In trying to teach my honors calculus course, it's occurred to me that part of the problem is that there are two directions of time at work in most simple epsilon-delta proofs: the time in which the choices are made and logical implications take their inexorable course, and the order in which the steps are most helpfully explained. It seems to me that a two-dimensional style of proof might make this clearer to the students, and keep them from making certain sorts of mistakes. Has anyone seen anything like this, as a form of training-wheels for the mind? (I could create some for my students, but I'd rather not create teaching materials that already exist.) As background for my question, I should mention that there *is* a way to present proofs that sweeps the two-different-timelines issue under the rug: namely, omit all motivation, and simply say things like "Given epsilon > 0, we may take delta = epsilon / 2, and then 0 < |x-1| < delta implies |2x - 2| < 2 delta = epsilon, as required." But proofs in this style, especially when the details are complicated, may leave some students wondering "How could an ordinary human come up with such a proof?" What is more common, in books like Stewart's calculus, is to embed within the proof, starting from the beginning, helpful words about where we're trying to go, and what choices will enable us to get there. But the students who read such "motivated proofs" then sometimes write proofs that say things like "|2x-2| = 2|x-1| < epsilon, therefore delta = epsilon / 2," which confuses motivation with implication. I'd welcome thoughts about how to make things clearer to students on such matters, be they two-dimensional diagrams of proofs, or helpful analogies between proofs and other sorts of texts, or catchy slogans about the nature of proof, or whatever. I should mention that I already make extensive use of the whole quantified-proposition-as-two-player-game gimmick in my teaching; my students are probably really sick of (universal) Adam and (existential) Eve by now... Jim Propp