Wednesday, February 25, 2009

Why teach the traditional long division algorithm?

There are two general long division algorithms that I'm aware of: the Partial Quotients algorithm as used in Everyday Math and other Reform Math programs, and the long division algorithm that many of us grew up with (including its European variations).

The partial sums algorithm has the advantage of being much more conceptually transparent to students.

But the long division algorithm has its own unique advantages. First, it is the most complex, multi-step algorithm that students encounter in all of arithmetic: divide, multiply, subtract, bring down; divide, multiply, subtract, bring down. As bky points out, unlike the partial sums algorithm, the steps are precise and predetermined, so that it's an algorithm in the strictest sense. All this, among other things, is good exposure for our future computer programmers--a concrete reference point, for example, for the principle of Recursion.

Also, because the traditional long division algorithm is not at all transparent, students, in order to remember what to do when, need to think hard about what they're doing and why. With enough practice, they start to grasp the underlying principles, including some subtle aspects of place value, and the inter-relationships between place value, division, multiplication, and subtraction.

Indeed, in the spirit of true Constructivism, a child who does the long division algorithm is experientially constructing his or her knowledge of these higher level concepts.

Why discourage this?

Third, the traditional long division algorithm lends itself more readily than the partial quotients algorithm (which tends to deal in integers) does to converting fractions into decimals. As David Klein and James Milgram argue in this paper, referred to me by Niels Henrik Abel, repeated execution of the long division algorithm helps you understand why all fractions convert to repeating decimals, and helps you see connections between the length of a given repeating decimal and the properties of the corresponding fraction. More generally, the authors suggest, performing the long division algorithm helps students build conceptual understanding of what a real number is.

Indeed, anyone who's considering ditching the traditional long division should read Klein and Milgram's paper first... and make sure they understand it.

6 comments:

Brendan said...

First, why teach one method over another. Both methods are simply two sides of the same coin. Learning both would lead to greater understanding.
Second, it has been my experience that division algorithms are usually taught as processes. The traditional algorithm simply is more foreign thus more difficult.

lefty said...

I agree with you: teach both; and, yes, one is harder than the other.

Blank said...

I agree. I even prepared these long division worksheets for my students.

Lsquared said...

I mostly agree with Brendan. The teacher I first got partial quotients from (several years before the reform math curricula came out) used it as a "scaffolding" algorithm. That is: she taught partial products first, used it to teach the standard algorithm, and then had students practice the standard algorithm (something that works pretty well, since the main difference between between partial quotients and the standard algorithm is whether you correctly come up with the next decimal place of the quotient in one step or several steps). Living in a school district here students mostly don't do long division by two digit numbers ever, I'd rather have partial quotients than nothing.

Kent Pilkington said...

Long division may have some conceptual benefits to being *taught*, but it is nothing short of a train wreck when actually worked out on paper. Even moderately simple problems tend to involve "borrowing" from the next most significant digit, in which case you end up with marked through and squeezed in numbers, and if the child's spacial, attention, or penmanship skills are not up to the task, they lose track of which numbers are in which columns or forget their place in the overall procedure and mass chaos ensues.

I found this blog entry because I'm searching for an alternate algorithm for division that doesn't involve the massive organizational and spacial overhead of long division. In one hour, my daughter (11) was able to complete 2 problems (1 digit into 3), one of them incorrectly, largely due to the confused and rambling nature of long division and it's editorial overhead.

This is a child who, at 5, understood supply and demand well enough from a 10-minute description to project how wholesale supply chain issues would impact retail pricing, and could even point out where you didn't know which factor produced a retail outcome.

If this is the best we can offer (as it was when I was in elementary school), then there are lot of brilliant minds out there that are totally screwed.

If you have any ideas (other than "practice!"), we're getting a bit desperate here!

Claria said...

My daughter is 10, she is quite brillant in math, most of the time, although she does make mistakes sometimes. But, at school, she's in fith grade so you know,she can't do this. In every math class, Iv'e heard she is doing wonderfull. But, when she got to this, she was lost.
I belive this is so because of how young their minds are, it's a bit too complicated for most of them. In maybe older grades like 6th or 7th, yes, teach it if you want. But I belive that is is still too difficult for most unless they srtruggle with normal ways. And, Iv'e noticed that my dauter always gat the lowest of a B in her report cards.I reallly think it's too hard, it should only be thuaght a a later grade. And I noticed something, the children who normally did bad at math seemed to do this eisly. It's odd.