> This is a beautiful description. I know as a teacher that I try to communicate this to my students, but also that I surely fail much more often than I succeed.

Thanks. There's an element of hopelessness in trying to explain the immensity of human knowledge to someone who's lived their entire life captured by mandatory schooling. It's just not an interesting thought to most of them. Their worldview has been so artificially limited that attempts to explain the limitations of their circumstances just appear to be more limitations. "Guys, there are more than six hundred thousand mathematicians working right now in the US, and they're working with concepts first thought of as long ago as 3000 BC and maybe earlier!" "Okay, will that be on the test?"

> Isn't it better if that's in the classroom rather than on the job?

If only we were having this conversation in a bar instead of a text forum. That's a huge question and it's clear you're actually interested in talking about your thoughts on it. It's also a subject that interests me.

When you say:

> But those topics have to be, or are believed to have to be, understood to be successful in the field,

I think that's where the big disconnect comes from. What's the point of school? Is it to know enough to be useful at a job, or to plumb the depths of knowledge? Is it some third thing? Ask any recruiter and they'll tell you the new hires aren't prepared to actually do anything useful, and ask any advisor and they'll tell you the new graduate students aren't prepared to actually do anything useful, so we can at least conclude that there's some kind of disconnect going on. Students spend thousands of hours and hundreds of thousands of dollars doing something that does not actually adequately prepare them for what people want them for.

I have a hundred different ideas about how to address this. One thought I've been mulling over recently is that a redefinition of grades is in order. From essay that we're commenting on:

"My colleague’s error consisted of believing that the more testable the material, the more teachable it is. A wider spread of performance in the problem sets and in the quizzes makes the assignment of grades “more objective.” The course is turned into a game of skill, where manipulative
ability outweighs understanding...

In an elementary course in differential equations, students should learn a few basic concepts that
they will remember for the rest of their lives, such as the universal occurrence of the exponential
function, stability, the relationship between trajectories and integrals of systems, phase plane analysis, the manipulation of the Laplace transform, perhaps even the fascinating relationship between partial fraction decompositions and convolutions via Laplace transforms. Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach."

At the undergraduate/introductory level, the only important question is "what awareness do you have of the breadth of this field, and what mastery do you have over the concepts that most agree are its "most fundamental"? You and I both agree that class is a "playground", and any "grade" is in fact meaningless. Rather than assigning As, Bs, Cs etc. as a percentile of subjective completion of arbitrary problem sets, As-Fs should be assigned at the discretion of the instructor as a holistic assessment of oral and written examination, completion of problems and problem sets, participation in the class, and general wisdom.

Students would of course resist this. They want to be graded on impartial, meaningless criteria. That's how they're taught from third grade, and it's the method that allows for the least possible interaction with the material. The only reason they want these grading criteria is so they can plan to spend as little time as possible on the class. This method of approaching learning simply has to be broken at every level of education. You shouldn't even have a GPA until college.