Xs of evil: America’s algebra crisis

Colby Cosh on why math stirs passions, raises fears, fuels debate

by Colby Cosh

The New York Times ran a deeply contrarian editorial Saturday about math education in the United States. In it, political scientist Andrew Hacker argues that the youth of America is being crucified on a cross of higher math.

A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

My question extends beyond algebra and applies more broadly to the usual mathematics sequence, from geometry through calculus. …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. (I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame.)

Hacker argues that the math used in the typical American workplace—even a technical, highly quantitative workplace—does not much resemble math as it is taught in the American classroom. Engineers, doctors, and bankers rarely use algebra as such. What we probably should be doing, Hacker thinks, is to foster mathematical intuition amongst students who can’t master higher levels of abstraction.

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.

It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted—and include discussion about which items should be included and what weights they should be given.

I’ve had some Canadians point out Hacker’s editorial to me in the spirit of “This guy’s crazy, right?” But his remarks have to be understood in an American context. There has been a strong push in the U.S. for pretty high universal national standards in mathematics, standards which have seen many states run afoul of what’s called “Algebra II” in curriculum circles.

American high school graduates are taking harder math and science classes, according to a recent report by the National Center for Education Statistics.

In September, Tom Luce, former CEO of the National Math and Science Initiative, said that the United States needs a “STEM-literate population” that starts by “convinc[ing] the entire country that every child must conquer Algebra II.” America has made steady progress toward that goal—in 1982, just 40 percent of high school graduates took Algebra II; in 2009, more than 75 percent did.

Keeping those figures in mind, here’s a short sample from an Algebra II exam: I note with some alarm that it features a question about complex numbers, which I don’t think I ever learned in the classroom despite having fought through Alberta’s Math 31 high-school course and a year of university calculus and statistics. This is pretty esoteric stuff to be expecting “every child” in a large, diverse country to conquer. (I think only physicists would ever actually use complex numbers at work, though I know electrical engineers are expected to master them as part of their theoretical education.) Insofar as Hacker is just pointing that out, his op-ed falls into the category of “man identifies patent, unaddressed insanity swirling around him” rather than “man quarrels with high educational expectations.”

Math has a special, awkward place in education. It is no wonder that it stirs passions and raises fears, for it is pure concentrated abstraction, and everybody senses on some level that how far you can go in math (speaking as someone who got the equivalent of a B in first-year calc) is a very precise, cruel measure of one’s cognitive separation from the cleverer beasts. America is pushing Algebra II because, of all high-school courses, it is “the leading predictor of college and work success”. But you don’t need Riemann curvature tensors to understand the logical flaw in the proposition “If Algebra II predicts success, making everyone pass Algebra II will make everyone successful.” Understanding the square root of minus one is no use to most of us, in itself; yet it is true that those who can be taught to understand it will, over time and as a group, earn and accomplish much more than those who don’t. This is true of any reasonably abstract concept, which is why there is always confusion over the actual value of learning to read a musical score or figure out a left fielder’s on-base percentage.

Andrew Hacker’s “algebra problem” is an interesting symbol of how rampant egalitarianism is in the American academy. Primordial America possessed an intellectual counterweight to the Jeffersonian faith in education; the minor Founding Father Fisher Ames is said to have responded to the notion that “All men are created equal” with the retort “…but differ greatly in the sequel”. Today’s American right, however, takes the tactical ground that no child must be “left behind”—that all can be educated for a STEM future, just as any goose can make foie gras in his liver if he is stuffed full enough. This happened because American public education became compromised by the teacher trust and its slovenly “easier-for-us” ideology: it became too tempting to whack the education industry over the head with standardized testing and with the excellent results of other countries’ education systems, as President Bush 2.0 did. It is probably not really realistic to expect the U.S. to compete in mass mathematics education with a small, homogenous Nordic country like Finland—but what American will admit to that? We built the Bomb and went to the Moon, man! (One supposes it would be unkind to note that the “we” in that sentence denotes, respectively, “a bunch of Europeans who fled the Nazis” and “a bunch of Nazis who fled the Russians”.)

Even Hacker, whose essential complaint seems to be that 25% would be a much better estimate of the number of American children capable of mastering Algebra II than 100%, won’t put it in such a direct, offensive way. But it doesn’t help that he conflates “algebra as taught in American schools” with algebra as such. Algebra is, above all, a single step up in abstraction. It’s a step that most children can take: once you give them the core idea of performing operations on “x” rather than on a particular number, the royal road is open.

And that much—the whole notion of a variable—is something ordinary people do require, if they hope to have the instincts, training, and mental safeguards Hacker agrees that they need. How the hell are you going to teach even the crudest statistics or concepts of probability to a student who doesn’t get what a variable is? How could an ordinary Cartesian graph be comprehensible? How, indeed, would one teach the difference between arithmetic and geometric growth—and hence the power of compound interest?

I suppose you do it, though Hacker specifically denies it, by sneaking the abstraction in through the back door: you give eleventy real-life examples of compound interest at work, until the student eventually realizes that the principle’s the same no matter what the actual principal and the particular interest rate are. He learns to identify a “variable” without being intimidated up front by xs and ys. There might be some merit in such an approach, for it is in fact the symbology that seems to frighten the math-averse. In this article about the rigours of Algebra II, a person claiming to be an accountant says “Most people I know who are lower income couldn’t solve 2x = 14 if their life depended on it.” And maybe he’s right. Yet there can’t literally be many people who couldn’t stumble into an answer to “Two times what is fourteen?” if it were presented that way and they were given a few minutes to wrestle with it.

Xs of evil: America’s algebra crisis

  1. There’s probably something to the argument that high school math is disproportionately advanced as compared to other subjects. That said, I wonder if there isn’t something of a cliquish culture among mathematicians. I know a few, and while they’re nice folks, they don’t seem all that interested in explaining their particular research or consultancy work to the uninitiated. Vonnegut’s remark about scientists who cannot explain their work to eight year olds comes to mind.

    Also, the way in which we teach math could easily be revamped. As it stands, teachers (who, let’s be honest, are rarely as comfortable with the material as their peers in other subjects) tend to toss the material out at a level that the math-gifted kids can easily digest, and seem to hope the rest of the class somehow follows along enough to pass – which ain’t really “teaching”. The texts are dense, and do less explanation than illustration, followed by rote practice (again, this works for the math-gifted quite well, but poorly for the rest of the class).

    As Cosh rightly points out, abstract thinking matters. And Hacker runs dangerously close to judging math by its ultimate pragmatic use (outside of French, writing, and typing classes, I’m hard pressed to say how most of my high school education proved directly useful). I’d argue that a rethought curriculum, more gifted teachers in the subject (and perhaps smaller class sizes), and maybe critically considering the entire culture of academic math, would make it less of a “huge boulder” as Hacker characterizes it, and more something every kid has a right to experience in a positive way.

  2. Advanced math was created by nerds, is only understood by nerds, and so those of us who aren’t nerds don’t do well.

    I graduated from university – with a degree in Arts obviously – but I would have been a high school dropout if forced to study algebra all the way through school. Math was my best subject in school until Grade 9 when we had to take courses in algebra or calc or trig and then it went all wrong for me. I remember arguing with my calc teacher about how you can’t have numbers less than zero, two negs make positive is nonsense – I thought mathematicians were pulling random numbers and symbols out of their arses.

    Grade 9 calc teacher kicked me out of class after two weeks – refused to teach me – and I was sent to general math class. My first class in general math I will always remember – we spent entire 80 min class studying probability by flipping one coin and there were students who still didn’t understand 50/50 after more than an hour of instruction and it wasn’t a special needs class.

    I went from one extreme to another when I just needed someone to explain math and science concepts in a different way. Luckily for me, my paternal grandfather saw that public school system was turning me into a simpleton and started teaching me maths and sciences in a way that made sense.

    • Tony, I do enjoy it when you put your own thoughts down on paper rather than throwing quotes out. Thanks for sharing.

  3. I believe our public school system is not nearly demanding enough when children are young – kids understand advanced concepts if explained to them in sensible way. I have niece and nephew both under 10 and I am helping my sister educate them outside school system because of our experiences in public school.

    We are teaching them math and science using real world examples instead of learning abstract equations in classroom. We had my niece addicted to bill nye science guy videos when she was 4 – I was so proud of her when she explained Newton’s 3 laws to me. About a year ago I came across Khan Academy videos online and it is wonderful site for teaching young people advanced concepts in a way they can understand.

    If I could design public school system I would make kids start school at age of 4 and there would be heavy focus on three R’s – it is appalling how many intelligent Canadians are functionally innumerate and illiterate – and this would last 5 or 6 years. When kids graduate from elementary school parents would be allowed to choose from a wide variety of schools that would specialize in maths, sciences, music, athletics, fine arts … etc. School system should be varied as possible to encourage kids to continue studying but we force everyone into one way of learning which benefits teachers/bureaucrats but no one else.

    • In Calgary where I live there are “science schools” and schools which cater to kids who are in high-level athletes and need a very flexible study schedule. There are also various “immersion” options such as Arabic, Spanish, Mandarin, German and of course, French as well as a Franchophone option. Parents of children in Alberta also have the option to use the money alloted to the education of their kids by the government to send them to ANY educational option they want whether it be private school or home-schooling should it better fit their criteria. Therefore, there is no “one way of learning” imposed on anyone.

      • I should have been more precise – I was ranting about Ontario, not Canada. I wasn’t aware Alberta top performer in education until a couple years ago when I heard UK MP Gove talking about what he wanted to do in Britain. You would think other Canadian provs would try some reforms that Alberta has done but they haven’t. Alberta is success story while many other provs are woeful.

        The Guardian ~ January 2011:
        Britain’s recent further slide down the international education league tables of the Organisation for Economic Co-operation and Development has provided Michael Gove with an opportunity for political capital. This follows on from comments he made almost as soon as taking office, in which he highlighted the achievements of Alberta, Canada, which regularly scores more highly than any other English-speaking region in the Programme for International Student Assessment (Pisa) rankings.

        • Thank you for the kudos to our oft maligned province, Tony. Some people actually believe we are stuck in the middle ages.

          • No….just the pre-industrial era. Before the industrial revolution of 1750. LOL

  4. Electrical engineers and others who do digital signal processing have to deal with imaginary numbers on a regular basis. It’s not just theoretical.

    • I seem to recall having to use imaginary numbers for plain old circuit analysis, never mind digital signal processing.

  5. Our educational system was set up for the industrial age.

    Standard product, efficiency, time clocks and bells ringing……schools became big, stern places with almost a prison atmosphere where people were ‘punished’. Students were never told where algebra came from, what it was used for….why it might be important….it was just one more boring punishment they were subjected to for the crime of being born. And we had to know math for the new age of machines.

    But there is no reason for everyone to learn algebra or any other math at the current high school level anymore. Tell students who invented it and why, what we’ve done with it, and give them the general idea of how it works. Then move on. Lots of people find it fascinating, and will follow it up as an interest.

    Math isn’t the only subject school has managed to ruin. Another one is Shakespeare. Anyone forced into learning endless plays in school is unlikely to touch it ever again. Teach one comedy and one tragedy…..tell them this was hot stuff 4 centuries ago…the Steven Spielberg of his time, etc …..and move on. Anyone intrigued by Shakespeare, or literature in general, has thousands of libraries to choose from.

    The difference is that once you left school in the old days….that’s all the education you got for life. If you couldn’t get much education when you were young……you were sunk ….permanently. School was solely for kids. We still have fall-out from that attitude, especially with boys….’school isn’t cool’, ‘school is for fools’….and they drop out ….to get a nice manly job in a factory, only to find the jobs aren’t there anymore….so they’re hanging out on streetcorners.

    It was only in later years that adults could go back to high school to pick up something they’d missed But it had to be in class with teenagers, which discouraged most people. Then we got night school.

    Anyway…..they taught you everything in school you MIGHT need to know for a lifetime because of this. The method is known as JUST IN CASE.

    Now….people can get an entire education….from grade 1 to a doctorate….online. At any time in their lives. And they can mix and match subjects to suit themselves. It’s DIY education, and has become extremely popular. And since you can pop in and pop out at various times you can choose to take a subject only when you need it. It’s called JUST IN TIME.

    It’s genuine Lifelong Learning. And you learn the latest…you don’t have half-remembered obsolete knowledge from grade 5 or grade 10.

    So the entire educational system is changing….and tossing algebra as a general subject is just one of those changes.

    • Lifelong learning is great. But those who’ve done more learning at an earlier age are likely to be better at it as their life goes on, so early training in things like algebra, other languages, the sciences, and making a historical/philosophical/literary argument is likely to pay off in the long run, and to pay off even more in the future than it does in the present.
      An online course is, in most respects, something like a high-tech version of a textbook — perhaps slightly more interactive, but still focused much more on content delivery than on analytic skills. Now anyone can take, say, an intro to microeconomics course online, and yes, that probably offers greater value than simply reading a micro textbook would. To the extent that learning is about amassing data, and about learning how to perform simple and repetitive processes, online courses are probably going to revolutionize things, but to the extent that learning is about mastering a particular set of critical and analytical approaches, facetime with an instructor and with fellow students will still pay off.

      • But humans learn….naturally….from the day they are born. School does it’s best to snuff out that urge to learn. LOL

        Online courses can involve discussing things in a classroom if you wish…but many students regard this as a waste of time and simply go with the video and interactive answers.

        • Stop saying “LOL” you twat!!! Arghh so irritating. You could use some upgrading yourself.

          • LOL indicates I’m laughing or joking…..some people can’t tell.

            Now find yourself some software on manners, because it’s obvious you have none of your own.

  6. Excellent article Mr Cosh. I’ll probably need to re-read it 4 or 5 times to really understand it, but really thought-provoking as I just finished nursing my 13-year-old through Algebra 1.

    • Have you heard of Khan Academy? Free, low tech videos that explain math and science concepts. For past year, my sister has my niece/nephew watch videos and practice basic equations. They are not learning this kind of math and science in school yet but they enjoy doing them on their own time – videos are great because kids can work at their own pace without peer pressure.

      http://www.khanacademy.org/

  7. PS: “Xs of evil”…..brilliant!

  8. Before I returned as an adult student to university I had to upgrade my mathematics. I took the course from the head of the math department at one of the universities. She told me that most highschool math teachers don’t know enough about math to teach it. She believed they should have a degree in math and then an education degree. I tend to agree. A math teacher should be able to teach a student many ways to tackle and solve a math problem. Failure causes a student to lose all of their confidence. We know that people learn differently…some orally, some through pictures….a teacher needs creative skills and needs to know their subject inside out. More than anything, a teacher needs to give a student the gift of confidence….that they are smart enough to grasp the concepts if the concepts are presented in a manner that they can understand. Tony talked about Khan Academy. Mr. Khan and Bill Gates have opened up a school and are bringing his special teaching talents to thousands and thousands of students kindergarten to advanced degree level. If we have good teachers there is no limit to what students can achieve, even in Algebra.

  9. Mr. Hacker appears to have a view of higher mathematics that can make learning it very frustrating — i.e. that higher mathematics is about pushing around symbols. If this is his view of mathematics then I can understand why it seems like a useless and meaningless activity to him. What if he had this same view of reading and writing? Would he argue that we should all just send smily faces and sad faces back and forth since words, sentences and paragraphs are too hard to deal with? Reee-dik-you-luss. Mathematics should be taught much more like reading and writing — as a way of expressing quantitative and logical ideas (many of them common sense, even to people who call themselves non-math types) using symbology in a particular way. I like his idea of working with more exciting examples like the consumer price index. I’ve seen too many current math examples that are uninspiring and contrived. If Mr. Hacker had been taught mathematics using a deeper and more holistic approach by teachers who had that too, then he might have found himself happily differentiating functions to get rates of change instead of complaining. Hacker’s frustration is his motivation for putting forward a position of quashing higher math for many others. He should seek a deeper understanding of mathematics to deal with his frustration, not try to stomp mathematics out by denying it to future generations. In a broader sense we need to deal with this frustrating view of mathematics by developing more successful ways of teaching mathematics to young people. This will not be an easy task since it involves communication between math-types and education-types (yes they do appear to be in separate camps). Yoiks!

  10. Good column, especially the last two paragraphs. If a sizable chunk of students are failing to grasp that the abstract subject of mathematics is in fact a part of everyday, concrete life, then it’s not being taught correctly, period. This is a fixable problem, but only if their teachers have a broader understanding of mathematics than most high school math teachers do.

    Much has been made in the original article about the fact that algebra filters out smart people, but as a post-secondary math/stats instructor at a (Canadian) polytechnic, I’ve found the converse to be true as well: that is, success in high school doesn’t even reliably indicate an ability to think quantitatively. Plenty of students who are more than proficient at pushing symbols around a page enter my classroom without a real understanding of what any of it actually means. They CAN solve a quadratic equation for x, but are lost when called on to model even a simple real-life situation that they haven’t encountered before word for word. If I had a dollar for every student who got an A in high school math who expressed shock when I explained to them, one-on-one during office hours, that the equals sign in an equation means “has the same value as”, I’d be that much closer to retirement. Far from being prepared to use math in real life, many top students seem to have learned in high school that if they shut off the non-math (ie, the “real life”) part of their brains in math class, they can still get A’s. I see this each term in my intro stats class, where in the middle of a lesson on discrete probability distributions, I ask: “Suppose I roll a die 60 times. How many 3′s do you expect me to get?” I tend to wait a minute or so while many students stare bewildered at me, and the bolder ones get out a pencil and paper and start plugging piles of numbers into equations. At this point I say, “This isn’t a difficult question. Pretend you don’t know anything about probability: don’t think too hard about this.” Freed from the notion that this is an abstract mathematical problem, they confront the question as-is, and most come up with the correct answer at once.

    Meanwhile, when a baker acquaintance told me, as so many people do, that she had never been any good at math, she was quick to assure me that OF COURSE she’d know how to adjust ingredients to bake 50 cookies if the recipe she had on hand only yielded 36.

  11. It’s interesting that a political scientist chose to take aim at algebra by noting that it’s rarely used in its pure form by professionals. By that logic, what justifies the study of much of English literature, history, or the physical sciences? Beowulf doesn’t usually figure into the day-to-day work of journalists or lawyers; doctors rarely have cause to reflect on the unification of Italy; accountants don’t often reflect on the Michelson-Morley experiment. For that matter, I don’t think political science plays much role in the day to day work of politicians.

    • I don’t see the contradiction you point out. High schools typically teach civics (aka. an applied dumbed down abstraction of political science relevant for citizens), not political science. You really only see a full complement of political science classes at the university level (and with good reason – understanding political science probably makes people worse citizens). Alternately, journalists and lawyers do use critical reading skills, even if they don’t gain from the specific knowledge of Beowulf (and I think a strong case could be made for teaching high school students books other than Beowulf).
      The purpose of high school is to give students the breadth they need to be productive/engaged citizens, and where possible, to identify strengths and weaknesses which may inform the next step in their development. Because there are many things people would gain from learning, and limited time, it makes sense to focus on those items which accomplish the modest objectives listed above.

      • All that time spent on critical reading skills, and you still answer a rhetorical question.
        Okay – my point is that those basic algebra skills are directly analogous to the benefit obtained from taking English, history, and the physical sciences in high school. Innumeracy is as serious a shortcoming as illiteracy in today’s world. How can one critically evaluate a news article citing statistics or financial trends without a functional understanding of algebra?

        • I’m a high school math teacher in Ontario. After reading Hacker’s article I looked at the grade 10 curriculums in math, English and history. I make frequent use outside my classroom of most of the curriculum expectations in English and history. I never use the majority of math curriculum expectations, other than to teach them to my students.
          A lot of responses to Hacker’s article conflate the definition of algebra –the use of variables to represent unknown quantities– with algebra as it’s taught in high school, which includes the quadratic formula, completing the square and factoring polynomials. Harder isn’t complaining about an expectation that all students know how to use variables. He’s complaining that all students are expected to be able to carry out more advanced, and far less useful, processes.

  12. This is the most American idea i’ve ever heard of. ” Seeing as nobody can teally grasp the higher forms of math, we have decided to just scrap it all together.” This has to be a hoax, for real. Who really actually uses algebra in the same form they learned it in? If you thought Americans were stupid enough before your in for a treat. Makes you feel hopefull knowing that there are so many idiots out there who cant understand highschool level education doesnt it. No wonder china is cleaning up at the games.

    • Try to not mess up basic grammar when you’re insulting someone’s intelligence. Also, I’m pretty sure that the algebra skills of Chinese athletes have little to do with their performance at the Olympics.

  13. I’ve personally benefited from being forced to take math classes I would have otherwise run away from. Later in life, I realized that I needed a stronger understanding of mathematics to answer the questions that interested me, and thankfully I had a base of knowledge to draw on. However, I was lucky – I had excellent high school teachers that either had mathematics degrees themselves, or higher degrees in the physical sciences. Despite lacking any native ability, I learned.
    That is often not the case, which is a serious problem. Teachers that don’t truly understand the material can copy what is in their lesson plan and grade assignments, but that can’t provide students with the intuition behind what they are teaching.
    Worse, there is a bizarre tendency to concentrate most of the math students learn late in their development. My memory is a bit rusty, but I recall learning addition and subtraction in grade 1, multiplication in grade 2, division in grade 3, long division/fractions/decimals in grade 4-5, some basic algebra in grade 6-8. Then all of a sudden we jumped into fuctions, geometry, etc. I suspect that those initial 8 years could have been used far more productively. My parents claim to have learned multiplication in kindergarten, and I’ve seen an old grade 8 math exam that I would have undoubtedly failed were I that age. Given that we teach math so late in a student’s career, we shouldnt be surprised that math is a second language to many.

  14. I’m an engineer and I use algebra every day.

  15. Without reading the NYT article, I agree with his assertion that we should try to put more practical math (like statistics, probabilities and logic) in our curriculum. I disagree that this is to make it easier. Math is a great subject because it forces students to think logically, creatively and to switch to a different perspective. Liberal arts are always talking about how they teach these, especially the last two. The difference with math is that it offers strong evidence at the end of whether your thinking was valid or not. In History and English, the ultimate arbiter is the teacher’s opinion.

    (BTW, the new western Canadian curriculum does push the majority of students in this direction. If “Foundations” is difficult, trades math teaches Cartesian graphs, statistics and compound interest with a minimum of variables. The process is slower. Higher algebra is reserved for ‘Pre-calculus’, intended only for future math and science majors.)

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