Teaching my nephew Geometry

dodecahedron
Originally uploaded by Geo & VO.

MY WIFE AND I SPENT the entire Summer teaching my nephew, Michael Trejo, Geometry, as a requirement to fulfill High School credit to begin Algebra II in Fall 2004.
I HAD TO RE-LEARN all the theorems, postulates, nit-picking definitions, and dreaded proofs in order to transmit that knowledge to Michael.
YOU KNOW, GEOMETRY IS A very beautiful subject. But geometry, as it is taught, is filled with com-plicated trivialities. As a result, the first four to five chapters offer little insight into geometric ideas that matter in the real world.
I CONTINUALLY FOUND myself assuring Michael, "Don't worry, Michael, we'll be getting to the geometry soon; we're almost there." Poor Michael, I had to teach him proofs of statements that were obvious to begin with.
AT THIS STAGE, who could doubt the truth of such a statement as: "Given any 3 points on a line, one is between the other two." This silliness is called the Definition of Betweenness. Can you believe this? When shown this, the natural reaction of an intelligent student (and his uncle), is irritation and impatience. All this tends to kill a student's interest in geometry long before they reach the meat of the subject.
THE ROOT OF THE PROBLEM is that High School math instructors are slaves --and they live on the plantation of Deductive Reasoning.
THIS IS THE notion that knowledge is somehow not valid until it has been organised into a highly formal system of theorems. All knowledge is deduced from something before it.
DEDUCTIVE REASONING is an interesting idea that educated people should know something about, just as they should know something about representative government; but everything in its place. This doctrine belongs somewhere else, not in geometry...
IF GEOMETRY IS CONSIDERED for its own sake, geometry students will study with greater efficiency, and they will love the subject!
I THINK HIGH SCHOOL geometry teachers know this, but they are stuck on the plantation.