Everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning. That level is different for everyone, and it's not a hard threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible.
If you’ve ever had to teach calculus in a classroom, then you’ve probably noticed that something weird happens when you teach students how to determine whether a function is continuous.
The Phenomenon: A Resistance to First-Principles Reasoning
You go through a whole zoo of different-looking examples with students, constantly emphasizing that while there are different “rules” for each example, the rules all stem from the same “first principle” that we are checking for equality of left and right-sided limits.
And you demonstrate this by going through exercises where you provide a new example (not already in the zoo) and work with students to solve the exercise from first principles.
But many students ignore the first principle and instead prefer to memorize the “rules” for each appearance of functions in each solved example. And if you ask them to solve a problem for which they don’t have a similar example, they give up easily: “How should I know? You never taught me how to do this.”
You explain to students that understanding that first principle will save them time and effort on homeworks, lead them to better scores on assessments (and hence a better grade in the class), allow them to more easily grasp math that they might have to learn in the future, and empower them to face any related problems they might face that didn’t appear in your zoo of examples.
But even after you provide all this support, nothing changes.
Why This Happens: The Abstraction Ceiling
In my view, the only explanation for this phenomenon is that everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning.
That level is different for everyone, and it’s not a “hard” threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible. I’ve seen this happen with my own eyes while working with many students longitudinally as they learned 4+ years of math.
We’re getting into a really touchy subject, and this may be an uncomfortable (maybe even unpopular) take. But surely there’s no issue accepting that not everyone can really understand Topological Quantum Field Theory at its core, right? Why should Calculus be any different?
In fact, it would be weird if there were a single level of math that marked the dividing line between “everyone who’s learned the prerequisites can do it” versus “reserved for geniuses only.” It’s likely a continuous thing that starts in fairly elementary math – the higher the level of math, the fewer the number of people are capable of really understanding it at its core.
Each student eventually hits their “abstraction ceiling,” the amount of time and effort it takes for them to succeed in math class begins to skyrocket, they find other subjects that they enjoy more (which may or may not use the math that they’ve learned so far), and they take an “off ramp” into those subjects.
Defense Against Misinterpretation
Of course, this is in no way an argument for giving up on students who are struggling. You support them as much as is feasible given the context of the class. But at the end of the day, when you’re reflecting on how well you’ve done as a teacher, you need to have realistic expectations.
As an analogy, if you run a summer basketball camp, you might have a couple really impressive kids who make you think “wow, maybe I’ll see them play professionally someday,” but you’re not going to turn all the attendees into future pro ball players. Most of the kids probably aren’t even going to get a basketball scholarship to college. You’re definitely not going to get the 5’2” kid to dunk. And that’s okay. Hopefully, you’ll get each kid to have fun and become the best basketball player that they have the potential to be.