The importance of the rough sketch
Sometimes, the rougher the better.
Our tutoring consultant deals with many levels of mathematics and science, including the dreaded standardized-test word problem. His advice in many, perhaps most, cases is the same: start by making a drawing. At the very least it helps to organize the information given in words (which can be confusing, all jumbled together in a paragraph). Very often it’s vital, to set out the shape and limits of the mathematical figure or to work out just what’s happening with forces and motion in Physics. Indeed, a decent Free-Body Diagram in the latter subject will earn the student a substantial part of the grade all by itself.
No doubt it’s due to the various ways people process information. We’re not education-theory specialists, but we have found that different channels (verbal, visual, words printed or spoken) work differently, carry different information more easily; using a variety makes it more probable that the important part will get through. And different people have their own preferred ways of learning and thinking. For most, a visual cue will help. (Not all: our tutor has one student for whom a drawing does nothing. But if the vital part can be put in algebra, she’s home free. Teaching has a variety of challenges.)
At the moment, his Geometry and Calculus students are both working in three dimensions. Visualizing objects in 3D is particularly hard. We’re not sure whether this is a product of a civilization of flat screens and printed words and pictures, or whether it’s inherent in a species that is mostly limited to the surface of the Earth. But even to set up a problem in three dimensions is almost impossible without a decent sketch, and that presents its own peculiar difficulties on a flat piece of paper. It’s very much a learned skill.
And a good bit of the skill is knowing what to leave out. A complete and accurate drawing of, say, a cylinder may only look like a flat rectangle with bowed ends. On the other hand, a couple of roughly elliptical curves with connecting lines can posses a startling reality. This is the basis of cartooning: somehow to suggest the important parts with a few sweeps of the pen. And as Guy Ottewell found in producing his (highly recommended) Astronomical Companion, a rougher drawing can be more effective than a precise plot. The latter can be distracting in its detail: the viewer may not understand what to focus on; and misleading in its precision, as many a Geometry student has found.
It seems to us that the apogee of the educational drawing must have been from the 1940s through the 1960s. The massive American war effort had to train millions of people in esoteric, technical subjects they’d never seen before. By the 1970s, certainly the 80s, computer-generated graphics took over. We’re not convinced of their greater effectiveness. Computers are excellent for many things, but especially poor at knowing what to leave out.
