I'm a big fan of the conceptual approach. One of the largest problems I see with math education is that we don't check if things are really clicking.
I graduated with an engineering degree from a great school, and still didn't have an intuitive understanding for i (the imaginary number) until I was about 26.
Go find your favorite tutorial introducing imaginary numbers. Got it? Ok. It probably defines i, talks about its properties (i = sqrt(-1)) and then gets you cranking on polynomials.
It's the equivalent of teaching someone to read and then having them solve crossword puzzles. It's such a contrived example! (N.B., this anguish forced me to write a tutorial on imaginary numbers with actual, non-polynomial applications, like rotating a shape without needing trig. See https://news.ycombinator.com/item?id=2712575)
Calculus needs these everyday applications and intuitions beyond "Oh, let's pretend we're trying to calculate the trajectory of a moving particle."
They're out there: my intuition is that algebra gives a static description (here's the cookie), while calculus describes the process that made it: here's the steps that built the cookie. Calculus is the language of science because we want to know how the outcome was produced, not just the final result. d/dx velocity = acceleration means your speed is built up from a sequence of accelerations.
I have tried (fitfully) three times to sit down and learn what calculus actually means. I still don't get it. And I suspect that clients and families will push it ever further away.
Yes, give me real world puzzles and applications for learning new mental tools and I will probably love them. Make it abstract enough and I cannot see the value in learning it.
I think the essentials on how to think with calculus can be conveyed in 1 minute. Let me know if the above helps :).
I think the key missing insight for me was that a continuous process and a discrete process can both point to the same result.
A pixellated word on a screen ("cat") conveys the same meaning as a perfectly smooth vector of the same word. The math idea, to me, is "can a discrete description/process" point at the same result that we get from a continuous one?
The idea of limits, is essentially figuring out when a discrete epsilon can still lead us to the true value of f(x) (if this trick works, we call it a continuous function, and can use calculus with it.)
Unfortunately, I doubt that these implications will be discussed in your average Mathematics classroom. Actually, my experiences with mathematics in primary, secondary and a lot of tertiary education, both as a student, teacher and observator, leads me to believe that conceptual discussion is rare, whilst procedural discussions (how to perform some calculation) are plenty and abound.
Regarding the Gaussian (imaginary) numbers, they have uses which have nothing to do with rotations. My intuition for them is that they are a way of factoring expressions such as A^2 + B^2 as the difference of two squares.
I don't know what age group you're targeting but I found solving problems in this space gave me a great intuition for imaginary numbers; I found it quite interesting too. Heck, AC circuits are great real-world playgrounds for differential and integral equations - Laplace transforms too.
I've been fretting about how to help my kids get a better understanding of math for years. I've been bouncing between too simple and overly complicated for too long. :-(
I think I'm just going to mandate they spend some amount of time per week using Mathematica and going through some introductory book. At least they will taking baby steps in a direction better than youtube. Grumble...
So I've been fretting on which book. Any suggestions? Heh heh...
For books, I don't have a huge amount of experience but I do like the Manga Guides to X (Manga Guide to Physics, etc.). A visual medium encourages the author to rely on metaphors vs. symbolic descriptions, which I prefer in an intro to get the concepts across quickly.