Meanwhile, I'm still teaching Elementary Geometry on Saturday mornings. One of the most important aspects of the course is teaching proofs. While there is a subset that will use the geometry itself, there is a larger group that will require training in the type of thinking that these proofs use. Anyone working in the legal field, for example, will need to know how to put together a proof. Yet, the traditional way of writing proofs seems backwards to me.
This is how I typically present on proof on a test:
Notice that the statements are on the left, while the reasons for those statements is on the right. That is backwards to me. Americans read left-to-right, and that puts the statements as something that comes before the reasons when you read a proof. However, the reasons are the connections between that line of the proof and the lines that precede it. Formally, you take the prior information and the reason, put them together using a law of logic or two, and produce the statement.
Of course, the design is traditional. One of my retirement plans is to write a good textbook for teaching Geometry at community colleges (they have good texts for Pre-algebra, Algebra I, and Algebra, but Geometry seems to be confined to high-school texts. In today's world, that would mean creating video segments and an on-line interactive homework system (our college uses MathXL, but they have no Geometry test listed) as well as writing the book with a little less flash and a fewer pictures of kids. When I do, I think I will write the proofs with reasons on the left, and maybe moved up a half-line.
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