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From The Faulty Logic of the ‘Math Wars’ by Alice Crary and W. Stephen Wilson.

At stake in the math wars is the value of a “reform” strategy for teaching math that, over the past 25 years, has taken American schools by storm. Today the emphasis of most math instruction is on — to use the new lingo — numerical reasoning. This is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms.

A mathematical algorithm is a procedure for performing a computation. At the heart of the discipline of mathematics is a set of the most efficient — and most elegant and powerful — algorithms for specific operations. The most efficient algorithm for addition, for instance, involves stacking numbers to be added with their place values aligned, successively adding single digits beginning with the ones place column, and “carrying” any extra place values leftward.

What is striking about reform math is that the standard algorithms are either de-emphasized to students or withheld from them entirely. In one widely used and very representative math program — TERC Investigations — second grade students are repeatedly given specific addition problems and asked to explore a variety of procedures for arriving at a solution. The standard algorithm is absent from the procedures they are offered. Students in this program don’t encounter the standard algorithm until fourth grade, and even then they are not asked to regard it as a privileged method.

I accept Crary and Wilson's argument, but what I find interesting is the divorce from reality that it seems to represent, though it is quite possible that they are playing a long game to a very particular audience.

To me, this issue of numeracy and maths preparation is a critical one in a modern, complex world. Mathematical competency is a necessary tool for a free mind and those incapable of basic quantification and calculation are ever on the back foot. I have been distressed to witness my three children progress through school without what I see as fairly rudimentary mathematical capabilities/concepts. In our case, the shortfall is (I hope) made up for by home instruction, but what about in homes not so liberally bestrewed with degrees in finance and economics?

Crary and Wilson describe the math wars as a conflict between those who wish to emphasize abstract and unstructured reasoning versus those who are insistent that there is a body of factual knowledge necessary before effective reasoning can take place and that the omission of that body of knowledge is a crippling blow to the intellectual preparation of a student. I am firmly with the latter over the former.

What is striking to me in the Crary and Wilson essay is that the entire point of their argument is not to establish whether there is a basis for believing one approach to produce better desired results than another but rather, they are arguing which party is to be allowed to wear the mantle of "progressive".

The closest Crary and Wilson come to an empirical assessment of the outcome of these two views of how to teach mathematics is

Reform math has some serious detractors. It comes under fierce attack from college teachers of mathematics, for instance, who argue that it fails to prepare students for studies in STEM (science, technology, engineering and math) fields. These professors maintain that college-level work requires ready and effortless competence with the standard algorithms and that the student who needs to ponder fractions — or is dependent on a calculator — is simply not prepared for college math. They express outrage and bafflement that so much American math education policy is set by people with no special knowledge of the discipline.

Fair enough. But surely there is some empirical, objective measure that could be used to identify the better approach? Not a peep.

So what this seems to come down to is not who best prepares children to use the tools and rigor of numeracy and mathematical reasoning but rather, who gets to strut under the banner of the term "Progressive". Oof! I wish I were misreading their argument but I fear not. And if I am right, then it explains much that is wrong with education today.