I then shared with my old friend how I had had an unusual experience in my undergraduate math education course yesterday when one of the students had asked me why young children need to know why there are 180 degrees in a straight line. "Can't you just tell them?" the undergraduate student had asked. "I know there are 180 degrees in a straight line and that's been fine for me", she continued.
A little taken aback I then asked the student how she would explain to a young child how there were 180 degrees in a straight line when there was no angle to measure. She didn't say anything.
So, with my old friend watching I then tore the three corners off a paper triangle and lined them up so that they made a straight line by putting the three points together. There, I said, 180 degrees in the three corners of a triangle and a straight line. Seeing a somewhat disbelieving glint in his eye I then took a 360 degree rotating protractor and rotated it through 180 degrees saying that degrees are a measure of rotation about a point; in this case the point where the three corners of the triangle come together. Here's a nice demonstration of the concept that angles are measures of rotation.
After a few seconds of thought he said "Hmmmm, why wasn't I taught math this way?" I wonder if he will say anything to his daughter.
The persistent perception of math as a process of telling and memorization seems such an enigma in this day and age.