seen a similar phenomenon with graduate school: students who did a non-technical undergraduate major, taking the minimum amount of mathematics they could (sometimes none), and then decided they wanted to go to graduate school in a program with stiff mathematics requirements. Extrapolating that to cover students whose public school mathematics training stopped at counting and basic calculator punching frightens me.

We can legitimately debate what the basic K-12 mathematics curriculum -- the foundation for both attending college and going into a trade -- should be; but if we are going to reduce it as much as Professor Hacker suggests, we'd better come up with a genetic test that can identify a fetus's future trade or college major by birth.

Update: To my utter lack of surprise, other people are weighing in on this. I'm not actively looking for all the commentary (and I'm sure it will be voluminous), but as I come across posts I'll link them here.


  1. Cannot comment on US education system, but this reads as spot-on to me.

  2. It often seems to me that there is a missing bridge between abstractions and real world: if you say that a given quantity refers to money, laymen interest arouses immediately. It might be the case that we waste too much time studying topics that are not as useful to daily life, but it is more likely that many teachers fail to explain how the knowledge of certain topics could help students afterwards.

    1. I believe that one of the commentators I added in the addendum (#4, I think) expressed a similar sentiment. People argue (with justification, in my opinion) that learning mathematics helps train the brain how to deal with abstraction. At the same time, I think you sometimes have to show students how a concept (mathematical or not) translates from one application to another.

      Earlier in my career I was involved in curricular decisions, relating to mathematics in particular, for business majors. I argued (and still believe) that if an undergrad business major sits in a math class and hears "block sliding down an inclined plane", they instinctively tune out. If they hear "rate of growth of an investment", they tune back in.

      I agree with your conclusion. At least here in the US, with college math a part of the problem is that instructors usually major in math and take science but not business courses, so their examples tend to come from physics or engineering rather than from economics or business (or something else entirely). In primary and secondary schools, teachers are more likely to be education majors, and I suspect that some of them have pretty much no idea of applications of math period.

  3. Hi Paul,

    do you read
    If not, this comic may be helpful for this topic.
    The one before that is also good. And some other too, especially this one:

    1. Yes, a (now former) doctoral student turned me on to xkcd a while back. I remember #1050, and you're right, it is pertinent to the topic.

  4. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies. It is also the foundation of the practical art of surveying.


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