Thursday, November 25, 2010

Straight Lines, Square Corners

When I was in middle school the boys were introduced to mechanical drawing using rulers, t-squares, triangles, and pencils. Later as a senior in high school I opted for a shop class in mechanical drawing. We were required to draw in units as small as 1/16 of an inch. Considering that all of the work in those days was done by hand, a lot of precision was expected and taken for granted.

In my computer studies classes here I have been teaching something called a "block diagram", an example of which is pictured. On the blackboard I just quickly sketch the diagram by hand. But our Tanzanian students invariably pull out their rulers and carefully draw boxes with straight lines and squared corners in order to make a copy in their notebooks. I would think, "why bother, all you need is a sketch to record the idea."

Maybe the answer is that in this part of the world people do not "naturally" think in terms of straight lines and ninety-degree angles. After all, how often do you see those in the natural environment?

However, thanks to the genius of the ancient Greek mathematicians we are thoroughly familiar with both the concepts of a (perfectly) straight line and a (perfectly) perpendicular angle as well as their (imperfect) manifestations. I am composing these words by writing on a table that has a rectangular top and is otherwise built entirely of rectangular pieces of wood. In the U.S. there is a strong preference for city streets to be arranged in a very regular grid pattern (with perturbations to allow for features such as steep hills -- how inelegant of nature). Later I will be typing these words into a computer which will display the text on a screen consisting of, say, a 1024 x 768 rectangular array of pixels.

So I suppose our Tanzanian students are being trained to think in a like manner. Why does this matter? Because straight lines, right angles, rectangles, and other geometric objects are abstractions that serve as very useful mental tools for building models. They are tools that enable us both to more readily create artifacts such as wooden tables and computer screens and also to impose structure, order, and manageability on the natural environment, such as in the form of streets, political boundaries, and coordinate location systems. They provide powerful leverage for spatial thinking.

Thanks to Euclid, to Mrs Hillebrand who was my tenth grade plane geometry teacher, and to the milieu and the intellectual legacy in which I grew up, when I use a rough piece of chalk to sketch a bunch of boxes I automatically, habitually "see" perfect rectangles. -Earl

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