Why Algebra

Math education is high on the list of both professional and lay “hot” topics. Everyone has an opinion. Andrew Hacker’s critique in Sunday’s New York Times  was a pleasure to read, given that I agree with him. But he also managed to base his argument on unassailable facts. For example, only 7% of the job positions in America actually require algebraic knowledge. But 57% of all college entrants fail to finish college largely because they can’t pass algebra! The same goes for high school diplomas. The price they pay for this failure is disproportionate to its importance Hacker argues.

Such weighty and critical decisions, which influence the life careers of millions and the future of America (so we’re told), are rarely based upon a serious look at the “evidence”–mathematically objective data. When it comes to “evidence-based” judgments we pick and choose our evidence. We fail to use precisely the kind of thinking algebra is supposed to teach.

Arguing against Hacker, rknop makes an interesting argument in his Galactic Interactions blog—but one I suspect Hacker would agree with. He defends algebra on the basis of the importance of the Liberal Arts and citizenship vs. the economy. But  both Hacker and I would then ask, Hacker and I would then ask: what’s the evidence for believing Algebra is good for the Liberal Arts?

Somewhere along the way it behooves us to challenge the 1893 curriculum. At Central Park East and Mission Hill we challenged HOW the required disciplines were assessed, but we didn’t go far enough in challenging why. For example, see how Marion Brady tackles the subject in a dramatically different way in his Washington Post article.

My proposal? For starters, every community and faculty should come up with a list, in order of importance, of the knowledge and habits of mind that they believe are critical to the exercise of good citizenly behavior. Then let the arguments begin. It might focus the ferocity of the arguments about algebra not to mention the value of a lot of the other requirements of formal education.

22 Responses

  1. And this is exactly wrong (I know it borders on arrogance taking on a progressive educator of your stature, but here goes…). Removing algebra from college curriculum with the intent of allowing more people to have the credential won’t change anything. The logic of credentials is that when they stop providing their “gatekeeping function” they are replaced. This has has already started to happen, the master’s degree is becoming what the bachelor’s used to be in terms the “must have” entry into the job market. So the goal of increasing social equality is not served and the notion of a broad and empowering liberal arts education is eroded. I agree that not enough people graduating algebra is a problem, especially when it is a requirement for graduating from college (or not graduating for any other reason), but getting rid of it just buys into the credentialing logic to no one’s benefit. Many colleges have taken a pretty aggressive stance in promoting writing skills and integrating that goal throughout the curriculum, why not math. It is not just that algebra is important for 7% of jobs (for the sake of argument) but that they are more highly skilled and renumerated jobs. Giving up on that broad idea of liberal education (which is exactly what is being advocated) is not just inviting a deeper segmentation in term economic terms but also (because of a decline in scientific literacy) in terms of democratic participation . I am not in favor of credentials and would welcome a more diverse, flexible and life long learning process but as long as credentialing is one of the functions the education system provides, our collective definition of what is important in terms of meaningful social participation, another name for purpose of liberal arts, is going to be tied up in “requirements for graduation”. Get more people passing, and appreciating algebra, but definitely keep it.

    • Thanks! There’s so many courses we don’t require that I think are ore importaznt. So one is always making choices–often between equally good possbilities. But tell me more about why you think algebra is more important than probability and statistics, for example.

      Deb

      • Algebra is not more important than probability and statistics, but basic algebra is helpful in understanding the notation and calculations of probability and statistics. Even so, I agree that the obsession with algebra is unfortunate. There are many other paths to numerical literacy — “numeracy” as John Allen Paulos calls it — and it would be better if students were allowed to follow other paths to numeracy which better capture their interest and motivation. A sufficient number of algebraic concepts will find their way into alternative courses just because the concepts are useful and necessary. I think that Paul Meyer’s comment (above) is more a plea for rigor in education than a defense of algebra. Let students take alternate paths — just make sure the alternatives require an equivalent of effort, self-discipline, and — yes — “habits of mind.”

      • Thanks— And maybe familiarity with some of those concepts, Jim, are essential to using and developing statistics on a sophisticated level, etc. But most of our high school an college grads are not at home with concepts that do not require algebra, and which as ..EBear?–notes, will use some algebra in context. Most have not merely little knowledge but largely mistaken knowledge about both probability and statistics that gets us all into serious trouble

        I’m of two minds about the term “hard work” (although it’s part of the Mission Hill “slogan”– think “work hard” may be is how it’s worded). The hard part troubles me. Some of the best work I’ve done was not hard, but required perseverance and challenged me–it was the excitement of the challenge that led me on, not its difficulty. (That sounds trite.) . Hard/easy seem inappropriate terms. Maybe that’s got a personal history? The books I’ve been most influenced by? Were they hard? No. Hard for me is often a synonym for being bored–which may sometimes have to do with lack of knowledge, over my head. But I’ve found that if a book seems too hard, I look for an easier one–that I get into quicker, can’t bear to put down. There are so many potential great reads out there that why struggle unless it’s necessary. I often also fund that the same book that once bored me, at some other point in my life became a favorite. And rigor!!!!! Look it up in any dictionary and I think you’ll see that 90% of all th definitions are definitely not what we should be looking for. When did it enter our language in this new way?

        I’ll do some more thinking about this–maybe you can help me.

        Thanks Coolit–thats a good summary of what I’m – at least in part – trying to get at.

      • Basic algebra is essential for probability and statistics.

        The algebra of quadratic equations is essential to probability, for example, understanding the central limit theorem (CLT) and the slow reduction of error as the sample size grows. The CLT’s normal (Gaussian) distribution approximates the sampling distribution of the sample-mean (for samples drawn independently).

        In statistics, the approximately normal distribution for the sample mean is the basis for confidence intervals on the population mean, and so for choice of appropriate sample-sizes. The appropriate sample-size depends on the variance of the population and the desired confidence-level—not on the size of the population!

        To understand such basic statistical results, and to demonstrate understanding by calculating appropriate sample-sizes, students need to understand square-roots and inverse functions.

        (There are other statistical ideas that do not demand algebra: The distinction between a population and a sample, the random assignment of subjects to treatments, controlled experiments versus observational studies, placebos, blinding of human subjects, blocking and stratification, etc.)

        Quadratic equations are essential for understanding Mendelian genetics, particularly for understanding why recessive genes that are harmful when paired survive. (Many are beneficial when paired with a dominant gene.)

  2. Expose kids to many different things, feed their passions, follow their interests, and the need for algebra, probability, statistics, will make itself felt by kids. Treat them as learners who have the will to learn already inside them, instead of coercing them into learning. Those with interest will come around to algebra because they need to learn it to pursue the learning they choose.

    Honestly, algebra is too important a subject to coerce kids to learn and then hate.

  3. “My proposal? For starters, every community and faculty should come up with a list, in order of importance, of the knowledge and habits of mind that they believe are critical to the exercise of good citizenly behavior. Then let the arguments begin.”

    This is an extremely radical and refreshing proposal given the current state (pun intended) of Common Core.

    Seth Godin, gave us his list for high school, in his Stop Stealing Their Dreams Manifesto, which would be a great place to start.

    Computer Programming
    Fine art
    Selling
    Presenting ideas
    Creative writing
    Product development
    Law
    Product management
    Leadership
    How Old is the Earth?
    What’s the right price to pay for this car?
    Improv
    How to do something no one has ever done before
    Design and build a small house
    Advanced software interface design

    in addition to knowing the following skills:

    Giving a presentation
    Handling a negotiation
    Writing marketing copy
    Shaking hands
    Dressing for a meeting
    Making love
    Analyzing statistics
    Hiring people
    Dealing with authority figures
    Verbal self defense
    Handling emotionally difficult situations

  4. Hi Deborah! I will keep this simple. What is your opinion on the value of Game Based Learning to the future of teaching kids maths?

    • Is it a book? I think games are a good way to build certain kinds of math skills. Ditto for any dice game, or spinners, et al. Or ones that require logic, thinking ahead, strategizing. Ditto for many card games!

      Best,

      Deb

  5. When I attended Euclid HIgh School (a working-class community near Cleveland, Ohio), I took Advanced-Placement calculus, physics, chemistry, American & European history, English literature & composition, and Spanish, My friends took AP German, French, and biology.

    My high-school teachers often taught summer or evening courses at nearby colleges or at Cleveland State. Dr. Ronald Powaski, when he retired from teaching history at Euclid, had time to write even more history books, some published by Oxford University Press.

    Americans have a national curriculum, which has been sensibly set by the College Board on behalf of an association of reputable universities and colleges.

    The AP curriculum allows working-class students access to quality courses and allows them to experience achievement after a year of disciplined work.

    I suspect that AP courses also attract academics to the teaching profession, some of whom become national leaders. For example, Mr. Charles Reno (M.A.T. Harvard) served on the committees writing AP curricula and tests in physics, calculus, and (I believe) computer programming.

    Your curriculum and model of community schools would have denied education and opportunities to my friends, and should be regarded as at best a symptom (and perhaps mephitic snake-oil) rather than a treatment for class inequality.

    • I have been thinking about your comments all day. I do not understand your comment that the curriculum and community school model discussed should be regarded as a symptom or mephitic.(btw, spell aid is not familiar with this word, I guess the software developer did not have AP English). I could conclude that AP courses and national standards are also a symptom of, rather than, a treatment of class inequality. Joe Graba, with Education Evolving, made this comment that I think applies:

      “I think one the prime obstacles to changing the schools we have is that generally the children of the influential people in every community are served reasonably well by the schools we have, and consequently
      they are much more inclined to want to make those schools better than they are to make them different.”

      AP classes and national curriculum are perfect example of making schools better and better always seems to mean harder for folks that were successful in the traditional public school model. If it is good enough for rich, white folks, it must be good enough for everybody else. My neighbor’s son is taking AP courses trying to get the proverbial 4.5 GPA so he can get into Georgetown. His family will do everything to make sure his grades are the best they can be, it’s a shame they are not putting as much effort into the rest of his life, which is the whole point of the curriculum and community school model Ms. Meier suggests.

      • My favourite thing? Just one? Okay, well I love that my kids can be thelvemses, free from the expectation to fit into a mold or act a certain way. They can lie on the floor to write, read upside down, take breaks to dance or pet the cat. They can be thelvemses and explore what that means, daily.

    • Hmmm. What leads you to that conclusion? VIrtually all the graduates of the schools I’ve influenced most (like CPE, CPESS and Mission Hill, ) and others like it send virtually all their kids on to college. I’m not getting the connection you’re making? Tell me more. Deb

      • Hi Debbie!

        College attendance is not a goal in itself, especially if admitted students cannot handle even remediation (“College algebra”, trigonometry, English composition).

        The following question is difficult and perhaps impossible to answer. However, you can state what you believe. Are the students of your schools enrolling mostly in sociology, (undergraduate) psychology, communications, etc.? Or do substantial numbers study nursing, economics, mathematics, electrical engineering, microbiology, (or end up in graduate psychology programs, after their tedious undergraduate programs), French literature, etc.?

        In turn, let me give some anecdotal evidence, which is the best I can offer. The Honors/A.P. curriculum prepared the working-class children of Euclid—usually the first generation to attend college—to perform well in the most challenging degree programs in the best colleges of the US.

        I know of at least five students from my Algebra & Trigonometry class who took the advice of Mr. Adam Pawowski and became attorneys after their engineering degrees (one outlier, the Harvard Law attorney, a son of a widower, studied economics at Swarthmore). I’m a statistician, the fourth child of a single mom (divorced). Our class had at least two physicians. Our class was a disappointment, because nobody went to MIT or CalTech. :)
        About 3 years after graduating from high school, I ran into the student who scored lowest in our A.P. physics and math classes. He was excelling at Cleveland State as an engineer. He didn’t complain that life was unfair, but he was grateful that he learned calculus and most importantly knew how to work hard.

        Debbie, do you claim that the curriculum (at at least one of your schools) would have prepared my classmates as well as the A.P. curriculum?

        Perhaps you and Todd Gitlin (B.A., mathematics, Harvard) could discuss high-school mathematics in a “Dissent” symposium?

        Sincerely,
        Jim

      • If we weed kids out of college math who haven’t done well in calculus or taken it in high school then of course it could be unfair. My son went to Hunter College as a math major. There were 70 kids in introductory math (most of whom had taken high school advanced algebra and many calculus!). By the end of the course there were about 25 students left. When he graduated with a math award 4 years later he reminded me–when I congratulated him–that here were only 4 math majors left and each got one of the 4 awards.

        I want math–taught well–thAt increases the number who want to take math in college, who find it one of their favorite courses, etc. Lots of kids in the high schools Im referring to loved math, and while they struggled with freshman math since we had not prepared them with what you might consider necessary, they persevered because they believed math could be fascinating.

        IT’S LIKE DOING WELL ON READING TESTS BUT NOT DOING MUCH READING–EXCEPT WHAT’S REQUIRED.

        I always got an A in my high school math but never took another course in math afterwards–I passed out of the required course because I was a good test-taker.

        It was when I became a kindergarten teacher that I became fascinated with math as I began to try to figure out what it was kids were up to.

        It’s not that I want to “drop this or that course”, but I want to reconfigure what constitutes math for everyone–in ways that will increase, not decrease, the number of students who arrive at college still eager to study math–as a minor or major.

        Deb

      • I neglected to note points of agreement:

        I agree with your concerns that “hard work” is not the best description of a good experience with schools.

        Nonetheless, time-on-task is still the best predictor of learning, according to Carnegie Mellon psychologists John Anderson, Herb Simon, Robyn Dawes, et al. Providing good exercises is an important part of the teacher’s art.

        It is also important to avoid stress and promote relaxation in a classroom and homework. Gary Marcus’s description of his teaching himself guitar (on his sabbatical) discusses this point, and other points of applied cognitive-psychology.

      • My dilemma with such studies is:-how do they measure “learning”?

      • Hi Debbie,

        I am trying to reply to your post below, :)

        1. Regarding your son’s experience as a mathematics major, this is a difficult topic.

        Most universities and colleges treat junior-senior degree courses as preparation for research careers in graduate programs. Such courses are useful for the majority of students who will not pursue research in mathematics—just as courses in history or philosophy or psychology discuss research topics and prepare students for possible research—in part by discussing design of experiments and observational studies, for example, the historian’s art of finding telling documents).

        Mathematical courses are intrinsically difficult and require adequate preparation and commitment. Mathematical research requires the ability to solve problems and deal with frustration, often by sidestepping failed attempts and trying new paths.

        There are high failure rates in Sweden, also, for advanced mathematics. There is no shame in failing a M.A. course on measure and integration theory, and many leading mathematicians volunteer that they failed such courses repeatedly (although Svante Janson was 14!). Advanced mathematical study requires that students live and breathe mathematics and make the material part of themselves—not the whole being, but a living part.

        (Americans have trouble entering graduate programs. A Swedish or Polish student would have studied mathematics for 5-6 years in a demanding program, and would have written a research thesis, before enrolling in a U.S. graduate program. Most American students would do better to take 2-3 years in terminal M.A. programs before attempting to enter graduate study at a Ph.D. granting institution.)

        2. Measuring achievement is difficult, but experts can agree roughly on rank ordering of performance on tasks. The psychologists find that the best predictor of performance is time-on-task, and of course they find that greater time on task results in greater achievement. Gary Marcus has a fun review of this literature in his book about his learning of guitar.

        This emphasis on time on task is an important corrective to America’s emphasis on individual traits (especially IQ, usually in disguise, and obedience).

        Gary Marcus discusses the importance of challenging but doable tasks, often devised with the help of a more advanced person. Dull repetition is rarely helpful, and often harmful of course.

        My high-school teachers invited students to come in at 7:00 am or after school for help with calculus. (Jaime Escalante provided even more guidance!) Time on task is important.

      • Good ideas. And actually I’m not sure the math course my so took was “advanced”–it was a freshman course, I believe. I’ll find out more since I often use it as an anecdote I ought to get more background on it!

        More on other points later!

        But I’m still focused on what K-8 math ought to look like so more kids see it as an exciting field. And on how and when we teach enough statistics and probability so that even our reporters and media aren’t so dumb on the subject. Our politicians depend on our ignorance re large numbers–another tragedy. Millions, billions? The two are used “interchangeably as though they are more or less the same. Huge is huge.

        Lying with statistics is a daily phenomenon, due in part not to malice but ignorance on those reporting.

        Deb

  6. When we say Algebra, what levels are we discussing? Algebra I is definitely worth the time and effort for our students. Algebra II should be required for Math/Science/Engineering Majors, while Probability and Statistics should be required for liberal arts. MSE majors should also have to and often do take Prob. and Stats. A liberal art major will run into probability/statistics more often than they will run into a quadratic equations in their future.

  7. Tim;

    I mentioned my hometown of Euclid, Ohio and noted that it was a working-class community precisely to avoid the reflexive, race- and class-baiting that is endemic to discussions of curricula.

    You can read demographic information here:

    http://en.wikipedia.org/wiki/Euclid,_Ohio

    Briefly, Euclid was formerly a majority of working-class whites, and now is a majority African American. It has been the home of the Slovenian Polka Hall of Fame for decades.

    The point is that the College-Board’s informal college-preparation curriculum has served working-class communities well, when there has been leadership from academic teachers. The curriculum is similar to that of state-funded gymnasia in Northern Europe (e.g. Poland), lycees in France, colegios in Spain, etc.

    The book “Jaime Escalante: The Best Teacher in Ameria” (and its iil-named film “Stand and Deliver”) is a better known example of an academic teacher who helped low-income students achieve excellence. Another example, Euclid’s Honors/A.P. program deserves to be better known.

    A third example: New York City’s Bronx Science, Stuyvesant, and Brooklyn Tech high schools also serve their communities well—in part by providing motivation for working-class students to study, I lived in Fort Greene. The Brooklyn-Tech students who would hang out on my stoop (sometimes smoking cigarettes) did not look like rich white kids.

    • I prefer to consider myself as reflective rather than reflexive. I have spent a lot time hanging out on my porch with rich, white kids and they could use “the knowledge and habits of mind that are critical to the exercise of good citizenly behavior.”

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