How much deeper understanding do students really need, Dept.

In a recent op-ed in the LA Times, Dan Willingham, a professor in the department of psychology at the University of Virginia, addresses a particular aspect of math education in the U.S.  Blaming poor math performance on bad curricula, he argues, overlooks that elementary school teachers may not have the deep understanding of math that is required to teach it. In fact they may actually fear math.

Students without deep understanding, Willingham argues, may be limited to inflexible thinking. That is, their math knowledge is limited to performing specific operations for certain types of problems but they may falter when presented with problems in new settings or with slightly different wording. The result is an increasing number of high school students floundering in math “because the groundwork of understanding was never laid in elementary grades”.

Willingham suggests that the solution might then be to find and hire those teachers who have “deep math knowledge” and who know how to convey it. I have no problem with hiring teachers who have a thorough understanding of math. What troubles me are the premises that students are doing fine with math facts and standard algorithms. Also I question the disturbingly prevalent belief that student outcomes in math would improve if only they had a deeper understanding of math.

Students are not doing fine with basic math

While high school students may indeed be floundering, I disagree that it’s because it was not taught with understanding in earlier grades. In my opinion the contrary is true; there has been too much of an emphasis and an obsession with understanding in math in elementary schools.

Over the past three decades—in large part propelled by NCTM’s standards that came out in 1989—the preoccupation with understanding has manifested itself with a de-emphasis on learning math facts. Also, standard algorithms for the basic operations are delayed while students are presented with alternate strategies that require making drawings or using convoluted methods. Such methods are nothing new; they were taught in the past, but after students had learned and mastered the standard algorithms. Now, however, they are taught first in the name of providing the conceptual understanding behind why standard algorithms work as they do. Simple concepts are made more complex under what passes as “deeper understanding.” Students I have seen entering high schools do not know their math facts, and use alternate inefficient strategies for simple operations such as multiplication.

 The codification of reform math ideology

The Common Core standards have effectively cemented in the math reform ideology that is increasingly incorporated in today’s elementary school textbooks. Adding to that are the bevy of ineffective teaching methods (inquiry- and problem-based learning, group work, so called differentiated instruction) pushed upon teachers in ed school and in professional development seminars.

Furthermore, they are told in ed schools, in professional development seminars and other sources that memorization is bad and that teaching standard algorithms “too early” eclipses understanding. Teachers who elect to teach standard algorithms and teach in traditional manners are sometimes told to teach their lessons with “fidelity” to textbooks they are required to use. Young teachers who fear for their jobs will do so. Older teachers who may have the understanding that Willingham would like to see are sometimes told the same. Unlike the younger teachers, however, the older ones can simply retire.  And unlike the older teachers, the younger ones are likely the products of the ineffective math teaching and are probably just as confused about math as many of the students we are seeing today.

Given all that, I would agree that having more teachers with a better understanding of math might help the situation. The improvement would come from not only from a larger knowledge base. Specifically, in greater numbers such teachers are more likely to be able to reject the nonsensical approaches foisted on them and use the resources and methods shown to effectively teach students math.

What is “understanding” in math?

Willingham admits that hiring teachers with better understanding might be difficult because state tests have been shown to be inaccurate predictors of who teaches well. Adding to this, is confusion about what constitutes “understanding”. What educationists believe is understanding is in most cases visualization—drawing diagrams that demonstrate what two-thirds divided by three-fourths looks like.  That is not at all what a mathematician means by understanding.  Also, being made to use formulaic “explanations” and dragging work out far longer than necessary with multiple procedures and awkward, bulky explanations is not a sign of understanding. Forcing students to continually stop and explain becomes nothing more than “rote understanding” in the end.

There are levels of understanding that vary depending on where a student is on the novice/expert spectrum.  Novices do not learn like experts; it takes time for knowledge to accumulate with procedures and understanding working in an iterative fashion to support each other. Insistence on understanding at every point where students should be learning procedures while working effectively with their beginning levels of understanding lacks educational value.  With the prevalence of misunderstandings about what understanding is, the criteria for hiring teachers who possess understanding might well result in hiring more teachers with the same misguided views.

Another misconception of understanding is the notion that students who know why a procedure works are in a better position to solve problems via enhanced flexible thinking. In an article by Greg Ashman, he describes a study which suggests that it is a mistake to assume that students in possession of conceptual understanding will use it.

While I disagree with Willingham’s point on understanding in his recent op-ed, he provides, in my opinion, a more apt characterization of the interaction between understanding and domain knowledge, in an  earlier article on Inflexible Knowledge . In it, he says:

“Understanding the deep structure of a large domain defines expertise, and that is an important goal of education. But if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots. There is a broad middle-ground of understanding between rote knowledge and expertise. It is this middle-ground that most students will initially reach and they will reach it in ever larger domains of knowledge.”

Simply put, no one leaps directly from novice to expert. For sure, teach math with understanding, but don’t obsess over it. Teach the math students need to know.



11 thoughts on “How much deeper understanding do students really need, Dept.

  1. Totally agree. The scary fact that I think teachers don’t like to acknowledge is that students remember very little of what you teach them; they certainly won’t remember the “deeper understanding” aspect. That only comes with time, maturity, and practice.

    The main benefit to teaching the “why?” is that failure to supply the answer to that is that some students won’t buy into what you’re selling if you don’t. Once you tell them why something works, that obstacle disappears and they’re willing to learn what you’re teaching, although they later can’t remember your explanation.


  2. That was my reaction to Willingham’s op-ed as well; he has a history of being sensible and correct but it is too easy to take this one as an endorsement of Common Core or the NCTM non-standards. The deep understanding that teachers need to have and to develop in their students is not a bunch of inefficient approaches or alternative algorithms instead of the standard algorithms of arithmetic and their usage. The correct deep understanding is as described by Liping Ma in her wonderful little book, Knowing and Teaching Elementary Mathematics. She called it PUFM, Profound Understanding of Fundamental Mathematics, and assessed US elementary mathematics teachers negatively in comparison with those of her native China. The Chinese teachers were not only competent in the algorithms of arithmetic (that could not be consistently said of the US ones) they were able to “on the spot” create little word problems for which their solutions required the arithmetic. In this, the US teachers were almost at a complete loss. The Chinese teachers’ years of preparation, by contrast with ours, reflected that PUFM already at their early professional preparation. They were recruited at roughly our high school junior year followed by only a year or two of something akin to our old “normal schools”, the one year of college my mother had in preparation for teaching elementary school. New teachers are not then simply dumped into their own independent classrooms but work closely with experienced teachers who have been deemed particularly effective. What a concept.

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  3. For a lot of people, it’s all about them – their area of specialty. It’s fine that he raises the math “anxiety” issue for elementary school teachers, but he then uses that to push his own (non-math expert) idea of what the solution is. His only real contribution to the debate is that because of the average “math anxiety” of non-math-certified teachers, then schools should hire specialists. I have no argument with that.

    But then he goes off his area of expertise to discuss the “mediocre predictors” of using math teachers who have (Ed-school) specialized knowledge. Um, how about trying to dig a little bit deeper and reviewing your assumptions? How about asking us parents of the best (STEM-prepared) math students what we now have to do at home. Rather than using guess and check when state test averages come out, one can dive deeper into what’s going on – look at success cases rather than average failure statistics. He wants more vetting from the educational world without (after ALL OF THESE YEARS OF INVOLVEMENT) listening to what practicing STEM parents and experts say. He is not new to our views.

    Many dislike anecdotal knowledges and cases, but anecdotes contain all of the information of what’s going on. If you study enough of them, then you will be able to split the issues and see patterns. If you just look at yearly state test results to decide “one thing” to fix (“additional vetting … in the classroom”), then you will always fail. Those who do the vetting are part of the problem, AND there is no ONE problem. You can’t collect “big data” and assume that all of the important information remains or that your biases won’t make you to see only what you want to see.

    Can we make non-math people understand what we know about math understanding? Can we make them know that properly mastering skills at all levels is never rote – that there are many levels of understanding? Can we make them see that there is a problem with a low CCSS slope to no remediation in college algebra when it starts in Kindergarten? What percent of the kids who get to a proper Pre-Algebra STEM track in 7th grade get help at home with mastery of skills? How difficult is that to “research?” How difficult is it for people to question their assumptions and see the world as it is rather than how you want it to be? STEM people have a lot of experience in that area.

    Willingham should know that this is a big systemic and psychological problem, not just one about “understanding.” Maybe he thinks the problem is us math experts, but we’re the ones who have lived it day-by-day, have done actual teaching in class, and have ensured that our kids are STEM ready. However, I had my son’s Kindergarten teacher lecture me about “understanding” in math. I’m not kidding. His first grade teacher told me that “Yes, he has a lot of superficial knowledge.” Anecdotes bite, but big data hides. Yearly state tests pass the buck so that parents are left a year late and many tutoring dollars short, and the big 7th grade math tracking decision ends up as a big whack in the head by a brick. I’ve talked to those capable kids and their parents. It’s too late for most. Their approach has been going on for at least two decades. Where are the results? When will reality break through assumptions? Willingham should study that.


  4. Also, I wonder if there is any bottom to deeper knowledge (a.k.a., “depth of knowledge” (DOK)) or deeper understanding. If there is no clear end to the process, it may be no more useful than an infinite programming loop or a Rube Goldberg machine. When I hear calls for deeper thinking or deeper understanding, my mind pulls up a picture of a child repeatedly asking “why?” after every explanation a parent gives them. Every why could be a valid question that could evoke a reasonable, substantive response. But, annoyance factor aside, at some point the next additional answer is not adding a whole lot to a child’s understanding of the initial concept. At some point, the next tiny possible addition to the child’s understanding is outweighed by the possible subtraction from understanding from the loss of clarity (due to the amorphousness and weight of all possible relevant explanations, no matter how tangential) and from the loss of time–there are opportunity costs. Conceivably, a student could spend twelve years learning basic arithmetic very, very well. By, in doing so, they will not be exposed to all the other math topics. Isn’t this why the allegedly “deeper” and “more rigorous” Common Core math is so slow?


    • As a teacher in the CA public ed system for 31 years, I totally disagree with the ill-conceived DOK approach. Many teachers stop and dig down for months on one concept. I do not see the effectiveness of this approach. I feel it is best to expose concepts by dipping their toes into the water and moving on. I trust that they will see this “stuff” again next year assumming their teacher gets to it.

      Making a single Math concept so impotant translates to the students that it is something so difficult that it requires spendimg months on this one concept. Oddly, teachers “teaching” these concepts can’t produce any DOK beyond what they read in the teacher manuals.

      Sadly, by the end of the year, other teachers at my grade level are still on beginning-of-the-year concepts while finding themselves skipping around lessons to the end of the textbooks in panic mode to prepare for the state required testing. It is so sad to see kids in other classrooms with a horrific look in their eyes because their teacher “did not teach this to us.” And my kids feel good that they “know” stuff the other kids don’t.

      And the students are correct!


  5. My view is that the understanding meme in K-6 is just a cover for lower expectations. The big change since I was young is full inclusion. In high schools, this is done in a full inclusion environment, but in K-8, it’s done in an age-tracking full inclusion academic environment for social reasons. They think they can deal with it using differentiated instruction, or more usually, differentiated learning with the onus on the child and parents (Come to the open house on MathLand. Been there. “Explain why 2+2=4”)

    However, this requires a deemphasis of quantitative mastery skills. They have to “fuzzify” the difference between the best prepared students and the worst prepared students. In comes vague conceptual understanding ideas as the basis for proper development in math. Add to that their love of mixed-ability group learning that assigns too much importance and value to engagement. Perhaps they see our kids doing well, but don’t ask us what we do at home. We STEM parents will not let our kids fail, so we ensure proper mastery at home. That’s what I had to do for even my “math brain” son in K-6. I have so many examples of K-6 teacher-to-parent push-backs in terms of their assumptions and how they do things. We would never dare to challenge that. Thankfully, that all went away in high school.

    I like your “why-why-why” explanation and the issue of opportunity costs. Should students need to understand base systems, like octal, in K-6? I also like to talk about how much understanding goes into learning to master the basics of telling time, counting change, and even memorizing (and using!) the times table. Many educators take that for granted so they don’t see the low level understanding that goes on. At the other end, when I had to prepare to teach a course in linear algebra, I found myself understanding so much more in terms of generalized linear spaces. I tried to teach that to my students, but knew that it wouldn’t sink in very far.

    I think a lot of this is their idea of natural learning where they present the grand ideas and provide engagement so that the rote skills will flow automatically. Just look at the music world. It ain’t gonna happen. Practice your scales, exercises, and etudes. Those are not just musically rote. There is subtle musical understanding going on. Educators have to deal with reality and not hide behind the vagueness of yearly tests. I was on a parent-teacher committee once where we discussed the school’s lower “problem solving” scores. Their solution? Spend more time on problem solving. It’s not rocket science (or STEM) thinking that’s going on here. It’s much simpler, but more difficult, than that.


    • Had to respond to the part about “counting change” as seen as too low-level!
      True story, my assistant teacher was at a counter restaurant a few weeks ago. She actually paid in cash, and the cashier was flustered and called a manager. So as she and her husband were waiting for their change, the manager finally appeared to tell the cashier that she needed a quarter, a dime, a nickel and three pennies. The cashier didn’t even know money, even though the computer told her the change!


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  7. I love the term “rote understanding!” That’s is exactly what the “deep understanding” in elementary school has become, without a doubt.
    I teach in a parochial school, so my experience is much different than that of a public school teacher. But I agree with you, Barry, that I am not quite convinced that students are fine with the basics – the facts and the algorithms. Most of the 9-12 students at our K-12 campus are, thanks to the fact that our elementary school ditched the EveryDay Math program about 7 years ago.
    As the department chair, I do the placement for the incoming high school students. Without fail, almost every student (with the exception of those with diagnosed learning disabilities) from our elementary feeders and middle school all place into Algebra 1 or higher. Almost every student coming to us from public school is very far behind in terms of mastery of the basics. There are exceptions of course; but I teach the Foundations class for 9th graders who aren’t quite ready for algebra. And of 31 students, only 4 are from parochial or private schools. And I have at least 8 students in this class who still do not know basic facts. I spent the first 2 weeks of school doing fact families, and trying to get students to “see” the relationships between the numbers and positions for addition/subtraction and multiplication/division.
    I think the other angle to this situation is the problem of grade inflation. Many schools assign grades based on good habits, like turning in all work on time (for completion, not correctness) and weighting assessments rather low compared to this. I have talked with 8th graders at registration night who say, without any sign of cognitive dissonance, that they have an A in math at their public school despite not having a clue as to what is going on. Of course when they get recommended for foundations, the first thing parents point to is “Junior has an A in math now! How can this be?”
    So grades assigned with good intentions, because the students are still “learning” the material, can add to a student’s and parent’s feeling that everything is fine…until the 4.0 student gets a 21 on the math section of the ACT.


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