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A defense of the long division algorithm

March 13, 2011

I recently had a short online debate with an economist about the value of teaching long division. I then tossed the discussion to my students in my Math for Elementary School Teachers class. A lively discussion ensued and here are my final thoughts.

“I promised to tell you where I ended up focusing my defense. The economist made many very good points, as some of you also did. What’s the impact of new technology on traditional learning? Is there any value in traditional technology-free learning? etc….I wish I had the skills to incorporate more technology into my teaching. It’s something I might think about doing in the future. But first I want to defend the long division algorithm, the other algorithms, mental math, estimation, and conceptual thinking in general.

The first point is that a good conceptual understanding of algorithms and estimation in the simple context of early mathematics is a stepping stone for things to come, for more abstract and scientific endeavors. In fact, even simple things like the multiplication algorithm and the fact that it’s based on the distributive property sometimes make unexpected come-backs. As computers are asked to do more and more complicated computations they for instance are faced with multiplying two huge numbers, as in

113897630764 * 289073454.

One method, called “divide-and-conquer”, consists in splitting the two numbers in half and do four multiplications involving less digits and then using the distributive property. As follows

(113897000000+63074)*(289070000+3454)

To appreciate this trick. Try to do the following multiplication using the mental math tricks we have learned and the “divide-and-conquer” trick (you’re allowed to scribbled down partial products along the way):

249 *598=(200+49)*(500+98)

The second point is historical and evolutionary. These algorithms, these concepts, have been trimmed, compressed, played with, and handed to us by previous generations. They are worth cultivating as a cultural phenomenon. Sometimes I use the word “beautiful” about a simple trick. People are not used to associate beauty with math. It’s all drill and trudgery. It doesn’t have
to be that way. Ask yourself the following question: are you going to be able to communicate the beauty of elementary mathematics in your teaching? One of my deepest conviction is that passion drives learning. Is it possible to make kids passionate about elementary math?

The final point, and the main point I fell back on when confronted with the economist points, is NUMBER SENSE. In the end, this is what I want to be able to teach: a certain confidence and ability in manipulating numbers algebraically. This I think has broad enough scope to be useful in many different contexts. So are we learning the long division algorithm to compete with calculators? No! Are we doing it so that if we’re ever stranded on a desert island without one, we can whip out pen and paper? No! The real reason to do this is to play with numbers, to build confidence with them, to build algebraic skills and increase one’s number sense.”

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