Math: Old, New, and Equalitarian — A Review

I have a new paper out in Academic Questions, a review, “Math: Old, New, and Equalitarian.”

Springer is charging a mere pittance for viewing it. Only $39.95. But I think you can get a PDF from this link, through something Springer is called “SharedIt“. I’m not at all clear what it means, and I have no interest in discovering. As long as you can read the article, I’ll be happy.

Here’s the start…

There are three ways to teach math to the young. The old way forced rote memorization of basics and then, for most, stopped the lessons, continuing them only for those who had the inclination or ability to advance. The “new” way was to “expose” every student from the beginning, no matter their age or inexperience, to the highest, most difficult mathematical concepts, so that all might know how wondrous and astonishing math is.

The modern way, which may soon be upon us, is to let students define what math is to them or their culture, to let them discuss their feelings about what math means, and to work toward the goal of equality, that happy state when all are satisfied with their level of (self-defined) mathematical understanding. Two new books—The New Math: A Political History, by Christopher J.Phillips, and Critical Math ematicsEducation: Theory, Praxis, and Reality, edited by Paul Ernest, Bharath Sriraman, and Nuala Ernest—bring these distinctions to the fore.

People were long happy with the old way and for the happiest of reasons. It worked. Nearly every child eligible to attend school could be made to learn, or at least to memorize, that 8 x 7 = 56 and that triangles encompassed half as many degrees as circles and what simple consequences flowed from these facts. Not every child could advance beyond these basics, but few thought that all should.

That attitude began to change mid-twentieth century, a time in which greater proportions of children were enrolling in all levels of schooling.Because of the Cold War and the impression that America was falling behind, the concern was that kids weren’t learning enough and that they needed to be better thinkers. “New Math” was the result.

In The New Math: A Political History, Carnegie Mellon University assistant professor of history Christopher J. Phillips tells of its rise and fall, centering the tale on the School Mathematics Study Group (SMSG), an entity created in earnest by government money during the Sputnik era: “Although originally funded to work on textbooks for the ‘college capable’ students in secondary schools, SMSG gradually expanded its operation, producing textbooks for every grade and type of student, including material for elementary schools, ‘culturally disadvantaged’ children,” etc. The ascension of New Math was thus partly due to routine mission creep found in well-funded bureaucracies.

At its onset, parents were more or less happy with the status quo. But education theorists and others were not. Students taught in the old way could cipher to the rule of three, but they didn’t know the why behind the how: “[O]ne generally accepted axiom was that math textbooks’ and teachers’ traditional reliance on memorization and regurgitation gave students a misleading sense of what mathematicians do and what mathematics was about.”

Yet is it really of interest what professional mathematicians do? Filling out grant requests, for instance? At any rate, what mathematics is about is something argued over by mathematicians themselves. This was true during the time of New Math…

Click the above link to get the PDF and read the rest (I think).


  1. 1. The link works.
    2. Part of the problem with learning math is that the subject is taught by people who are intuitively adept at handling its logic and, most importantly, its symbolism. For the non-adept student, math is a foreign language with an unfamiliar grammar. The adept have a major inability to understand the bafflement of the non-adept. The non-adept soon recognize they lack something their teachers have and react to this deficiency emotionally as personal failure rather than as opportunity to learn. They soon hate math. Yet it’s the way math is taught that should be hated. The book review illustrates how math educators have been blind to the typical student’s predicament. I would argue that the skill every good teacher (at least at introductory levels) should master is to see the subject with new eyes as a stranger in a foreign land. Then as tour guide, take the student from rote memorization through seeing and appreciating the culture to eventually speaking the language, adequately if not fluently.

  2. Back when the dinosaurs were young, I was deeply involved in ‘School Reform’. One of my efforts back then was an 87-page monograph in which I took a look at the different approaches to mathematics education in various countries, and the relevant research in cognitive psychology.

    After reading Matt’s piece, the first thing that strikes me is all parties’ obliviousness to whether anybody’s model of how mathematical education should proceed actually works. There’s a lot of airy-fairy theorizing, setting up implicit (if unconscious) models of mathematics, schooling, how the brain works, how children’s brains work, a lot of setting up ideal models that are just ‘obviously’ better, and not a lot of defining model skill well enough to throw out models that have less skill. Even ‘back to basics’ people just assume that (for example) 35 long-division problems a night for 2 weeks in the fourth grade ‘works’. Compared to what? And what about long-term?

    Another thing that strikes me immediately is the studied provincialism of American thinking about math education. It doesn’t occur to people even to look at teachers and textbooks in China, France, Korea, Germany, and Japan, for example, and see what might be gleaned from them.

    Now, I haven’t read any of the authors mentioned in Matt’s piece. But I can say that I can find absolutely nothing in Matt’s review of those authors that indicated the slightest knowledge of, or even interest in, what researchers in cognitive psychology have learned about what goes on in children’s brains as they learn math. Sadly, that field has become increasingly converged in recent decades, so I’ve learned not to trust too much coming out of it without a darned good reason.

    But what we used to know, at least, was that Sympathetic Magic doesn’t work. That is, you don’t produce inventive, creative mathematicians using inventive, creative initial pedagogy, and you don’t produce boring, rote-bound mathematicians using an initial pedagogy that is heavy on memorization and other rote procedures. (Sympathetic Magic has been the central conceit of every mathematics ‘reform’ that I know of).

    To the contrary, children love rules, and they instinctively memorize things every day. Even more importantly, while Richard Feynmann might have had his fill of 35 long-division problems a night, the stuff he memorized in grammar school still served him as the fundament of all his later mathematics education.

    No doubt, there are better ways to teach ‘figurin’. Examining the textbooks, and teacher pedagogy, in other countries, would help, as well as taking a look at the Saxon Math texts in the US.

  3. “the New math would promote “mental discipline”.” Must be another 1984 term “mental discipline”. Until the insanity of double speak, discipline meant rigid learning and behavior (rote learning, much of it). I guess math isn’t the only thing we “advanced” fools in America have completely and utterly destroyed, along with ourselves.

    Not understanding how math works is why climate change preachers don’t have a clue why their statistics and mathematics are so far off base. They don’t have any understanding of what they are doing—the computer program does it and they just adjust the variables till they get the desired answer. If questioned, they scream you know nothing of math and hope you don’t post a calculus equation for them to solve, especially one they can’t find an answer to on the internet. If electricity ends, America becomes a land of illiterate dolts.

    Bach and other great pianists lived through the “torturous” time where rote learning was the only way. They wrote incredible music. Funny that can’t be done now….We must be more stupid and unlearned than the middle ages were. We have devolved.

    I worked for a credit counseling agency that was bothered by the exactness of math, so went with creative accounting. They are bankrupt and out of business. I learned all I needed to know about “creative accounting” through their failure.

    I was just running through what math illiteracy gives the USA: Insurance companies that “don’t raise your rates for your first accident” (they raise EVERYONE’s rate and you pay through the nose for everyone’s accidents), climate change believers, credit card debt (it’s not really money, right?), kids who say “I don’t need to do addition, etc, my calculator does (asking who will program the calculators gets a “deer in the headlights” look from the speaker). We are truly a math illiterate country full of fools.

  4. The underlying assumption is that those who teach math are unusually smart or passionate about math is pretty far from the reality. It may be assumed that if math lovers taught math, they would do a better job teaching than those who have a bare grasp of the subject. It is fairly common for math teachers to confide or boast to her students about her poor math grades in college. This is troubling, because the message isn’t that hard work is essential to success. The message is that any loser can teach math, which hardly inspires the thirst for learning in her students.

  5. I found the study that had been in the back of my mind while writing my previous comment. [Geary DC, Hamson CO, Chen G, Liu F, Hoard MK (1996). A biocultural model of academic development. in Paris S, Wellman H, eds. (1996). Global prospects for education: development, culture and schooling: a festschrift to honor Harold W. Stevenson. Washington, DC: American Psychological Association.]

    Here’s my summary of the results. I’ve bolded a few things for those just interested in the take-aways.

    American high-school seniors currently enrolled in their fourth year of high school mathematics at a high school which received a National Excellence in Education Blue Ribbon Award the year the study was conducted, were tested on their ability to do quickly and accurately the kinds of basic arithmetic operations known to be related to employability, wages, and productivity.

    Ninety-eight percent of college-bound Chinese high school seniors in the same study made a better score than the U.S. students’ average score; in other words, the average elite U.S. high school senior would rank at the second percentile in arithmetic computation in a Chinese high school class of college-bound seniors. Nor did the American students counterbalance their relatively poor computational ability with greater reasoning skills, as American mathematics education experts sometimes contend. These were merely slightly less bad than their computational skills, and nowhere near the abilities of their Chinese peers.

    The researchers also studied the computational skills of sixty- to eighty-year-old American and Chinese adults, people who could have been the grandparents or the great-grandparents of the American and Chinese high school seniors. Despite the fact that these computational abilities are known to decline noticeably with age, the older Americans were able to do operations such as addition with carrying and subtraction with borrowing just as well as the older Chinese adults, while both older groups performed as well as present-day American college-bound high school seniors decades younger than they, strongly suggesting that American schooling in these fundamental operations used to be as effective as Chinese schooling but has become markedly less effective over the last decades.

  6. The history of the Late Modern Age is one of replacing stuff that works with stuff that sounds good.

    Teaching reasoning sounds good; but students must first have something to reason about. As it stands now, students graduate knowing neither facts nor reason. Who has the time to reason their way to 5×3=15? That’s like a pianist reasoning their way to a chord progression when their muscle memory should already have taken them there. Ditto, a jet fighter pilot who has practiced maneuvers until he has them by rote precisely because when he needs to call upon them he has not the time to derive them from first principles.

    Once you have laid down this bedrock of facts, you can if you are interested delve into the why and wherefore; but most people have neither the time, need, nor interest in doing so.

    Back in the 70s — This was before Disco! — I was a graduate assistant at Marquette and had an office with two other TAs on the second floor of what was then the Math building and is now not there at all. They were building something called an “interstate highway” at the end of the block. Three other TAs had the office next to us. The rest of the math department was on the lofty third floor, where they needn’t associate with us peasants. The remainder of the 2nd floor was given over to something called the “Education Department.”

    One day, we math grads fell into discussion with one of the Education professors regarding the New Math, by which little kids were taught abstract set theory. (Naturally, to us it was not New at all; but to Ed Profs it was as a shiny city on a hill.) At one point we objected that not even the teachers understood such abstractions, let alone the kiddie-poos. “The teacher does not need to understand the material,” the professor of education informed us. “He only needs to understand how to teach the material.”

    To this day, I sometimes amuse myself by displaying my superpower in checkout lanes by telling the clerk how much change I will get before s/he enters the amount in the machine. They stare in befuddled bemusement when the machine tells them the same result I gave them a priori. Cheap thrills, I know; but I takes em as I finds em.

  7. YOS:
    When were you at Marquette? I was there 67-72. Talakos (sp?) got me interested in Stats and suggested UNC-Bios for grad school. Markowski(sp?) taught probability theory at the time.


  8. 69-71. I studied general topology under J. Douglas Harris. Madukhar Deshpande was abstract algebra. Bronkowski was chairman. Eduard Cech was there, too, and helped me get my master’s thesis published. I assisted on an NSF grant for him one summer.

    I taught a few study sections of freshman calculus — once with a hangover the size of a bottle of ouzo, proving it is possible to do calculus in your sleep — but I gather you were past that while I was there.

  9. The last batch of posts on Math Education (especially the SJW ones) brings to mind the 1988 movie “Stand and Deliver” [Edward James Olmos portrays Jaime Escalante]

    I think the SJW’s would have kittens over that movie.

  10. “The theory of structuration is a social theory of the creation and reproduction of social systems that is based in the analysis of both structure and agents (see structure and agency), without giving primacy to either.” (from Wikipedia, for what that’s worth)

    – “structuration” is a term used, relative to some vaguely referenced issue, in the full essay to which a link is provided above. Accepting the above definition leads to the conclusion the intellectual that used “structuration” didn’t seem to be saying anything, but strived to say it in a lot of syllables.

    Which brings us to Eric Hoffer’s observations about intellectuals:

    “There is not an idea that cannot be expressed in 200 words. But the writer must know precisely what he wants to say. If you have nothing to say and want badly to say it, then all the words in all the dictionaries will not suffice.”

    “One of the surprising privileges of intellectuals is that they are free to be scandalously asinine without harming their reputations.”

    “Those who see their lives as spoiled and wasted crave equality and fraternity more than they do freedom. If they clamor for freedom, it is but freedom to establish equality and uniformity. The passion for equality is partly a passion for anonymity: To be one thread of the many which make up a tunic; one thread not distinguishable from the others. No one can then point us out, measure us against others and expose our inferiority.”

    “The awareness of their individual blemishes and shortcomings inclines the frustrated to detect ill will and meanness in their fellow men.”

    “It has often been said that power corrupts. But it is perhaps equally important to realize that weakness, too, corrupts. Power corrupts the few, while weakness corrupts the many. Hatred, malice, rudeness, intolerance, and suspicion are the faults of weakness. The resentment of the weak does not spring from any injustice done to them but from the sense of inadequacy and impotence. They hate not wickedness but weakness. When it is their power to do so, the weak destroy weakness wherever they see it.”

    “To most of us nothing is so invisible as an unpleasant truth. Though it is held before our eyes, pushed under our noses, rammed down our throats — we know it not.”

    “The weakness of a soul is proportionate to the number of truths that must be kept from it.”

  11. As a point of reference, medical school is pretty much pure rote memorization, as far as I can tell. An incredibly astonishingly vast amount of memorization. Which is one reason I’m an engineer and not a doctor, as I have a poor memory.

    When I was taught a subject via rote methods, I had to try to find some underlying concepts to tie things together in my mind, or I would do poorly on the test. At the time I considered this a form of cheating, and felt a little guilty about doing it.

  12. YOS – as far as cheap thrills in the checkout line, I find it entertaining to give the kid in the drive-thru window, e.g., $22.15 for a $16.65 total (since I don’t like ones filling up my wallet, and I prefer quarters to dimes and nickels in the change cup as they are easier for me to grab with arthritic fingers). The kid then eyes me suspiciously, but proceeds with the transaction when I don’t blink, and looks surprised when the change total pops up. Then my wife scowls at me and punches me in the shoulder. Rinse and repeat.

  13. YOS,
    Yes, I started as a math-bio dual major, then shifted to pure bio (pre-med) in 69. 1968 was a lost year. In ‘70 I decided for grad school and decided to pick up a math minor (for systems biology and to boost my grade point). Had Campbell for Advanced Calc and Talakos/Markowski. Statistics for math majors was far more interesting than stats for biologists. Talakos encouraged me and it was off to Chapel Hill.
    We must have crossed paths in the math building. Bronkowski was a neighbor of my distant cousin, who was there at the same time.

    How does a topologist end up in Industrial Statistics? Doug Bates at Madison was a differential geometer before he switched to non-linear models under Watts, but that is fairly close. That would be a story in itself….

  14. How does a topologist end up in Industrial Statistics?

    The short answer is “better pay.”

    A somewhat longer answer is that confidence intervals can define a metric on the space of parameters, defining “close” as “within the interval.” I had been working with Harris on proximity spaces and I thought I could make this work as a proximity, but I never quite made it jell before the need to secure an income trumped everything else.

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