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.

I don’t really know how one can understand something without first doing it multiple times. If we “claim” to understand how to pitch merely by reading “Shoeless Joe”, we all know what happens when amateurs get up on the mound and start throwing for the first time (that’s why there are really big backstops behind the catcher). We may “claim” to understand ballet, but having witnessed thousands of hours of practice at my daughter’s dance studio, it’s very apparent that until hundreds of hours of practice at the barre are performed, kids really have no clue what they’re doing.

Mathematics is a very precise subject. It is hierarchical in nature, much more so than any other subject in school. I am not interested in creating mathematicians for the general populace because that in itself requires thousands of precise hours drilling and a mindset that precludes most; one needs to think beyond the obvious and, quite frankly, it’s a very difficult profession to achieve. But what I AM interested in, is ensuring ALL kids have a very strong foundational understanding of arithmetic which has been proven to aid kids in obtaining good careers in the workforce. We may not all need Calculus, but we most definitely need a mastery of fractional arithmetic and times tables. And Algebra. And Geometry.

Despite what “experts” claim, ALL kids require this knowledge in order to help them later on in life. And the solution is so simple. So why is it that so many are focused on turning the majority of students into such a narrow subset of our workforce? And all based on an inaccurate premise?

Mathematicians will continue to enter the workplace as they always have, but based on the direction of basic, general math education in our schools across North America, our future mathematicians will be coming from other countries where they value a very strong,fundamental education of mathematics…something that in the name of progress, has escaped our educrats here in North America.

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The purveyors of the “dual intensity” approach, believe that they are combining the “doing” of which you speak with the “understanding” at the same time. I describe the results with a group of fifth graders in the post above. Naysayers will say that what I experienced was because I was doing it wrong. Of course. That must be it.

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Dual intensity isn’t either as dual as it is promoted to be nor as intensive. The examples I’ve seen force students to dwell for long periods of time on each problem, so fewer examples are covered and thus less practice. While this might — when done well — reinforce concepts it does not lead to what is the more valuable outcome for the truly elementary skills, namely proficiency to the point of being able to execute them with minimal conscious effort.

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“When done well” it’s used as the means for explaining what the procedure represents and why it works as it does. Maybe one can have students do one or two that way, as Singapore does. But as I saw with the fifth graders I was working with, it was not sinking in, and it was just drudge work. If a kid gets it fine; if they don’t move on. Eventually through experience with the procedure and with solving various problems, they will develop the understanding, and may better be able to grasp the area model approach better–particularly after they have had to solve problems with finding areas of rectangles. Sometimes it’s better to let procedural fundamentals sink in first before getting into the conceptual underpinnings–like the formal definitions of limits and continuity in calculus as an example.

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If we spent the first part of teaching a kid something the understanding the understanding the understanding…they would quickly lose interest and just give up. Don’t blame them. They’re kids. They need to be “doing” something constructive, and see the success in what they’re doing, in order to keep them interested. It’s called baby steps. If you take this away, and replace it with an adult mindset, kids tune out.

Why this has to be explained to grown ups…constantly, is beyond me.

But what do I know anyways? I’m just a parent…

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As a teacher, I would agree with you Barry.Many find this ‘drilling of understanding’ to be very tedious; for others it is meaningless busy-work that leads nowhere.

Experience has certainly shown me that understanding and procedure reinforce each other and is the best approach.

Ontario seems to use even more convoluted methods than you illustrate here, but does seem to be prioritising a fruitless drilling of understanding. Even though the word drill is much despised here, it is alive, but not doing well, as you have explained.

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And the purveyors of “dual intensity” seem to think that this combines understanding with the “doing”. It doesn’t. It bogs down the doing so that kids end up hating it. The fifth graders kept asking me “Do we hafta draw the boxes?” When I finally said “No, you don’t” they cheered. So shoot me.

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And, even if they do “understand”, the understanding is false anyway.

Sure you can draw lovely diagrams that show that 2/3 x 4/5 = 8/15. But that breeds a horrible limited sense of fractions.

The result is students who:

1) are totally uncomfortable with improper fractions — how do you draw 7/5? The result is students are taught years of fractions and never see an improper one (some get quite shirty that I decline to “simplify” from 7/5 to 1 2/5, since their teachers insisted that 7/5 was not a suitable answer.)

2) bound totally to arithmetic, since drawing x/2 times 3/4 = 3x/8 is a bit of a challenge. All that drawing of fractions, and the most powerful use of them cannot be drawn.

3) understand fractions as a physical thing only, and hence are unable to leap to fractions as a concept. I suspect a fair number of my entrant students arrive quite uncertain about the difference between 0.5 and a half. One is a number, the other is a fraction — to them.

I refuse utterly to draw any diagrams when doing fractions, as I try to get them to learn that the real power of Maths is about abstraction. I also move to algebra with fractions as quickly as possible — since the rules are identical, fraction algebra isn’t particularly tricky. Provided you don’t insist on diagrams, of course.

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The people pushing the “dual intensity” approach would likely say that “Of course we want them to work with numbers rather than diagrams. But it’s important that they ‘understand’ before the proceed to the standard algorithm.” I had such a dialogue with the person I alluded to, who plays a prominent role in EngageNY. I kept asking “What about if the student doesn’t understand and can’t work with diagrams? Do you not allow him/her to proceed to the standard algorithm?” No direct answer except finally to say that “prerequisite understanding” is necessary before moving on to standard algorithms.

As to using diagrams for representing, say 1 2/3 x 5 3/4, there are ways of doing it as I’ve seen in the textbook my school is using. Not too much value-added in my opinion. As I say, the diagram using the area model is fine for introducing the topic and explaining why it works. Sure, teach for understanding, and then let the students move on to the procedures–quickly.

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I object to the word “improper” in this context for some of the reasons Chester gives. Not, by the way, because there is anything improper about it — the use of the word “proper” for fractions is in the sense of “strictly belonging to”, as in a “proper subset”, which denotes a subset which is strictly less than the entire set.

My problem with it in the elementary school classroom is that such subtleties of language are no longer taught and indeed are no longer understood by teachers. There appears to be an inference by most teachers that the word “improper” indicates that this is not an acceptable answer. That’s nonsense, of course — there is every reason to accept improper fractions as a properly simplified form. This error in taking the wrong connotation is similar to that of taking “irrational number” to mean a number that “does not make sense” rather than one which “cannot be expressed as a ratio”

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The ‘child must understand first’ is very common in UK primary schools too which makes it very difficult for us mathsy people to put forward the case for ‘everybody learn the algorithm and be successful, assimilating understanding along the way’ approach. This situation is incredibly frustrating, especially for children with SEN. A really good example is with short multiplication: ‘understand first’ dictates we all teach grid method which is both limited and convoluted, and those with SEN frequently end up with the wrong answer (and no understanding either). With the proper method, not only are they very successful when the algorithm is learned, but it helps with fluency of recall of multiplications too.

Another downside to the ‘child must understand first’ is that with skills-based curricula (overwhelming majority of primary schools in UK), you will find the subtle coercion in learning objective design/policy actually ‘signs off’ a child on a method well before they have learned it off by heart. ‘To be able to understand how multiplication of large numbers works’ means that, again, those with SEN who need to practise more than other children are moved on way too early. Not being able to use the word ‘know’ in a learning objective prevents us traditionalist teachers from ensuring that children can perform larger calculations quickly, efficiently and accurately enough to access their secondary maths lessons years down the line.

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Let me premise this by saying I am by no means a mathematician or math teacher. I was just a regular student who received high marks in math, saw fellow students floundering and sought to assist them as best as I could. First, when I was taught multiplication of fractions in the late 80’s we never used drawings or doodles. To me this concept is dumb. Beyond so. In case you’re wondering my location, southern Arizona. I always went by the premise don’t try to ‘get it’ just do it. Math just is. To me it’s always been the universal language. It’s never needed to be understood. For example if you spend your entire time trying to explain why 2 minus 1 is 1, you’ll never go anywhere. It just is. And as far as the ‘they are just kids’ theory… For just over one hundred years we’ve been dumbing our kids down and telling them they couldn’t do things. You do realize that predominantly the Revolutionary War was fought and won by kids. If it weren’t for our ancestors realizing kids could do more than what we now give them credit for we’d never have progressed passed living in caves. A person’s mind never is as active and learns as much as PRIOR to pre-school. We start school too late now, it wasn’t always this way. We used to be graduating college by the time we were 14.

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Seems to me worth pointing out that the ‘progressive reformers’ I know all hate the EngageNY curriculum. They hate the drill, they hate the understanding stuff, they just hate it. And I’m with them: it’s a weird, sloppy piece of work.

Speaking personally, I agree that it’s hard to figure out the relationship between understanding and procedures. I was recently teaching the standard algorithm for multiplication to my 4th Graders, and I noticed how, in the course of working on that procedure, inevitably, students had to put some of their understanding aside.

Here’s where I think we’d disagree, though. I had previously done a lot of work with

partsof the standard algorithm for multiplication: multiplying by 10s, mental computation with a strategy that mimics the standard algorithm (i.e. 35 x 27 = 35 x 20 + 35 x 7). It seems to me that, because my students were proficient in these parts of the algorithm, it was much easier for them to put the pieces together in the standard order.So it seems to me that it’s valuable for students to learn the components of an algorithm before being asked to learn the entire thing, whenever possible, simply to increase the efficacy of the standard algorithm instruction.

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Thanks for your comment.

Actually, the description you provide of the multiplication by 10’s was how double-digit multiplication was introduced in my 4th grade math book (in 1959) and in others of that era.

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Great!

To me, that’s a kind of ‘understanding before the algorithm’ that is important. But should that be called understanding? Or is it just some other procedure?

I don’t know. The distinction between understanding/procedures gets a bit fuzzy when I think about it too much.

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As I said in the article “.Understanding sometimes comes first, sometimes later.” To insist as a policy that it always come first is what I was focusing on.

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Yeah, for sure. I agree with that, I think.

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My explanation is that understanding comes in levels, abstraction, and variations. There is no such thing as understanding first if one uses understanding in a generic sort of way. Conceptual (low level) understanding may motivate one a little bit, but that’s always happened forever. Every unit in a proper textbook introduces new concepts built on top of old ones. All of my traditional teachers from decades before did that, even for simple arithmetic skills. That’s just the start. What is missing these days (even with CCSS so-called emphasis on skills) is practice on all variations and depth. K-6 teachers, and therefore students, do not value any proper level of practice. They think repetition is just about speed and not understanding. That’s completely false. All proper textbooks provide problem sets with variations and depth, and all competent teachers assign and expect individual completion of non-trivial sets of homework problems. Each problem is really NOT the same. There are subtle differences that challenge one’s understanding. Individual practice still does not happen properly in K-6. Educators seem to only concern themselves with what goes on in class – often still in mixed ability groups where many kids fall though the cracks. Words are not understanding. Understanding comes from doing problem sets individually. I had so many nightly “discoveries” from that process. Modeling that process with a group in class does NOT transfer to many other problems. It has to be done individually and in quantity.

My son had a lot of drawing pictures for all subjects. Each child is supposed to have a different learning style, but many of his teachers forced all kids to do things with pictures. So much for translating educational philosophy into practice. I remember his crayon pictures of science terms in sixth grade when he had already memorized them. I promote anything that works, including memorization (which often includes a lot or understanding), but I didn’t see much of that in K-6. What I push for are math curricula that set high STEM expectations and push a lot of individual mastery of proper textbook problem sets. This is what math is all about from pre-algebra onwards through college. That’s what works and enforces the most depth, variation, and abstraction of understanding. Unfortunately, K-6 is now officially a NO-STEM ZONE and for many kids without skill practice at home or with tutors, it’s all over by 7th grade. The fundamental systemic flaw is that CCSS only defines a one-slope curriculum track to no remediation in a college algebra course and PARCC calls that “distinguished.” I’ve noted many times that I got to calculus in high school with absolutely no help from my parents. With my son, it could not have happened without my help, and we parents got notes telling us to practice “math facts” at home. I can’t tell you how incompetent I think that is.

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This is my experience as well, and that of hundreds of others who have taken the time to contact me and tell me their stories. The game is all about talking about understanding, but there is no action at the primary level to support their claims. We now have parents being told THEY have to teach their kids times tables at home, or have kids (like mine and countless others) sign in on boring computer games to “practice” their math facts at home. We are being told there isn’t time in the classroom to do these things. And this is such a missed opportunity! What it indicates, is that there is very little proper instruction being given to kids to ensure understanding and mastery of their math facts will occur in the classroom.

Teachers aren’t to blame, as dumbed down curricula and current teacher workshops led by School Districts and ed consultants now try and wow teachers about the importance of 21st century learning and how drills kill. So although there is great discussion about how our kids nowadays are experiencing a much better understanding of arithmetic, this is just false advertising. Just look at the proliferation of tutoring rates over the past 15 years. Nothing against tutoring centres, but i take exception that kids must receive additional support either from parents, or from tutors, if they want to master any arithmetic procedure. This is wrong. And it speaks ill of those in charge of running the public system for ALL kids.

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Well put Steve!

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Its a good thing kids learn to speak before getting to school. Can you imagine teaching spoken English like this. You must have deep understanding of phrase structure, tenses, possessive pronouns and so on before you can practice speaking in sentences.

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Its a good thing kids learn to speak before getting to school.I don’t want to carp too much, but speaking language is not the same. Kids are programmed to learn to speak, which is why the stupidest kid you will ever meet will still be able to speak. Kids are not programmed to understand fractions.

Instead imagine football coaches teaching this way — you can’t kick until you clearly understand the concepts of transfer of momentum. You can’t pass until you can clearly explain that the path is a parabola. Instead we get them out there and playing. And while they are doing we intersperse the teaching so they understand what they need to be doing. And we drill, drill, drill the key skills. No sports coach worth their salt would think that drill was a bad thing.

Nor would teaching a kid to drive be much good if they had to pass a complete theory of driving exam before getting behind the wheel for the first time.

No-one outside of schools teaches the way that school teachers are meant to teach. That should be an absolute giveaway that it is a rubbish way of teaching.

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Agreed sports is the better analogy and the point that even the least enthusiastic math student will understand that to do well in sports means going to practice and doing drills.

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