I do agree with the part about the quantifiers. This is, at least in theory, one of the reasons that we are supposed to teach the epsilon-delta definition of limit in college calculus courses. I generally try to frame it as a game between the prover and the skeptic, see for instance the description here. One of the main difficulties that students have with the definition is staying clear of whose strategic interest lies in what, for instance, who should be the one picking the epsilon, and who should be the one picking the delta (the misconceptions on the same page highlight common mistakes that students make in this regard).
Incidentally, this closely connects with the idea of steelmanning: in a limit proof or other mathematical proof showing that a definition involving quantifiers is satisfied, one needs to demonstrate that for all the moves of one’s opponent, one has a winning strategy to respond to the best move the opponent could possibly make.
The first time I taught epsilon-delta definition in a (non-honors) calculus class at the University of Chicago, even though I did use the game setup, almost nobody understood it. I’ve had considerably more success in future years, and it seems like students get something like 30-50% of the underlying logic on average (I’m judging based on their performance on hard conceptual multiple choice questions based on the definition).
I do agree with the part about the quantifiers. This is, at least in theory, one of the reasons that we are supposed to teach the epsilon-delta definition of limit in college calculus courses. I generally try to frame it as a game between the prover and the skeptic, see for instance the description here. One of the main difficulties that students have with the definition is staying clear of whose strategic interest lies in what, for instance, who should be the one picking the epsilon, and who should be the one picking the delta (the misconceptions on the same page highlight common mistakes that students make in this regard).
Incidentally, this closely connects with the idea of steelmanning: in a limit proof or other mathematical proof showing that a definition involving quantifiers is satisfied, one needs to demonstrate that for all the moves of one’s opponent, one has a winning strategy to respond to the best move the opponent could possibly make.
The first time I taught epsilon-delta definition in a (non-honors) calculus class at the University of Chicago, even though I did use the game setup, almost nobody understood it. I’ve had considerably more success in future years, and it seems like students get something like 30-50% of the underlying logic on average (I’m judging based on their performance on hard conceptual multiple choice questions based on the definition).