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If you are worried about epsilons and deltas being too hard, then you can write your calculus text to not be rigorous rather than trying to make it rigorous wit

by rsj_hn 5y ago

If you are worried about epsilons and deltas being too hard, then you can write your calculus text to not be rigorous rather than trying to make it rigorous with non-standard analysis. I think giving up on some of the formalism while still sketching the ideas of proofs is the way to go from a pedagogical point of view.

bgorman 5y ago
Have you compared the approaches? I learned with epsilon delta, but I would be interested in seeing data about learning outcomes with alternative approaches. In my experience, epsilon delta understanding was pretty much unrelated to passing calculus, where tests consisted of formula/substitution crunching.
- rsj_hn 5y ago
  > In my experience, epsilon delta understanding was pretty much unrelated to passing calculus, where tests consisted of formula/substitution crunching That's a shame. There are so many exciting things to learn in calculus that you can skip the epsilon delta stuff and still do so much more than formula/substitution crunching. Calculus is the gateway to differential geometry, topology, mathematical physics, differential equations, taylor series which are useful for numerical approximations and so many other things, analytic number theory, complex analysis, many forms of statistics. So many wonderful things you can do with it. Imagine using your new found calculus knowledge to show that the orbit of the planets about the sun is an ellipse -- which is what Newton used calculus to do -- OR learning all about epsilons and deltas. Which would be more fun as a student learning calculus. Make the math exciting enough and people will put up with the drudgery of calculations.
mcguire 5y ago
"...you can write your calculus text to not be rigorous..." Oooh, the comments you'll get from other mathematicians...
- rsj_hn 5y ago
  I can take the heat. School texts don't need to be rigorous and there are different levels of rigour. I think Bourbaki did more harm than good.
ska 5y ago
> you can write your calculus text to not be rigorous rather than trying to make it rigorous with non-standard analysis. e.g., many engineering calculus courses, and explicitly some of the texts.

by rsj_hn 5y ago

bgorman 5y ago

Have you compared the approaches? I learned with epsilon delta, but I would be interested in seeing data about learning outcomes with alternative approaches. In my experience, epsilon delta understanding was pretty much unrelated to passing calculus, where tests consisted of formula/substitution crunching.

rsj_hn 5y ago
> In my experience, epsilon delta understanding was pretty much unrelated to passing calculus, where tests consisted of formula/substitution crunching That's a shame. There are so many exciting things to learn in calculus that you can skip the epsilon delta stuff and still do so much more than formula/substitution crunching. Calculus is the gateway to differential geometry, topology, mathematical physics, differential equations, taylor series which are useful for numerical approximations and so many other things, analytic number theory, complex analysis, many forms of statistics. So many wonderful things you can do with it. Imagine using your new found calculus knowledge to show that the orbit of the planets about the sun is an ellipse -- which is what Newton used calculus to do -- OR learning all about epsilons and deltas. Which would be more fun as a student learning calculus. Make the math exciting enough and people will put up with the drudgery of calculations.

mcguire 5y ago

"...you can write your calculus text to not be rigorous..." Oooh, the comments you'll get from other mathematicians...

rsj_hn 5y ago
I can take the heat. School texts don't need to be rigorous and there are different levels of rigour. I think Bourbaki did more harm than good.

ska 5y ago

> you can write your calculus text to not be rigorous rather than trying to make it rigorous with non-standard analysis. e.g., many engineering calculus courses, and explicitly some of the texts.