A couple of weeks ago, I was asked to speak on a panel about teaching during Union's new-faculty orientation. We had one person from each of the academic divisions (arts and literature, social science, natural science, and engineering), and there was a ton of overlap in the things we said, but here's a rough reconstruction of the advice I gave them:
1) Be Wary of Advice
Because it's always good to start off with something that sounds a little counter-intuitive... What I mean by this is that lots of people will be more than happy to offer advice to a new faculty member-- often without being asked-- but a great deal of that advice will be bad for the person getting it. This isn't the result of active malice, just that teaching is a highly individual endeavor.
The relatively harmless example I use to illustrate this comes from my first year of teaching, when I went around and asked my new colleagues for advice. One very successful guy said that he made an effort to maximize the effect of our small classes by "breaking the fourth wall" and walking away from the chalkboard out into the middle of the room.
That sounded good, so I tried it for a while. And quickly found that while it worked well for him, it was a disaster for me. I'm multiple standard deviations above average height, and several years later a student wrote on a course evaluation "He is loud and intense." That combination meant that when I would walk out into the room while lecturing, the students closest to me were basically cowering in fear. Once I noticed that, I made a point of staying back near the blackboard unless I needed to go out into the room for a demo or activity, and everybody was happier.
So, my advice to new faculty is: be wary of advice from older faculty. You'll get lots of it, but much of it will be bad for you. You need to be independent enough and self-aware enough to recognize and use the bits that will work for you, and discard the bits that won't.
2) Don't Be Afraid to Try New Things
One of the big categories of well-intentioned bad advice that most new faculty can expect to hear is some form of "don't rock the boat until you have tenure." This particularly comes up in the context of teaching-- I had people who were speakers at a workshop on improving introductory physics courses tell me not to try to implement any new methods as a junior faculty member. The logic is that changing anything will have a short-term negative effect on your teaching evaluations, and you shouldn't risk that before a tenure review.
I think this is bad advice, because if you're going to go with that, there's always some reason not to change things-- you'll be up for promotion to full professor, or looking for a fellowship, or an endowed chair, or something. There are always good, logical-seeming reasons to keep your head down and do something functional and not take risks that could improve your teaching.
So, my advice is that if you look around and see something that you'd like to try that would improve your teaching, go for it. Again, you need to be independent and self-aware enough to recognize what's likely to work, and make adjustments when needed, and you need to be prepared to defend the choices you make should it become necessary: "My evaluations went down when I changed teaching methods; this is a well-known effect of change, but they've improved since, and student learning is better by these metrics."
3) Don't Assume Your Students Are Like You Were
I count this as the best one-sentence piece of advice I got as a junior faculty member. It's really important to remember that people who become college faculty are necessarily unusual-- we're the ones who had enough interest in our subjects to continue into graduate school, and who were sufficiently passionate and self-motivated to succeed there.
That's just not going to be the case for the vast majority of the students we will encounter as faculty. There will be a few, and we should cherish them, but most of them are not going to find the subject as intrinsically fascinating as we do, and won't put in the same level of independent effort.
So, my advice is to remember that (paraphrasing another famous comment) you go to class with the students you have, not the students you wish you had or the students you were. If you go in expecting students to react the same way you did, you're going to end up disappointed and frustrated. This doesn't mean you can't ask them to become better than they are, it just means that you need to make an effort to meet them where they are, and move them toward where you'd like them to be.
(For much of my college career, I was probably closer to the mean attitude of our students than many of my colleagues were-- I played rugby and partied a lot-- but I still regularly need to remind myself of this...)
That's the advice I offered, with the repeated caveat that they should remember the first item on my list, and be wary of all advice, including mine... Happily, many elements of this were echoed by my fellow panelists, so I think I was basically on the right track, but still, it's impossible to succeed for everyone...
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I am a bit bemused by your implied equivalence of going to grad school and being faculty. That may still be accurate for some disciplines but since the 1960's has become increasingly less so for STEM. None of the physics PhDs in my year group (5) became faculty initially. One did so mid-career briefly as the result of a down-sizing, and two of us have become research fellows since (primary) retirement but only part time. I agree with your basic advice about students being different (probably,) but the idea of going to grad school and not then being faculty needs to become part of the world view as well.
I'm not trying to say that going to grad school necessarily leads to becoming faculty-- obviously, it doesn't, given the persistently awful job market in academia. But going to grad school is a prerequisite for becoming faculty, which means that the small number of folks who DO become faculty are drawn from the small (and unusual) population who chose to go to grad school and were successful there.
The framing that works for me is "mathematics is to most students as foreign languages were to me." Most of us had SOME subject we found completely dull and unintuitive, although it wasn't the subject we became professors of.
Speaking of the students you have, one of the biggest surprises when I moved into teaching is that so many of the teaching techniques that were not just useless but actively damaging to my education are actually helpful for the majority of students. Giving, say, 20 examples and expecting them all done out neatly isn't punishment or handwriting practice, it's a good way to master the topic for students who learn math slower but learn handwriting faster than I did.
The bit about not assuming your students are like you goes double if you are ever teaching a service course, whether it's a basic intro mechanics/E&M sequence (taken by most engineers, chemists, and geologists as well as physics majors; even more so if it's the non-calculus version that some universities offer to biology and pre-health students) or a general education English class that everybody who wants to graduate has to pass at some point. At least the overwhelming majority of students in most upper-division courses will be majoring in that department (or one of the departments involved if the course is cross-listed). That makes them more like you than the typical service course student.