Do you remember your high school geometry class? I sure do—and I remember it fondly.
Mr. Brooks was in his first year at my high school, and we thought he was kinda cool with his full red beard and professorial glasses. In the first five minutes of our first class with Mr. Brooks, he told us that we were going to do proofs, and lots of them.
He didn’t expect us to know all the definitions, theorems, and corollaries by heart, so he let us have a 3″x5″ card on which to write down as much geometric knowledge as we could fit. We were even allowed to use this card while taking tests. To make it even easier, we didn’t need to write down the whole definition, theorem, or corollary; we could simply write our own shorthand codes like “Def of Sq” for definition of a square, or “Th. 3-5” for Theorem 3-5. (To this day, I can’t recall what Theorem 3-5 said.)
He taught us a few trivia items, like using the symbol for Therefore: ∴ . We learned that the acronym QED was Latin for “quod erat demonstrandum,” which means “that which was to be demonstrated.” Using a Latin acronym made us feel instantly smarter. (We soon fell back to our juvenile selves when we conjured up other meanings for QED, like “Quite Easily Done” for easy proofs, or “Quit. End. Done.” for the seemingly impossible proofs.)
All year long, for 180 days, we studied two-column proofs. That was it. That was my World of Geometry. Everything was a T-chart; nothing else. It wound up being the easiest “A” I received during my entire high school math career, all thanks due to that magic 3″x5″ note card full of theorems, definitions, and corollaries.
And in retrospect, it was the most disappointing math class I ever took.
Here’s why: During my third year of teaching, I was given the task of teaching honors geometry with the Discovering Geometry textbook. I soon realized what a real geometry class looked like. And I think I had even more fun than my students did.
(Disclaimer: It’s little wonder that, due to Discovering Geometry’s effect on my teaching experience, I now work for the company that publishes it.)
In class, we argued over which group had written the best definition for supplementary angles. Constructions with compass-and-straightedge and patty paper were the norm. Making conjectures was the fun part about class, and trying to find counterexamples to those conjectures was the challenge. Most enriching of all: we were building our own knowledge and understanding of geometry, and the great payoff was being able to apply what we learned in class to the world around us. Through the investigative approach, our class was opening up to a new world of mathematics—much like geometry teacher Jai Maldonado’s class, in the video below, understood math concepts more fully through the use of patty paper:
The Common Core State Standards for Geometry ensure that all students will have this rich learning experience. While learning the core of Euclidean geometry, students will also learn how to use tools like “compass-and-straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.” to build an understanding of geometric constructions. Students will also apply their learning in modeling situations. Their world will be opened up to rich problems, such as “designing an object or structure,” in order to optimize an attribute (e.g. cost) under numerous constraints.
Many students’ eyes will be opened to a new, geometric world…no 3″x5″ card necessary.