Day 1 of the class: The derivative calculates the slope of a function
Day 2: The integral calculates the area under the curve of the function
Days 3-89: Rote exercises deriving and integrating increasingly obscure functions
Day 90: Final Exam
Spending a few days at the end re-exploring the "big picture day-1" to tie together all of the various strands of knowledge you accumulate over the semester would have made all of it so much more effective.
See, I loved math. So all through Calculus I could easily remember the big picture. Every time I practiced an Integral I imagined curves and calculating the areas under them and the visual problem and the relevance of what the curve represented in real life and the massive amount of applications it could be used for in the real world lit up my neurons like fireworks in the sky.
(Then I went into computer science, and wound up never needing calclus again, based on the type of work I happen to be doing, but alas)
But where I really would have wished a constant reinforcement of the "big picture" is History. Cuz it always seemed so pointless and useless. Why are we studying these old dead people, and everything they did. Who cares? They're old, and they're dead, and nothing they did matters to us anymore.
Until you grow up, and go from your 20s to your 40s, and suddenly realize oh shit we're living THROUGH history. We're creating history NOW. We're making choices, and we're making mistakes just like those old dead people in history. Old dead people that weren't really any less developed or evolved primates than us. Just equally victims of their circumstance like us, and also agents of change like us.
Suddenly history seems much more significant.
Except it's not funny because then it leads to people both-sidesing genocide and actual nazis because the allies also committed war crimes.
History is a mess because humanity is a mess. But we like to think that we aren't. Probably as a defense mechanism.
"History" is the study of different narratives to TRY to come to a semblance of truth, but even for recent events this is almost impossible.
Pick an example like "Did the US dropping the nuclear bomb on Japan ultimately save lives, or was it unnecessary" and it's impossible to find the truth between 2 conflicting narratives, each fairly justifiable.
Another one is, most Soviet Anti-American Propaganda was true.
Not only would "narratives matter" be an uncontroversial statement among historians, they'd tell you that all historical writing is narrative construction. And they'd hand you a book on historiography and teach you about the methods that historians use to understand an honestly present narratives in their writing.
Note that this does not mean that historical writing is bullshit. A lot of engineers seem to come up against these observations in the humanities and then just assume that nothing can be done and that entire fields must be discarded while the people working in those fields have been living with this stuff for their entire careers.
Seeing graphics animate according to a derivative that you just plotted yourself is really useful to develop practical intuition about what it means.
After Effects is too expensive and complex for high schools, but maybe some kind of modern Logo-style environment that combines coding and animation could be useful for calculus beginners. (And linear algebra too — another field where the basics have a direct intuitive application in computer graphics.)
(The last time I taught math was vector calc 4 years ago. The last time I taught the first semester intro to calc was 30 years ago OMG-LOL.)
I think most of these questions are not measuring intuition per se, but rather has the tested person previously seen such functions plotted on a graph.
Either that, or my mathematical intuition has got rusty from years of code monkeying.
Perhaps (regularly) seeing functions plotted on graphs is a necessary precondition to maintain intuition :)
Instead of sketching the derivative based on the graph of a function, we had to sketch the function based on a table of data which described the function as well as its first and second derivatives in terms of value, existence, and sign at various points and intervals.
(note: this was not in the US, but in the early 2000's in a small European country)