One of my beloved professors told me a story about 30 years ago that has stuck with me, and has informed (not enough, alas) my own teaching practice. He was taking a full-year graduate course in C*-algebras with (I believe) Israel Halperin. By the end of the course they had covered 16 pages (!) of a textbook that had several hundred pages. Sixteen. That’s it, that’s all.
However, they covered those 16 pages so thoroughly, with so much detail, exploring all the points in such great depth, and looking at so many different perspectives, that he was able to read, quite successfully and on his own, the rest of the textbook over the summer.
At the time this story was a revelation to me. What is remarkable to me now is how little things have changed over the years, in the sense that this wonderful experience is still so rare. Our curricula are still based on “covering” a lot of material. Typical university courses are constantly rushing through material, as fast as possible, because we have to “cover the material.” So in class we rarely have time to explore anything in any kind of detail; instead, we scratch the surface and trust that students will fill in the details. But students are beginners, and are ill-equipped to go into the depths; they just don’t know what to do, much less how to do it, and besides they are just running as fast as they can on a treadmill, trying to survive, and so have no time for reflection.
And textbooks typically also don’t go into depth … they’re too busy covering lots of material, too, in an attempt to satisfy as many potential adopters as possible, so as to sell as many copies as possible.
But covering material is just a cop-out, giving us a convenient excuse when our students have not learned very much. (“Well, at least I did my job … I covered the material.”)
What will make a huge difference to our students is if we can manage to change our approach so that we cover less material, and guide our students to master a smaller amount. Having a vague idea of a lot of stuff turns out not to be of much use (by itself) in mathematics; sure you want to read around various subjects for breadth of knowledge, and you can’t be expected to be an expert on everything. But each student ought to have mastered something.
Only by mastering something can a student build a good foundation upon which to effectively learn other mathematics. But our education system is not structured to require or even encourage mastery. Rather, “everyone” (i.e., the department chair, the dean, etc.) seems to be happy as long as the class average is between 60 and 70.
The best place to begin requiring mastery is in elementary school and high school, where classes are relatively small. Students should move on to the next level when they demonstrate that they are ready to do so, much like in a karate dojo.
As for universities and colleges, the large classes and minimal resources available mean that under current circumstances it’s hard to imagine how any meaningful improvement will take place any time soon.
ps. Just to be perfectly clear, I don’t blame individual professors or students for this state of affairs. Practically everyone means well and is working very hard, just trying to stay afloat. The problem is systemic, and the solution requires re-thinking our current model of undergraduate education.