Drilling “rote understanding”

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”.)

fraction of fraction

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.

Understanding and procedure work in tandem—you need both. Understanding sometimes comes first, sometimes later. As evidenced by EngageNY/Eureka Math, and other programs making inroads in school districts across the US, the Common Core way is “understanding first, procedure later” which aligns with the reformers’ view of math education and their mischaracterization of traditional approaches being a set of “meaningless drills”.  So instead, we now have a nod to both camps. In the reformers’ view, students are made to use procedures that supposedly impart understanding. Ironically, as much as the reformers disparage “drill and kill” they have no qualms about “drilling understanding”.  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.

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.


Not Much to See Here, Dept.

When I read the headline “A Passion for Making Math Make Sense to Kids” I thought it was another of those “students need to understand not memorize” type of articles, with homages to growth-mindsets, memorization is bad, and traditional-math-never-worked-because-it’s-taught-as-rote memorization-and-a-bunch-of-tricks-and-look-at-all-the-adults-who-say-they-don’t-like-math.
But no. It was about Susri Anuradha, a woman with an engineering degree who opened up a Mathnasium franchise. Mathnasium is one of many of the learning centers to which parents send their kids to learn the math that isn’t being taught in schools, in the manner held in disdain by math reformers of all stripes.
“In 2015, she opened her first “Mathnasium”, a math learning center for children ages 5 to 18, in Acton, MA. Earlier this month, she opened opened her second franchise location in Burlington, MA. Anuradha holds a bachelor’s degree in engineering and a M.S. in Information Management Systems from Harvard University. After a rewarding career in Information Technology, Anuradha followed her passion of “Making Math Make Sense to the kids” and opened Mathnasium of Acton. Her decision to open and expand Mathnasium is driven by her desire to help children build confidence and excel in the subject she enjoys most – math.”
Of note was this quote from Ms Anuradha: “I love it when I meet my students in social gatherings and they come and hug me. I love to see kids getting empowered by the gift of education.”
I have to say I love it too. For the record I teach math in a traditional manner. Not to brag too much, but I just received a card from the mother of one of my students during teacher appreciation week. The card said “Thanks for bringing math back.”

Everyone’s Happy in Happyland, Dept.

I did a long-term sub assignment in San Luis Coastal school district in California a few years ago. I wrote about it in “Confessions of a 21st Century Math Teacher”.

I taught during the year in which California was in transition to the Common Core. We were told quite often that “next year would be different”. No more teacher in front of the class saying to open books to such and such page and do the following exercises. Teachers would facilitate learning, students would learn to “problem solve” and to “think” and “understand”. This assumed that the status quo was rote memorization and teaching without understanding or conceptual context.

The Superintendent of San Luis Coastal who was in charge then and still in charge today has a personal philosophy that aligns with the above bromides.  He wrote about his personal philosophy at length here.  An excerpt follows:

I believe students in the 21st century are different. They are digital natives and live in a world where “any knowledge” can be found immediately on Google. Therefore, why regurgitate knowledge (like an “academic rationalist”) when it is far more reasonable to expect a student to apply this knowledge and to make new meaning from this knowledge. (This is my “cognitive processor” or “social reconstructionist” coming out.) Relevance is critical among this generation of students in order to motivate them to move beyond what I see as low-level thinking.”

His constructivist viewpoints are bolstered by the school district’s hiring policies which use the Danielson Framework for evaluating potential new teachers. The webpage for this framework states right at the beginning that “The Framework for Teaching is a research-based set of components of instruction, aligned to the INTASC standards, and grounded in a constructivist view of learning and teaching.” What then follows is a description of 22 components (and 76 smaller elements) of what they consider teaching.

This framework, coupled with the Superintendent’s philosophy lays the groundwork for hiring and firing. If you are  an advocate of student-centered, inquiry-based, project/problem-based learning, c’mon in.  Traditional type teachers need not apply.

If you wish to teach skills, they better be learning, critical thinking and problem-solving skills.  Top-down, open-ended, ill-posed problems with many possible answers are preferable to the stuff that this particular cadre of educationists hate; i.e., distance/rate, work, mixture, and number problems.  No relevance to what kids really care about.

To my knowledge there has been little to no parent, teacher, or student backlash in this school district. So it appears that everyone is happy in happy-land. That said, I refuse to teach there. Not that strong a statement considering what their response would likely be.

A Comment Worth Reading

SteveH, a frequent commenter on issues relating to math education, left a comment on the piece below on “Hidden Figures” that is worth reading. So I’ve reproduced it here:

Traditional education pushes and values incremental mastery of skills along with understanding. That still happens in high school AP Calculus tracks, but not in K-6. Facts are “mere” and skills are “rote.” Add to that the use of social promotion and full inclusion where curicula like Everyday Math “trust the spiral” and assume that kids will learn when they are ready. Meanwhile, STEM parents and those who know better hide the tracking at home and ensure mastery of basic skills so that their kids are ready for a proper algebra I class in 8th grade. CCSS has now officially made K-6 a NO-STEM zone (PARCC actually states this) and educators claim that students can catch up by taking summer classes or doubling up in math in high school. Right. I had to work with my math brain son in K-6, but didn’t have to do a thing for his traditional AP Calculus track high school classes. Not a thing.

Is math a natural learning process in K-6? Does reform math provide a better “understanding” base for faster improvement later on? There is absolutely no proof of that. In fact, after 20+ years of reform “understanding” math, quite the opposite is shown – that if one fails to get on the advanced (algebra in 8th grade) math track, then any sort of STEM career is all over. I got to high school calculus in the old traditional K-6 days with absolutely no help from my parents. I had algebra I in 8th grade followed by geometry, algebra II, trig, and calculus. What’s different now? K-6. The women in the movie would have a much more difficult time of it now.

CCSS officially increases the academic gap. Parents who make up the difference at home and with tutors hide this systemic K-6 failure and those educational pedagogues never, never ask us parents what we had to do at home even though it would be a very simple task. (All of my son’s STEM friends had help outside of school.) They just claim that their process works, point to our kids as examples, and then blame the other kids or claim that they just need more hands-on real world engagement. They do not understand the importance of pushing and nightly individual success on homework problem sets. That’s the fundamental problem I see with the students I tutor. They don’t value homework. When you get to college, it’s ALL about the P-sets. My son stays up all night to finish them if he has to. This is likewise true for programming classes. It’s ALL about doing everything you can to finish your individual (not group) program with no errors. THAT is where true understanding is achieved. Back when I taught college math and CS, it was NEVER about engagement or any sort of group or class work. It was about the hard, individual work put into P-sets and programs. Success on homework and tests REALLY helps engagement, not in-class group work that does nothing for grades.

In this age where we can’t have any sane, fact-based discussion on health care choices, let alone understand even what insurance means, I have no hope for change in education, especially when some claim that it’s a liberal/conservative issue. Some of us are actually unaffiliated and quite willing and able to separate issues from political party ownership. I push educational choice, but that apparently means that I believe in all sorts of other baggage. Some people alter reality to fit their simplistic view of the world. You can’t argue with these people. We can only appeal to parents who want to understand what’s going on.

I Must Be Missing Something, Dept.


The movie “Hidden Figures”, about three African American women who were instrumental in the space program in the 60’s, has garnered lots of “STEM is not just for white guys” types of promotions.  (See this, or this or this.  There are many more.)  Also there are many blog posts by various progressivist/educationist types, praising the movie.

It seems strange to me then that in all this “conversation” about the themes of the film, there is nary a mention that the three women rose to their prominence based on the traditional math education they received. This is the type of math teaching, prominent in the era the women were from, that is so derided and despised by math reformers and given blame for “failing thousands of students”.

Is there an unstated progressivist narrative going on? How does it go again? The women were interested/gifted/talented in math and science to begin with and thus were destined to succeed in it no matter how it was taught? Have I got the right words? Or am I missing something?

I ask this because of this excerpt from an article in The Atlantic about the film:

“Math, in that sense, is in Hidden Figures a tool of meritocracy. It is a symbol of the power of education (chalk being handed from one person to another is a recurring motif in the film), but it is also, more broadly, a metaphor for a world that could be so much better if we would just let everyone, equally, have a say in its improvement. Math’s equations double, in Hidden Figures, as a hope for equality.”

I agree. Interesting that the reform math that passes as education and brings people flocking to NCTM and other conferences to adulate various math reformer as if they were rock stars actually penalizes the very people the reformers/progressivists think they are helping.

Stop me if I’ve said this before: The inequity arises from those who can afford to do so paying for the appropriate education offered at learning centers and the like.  And those who cannot afford it being deprived of what they need.

OK. I’ll stop.