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We are teaching a real analysis course each week for two hours one day and one hour another day. We use a blackboard and we upload most of the class notes to encourage students to participate instead of furiously taking notes.

We are trying to figure out a good way to gauge how much time a given proof or example will take to present in real class time (including time for student questions).

  1. of course, one can just spend an additional three hours teaching the material to an empty classroom, but we are trying to find more time efficient ways.
  2. keep a list of how much proving some theorems took in actual class time. Then use that list to help decide how much time to devote for the current proof, if it is similar to any of the listed ones.
  3. Split the lecture into time blocks and add as much material as possible to each of these blocks. For example, we can split a two hour class into 30 minute blocks and then devote each block to some theme/topic/theorem. When we hit the 30 minute mark, we must stop and move to the next topic. The price of this is that you might have rush the proof, which is bad, so we might have to slip into the next block.
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    $\begingroup$ What kind of textbook do you have/how do you prepare lectures? Usually, you shouldn't need to think too much during a lecture, as the whole proof and how you want to present it was previously prepared in your notes. Then you only need to ask yourself "how long does it take to write X pages of paper on the board?" Add to that some minutes for questions and you should have a good time estimate. If that doesn't work, maybe edit the question to say why not (no properly prepared notes, too many unexpected questions,...). $\endgroup$ – Dirk Jul 2 at 8:00
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    $\begingroup$ You write "we" but do you actually mean "I" or are you teaching this course together with others? $\endgroup$ – J W Jul 2 at 11:45
  • $\begingroup$ @Dirk Thanks. Well, we are trying to figure out what are better ways to prepare for lecture. We use an analysis textbook but we also type our own notes. It is not as simple as transferring pages to blackboard. Some parts of the proof are new and so they require extra verbal explanation and figure-drawing, even though they don't take much space. We are trying to quantify how much time that can take. $\endgroup$ – Thomas Kojar Jul 2 at 19:00
  • $\begingroup$ Take several topics you already taught that are taught by others. Compare times, calculate the coefficient. As you go on, refine the coefficient. $\endgroup$ – Rusty Core Jul 2 at 19:20
  • $\begingroup$ @J W We are a team of instructors each with their own session. $\endgroup$ – Thomas Kojar Jul 2 at 19:31
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I also tried to estimate this for my lectures in advance for some years but couldn't really figure it out. However, after a several years of teaching math I got pretty good at estimating. Here is what I got:

  • If I prepare lectures by hand on paper, I do roughly one page per 20 minutes for introductory courses (first year students) and up to one page per 15 minutes for advanced students. (I figured this out by putting time markers in my handouts during lectures.)

  • If I prepare TeXed handouts, I take a little more minutes per page, but not much.

  • This needs adjustment: proof-heavy pages take a little longer, matrix-heavy pages take much less time.

  • I always plan conservative, i.e. a little less material than I would expect to cover at first guess.

  • If I am very unsure, I plan even less material but bring some additional examples to fill up the lecture.

  • It is possible to explain the same thing/proof/example in different time without actually just speeding up or slowing down, but by taking a different route. This is quite difficult to do on the fly at the blackboard, but it is a skill that can be practiced (with good planning).

A final note: It does not really matter, how much material you can cover, but how much the students learn. So do not only track how long it takes to cover the material, but do an effort to try to figure out if the students are still up to date with the stuff you are teaching.

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As already suggested in the comment:

Type out everything in the notes. If you want to draw a figure, have it in the notes and know how long it takes to draw. If you want to explain something, have it in the notes and write it down.

Also one thing that I myself also only learned after multiple lectures: Try to not explain verbally if possible, write everything down. Yes, a verbal explanation might seem nicer at first, but a considerable number of students is writing down everything and only later starts to process and understand it. If you do verbal explanations that are neither on the board nor in the textbook, that might be nice for the good students that already fully understood the topic and listen carefully. But the average or below average students, that are still struggling, will have a lot of trouble reproducing it later at home if part of the explanation is missing.

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  • $\begingroup$ Thanks. We post the lecture notes ahead of time, so in class we are trying to create a dialogue. But yes for lectures who don't, the verbal explanation will be a nightmare for the students as they are trying to take notes and listen. Someone told me there was a study that showed that students were simply not processing both listening and note-taking. $\endgroup$ – Thomas Kojar Jul 3 at 14:55

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