20Aug 2012

# Should we teach algebra?

Paul Raeburn
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Andrew Hacker's New York Times Op-Ed--Is Algebra Necessary?--ran more than three weeks ago, but it is still generating blog posts and comments in a wave that could probably be described with some algebraic equation, but not by me.

I thought the wave had crested, but when Jennifer Ouellette posted on Scientific American a few days ago, I decided to say something about this. "Today we are defending algebra," she writes. "Again."

Hacker started this discussion by suggesting that algebra was too hard for most students and was a chief reason why many dropped out of high school. Further, it's unclear whether algebra helps many of us at work. In short, it doesn't do any of us much good later on, he argued.

Any science writer who has worked in a newsroom knows that few reporters--most of whom, presumably, have had algebra and finished high school--can calculate a percentage. I could have had a brilliant career if I wasn't spending half of my time at the AP computing percentages for political science majors or law school grads.

Whatever benefit they were supposed to have received from algebra, they didn't receive it.

But we're here to discuss the journalism, not whether Hacker was right or wrong. Some of the discussion revolves around whether algebra is an important reason why students drop out of school. That's an important issue for society and for education professionals. But I'm more interested in the broader question: Is algebra worth knowing?

Ouellette, who covers mathematics, argues strongly for algebra as something we do use in real life, as something worth knowing in itself, and as an important preliminary to careers in physics, engineering, biology, and so forth. She talks about a Japanese education experiment that left 1 in 4 college students unable to calculate an average. As a confessed former math-phobe, Ouellette makes a strong argument for teaching algebra. I'm not always entirely clear on what she means by algebra. Certainly averages are something we use in real life. Quadratic equations? Maybe less so. Anyway, be prepared if you want to take on Ouellette; she's a tough campaigner.

She also links to some interesting pieces by others. One of them, by Mark Chu-Carroll, a computer scientist, at Bad Math, argues that of course we need algebra in real life, even if we are not scientists. His example? Calculating which of two mortgages is the better deal. Here's how that goes:

How do you figure that out? Well, the amortization equation describing the mortgage is:

Where:

m is the monthly payment on the mortgage.

p is the amount of money being borrowed on the loan....

I'm not persuaded. I've had more than my share of math classes, and I don't have that equation in my head, I wouldn't easily know where to find it, and I don't have software that would run the calculations for me. I'm guessing there are plenty of scientists and engineers who ask for help when it comes to sorting out their financial investments.

You can find more links in a rather angry post by Blake Stacey who dismisses Hacker as bizarre and completely without merit. One such link went to a piece by Cathy O'Neill who takes a thoughtful look at why algebra is such a stumbling block for students, and how it might be better taught. Over at The Washington PostDaniel Willingham, a cognitive scientist, argues that math and science are good predictors of individual income, which is not quite the same thing as saying everyone should know algebra.

Evelyn Lamb, also at Scientific Americanfinds a great example to explain what it is that people need to know from the algebra they are learning now:

Algebra and geometry, another subject Hacker could do without, help develop logical skills and abstract reasoning so we can understand why we are making less money than before if we get a 20 percent pay cut followed by a 20 percent raise (or a 20 percent raise followed by a 20 percent pay cut—hello, commutative law of multiplication!) or how much merchandise we can purchase if we have \$100 and a 25 percent off coupon.

The missing piece here, for me, is that a whole lot of people who have had algebra don't get this, in my experience. Sure, scientists get it, and business analysts get it, and engineers get it, but they've had more math than high-school algebra, and a whole lot more practice.

This debate isn't likely to end soon, and it's spawned what I think is a fascinating discussion on the blogs.

Dip into it, and enjoy.

-Paul Raeburn

Thanks for the comments. I'm thinking I should have hit harder on the point that when we say "algebra," in this discussion, we could be talking about any number of things, from the simple math carpenters use to relating trigonometric functions to the unit circle. Maybe the question should not be whether we need algebra, but, rather: What algebra do we need?

I've written about this on another blog like yours, arguing that we don't "need" algebra in everyday life.

I'm going to approach this two ways. FIrst, there's the issue of whether people use algebra in their daily lives. I would say unequivocally the answer is yes, even if you aren't conscious of it. Hacker and a lot of other people seem to think "all those funny equations with symbols and such" when they think "algebra." But being able to do a quick calculation in your head to check whether your hourly rate is enough for rent that month is something which many working people do. And many professions, carpentry and plumbing being two, run into simple algebra all the time. No, it isn't a cubic equation. But the guy who built your stairs applied some pretty simple algebraic concepts, even if it was 100 years ago. This is especially true if he had to figure out how much room he had for the stairs to extend and it wasn't a simple 3/4/5 combination. Masonry uses algebra, when you decide how much brick you need and how many courses you will lay down, or how many boxes of floorboards you need to cover the square footge in a home.

This is all pretty simple math, of course, and it isn't the same as more advanced analytic geometry. But it involves algebra nonetheless.

And to Steve Miller's point, I can make lots of really simple agebraic operations look complicated if I use "advanced" notation. But if you say to someone "Multiply the loan by the rate by the term in years and you get the total interest you will pay" that sounds a lot simpler, right? And that latter calculation is much easier to do. Steve is right, though: you have to have the whole concept of plugging numbers into a formula to make not only that loan equation work, but a zillion other things (like hourly rates of pay). And we do it all the time; it's just not called algebra and people make a completely arbitrary distinction between "complicated" and "easy" math.

But the use case isn't the only thing that you need to consider here. There's the knowledge case, and that's where I think poeple make two mistakes. First, assuming that if you don't use knowledge in the way I use knowledge of a recipe to make a cake it isn't terribly useful. The second is assuming that education only serves that purpose.

What do I mean? Let's take your essay and replace all the math terms with history. "But we're here to discuss the journalism, not whether Hacker was right or wrong. Some of the discussion revolves around whether history is an important reason why students drop out of school. That's an important issue for society and for education professionals. But I'm more interested in the broader question: Is history worth knowing?"

Hopefully you wouldn't be arguing that  because many people who have hd history classes don't understand why it is important, it isn't worth teaching. But we all know it is important, even if not everyone will be a historian or poli sci major.

And this is what makes the whole debate so goddamned ugly. The assumption is that education is for making people minimally useful workers, as opposed to citizens. The whole reason we have any public education at all is so that people can take part in their political lives. Does everyone do it? Hell no. But can you imagine going to a PTA meeting with a bunch of people who knew no history at all? Or had never read anything except a Dick an Jane book? After all, Dickens isn't useful either, and most people never analyze literature once they are out of school. Why bother? Math isn't the only class that makes people drop out.

And it's this attitude that leaves the door open for the anti-science movement that has made itself felt in many school districts. After all, science isn't useful in many areas of life either -- I never "needed" the cell biology from 10th grade. So why bother about evolution? Why teach the scientific method at all? Lots of scientists have completely forgotten the simple cell biology they took in high school. So why bother with it? Why care about teaching proper evolutionary theory?

The issue you bring up about people not "getting" the use of abstract skills is worth noting here. I've known people who had little formal education but wanted it for their children -- in fact, my admittedly anecdotal experience has been that people with lower socio-economic status often value education *more*.  They aren't ignorant of the fact that there's all sorts of abstract skills that are involved in "working" the system. They all know that they don't have them, and aren't always aware of how to get them, but they know it's important. After all, you don't have to be a financial analyst or a lawyer to know the landlord is scamming you. But being able to articulate how that scam is happening and understand how it works --  that's power.  Every working class kid knows *that* instinctively.

That might be the disconnect you're seeing. Knowing how to acquire skills and knowledge is not instinctive -- the "learn how to learn" part. That's the big failure in American education, if I were to point to one.  But then we're getting into how stuff is taught and not what the subject matter is -- a distinction people like Hacker ignore.

One reason to study algebra is for cultural awareness:
for example, to be able to tell when arithmetic suffices and algebra is overkill, even if you then avoid any problem which needs algebra.

It's not so different from a functional adult knowing when dry cleaning, arc welding, or hip surgery are apropos, even though they never actually practice any of those fine arts.  What is different is that many people get right up to the edge of using algebra before they chicken out.
Few of us wonder if we should set up our own orthopedic sugery or hire it out. Too many of us just throw all the clothes in the washer: even algebra won't help with THAT decision.

I would say that you don't need to have the amortization formula at the top of your head.  It is easy to find and easy to use - if you understand how to substitute numbers into it and calculate the result.  I think that is where having taken algebra makes the most difference. I would venture that most people who did not take algebra could use that formula but would be too intimidated by the appearance of the equation to realize that it is actually not that hard to work out the numbers and make the comparison.

Steve Miller