Wednesday, December 16, 2009

Favorite comments of '09: bky on why teach long division

Re: why teach long division?, bky writes:

There are two good reasons for teaching the standard algorithm for dividing integers, if that's what you mean by long division:

(1) so kids can calculate the quotient of two numbers(and have an exact form for the remainder if that is wanted, rather than a decimal expansion), and

(2) it is an introduction to algorithms. It is odd that many people who deride the teaching of the traditional algorithms cite the availability of calculators as a reason not to learn the algorithm. I find it useful to regularly (every 6 months or so) have my kids (homeschooled) go over the operations of it, with the idea of helping them not only have confidence that what they are doing makes sense but also preparing them for them to understand the concept of algorithm by familiarity with a few specific examples (also on the list: standard stacking algorithms for multiplying, adding, subtracting).

A good idea is to practice occasionally with something like money: show 734 as 7 dollars, 3 dimes, 4 pennies. If you divide by, say 3, you start with 3 piles, each with 2 dollars; the left-over dollar is exchanged for dimes; etc. It is also useful to do the same problem on paper based on writing out the expansion 734 = 700 + 30 + 4 and then successively dividing each place value with remainder, and throwing the remainder downhill:

734 = 7x100 + 3x10 + 4
= 3x(2x100) + 1x100 + 30 + 4
= 3x(2x100) + 13x10 + 4
= 3x(2x100) + 3x(4x10) + 14
= 3x(2x100) + 3x(4x10) + 3x4 + 2
= 3x(2x100 + 4x10 + 4) + 2
= 3x244 + 2

This is "doing it by hand". The algorithm is a formalization. The algorithm is based on repeated division-with-remainder; you never really need to know what place value you are working with, or which side of the decimal point -- just divide and throw the remainder on the next lower place value. If kids understand this for integers, then dividing in the presence of a decimal point is just as easy. Note also how distribution is vital to long division, and since distribution is difficult for grade school kids this also gives practice recognizing and using that vital aspect of arithmetic.

Some critics say that the LD algorithm doesn't teach place value. Of course, it's not suppose to: it's supposed to divide numbers. But looking at it as an algorithm does indeed reinforce the concept of place value. Long Division is a keeper.


Elaine C. said...

... it's also FANTASTIC for teaching kids how to divide polynomials.

Substitute places for terms, and voila... dividing polynomials made easy!

And once they have long division down, you can then teach synthetic division.

Catch a calculator doing any of that!

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