### Is Our Kids Learnding, Algebra Edition

#### by evanmcmurry

Thoughts on Andrew Hacker’s op-ed in Sunday’s *New York Times* arguing that we should 86 algebra from core high school curriculum:

1. I definitely think Hacker’s talking more about algebra II, and higher-level stuff like trig and calculus, than he is straight-up freshman-year algebra. He makes the distinction, in a blink-and-you’ll-miss-it way, but the article explicitly talks about algebra for the most part.

However,* if* we confine Hacker’s article to algebra II, then he’s got a point: a mathematician friend of mine with experience teaching at Hacker’s exact level recounted

*a foundations class that (college) freshmen could take in lieu of an algebra II-type thing. It had *some* algebra but mostly combinatorics, probability, and logic. It will be infinitely more useful to them in their lives, and I think some research needs to go into whether those types of courses should be offered in high schools FOR THOSE TYPES OF STUDENTS.*

That sounds perfectly reasonable to me. It’s a much narrower reading of Hacker’s article than his own thesis allows; whether Hacker meant to make the broader argument, or the *Times* thought the more sensational headline would increase traffic, I have no idea. But I wonder if this debate would be more helpful if he’d tailored his argument to be less risible.

2. This piece seems missing a lot of context. Like, all context. Are the problems students are experiencing with algebra new? Were students having just as much trouble with the subject ten, twenty-five, fifty years ago? If they were, but graduation rates were higher then, what were we doing differently, if anything? If the curriculum hasn’t changed, but the results have, wouldn’t that suggest that the problem is not algebra, but something extrinsic to schooling? Along the same lines, are other countries having this problem? (Dana Goldstein suggests not.) If not, why not? If they’re teaching algebra and getting different results, again, that would suggest the problem isn’t algebra but something either endemic to our education system as a whole, or to our society’s relationship to it. In short, Hacker seems uninterested as to *why* we’re doing poorly at algebra; he seems content to point at it and say “there’s the problem.” This strikes me as classic mistaking-the-symptom-for-the-disease.

(Hacker tags the international argument, briefly: “It’s true that students in Finland, South Korea and Canada score better on mathematics tests. But it’s their perseverance, not their classroom algebra, that fits them for demanding jobs.” Anybody who knows what that means, lemme in on it.)

3. Check out this paragraph:

*There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong — unsupported by research or evidence, or based on wishful logic.*

That’s called reasoning. Good thing somebody, probably a teacher, taught Hacker how to do it. I doubt that teacher taught Hacker reasoning by using this exact example (“Suppose that X believes P, wherein P is the proposition that algebra is valuable”) but probably used similar examples from which Hacker eventually assembled the skill. Hacker may never have encountered the exact problem his teacher used, but that doesn’t make the problem irrelevant to Hacker’s life; in fact, it enabled him to get published in the *New York Times*.

So when Hacker writes, “But there’s no evidence that being able to prove (x² + y²)² = (x² – y²)² + (2xy)² leads to more credible political opinions or social analysis,” he’s being too clever by half. No, knowing how to solve that equation probably doesn’t help you project the funding effects of the Affordable Care Act on the deficit. Knowing how to solve complex equations with multiple variables sure does, though. Which is to say, learning to solve (x² + y²)² = (x² – y²)² + (2xy)² is valuable, even if it’s never encountered in real life. In this case, it’s the skill, not the knowledge, that matters.

4. John Locke, Book IV of *An Essay Concerning Human Understanding*, hit it:

*There could be no room for any positive knowledge at all, if we could not perceive any relation between our ideas, and found out the agreement or disagreement the have one with another, in several ways the mind takes of comparing them. (IV.v)*

That reads like a simulacrum of algebra to me. I know, Locke is dead ol’ white guy, and his formulation of knowledge, of which the above is 25%, was by definition elite. But still. There’s something foundational about algebra, no matter how unpleasant it is, something that girds it against passing educational fads, shifting socio-economic concerns, and, well, popularity. Or, as my mother used to say when I complained about doing problems 1-99 from my Algebra II textbook, “Math doesn’t care whether you like it.” She was right; the quadratic equation persists whether we learn it or not, and the laws of mathematics participate in the governance our lives whether we enjoy them or not; refusing to learn math because it’s obtuse only disadvantages you. Hacker knows this:

*Mathematics, both pure and applied, is integral to our civilization, whether the realm is aesthetic or electronic. But for most adults, it is more feared or revered than understood.*

But if this last part is truly the problem, it seems like *more* education is called for, not less, right? (h/t Kasia for pulling that quote out.)

5. If I were a nineteen year old who’d just read Marx (or a 24 year old who’d just read Althusser) I’d point out that this essay sure seems to be calling for the disabling of the proletariat of any ability to navigate an increasing complex economic system of which they are very much the victims. While I normally think that those who describe Republican attempts to remove humanities and such from the curriculum as “preparing new Republican voters by sowing ignorance in the populace” are assigning too much intention to the GOP, a call for not teaching complicated mathematics just after an historic financial collapse involving derivatives that were commonly described as “complex” sure seems a little off. In short: if you wanted to make sure nobody could understand your crimes well enough to be outraged over them, this would be a good way to do it. That’s IF I were a nineteen year old who’d just read Marx.

Which brings me to my next thought:

6. The two states Hacker highlights as having the worst attrition problems are Nevada and North Carolina. Those two states have some of the worst unemployment rates, with Nevada holding the ignoble record. I have no idea what the causation would be between those two facts, if there were one; something tells me there’s a chicken-and-an-egg thing with education and unemployment. But this ain’t the causation:

*Toyota, for example, recently chose to locate a plant in a remote Mississippi county, even though its schools are far from stellar.*

Huh, it’s almost as if there’s another explanation for that or something. Either way, that a manufacturer decides to bless a municipality producing poor education rates with a factory is not an endorsement of that area’s education policy, nor is its long-term consequences beneficial to that region’s educational performance. Striking algebra as a requirement because it’s not necessary for the local economy is a good way to ensure you produce workers who are able only to satisfy the needs of the current economy, not the potential needs of a future one. In other words, you don’t want to have a city full of factory workers when your economy suddenly goes high-tech.

Hacker also makes this argument in reverse:

*An equally crucial issue is how many available positions there are for men and women with these skills. A January 2012 analysis from the Georgetown center found 7.5 percent unemployment for engineering graduates and 8.2 percent among computer scientists.*

But will the unemployment rates for those jobs now be the rates forever? What happens when our economy rights itself and we suddenly find that we need high-tech jobs to compete? I think the internet bubble is a good example of the unpredictability of of required skills in an economy: who knew people would one day be making billions over competing search algorithms? If you had been designing an education system in the 80s using Hacker’s article as your guide, you would probably conclude that the America is headed towards a service-oriented economy, and needs real-world math skills along the lines of working a register or a database. All well and good, but there goes your job force capable of staffing the high-tech industry. Education is by definition a long-term enterprise; trying to game the skills of the future based off the job requirements and unemployment rates of the present seems like a losing tactic.

The rest of you, thoughts?

God. I should be doing homework for class but instead I’m about to write a ginormous comment on your blog. I hope you’re happy.

Anyway, two caveats: 1) I am not a mathematician and 2) I haven’t read the Hacker piece. So I’m going blind on this one.

Part the First: What is Algebra?

Algebra is a technique for solving problems that involve variables, ranging from the absurdly simple (for instance, x+4=5, x is obviously equal to 1) to the horrendously, mind-numbingly complex. While many of the techniques of algebra (and, in fact, the name “algebra” itself and even our current number system) were developed and perfected in Arab lands, we mostly teach algebra here as starting with the Cartesian plane, which allows us to graph points, lines, curves, et cetera, and allows us to solve geometric problems numerically. As such, it is an incredibly powerful tool.

What we call Algebra I today basically covers everything involving linear equations, that is equations that form straight lines and have a variable raised to a power of 1: y=mx+b. These problems are typically very easy to solve but have a great deal of use in the modern world. Algebra II and Algebra III (sometimes called Precalculus) cover concepts with higher powers of x, i.e. x^2, X^3, et cetera, functions divided by other functions, exponential, logarithmic, and polynomial functions. While these problems are typically not incredibly complex conceptually, the number of terms (especially in large polynomials like (x^3+2x^2+3X+2) times (x^4+2x^3-4x^2+9x-1)) gets incredibly cumbersome very quickly, and it’s very easy to make a simple mistake that makes the entire problem impossible to solve.

Mathematicians in the middle ages up until the seventeenth century developed all kinds of one-off tricks that made it possible to solve particular classes of algebraic expressions more quickly than they would seem, and in fact mathematicians often neglected to publish their results because being able to solve certain classes of problems was more likely to get you hired by the next wealthy noble needing a solution to a problem of how to maximize the area of the stained glass windows in a church, for instance, or how best to run a drainage ditch. Mathematicians of the day often held public competitions to see who could solve certain complex problems the fastest, and if you’re imagining a bunch of guys in pantaloons and funny hats quick-drawing polynomials expressions “Man With No Name” style, you’re not far off.

And then, in the seventeenth century, Newton and Leibniz independently invented calculus. Instead of needing a myriad of formulae to solve particular kinds of expressions, huge swaths of problems could be solved with a simple derivative that any mathematician could do. Aside from allowing us to solve entirely new classes of problems that had hitherto been completely opaque to numerical reasoning, calculus basically made great amounts of the work done my mathematicians of the middle ages completely unnecessary, a bit like the automobile would rid the world of large-scale need for the horse and buggy a few centuries later. All of those min/max problems I mentioned above with the need to figure out how much area you can get out of a set amount of glass, for instance, can be solved instantly with a single formula.

Part the Second: So Why Study Algebra today?

Well, first of all because a lot of it is still useful in solving simple problems. In particular, that Algebra I stuff I mentioned above about linear equations is still 100% useful to anyone solving any kind of numerical problem — indeed, part of the power of calculus is that it allows us to reduce the larger, more complex problems into linear equations that are a snap to solve. Also, Algebra I gives us our first step towards abstraction in quantitative reasoning, a step away from numbers and towards understanding values in relation to other values. Understanding, for instance, that set monthly costs represent the y-intercept value in a linear equation that determines profitability should be intuitive quantitative reasoning that any business owner should just instantly have on hand. For that matter, anyone who works for a business that needs to be profitable should know it. You know, basically everyone everywhere?

As far as those big messy equations of Algebra II? While Calculus allows us to solve most of those very readily, there’s a lot of value in having experience manipulating polynomials for those of us who are going on to higher mathematics and other STEM (Science, Technology, Engineering, and Math) fields. A lot of calculus problems tend to have the form of looking at an equation, fiddling with it using the techniques of algebra to get it into a form that you can easily solve with calculus, then taking the thing you get out the other end of the calculus and simplifying it with the old algebra stuff.

So what does this mean? Basically, I’d argue that Algebra I is essential, in a nuts-and-bolts practical person-in-the-twenty-first-sense, to pretty much anyone’s basic education. Everyone should be able to solve the equation 2x+3=5 who would even begin to consider themselves educated. Algebra II and higher, though, again in that relentlessly practical sense, is less immediately useful to those without the need for the tools of mathematics in their career. Engineers, computer scientists, physicians, and the like need those skills, but lawyers, bakers, and political writers have a much less obvious direct need.

Part the Third: Morlocks and Eloi.

But there’s more to education than the relentlessly practical, and not for the fuzzy-headed “I’d like to buy the world a Coke” reasons you might expect me to give. Exposure to the techniques of higher math (even as much as we can say solving simple polynomials is “higher math”) is an essential part of having an understanding of the modern world and deeply enriches the inner life of those exposed to it, at least as much as understanding, say, the use of the Oxford comma or the causes of the Second Punic War.

This stuff Matters, in a real but less overtly practical sense, because by taking away the need for higher levels of numerical abstraction from certain social/educational/political classes, aren’t we taking away opportunity from those not exposed to these concepts? For that matter, since our ruling classes tend to come from the ranks of the law and humanities, we do ourselves a grave disservice if we aren’t requiring them to have at least some grounding in numerical reasoning, for reasons that should be obvious in our highly technical world.

Basically, saying to some students as young as thirteen or fourteen that of course they don’t “need to” know any of the techniques of numerical reasoning puts us on a very dangerous path. For instance, I’m an advocate of increasing the funding and prestige of trade schools that will help give persons without the scholastic abilities to go on to college a marketable and practical skill, but I worry that such schools will help create (or further!) a social, cultural, and wealth divide between the haves and the have-nots. The creation of another tier, with tradesmen on the bottom, techies in the middle, and cultural elites on top fills me with even greater dread. Who decides who goes where? Even in the absence of enforced roles, the encouragement of family and social class can be powerful, and I think we need less stratification in our society, not more.

Epilogue: Financial Instruments

This is an aside, but in your piece you mention the complex financial instruments vital to the collapse of Wall Street. I Am Not a Banker, but it’s my understanding that the mathematics involved isn’t particularly difficult (no more than compound interest), but that the complexity arose from the extensive use of jargon in the world of high-end finance. Charitably I’ll admit that every field has terminology that makes professional work simpler, but I suspect the degree of obfuscation in the financial world is often intended to swindle the mark. I’ll give them some leeway on “derivative,” which at least means something like what the naive reading of the word would entail, but parsing concepts like “credi default swap” without a finance-specific dictionary?

At least when STEM people derive terms like “polynomial” or “parthenogenesis” they use Greek and Latin roots to define their terms, deliberately setting them up as alien concepts that need specific definition. The reason I say that the financial world is deliberately obtuse is that their terminology “sounds” like it should be readily understandable, that their concepts should be fairly clear, but in reality they’re using common terms in foreign ways. Just a personal hobbyhorse.

(Wow. 1400+ words. Maybe I should go back to blogging about this stuff in my own space.)