In Joe Braun's hands, the elements of Euclidian geometry weren't a collection of ancient, mere facts; rather, they were a set of timeless truths that could be derived anew and exploited anew by any student willing to absorb Euclid's axioms, methods of thinking, and habits of thought.
In Joe's classes, exercises assigned the previous day were passed from one student to another for review and correction. A student who completed an exercise successfully was often called upon to explain the exercise to students who had experienced problems with it.
When beginning a new lesson, Joe spent relatively little time standing at the front of the classroom, lecturing. Instead, he sat at his desk with his textbook open and guided the class in its confrontation with the lesson. Sometimes he invited a student to read the lesson aloud. Sometimes he himself read the lesson aloud and called on one or more students to explain key terms or concepts.
When an illustration was needed, Joe left his chair with cat-like grace and drew what was needed on the chalkboard but then returned to his desk and to his textbook. The implicit message was that careful reading and thoughtful contemplation of the lesson were the first steps in achieving mastery of it.
Most students who entered Joe's geometry class with a skull full of mush left it thinking like Euclidians. Those who demonstrated a special flair for Euclid received extra challenges involving trisection of angles, squaring of circles, and consideration of parallel lines that converge -- challenges delivered with a twinkle in Joe's eye, and which sometimes involved trips to the public library and late nights of independent study.
The class I took from Joe was attended by a number of star athletes, who spent more hours with Joe on the football field than in his classroom. As far as I could tell, Joe afforded no special privileges to such athletes, unless you count as a privilege the opportunity to receive occasional help from students less athletically gifted.
I didn't fully appreciate how thoroughly I had absorbed Joe's lessons until nearly twenty years after I left his classroom, when I found myself using his methods to help my children with their geometry lessons. When I called Joe in 1982 to express my appreciation for what I had learned from him, he was spending some of his time in retirement tutoring students in English.
Euclid's great accomplishments was to organize the elements of geometry into a logically coherent framework supported by rigorous mathematical proofs. One of Joe's great accomplishments was to help preserve that part of our culture and pass it along in trust to those who came after him. Thank God, I was such a one.