# 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.

### 15 Responses to Algebra is necessary!

1. DrQz says:

Slightly longer short answer http://is.gd/kNx7MG supported by UCLA psychological study.

• Great, article. The point that studying math hones your ability to resolve problems logically should always be a main argument for why we teach math in the first place.

With that said, I disagree with your last sentence. I believe that Math is one of the few courses in public education taught with integrity. By that I mean we truly want the students to know what they’re doing. Kids often turn to something like English because they find the amount rules and mechanics of math to be overwhelming and frustrating. The truth is there are also rules and other technical aspects of English; for some reason they are just rarely taught. Actually, the explanation isn’t even that obscure. Schools can get away with it. English taught with minimal emphasis on grammar can still be evaluated in a way that brings results that fit neatly on a bell curve. Still, most high school students can identify the past participle of a verb more easily in their foreign language than in English. Math can’t really afford that sort of luxury. You can’t skip the basics and go straight to the fun part.

The problem here shouldn’t even be with math. We are at least setting the bar where it should be. We might be struggling with reaching our goals, but it seems like there isn’t even any effort to take other areas seriously. That scares me more. A sound education in the arts is crucial to culture, and I’m terrified of the idea of American Culture getting any worse.

2. Chao Xu says:

“The point is to teach critical thinking and problem solving skills, so we don’t have a scarily irrational and illogical populace.”

It is inherently difficult to find theorems to prove in the set of algebra taught in high school. What are there to prove? The most sophisticated thing in those algebra text is the quadratic formula, and that proof is no more than a few lines of equations.

I would think some other math topic should replace the study of algebra.
For critical thinking, set theory works well. It has no pre-reqs. There is no algebra involved, there are lot of definitions and prove theorems about things partial order, ordinals and other things that goes from intuitive to counter intuitive.

For problem solving: graph theory. There are many graph theory problems that doesn’t rely on algebra at all, like proving “A simple graph with n vertices (n ≥ 3) is Hamiltonian if each vertex has degree n / 2 or greater.” you need to know nothing more than n/2*2 = n. All the problems are tricky and require one to see connections between all kind of graph invariant.

Both of the above actually teach student to do something no algebra course actually do: write proofs.

• Joe says:

Because writing proofs is more important than understanding the basics of algebra. Okay.

• I hate proofs. I loved geometry and calculus (I ended up with an English degree) but proofs were the bane of my existence. I also teach special ed, and I can tell you with certainty that a kid who doesn’t get Algebra is definitely not going to get proofs. No way.

• soffer801 says:

I have to disagree with you on this one. A proof is nothing more than an argument establishing, undeniably, the veracity of some claim. There are many parts of mathematics where algebra is not required prior knowledge to comprehending proofs. In general, anyone that can recognize logical thought should be able to recognize a proof.

That being said, you’re probably correct about proofs, as they are currently taught. I think they’re made to be artificial and unmotivated when they in fact should be very natural. In fact, nearly every other discipline has an analogous concept. You can’t just state your ideas, you need to provide evidence. It’s just that in mathematics, the rules about what constitutes evidence are extremely strict. But somehow this is lost in the geometry classroom (or at least was in mine).

3. Marc says:

I think the NYT article was an excellent example of an all-too-common reaction in this country lately. Aside from Hacker’s blatant attempt to draw unfounded causation, as you mentioned, I wonder what exactly the purpose the purpose of handing a student a diploma would be if we did not require some sort of of education and learning to have taken place to earn said degree. I agree that math could be taught in a much more meaningful and useful way even to those who not be regularly working with mathematical equations, but I found his idea, that because something is difficult and that students are giving up rather than persevering in this subject, it is of sound judgement to simply discard it from the core curriculum. As someone who struggled with algebra and physics, I can say without question that I use skills I learned in both courses at least once-a-week.

Solid education reform has to start with an acceptance among the populace that 1) our students are not performing – as a whole – as well as they should, and that 2) we can do better. Dropping math takes us in the opposite direction. Thanks for writing this.

4. Nathan Strenge says:

As a math teacher who has taught AP Statistics, Algebra II, and Precalculus over the course of my first three years of teaching, I can tell you two things: 1) Some level of algebraic understanding is absolutely vital for every high school student in America, and 2) There needs to be measures put it place to make high school math more applicable to students. As many others have already alluded to, we cannot discard Algebra simply because not everyone uses it in their future work environment. When taught properly, Algebra develops problem solving and quantitative reasoning skills that no other core high school course does. That said, teachers across the country do need to make an effort to find applications and connections between math and other disciplines. These connections are everywhere, if you know where to look. Connecting math to today’s high-tech, statistical world can bridge the learning gap and get student’s excited about math class. Should we completely throw out Algebra because too many kids don’t understand it? Absolutely not. Do we need to make a structural change to the course to make it relevant in today’s society? Absolutely.

5. Lynda B. says:

In my teaching experience, I have known many hard-working, dedicated and selfless individuals who put students’ education first. Are hands are tied by the College Board. Simply go to the website and look at practice tests for the ACT. Nothing but unrelated skills and symbol manipulation. As educators, we want to change curriculum. If our students do not achieve, however, on these standardized tests, we are seen as incompetent and our research and efforts not valued.

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