I tutored some students in geometry recently. Here is a note I sent to their parents to let them know what their children were doing and how their children were progressing (I will call one student A and the other B):
A and B did good in tutoring today. A and I covered six proofs — that’s a first. We have not done that many proofs in one sitting before. He’s doing good! We also discussed the nature of definitions (as having a genus and differentia, which we had discussed weeks/months ago) and their importance; we discussed and analyzed the definitions of polygon and triangle, and from there discussed quadrilateral and pentagon, midpoint and bisector, and, to discuss the concept “genus”, capitalism, democracy, communism (genus political system), and then dog and cat (genus mammal). And we discussed how, in learning a subject, it is important to review material that came before what one is doing now; A had forgotten what a polygon was, what a midpoint was, what a definition was, and what some of the axioms were, and so could not use them when he needed to. This was a great discussion for learning how to learn and how reason works. And we discussed Mill’s Methods of induction: the method of agreement, the method of difference, the joint method of agreement and difference, and the method of concomitant variation. These are very important for science — and every day life.
B and I analyzed four theorems in the book, and went over one proof (two proofs?). The proof was several steps more complex than others he had done, so it was one I needed to help him on. We had to do a bit more reasoning and add more to the diagram than what he’s used to. Most diagrams so far had given him all the information he needed. One theorem we covered proved that the hypotenuse of a right triangle (with a 30 degree angle) is twice the length of the leg opposite to the 30 degree angle. This triangle is one that we build on and use in trigonometry — which fact I pointed out to B. We find 30 degree angles and 60 degree angles used around the unit circle; now B will know why we use those angles in particular.
Very good sessions today. A and B are both making clear and definite progress; they have come a long way.
Geometry is used — as it should be — as a means of teaching reasoning. These students are learning how not just geometry but all knowledge has structure. They are learning to make connections, i.e., to integrate knowledge, and are being trained to take integration as the norm. They are learning how to think independently.
A good teacher is invaluable; a good teacher opens a student’s eyes to connections and heights the student would not see himself or herself. And if reasoning is important in life then so also is a good teacher important to a student’s adult life.