My biggest “Screw everything” moment about math was the first lecture of my numerical methods class in college when the professor said: “All that calculus you’ve been learning your whole lives? It’s useless. Carefully curated set of a few dozen problems that are doable by hand. Here’s how it’s really done for anything remotely practical”
And then we learned a bunch of algorithms that spit out approximate answers to almost anything. And a bunch of ways to verify that the algorithm doesn’t have a bug and spat out an approximately correct answer. It was amazing.
But the most long-term useful math class (beyond arithmetic and percentages) has been the semester on probabilities and the semester on stats. I don’t remember the formulae anymore, but it gave me a great “feel” for thinking about the real world. We should be teaching that earlier.
When I took 400 level Real Analysis: “All that calculus you’ve been learning your whole life? It’s a lie. Those epsilon delta proofs? They were fake - none of you were smart enough to challenge us on ‘limits’. And now we’re gonna do it all again only this time it’s really gonna be rigorous.”
Is there any somewhat simple explanation of what are the limitations of the epsilon-delta definition of limits that make it non-rigorous? I've been trying to find some information about your comment, but have so far come up empty.
I'm shaky on this - it's been thirty years - but I believe the Calc I epsilon delta proofs relied on the notion of an open and closed intervals on the real line, which we all intuitively understood.
The upper level Real Analysis made us bring some rigor as to what an interval on the real line actually meant going from raw points and sets to topological spaces to metric spaces, then compactness, continuity, etc. all with fun and crazy counterexamples.
And then we learned a bunch of algorithms that spit out approximate answers to almost anything. And a bunch of ways to verify that the algorithm doesn’t have a bug and spat out an approximately correct answer. It was amazing.
But the most long-term useful math class (beyond arithmetic and percentages) has been the semester on probabilities and the semester on stats. I don’t remember the formulae anymore, but it gave me a great “feel” for thinking about the real world. We should be teaching that earlier.