An intermediate approach I've been taking with my own son is to use calculus and initially not bother with either the manipulation OR the formalism. You have something moving, have the position and want the speed. Or vice-versa. You use it in a bunch of places to build out vocabulary and fluency, and build up both the mechanics and the formalism later.

And we derive a few formulas along the way. integrals of x and derivatives of x2 (x^2, depending on your programming language) pop up all over the place.

I don’t know how old your son is but for a high level and amusingly written book was How to Ace Calculus, The Streetwise Guide. I know it sounds a bit gimmicky but even after years of calculus classes I found it helpful for the higher level view. I always found from engineering school there being a stark difference between the classes taught by engineers and those taught by mathematicians. It always made more sense to me to have a concrete example of why I might use something before diving in than to have all this abstraction and be expected to abstract it to other problems.

I appreciate your perspective, especially around motivation, and I think you're right that students learn differently.

I do wonder though -- for heavily skill-based endeavors (say swimming, sushi-cheffing, curling, etc.), the best way to learn is to do. First principles/mental model approaches tend not work so well in these domains, at least not isolation.

The best approach is probably a blend: learn by doing and understanding first principles.

A difference, though, is those domains are almost entirely about doing: you don’t have to understand buoyancy or the biology of the tongue to swim or cook. Math is almost the opposite: this day, the mechanics of math are nearly entirely automated: there’s no real reason to manually work out answers except insofar as it helps you understand the principles. The main reason to study math now is so that you understand what your computer is doing, so you can instruct it more efficiently. So, first principles in math are significantly more important: someone who can do the mechanics of calculus or long division but doesn’t understand what their doing is just doing slowly and unreliably what a computer can do instantaneously.

You don’t necessarily have to start from the bottom, though; you can drill down and get to first principles instead of starting without context at low level abstractions. Both of the angles you two are talking about (providing context and working through the mechanical applications vs explaining fundamental principles that makes things work) seem addressable in the same course if you start with understandable problems and drill down until you get to the principles that allow you to solve those problems.

I think it would be quite difficult to do in a reasonable amount of time, but using history as a course outline would be a cool way to explain the motivations/context behind the principles being discussed. That might better illustrate how math research works by explaining the gaps/interesting questions that existed prior to certain ideas and how those ideas evolved and lead to other gaps/questions.

The type of math courses I’ve gotten a lot out of already kind of did that, they just usually used more of a condensed, logical narrative than a historically based one.

Definitely, although I think where to start really depends on the context.

I see what you're saying, but I would say there's a strong "doing" component in math.

It's still really important to work through problems mechanically (often with only partial understanding) if only to become familiar with patterns. Whether this exploration is done via a CAS or by hand, acquiring this mechanical familiarity is important if one is to go deep. Math is often not merely read but done.

Take something like automatic differentiation (AD) for instance. It's just the chain-rule in principle right?

One could try to study the principles of AD in textbooks/journal pubs and then try to implement it in code. Chances are one is unlikely get it right the first few times because anyone who's ever tried to implement anything from numerical computation papers knows that published papers regularly omit important implementation details (unless they also publish code). There's so much heuristic/unintuitive behavior [1] in the world of numerical computation that first principles alone is rarely enough to get you a decent implementation in code.

On the other hand, if one mechanically works through AD derivations for a bunch of functions and observing the patterns that emerge (without necessarily grasping all the principles at first), one begins to notice things, like difficulties posed by corner cases like non-smooth/discontinuous functions, non-n'th differentiable functions, etc. A CAS could be used for this exploration (manually doesn't necessarily literally by hand) but it's so important to actually try stuff out and really grasp how mathematical objects work in practice. Once this contextual understanding is acquired, going back to AD journal pubs and reading about the principles is likely going to make way more sense because one has seen the behavior "in the field".

I do think the pure mechanical repetition is often underrated as a autodidactic mechanism. (again I don't mean literally by hand, but rather "doing" instead of just "reading and thinking").

[1] non numerical comp folks are often surprised to hear this because they think everything's super deterministic -- it's not. Tiny random perturbations in matrix can change things wildly. Take naive Gaussian elimination -- the ordering of the rows of equivalent linear systems can vastly affect solution efficiency and numerical stability, but one can't appreciate that unless one understand how GE works mechanically and has worked through examples.

Yes, I think doing both in parallel, or alternating between periods of theoretical study and periods of applied learning (projects for instance) could work very well. I do, by the way, fully agree that just first principles based learning doesn't cut it either. Practice is necessary, I just didn't feel I learned from practicing things which I didn't understand at all.