or How GeoGebra and the standards are making geometry exciting again
I learned about proofs in my high school geometry class the way most people probably did: two columns; statements and reasons; lots of arcane symbology and abbreviations; notebooks filled with reasons; and a lot of this from the textbook:
I don’t want to get into the reasons I found (still find) this boring and overly formal; let's just say it’s been one of my least favorite subjects to teach. Honestly, my response to this in the past has been to gloss over proof in favor of applications and activities or problems that make use of all those theorems. So I haven’t ever really taught it well. This year as I approached congruence (8th grade Integrated 1) and similarity (9th grade Integrated 2) with dread, I took some time to look at the standards and think about the progression I was going to use, the reason we teach this in the first place, the tools I like to teach with, and the results I was looking for. I slowly began to piece it all together in a way that made sense to me (and kind of makes me feel silly for not realizing this earlier). Integrated 1: Construction → Rigid Motion Transformations → Congruence Proofs I spent more time than usual on construction this year, but less time than usual on hammering the prescribed constructions (copying, bisecting, perpendicular and parallel, etc.). Instead, I treated it more as an avenue towards exploration and creativity with this construction project (thanks Pam Rissmann for the idea).
This was a good way for students to get used to the tools (both compass/straight edge and GeoGebra). I didn’t really formally assess the classic constructions; I figure they’ll need to be able to do most of those things to get through the transformations sequence. So I set up transformations as a series of construction tasks, using paper constructions to bring students to an understanding of how imprecise they are, then moving to GeoGebra. We start off with the whole toolbar, and eventually whittle it down to basic tools to show we know what we’re doing (inspiration from euclidthegame.com). The big change in mindset for me came with really focusing on the definition of congruence as the result of rigid motion transformations. So I decided that that’s how we’re doing proofs this year. Instead of two column proofs, the students are moving triangles around, and learning a lot of GeoGebra tricks in the process
Integrated 2: Review rigid motions → Add dilations → Similarity proofs In reviewing the previous year's work on transformations, we made a big deal about the link to congruence and similarity, and defined these terms solely using transformations. They walked though a couple of these with some assistance, and today they started on their partner projects, working on visual proofs of the angle bisector theorem, and narrating them in a screencast.
Am I missing anything? Seriously, I'm trying to think of a time after my high school Geometry class where I needed two-column proofs, and I can't come up with anything. If there's something I'm overlooking that my students are really going to miss out on, please let me know!