Not This Again, Dept.

In a recent article in Huffington Post, Keith Devlin (a mathematician from Stanford who writes about math education) says that math as it was taught in the past focused largely on computational skills. But it should be about “number sense” since computations can largely be done by computers, calculators, “the cloud”, etc, these days.

He waits until the end to tell us what “number sense” means–which in his case means “conceptual understanding” as opposed to procedural fluency.  As if he has heard the arguments before (and he has), he takes a pre-emptive strike against potential criticism from people like me by paying lip service to the need for learning procedures and then goes into his case for “understanding”.

Make no mistake about it, acquiring that modern-day mathematical skillset definitely requires spending time carrying out the various procedures. Your child or children will still spend time “doing math” in the way you remember. But whereas the focus used to be on mastering the skills with the goal of carrying out the procedures accurately — something that, thanks to the learning capacity of the human brain, could be achieved without deep, conceptual understanding — the focus today is on that conceptual understanding. That is a very different goal, and quite frankly a much more difficult one to reach.

This, by the way, is the new state of affairs that the mathematics Common Core was created to address. Outsiders, including politicians in search of populist issues to incite voters and others with an axe to grind, have derided caricatured portrayals of this important new educational goal, by describing it as “woolly” and “fuzzy”. But I disposed of that uninformed red herring already. The fact is, number sense is (rightly, and importantly) the primary focus of 21st Century K-12 mathematics education that millions of children around the world are receiving today. Children who are not getting such an education are going to be severely handicapped in the world they are being educated to inhabit.

While he may have disposed of the “red herring” of the educational goal of understanding, I’d like to revisit it.  As I’ve written about in numerous articles (thus disposing of the “red herring” of traditional math done poorly being the definition of traditional math), a glance at textbooks from the past shows that in fact the conceptual underpinning of various procedures has in fact been included and was a part of math education. Granted, the explanation of why the invert and multiply rule for fractional division works was not provided. I knew how to use it as well as what fractional division is and how to solve word problems with it, but as I said in my last missive, I didn’t learn how the rule is derived until 10 years ago. Nevertheless, I did manage to major in math despite the omission of something that brings gasps  of astonishment and “tsk tsks” from the education establishment–and largely from people who themselves have benefitted from traditional modes of math education.

As Prof. Devlin asserts, in many US classrooms today,students must  demonstrate an “understanding” of computational procedures.  How this has played out in the last 25+ years, and now even more so under Common Core, is that before they are allowed to use standard algorithms, they must learn alternative procedures to provide the conceptual underpinning in the belief that teaching the standard algorithms first obscures what Devlin calls “number sense” or “understanding”.  The teaching of mathematics has thus been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations.

In so doing, students are forced to show what passes for understanding at every point of even the simplest computations. Instead, they should be learning procedures and working effectively with sufficient procedural understanding. But this “stop and explain” approach to understanding 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 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.

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. While students undergoing instruction under the prevalent interpretations of Common Core may be able to recite an explanation they have been told–um, that is, “discovered” with some “guide on the side” guidance from the teacher on how best to phrase such explanation–that is not the test of effectiveness of a math program. In many cases, what passes for “understanding” or “number sense”  is “rote understanding”.   In so doing, the math reformers have unwittingly created a body of students in which “understanding” foundational math is not even “doing” math.


13 thoughts on “Not This Again, Dept.

  1. Concept: an abstract idea; a general notion.

    “… deep, conceptual understanding”

    This is an oxymoron.

    True mathematical understanding (and the ability to do problems) comes from mastery and scaffolding of skills directly taught and individually mastered using traditional math textbooks – like those in most high schools that prepare students properly for STEM degree programs. Concepts are superficial and don’t solve problems. Numeracy is vague and meaningless.

    “So what, then, remains in mathematics that people need to master? The answer is, the set of skills required to make effective use of those powerful new (procedural) mathematical tools we can access from our smartphone.”

    Baloney! This defines a path that guarantees that ALL students will not have the ability to get into a STEM degree program in college. That’s what a CCSS curriculum officially defines. The highest level (“distinguished”) in PARCC is only a 75 percent likelihood of passing a college algebra course, and this curriculum starts in Kindergarten. This is really nothing new. It’s been going on for over 20 years with curricula like MathLand and Everyday Math. Arguments against traditional math in the lower grades are fundamentally wrong because they haven’t been seen for 20+ years. You don’t hear those complaints about traditional math in high schools because that’s where all of the STEM-prepared students are. With full inclusion in K-6 and the need to lower expectations, the cover becomes blather about how bad traditional techniques are and the wonders of conceptual understanding and problem solving – neither defined in any way that leads to the traditional AP calculus track in high school. Education should be about keeping doors open, but CCSS slams them shut starting in Kindergarten. K-6 is now officially a NO-STEM zone.

    Who are the kids who now have any chance of getting into a STEM degree program? Only those who are supported by parents and tutors who don’t fall victim to this conceptual understanding crap. I got to calculus 45 years ago with absolutely no help from my parents. That’s impossible now. Just ask the parents of the best math students. Ask us. It’s not difficult. My son got “number sense” crap in K-6, so I had to ensure mastery of basic skills, on which the foundation of all “deep understanding” is built. When he got to high school with proper traditional teaching and textbooks, I had to do nothing. He just took his abstract algebra final at Yale with no thought of how that was useless in this age of computers and the cloud. Perhaps Kings College London taught only vocational courses.

    Devlin is fundamentally wrong in so many ways. Just ask the students who got accepted into Stanford what they had to learn in math. Look at the admission requirements for any college of engineering. The fundamental flaw in Devlin’s argument (and those of modern K-8 educational pedagogues) is that less is somehow more – that concepts and vague numeracy can provide all that is needed for STEM degree preparation. I could argue for higher level understanding (not conceptual) in math, but that can only be built on top of lower level skill understanding. It’s bottom up from skills, not top-down from conceptual understandings and vague cover talk of numeracy. More understanding can be added on top of mastery of skills, but conceptual understanding with poor skills is nowhere and can’t be fixed. You can be a user of apps, but you won’t understand the basis of the calculations and you will surely never be able to create them.


  2. Sure. Let’s just throw a novice into the pool and yell, “SWIM!!” That’s understanding before knowing how to do it properly, right? There is a specific pattern to learning how to swim. Ever notice that? Because if they didn’t learn, they’d die.

    All this crap about learning with understanding…anyone who’s actually learned anything properly, will attest to the fact that the understanding ALWAYS comes along with the actual learning part. It’s only recently, when ed consultants and parent gurus have decided to make a career for themselves by marketing themselves as “child experts” that we now have the common misconception that the previous way we learned math was all wrong. It’s human nature to want to improve the way we learn or how we do things, but shouldn’t we also recognize when something is efficient? When something needs to be taught a specific way? Do we hand over our car keys to our 16 year old and say “get in and just drive!” Why is education so different? Perhaps parents and others need to remember that we’re actually pretty good at discerning how kids learn, and give ourselves some credit about how we know what works best for our kids, rather than allowing others to do the thinking for us.

    If there were specific cognitive studies that Devlin cited in his article to support his claims, citing how children have magically changed throughout the past 20 years to process thoughts and automaticity differently than previous generations, he might have a point worth considering. However if this is just another sales pitch for a book he’s trying to sell, or have more sheeple hop on the Boaler bus, I’ll give this one a pass.

    “Experts” like Devlin need to stop thinking they know how kids think, and what they need. The information is already out there. We just need to acknowledge that our forefathers, and previous generations of teachers actually knew how to do their job when it came to teaching kids. Enough with the sales pitch Mr. Devlin. I’ll stick to what I know works for my own kids.


  3. “As I said in my last missive, I didn’t learn why it works until 10 years ago, though I did manage to major in math … ”

    You didn’t understand an operation for dividing fractions and managed a math major. Great! Jim Abbott was born without a right hand and he managed to pitch a no-hitter. Great! I still think we’d prefer for students to be born with both hands. We might even try to design life experiences to maximize the chances they’ll keep both hands.

    While both your experience and Jim Abbott’s could have been much worse, neither seems like anything we should wish for our students.


    • Wow, we now have someone who wants to change the way the entire way that math has been taught for 2 millennia…just cuz he says so!

      Unless Dan has proven expertise that he can cite regarding new and innovative ways to teach our kids mathematics, stay on the progressive Boaler bus. Experimenting on our children is not welcomed, nor is it supported.


    • Having understanding come at a somewhat later date is not an inferior type of learning. Probably part of normal human development. I have always loved history. I grew up in England and was immersed in history. I majored in history, loved it and did really well. When did I start to ‘understand’ history. About 20 years ago.
      Quite acceptable.


  4. “While both your experience and Jim Abbott’s could have been much worse, neither seems like anything we should wish for our students.”

    You’re cherry-picking to try to discredit all issues people have raised. What exactly is your proposal for “understanding” of dividing rational expressions, or do you only care about some sort of understanding for simple fractions? I was taught how invert and multiply worked 50 years ago and why it works. That’s a good goal, but it’s not sufficient. What about dividing 5 by 3/5? What understanding is required for that? Does it magically trickle down from concepts? What, exactly, are those concepts? How about 2 1/2 divided by 5? Do words (concepts) teach the understanding that comes with individual students having to actually do a large variety of homework problems? Does very limited and time wasteful in-class group work teach all of the understanding required for every unit of material? Anyone can reduce coverage and expect to do better even if they use hand puppets.

    For in-class student-centered groups, some get the light bulb effect, but the rest get directly taught (badly, perhaps) by those students. Also, do those epiphanies or the engagement actually translate to all of the other units that don’t have time for that kind of treatment? What proof do you have? How does it all fit in with homework and tests and where each student is headed in the future? No, we have people like Devlin redefining math and advocating low expectations for all students starting from Kindergarten, while at the same time claiming some sort of better understanding than what is achieved with the traditional high school math track. There is no such thing as rote learning where students can be successful in math with little to no understanding. That’s only possible if you are a really bad teacher. I taught math for years and nobody could do that. They might have gaps or need more work in certain areas, but the solution is not top-down from concepts as if skills and the ability to solve problems derives from that. It’s the other way around. Silly talk of math zombies is a vain attempt to filter reality to justify one’s beliefs.

    Mathematical understanding comes in levels and goes hand-in-hand with scaffolding and mastery of units carefully developed and tested with traditional math textbooks. You start with simple explanations of dividing fractions and then progress to dividing rational experssions. There are far too many of these concepts and understandings to cover them all with vain in-class epiphany sessions. Those ideas and processes do NOT translate to all other topics. That’s not how students prepare for the SAT, AP calculus, and the AMC tests – success is not some sort of magical top-down conceptual process. It comes with hard work from the bottom up on a whole lot of individual (not group) work. Full understanding comes from mastery of individual homework problems – problems that cover a LOT of variations that could never be solved from concepts. Understanding comes from the mastery of many layers of skills. Rote math success is completely wrong and made up to justify all sorts of educational silliness that tries to get something from nothing.

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  5. “One handedly”, I too managed to make it through the calc series, diff eq, linear and abstract algebra, and a bunch of stats classes without really understanding why invert/multiply works.

    I don’t know how I did it with such a handicap. I know that most of my “oh, I get it now!” moments came after hours of practicing familiar concepts until my brain managed to connect the dots. The invert/multiply dots connected when I started teaching.

    I’m not sure any lesson or teacher could have helped or motivated my brain do that.

    But maybe if Mrs. Pisauro had taught me a conceptual understanding of dividing fractions when I was in 7th grade like she should have, I would have already derived the Grand Unified Theory.


  6. Understanding is never simple or single layered. Most modern math “reform” pedagogues like to haul out the “invert and multiply” chestnut to claim the high ground of understanding, but we never hear what their proposed understanding is. A proof is not even sufficient for enough understanding to do all problem variations. And what happens to that so-called deep conceptual understanding when it has to be applied to rational expressions, like dividing 3(x-5)^3/(x-1) by 9(x-5)? What “conceptual” understanding is used to solve that?

    When I took algebra long ago in the real traditional era (not the last 20+ years of fuzzy curricula like MathLand, TERC, and Everyday Math, we had to do only one equation manipulation per line and list the algebraic identity or rule we used to make the change. We wanted to combine multiple steps into one, but we were not allowed to do that. I never hear modern pedagogues talk about how that is explaining your process. Perhaps it’s not wordy enough or perhaps it’s not new and different enough to be used as their claim to fame.

    Back when I taught college algebra, I used to require students to do that because there is a new form of understanding in the lower grades that uses words. Given an equation, I would have students start by circling all terms and include the sign with each term. I would then have them change all terms to rational expressions because a = a/1. Then I would have them circle all factors in the numerators and denominators. If the factor did not have an exponent, I would have them put a 1 there because a = a^1. Then I would have them move the factors around because a*b = b*a. Then I would have them move the factors up and down because a^-1 = 1/a. They would understand these manipulations because of the identities, not silly general words. I would have them explain with identities why a leading minus sign could be either a (-1) factor in the numerator or the denominator. I would have them “understand” that order of operations has a very limited meaning and does not indicate what can or can’t be done (a+b)+c = a+(b+c). I don’t hear modern reform pedagogues talking about this kind of understanding – the type that comes from working on and mastering all sorts of problem variations. Manipulating equations successfully is never rote. General conceptual words and even proofs don’t give you enough understanding to actually do problems. You get understanding from completing a long series of INDIVIDUAL homework problems (not time-wasting student-led group work in class) that are based on really knowing why you do each step of a solution – with identities, not vague words.

    The trashing of “invert and multiply” to claim a higher ground of understanding in math education is really simplistic and pathetic, especially since they never explain what a proper understanding is. They are not being honest. There is a whole lot of understanding that goes along with traditional math. They choose to ignore it and claim that those students are just “math zombies.” Meanwhile, all of the best math students are created by a traditional AP Calculus math sequence in high school. Unfortunately, it’s all over for many kids by the end of Everyday Math and CCSS NO-STEM math in sixth grade. Having no STEM curriculum path or options in K-6 is incompetence. You can’t increase the range of students in K-6 with full inclusion (also increased by parental help at home and with tutors) and then magically claim that you can do as well or better than before with more understanding. Garbage! Just ask us parents of your best students what we had to do at home. You don’t need research. ALL of my help for my math brain son was in K-6 with enforcing mastery of basic skills, not high school or college. Don’t you dare claim my son as an example of silly and low expectation reform math success and don’t call him or his STEM friends “math zombies.” Show me an example of a student at that level who is different and why that is the case.

    BTW, invert and multiply takes the denominator, inverts it and then divides it by itself 1 = a/a. Then it is multiplied by the original fraction since a = a*1. So a better question is how can you justify multiplying two fractions? We don’t hear about that nugget of understanding, just the invert and multiply. There are so many levels of understanding and it is really pathetic when the invert and multiply example is used to justify understanding they never explain or apply to all units of math.


    • I take your example of invert and multiply. Just show why, and how it works, and how it will always work. Explain that carefully and thoroughly. Best case scenario. Doesn’t take much time.

      Same with standard multiplication algorithm. Show kids what it’s doing. That will take 1/2 hour. Then they will know how and why it works, and they will and should use it, ever after.


      • Thanks for your comment, Matthew. In fact, I and many other teachers I know, do what you advise; I teach my students how it works and why. We teach many other concepts as well, despite the ongoing criticism that most teachers do not do that. With respect to the why, I do it in the context of solving equations in the form 3/4(x) = 5/7, say. I’ve explained this in other articles I’ve written. Some kids get it, some don’t, many forget it; they are anxious to do the procedure.


      • One-half hour will explain and create mastery of all variations “ever after?” Nope. try again. Mastery and understanding require individual daily homework sets and continual hard work. Ensuring that mastery is job number 1 for math and it is the goal of the traditional AP calculus track in high school – the path of all of the best math students. Many students never get there in the first place because of the silliness of K-6 math education – viz. MathLand, TERC, and Everyday Math which have ruled for the last 20+ years. EM “trusts the spiral” and takes no responsibility for any sort of learning other than the NO-STEM CCSS level. Traditional math has been gone from K-6 for decades, and integrated math lost the battle in high school to traditionally taught IB and AP math tracks. Still, many choose to trash traditionally-taught math even though they offer NO EXMPLE of success of their ideas. I would be their biggest supporter if they could do that. When my son applied to college, he had to show his math grades, his SAT I and II math scores, his AP Calculus score, and his AMC/12 and AIME scores. Success on those tests had nothing to do with fuzzy constructivist or discovery ideas. They were built on top of a traditional math education and individual nightly homework sets.

        All understanding is built on top of mastery of skills, but constructivist math pedagogues think that a very limited number of top-down in-class student-led group discoveries will get the job done. Nope. It all changes in high school because colleges and the real world drive reality down to the freshman year and the teachers are (at least) certified in math. The best math teachers my son had came from industry, not ed school. You would think that ed school pedagogues would hesitate just a little when we STEM parents complain or raise issues. We’re not stupid and only want what we had when we were growing up. Lose that argument along with the invert and multiply one. I once had a first grade teacher lecture me about the wonders of explaining why 2+2=4 in MathLand. It was astonishing!

        There is something wrong when we parents get notes telling us to practice “math facts” at home. Well, are they important or not? Why hide the skill tracking at home if it’s so important? We STEM parents create the best math students and educators use them as cover for their silly ideas. Low expectations and fuzzy NO-STEM math in K-6 is a fundamental systemic problem.


    • This comment is right on. It is not all/or nothing and the same chestnut examples are always given. For the most part. I was always given thorough explanations throughout my schooling, though they may well have missed the explanation re division of fractions. Yet, that is the example that is always pulled out. Unfortunately,all the other explanations that have always been routinely given are forgotten and the extremely good math instruction that has taken place is dismissed. And all this was done with much less money and much less fuss. And thank God, I now understand division of fractions.


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