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.

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