,, additionally, moving limits to the end of a first-year course allows students to develop intuitions around the derivative first before seeing the formal proof of their validity’’
Waiting a year to get from intuition to theorems is a perfect way to ruin math.
Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking.
At the same time there could be more examples taught on why these building blocks were historically needed and what they are used for solving nowadays.
I'm curious if you've taught calculus? Do your students remember limits by the end of calculus? Most studies show that students DO NOT RETAIN limit concepts (ESPECIALLY epsilon-delta ones). It is used as a crutch and then largely discarded before anyone is actually comfortable/familiar with them.
By being a little handwavy at the beginning, you can then tell Dorothy that she's had her ruby slippers with her all along, and by this time they recognize the power and importance of the concept.
In some ways, it is good to be able to explain, from the ground up, why each piece is in place. But, sometimes, understanding how to build a student requires knowing when we need to temporarily handwave something away so that it is actually meaningful when they get it. And then, doing that so they don't maintain a false conception is also important, which is why handwaviness is often helpful ("this is kind of like dividing by zero, but kind of not, and we will get to the distinctions later so just trust us for the moment").
> Most studies show that students DO NOT RETAIN limit concepts
Is that so ? I would not have guessed. I am not being sarcastic. Going by experiences of my own high school cohort I would have claimed that limits had a more lasting impression.
BTW I enjoyed your arxiv paper on 2nd order derivatives.
> Waiting a year to get from intuition to theorems is a perfect way to ruin math.
It's interesting that you chose to make that point in a thread about calculus specifically. It had pretty shaky foundations for most of its history, and even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result.
> Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking.
Math is a means to an end. Making the tool easy to use is a desirable property. I've heard "it's not supposed to be easy" applied to many disciplines, from film photography to software engineering, and I think it's mostly gatekeeping.
> even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result.
I think this is practically fixed by Robinson's NSA when it's combined with big/little O notation:
δy = f'(x) δx + o(δx)
A (nonstandard real) quantity is o(δx) when it's infinitesimal relative to δx, i.e. s ∈ o(δx) whenever s/δx is infinitesimal. So δx² ∈ o(δx) but δx ∉ o(δx).
Waiting a year to get from intuition to theorems is a perfect way to ruin math.
Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking.
At the same time there could be more examples taught on why these building blocks were historically needed and what they are used for solving nowadays.