He waits until the end to tell us what “number sense” means–which in his case means “conceptual understanding” as opposed to procedural fluency. As if he has heard the arguments before (and he has), he takes a pre-emptive strike against potential criticism from people like me by paying lip service to the need for learning procedures and then goes into his case for “understanding”.

*Make no mistake about it, acquiring that modern-day mathematical skillset definitely requires spending time carrying out the various procedures. Your child or children will still spend time “doing math” in the way you remember. But whereas the focus used to be on mastering the skills with the goal of carrying out the procedures accurately — something that, thanks to the learning capacity of the human brain, could be achieved without deep, conceptual understanding — the focus today is on that conceptual understanding. That is a very different goal, and quite frankly a much more difficult one to reach.*

*This, by the way, is the new state of affairs that the mathematics Common Core was created to address. Outsiders, including politicians in search of populist issues to incite voters and others with an axe to grind, have derided caricatured portrayals of this important new educational goal, by describing it as “woolly” and “fuzzy”. But I disposed of that uninformed red herring already. The fact is, number sense is (rightly, and importantly) the primary focus of 21st Century K-12 mathematics education that millions of children around the world are receiving today. Children who are not getting such an education are going to be severely handicapped in the world they are being educated to inhabit.*

While he may have disposed of the “red herring” of the educational goal of understanding, I’d like to revisit it. As I’ve written about in numerous articles (thus disposing of the “red herring” of traditional math done poorly being the definition of traditional math), a glance at textbooks from the past shows that in fact the conceptual underpinning of various procedures *has* in fact been included and was a part of math education. Granted, the explanation of why the invert and multiply rule for fractional division works was not provided. I knew how to use it as well as what fractional division is and how to solve word problems with it, but as I said in my last missive, I didn’t learn how the rule is derived until 10 years ago. Nevertheless, I did manage to major in math despite the omission of something that brings gasps of astonishment and “tsk tsks” from the education establishment–and largely from people who themselves have benefitted from traditional modes of math education.

As Prof. Devlin asserts, in many US classrooms today,students must demonstrate an “understanding” of computational procedures. How this has played out in the last 25+ years, and now even more so under Common Core, is that before they are allowed to use standard algorithms, they must learn alternative procedures to provide the conceptual underpinning in the belief that teaching the standard algorithms first obscures what Devlin calls “number sense” or “understanding”. The teaching of mathematics has thus been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations.

In so doing, students are forced to show what passes for understanding at every point of even the simplest computations. Instead, they should be learning procedures and working effectively with sufficient procedural understanding. But this “stop and explain” approach to understanding undermines what the reformers want to achieve in the first place. It is “rote understanding”: an out-loud articulation of meaning in every stage that is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill.

The approaches to math teaching in the lower grades in schools is a product of many years of mischaracterizing and maligning traditional teaching methods. The math reform movement touts many poster children of math education. Their views and philosophies are taken as faith by school administrations, school districts and many teachers – teachers who have been indoctrinated in schools of education that teach these methods.

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. While students undergoing instruction under the prevalent interpretations of Common Core may be able to recite an explanation they have been told–um, that is, “discovered” with some “guide on the side” guidance from the teacher on how best to phrase such explanation–that is not the test of effectiveness of a math program. In many cases, what passes for “understanding” or “number sense” is “rote understanding”. In so doing, the math reformers have unwittingly created a body of students in which “understanding” foundational math is not even “doing” math.