Grading on a Continuum

Anecdote: I had a social-sciences teacher in high school who didn't understand that real numbers are a continuum.

On the first day of class, he tried to present how grades would be computed at the end of the course. So on the board he wrote something like: D 60-69, C 70-79, B 80-89, A 90-100% (relating final letter grade to weighted total in the course).

Then he looked at it and said, "Oh, wait, that's not right, what if a student gets 89.5%?". So he starting erasing and adjusting the cutoff scores so it looked something like: D 60-69.5, C 69.6-79.5, B 79.6-89.5, A 89.6-100%.

And of course then he went, "But, no, what if a student gets 89.59%?", and started erasing and adjusting again to generate something like: D 60-69.55, C 69.56-79.55, B 79.56-89.55, A 89.56-100. And then of course noticed that there were still gaps between the intervals and went at it for a few more cycles.

I think he took about 10 minutes or more of the first class iterating on this (before he gave up he'd gotten to maybe 4 places after the decimal point). I remember myself and a bunch of other students just looking back and forth at each other, slack-jawed from astonishment. It raises a couple of questions: Did he not know that real numbers are dense? And had he never thought through his grading schema until this very moment?

I always think about this on the first day in my statistics courses, when we are careful (following Weiss book notation) to define our grouped classes for continuous data using a special symbol "-<", meaning "up to but less than" (e.g., B 80 -< 90, A 90 -< 100, so that any score less than 90 would be unambiguously not in the A category, leaving no gaps). As I present this, my social-science teacher embarrassing himself is always at the back of my mind -- and I'd like to share it with my students as a case-study, but frankly the anecdote would take too much time and distract from the critically important first day of my own class.

But the initial reaction we got for that teacher was accurate; although he couldn't perceive it, he was about as dense as the real numbers all semester long.