In working with a group of fifth graders in need of math remediation at my school, I had them do the exercises in their book. It involved multiplication of fractions, and it used the area model of a square as the means to illustrate what multiplication of fractions represents, and why one multiplies numerators and denominators. A problem like 3/4 x 2/3 then is demonstrated by dividing a square into three columns, and shading two of them, thus representing the 2/3. Then the square is divided into four rows, with three of them shaded–the 3/4. Thus 3/4 of the 2/3 have common shading, and the interesection of the shading of the 2/3 and 3/4 portion yields 6 little boxes shaded out of a total of 12 little boxes which is 6/12 or 1/2 of the whole square. The students see what 3/4 of 2/3 means in this model in terms of area and the reasoning behind multiplying numerators and denominators.
This was the explanation used in my old arithmetic book from the 60’s (and in other textbooks from that time in an era denigrated as being about “rote memorization” without understanding”.)
Source: “Arithmetic We Need” by Brownell, Buswell, Sauble; 1955.
It is also the method used in Singapore’s math textbooks. 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.
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). There is nothing wrong with that. But they are also combining fluency with understanding: “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 had a conversation recently with one of the key writers and designers of the EngageNY/Eureka Math program that started in New York state and is now being used in many school districts across the US. I noted that on the EngageNY website, the “key shifts” in math instruction described on the CC website, were broken out from the original three (Focus, Coherence and Rigor) , to six. The last one, called “dual intensity” was, according to my contact at EngageNY, an interpretation 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 rigorous 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 (use the definition of “rigor” to increase the emphasis on procedural fluency) it appears to me that he was 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.
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. Robert Craigen, a math professor at University of Manitoba who is active in the issue of K-12 math education has described this process as “the arithmetic equivalent of forcing a reader to keep his finger on the page sounding out every word, with no progression of reading skill.”
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. I couldn’t help but picture reformers shaking their heads in dismay, believing that I was leading the students down the path of ignorance, destined to be “math zombies”. But in making my decision and announcement (which was met by cheers from the fifth graders), I believed that had I continued, I would just be giving them little more than a “rote understanding” for procedures they would not be able to perform for problems they would not be able to solve.