Undoing the ‘Rote Understanding’ Approach to Common Core Math Standards

by Barry Garelick

A video about how the Common Core is teaching young students how to do addition problems is making the rounds on the internet: http://rare.us/story/watch-common-core-take-56-seconds-to-solve-96/

Much ballyhoo is being made of this. Given the prevailing interpretation of Common Core math standards, the furor is understandable. The purveyors of these standards claim that they neither dictate nor prohibit any pedagogical approach, but the wave of videos and articles sweeping the internet like the one above suggest the opposite may be true: that, in fact, the Common Core math standards are dictating how teachers are to teach math.

The method of "making ten" is not unique to Common Core. It is how it is implemented that's the problem. The method is embedded (and explained by way of example) in Standard 1.OA.C.6: 

"Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as… making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9)…."

The "making ten" method is included in the math program used in Singapore—a nation whose fourth and eighth graders have consistently obtained the highest scores in international math tests. Specifically, in Singapore's Primary Math textbook for first grade, the procedure for adding by "making tens" is explained. Of particular importance, however, is that the procedure is not the only one used, nor are first graders forced to use it. This may be because many first graders likely come to learn that 8 + 6 equals 14 through memorization, without having to repeatedly compose and decompose numbers in order to achieve the "deep understanding" of addition and subtraction that standards-writers—and the interpreters of same—feel is necessary for six-year-olds.

"Making tens" is not limited to Singapore's math textbooks, nor is it by any means a new strategy. It has been used for years, as it was in my third-grade arithmetic textbook, written in 1955 as shown in the figure below:

(Brownell, et. al., 1955)

The book did not insist on this method; it was introduced as a possible help. Students were required to use it in one set of exercises in my old textbook. Then it was up to the student whether to use it or not. As such, it served more as a side dish than the main dish it is turning out to be; in some cases, students discovered the method on their own. (I have written about this and other methods in a series on common sense approaches to the Common Core math standards: see here and here.)

What has been used as a help in older textbooks and in Singapore, is turning out to be a hindrance in the U.S. under the current interpretations of Common Core. Insisting on calculations based on the "making tens" and other approaches are in my opinion not likely to prove useful for all first graders. Teachers should be free to differentiate instruction so that those students who are able to use these strategies can achieve those goals. It is unrealistic and potentially destructive to interpret the Common Core math standards as requiring that all first grade students use these strategies in the name of "understanding". That should be the real objection voiced to demonstrations of this method under Common Core—not the method itself.

The mantras of "students shall understand" and "explain" are what Tom Loveless of the Brookings Institution calls the dog whistles of Common Core that are picked up on and responded to by the math reform movement. In my opinion, while not dictating particular teaching styles, the CC math standards have given the math reform movement that has been raging for slightly more than two decades in the United States a massive dose of steroids. Reform math has manifested itself in classrooms across the United States mostly in lower grades, in the form of "discovery-oriented" and "student-centered" classes, in which the teacher becomes a facilitator or "guide on the side" rather than the "sage on the stage" and students work so-called "real world" or "authentic problems." It also has taken the form of de-emphasizing practices and drills, requiring oral or written "explanations" of problems so obvious they need none, finding more than one way to do a problem, and using cumbersome strategies for basic arithmetic functions.

The reform math ideology is in fact encouraged in the Common Core math standards in some suble and not-so-subtle ways. In particular, CC's own documentation of the standards states that understanding is a major shift in how math should be taught:

"The standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures."

Such philosophy plays into the math reformers' unwavering beliefs that requiring students to master practices such as "making tens" will result in students understanding how numbers work—as opposed to just "doing" math. In fact, reformers tend to mischaracterize traditionally taught math as teaching only the "doing" and not the understanding; that it is rote memorization of facts and procedures and that students do not learn how to think or problem solve.

Because CC emphasizes understanding rather than just doing, it has become the way to teach math and has become synonymous with Common Core. Schools and administrators may resist an approach that does not require students to master a supplemental approach to help add numbers. This is because standardized tests—the mechanism of accountability for many in education—may require reform math approaches to math problems. The fear is that students will do poorly on such tests because they will not know how to write explanations that demonstrate the so-called understanding. But such thinking confuses cause and effect. Forcing students to think of multiple ways to solve a problem, or using "making tens" as a method to explain why, for example, 9 + 6 equals 15, does not in and of itself demonstrate understanding. Those who believe it does seem to be saying: "If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians." It is an investment in the wrong thing at the wrong time. The "explanations" most often will have little mathematical value and are on a naïve level since students don't know the subject matter well enough. The result is at best a demonstration of "rote understanding."

Interestingly, the nations that teach math in the traditional fashion seem to do quite well on tests like PISA, the international exam that is essentially constructed along reform math principles. Perhaps this is because basic foundational skills enable more thinking than a conglomeration of rote understandings.


Brownell, W.A., Guy T. Buswell, I. Saubel. "Arithmetic We Need; Grade 3"; Ginn and Company. 1955

09 15, 2014
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