Common Core math: deep or dull?

According to a New York Times article by Motoko Rich, parents and students are finding Common Core math not only confusing but tedious and slow.

To promote “conceptual” learning, many Core-aligned textbooks and workbooks require steps that may be laborious for students who already get it. A second-grade math worksheet, pictured in the article, includes the question: “There are 6 cars in the parking lot. What is the total number of wheels in the parking lot?” To answer the question, the student drew six circles with four dots within each. (Actually, this doesn’t seem new; it reminds me of “New Math” and “constructivist” math.)

One nine-year-old, apparently weary of this kind of problem, stated that she grew tired of “having to draw all those tiny little dots.”

Students with good understanding may be put through steps that seem redundant to them. If they skip those steps, they may be penalized.

“To make a student feel like they’re not good at math because they can’t explain something that to them seems incredibly obvious clearly isn’t good for the student,” said W. Stephen Wilson, a math professor at Johns Hopkins University.

One reason for emphasizing “conceptual” learning is that employers apparently are demanding critical thinking. Several questions remain to be answered, though: (a) whether Common Core math–in its current forms–really is promoting conceptual learning; (b) if so, whether it also promotes math proficiency; (c) whether the current approach is benefiting students at the upper and lower ends–and those in between, for that matter–or holding them back; and (d) whether this is the kind of “critical thinking” that will serve students well in college, the workplace, and elsewhere.

I will comment briefly on the first question; I welcome others’ insights.

Tedium and depth are not the same. One can go through a long explanation of a problem without gaining any understanding; one can solve a problem quickly and come to understand a great deal.

In sixth grade, in the Netherlands, I learned mental arithmetic: I learned to add, subtract, multiply, and divide double-digit numbers in my head, using all kinds of tricks that the teacher taught. Those tricks enhanced my understanding of what I was doing. I enjoyed the swiftness and ingenuity of it; I would have detested it, probably, if I had to write it all out, step by step, and illustrate the steps with circles and dots.

Detailing and explaining your steps is a worthwhile exercise. But part of the elegance of math has to do with its mental leaps. Sometimes, when you do steps in your head, or when you figure out which steps in a proof are assumed, you not only understand the problem at hand, but also see its extensions and corollaries. Sometimes this understanding is abstract, not visual or even verbal.

There seems to be an unquestioned assumption that one comes to understand math primarily through applying it to real-life situations; hence the Common Core emphasis on word problems. While word problems and practical problems can lead to insights, so can abstract reasoning, and so can models that bridge the abstract and the concrete, like the multiplication table.

Yes, the multiplication table–horrors, the multiplication table!–abounds with concepts. If you look at it carefully (while committing it to memory), you will see patterns in it. You can then figure out why those patterns are there (why, for instance, any natural number whose digits add up to a multiple of 3, is itself a multiple of 3). (Something similar can be said for Pascal’s triangle: one can learn a lot from studying the patterns.)

In other words, conceptual learning can happen in the mind and away from “real-life situations”; it need not always be spelled out at great length on paper or illustrated in terms of cars and wheels. Nor should students be penalized for finding shortcuts to solutions. Nor should memorizing be written off as “rote.” Yes, it’s good to understand those memorized things, but the memorization itself can help with this.

In ELA see a similar tendency toward laboriousness (that likewise long predates the Common Core). Students are required to “show their thinking” in ways that may not benefit the thinking itself. For example, they may be told to explain, at great length, how a supporting quotation or detail actually supports their point–even when it’s obvious. Students with economy of language (and, alas, clarity of thought) may lose points if they don’t follow instructions. Instead of being at liberty decide whether an explanation is needed, they receive a message along the lines of “Explain, and explain again, and then explain that you have explained what you set out to explain.”

Critical thinking is important–and one should think critically about how it is conveyed and taught.


  1. The intro scenario is just another case of a child who is not placed by instructional need, and is instead forced into a situation with children who are developmentally delayed.

  2. Roger Sweeny says:

    It would be really nice if we knew what we wanted (what exactly is this “critical thinking”?), if we then had a good way of measuring whether young people had it, and if we then ran experiments seeing how different curricula and techniques affect student’s getting it.

    None of those “ifs” are true. Is it any wonder there is so much change in this business and so little progress?

    • When I was young and naive, I thought that people would be influenced by evidence and reason. So when I read about that “horrible” Everyday Math I decided to check the research. What I found out is that in a wide variety of gold standard random assignment studies, Everyday Math has been found to have a positive effect on student achievement. Far from failing to teach basic facts, Everyday Math spends every fifth lesson on nothing but reviewing math facts and committing them to memory.

      I posted this information, foolishly thinking that people would be influenced by actual evidence. Instead I was wildly vilified and denounced. I am confident that is what will happen now. There will be a flurry of angry posts denouncing me as an evil constructivist. What there won’t be are any references to scientific random assignment studies that show Everyday Math doesn’t work.

  3. It seems that, as popular constructivist curricula like Everyday Math, that the goal is to make math less “mathy” (ie. show your work – in appropriate situations) and more wordy; thereby making it more difficult for those kids whose verbal ability lags behind their math ability (often boys). Also, many of the test questions I have seen are poorly worded, even to adult perceptions, to the point of total confusion for kids – even those who are able to do the math. That said, much of the problem (as in school in general, esp ES-MS) is the refusal to group by instructional need. Even twins may need very different instruction/grouping.

    • Less mathy – and more writing about math – written explanations for even basic fact work. Also – no emphasis on mastery.

      • In other words, an emphasis on the development of verbal skills rather than analytical skills. It is much the same as trying to teach French by having the student write out an explanation of how they developed the reply

        • in English.

          (Sorry, the comment box disappeared on me before I completed the sentence. )

  4. I’m sorry to be so slow, but could someone please explain why it’s even remotely reasonable to rethink the way we teach mathematics or arithmetic. We’ve been teaching arithmetic (with Arabic numerals, zero, positional notation and the basic algorithms) for about 1,000 years. Shouldn’t those who want to change the plan have an incredibly heavy burden to show that they’re smarter than anyone in the last 1,000 years?

    Most of this new math (or new, new math, if you like) seems to come with little in the way of serious, peer-reviewed study. Why do we even give these people a hearing?

    • Rob – Perhaps not quite a thousand years but I think that basically the elementary arithmetic taught in the schools was more or less codified by Simon Stevin (1548-1620).

    • wahoofive says:

      Rob, there’s a general consensus in America that what we’ve been doing in schools is not working, that far too many students are coming out of elementary school (even high school) not really able to do arithmetic. You don’t have to agree with this, or agree that the new ideas are any better, but that’s the motivation behind these changes.

    • That’s easy. No one has ever gotten tenure or research funding promoting the continuation of tried and true methods. It is a flaw in the production of Ph.Ds in education vice the traditional Master Teacher. The latter promotes sound, time-tested methodologies, the former only gets ahead if they discover “new” methods.

  5. wahoofive says:

    This topic strikes me as a specific example of a general principle which is endemic in all subjects: the tension between *learning* a subject and *demonstrating to someone else* (i.e. a teacher) that you’ve learned it. It’s the same thing with reading a novel and understanding it, then having to write a paper about it. Writing the paper may or may not contribute to your own learning process, but it proves your understanding to others.

    The good news when you learned “mental math” is that there was an easy way to test your mastery of it, by giving you arithmetic problems and see if you got the right answers. But “critical thinking” isn’t as easily assessed. Even if we all agreed that conceptual learning was an important educational goal (which I realize we don’t), it’s always going to be difficult to test for, and inevitably lead to some duplicated effort.

    But arguably that, too, has some merit. No matter how well you understand something as an adult, you have to find ways to communicate that understanding to bosses, clients, judges, or members of Congress. All math students hate having to “show their work” but there are good reasons to require it.

    • J.D. Salinger says:

      If “explain” means “showing your work” I have no problem with it. If it means writing an explanation or giving an oral explanation, I have problems with it. If a student can solve the problem “How many 2/3 oz servings of yogurt are there in a 3/4 oz cup of yogurt” by dividing 3/4 by 2/3, and doing so correctly, but cannot explain why the invert and multiply rule for dividing fractions work, would you say the child lacks understanding? And why is it absolutely necessary that understanding of conceptual underpinnings be part of procedural mastery at every point? Sometimes understanding comes first–sometimes later.

  6. If I had to waste time explaining how I reached a result in math and I had shown all the work on the paper, i’d call the instruction a idiot.

    In the real world, which academia isn’t, you don’t have time to sit there and explain how you got a result, but then again, I’ve seen people who can’t figure out sales tax on a sale when the register is on the fritz, or who struggle to make change when the sale amount is $16.80 and I hand them $17.05 (I would like a quarter back), but the cashier has the deer in the headlights look.

    Making math more ‘wordy’ isn’t going to help kids learn it any better, which is by lots and lots of practice. Kind of like playing a musical instrument, you won’t get better at it unless you practice (unless you happen to be a savant who can pick it up and play like a superstar).


    • Ann in LA says:

      The explanations also slow down instruction. If you are constantly stopped from doing math so you can take a time out to do ELA, there will be less time for math. To compensate, do curricula simply lop off topics to fit the time available?

  7. Indeed, it seems like a student almost gets punished with this approach if s/he can synthesize and apply their thinking too quickly. Then again, we don’t have Comnon Core in Virginia and I don’t know as much about math education.

    As for reading, we have the same problem under the Virginia ELA Standards of Learning–tons of reading strategies and having to show your thinking when it’s not necessary and tedious. I am really hoping that in middle school that the reading strategies curriculum gives way to a more rich and meaningful curriculum.

    • I’m cynical enough to think that giving credit for talking/writing about the process/answer (in the preferred way, of course) is a way of giving kids who can’t GET the right answer/can’t DO the math good grades – the same goes for group work and “creativity” (in ES, often meant as artsy/crafty – ugh).

  8. i wonder if they give credit for 30 wheels as an answer if the student draws a steering wheel on each car.

    • Nope. If the teacher has done her job differentiating for her gifted students, those students will know not to add in the steering wheel, the spare tire, and the hot wheels car the toddler is waving around if they want full credit. If not, she will learn from those low scores and decide if she wants to differentiate appropriately the following year. She very well may let it go, as these types of boys are not teacher pleasers, and a low score will mean no one will question the decision to leave them out of honors classes in the future.