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I recently finished my third semester of teaching calculus to freshman college students. This means I was drawing the same pictures, solving the same example problems, and discussing the same techniques as I had in two previous semesters. With multiple sections per semester, review sessions, and office hours, it is possible that in one semester alone, I'll teach the same idea/say the same sentence/do the same problem 5-10 different times.

Due to all of this repetition, this last semester I felt myself growing tired of calculus. I know that for my students, the material is new and (hopefully) interesting and exciting, but it was none of these for me, and I could feel it affect my teaching. I would battle to maintain enthusiasm and joy for the material as I taught my students, but I often lost this battle, as is demonstrated by the following comment I received for instructor evaluations:

"Jared was a great TA who could possibly improve on enthusiasm"

I completely agree with this student, and since I hope to be able to teach mathematics for many more years than just 3 semesters, this is a problem I should begin to address now.

For teachers of mathematics, how do you maintain enthusiasm and joy in teaching the same material year after year?

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Respect for addressing this issue. – Git Gud Jan 15 '14 at 23:06
I'm also a grad student, but if you can manage some classroom interaction, that helps me to keep things fresh. Sometimes I give challenge problems and sometimes I'll have a bright student do them, which can also be helpful. Once in a while (eg. right after an exam) there won't be enough calculus material to go over and I'll just pick some other (math) topic to talk about for a while. Hopefully your situation will allow you to teach other thinks like linear algebra. – J Swanson Jan 15 '14 at 23:08
Calculus is one of humankind's greatest achievements, allowing us to make strides towards understanding the infinitely large, infinitely small, and infinitely often. Having a sense of awe for that has been really important to my teaching. Helping students feel the same sense, and realize they're experiencing a once in a lifetime eye-opening experience that will help them ask and answer questions they couldn't have hoped to previously is how I approached it. I realize I'm basically saying "do a great job!", but I think that emphasizing the feeling of great discovery is really helpful. – Callus Jan 15 '14 at 23:13
@Jared : My experience is similar. If you have taught the same material many times, you probably have many notes that you can recycle and in general it should take you less time than it used to to do what you absolutely have to do to prepare. Maybe you could take some of that extra time and write some Maple worksheets to show off, or some problems for students to do in class, give the students online surveys, find YouTube videos for them to watch and assign them problems from them, etc. That may keep you and your students from getting bored, and your students may appreciate it. – Stefan Smith Jan 16 '14 at 0:37
@Jared : I also agree with what JSwanson wrote about classroom interaction. Students seem to like it. – Stefan Smith Jan 16 '14 at 0:38
up vote 20 down vote accepted

In operational human terms, I think it is best to avoid teaching/TA-ing exactly the same thing every semester. At the very least, go through the whole year's cycle, for example, of Calc I and Calc II. Then, with the summer to further "forget", starting again with Calc I in the autumn may not seem so bad. One's mind has time to forget a little, to romanticize, especially to fool you (constructively) into thinking that "this time I'll do it right!" (and everyone will understand perfectly...).

Even better is to go through a longer cycle-time of two years, perhaps Calc I,II,III, IV, and then repeat. Such non-repetition does entail a bit more effort to "prepare", but this is the cost of avoiding staleness.

The same sort of issue exists, perhaps even more poignantly, for the basic grad courses, which most likely you'll find yourself teaching at some point. It is very important to not become jaded, to not lose a grip on how "obvious" things are, simply because one has thought it through so many times. Knowing how long it takes for one's own head to "forget" the short-term details is important, since it seems best to out-run the short-term memories to have "freshness" and enthusiasm.

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"At the very least, go through the whole year's cycle, for example, of Calc I and Calc II." This assumes that he has a choice in what he'll be teaching – Bey Jan 16 '14 at 2:57
@Bey Sure, but the question is also about the many years to come, and choice usually increases somewhat. – Phira Jan 16 '14 at 9:25

I also teach Calc and Linear Algebra courses several times a year. I have been doing it for several years in a row.

This may sound shocking, but one thing that works for me is preparing my mind before every class as if I were a "Math jazz musician", i.e. I try not to "act" or "intepret" but, to some extent, improvise or "rebuild" the material, adapting the exposition to the audience mood. In my view, you cannot teach anything if you do not make a connection with your audience. See every class as a "Math jazz-session". And listen to more jazz...

This may help too.

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This is very helpful. I often aim for uniformity in my lectures, so as not to give one section any advantage over another. So it does feel like I'm playing the same piece of music, exactly as I played it many times before. This is a helpful paradigm. Thank you. – Jared Jan 16 '14 at 16:27

This is purely from my experience. If the material grows old, then hopefully you are more familiar with it, and so you can actually concentrate on teaching. What I do is ask a lot of questions to the students, and try to make them come up with some answers. Also, if you are TAing, that means that students often ask you heir "doubts". You should first ask them what approaches did they try. For example, suppose the question is: when is $sin(x)$ incresing? You can tell them- the first thing to try may be to look at $sin(x)-sin(y)$; can we use some trig identities? What about standard method of differentiating? What about drawing an actual graph, etc etc. Also, talk about related problems.

If you are teaching, try to motivate the students with Historical examples, and again, ask them a lot of questions. Even if the material is old- the students are all new, and so teaching classes in different semesters could very well be very different experience. The better you get at asking "leading questions", the better will be your teaching, and the joy that you get out of it.

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This. The more interaction there is between the students and you, the less uniform are the same classes in different years. – Roland Feb 4 '14 at 18:37

Francis Su's article on teaching is a good read: As people have noted, it's good to loosen up control in the classroom, leave loose threads for students to follow, and go on tangents based on student reactions during class. As Su writes:

I have often started off calculus lectures with 5-minute “math fun facts” that have nothing to do with calculus, just to get students excited about mathematics.

Additionally, think about how you might give open ended questions, to motivate students to create something that you did not expect, so that they are creating value for you, not just the other way around. Su writes:

I have often given fun exam questions: students can earn some easy points just by sharing the most interesting thing learned in the class, or a question they’d like to pursue further. Or “write a poem about a concept in this course.” Or “Imagine you are writing a column for the newspaper ‘great ideas in math’. What would you put in it?”

Of course, you can adapt these kinds of questions to be more rigorous, if you prefer. Su also encourages developing personal relationships with students.

If you find yourself repeating (for example, answering the same problem 10 times), that's a good time to put it in a format so that you wouldn't have to repeat it again--for example, putting it in the lecture notes, or if you're up to it, recording videos and "flipping the classroom."

Teaching is an art and a good way to reinforce this for me is to look up how other people have taught similar material, the explanations and media they use, and think about how I might "remix" them in my own teaching--for instance, I know the stuff on but I like looking at it from time to time, because the way he explains things is something I can learn from.

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I don't work in mathematics directly, but in engineering, whenever I'm reviewing something that has a strong theoretical component, I emphasize multiple interpretations of the theory. There's always more than one way to state a core theorem, and more than one way to prove it. Nothing stands on its own--a theorem or property justifies or implies many other things, leads to numerous corollaries. If the immediate topic is getting boring, try leaving a loose thread that leads somewhere more interesting, so that students don't feel like their imaginations are limited (and so that you remember that yours is not either).

Maybe it's not so helpful in calculus alone, but theorems and abstract formulations typically have a plethora of applications. Inner product spaces, for example, are the perfect framework for thousands of concepts in signal processing. I can go on for hours without repeating myself on this topic, and still come back to something that re-illustrates the basics.

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how do you maintain enthusiasm and joy in teaching the same material year after year?

The material is not exactly the same. You get some kind of feedback from students and try to improve the course. This, of course, works only if 1) you have the right to make changes in the course and 2) you see how the students try to use the knowledge in practice.

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I am not a teacher by any chance but I am a first year undergrad studying applied mathematics and the same question pops into my head. I say how can these professors not get tired and bored of what they are teaching year in and year out. I would say to change the example questions, problems,... etc. Try constructing lessons of your own without having to look at the textbook. Try teaching the students the new materiel in ways they will find interesting and fun. Try to give students a chance to be professor for a day and see how that pans out. As I am a student myself I believe it would be pretty fun. As I said I have no experience in teaching but I thought I'd have my say.

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I have to disagree with that. The professor and the topics might be the same, but the students are not. And semester after semester the new students will have the same doubts and questions than the previous batch. So there is no reason to change the material so that the prof does not get bored. The solution is simply to rotate people around several courses, so you don't get bored by teaching the same stuff all the time. Most departments I know do this. In my case, over the last 11 years at my current job I have taught 40+ classes; the one I repeated the most was 5 times. – Martin Argerami Jan 16 '14 at 4:16

I've TAed for 4 semesters of calculus, though one semester of calc I, another semester of business calc, and two semesters of calc III. Every semester I've used a different textbook which has made every semester different, including the calc III semesters.

I haven't gotten bored yet, but I'm actually looking forward to it. There are a couple games I hope to play with myself.

1. Memorize your notes. Try to learn your notes completely as quickly as you can. This includes memorizing what problems that you plan to cover. Once you've memorized your examples, start tweaking the problems on the fly.

2. Use really strange analogies. I know of a professor in undergrad who was notorious for using strange examples in his differential equations class, such as talking about a differential operator acting on $e^x$ being alike a monkey throwing a coconut at a cat. Yes, very strange, but everyone remembered him because of this. I don't think you should overdo it, since it can actually detract from understanding the material correctly, but if done right it can make things more entertaining for everyone, yourself included.

3. Restrict your speech. Try to avoid using a particular word, such as "obviously", "therefore", or "now"--whatever you might frequently say. Alternatively, choose a catch-phrase per semester to use in class, perhaps something as outlandish as "golly-gee-wilikers".

4. Tell jokes. Come up with a list of jokes, math or non math related, to tell at the beginning of class, at the end of class, or exactly halfway through.

5. Think of multiple ways to do a problem. Try to think of as many ways you can do a problem, no matter how trivially different. Try to guess which path your students will take--exams are always surprising, perhaps one can take this as a challenge so as not to be surprised. Try to think of incorrect ways, including horrendously incorrect ways. Think of the points a student might get stuck at. Do a problem multiple ways, or incorrectly at first and then do it correctly in class.

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