Asilomar #3: How To Teach Geometry

Session Title

Don’t Just Cover Geometry, Discover Geometry


Michael Serra, Teacher / Author


Serra took the participants through his well-worn strategies for teaching Geometry, emphasizing manipulatives and induction. His students develop definitions and terms together. At whatever point they see a pattern in their experimentation (ie. “okay … all the angles in a triangle seem to add to 180°”) the students write the conjecture.

This approach requires of a teacher a rare mix of knowledge, confidence, and humility and it’s no surprise to me that operationalizing his approach into a textbook has yielded unpredictable returns. Whereas other books will cite and define the parallel line postulate, his book cites it and starts the definition but leaves the critical parts blank. They aren’t defined farther down the page. They aren’t defined in the back of the book. They’re just blank. Some teachers will simply fill in those blanks for their students who then dutifully record the conjectures in their notes. Others will have the students develop those conjectures through observation and experimentation, as intended.

Serra demonstrated the latter approach over the course of ninety minutes, developing 75% of Cartesian Geometry with nothing more than induction and patty paper.


Document camera for modeling the experiments.


Pre-printed shapes and forms. We operated on them in groups with scissors and rulers.


  • “This is all of Geometry in one class. Finish up in a day. Let’s watch movies now.” — Ian Garrovillas.
  • I wish I had something insightful here linking Serra’s nascent web presence to Cringely’s Burn, Baby, Burn, which speculates on the fate of higher education in the Internet age. Serra’s name carries a lot of currency in the business of math education. Converting that currency into imaginary Internet money and then that imaginary Internet money into real money is the necessary journey but there isn’t a road map. I don’t know what I’d do in his position.
  • Ian asked if I had ever thought about monetizing my blog, by which I think he meant sidebar ads or something. I’m so lost in that world but my creeping sense is that sidebar ads would cheapen the experience of publishing content (for me) and interacting with content (for you) in a way that would be zero-sum or quite possibly negative-sum. I know for certain, however, that writing content here has been a financially profitable experience, though that profit has come about in some strange, indirect ways that I don’t know how reproduce with any reliability.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.