What math should be taught?

The Politics of Math Education
New York Times Op-Ed, 2015-12-03

Pittsburgh —

AMERICAN children have been bad at math for well over a century now. As early as 1895, educational reformers lamented Americans’ “meager results” in the subject. Over the years, critics of math education in this country have cycled through a set of familiar culprits, blaming inadequate teacher training, lackluster student motivation and faulty curricular design. Today’s debates over the Common Core mathematical standards are just the latest iteration of this dispute.

Although these issues are important — no reform can ever succeed without considering teacher training and textbook design — resolving them will never make the underlying question of how to teach math “go away.” This is because debates about learning mathematics are debates about how educated citizens should think generally. Whether it is taught as a collection of facts, as a set of problem-solving heuristics or as a model of logical deduction, learning math counts as learning to reason. That is, in effect, a political matter, and therefore inherently contestable. Reasonable people can and will disagree about it.

Perhaps no reform has illustrated this point as clearly as the wide range of mid-20th-century curricular changes known as the new math. Many of these reforms promised that the introduction of sets, nondecimal bases and formal definitions would lead students to think of math as more than just a bunch of dusty facts: It was a powerful and rigorous way of approaching complex problems.

The new math was widely praised at first as a model bipartisan reform effort. It was developed in the 1950s as part of the “Cold War of the classrooms,” and the resulting textbooks were most widely disseminated in the 1960s, with liberals and academic elites promoting it as a central component of education for the modern world. The United States Chamber of Commerce and political conservatives also praised federal support of curriculum reforms like the new math, in part because these reforms were led by mathematicians, not so-called progressive educators.

By the 1970s, however, conservative critics claimed the reforms had replaced rigorous mathematics with useless abstractions, a curriculum of “frills,” in the words of Congressman John M. Ashbrook, Republican of Ohio. States quickly beat a retreat from new math in the mid-1970s and though the material never totally disappeared from the curriculum, by the end of the decade the label “new math” had become toxic to many publishers and districts.

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 was a claim both about who controlled intellectual training and about what forms of mental discipline should be promoted. The idea that the complex problems students would face required training in the flexible, creative mathematics of elite practitioners was replaced by claims that modern students needed grounding in memorization, militaristic discipline and rapid recall of arithmetic facts.

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.

We need to get away from the idea that math education is only a matter of selecting the right textbook and finding good teachers (though of course those remain very important). The new math’s reception was fundamentally shaped by Americans’ trust in federal initiatives and elite experts, their demands for local control and their beliefs about the skills citizens needed to face the problems of the modern world. Today these same political concerns will ultimately determine the future of the Common Core.

As long as learning math counts as learning to think, the fortunes of any math curriculum will almost certainly be closely tied to claims about what constitutes rigorous thought — and who gets to decide.

[I (the author of this blog) went to high school in the early 1960s,
majored in math in college in the mid 1960s,
and studied math as a graduate student in the late 1960s and early 1970s.
I was steeped in the ideas of the "new math".
I thought it was an excellent approach.
My only complaint, which may reflect my own shortcomings,
was that what was taught often seemed too abstract,
without examples.
For example,
somehow I lacked (maybe it was taught but I ignored it)
a thorough grounding in the group of isometries of the Euclidean plane (and 3-space),
showing the relations between reflections, rotations, translations, and glide reflections
and the subgroups that they generate.
Such would provide an ideal (in my opinion) introduction to group theory,
together with enhancing ones understanding of geometry.
See, for example, the Springer UTM texts
Martin, George E. (1982). Transformation Geometry: An Introduction to Symmetry. ISBN 978-0-387-90636-2.
Armstrong, M. A. (1988). Groups and Symmetry. ISBN 978-0-387-96675-5.
This subject shows how the ideas of the "new math"
(in this case, group theory which was introduced in the 19th century)
can be used to analyze Platonic solids,

which surely are among the most classical ideas in mathematics.
An interesting, in my opinion, combination.]

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