There’s a debate about the relevance of geometric proofs and then, about the importance of writing them formally. For me, geometric proofs are more about the analysis, the support and the organization of an argument. I generally find that my strongest writers are usually the first to see how a proof ought to flow, with the rest of the class eventually catching on. After that, their written responses become more direct and they include more evidence to substantiate their answers. Even in other subjects, several of my proof-writers have begun to write more coherently. In that way, proofs are very useful.
What I don’t find useful is forcing students to memorize postulates or theorems. In years past, less invested students never had the proper names at their fingertips; without the correct property, they often wrote little more than the ‘givens’.
Last year, in a sudden burst of frustration, I decided to allow open note assessments for the unit. I even went one step further by purchasing 1-subject notebooks for the class that we would fill in together. At the front of the notebook, we made lists of theorems, formulas, angle properties, and vocabulary. Starting from the back of the notebook, students would include the proofs we practiced in class. Eventually, I learned to photocopy notes if I needed to move the class along quickly (since some of the students took forever to make their notes beautiful). Students also needed diagrams printed out for them so that a poor illustration wouldn’t throw them off as we interpreted proofs together. Towards the end of the unit, we began several classes with 5 minutes of cutting and pasting, with much more time for productive discussions then onward.
Looking back, the geometry notebook wound up being my greatest success in organizing and teaching last year. I don’t remember a class ever picking up proofs so quickly – and most of them took pride in informing me that they liked writing them. The notes lessened the stress at test-time. They also helped students focus on interpreting diagrams, gathering support, and organizing a proof – which were my key goals for the unit anyway. Better still, a majority of the class hardly used the notebooks on the final assessment, since they were so comfortable using triangle properties and the like. If I taught an advanced class, I probably would expect them to memorize theorems, organize themselves and demonstrate their understanding without any help. But for my regular level, the notebooks proved to be a great resource.
Note: This year, I will have an origami-lover in my geometry class. It’s a great opportunity to do some paper-folding proofs, but I really don’t have much experience with them. Any suggestions?