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On Klein & Milgram, continued
Posted:
Aug 8, 2000 12:54 PM
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In a previous post, I addressed the first of Klein and Milgram's arguments
in favor of long division, which was an appeal to the authority of eminent
mathematicians. Here I would like to comment on some of their other
arguments:
* mutual understanding
* power
* preparation for algebra
* efficiency
* understanding real numbers
* preparation for polynomial division
The first four of these arguments are not specific to long division. The
first three are given in a long passage K & M quote from an AMS report
published in the February 1998 Notices. These arguments apply to 'standard'
algorithms for all four operations.
* Mutual understanding. Basically, the AMS report says kids should learn
standard algorithms because they promote mutual understanding and
communication.
* Power. The report says standard algorithms have the advantage that they
are universally applicable, as opposed to students' 'invented' methods,
which may not be.
* Preparation for algebra. The report says that the arithmetic of whole
numbers is analogous to the algebra of polynomials, so being good at
arithmetic will lead to being good at algebra.
I find the first two of these a bit stronger than the third. But even
granting all three is not equivalent to agreeing that the standard long
division algorithm should be taught. For example, the 'ladder division'
algorithm meets the criteria for mutual understanding and preparation for
algebra, arguably even better than the standard algorithm, and is powerful
enough for most practical situations requiring long division. The van de
Walle algorithm I have mentioned before meets all three criteria as well or
better than the standard algorithm.
The efficiency argument is similar to the power argument and my response
would be similar. Basically, other paper-and-pencil algorithms exist that
are more or less as efficient as the standard algorithm in most situations.
Besides, if efficiency is the goal, why are we talking about paper and
pencil anyway?
This leaves what K & M would probably agree is the core of their argument,
namely that long division builds understanding of rational vs. irrational
numbers and also prepares students for polynomial division.
To take the second of these first, K & M claim that kids should learn the
standard long division algorithm so that they can have an easier time later
with polynomial division, partial fractions, Laplace transforms, and linear
algebra. I agree that polynomial division is important, but why not teach
it first and then we could do long division of whole numbers as a special
case where x=10? Seriously, I am not convinced that we should spend months
on a topic in grades 4-6 for the sake of probably modest efficiencies in
the undergraduate mathematics curriculum.
That leaves the utility of the standard long division algorithm for
building understanding of irrational vs. rational numbers as justification
for teaching it to fourth graders. I'm not sure if this is what K & M mean
by a 'core application of mathematics in our society today', but again I am
not convinced the benefit is worth the cost. The distinction between Q and
R is, of course, crucial in mathematics. Whether it matters much in
elementary school mathematics is not clear to me. And if you are really
attached to making the distinction, the van de Walle algorithm works just
fine and is easier to learn.
Apart from the polemics against math educators, which I don't take
personally, most of the rest of K & M's paper consists of fairly standard
expositions of the mathematics of long division. I didn't find anything
either very interesting or very objectionable in it. I would, as I
mentioned before, be interested in how the approaches outlined would
actually work with kids and teachers. My guess is that they would be a
fairly hard sell.
I will post some concluding remarks in another email.
Andy
PS
In the section called "The Long Division Algorithm with Remainder Explained
Algebraically," there was a reference to estimating as a crucial part of
the process. I think there may be an editorial lapse somewhere because the
reference seems to indicate that estimation had been mentioned before, but
I can find no earlier mention in the paper as posted.
In any case, M has argued elsewhere that the standard long division
algorithm promotes estimation skills. This is no doubt true since failure
to estimate accurately makes carrying out the algorithm very painful. And I
agree that estimation is a crucial skill. But why not simply teach it
directly? Teaching topic A because it helps with topic B is probably less
efficient than just teaching B.
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