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.
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.