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I am a teaching assistant in mathematics at a university in Continental Europe. The course for which I am an assistant is a third year course, so usually the students are expected to know the basic notions from calculus and linear algebra. However, I often notice that students are not as familiar with the basic material as I would wish them to be$^1$. So, the natural question to ask is:

What should be my teaching strategy?

There are several possibilities. Each of them has its own pros and cons, which might contradict each other; but in fact, they don't, because you can regard it as a pro or a con, depending on your general attitude towards teaching.

1. Go very slowly and repeat everything.

Pros:

  • They can actually learn a lot from the exercise class.
  • Everything I do in the exercise class is self-contained; they find everything relevant to a certain topic concentrated on a few sheets of paper. Whence:
  • Learning the material is made easier for the students (at least in a certain way).

Cons:

  • People who already know the basic material get bored easily.
  • I have trouble finishing; I lose plenty of time repeating material that should be known.
  • When learning the material, they lose focus on what's important and what's not. This goes hand in hand with:
  • Due to me repeating everything, the density "content per time" seems to be very high for the students, although they should already know a lot of things I say.
  • A funny aspect which I encountered is the following: If I repeat basic material, students do not seem to understand the material, but if I skip the repetition and just use the results (e.g., I say "follows from linear algebra"), they think that they understand. But in fact, they don't, because they do not really understand what happens at a particular step in an exercise.

2. Just assume that they know the material.

Pros:

  • I can actually finish in time, so they get solutions for every exercise.
  • The students who know the material can follow and don't get bored.
  • The students should look up the material on their own; in fact, one goal of the university is to make people work independently and scientifically.$^2$

Cons:

  • They don't really understand what happens during a particular step of an exercise.
  • As a TA, I somehow feel bad when using this method, because I know that there are a lot of students who don't understand the material. Of course, I want them to learn as much as possible.
  • At the latest they should be able to apply the material in the exam. If they don't learn it fast, they never will, and they will most likely fail the exam. (See footnote 2: With this method, I don't encourage them as much as with method 1 to really learn the material.)

3. A mixture of 1. and 2.

Somehow combines a few pros and cons of each 1. and 2.

Which method do I apply?

Usually I go by method 1., because my aim is to make the students (all students! - even if this is utopian) fully understand the material. However, I am a bit sceptical because of the cons.

Related questions:

  • Should I even care about students as in footnote 1?

  • Assume that the students would be interested in the material but forgot a lot of basic notions. Does this change the answer to the main question?

  • Would it be a good idea to go by method 2. and give a sheet of paper with the used basic theorems to the students?

$^1$ This is actually an euphemism. Unfortunately, the students I talk about do not care about grades or even understanding the material. They just want to pass the exam.

$^2$ ... but this does not work very well in practice. The kind of students mentioned in the footnote above just don't care and blame the TA for giving bad exercise classes. They don't understand that at a university not everything is served up on a plate so that they only have to eat the meal - they also have to cook. But it is a matter of the TA's attitude if one should prepare the ingriedients for them (like in a cooking class) or just give them the full recipe.

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    $\begingroup$ This is exactly the classic conundrum of teaching, probably in all subjects, but arguably worse in mathematics (and maybe other "hard" sciences) due to the notion that students have nearly perfect recollection of all prior math classes they ever took. No easy answers, and often a teacher's strategy will (as noted) be wildly misunderstood, etc. And heterogeneous student populations... $\endgroup$ – paul garrett Dec 16 '16 at 21:55
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    $\begingroup$ "they think that they understand" - this is well-understood and not surprising, see youtu.be/eVtCO84MDj8 $\endgroup$ – kcrisman Dec 16 '16 at 21:58
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    $\begingroup$ Can you say what an "exercise class" is for those of us not in your context? Thanks! (It seems like it is not the same thing as just a quiz section for content they are seeing in a separate lecture.) $\endgroup$ – kcrisman Dec 16 '16 at 21:59
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    $\begingroup$ Is it at all possible for you to do a video lecture for each individual concept from previous courses that you want the students to be able to complete? If so, do that. Then on each assignment, put links (or QR codes, or something similar) to the videos that they will need to remember for that particular assignment. Students can then self-select themselves if they don't remember particular concepts while higher-level students can move forward at a quicker pace. This allows students who remember to have less work to do than ones who don't (and provides possible incentive to remember). $\endgroup$ – Joey Kramer Dec 17 '16 at 6:59
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    $\begingroup$ @JoeyKramer that sounds like a nice idea for a few reasons. A more practical approach may be to link to existing videos by other people. $\endgroup$ – Pat Devlin Dec 17 '16 at 7:50
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This is a good question, and I hope my answer is just the first among other (likely better!) ones. [I'll try not to rant, but I'm not making any promises.]

Let me start with a few general remarks.

  • This is a good question, and the fact that you are thinking about teaching in this way is (obviously) a very good sign. Keep reflecting, keep asking, and keep experimenting! (Try incorporating new ideas into your teaching approach, and see what happens. Reflect, discuss, experiment, reflect, discuss, ...).
  • In general, there's a dangerous trap in all questions like this. Namely, we need to avoid the temptation of thinking that there is a best answer. What works well is highly dependent on the individual students, the instructor, the material, the course format, et cetera. And even with all things being equal, there are still many different effective approaches. So going along my first point, take my answer (and any others) with a grain of salt, and be sure to develop a style that works best for you.

Now let me address some of your questions one at a time.

Should the course be too fast or too slow?

Given the choice between those, I (like you) prefer to err on the side of too slow; your job is for them to learn, not for you to prevent boredom. That said, naturally the best is somewhere "just right," and if you're close enough to that sweet spot, then it wouldn't matter whether you're a little too fast or too slow. [Opinions differ on this point, but we can all agree that the course should ideally be at just the "right" pace]

How do we know how fast is too fast?

I wondered this question for a while until one day I realized the answer. Although I may have some idea, I ultimately don't really know what's too fast---but the students do! To say it differently, you may be an expert in the material, but only the students can learn it.

In my opinion, the best thing you can do is (1) ask plenty of questions [in class, quizzes, take-home assignments] so that you [and ideally the students!] can discover what the students do and do not know, and (2) let the students dictate the pace.

As an example, you might raise a question for the class and have them work on it individually or discuss it in groups. You'll be able to see how quickly they work through things, and you can walk around and help out students individually [this lets you get a finer understanding of each student's background, and it also lets you have really personalized instruction]. If you see (perhaps to your surprise) that many students are struggling with some concept that they should have known, then you can address this as a class or give some extra homework on it.

How do we teach students who don't know the pre-requisites?

So this is tricky, and depends on many things. Let me give some scenarios to illustrate what I might do.

Scenario 1: Say you're teaching differential equations. I would assume the students are all proficient in the mechanics of first-semester calculus (otherwise I couldn't possibly teach everything), and on the first day I would probably say something to them along the lines of "To do well in this course, you'll need to know topics X, Y, and Z. Here is a quiz on these topics that you should already know. [they take quiz, then perhaps we discuss the answers] I don't really care if you know this material right now, but I will give you another quiz on this material at the end of the week. If you don't ace that quiz, you should change out of this class because I literally cannot imagine how you could possibly pass the course otherwise."

Scenario 2: Suppose you're teaching some class, and there's material from a few classes back that they maybe forgot [for example, trigonometric identities used to compute integrals like $\cos^2 (x)$]. In this situation, I would spend some time reminding the students of the facts we need, but I wouldn't address it nearly at the level that would be expected in their previous class.

What about individual students who are falling behind?

In office hours, if a student asks me to go over section 4, I usually start by asking them a question about section 3. I then go back in time until we can put our finger on what it is they don't know. Students almost never like "relearning" things that they "already know," but they need it. After you figure out what they need to work on, ask them plenty of questions and let them drive the pace entirely.

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    $\begingroup$ +1. Re: Scenario 1 (a diagnostic quiz, and advising to switch out of class if unacceptable), the experience at our institution is that the weakest students will almost never take that advice. (See: Dunning-Kruger.) Do you find that students actual act on that recommendation? $\endgroup$ – Daniel R. Collins Dec 17 '16 at 1:20
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    $\begingroup$ No, actually. A bunch don't. But there are two kinds of students who might do poorly, and this "scary advice" helps the students who erroneously think "I probably don't need to know that stuff." It also sets the tone for the course (I can always tone it back later, and the diagnostic doesn't have to count for their grade), and it lets me start at the level I want to end up at. [In my experience, most of the students in danger of failing are those that somehow decided they don't need to show up for any class sessions or do any homework.] $\endgroup$ – Pat Devlin Dec 17 '16 at 4:26
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    $\begingroup$ @Daniel I think at least the quiz makes the students feel responsible for choosing to stay in the course. The mind set of the "weakest students" is often quite admirable. What may seem an irrational choice makes some sense in their minds. If their previous teachers have not always taken them seriously, at least I can help them understand the challenge they are taking on and help them understand what they have to do to meet it. My point is that I'm for such quizzes when they clarify the situation of the student, even if the quiz does not eliminate all the students who end up struggling. $\endgroup$ – user1527 Dec 26 '16 at 17:01
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First: you should coordinate and discuss this with the teacher giving the lecture. You still have some freedom, but you should at least know what he or she expects.

Second: even if the precise students, level, and institution matter, I would broadly advise to mix 1. and 2. but with more emphasis on 1. in the earlier sessions, much more emphasis on 2. on the latter sessions, some pointers to learning material all along the way.

Third: make all you can to make very very clear that the prerequisite must be worked out, that it is absolutely ok (for you) if they need to work a lot for this, and that failing to master them as they should will give them an automatic fail. This is hard, especially if you feel uncomfortable with the "full metal jacket" model of teaching. If you have a say on the final exam, you can introduce a punitive part on the prerequisites and warn students abundantly. Maybe punitive is not the best word, but to explain what I mean, here is what I did once in a Freshman course (I was in charge of the lectures and the exam, not the exercises sessions).

The exam started with 6 (very simple) multiple choice questions on the prerequisites. They gave a small amount of points, but also a multiplicative coefficient to the grade, from 1.1 for 6 correct answer down to zero for 2 or less correct answers (no negative points for incorrect answers to avoid stress-motivated non-answers, but answering randomly would have given the coefficient 0 with large probability anyway). To limit cheating, I used AutoMultipleChoice to mix randomly the order of questions and answers (I had one pdf file of exam sheets consisting of more than 300 different versions - one for each student).

It turned out ok, but I didn't had the opportunity to run this on several years due to changes in my teaching. Also, one should make a dozen version only to be able to check them thoroughly: my pdf has been ill-printed and some part went missing in about on sheet out of ten, complicating things to an absurd point.

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  • $\begingroup$ I thought of doing this one time, but I ultimately decided against it. $\endgroup$ – Pat Devlin Jan 3 '17 at 13:57
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As a student, I think I can give an answer that could possibly be useful and complementary to the other ones. It is mainly based on my personal experience, so it could be irrelevant. If people think so, don't hesitate to downvote.

As I understand your question, you are more concerned about prerequisites not being well known by the students rather than them not knowing the material taught in the lectures your exercise sessions are intended to illustrate, if so.

Firstly, I believe it depends a lot on what is the philosophy in your institution. In mine, exercise sessions are not mandatory and they are specifically addressed to students who are motivated to understand the material taught during the lectures. I personally used to not go to the lectures neither the exercise sessions and this was probably an awful decision at the time, but it's not the point. The point is that when I decided to go to these sessions, I did not expect the teaching assistant to remind me the properties I didn't remember because of lack of work as I was perfectly aware that I should have known them. Instead, the fact that the teaching assistant did rely on the assumption that the students should know the prerequisites, acted like a wake-up call, sending the important message that the students should work if they want to keep up. Of course, if on the contrary your institution makes the exercise sessions mandatory, this changes the all perspective, but in that case, I agree with the fact that reminding prerequisites may still lead the students think they know while they don't.

Secondly, as you say, repeating prerequisites slows the session down and may bother the students who know them and may give a false impression of understanding that will surely backfire sooner or later.

However, despite being convinced that you should not repeat what should be known, I think you could say things like "I use theorem X you learned last year. You should check if the hypotheses are respected". Another solution used by some of my professors is to devote the first exercise session of the year/semester to quick exercises on concepts and results considered as important (but reasonably complex) prerequisites, as it is useful for all students in general (even the best ones, but it is only based on my (narrow) observations and they could naturally be biased) and addresses the problem in an intellectually interesting way, without slowing down the more important sessions to come.

As a final note, I think that the exercise session covering a new material X should always be preceded by some short reminder of the main new (or recent) definitions and results useful for the material X. For example, if your exercise session is about the introduction of Lebesgue integration, I think it is useful to quickly remind the students how the integral is constructed and what are the main tools (e.g. Lebesgue dominated convergence theorem,...) even if they have seen these in the lectures the exercise session is intended to cover. It may seem obvious but not all the teaching assistants I have had began by doing so and it can be very unproductive, in my humble opinion.

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