Drilling “Rote Understanding”

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.)

Source: “Arithmetic We Need” by Brownell, Buswell, Sauble; 1955.

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.

11 thoughts on “Drilling “Rote Understanding”

  1. This is the sort of thing that happens – especially in the early grades of elementary school – when people with zero understanding of how learning works OR how it’s different in the Early Childhood (up to 8YO) window are in charge of setting out curricular expectations. The early years are where kids are hard-wired to learn, experientially and at their own pace (as opposed to lockstep), the “what,” while the “why” comes more naturally from 8YO onward as children are more capable of abstract thought and of holding multiple ideas long enough to manipulate them. Making things concrete through drawings is still artificial, though, because the teacher is leading students step by tortuous step through long and convoluted processes – multiple algorithms, too! – before they have the mental stamina to really manipulate the numbers (or words and literary concepts like inferences, in the case of the ELA standards) naturally, so of course they’re reverting to rote memorization of the new algorithms instead of deep understanding of the processes at too young an age for a large percentage of them. We don’t ask children to read words before we allow them to speak those words, but we’re approach math instruction in a similar way; can you imagine how ludicrous it would be to forbid our children to say their names before they could recognize them in writing? That’s not how that window of learning WORKS.

    Reminds me of the pushback I got from high school instrumental music teachers when I suggested starting to teach jazz style in elementary school: I was told that no, kids shouldn’t be learning jazz until they could read the more complex syncopated rhythms, when in fact many of our jazz greats in fact learned jazz by imitation as younger musicians and learned to read music later. (And yes, I taught jazz style by rote and then went back and showed them how the rhythm they were looking at worked, and showed them how to apply it to other rhythms down the road, BUT I didn’t start with the notation first.)

    1. People forget that the teacher is in charge of determining when learners understand. They decide how much drilling is required & how many diagrams must be drawn before concept is understood.

      1. Sometimes the concept will NOT be understood even with explanation and drawings. Should the student then be forbidden to use the procedure for fear he/she is “doing without knowing”, which is the anathema of the “Students must understand or they will die” crowd? In Singapore Math, students may be required to do two diagrams at most. In today’s books, there are many. Teachers may decide otherwise, but the current interpretation of Common Core is that the “instructional shifts” require the “dual intensity” approach of drilling understanding.

      2. Not necessarily. In NH for instance, if your school is in the lower 5-10% among all of the schools, there is a method for remediating those schools. That’s also coming from ESSA when they label schools as either Comprehensive or Target schools.
        Then they begin a remediation which is provided by the State Dept. of Ed.
        I don’t believe NH’s approach in the past has been helpful at all. It simply does not focus on what really needs to be fixed.

        Teachers are repeatedly told how to teach and what to teach. The professional development drives failed teaching methods that are pushed by many who do not work in the classroom, Same with “consultants” who do not teach.

        Add to the grant funding that comes into the school that requires teachers to use failed teaching methods and you have a situation where the teachers has little if any autonomy left. (Constructivism, discovery, inquiry, student-centered, etc)

        Who chose the curriculum? Program? Not the teacher.
        Who chooses the teaching methods to use? Not the teacher.
        Who chooses to incorporate non-academic dumbed down competencies into the classroom? Not the teacher.

        New teachers are also fed fuzzy math practices and failed pedagogy in the Schools Of Ed by those who do not teach in the classroom.

        This all adds up to the inability of a teacher to change up curriculum and methods if they want to improve the quality of education in their classroom.

        Fortunately there are some, AP math teachers for instance, who might be able to avoid some of this. But I even hear from them that they are supposed to be a “guide on the side” instead of the sage on the stage. Math teachers know discovering math is not a good way to learn math. But unfortunately there are so many factors pushing these failed teaching methods that they are now in a position where they have to comply.

        Barry manages to explain what is going on so everyone understands it
        He does it again with this article.

    2. I like to use music as an example for learning. All of the kids who get to All-State have had private lessons in addition to the in-class band and orchestra group work. While group performance is very important, what opens up individual doors of opportunity are skills, not the engagement of class work. Skills drive engagement, not the other way around.

      When choosing a private piano lesson teacher for my son, I was confronted with three types of approaches. One was an informal, fun, natural approach that assumed that skills would automatically be developed. In the piano world, this often starts with fun, jazzy pieces that do not emphasize proper hand positioning, fingering, and scales. At the other extreme are pedagogues who enforce an exact process using leveled books or systems. They have a better chance of working than the first approach, but at the risk of losing the student.

      The solution I found was somewhere in the middle, but much closer to the latter approach. There is no easy, natural path to achieving a high level in any field, so for piano, there is no escaping Hanon or Czerny for finger exercises, Bach for “Inventions”, and Chopin for etudes. You do not need a strict leveled system, but students do need pushing and expectations from the instructor. The homework has to get done and recitals (tests) have to be given.

      El Sistema shows how this works in Venezuela for kids from the barrios. It starts from pre-school ages and offers all students private lessons. Individual opportunities come from auditions, not tests using words. Those who do not reach the top levels still get to learn the value of skills and hard work. That translates to everything else, not words or talk or blather.

  2. Explain why the teachers decisions are ignored? They decide when learners understand, how much drilling & how many pie charts need to be drawn.

    1. The current interpretation of CC’s “instructional shifts” are that students must do the “drilling for understanding”, and that understanding must come before doing the procedure. The exercises in the books require the drawings to be done. Yes, teachers may ignore this as I did to the cheers of the 5th graders I was in charge of. I have seen this approach also in the online instructional programs used in schools, where there is no choice BUT to do comply with the more inefficient pictorial methods.

  3. Dr. Raj Shah is the founder and owner of the Math Plus Academy. He says in his video that “In this day and age, being able to execute an algorithm in arithmetic to get an answer is a fairly useless thing to do.”

    Blah, blah, rote blather. Rote, rote, rote, rote, rote! Being able to consistently get an answer to all sorts of problem variations defines understanding. This process forms the basis for all homework and college problem sets. Individual, not group work. Words and even proofs are not sufficient. None of these turf and entrepreneurial pedagogues show exactly how their top-down understanding-to-skill process works. Transference of their form of understanding? NO PROOF WHATSOEVER. It doesn’t work.

    On his web site he says that “They develop a GROWTH MINDSET”

    What the hell is a GROWTH MINDSET? It’s a solution to a non-problem. It’s an advertising term.

    This is their special academic turf and their claim to fame and fortune. It’s all about them and not the math. They claim to love the balance of understanding and skills, but how do those skills get done? How do they test for them? Where is the proof? Fine. Go after skills from an understanding-first approach. Does it work?

    Traditional math in K-6 hasn’t been around for 20+ years, so why the need for an after school program? I got to calculus in high school with no help from my parents and no after school program or summer camp. Why do schools have traditional math in high school? What other path produces better results? Integrated math? It lost. The only bastion of fuzzy math thinking left is K-6, and we all know how much they know about math understanding. Parents finally forced CMP out of our middle school, so the only place left for this fuzzy thought is the non-subject-certified fairyland of K-6. Students may survive for most subjects, but not in math. It’s all over by 7th grade.

    His “academy” has after school classes, workshops and camps. This is not an in-school solution, so more is better, right? Is it the best use of that time? What sort of skill help or expectations are provided at home to those students? Fuzzy, happy math works best for students who already have the skills.

    This discussion is not about critical thinking and problem solving. It’s about philosophy, academic turf, and money. Nobody asks us parents of their best students what we had to do at home and with tutors – and what kind of work and tutoring was done.

    Everything would be fine in K-6 if they really valued enforcement of skill development no matter what the approach. Relying on process and presumed transference is incompetence. If they give up after struggling to get to CCSS-level math skill proficiency, they’ve failed. Having differentiated instruction (self-learning) with in-class leveled groups only hides the skill tracking at home. It increases the academic gap.

  4. Why not look at the science? Looks like Barry understands all of this but the education reformers (and snake oil salesmen like Dr. Raj Shah) want to keep ignoring it.

    Between 1995 and 2010, most U.S. states adopted K–12 math standards which discouraged memorization of math facts and procedures. Since 2010, most states have revised standards to align with the K–12 Common Core Mathematics Standards (CCMS). The CCMS do not ask students to memorize facts and procedures for some key topics and delay work with memorized fundamentals in others.

    Recent research in cognitive science has found that the brain has only minimal ability to reason with knowledge that has not previously been well-memorized. This science predicts that students taught under math standards that discouraged initial memorization for math topics will have signifcant difculty solving numeric problems in mathematics, science, and engineering. As one test of this prediction, in a recent OECD assessment of numeracy skills among 22 developed-world nations, U.S. 16–24 year olds ranked dead last. Discussion will include steps that can be taken to align K– 12 state standards with practices supported by cognitive research.

  5. To bring this to real world application I am in the process of trying to re-enter the workforce after years of being a stay at home mom (strangely I was often not at home during these years, but I digress). I had to take a math competency test that looked at both accuracy and speed. I scored very well but used traditional math teachings (e.g. invert and multiply when dividing fractions) to solve the problems. Keep in mind this is gate #1 for ppl entering the workforce and yet we want to lay a lengthy constructivist approach on children to be relied on in these screening tests later? This serves neither the candidate nor the employer.

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