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First, I am not a professor, but I was a teaching assistent for a couple of courses. One time I took over a few sections for a friend who was also a TA. The course was 'math for chemists' (I think it was the second quarter or third quarter of the course).

I was explaining how to get a linear approximation for a differential equation and draw a phase plane from that (I hope I'm using the correct English terms here...). So you have: $$ \frac{dx}{dt} = f(x, y) $$ $$ \frac{dy}{dt} = g(x, y) $$

You seek for the fixed points where $f(x, y) = 0$ and $g(x, y) = 0$ at the same time, then take the linear approximation at that point. When I explained this (of course I did a more thorough explanation than these two sentences...) I got very little feedback from the students. I felt like the students were not sure how to handle the linear approximation so I pointed out that it is analogous to taking the linear term of a Taylor series, but in two variables instead of one. When I asked if the students understood this, I got no reaction at all. This made me very uncomfortable: it was clear that not everybody fully understood the topic, but at the same time I had no idea what to explain, for from my point of view, the explanation was very clear. I'm more used to teaching math in a 1-1 setting, in which people can't really ignore your questions, so I didn't know how to handle this.

I made the remark 'it's better to ask for clarification now than to discover you don't understand the material during the exam', and thought about asking a specific student (so it would be harder to ignore my question) but I didn't do this in the end (most of the students in that section seemed pretty shy and I didn't want to make them uncomfortable).

Are there any other (better?) tricks or strategies you have used or you would use to avoid these situations?

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Are all your examples thinks that chemists students will care about? Or are you trying to force maths on them that they don't yet know is of value? –  Ian May 6 at 11:34
    
No one is going to fully understand something after an explanation. Even if each piece is acknowledged in isolation, it takes time for a true understanding to take root. –  Tac-Tics May 8 at 20:10
    
To clarify: replacing another TA for two weeks, so I knew neither the students, the other TA (which was not the one I was replacing) or the math handled in the course very well. The course was 'mathematics for chemists', and we divided some topics between the three of us. This all made it very hard to judge if the students didn't understand and were too shy to ask, or did understand and were too shy to let me know (or they just didn't care). So Ian, no, and no. Tac-Tics: the explanation was meant to be a recap of something they should have seen before. –  Mon Kee Poo Sep 7 at 19:49

7 Answers 7

up vote 22 down vote accepted

It's perhaps worth noting that there are (at least) two distinct reasons why your students might offer no reply when you ask them if they understand what you've just explained:

  • They might not understand what you've told them, or even understand enough to know what it is that they don't understand. This typically happens when the students lack (or have failed to properly internalize) some prerequisite knowledge that you're assuming they have.

    For example, if your students have never seen a differential equation before, and have no idea what the notation even means, asking whether they've understood your explanation of phase plane analysis and Jacobians is, most likely, going to be met by blank stares. They may even assume that, since they don't know those things that you're clearly assuming they should know, the class is clearly not meant for them, and they should just keep quiet and not interrupt the other students' learning with stupid questions.

  • Conversely, the other possible reason for silence is that your students already know everything you've been telling them, e.g. because you're (intentionally or not) repeating something that was covered in an earlier course. They could answer your questions (at least with "yes, I know all this"), but they won't, because they assume that the questions are really meant for some other students who don't know the subject already, or that you're only asking to make sure you can safely move to the next topic (and that you'll do so if nobody speaks up).

In your case, I do suspect that the first explanation is more likely, but you should keep in mind that both are, in fact, possible. It's also quite possible for both reasons to occur simultaneously in the same class: some of your students could be silent because they have no idea what you're talking about, while some others might be silent because they already know all of it and don't want to interfere while you explain it to the rest.

This can also be highly cultural. Here in Finland, for example, I've heard several lecturers coming from abroad (especially from the U.S.) complain about getting "nothing but blank stares" from the class, because Finnish culture is traditionally a lot less outspoken than, say, in the U.S., and people coming from other cultures may not immediately recognize the subtle body language cues that can, sometimes, signal the difference between "I'm not asking questions because I want you to move on" and "I'm not asking questions because I have no idea what to ask".


OK, so what to do about it?

First of all, the situation you describe seems like a perfect opportunity to take a few minutes to explain that you really do rely on feedback from your students to know whether you're going way too slow or way too fast (or both), and that you really want them to speak up even if (or especially if) they have nothing to say but "I don't get any of this" or "I already know all of this". Try to create a friendly and relaxed atmosphere, let your students know that it's always OK to ask question or speak up, and, when they do, make sure to respond positively.

Second, if you're still getting no response by asking the entire class, don't be afraid to pick a random student and address your questions to them personally. As uncomfortable as it might, initially, feel to the student who ends up in the "hot seat", it at least ensures that they can't just "let someone else answer", and so gives you a single direct data point about that student's level of understanding.

Also, since you say that you feel more comfortable teaching in a one-to-one setting, you can always just pick one student out and work it out with that student until they get it. The odds are that, by doing this, you're also helping all the other students who may have the same or similar difficulties. Besides, by taking time during class to specifically engage one student in dialogue, you're showing the other students that they can also ask specific questions without worrying about "wasting time" or disrupting your lesson plan.

Jsor's suggestion of picking a "stooge" to sit in class and speak up when the other students won't is also excellent, and something I would've suggested myself. If you're a professor, it can be a TA who sits in class; if you're the TA yourself, it can be any student you trust to do the job. If necessary, ask a friend who's done the same class recently to join in and ask basic questions if your students won't. I've been that stooge before, and I can testify that it really does work.


Finally, and somewhat tangentially, it occurs to me that you might need to simply lower your expectations a bit. You're TA-ing a course on "math for chemists", presumably aimed at undergrad chemistry majors with no prior math background since high school. I don't know about your chemistry students, but back when I did my minor in chemistry, even using the quadratic formula to solve second-order equations was considered "advanced math" — i.e. something that the students would not be generally required or assumed to know — in first or second year chemistry courses.

Yes, courses like "math for chemists" exist precisely to provide some of those missing math skills needed for certain topics in chemistry, but the point is that you really can't assume in such a course that the students know anything past the minimum high school math curriculum — and they may have forgotten much of that. It probably doesn't include differential equations, and I'm pretty sure that it won't include Taylor series. You're going to have to start from the basics — equations, variables, constants, basic algebra — and work up from there, brutally distilling each concept down to the bare minimum necessary to work up to the more advanced topics you're really supposed to be teaching.

Or course, as a TA, that's primarily your professor's responsibility, but your job is to fill in the missing gaps for each student on an individual level. If you've got a good textbook (or a good set of lecture notes) for the course that presents all the preliminary material in a clear and concise form, I'd suggest reading it until you know most of it by heart.

If you don't have one, try to find one; even if it's not official material for the course, having a good introductory textbook aimed at students like yours to fall back on is worth its weight in gold. It means that, when a student asks you "what's a differential equation?", you can just paraphrase what the book says about them, instead of having to make up your own five-minute explanation of differential calculus on the spot.

(Of course, it should go without saying, but you should not follow any book slavishly. Rather, use the book as a "cheat sheet" for your own teaching, as and when needed.)

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1  
Good answer. I think I agree with everything. I feel like I have to add that I was just taking over one section, and I (and another TA) was supposed to give some explanations on some topics the students had already seen. So, I didn't know too much about the students (besides that it was not the first quarter of the course). Maybe they already had so much explanation about the topic I was explaining that they just faded out... I don't know :P –  Mon Kee Poo May 5 at 17:28

When I think of the what caused my students (or myself, back when I sat in lectures) to get to the point of an empty stare, it's that they were so confused that they didn't even know what to ask. They know they don't understand, but they lost the thread a while ago.

So what can you do? Two things:

Make sure students feel comfortable asking questions, so they will ask questions before getting so utterly confused they can only stare blankly and hope that the lecture will end soon. To make sure your students ask questions: acknowledge, by name, good questions--"Good question, Jane"--and be gentle when students ask questions that reveal their ignorance.

Make sure your students trust you enough to keeping working even when they're confused. Math is difficult, and some topics won't be understood hours, days, or even weeks after a lecture. Tell your students that confusion is often a step on the road from ignorance to knowledge. Encourage your students to work and play with the mathematics until it is understood.

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In fact, if you have a really good student or a TA who sits in on lectures, you can take them aside and ask them to be a stooge. Whenever I TA and it seems like the class is totally lost I start raising my hand and asking basic questions about where I thought the instructor may have been confusing or misleading, or a point that I remember being difficult when I took the course. –  Jsor May 5 at 6:33
2  
In such a case I often ask where they got lost (usually the first answer is "when you came into the room," but it moves on from there). Retake the thread (perhaps slower, perhaps taking a different route, perhaps do a few examples more). –  vonbrand May 6 at 1:44

It's hard to admit you don't understand if you think you're the only one in the class that's lost (rarely actually the case, but individuals get self-conscious).

I employed a system where if I wanted to check perceived understanding, I would do an "understanding check". Students would hold up 1-4 fingers:

1 - I'm so lost I can't even ask a good question.

2 - I'm confused, but I kinda get it a little.

3 - I pretty much get it, but there are some fuzzy bits.

4 - I understand it well enough to teach it to someone else.

A quick straw poll, then I could say something like, "Looks like I saw a lot of 2's--can anyone ask a specific question about what I didn't explain well?" which both removed the fear of being 'the dumb one' while shifting the blame back to me.

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Welcome to the site! I really like this answer and look forward to hearing more. –  Chris Cunningham May 6 at 14:40
    
I do this all the time. It works amazingly well! Good answer. –  MHH May 6 at 18:58
    
You can even have them close their eyes when holding up the fingers - that way they don't have to admit not knowing things to their classmates. –  James S. Jun 26 at 1:05

As I imagine your interaction with the students, each of you had the following mental picture of what is going on. This is obviously a big assumption about the nature of the class and of your own personality, but the gap I describe is definitely there regardless of the details.

  • You: A differential equation (which is synonymous with a system thereof) is a general gadget that can have multiple constituent scalar equations involving arbitrary anonymous functions. As functions (in this case differentiable ones) they have lots of properties and methods, such as Taylor series, at various points of their domain. All of these methods are available as tools for modifying, analyzing, or simplifying the gadget of a DE containing them, and there is a relatively conceptual heuristic (the one encapsulated by the precise concept of "linear approximation") that allows you to say how these alterations relate to the original.

  • Them: A differential equation is something like $x'' + 3x' + 2x = 0$. Worse, it could be $x'' + 3x' + 2x = \sin(t)$, and at the absolute outside, maybe something like $x' + t^2x = 1$, with a variable coefficient (but only in the first order). A system of differential equations is something like

    $\begin{align}x' &= x + y \\ y' &= 2x - 3y\end{align}$

    and it is meant to be solved by a process involving linear algebra, which is a lot of work and has very specific steps. If you were to change any of the numbers, the steps would be completely different, though the process is the same. The solutions to the two equations (original and changed) are pretty much unrelated.

So you see how they could become confused: your explanation has nothing to do with their experience. It doesn't mean anything to them to "expand a DE in Taylor series", they don't even think of functions $f(x,y)$ and $g(x,y)$ appearing in a nonlinear equation (maybe in their examples they've seen $x(1 - y)$), and the very concept of solving a different equation to analyze the first is fundamentally illogical.

In order to connect with them, you need to be gradual, gentle, and totally explicit. What they really want to know (and you know this because you threatened them with it) is how to do this kind of problem on the test; so, show them that. Do a really simple example with 100% detail and explain what each step is. Do another one. That will satisfy almost everyone.

It seems like you want to give them a deeper understanding: that's admirable but you can't make the horse drink. In fact, since you are not the professor, you can't even lead them to the water: as the TA you really can't change the tone of the lessons. If they weren't taught the phase plane method conceptually, then you aren't going to get them to care and they will be angry if you make their understanding of the "real" material dependent on an abstract concept.

Supposing it were your job to teach them the ideas you want them to learn, you would still have to set them up. This depends a lot on the class: for example, I last taught ODEs to an honors class. With them, I could assume some readiness to take me on faith until the payoff. But even then, it's not a good idea to build a huge conceptual structure before showing them the application. In my class, this was the application, so that's okay. But you are teaching a service class: math for scientists; definitely applied, probably purely practical. In this class, no piece of theory counts as an application, so you have to show them a real computation first. Then, maybe, you can begin the process of abstraction.

In short, the explanation you tried to give presupposes a pretty strong math background, a fair level of mathematical maturity and intuition, and is in itself worth about two days of lecturing when you are not the lecturer and probably tried to give it in only a few minutes. You got blank stares because you went over their heads.

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I like your answer but it's a bit too specific to the situation I described. I gave the situation just as an example (else I would have included more details), and my question was not about the situation. The explanation I give in the post is of course not the explanation I gave them (it is an extremely short recap for the more experienced people here). –  Mon Kee Poo May 6 at 0:42

Tutoring is complicated and hard! Here is how I conceptualize it, as a program.

function Tutor(Problem) {
  print('It is okay to not know how to do this problem.')
  result = askStudentToSolve(Problem)
  if (result == 'no idea'):
    newProblem = createSimplerVersion(Problem)
    Tutor(newProblem)       // Notice this can create a recursive loop of problems!
  else if (result == 'correct first step'):
    recognizeSuccess()
    guideStudentThrough(Problem)
  else if (result == 'success'):
    recognizeSuccess()
}

First, make sure that the students know it is okay to not understand, and it is okay to tell you this. Then, present the problem and ask them to get started. If they cannot even get started, create a fundamentally simpler version of the problem, either by reducing the complexity of the algebra, reducing the number of variables, or making the problem seemingly trivial. Creating a simpler version of a problem is, in my opinion, the most important skill for a tutor to develop.

Present them with the simpler problem -- on a blank sheet of paper with plenty of space for them to write. Repeat the process. (Make sure it is okay to say "I don't know;" if they don't know, then create a simpler version, etc).

In the end you may have created 4 problems -- the simplest one being something trivially easy, but they will solve that one. Then you will have them complete a simple version of the problem. Get out the next-easiest problem and see if they can do it now, with the benefit of the easier version being in their head. They will gradually work their way up to the main problem at hand.


Implementation Notes:

I'd say the createSimplerVersion() function is the hardest 
one to write, while many implementations of Tutor unfortunately 
only have an implementation of guideStudentThrough(), totally 
missing the recognizeSuccess() and the main if/else loop.
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2  
This is a good reflection for one-on-one tutoring, but doesn't really address the question of how to handle the group situation being asked about. –  PurpleVermont May 12 at 23:29

Was going to add this as a comment but...I think you have to create a culture of questioning right from the beginning. You might be dealing with student who come from places where asking questions is not expected.

Three things I say on the first day:

  1. "Think of me as a resource, like a book or the internet. I can't help you if you don't tell me what you don't understand."
  2. "Your most powerful weapon in this class is this:" (point to raised hand)
  3. "If there's one thing I know as a teacher, it's if you have a question it's more than likely someone else does too."

If you've said all of these things from the beginning instead of waiting until the class reaches the impasse then it's easier to say things like, "OK, I bet at least two people have a question about what I just said," or "Everyone who truly understood what I just said, raise your hand." Gives you the chance to receive more feedback.

As for putting yourself in the "beginner's mind," they don't know, and they don't know what they don't know. You can't go slow enough. If you have to repeat something again from the beginning then do it.

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I didn't see anyone mention this specifically so here's my input.

I've sometimes found it difficult to get students to answer certain questions in class. Either they're lost, tired, bored, etc. Also, when you pick a specific individual from the class you might not get an accurate representation of what the class knows on average. That student might know the answer perfectly, might know nothing, or might not fully explain themselves in hopes of sitting down faster.

I suggest taking a step back and asking a slightly easier but more specific question and having the students write their answer on their sheet of paper. I usually tell them to answer on the corner of the page of to circle their work that way I can quickly browse over a few students' work and get a feel for what the class knows. This seems to work well and I think the reason is because they don't have to answer in front of their classmates.

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