Over the last several years, the press and television have publicized many parents’ frustration with how math is being taught in the lower grades. On the internet, videos abound with examples of how procedures such as addition and subtraction are being taught to students using convoluted methods that are leaving students and parents baffled as to 1) how to do the procedure and 2) angry that the standard methods are delayed. (This video is one of many examples of parent concern over how math is taught under Common Core.)
The current interpretation of Common Core by publishers, instructional coaches, professional development vendors, and other educational entities, maintains that teaching the standard methods (known as standard algorithms) for various procedures too early can eclipse the conceptual underpinning of why the algorithms work, and can lead to students being confused. A video by one instructional coach argues that teaching only procedures 1) has only worked for a small group of students and 2) that the reason students have a hard time with math is “No one taught them to understand the concepts and why we’re doing what we’re doing. We didn’t teach them how to think; we just taught them how to ‘do’ and execute…” The premise stated by this coach and others, contains the usual mischaracterization that procedures were taught in a void without contextual understanding. He also maintains that Common Core’s main focus is on “understanding”. This article explores this notion, and how and why Common Core is interpreted and implemented in the ways we are seeing.
A case in point
A case in point has presented itself in my recent work with a group of fifth graders in need of math remediation at the school where I teach. The students were doing exercises from their textbook on multiplying fractions. Instead of applying the standard method (or algorithm) in which numerators are multiplied by numerators and denominators multiplied by denominators, students first had to draw diagrams for each and every problem.
The diagrams I speak of have been used in many textbooks as a means to motivate the particular procedure for multiplying fractions. Such diagrams use the area of a square as the means to illustrate what multiplication of fractions represents, and why one multiplies numerators and denominators. For example, a problem like 3⁄4 × 2⁄3 is demonstrated by dividing a square into three columns, and shading two of them, thus representing 2/3 of the area of the square. Then the square is divided into four rows, with three of them shaded–this is 3/4 of the area of the square. Where the two shaded areas intersection therefore represents 3/4 of 2/3 of the square. The intersection of the two yields 6 little boxes shaded out of a total of 12 little boxes which is 6/12 or 1/2 of the whole square. This is done as the reasoning—the conceptual understanding—behind multiplying numerators and denominators.
The students see what 3/4 of 2/3 means in this model in terms of area of a square.
Nothing New Under the Sun
This was the explanation used in my old arithmetic book from the 60’s (and in other textbooks from that time and earlier times thus belying the notion that traditionally taught math ignores understanding and focuses only on rote memorization.)
The method used in my old textbook is also the method used in Singapore’s math textbooks. It is an effective demonstration of what fraction multiplication represents and why one multiplies numerators and denominators. In Singapore’s textbooks (as in mine), students are asked to use the area model for, at most, two fraction multiplication problems. Then students are let loose to solve them using the algorithm.
But in the current slew of textbooks claiming alignment with the Common Core, after the initial presentation of the diagram to show what fraction multiplication is, and why and how it works, students are then required to draw these type of diagrams for a set of fraction multiplication problems. The thinking behind having students draw the pictures is supposedly to “drill” the understanding of what is happening with fraction multiplication, before they are then allowed to do it by the algorithmic method.
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.
Such topsy-turvy approaches to math education have been around for more than two decades, but the interpretation and implementation of Common Core have made them more popular. To compensate for what reformers believe is a lack of understanding, the teaching of mathematics has been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations.
What ultimately happens is that these exercises in understanding simply become new procedures, which small children attempt to learn and memorize because that is what many small children do. On top of all that is that these methods are not efficient and very confusing, resulting in frustration and feeding into children’s dislike of math—something this method was supposed to cure.
The Instructional “Shifts” of Common Core: The Source of Much of the Hidden Pedagogy
Where is this interpretation coming from? One possible source are the “shifts” in math instruction that are discussed on the website for Common Core. One of the shifts called for is “rigor” which the website translates as: “Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity”. Further discussion at the website mentions that students should attain fluency with core functions such as multiplication (and by extension, multiplication of fractions): “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.”
I learned of the connection between these “instructional shifts” talked about at the Common Core website, and the current practice of drilling understanding in a conversation I had with one of the key writers and designers of the EngageNY/Eureka Math program. (EngageNY started in New York State and is now being used in many school districts across the US.) On the EngageNY website, the “key shifts” in math instruction went from the three that were on the original Common Core website (Focus, Coherence and Rigor) to six. The last one of these six is called “dual intensity” is, according to my contact at EngageNY, an interpretation of Common Core’s definition of “rigor” and states:
Dual Intensity: Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year.
He told me that the original definition of rigor at the Common Core website was a stroke of genius that made it hard for anti-intellectuals to speak of “rigorous” in loosey-goosey ways. And, in fact he was able to justify the emphasis on fluency in the EngageNY/Eureka math curriculum. So while his intentions were good (using the definition of “rigor” to increase the emphasis on procedural fluency) it appears to me that he may have been unwittingly co-opted to make sure that “understanding” took precedence.
In our discussion, I pointed to EngageNY’s insistence on students drawing diagrams to show place value in adding and subtracting numbers that required regrouping (aka “carrying” and “borrowing”–words that are now anathema in this new age of math understanding). I asked if students were barred from using the standard algorithm until they acquired “mastery” of the pictorial procedure. His answer was evasive, along the lines of “Of course we want students to use numbers and not be dependent on diagrams, but it’s important that they understand how the algorithms work.” He eventually stated that Eureka “doesn’t do standard algorithms until students know the prerequisites needed to do them”.
Thus, despite Common Core’s proclamations that the standards do not prescribe pedagogical approaches, it would appear that in their definition of “rigor” they have left room for interpretations that understanding must come before procedure.
“Understanding” Coexists with Procedural Fluency
Understanding and procedure work in tandem. Sometimes understanding comes first, sometimes later. As evidenced by EngageNY/Eureka Math, and other programs making inroads in school districts across the US, the interpretations of Common Core have resulted in an “understanding first, procedure later” approach. That interpretation makes it appear as if both sides have reached common ground. Reformers can now say “You see? We’re not against drills”—provided such drills are drilling understanding.
The major problem with this approach is that not all students take away the understanding that the method is supposed to provide. Some get it, some don’t. And while it may work to give the adults who design such programs a mental visualization, they’ve had the advantage of many years of math experience (and brain growth) that students in 5th, 6th and even 7th and 8th grades do not have.
Students are forced to show what passes for understanding at every point of even the simplest computations. This drilling of understanding approach 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 Seductive Nature of a Bad Idea
The scary part about all of this is how easy it is to get swept in to the recommended methods. I was working with the fifth graders and insisting that they draw the diagram to go along with each problem, when midway through the period I realized that I was forcing them to do something that I felt was ineffective. The next day, I announced to them that instead of them having to do the rectangle diagrams, they could just do the fraction multiplication itself. While my decision was met by cheers from the fifth graders, I couldn’t help feeling guilty in spite of my own beliefs. I imagined reformers shaking their heads in dismay, believing that I was leading the students down the path of ignorance, destined to be “math zombies”.
The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. These beliefs are now extending to well-meaning mathematicians who had publicly opposed much of the reform philosophy. The math reform movement, in trying to overturn students “doing but not knowing” have unwittingly created a new poster child. While the reformers believe the new poster child represents one with “deep understanding”, they have instead created a child for whom “understanding” foundational math is not even “doing” math.