If you were a rich Greek in 100 B.C., it was not uncommon to pay a tutor to come to your villa and teach you math. When it cooled off later in the afternoon, you might gather with the other men in the center of town and talk. Among the academic topics they discussed were philosophy, politics, and math. The image of a Greek drawing circles in the sand is not so much a stereotype as what some people did with their free time.
Math was a hobby. Before the Greeks, people only used math when necessary. To count, to design, to build. The Greeks were by all accounts the first to write down the rules of math in an effort to generalize them. Look at a door frame. How do you know those are four right angles? I suppose you could hold a square block to the frame to check. How do you know your square is accurate? It could be a parallelogram, which also has four sides of equal length. How do you know your angles are right angles? The Greeks proved how to prove a square was, in fact, a square. (An easy way to check is to inscribe a circle in the square. The radius should bisect the sides.)
But once you prove why something you can obviously see is true, what comes next? Mathematicians whose names we know – Euclid, Pythagoreas, Archimedes – were among the most famous Greeks who extended math beyond the door frame. They helped develop trigonometry to study astronomy long before it was needed to steer ships and fly airplanes.
We do very little of this type of math in school. In the past, to study math was similar to that of philosophy; unsolved problems prompted study until a one found a logical argument in support of an answer. Math was regarded with the same mysticism and authority. Boiling it down to a series of problem sets somehow assessed by a timed test runs counterproductive to the Greek model of study, discussion, argument, and proof. If tests are the way we evaluate a student’s mathematical reasoning, we ignore thousands of years of history in favor of the last hundred.
While Algebra 1, Algebra 2, and Geometry largely have been warped because of their presence on standardized tests, there could be some flexibility in Precalculus. After a long week of trigonometric book problems, I am wondering whether the study of trigonometry could feasibly be framed in the same manner as a Greek discussion. Could we start with a unit circle and spend class in groups developing a set of rules and proofs? Could we emulate the Greek town center in a classroom? We wouldn’t “get through” as much. Assessment would look radically different. And while we still might not satisfy the annual “why are we doing this?”, perhaps students wouldn’t be bored enough to ask.