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Division algorithms

September 23, 2011

Julie asked in the comments recently (slightly edited here):

I see that the CCSS has 5th grade teachers teaching the partial [quotients] method of division and then 6th grade teachers are to link this understanding to the standard division algorithm. The kids don’t understand it nor do they want to. Nowhere in the 6th grade CMP2 does it show teachers how to make this transition. It seems like a small detail but it causes major problems for our students, teachers and parents at grades 5 and 6. I have in the past asked the teachers to continue w/ partial [quotients] but now w/ the new CCSS it clearly says to fluently divide multi-digit numbers using the standard algorithm. Do you have suggestions?

As you can imagine, the issue you point to caused some consternation at CMP. Common Core calls for sixth-grade students to:

Fluently divide multi-digit numbers using the standard algorithm.

This occurs frequently in CCSS, I’m afraid. The standards seem unusually concerned with which algorithms students are using at which grade levels, and there is little justification for why a particular algorithm might be the right one. (For further extended discussion of related matters, see Christopher’s blog, Overthinking My Teaching).

We are of two minds on the particular standard Julie points to, and on wide range of other similar issues.

1. Spirit of the law

Feedback from the field has been quite strong indicating that high-stakes assessment is what is going to drive instructional priorities in the roll out of Common Core.

We have tried to imagine a principled test that would be able to sort out whether a student knows the standard algorithm. This test would have to have one of two types of questions: (1) an extended response scored by a human being who had been trained to look for the standard algorithm, or (2) a multiple choice question in which a student is supposed to fill in a missing digit somewhere in the algorithm but not in the final quotient. The first seems really unlikely and the second is unconscionable.

So in a version of Bits and Pieces III (with a tentative title of Decimal Ops) that we are field testing this year, we have split the difference. We are developing sense for the standard division algorithm by connecting to division of fractions, as we did in CMP2. Then we have this:

And we have some text and problems around this that support connecting to the thinking behind the partial quotients algorithm. But in the student edition, we don’t go so far as getting to the digit-by-digit standard algorithm.

There is a principle at play here. The standard algorithm is efficient (although not a lot more efficient than a well-executed partial quotients algorithm, and certainly not much more efficient than the algorithm depicted above). But the efficiency of the algorithm comes from ignoring place value and operating on each digit, or subset of digits, as though they represented units. The standard algorithm does not develop children’s number sense.

Furthermore, a common justification for teaching the standard division algorithm is that it is necessary for algebra-it generalizes to polynomial long division. But this argument is not quite right. Polynomial long division-to the extent that it can be justified as a necessary algebra topic-is more closely tied to partial quotients than it is to the standard algorithm.

2. Letter of the law

So even if sixth-grade CMP3 doesn’t go all the way to digit-by-digit standard long division algorithm in the student materials, we want to be supportive of teachers who need to meet every detail of the standards. We will have support for that in the teacher materials. The student book gets us awfully close to the standard algorithm. We will write teacher materials to help teachers build to the standard algorithm by going further with the ideas in the student edition.


From → Grade 6

  1. Jackie Brown permalink

    The standard algorithm does not necessarily mean “the traditional algorithm in the US”. A standard algorithm is a procedure that always works. Lattice multiplication for example is standard because it always works, and it just happened to be the traditional algorithm used by Fibonacci and others throughout Europe & Asia – sometimes called Napier’s bones.

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