Count the Tropes, Dept.

Another in one of an infinite series of articles that has appeared at Forbes and other like publications about how the Common Core math standards are being maligned has surfaced. I gave up counting the tropes in this one.  The author makes the assumption that because students were unable to apply standard algorithms or “rote” procedures to “advanced math” (left undefined in the article), that teaching procedures is tantamount to “no understanding”.  There is no mention of the difference between how novices or experts think–and in fact, going out on a limb here–I would venture to say that the author of this piece probably learned things in the manner that he now derides.  He makes this statement:

There is nothing in the standards that specifically requires any instructional strategy to be taught. There is a great deal of academic freedom for teachers to teach in whatever manner they wish. The CCSS does not require this kind of new “number sense” methodology.  What it does require is that students learn multiple ways to attempt to solve a problem, to promote strategic critical thinking and creative problem solving.

I might ask where in the Common Core standards does it require that? The Standards of Mathematical Practice?  Maybe, if you interpret it that way.  The CC standards also have a standard requiring that students learn the standard algorithm for multidigit addition and subtraction by 4th grade. This has been interpreted to mean that the standard algorithm should not be taught until then, but both Jason Zimba and Bill McCallum, lead authors of the CC math standards have stated publicly that the standard algorithm can be taught earlier than 4th grade.  In fact, Zimba even recommends that students in the first grade be taught the standard algorithm.

Once the standard algorithm is taught, then alternatives to it make more sense than the other way around. In that vein, multiple methods can occur if the sequence of topics is done in a logical order.  I’ll go out on another limb and say that this is mainly because procedural mastery is more easily obtained via the standard algorithm than by pictorial or inefficient methods.  Ironically, students who have mastered the standard algorithm often find short cuts themselves.  Or they are taught the short cuts after mastery of the standard algorithm. In any event, teaching “number sense” and top down understanding before the foundation of procedural mastery is obtained seems to be the go-to advice of reformers.  Such approach makes the same mistake twelve-year-olds make when they assume they can achieve stardom in whatever field they daydream about, without the requisite practice and work.

Greg Ashman gives a good example of this in his blog:

If you have been practising column subtraction and you are presented with 2018-1999 then, even if you could work it out by counting up, you will probably draw on the most familiar strategy because you lack the bandwidth to analyse it at the meta level. If I really felt it was important for students to be able to determine when counting up is more efficient than column subtraction then I would teach this to students explicitly. I would use contrasting cases where I presented two examples such as 3239 – 2747 and 2018 – 1999 and I would work both problems, both ways, drawing a distinction between the two.

…[T]he idea that there are teaching methods that are better for conceptual understanding than explicit teaching is equivalent to the idea that there are teaching methods that can short circuit the novice-expert continuum and teach expert performance from the outset. This is the progressivist dream and one that, after a couple of hundred years, is yet to be realised.

This article originally appeared in Quora, and was picked up by Forbes. If you are interested, I wrote a comment on the Quora piece which addresses other aspects of his arguments.

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One thought on “Count the Tropes, Dept.

  1. Whether or not one learns to count up, a general technique still has to be learned and mastered. Instead of multiplying by 5, divide by 2 (which is easy to do left to right) and move the decimal place over one. You still have to learn how to multiply numbers with no short cuts. There are many of these things. Do they teach them all or do they think there is some magic transference? What about in algebra? Some of them go out of their way to find problems that seem to be a solution of N equations in M unknowns, but have a simpler way to eliminate unknowns. The problem is that these things do not eliminate the need to master the general methods. However, the fuzzies are now declaring them as some sort of litmus test of understanding or whatever it is that they want. They want to claim the high ground of understanding, even over AP/IB, but can’t really explain what it is. Go ahead and give tests with both 3239 – 2747 and 2018 – 1999 as problems. Compare students from a traditionally taught math sequence with those using their sequence.

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