There’s a pretty compelling argument in the *New York Times* this morning that requiring algebra (and higher math more broadly) of all students is unnecessary and even detrimental. The author reports that discouragement about math is the top academic contributor to drop-out rates, and I’m sure all of us have learned the dirty secret by now: virtually no one solves quadratic equations in the workplace.

I say this as someone who was always good at math — in fact, I often regret not continuing on with Calc II in college, and I’ll occasionally read up on math for fun. (A few years ago, for instance, I read David Berlinski’s *Tour of the Calculus*, which tied up a major loose end for me: *why* integrals and derivatives cancel each other out. I guess we would’ve gotten to that in Calc II.) To me, it seems to make more sense to make such topics available and let people with natural aptitude and inclination find their way to it, rather than drilling it into everyone.

What do *you* think, readers?

Sunday, July 29, 2012 at 10:23 am

So if a topic causes discouragement, we shouldn’t teach it. The argument isn’t actually compelling. If an educational system is incapable of teaching algebra to students without severe learning disability, there is something profoundly wrong with the system. Jettisoning the subjects it fails to teach will not solve its problems. The other issue is that you can’t fake math. It is likely that if algebra isn’t being taught effectively, not much else is either, the latter just being more difficult to assess.

Sunday, July 29, 2012 at 10:24 am

It seems obvious to me that the problem is poor pre-algebra education, and by extension terrible ideas that predominate in elementary education in general. By the time many kids get to the point in high school when they’d be taking algebra, I suspect their basic computational skills are so lacking that trying build on such a weak foundation all but guarantees failure.

Sunday, July 29, 2012 at 10:25 am

Do they need it? I don’t know that they “need” lots of things, but I’d think at least basic algebra would help. It doesn’t mean much to know the quadratic formula, which indeed few would use in real life, but certain basic “solving for x” skills can’t help but be good have.

Sunday, July 29, 2012 at 10:26 am

If you want to foreclose a huge number of career options to people before they even have an opportunity to understand what those career options entail or even develop the maturity to try to pursue them, stop teaching them math. You can do better than suggesting this is a compelling argument. It’s appalling.

Sunday, July 29, 2012 at 10:31 am

Just checking that you both read the actual article rather than just responding to my very partial summary.

Sunday, July 29, 2012 at 10:34 am

I don’t know – or more precisely don’t remember – what it’s like to not know algebra. I’ve never known what it’s like to be able to climb a rope. At school it always seemed reasonable to me that I should be excused from rope-climbing activities, on the basis that I just couldn’t do it, and that it would be a fair quid pro quo to let those students who found abstract symbolic manipulation similarly impossible go and do something with a better effort/success ratio for them.

Apparently such permissiveness was incompatible with the goal of forging “well-rounded” students. I have never been, and never will be, “well-rounded”, and I consider it the height of bullying obtuseness to expect it of others. Let us be what we are, to the best of our abilities.

Sunday, July 29, 2012 at 10:38 am

I read half the article, which was terrible. The default mode is jobs jobs jobs, etc.

Sunday, July 29, 2012 at 10:45 am

I feel like I’ve been trolled :)

Dominic, you can climb a rope. The problem is that in gym class, they don’t teach you to climb a rope. They put a rope in front of you every once in a while and tell you to climb it, the inability to do so on the spot resulting in profound shame. This is probably comparable to a lot of mathematics education in the country, with similar results. I was fortunately spared rope-climbing, but I wouldn’t have been able to do it. I didn’t successfully do a pull up until I was in college and joined the crew team… receiving an actual physical education. (Disclaimer: I’m back to being a fatass now.)

Letting us “be what we are” bespeaks a kind of essentialism that would leave behind all sorts of people that would otherwise be able to become something they aren’t… a far more exciting prospect. It’s like allowing children to eat Kraft macaroni and cheese because that’s easiest on the adults caring for them.

Sunday, July 29, 2012 at 10:48 am

No one really uses arithmetics anymore either, we have calculators and cash registers for that. Why not abandon math altogether? It would be so much easier to cheat the workers out of their pay!

Sunday, July 29, 2012 at 10:48 am

I think the article makes more sense if your high school, like mine, had Pre-Algebra, Algebra I, and Algebra II. I think the argument is really directed towards making the advanced versions of algebra instruction optional; I don’t think there’s any indication that it’s an argument for cutting out basic “solve for x” instruction.

If we replaced Algebra-II-style instruction with the math-for-capitalist-living course he spitballs on page 3, a lot of kids would probably be better off. You shouldn’t be able to get out of high school without understanding how insidious compound interest is, for instance, or without a basic understanding of how statistics inform and misinform us.

Sunday, July 29, 2012 at 10:52 am

If students struggle with math in the manner he describes, good luck teaching them statistics. The bottom line is that teaching math can’t be faked, because assessing comprehension is a pretty straight forward process, and so in some ways, math performance is the most accurate measure of our educational system.

Sunday, July 29, 2012 at 10:55 am

You can teach people to be literature in basic statistics without getting deep into the weeds of formulae…

Sunday, July 29, 2012 at 10:57 am

Take two: You can teach people to be literate in basic statistics without getting deep into the weeds of formulae…

Sunday, July 29, 2012 at 11:00 am

But I do agree with gerry in the sense that teaching, e.g. matrix arithmetic (probably the most painful part of “Algebra II,” which is a weird mish-mash of topics) to high-schoolers is likely a lost cause. I actually have reasons to use matrices, and I definitely didn’t benefit from anything I saw in high school. This isn’t because students can’t learn matrices, or that they aren’t useful, but because it is taught as a kind of digression and likely with even less enthusiasm than one typically encounters in a math class.

Larger point with regard to the utility of math: I have difficulty with the procedure for integrating functions. I just forget the details. Regardless, think about functions and their integrals on a near daily basis, both in and out of my profession–likewise with derivatives. I simply can’t imagine thinking without the conceptual tools of algebra and calculus, even though when I need to integrate or differentiate something, which is regularly, I do it numerically using a computer, often out of necessity because they are data sets and not symbolic functions. I still need to understand what those processes *mean*. I would certainly be in favor of emphasizing these aspects in teaching math.

Sunday, July 29, 2012 at 11:20 am

This might brake some of your commenting rules, but I think your argument against maths from some time back was much better… (as another who was good at maths and never had any use for it)

Sunday, July 29, 2012 at 11:21 am

The author of the NYT article dismisses the claim that mathematics hones critical and abstract thinking abilities in a few sentences. I find that to be a glaring oversight. Moreover, Hill makes a crucial point when he or she notes that not taking mathematics closes many doors later in life. Perhaps we need to adjust mathematics curricula, but I suspect the problem lies not in math education per se, but the whole of K-12 education and the role of education in our culture. I am an advocate of a tracked education system, which requires a far larger change than just math education.

Sunday, July 29, 2012 at 11:28 am

I’d be interested to hear how he squares his support for just requiring arithmetic and “citizen statistics” with his claim that he doesn’t “advocate vocational tracks for students considered, almost always unfairly, as less studious”. A second-tier track would be the main effect of his proposal. (Not even a properly “vocational” track, mind—just a second-class track.) Basic financial literacy, especially about interest rates, an elementary grasp of statistics—all good skills everyone should be taught. But not as a replacement for algebra, I think.

His line about how “it’s not easy to see why potential poets and philosophers face a lofty mathematics bar” is charming, given the content and methods of mainstream English-speaking philosophy.

The article irritated me mostly because it read as another in a long line of People Who Feel Empowered To Say What’s Wrong With Education. The main qualification for such people is to not have any specialist knowledge about or substantive experience with teaching children in schools. Opinionated billionaire? Let’s hear your views! Computer programmer who lucked out? Tell us what to do! Elderly political theorist who hung out in some classrooms? I want to learn more! Such people aren’t automatically wrong about everything, but it’s a consistently crappy genre.

Sunday, July 29, 2012 at 11:43 am

“Dominic, you can climb a rope”

At 8 I lacked the necessary gross motor co-ordination to perform a forward roll. I can do one now; the fog cleared somewhat as I passed through my teens. I imagine that some of those who found themselves incapable of algebra at school might have a better chance if they came back to it at some point in adulthood, having acquired over time the ability to settle down and concentrate in the right sort of way.

School co-opts your time, and enacts decisions on your behalf about how you should use it and to what ends. Some of these decisions are reasonable, and some are not. Adults in authority do not take kindly to being told by school children that their decisions are unreasonable. Nevertheless, I am entirely on the side of any child who wishes to insist that there is in fact – for them, there and then – some such word as “can’t”.

Vignette from a failing marriage: we are out with my daughter on a bicycle, which I am supposed to be teaching her how to ride. It’s evident that she’s just not ready; we’re getting nowhere; I get no end of stick for being reluctant to “encourage” – that is, bully – her into trying and failing again and again. A year later she is riding a bicycle with reckless confidence, having mastered it quite abruptly. No amount of extra cajoling from me on the earlier occasion would have helped her obtain that mastery a moment sooner.

I don’t believe that “good teaching” has in all cases the power to overcome incapacity. Teachers are force-multipliers for the student’s willingness and readiness to learn. Good teachers are skilled opportunists, on the lookout for signs of readiness, hooks to hang things from. They have to improvise with, and around, the complex and uneven process of the student’s own development. At times they are hardly needed at all; at other times, there is nothing they or anybody else can do to help.

Sunday, July 29, 2012 at 12:16 pm

You couldn’t climb a rope at 8 for obvious developmental reasons, the same reasons I couldn’t do pull ups. My body was pretty wildly mis-proportioned until high school, at which point I didn’t have to take PE anymore. The same factors are present in teaching things like math and language, except the clock is ticking backwards. If you miss the window to teach these things (a window that is probably more different than our “grade” system captures), the ability to learn them at a practical pace may be gone forever.

I simply disagree that there is a widespread incapacity for understanding algebra present in the US populace, which your arguments seem to presume. Many people I know began to learn it at the age of 10 or 11, and they certainly aren’t prodigies. They just had teachers that didn’t assume they were stupid. If algebra is difficult to learn for current students, it’s because the math education they have received up to that point has been poor, and certainly there isn’t a widespread incapacity for learning basic arithmetic. History and the performance of the rest of the developed world seems to bear this out. Math performance is especially bad now, and that’s a fairly recent thing. Your general comments about teaching are of course true, generally, but they don’t account for how poorly the currently generation of young people perform at math relative to their forebears or other young people in the world with comparable socioeconomic environments. We aren’t talking about particle physics here. We’re talking about something that many people learn, and in other countries, this is the norm, before they are even teenagers.

Sunday, July 29, 2012 at 12:22 pm

I believe this is the post Patrik is referring to.

Sunday, July 29, 2012 at 2:39 pm

This may be a tangential point, but still: if a common argument (strong or not) is that learning lots of math strengthens one’s critical reasoning skills, then why not cut out the middle man and have required courses in critical reasoning? Constructing arguments, recognizing fallacies, basic propositional logic. This is stuff that could probably be taught in junior high or earlier, and if taught in the right way it would be both immensely “relevant” or “applicable.” It would probably even help many students with their actual math classes.

Sunday, July 29, 2012 at 3:19 pm

Michael, the sorts of things you mention ought to be a part of all of the humanities courses taught, indeed possibly every course taught, full stop. The thing about math is that critical thinking is a sine qua non of the subject. This goes back to my argument about not being able to fake math. I found the construction of proofs in my tenth grade geometry class to be one of the most useful aspects of my pre-college education. Of course, “constructing proofs” is what one should be doing, in principle, in all of the essays required for the other classes I took in high school, but I failed to appreciate it at the time.

Sunday, July 29, 2012 at 6:50 pm

“I often regret not continuing on with Calc II in college”

I am glad I’m not alone in this. I actually tried taking it my senior year, but it turns out doing no math for four years makes the homework really hard (and not in any interesting way — I’d just forgotten the practical skills that I only used in solving math problems). I did get to the fundamental theorem of calculus before dropping the class, though, so I can confirm you would’ve gotten it in Calc II.

Without reading the article or any of the above comments, I dogmatically pronounce that the solution is to stop letting people drop out of school. And to require everyone to finish calculus.

Wittgenstein reportedly was teaching calculus to his elementary schoolers when he quit, which makes me think that the solution may be to hire teachers who will beat their students, then lie about it and want to kill themselves.

Sunday, July 29, 2012 at 7:37 pm

I managed to finish Calc III in college, and it was definitely a formative and indispensable experience. It was the worst grade I received in college, and I hung it up afterwards, but expanding calculus to many dimensions was extremely powerful even in purely conceptual terms. Change of variables and the Jacobian absolutely wrecked me on the exam. That was the one homework assignment they didn’t grade (i.e I didn’t do it).

Sunday, July 29, 2012 at 8:23 pm

I think it’s precisely because logic and proper argumentation are part of every subject that they deserve a class to themselves. You can do half-assed argument analysis in an English class along with sentence diagrams or spelling or whatever, or you could instead do it properly thoroughly on its own.

I suppose many people may have received that kind of instruction in normal classes – what a syllogism is, why ad hominems are bad, etc – but I’m pretty sure I didn’t. Not to slight my teachers, who were by and large good and occasionally wonderful… especially in math and science, at which I was pretty good as long as I didn’t have to build anything. (and I’ll admit that I burned out on math after passing the Calc AP test)

Monday, July 30, 2012 at 4:44 am

@Hill “I simply can’t imagine thinking without the conceptual tools of algebra and calculus, even though when I need to integrate or differentiate something, which is regularly, I do it numerically using a computer”

Yes, this is what I’d most like to see a change in. Many students end up getting the impression that what’s important about calculus or differential equations is a whole pile of techniques to compute integrals and solutions: integration by parts, the chain rule, separation of variables, etc. But actually working symbolic integration isn’t anywhere near the most important part of calculus, and isn’t something I’d teach (if at all) until after someone understands what it’s useful for as a conceptual tool. If someone is fluent in calculus but has to use Maple or Mathematica to actually compute integrals, that’d be a success.

Monday, July 30, 2012 at 11:00 am

I would second that Mark. Having both a degree in mathematics and having spent 4 years teaching students Mathematica, I believe that the concepts and purposes are important for most even if relatively few can do the calculations themselves. Only STEM majors need be able to do the computations themselves. That said, most people here are focusing too much on calculus and not, for instance, geometry, discrete mathematics/number theory, linear algebra, etc. Calculus is more used in engineering, whereas discrete mathematics is used more in computing, etc. So, to get back to the article, we might note that different branches of mathematics are more commonly used in different fields. Moreover, I have noted that students who struggle at one kind of mathematics often do much better with another. Consider: calculus vs. algebra vs. geometry vs. linear algebra vs. abstract algebra vs. statistics, etc.

Monday, July 30, 2012 at 1:40 pm

There seems to be a nearly overwhelming consensus that the article is in fact stupid. Who am I to deny it? I repent in dust and ashes.

Monday, July 30, 2012 at 4:02 pm

Adam, I think the article has a good point. Yet, changing math curricula without other substantial changes will be detrimental. So, less math, but more logic and critical thinking, and I would be entirely ok with it. Especially if we acknowledge the need for vocational tracking.

Monday, July 30, 2012 at 5:24 pm

Side point: The article asks how many people “remember what Fermat’s dilemma was all about” from their mathematics courses. Google turns up that phrase in only: 1) this article; and 2) people puzzled about this article.

Does anyone have an idea what the author might’ve meant? The only well-known “dilemma” I can think of Fermat having encountered was having a proof but lacking sufficient space in the margin to write it. But that’s more math trivia (or perhaps a joke) than something that’s actually taught in mathematics courses; and I’m not even sure it qualifies as a proper dilemma.

Monday, July 30, 2012 at 11:35 pm

So I live in Korea, teaching middle school students (aged 13-15). They apparently study the basics of set theory. The smartest kid in my class, at least, knew what ordinal sets were. He was actually using his spare time to read a book about Cantor.

Education is a whole different game over here. There is a huge industry built around teaching math to students.

So it’s possible.