Shaking up traditional geometry
My proof-based geometry course is a proof-based geometry course. But I can still shake things up: Logic, Non-standard Theorems, Construction, Construction, Student-generated reference materials (for use on tests)
Were I to stick closely to my text (Jugensen/Brown/Jurgensen), most of my readers would recognize the course instantly as the more or less standard geometry course that’s been taught in the United States for a century. Of course the amount of proof has been substantially reduced from fifty years ago, but the idea, the sequence, they are the same. This is a course in proof, but also in reasoning. It is the only axiomatic system that most high school students explore.
And I hate teaching it. Ugh. So I “innovate” – though I suspect that all of my innovations are quite old, and have been done before.
1. Open with a unit on logic, and logic proofs. For those of you from NY State who recall the proofs in Course II, no, more, harder. Include extraneous statements. Teach more rules of replacement and rules of inference, and prove the rules before using them. Venn Diagrams and Euler Diagrams and truth tables. Consistency. And indirect proofs. This was a big unit.
2. Have students create their own glossaries/reference sheets. Allow/insist on constant revisions and updates. Allow/insist that the students bring their reference sheets to each quiz and test.
3. Construction. Fully one quarter of the class periods devoted to construction. Some standard construction. A lot of more creative stuff. We have a set of Michael Serra’s geometry books, and his opening chapter has been a nice resource.
4. Construction. Students must have the tools with them at all time. Quick constructions often become parts of ordinary non-construction lessons.
5. Oddball theorems. There are two types of deductive proof that students encounter.
The kind of deductive proof we more often associate with high school geometry presents a diagram with some given information and asks the student to prove another piece of information. What is being proven is usually already clearly true to the eye.
The other kind is to prove a theorem. The book does this for the students. Or I do it in class. And then we use the theorem. Sometimes the proof of a second version of the same theorem is offered as an exercise. And then, if this were the 1970s or earlier, we would ask the students to memorize these proofs, and recite them on a test.
But this is wrong! Proving theorems is at the core of what mathematicians do. The students need to be asked to prove theorems. But all the good ones are taken. So I will ask students to prove less-known, less-useful theorems. Practice doing the real thing.
(Getting to oddball theorems came out of discussions with math bloggers 2 – 4 years ago. Don’t remember exactly who, and exactly when, but the list of helpful suspects includes Ben-Blum Smith, Pat Bellew, f(Kate), PO’ed Teacher, and the Math Curmudgeon. I think Ben is the likeliest to have hosted this sort of discussion.)