The “More than one way to teach math ” gambit

A Dec 3, 2016 op-ed by Christopher Phillips in the New York Times, talks about how Americans have been bad at math since 1895.  This in itself is an interesting claim, given that for a time period from the 40’s through the mid 60’s, math scores on the Iowa Tests of Basic Skills made a steady climb in the states of Iowa, Indiana and Minnesota.  See here for further information. It’s also interesting from the standpoint that a glance at older textbooks from the 1890’s into the 1940’s and 50’s, shows the types of math questions that students were expected to master, which an alarming number of students today have difficulty doing. Furthermore, going back to the 1980’s, many of the students in first year algebra classes had fairly good mastery of fractions, decimals, percents and general computation essentials. In today’s first year algebra classes it is not unusual to see alarming deficits in such knowledge and skills.

Interestingly, the drop in scores around the mid 60’s coincided with the student cohort population that was getting the brunt of what is termed the 60’s new math, which Phillips’ op-ed also focuses on.  He states:

Though critics of the new math often used reports of declining test scores to justify their stance, studies routinely showed mixed test score trends. What had really changed were attitudes toward elite knowledge, as well as levels of trust in federal initiatives that reached into traditionally local domains. That is, the politics had changed.

Whereas many conservatives in 1958 felt that the sensible thing to do was to put elite academic mathematicians in charge of the school curriculum, by 1978 the conservative thing to do was to restore the math curriculum to local control and emphasize tradition — to go “back to basics.”

This is fairly accurate, though there are some things that he neglects to say which are worth mentioning.  One is that the new-math era was one of the only times that mathematicians were given an opportunity to make proper math education available to the masses. The difficulty with the program (specifically in the lower grades) was due in large part to its formal approach which was steeped in set theory and logic and which placed less emphasis on basic computational skills and procedures. The general public, the education community, and even mathematicians themselves judged the new-math programs a failure.

The second and not insignificant issue is that mathematicians were assigned the blame, and the education establishment took back the reins. That establishment received an inadvertent boost in 1983 with the publication of A Nation at Risk, the shockingly pessimistic assessment of the nation’s schools by the National Commission on Excellence in Education. The report sounded another alarm about student math performance, and the National Council of Teachers of Mathematics (NCTM), an organization that became increasingly dominated by educationists during the 1970’s and 80’s, took advantage of this new education crisis to write revised math standards. The Curriculum and Evaluation Standards for School Mathematics, published in 1989, purported to put the country back on the math track. But because it was, in part, a reaction to the new math and those believed responsible for it, NCTM did not promote a lively public debate, as had the creators of the new math, but suppressed it.

Phillips concludes that “The fate of the new math suggests that much of today’s debate about the Common Core’s mathematics reforms may be misplaced. Both proponents and critics of the Common Core’s promise to promote “adaptive reasoning” alongside “procedural fluency” are engaged in this long tradition of disagreements about the math curriculum. These controversies are unlikely to be resolved, because there’s not one right approach to how we should train students to think.”

While the controversies he mentions have been going on for a long time, so also have there been people who learn math via the methods held in disdain by math reformers. Such methods include memorization, as if memorizing is tantamount to “rote learning”. To reformers, “fluently deriving” the math facts (such as 7 x 4 = 7 x 2 x 2) is superior to simply memorizing because it includes “understanding”–the holy grail of math reformers. In all of these polemical debates that have been circulating in the press and in the literature over the years, there is a dearth of studies to see how people who are successful at math have learned it. The methods used by tutors and learning centers seem to focus on those methods said not to work–and the success of the students who use these techniques are appropriated by the schools and school districts who claim that their reform-based programs have worked.

There may not be just one right approach but I tend to disagree with his framing of the argument–in particular his claim that teaching math is the same as teaching students to think.  He also neglects to mention studies that follow established research practices that do show some practices to be effective and some to be ineffective. Shall we continue to rely on statements such as “research shows”, in which the research is flimsy, questionable, and unscientific, with newspapers taking so-called experts’ word for it? Or should we start relying on a growing body of verifiable and peer reviewed research that is seeing more reliance on cognitive science? (See, for example, this report by Anna Stokke, a mathematics professor at University of Winnipeg.)

Phillips is correct that the pathways selected have often been political. But leaving it that “there’s no right way” is not going to solve very much, when in fact there are ways that have proven to be more effective than others.  And some ways that are not effective at all.


7 thoughts on “The “More than one way to teach math ” gambit

  1. Step one – look at the author. He fits a classic category of critical (or lack thereof) analysis – it’s all about me.

    Christopher J. Phillips teaches history at Carnegie Mellon University and is the author of “The New Math: A Political History.”

    Forget the fact that he is pushing a book, the key is that he sees only what he wants to see. Reality is adapted to fit his thinking and content knowledge. Psychologists think its all about their area, like how the brain works. He then makes this miraculous statement:

    “These controversies are unlikely to be resolved, because there’s not one right approach to how we should train students to think.”

    Um. This is math, not a course in learning to think. Kids can’t even do the simple basics no matter which magical inductive engaging process is used. An historical perspective can’t help if you have no idea of what you’re talking about. It apparently can’t be just a really dumb-ass reason like K-6 math sets very low expectations and covers it up with a huge load of top-down thinking and understanding crap. These people really, really, really have to ask the parents of the best math students what they did to enforce proper mastery at home. Ask us whether we worried about training our kids specifically to think as if that, and not mastery of skills, is the path to success. We don’t have any issue with knowing what works and what doesn’t. Ask us. It’s not difficult. Don’t just filter reality through your own background and bank account. When you want to know what works, look at the success cases. Duh!

    Also, as a child of the 50’s and 60’s, I had a lot of expectations (including the threat of summer school or staying back a year) and a lot of mastery of basic skills. In 6th grade, we had interpolation contests because that was pre-calculator days and we had to be able to estimate a middle value between tables of numbers separated by 10 units. We knew how to manipulate fractions, and amazingly (!), we even mathematically understood invert and multiply. This was done with no help from parents at home. Now, we parents are expected to go to their math open houses and we get notes telling us to practice math facts at home. To us parents of the best math students, there is no confusion here. There is a systemic problem in K-6 and parents have to do their work at home. I really should have kept a notebook of help I gave my son – work that completely ended when he got to proper textbooks and the AP Calculus math sequence in high school. Success is based on content knowledge, a proper textbook, and enforcement of basic skills from the bottom up. Learning to think will take care of itself. The biggest historical change I’ve seen is the change from higher expectations to full inclusion, differentiated IQ-based learning. How can you increase the range of abilities in one classroom and expect to do a better job? You do it with a lot of blather about engagement and top-down thinking and understanding while demonizing “mere facts” and “superficial knowledge.” I’ve had teachers tell me these things straight to my face. This is a blatant, fundamental, and systemic flaw that is incredibly obvious.


  2. There is also no confusion by most high school AP Calculus track teachers who use traditional textbooks and are driven by the need to ensure student success and success on the AP tests. The fuzziness is ONLY driven by those with no math content knowledge and skills. We STEM parents have no issue at all. Even though I wanted something more for my son in math than what I had with my traditional math schooling, the MathLand and Everyday Math curricula that I saw went completely in the wrong direction. Full inclusion and social promotion (low expectations) are fundamentally flawed. Tracking is hidden at home and the academic gap is increased. Their only solution it to claim that their “trust the spiral” process works by definition and to blame IQ, parents, peers, and society. Oh yes, and now conservatives even though the anti-fuzziness crowd crosses the complete political spectrum. Many, however, see only what they want to see.


  3. I quibble about Phillips’ use of the term “conservative”. Like “liberal” today, this political label is so elastic it can mean many things. And as used in his essay it does *not* mean what it generally means in today’s political discourse in North America (which does not correspond to the way the term is used in Europe and also differs markedly from the general use of the term in the phrase “conservative cleric”.) The same is true of “liberal” — there is a tendency for those labelled “liberals” in today’s discourse to want to police language, attitudes and certain types of behaviours and make certain conventionally standard points of view either beyond the pale and so socially illegitimate or literally illegal. That is, today’s liberals are not advocates of liberty in everyday human affairs; today’s “conservatives” are in close convergence with “libertarians”, which emphasize that particular aspect of classical liberalism.

    In the essay, Phillips tosses out the term in a sentence that seems to imply that a certain tribe or even people within that tribe once held to one position and today hold to a differing position. I think that’s a misuse of the label — if one uses “conservative” to mean “that which leans toward preserving the status quo” then this is perhaps arguable — but the term is not used that way nowadays. And if the term is used in this sense, there is no sense in which these are the same people or even the same tribe.

    Think of how absurd that would be in political terms. In a communist dictatorship, who would be a “conservative”? Obviously, if it means preserving the status quo, that would be a hard-left communist ideologue. In a completely open free market, who would be a “conservative”? An advocate of the free market and opponent of centralized economic measures. Are these in any meaningful sense “the same tribe”? I think not.

    Now in terms of Phillips’ statement, if professional mathematicians are in charge of the curriculum, then it would be “conservative” (in that sense) to want them to remain in charge. If they are shut out of doing so, then it would be “conservative” not to want them to remain in charge. This is not a matter of the conservative “tribe” wandering all over the map to contradictory positions — it merely means that the label is slapped on a different group of people with a different view, due to a changing context.

    But finally, I think it’s evident that there is a false dichotomy in the two choices he gives, leaving out the third alternative — which happens to be the status quo in North America today: namely that NON-mathematicians have central control over curriculum, and professional mathematicians and grassroots, “local” interests such as parents and many teachers, are united (largely) against this central force. Thus his analysis (as cited here) does not even describe the current dispute.

    Perhaps I’m not old enough to know different, but in my adult life I’ve never seen any great dispute between professional mathematicians and the general, “local” public regarding what should be the content of the curriculum. You’d likely have to go back to the introduction of the New Math in the 50s/60s for that, and even then I would argue this represented a minority of mathematicians, who had starry ideals about education. We’ve still got a few of those around …


  4. The goal for many is to push people’s hot buttons to get them on their political bus where they are fed their version of truth. Many hop on because it’s easier than doing their homework on each issue. They own you and you help them push their agenda. Everything is politicized and the focus switches to a battle of sides and personalities – the fundamental flaw of media. They are driven by ratings and hot buttons click-throughs. Phillips sees this history and can’t separate what works from what doesn’t because he doesn’t know math. He thinks that an historical analysis gives him insight into the needs of the content, but he sees only what he wants to see. He really needs to read Feynman’s “Cargo Cult” lecture and his work on selecting math textbooks. Phillips’ hypothesis fails.

    “These controversies are unlikely to be resolved, because there’s not one right approach to how we should train students to think.”

    This statement is not even wrong. Some people actually know what works and have been directly involved with making that happen – both teachers and parents. You just have to find and ask them. If you define the problem as learning to “think”, then you can make up anything you want. Top-down conceptual understanding and general (no one right answer) inductive problem solving is just a modern educational pedagogue’s rote justification for low expectations. It allows them to justify full inclusion and the role of teacher as potted plant guide-on-the-side. History alone can’t give Phillips any insight into the content and pedagogy war that’s been going on for decades or why there is a complete change in approach from K-8 to high school. He sounds too much like someone who claimed that my views on math education made me a racist. Life’s tough when you filter reality through a very narrow content or belief tea strainer.


  5. Hi Barry,
    I really enjoy your work.
    Your link for the Anna Stokke paper has ellipses in it and doesn’t connect to the report

    It’s heartening to see the Canadians push back on the “discovery learning” canard.

    I told my uncle about discovery learning back in 2003 – he said it sounded like Professor Harold Hill’s “think method” of learning music in The Music Man!


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