How did last Spring’s Geometry turn out?
I shook up the curriculum for an off-track Geometry last Spring. It’s worth looking at how it went.
In this post I review what the changes were, and summarize the results. I will follow up with more detail in the coming weeks.
In my school, the “advanced” math group, as freshmen, do one term of algebra (usually harder stuff) in the Fall, and take the first term of geometry in the Spring. 2012-2013 I had both of the off-track sections, and rewrote chunks of what I was doing. More significantly, I restructured the course in a way that seemed to me to be a little radical.
(I clearly miss the classroom. My mind instantly goes to little radicals: me, the square root of two…. puns are better with an audience)
1. Open with an extended logic unit, with proof. Much more than the old Regents Logic. Include extraneous statements. 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. 4 weeks.
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 (every Friday) devoted to construction. Some standard construction. A lot of more creative stuff. A set of Michael Serra’s geometry books – a good resource. Students required to have the tools with them at all time.
4. Oddball theorems.
a. Most high school geometry proof is 1. diagram + 2. some given information = 3. prove something that is already obviously true.
b. The other kind of proof 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. If this were the 1970s or earlier, the students would memorize theorem 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. And all the good ones are taken. So I ask students to prove less-known, less-useful theorems. We practice doing the real thing. We talk about the difference between proving a theorem, and doing a proof-exercise from the book. We approach them slightly differently. And we write them differently.
So how’d it go?
1 We covered all the material I intended to cover. Some of the time given over to construction embedded other topics. At other times the experience with construction allowed the students to move through material more quickly.
2 Most students experienced success writing proofs. Students recognized the difference between theorem proofs and ‘exercise’ proofs. Some students were taken with proof by contradiction. One asked if he could use it all the time. (Irony here, on homework in a graduate course earlier this month, the professor asked me to not to use indirect proof where direct proof was easily available)
3 The construction experience was overwhelmingly positive, and added to course, without causing us to skip material. Most students were pleased with what they were able to produce.
4 The logic unit did not detract from the course. However, not all students ‘felt’ the connection between the logic proofs and the geometry proofs.
5 I got resistance from a small number of students to learning things that would not be on the Regents Exam, exacerbated by the difficulty of the material.
6 I polled both sections at the end of the term about what they liked best: Logic, Construction, or “Proof Geometry” – and was surprised to find that one section overwhelmingly preferred geometry, with logic second and construction last, and the other section was divided between geometry and proof, with construction last. This did not seem consistent with how engaged they were during the construction periods.
I will follow up, with much more detail, in the coming weeks.