Essay — From the September 2013 issue

Wrong Answer

The case against Algebra II

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In 1545, Girolamo Cardano, a doctor, a wearer of magical amulets, and a compulsive gambler, published a math book in Latin called Ars Magna. The “great art” of the title was algebra. When Cardano was done, he knew he had come up with something huge and powerful and timeless; on the last page was the declaration, written in five years, may it last as many thousands. The equations in Ars Magna looked very different from the ones we are familiar with — here, for instance, is how Cardano wrote the solution to x3 + 6x = 20:

Rv : cu. : R108 p : 10m : Rv :cu. R108m : 10

But the algebraic rules Cardano described and codified are variants of the techniques that millions of students are taught, with varying degrees of success, today.

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is the author of fourteen books. His new novel, Traveling Sprinkler, will be published this month by Blue Rider Press.

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  • Susan Ohanian

    As a longtime teacher, Nicholson Baker’s elegant, witty, profound, and wonderfully-researched article has raised my spirits immensely. His take-down of Algebra/Duncan/Gates/Common Core/Achieve gives me hope, and godonly knows, those of us who care about the kids in public schools are desperately in need of hope.

    I just sent this article to every member of the Vermont Board of Education. And I didn’t even brag to them that Baker uses my coined word Standardista twice. That’s just icing on the cake for me. I urge everybody else to send this article to their boards of education. It puts algebra in its place. About time!

    Thank you, Nicholson Baker! Thank you, Harper’s!

    • Patricia West

      I’m not mathematician, but even I can spot the error on the accompanying cover illustration. A through D are supposedly foils (or alternative answers) to a math problem written higher up on the board. However, they are identical. (At least those not obscured by the magazine’s name.) Nevertheless, I agree with the author’s point that it’s better to let students skip Algebra II than drop out of school.

      • A Mathematician

        The answers aren’t identical. The signs are different. Answer C has “-108y^3″ but answer D has “+108y^3.”

    • sgustard

      Actually Nicholson Baker is not a longtime teacher, but maybe that’s not what you meant to say. Also dropped from our curriculum is “grammar.” I look forward to school being a feel-good, non-challenging, self-esteem-boosting experience for all our future cashiers.

      • vrz

        She did not say Nicholson Baker was a longtime teacher, but that she was, and she appreciated his article. No need to make “grammar” insults if you are not reading carefully.

  • katewillette

    I spent 12 years teaching geometry, alg II, and pre-calculus in a tiny private school in Washington state. My students (now in their 30s) will all tell you that they liked my classes ONLY because I arrived each day believing with all my heart that they shouldn’t have to be there. They should have been given a choice about those classes. We were going to avoid taking those hideously written textbooks seriously at all costs, and nobody was ever going to weep or sweat over their homework if I could help it. The ones who loved math got to do it happily, and the ones who needed the credit to make it into college were not tortured for being in a stupid system.

    This article captures my sentiments perfectly. I would only add that my standard response to those who insist — a la Arne Duncan — that algebra II teaches logical thinking goes like this: SO DOES EVERY COMPUTER LANGUAGE, only you can actually make things with C# and java.

  • Gordo Flimnap

    Thank you Mr. Baker. The most advanced math I have used to date (I’m 63) is the area formula for circles, which I use to compare pizza prices.

  • RoyceGW

    Yes, well, everyone should have enough math to understand the twisted minds of those who write Harper’s Index. My mind was sufficiently twisted two years beyond Algebra 2.

  • BitDreamer

    I appreciate Baker’s essay calling for the end of the Algebra II
    requirement, but I feel like you have the wrong guy. The Algebra
    in this article bears no resemblance to the algebra I know (and love).
    I had no idea what a “Rational Function” was before I read this
    article. I cringe at the apparent term “cross multiply”. I am
    horrified by students who say they hate math “because of all the
    memorization”. Seriously? You have to memorize body parts and species
    in Biology, vocabulary in English, dates in History. But Math is
    supposed to a very small set of rules (6? 3?) and the rest is practice.
    Once you have the foundation rules, you can do anything. It’s like
    music — I can’t play the Moonlight Sonata, but when I look at the
    sheet music for it, I understand it. Any first year music student
    could say that. Math can be like that.

    When my closest brother and I were in high school, I was in the upper
    track and he was in the lower (despite comparable IQs, go figure). His
    homework was harder than mine. The theory was that the lower lever class,
    in the interest of proficiency, would be spared the explanation of WHY
    math works and just given tons of formulas to memorize and crank. He
    hated his math; hell, *I* hated his math. Have we now put EVERYONE on
    the lower track? Students would hate anything taught this way.
    Math should be empowering, the one subject that a student, with a few
    solid tools, can figure out problems, without having to take the
    teacher’s word for it, or read the book — I never read my math
    books — or memorize tons of stuff. In fact, for me, it’s the
    cornerstone for the concept that there IS truth I can verify myself,
    and that I don’t have to believe whatever is in the news, like the
    story that the economy is getting better when 99% of us know it’s not,
    that I can question the first answer that pops up in Google. Math
    makes sense if it is taught right. Can we please try THAT
    Math/Algebra before we throw the whole thing out?

    • Michiel Bijnens

      Exactly. I discovered when I was young that all mathematics is deduction, and just copy-repeating from memorized formulas is an act that requires no understanding. Once I mastered the art of deduction, I skipped all my mathematics classes so all my tests where actual tests of my ability to do mathematics (in a 90 minute time window.) Although I lost frequently points for “using the wrong method” – the travesty is that some test are designed to test memory! and then give you enough time to struggle with moving numbers around – instead of finding the solution to a problem. During high-school I already experience the result of “learning” mathematics this way when I had young stand-in teachers come up to me and say “I don’t understand your solution (which was written down step-by-step) but I ASSUME it’s correct.”

      • Jan Priddy

        My son did that all the way through Calculus in high school—memorized the formulas and rules and then came to class to take the tests (and often set the curve doing it). All of this is irrelevant to the point of the article that we are failing to teach basic day-to-day application in the interests of abstract processes that discourage students.

        I have students who want their calculators to figure out the percentage that a score of 42/50 earns them. I can still do math in my head faster than my students who are taking Calculous. And I am a high school English teacher.

        I don’t remember finding math hard and I started Algebra when I was 11 years old. I simply “got” it, but some people really do not get it when they are 11 or even 16. And many would “get” higher math if they studied it within the field they do get.

  • Paula

    How can we stop this juggernaut called the Common Core? Impossible. But I’m thrilled that someone is trying. People wonder why high school students are stressed. They are stressed because their situations are impossible! Those who want to go to college have to meet higher and higher standards in all areas–not just those they plan to major in. Those who may not want to go to college are forced to meet higher and higher standards just to graduate. Most can’t meet these standards, they aren’t offered alternatives and all of this is done in the name of giving everyone “access to the standards.” I have taught and worked in various capacities in schools at all levels and what is scary is that teachers buy into this! This idea that if you just teach it right, everyone will get it.

    Have you seen the writing standards? Kindergarteners are taught how to organize “hamburger” essays (topic sentence and conclusion are the bun…etc.) I finally had to leave teaching for good because I couldn’t bear it. We aren’t serving students. We are serving the publishing companies that make a fortune by promising that their texts have the magic formula to make the standards accessible to all.

  • Commuted

    I just want to say fractions should be replaced with simple algebra. Also I was stopped by the pay-wall so I only read the first paragraph. You should allow one article. Poor people are going to be uninformed. Some things never change…

    • LilMoby

      It is available in my library and local bookstore. You might check yours.

  • Superman

    Mathematics comes from two words that mean to do and to
    remember. Most people find mathematics a hard topic and a very abstract topic.
    Although, I cannot make mathematics easy, I can try to help you to do and to
    remember and I can make mathematics practical, especially algebra II. You will learn that if you master
    some of the rules of mathematics by doing and remembering then you will be able
    to solve problems (that occur on a regular basis) in your life. You will also
    learn that many public policies are easily shown to be a ‘bad idea’ by doing
    some mathematical analysis including those policies that were instituted heavily over the
    last 100 years and still believed by a majority of (progressives, conservatives and liberals) to be good
    ideas even though it helped to turn the Earth into a blacktopped and concreted
    toxic cesspool.

    • basilissa

      Mathematics comes from the word to learn, “manthano.” A mathema is a thing learned. Mathemata is the plural, which is why our word is also plural.

  • vajra78

    The joke is that algebra II is barely an ABC course for real mathematics it’s not even anything complicated. Algebra II should be required to pass elementary school and Abstract Algebra should be required to get into college. Haha, just kidding : ) Great article.

    • vajra78

      I wanted to ad that I think the biggest problem with math in general is its always taught out of application or as the author points out with the Gecko is taught with a false concept. Algebra truly explode in the second half of the nineteenth century as a product of applications desired by the department of defense. Furthermore, calculus is a product of Newton’s investigation and explanation of natural phenomena.

    • You’d be surprised.

      Actually, I have experience teaching algebra to elementary children. The trick is to teach the language and the concepts early, and without penalty. Then parallel their arithmetic with the algebraic equivalent. Using this method, third graders can master 2-step algebra. 2-step algebra is the median accomplishment of the average 11th grade student. Guess what my third graders score as on standardized exams…

  • Middle School Math Educator

    With 27 years of experience, I have been through several math education “revolutions”. Math came easy to me but I quickly learned that in teaching it, that meant something but not much. I am right brain person which means (to me) that I like to see wholistically, look for connections. Certainly, the textbooks that I learned/taught from are not organized that way. Step by step, things are presented and if you missed a step, you might as well be stepping off a cliff! I found myself innately alterating lessons to help embed answers like “why do I have to learn this” and “how does this make sense”.
    I BELIEVE
    Math does make sense, and thank goodness I sought out and found professional development 5 years into my career that showed me this. It permeated my teaching for the next 22! Math is very interconnected in its topics, but one might never know it from the Standards, Expectations, textbooks I used.
    I am of that school of “if it is taught right, anyone can see it.” But after 27 years, I am realsistic in that public education 1) does not organize their resources to support students who don’t get it the traditional (and most efficient way) 2) don’t demand teachers to be trained and certified at a level of math content necessary to adjust modes of instruction to meet learning needs. (Three math courses in college don’t cut it!)
    Unfortunately, some potential teachers have chosen K-6 certification to “avoid math”. This does not bode well for their students when they are assigned to be math instructors. Don’t get me wrong, there are some GREAT elementary/middle math teachers, but I also have worked with my share that are limited in their own understanding and thus unconfident in deviating from assigned paths. In the end, “some kids get left behind” to borrow part of a reform phrase.

    (WARNING: NOT OFFICIAL NUMBERS, ANECDOTAL ASSUMPTIONS BELOW)
    Efficiently taught, math can be taught by less than mediocre teachers and successfully mastered by 20-40% (who would figure it out anyway) of students. Taught with many learning styles in mind, I think 90% of those with an average IQ or better can learn (AND APPRECIATE) the power of mathematics. My goal was develop a confidence in each child that left them saying “This makes sense, I can figure it out and really know it.”

    For 50% or less they will use up thru 8th grade math all their lives (percentages, conversions, ratios, basic geometry), but I am not an advocate of dropping Algebra II. I am advocate of showing the richness of the topic and showing students how it is embedded in the “languague of development” of every technology they enjoy in their life. Some may be inspired to choose careers that previously said “that has too much math in it for me to pursue.” Let’s not let students close doors, before they ever peeked inside to potential life paths.

    (REAL FACTS from a recent Broookings Institute Report)
    STEM careers make up 20% (or 26 million people) of the work force. 50% of these call for a college degreee. 50% call for an associates or certifcation past HS. Both categories pay at least 10% better that people with similar educations. Finally, there are massive openings in these fields (Great Recession time being the reception) for those students prepared to take them. Baby Boomers are retiring.

    A common thread is a need for higher level math and science through HS to choose one of these fields. Is this report (Brooking Institute, June 2013, “The Hidden STEM economy”) this an argument for Algebra II? You will have to decide.

  • snowyphile

    Cardano is said to have acquired the solution to the cubic equation for cash.

  • Mark McDonough

    By a weird coincidence, I was just talking about this subject today with my boss before coming home and reading this article. We both agreed that except in a few professions, most people (ourselves included) rarely need anything beyond basic math and a good grasp of fractions and percentages. Ironically, this everyday math is often taught before people are really ready to understand it – so that high school can be reserved for geometry, algebra, and calculus.

    What sparked this discussion was an internet outage at a local mall – my boss had to stand in line for half an hour because with the checkout terminals down no one was capable of calculating 6% sales tax. When she showed a clerk the magic of multiplying by 1.06, the clerk asked if she could stick around for a few minutes and train the rest of the staff!

    So we’re torturing people with math they’ll never use, and failing to teach them math that’s actually useful. Wonderful article – thanks.

    • Nicole White

      My experience as an Algebra teacher was exactly the same. I had students who did not know their times tables or what the little marks on a ruler meant being forced to learn Algebra I and II. My school had adopted Common Core Standards, and I was not allowed to spend time “remediating” basic skills if my curriculum guide said I was supposed to be teaching polynomial equations. So I quit. I felt like I was failing the students as a human being by not spending time on the math skills that I knew they would need in real life. By the way, my school had a 50% drop-out rate.

  • tfarnon

    Judging by what I experience when I’m shopping or engaging in other monetary transactions with a live human being, even simple arithmetic hasn’t been taught in schools for at least 20 years. Call me old-fashioned, but I’m used to working simple problems in my head, and more complex problems on paper, usually without a calculator, slide rule or abacus. I believe that students need even more mathematics prior to High School graduation, not less. I also believe that arithmetic should be taught in elementary school without the use of calculators, computers or other electronic devices.
    Too many people have no fluency with numbers, and too many can’t see useful relationships between numbers because they were never forced to acquire those skills in school. Just as literacy is essential, numeracy should be considered equally essential. Numeracy is critical to giving back accurate change to customers. Numeracy is critical to balancing your checkbook. Numeracy is critical for comparing prices at the grocery store, for doubling or halving a recipe, for figuring out just how much of a given expensive material you need to purchase at the hardware store. In STEM disciplines, numeracy is critical. It’s essential to know the difference between 2 x 10 and 10^2.
    If the study of mathematics is “torture”, tough! I believe that an aversion to mathematics is partly a natural human tendency to want to avoid anything that requires effort, and partly a long-standing social phenomenon called “Physics Envy”. Lots of required subjects in school are “hard”, but nobody complains when students are compelled to learn to spell, to use standard grammar, to study major literary works, to study history, to learn to play a muscial instrument, or to learn a foreign language. Mathematics is really no different. Even if the specific subject material has no immediate application in a student’s life, the discipline and understanding acquired in studying any given subject carries over into other areas.

    • Terry Fondow

      As an engineering student, I took three years of college mathematics and I love the subject. I did not pursue engineering as a profession, preferring to teach math and science. Although I worked as a navigator at sea (before GPS) and also worked for two fortune 500 companies in addition to serving as a teacher and principal, the only professional use for all of my math learning was to teach, take grad classes, or judge the merits of research articles. To force all young people to take three years of math in high school and to require Alg II for college entrance is useful only for institutional gatekeeping.

      As as side note, I am always amazed that otherwise smart people will assume that because they have met young people who don’t know something that whatever that something is must not be taught in school.

  • hilbert90

    I wrote a response here. Mostly agreement, but some preliminary thoughts on why it seems possibly a bad idea as well. I haven’t thought too deeply about this, so be gentle.

    • Terry Fondow

      Three of your premises bear further consideration. First, Alg II is easy. Easy for you to say, but not an assumption based on facts. Second, most of the people who fail Alg II are not trying. Again, no facts to support your assumption. Third, failure at Alg II indicates lack of readiness for college. Many private colleges do not require Alg II for entrance and Baker’s article cites the logical fallacy of your third premise.

      A more sensible approach is to require all students to take Algebra I before they leave high school, but do not make passing the class a graduation requirement. It is far more important to ensure that students have mastered measurement, the ability to estimate the reasonableness of results and to understand the relationship between decimals, percent and ratio.

      • hilbert90

        As to number 1, it isn’t really an assumption. I give a sketch of an argument for why it is “easy” (I actually say “not that hard”). Namely, it is very old historically, it is not as abstract as much of what you will encounter in college even if you major in the arts, and other cultures don’t seem to struggle with it.

        Sure, trying to turn this into a more solid argument could be a whole blog series, but it is disingenuous to say I’m just taking this as an assumption. Also, the stats merely tell us how well students do on tests about algebra. It is a fallacy to assume this is equivalent to the “difficulty” of the subject.

        As for 2, I point out that I’m just making that part up. No facts to support it, but no facts to contradict it. I understand this doesn’t account for “most” students, but surely a non-trivial amount do. It would be interesting to get stats on this. Have you ever been in a high school classroom? This assumption seems wholly merited.

        As for 3, I was extraordinarily careful in my wording so that I didn’t make his fallacy because I agree that teaching Algebra II does not necessarily teach logical thinking and such skills that will be needed in college. What I say is that a class such as Algebra II can serve as a litmus test for whether or not students can handle classes that require a little abstraction. These are definitely not equivalent. I also point out that other classes could serve this purpose.

        For your second paragraph, I agree. I even say “Sure, give them a high school diploma if they can’t do it, but college may not be the best fit.”

        • Terry Fondow

          I thought you had invited a critique of your ideas. My mistake. In answer to your question, yes, I have been in a high school classroom – serving as a science and math teacher and principal for 29 years. I have seen first hand the damage that our one size fits all approach to graduation requirements has done to about 15% of our students.

          • hilbert90

            I do invite critiques. This doesn’t mean I’m required to remain silent if I see fallacies or misunderstandings in a critique.

            I agree with some of what you say and point out some fallacies. I also clarify some things which were probably my own fault for not being clear enough the first time around.

            Making a counter-response does not mean I didn’t invite the critique in the first place. In fact, it means I actually thought about what you said.

            Since you’re an experienced teacher, I’m sure you can point to tons of examples of perfectly capable students who failed a class because they were too lazy to do the homework or hated the class.

  • Walter Greatshell

    For a country that worships athletes for their specialized gifts, it amazes me that we don’t understand the importance of intellectual specialization. Taking an intelligent kid who loves writing and art (as I did, and now do for a living) and punishing him with Algebra is akin to taking a star quarterback and demanding that he succeed at gymnastics. It is idiotically counterproductive. The gifted should be given the means to develop their gifts to the fullest potential — whatever those gifts are. Otherwise we just encourage mediocrity.

    • Andy

      Why do right brained people have to sit through mind-numbing english classes? This is a two edged sword. I don’t know where this idea that creative people are being singled out and subjected to these cruel courses. I enjoy math, but not english, yet the requirement for English classes (at least at my school) are more extensive than those for math. This ‘punishment’ goes both ways.

  • Real Analysis

    Baker’s article is interesting, but it, and this discussion, conflates several different questions: (i) Is basic algebra useful and learnable, and by whom? (ii) Should algebra 2 be *required*, or just available in high school? (iii) Do (and should) colleges require algebra 2 for admission? (iv) Should mathematics be taught in “pure” or “applied” form? (v) Can the average high school-age student make wise curricular decisions on her/his own?

    Yes, algebra is hard for everyone, and very hard for many. But it’s among our species’ best achievements, right up there with good manners, literature, and arithmetic. A large majority of students, properly taught and with reasonable input of effort, can master Algebra 2.

  • Conrad

    I am the Chair of a college Mathematics Department and a Professor of mathematics. I know that children are born with natural curiosity about everything. It’s a shame we have ‘turned off’ that curiosity. Parents and teachers are part of the answer and bear part of the blame. Great instructors can teach virtually anything to anyone! Parents need to be supportive.

    • Jan Priddy

      Few elementary teachers are strong in math skills and that is something that must change. More than 50 years ago I lucked into an elementary school that invested in math and I have enjoyed my skill ever since. However, I don’t know that even the best math educator can teach everyone to work in abstractions, to imagine three-dimensional objects, and so forth. I’m not sure anyone “knows” this is true.

  • 0paul0

    I agree with Nicholson Baker that if a lot of students have difficulty with Algebra 2 (or any other subject) then we should look at how to teach it better. And we should also re-examine if schools are teaching students the skills they need.

    But I disagree strongly with his argument that sufficient “this is too hard!” complaints are proof that an activity is worthless. Should people beginning an exercise regimen give up if they feel sore after walking a mile? Should we stop expecting students to memorize vocabulary in a foreign language if they complain “this is too hard”?

    In the world of relaxed expectations that Baker seems to be advocating, maybe it’s too much to expect students in a literature class to read long books like War and Peace? Maybe pre-meds shouldn’t be required to memorize hundreds of chemical formulas in organic chemistry class? Think of the complaints if we force them to do all that work! The poor suffering students!

    For some reason, in America, high standards in athletics are admired but high standards in academics and intellectualism are sneered at. Why does our culture glorify the aspiring athlete that practices free throws for hours (and then chugs Gatorade and walks off in his Nikes) but fail to glorify the student studying rational functions for algebra class? When was the last time you saw a celebrity mathematician discussed on Entertainment Tonight?

  • Guest

    http://www.gallup.com/poll/164249/americans-grade-math-valuable-school-subject.aspx

    Kinda blows away the whole “kids are never going to use it” argument, huh? And if you’d apply this argument to Shakespeare, history, and phys-ed (all of which are undeniably less useful than math) as well as everything else in high school, before long you wouldn’t have any curriculum at all.

    • Jim Martin

      Shakespeare, history and P.E. are useless? After reading the Gallup results, I must confess that it looks like the less educated people are simply repeating what they’ve been told, and that is that math is important. I don’t believe it.

  • Steve

    “Kids don’t hate smelting, or farming, or knitting, or highway design, or portrait painting, or neurology, or juggling rubber balls, or sonnet-writing, because they don’t have to take three years of instruction in any of these arts.”

    I had to take four years of fabricating Jesus metaphors from English literature and all it taught me was to hate Harper Lee and John Steinbeck, should we stop teaching literature too? I don’t recall ever needing to know the history of the Protestant Reformation, should we stop teaching the four years of history I took? How about the decade of phys ed that just made me hate sports and everyone who plays them? Should we stop teaching phys ed because some kids don’t like it? Algebra is no different from any of these.

  • Howard

    One of the standard subjects in algebra 2 is SYNTHETIC DIVISION, a shortcut method of dividing a polynomial by a binomial. I am a retired physicist. I have never used it. I have asked other physicists, engineers, chemists, and mathematicians whether they had ever used it; none had used, and few remembered it. It offers no insight or understanding of mathematics, yet it takes several weeks of class time. Get rid of it!

    • Howard

      Sorry; I meant monomial(x+a), not binomial.

  • Pat

    This article is only available to magazine subscribers.

    Just thought I’d add that the words “only” and “available” are in the wrong order – just one of those grammar things. Tsk tsk. And I’m an algebra teacher.

  • DrMcCluney

    I finally had a chance to read “Wrong Answer” by Nicholson Baker. I have B.A., M.S., and Ph.D. degrees in physics and took a lot of math courses over the years. It was not until I got into grad school and finally developed some facility with geometry, algebra, and calculus, that I finally came to see the beauty of mathematics, especially algebra.

    As so many before me have, I suffered mightily through my first courses in geometry and algebra. Looking back on it I now realize why. The teachers and the textbooks presumed their jobs were to provide students with a first-principles introduction to the subjects and insisted on developing axioms and detailed, meticulous proofs of their absolute correctness, a process guaranteed to bore most students and engender hatred for the subject in many of them.

    If the approach had been to start with explaining basic algebraic (and geometrical) principles and how they could be used to solve useful problems, I think the average student would learn the subject (by doing it) more quickly and less painfully.

    Along the way, good teachers could occasionally give some asides (promised not to be on the tests) explaining the beauty and importance of crafting an absolute proof and what exactness means and how exactness generally fails when making measurements in the real world. Teaching the subject in this way, I think, could simultaneously help students develop a degree of mastery of the process of doing algebraic and geometric derivations to obtain an equation offering great utility in the real world while instilling in them an appreciation for the art of devising mathematical proofs of laws that are eternally correct within the limits for which they are appropriate.

  • http://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=2220942 Jamal Munshi

    I wrote a paper on using computers instead of scary equations to teach undergraduate courses in finance. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2512091

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