Another bite of pie . . . and common core math standards

Well, I ate far too much pie and sat stalled in traffic far too long waiting to get back into New York City via the Holland Tunnel. Happy belated Thanksgiving to my blog readers!

Now that I have Internet access again, I’ve been checking my various education sources and came across this interesting follow-up on the common core math standards. The author is a secondary school math teacher who writes frequently on education policy (and like me is a second career teacher).

I teach math only as needed for economics – which means I actually end up teaching a fair amount of math – but I don’t consider myself remotely expert in this area.

Still, one comment from the article resonated with me:

Let’s look first at the 97 pages of what are called “Content Standards.” Many of these standards require that students to be able to explain why a particular procedure works. It’s not enough for a student to be able to divide one fraction by another. He or she must also “use the relationship between multiplication and division to explain that (2/3) รท (3/4) = 8/9, because 3/4 of 8/9 is 2/3.”

It’s an odd pedagogical agenda, based on a belief that conceptual understanding must come before practical skills can be mastered. As this thinking goes, students must be able to explain the “why” of a procedure. Otherwise, solving a math problem becomes a “mere calculation” and the student is viewed as not having true understanding.

This reminded me of a ditty that helped me learn how to divide fractions back in the Triassic Era. I confess that I passed this advice along to my own children when they were struggling with elementary school math:

“When dividing fractions don’t ask why; just flip the sucker and multiply.”

Do I really agree with my own advice? Yes and no. It’s helpful to understand why dividing by a fraction yields a larger number – not least as a reality check against the answer that the calculator throws out. I also think that we can teach this concept with some of the manipulatives traditionally employed in math education. Grandma divided the cherry pie into 8 slices, but there are 12 people at the table. So each person gets 2/3 of a slice. 8 divided by 2/3 = 12, right? Try it with cookies you’ve baked for the class, and any teacher has a winner.

But 3/4 of 8/9s? That’s both difficult to demonstrate visually and difficult to grasp conceptually. I’m all for cutting up the pie or cookies once . . . and then flipping the sucker over and over again until the algorithm becomes second nature.

Teachers don’t have unlimited time. One colleague, who really is an expert math teacher, tells me that her biggest problem with the standards is not so much their content as their requirement that she teach twice as much material to students who are struggling with the material she’s already trying to present. If conceptualizing every problem really helped students learn more math more quickly or more thoroughly, I’d applaud this approach. But I worry that in practice “discovery” will drive out drill without producing much more conceptual understanding.

The seniors I’ve taught concurrent enrollment (college credit) economics would almost all have benefited from a lot more drill. Almost none of them, for example, can calculate percent change until I’ve retaught the algorithm . . . er, concept. On the other hand, my students would also benefit from a more solid conceptual understanding. I’m always startled at how few students can look at a linear graph and identify a direct or inverse relationship, based on the positive or negative slope – or explain what difference direct or inverse makes to the mathematical relationship they’re exploring. That should be second nature for students who all have a couple of years of algebra under their belts, right? But it’s not.

So what do you think?



  1. Trax

    I agree and yet I disagree with you. The level you are talking about is higher than when the (for example) concept of dividing fractions or similar thoughts should be taught. Little kids can get the ideas of fractions using manipulatives without knowing algebra and definitely before they are shown the algorithm. This can give them a working model in their heads before they are ever exposed to the abstract concepts and notation given to more mature students. I think this is what the Common Core is aiming at. Middle school or high school teachers should not have to teach the basics any more because the students got it much earlier in life. At least that is the thought. : )

    • Mary McConnell

      I am not one of the common core critics who assumes that the new standards represent a government plot to nationalize education. My concern is simply that past efforts to make math more conceptual have often (especially in the hands of less gifted educators) led to kids mastering neither the concept nor the algorithm. Note that my example included starting with a manipulative-based conceptual understanding . . . and then following up with LOTS of practice.

      The good, and bad, news is that math is more easily tested than many subjects. If students prove more able to solve math problems, and especially more conceptually difficult math application (story!) problems, then I’ll be cheering along with the common core supporters.

  2. Sara Davis

    If your students had been allowed to construct the concepts (construct has a little different connotation than discover) that you are using perhaps they would be able to do what you expect them to do. I suspect they were taught with a lot of drill. Teaching for meaning is one of the constructs for the CCSSM and teaching for meaning has been around forever, but few teachers understand the full implications. I have issues with some of the CCSSM, but the emphasis on math practices is good.

  3. Karlene Johnson

    It is interesting that when students find it difficult to do a mathematical procedure further down the educational journey, we automatically assume that they did not get enough practice with the algorithm. If find this a peculiar assumption because the vast (and I do mean vast) majority of our schools never have got on board with the reform movement and still use large quantities of skill and drill work. This also applies to other disciplines besides math. When I taught junior high, I was astonished that the spelling words my students were working on included ‘egregious’ but on a very regular basis I had students writing the word ‘does’ like ‘dose’. Upon investigation, I discovered that they had had weekly spelling tests since first grade, so spelling drills didn’t really ‘take’. While spelling (at least to me) seems far more arbitrary than mathematics, I think it is just as reasonable to assume that skill and drill without understanding is not the most efficient method of longterm learning that we imagine it is. I work with college age students now. They can add, subtract, and multiply very well and I imagine that they learned this with skill and drill, but they have also had very longterm experience and practice with the concepts. I have noticed, however, that the later the concept appears in the curriculum, the poorer they understand it AND the poorer they are at algorithmic knowledge. In discussing the situation with students, I have NEVER once had a student tell me that they had a chance to discover the knowledge for themselves. They always report that they did lots and lots of practice, but it didn’t really make sense to them so they have a hard time remembering it. Please do not misunderstand me. I am not saying that practice opportunities have no place in the curriculum. I am saying we need to examine our assumptions about ‘drill’ and I think it is time that we revamp our understanding of ‘practice’ to include sequenced practice that requires students to continue to be ‘mindful’ rather than to do it by rote which implies that it is so automatic that is it ‘mindless’. My older daughter was in Singapore Math and I loved how the exercises were set up in such a way that the practice exercises grew continually more complex so that you had to keep making sense with each new problem you encountered. This wasn’t drill, it was deep conceptual AND procedural development. It is time for us to think of these two ideas as ‘both/and’ rather than ‘either/or’.

    • Mary McConnell

      I, too, love Singapore Math, and agree with you that a combination of constant practice and conceptual teaching works best. My concern about a focus on conceptualization is that it often comes at the expense of practice (sure, let’s not use the word drill.) Singapore Math has another element that students need (and tend to hate as much as drill): lots of “story” problems that require them to apply their growing math skills.

      Online math teaching actually offers excellent opportunities to accomplish what you’re suggesting, since a good computer program will catch repeated errors and help students correct them, while enabling students who have mastered one concept to move on to more complex applications.

  4. Linda Arnold

    e “understand” problem does not require real understanding. For instance, if I said “understand that 10 /2 = 5 because 2 * 5 = 10”, the vast majority would approach this as “multiply the last number times the answer to get the big number” and “of is a keyword for multiply”. That is must memory without understanding. There is a great deal more to understanding fraction division that just doing sonmething that amounts to reciting “divisor times quotient = dividend”.

  5. Stan

    Kids are good at following rules. Rules and routines are pretty standard in the day to day so it may be no wonder. Literature will support the idea that when you are able to explain a procedure or concept, your knowledge is greater than than of just following rules. So it this then a method of assessment or a structure that should be enbedded in instruction? I do not know.

  6. Stacia

    My struggle with the common core is not that they want to require a more conceptual understanding of mathematics, it’s that they require that conceptual learning at too young an age/stage and require too much to be taught in a year.

    Students in first grade are introduced to variables. Why? Why is it not enough for students to finish first grade being fluent with simple addition and subtraction? Yes, teach them with manipulatives, teach them with a number line, then drill them to pieces. If a student needs to use their fingers for 2+7 then they are going to struggle with 12+17. Let’s require mastery of 2+7 before moving onto 12+17.

    Common core also requires too much material to be taught in one year. I’ve heard teachers say from the time I was young (I come from a teacher family) “We teach a mile wide and an inch deep.” So, while we should be teaching maybe a foot wide and a mile deep, common core says, “Nope! We want you to teach a mile wide and a mile deep to younger students in the same amount of time.”

    If drill is done properly, it will be beneficial to students. Doing a lot of problems, is not necessarily drilling one bite-sized concept. My suspicions are that students are asked to do a lot of problems, but not all of those problems require the same algorithm/process and so they are not being truly “drilled.” Math seems to be the only area in life where the proverb “practice makes perfect” is poo pooed. Piano teachers drill scales before Mozart, basketball coaches drill dribbling and passing before they play their opponents, etc.

    As a high school math teacher, I am really passionate about math education. I really desire that students learn math. I want to live in a world where math does NOT equal hard in 98% of the population’s mind! But it has to start with every student learning the basics. After all, I shouldn’t have students counting on their fingers… but I do.

    • Mary McConnell

      Ah, a defense of drill. Thank you.

      I, too, draw the analogy to sports and music when I’m assigning grammar and usage drills to my writing students. Why, I ask them, does their soccer coach or piano teacher force them to practice the same kick or scale over and over again. They invariably give me the right answer: muscle memory. When it comes time for the big game or the major recital, they want some movements to be automatic, so that they can focus on the hard stuff. Writing is like that. So’s math.

  7. Barry Garelick

    Thanks for the link to my article; I appreciate your thoughtful analysis as well as the comments. Stephanie Sawyer recently left a comment at the article that was also quite insightful. Fee free to contact me regarding issues on math education.

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