The Most Common Errors in Undergraduate Mathematics (2009)

> Guess what, when teaching only staff is asked to teach 6 courses in an year, they teach the same as a research prof who spends half their time in research and half their time teaching 3 courses. > ... > The problem is capitalism, and money-optimizations being the final decision maker rather than quality of teaching or research (which is also much worse than 50 years ago). > ... > You need more money from the state to go into education.

Absolutely, agreed on all points. Find people who are passionate about education and give them the time and resources to do it well. Build a culture of learning, mentorship, and open discussion. Let students and teaching faculty mingle as much as possible with research faculty while still keeping priorities straight.

Sure, in an ideal world, every student would get one-on-one tutoring from a brilliant researcher. But this doesn't scale well, and isn't necessarily a great use of the researcher's time. Teaching-only faculty are more than good enough until the student approaches the research level.

> I don't agree at all with lecture recordings. As much as you want people in society learning, you also want a lot of people in society learning to teach.

Oh I agree, perhaps I misrepresented myself. I'm not suggesting we replace traditional lectures with a big movie screen that plays pre-recorded lectures.

However, I think the current way of doing things -- where a professor inherits some slides she didn't create herself and reads them off in front of the class with no preparation whatsoever -- isn't the best either. This just goes back to finding passionate teachers and giving them the resources they need to be successful.

I do think pre-recorded lectures have their place alongside traditional textbooks, lecture notes, etc.. One downside of traditional lectures is that lecturers get very little feedback about their teaching style, and it's difficult to diagnose how students are really doing.

A handful of courses in my undergrad math program were run in an "Inquiry-Based Learning (IBL)" format. Rather than a traditional lecture, the professor breaks the class into small groups and asks a series of leading questions designed to help students discover a new concept on their own. The professor can adjust the pace and offer explanations as needed.

Here's an example [1] of some class handouts from a topology class. I borrowed them from a friend who took the class, and going through all the exercises on my own brought me a sense of clarity that I was never able to achieve from the standard lecture-based topology course I took. My friend felt the same way, and according to surveys done by the department, students overwhelmingly prefer the IBL format to traditional lectures. This format benefits professors, too, who get immediate feedback about their teaching methods and how well students are doing.

[1] https://benrbray.com/static/files/umich_math490_f16_sbray.pd...

[1] http://www.math.lsa.umich.edu/~jchw/teaching.html

I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin".

It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and it's also off by a multiple of pi, depending on that subinterval. "arcsin" is then defined as the inverse of sin, restricted to the interval [-pi/2,pi/2].

Of course, the bigger issue here is that f^n for any function f is inherently ambiguous, because it could refer either to the (pointwise) multiplication operation or to the composition operation.

And, I did not say it is impossible or even uncommon for research faculty to be good at teaching. However, being good at research does not automatically make one good at teaching, and there are other factors like time that prevent those who do care about education from giving their students a good experience in class.

In my own personal experience, and that of my classmates, the vast majority of research faculty simply do not have either the interest or the time to teach well. Out of the 30 or so math courses I took as an undergrad, I would say only about 4-5 of my professors put real effort into teaching.