Algebra is necessary!

In a recent opinion piece on the New York Times website, Andrew Hacker raises the question “Is Algebra Necessary?

The longer answer is that completing the square, or factoring quadratics is probably not going to be useful to most Americans when they enter the workforce, but that’s not (or at least should not be) the point of a math class. The point is to teach critical thinking and problem solving skills, so we don’t have a scarily irrational and illogical populace.

Hacker contends that mathematics is too difficult and that students who will never use it are held to unrealistically high standards regarding their mathematical aptitude.

Hacker cites the statistic that only 58 percent of those entering higher education end up with bachelor’s degrees. He claims that mathematics is the main stumbling block to graduation. Specifically at the City University of New York, 57 percent of the students did not pass the required algebra course. If these statistics are accurate, then one of two things is true:

• Hacker is trying to pull a fast one on us, and lie with his statistics, or more likely
• Hacker doesn’t realize that he’s proved nothing

These statistics, in tandem, are meaningless. Of course it’s impossible to tell from the single data point he gives us if these numbers are at all correlated. But even if they are (as I suspect), the implication is that mathematics is causing the students to drop out. I feel quite strongly that anyone who has earned the title “doctor” should better understand the difference between correlation and causation.

Hacker goes on to explain, to his apparent dismay, that many schools look for a 700 on the SAT math section (I’m a proud graduate of one of the schools he cites). On this page you can find the statistics for test scores of admitted students for WashU. I’ll summarize. The 25-75 percentile range for SAT scores in math in 2010 was 710-780. For verbal, it was 690-760, a negligible difference. The english and math ranges on the ACT are identical. It’s not that these schools are requiring abnormally high math scores. They’re as selective verbally as they are mathematically, a point which Hacker conveniently omits.

I would be remiss if I did not give Hacker the credit he deserves. He says “what is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey.” Hacker is absolutely correct on this point. It is then almost laughable that in the very next paragraph, he states that “there is no evidence that being able to prove

$(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$

leads to more credible political opinions or social analysis.” As Hacker said himself, mathematics is not about having memorized this result. What matters is what it conveys. I doubt Hacker knows this equation is exactly what one uses to generate Pythagorean triples.

To be fair, I don’t think we should expect non-mathematicians to recognize this formula. It’s beyond what I think is necessary to be a well-rounded citizen. But I do think we should expect the average American to be able to do the basic algebraic manipulations to prove this identity. This is the sort of mathematical literacy I would deem the rough equivalent of being able read, or being able to write in complete sentences.

To reiterate, being able to generate Pythagorean triples won’t help the average citizen become more informed, but that’s not really the point. Analogously, it’s not helpful to ensure that every student knows all of the president’s names and their dates of office, but that isn’t why we study history.

In fact, the problem with math education rears its ugly head exactly in this analogy. History teachers don’t care so much about our memory of dates, because whether the bill of rights was ratified in 1791 or 1792 isn’t super important. So why must math students memorize theorems and how to apply formulas? That’s the easy stuff. We should be testing them on their critical thinking and problem solving abilities. Admittedly, these sorts of tests are more difficult to write, but no one ever became a math teacher because it was easy.

Unfortunately we’re just now getting to the most appalling of Hacker’s ideas. He says that “instead of investing so much of our academic energy in a subject that blocks further attainment” we should start looking at alternatives. The implication here is that mathematics is the problem, rather than the manner in which it’s taught.

My favorite sentence in the piece is in the last paragraph: “…there is no reason to force them to grasp vectorial angles and discontinuous functions.” The implication here is that if the exact mathematical knowledge won’t be useful (and I’m not convinced that either of the things he mentioned are not useful) then it shouldn’t be taught. As a friend of mine pointed out: “most jobs do not require algebra just as most jobs do not require you to know that Barack Obama is president; that’s a really stupid argument against the value of knowing it.”

There is no doubt about it; math is difficult and often poorly taught. But that’s a pretty ridiculous reason to stop teaching it.