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I am an instructor in a STEM field. I am teaching an upper-division course where I try to interact with the students rather than just lecturing. I am teaching in the students' native language; there do not seem to be any language issues.

I am struggling because the students tend to give wrong answers to simple questions I ask during lectures. These really are simple questions; they should have learned this material during the first few weeks of their first year.

To make matters clear, let me give an example. If we are in the middle of a proof, I might ask "what is the result of log (a * b)?". The right answer is "log(a) + log(b)", however, they will say "log(a) * log (b)" as an answer. Then, I did not see any other way and would say "no guys, it is log(a) + log(b)."

Situations like this repeated over the entire semester. Students complained to my boss and in the student evaluation that I was demeaning them and that I was upset when they answered something other than the answer I wanted to.

I will teach some of these students next year in another course and I have a hard time trying to find a way to solve this issue. The only solution I see for this is to just lecture and not encourage participation in class. However, I was wondering if there would be any other wiser solution.

Tips from here are good, but instead of a colleague, I'm dealing with students.

Edit: This is actually an upper-division chemistry course. The situation above arises, for example, when I have to explain why pH + pOH = 14. Students should have learned that in general chemistry, but I like to derive it to remind them. When I perform the derivation, I start from the auto-ionization of water and eventually arrive at: 14 = -log ([H3O+][OH-]). Then I ask them how to simplify this in order to complete the proof. But they do not remember the properties of logs.

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    A bunch of information from the comments has been edited into the main post. There was also another example ("what shape is this? It's blue. No, it's a square.") -- I removed this from the post as it was causing some confusion, but some answers may still reference this. Also: comments have been moved to chat. Please continue the discussion in chat, but use comments below this one only to suggest improvements or request clarifications to the main post. We can only move comments to chat once.
    – cag51
    May 16 at 0:37
  • How common are these incorrect answers (most answers?, occasional?) and are they consistent year-to-year (e.g. log rules are usually forgotten)?
    – Esme_
    May 17 at 6:06
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    "I am teaching in the students' native language; there do not seem to be any language issues." - Is it also your native language?
    – David K
    May 17 at 12:10
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    It might help to tell us more about the level of the class. "Upper-division" needs translation for readers in other countries. May 17 at 13:47

18 Answers 18

88

Like others here have indicated, you can reframe the questions to avoid this, but there's another aspect to consider.

It can be really hard to foster an environment where people aren't afraid to speak up at the risk of being wrong. It's embarrassing enough to put yourself out there in front of a couple dozen of your peers and friends and be wrong, but if your professor responds negatively, whether it's in body language, vocal tone, and/or word choice, that only makes it worse. On top of that, the expectations that a professor sets up in how in the question is presented can also contribute to this.

Let's take your example:

A real example: If I ask "what is the result of log (a * b)?". Even though I'm expecting "log(a) + log(b)" because we are in the middle of solving a problem and need to use the properties of logs, I'd be prepared if they say "multiply a and b and take the log", however, I will get "log(a) * log (b)" as an answer. I don't see how I can help them with their self-confidence and try to make their wrong answer seems "ok". The only thing that I would say is "no, that's wrong".

And try approaching it differently:

Prof: Does anybody remember the math identity log(a * b)? Take a few seconds to think about how we would restructure this.

This does two things: it reminds them that it's something they've probably seen before, but it is also indicates to them that it's something you don't expect them to immediately be able to recall (if at all).

Prof: Does anybody want to take a guess on how to rewrite it?

Using the word 'guess' helps to lower the stakes and indicate to the students that it's okay to be wrong.

Student 1: log(a) * log(b)

Prof: That's a reasonable guess, but a very common mistake that comes up every year

Student 2: Peanut butter

Prof: Not quite the direction I was looking for; I might not have been as clear as I thought. What I meant was... (clarifications/rephrasing)

Instead of just saying they are wrong, reflect on their answer and why they might have gotten it. Use word choices that acknowledge their mistakes without making them feel like a fool. Even though a blunt "that's incorrect" might technically be a correct word choice, there's a good chance that there's more that could be said about their answer, which will help you come off as more empathetic.

But word choice is only a single part of this. There's body language and vocal tone. If you come off as disappointed or defeated or exasperated, it will negatively impact your word choice and how your students respond to it. It can really help to just maintain positivity and excitement through the whole process, and that will help to encourage students to feel better about taking risks and making mistakes.

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    Why would it be ok to lower the stakes on the retention of basics?
    – bukwyrm
    May 16 at 14:09
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    @bukwyrm because in the end a positive and encouraging environment facilitates learning better. Scolding will not make people who don't understand the basics any smarter
    – Some Guy
    May 16 at 14:40
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    Second: People are more likely to remember things the more they actually use them and the more recently they've used them. There are a lot of basics I know because I apply them on a regular basis, but there are a ton that I don't remember off the top of my head because I haven't used them in a long time or I just use them very infrequently and have to refresh myself when I need them. And just because a professor/researcher uses something day-to-day, it doesn't mean a student taking classes does; those are two different situations and you can't have the same expectations for both.
    – anjama
    May 16 at 15:16
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    @Zanzag Sometimes students get assigned to us and don't have all the proficiencies we expect. But since they're there, it's our obligation to try and teach them. Creating an environment where they're willing to try at least (and be wrong) is beneficial because it means that they (the ones most likely to struggle) are more likely to engage with the classroom, rather than just sit and stare off into space daydreaming. Not only are engaged students more likely to learn in class, but they are going to be more likely to seek help outside of class (e.g., office hours) if they need it.
    – anjama
    May 16 at 16:55
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    @Zanzag This is a philosophy of (modern) teaching matter. You seem to be thinking about this in a practical "would we do this in a real world job?" type of way. In that case, "guessing" and gently allowing errors can be a real problem. You certainly don't want your doctors and nurses doing that. But this is a school room, not an operating room. They are there to acquire new skills and challenge their abilities in a low-consequences environment. Much of the modern (European/US) school system has shifted away from a "trade school" mentality, and does not emphasize "how we do this at work". May 16 at 20:38
35

You are asking them to answer a question on the spot. They get flustered, say something stupid, and you don't respond in a very sympathetic way. Then they are upset.

Yes, you are right, log(a*b) is just not log(a)*log(b). But it is an understandable way to mess up, particularly with the pressure of answering a question in front of the class, particularly with the pressure of being asked by a professor who has a reputation for shaming people (that professor being you, apparently).

So you need a way to ask the basics, and get them to realise their mistakes, without feeling shame over it. I've seen people achieve that in two ways;

  1. Whenever you mistake yourself, highlight it, laugh at yourself, then move on. I was taught by a prof who, after making a sign error, came out with the wonderful line; "maths; it's like sex, good fun, but quite difficult, and you probably shouldn't do it in public", we all found that funny. And when we made a mistake, he would say "ah, no, that's exactly to sort of mistake I'd make too, but actually the answer is ....". You are making it clear that mistakes are normal, and so people don't take it so hard when they get things wrong publicly.

  2. If they are making a lot of mistakes, you might need a different approach. Firstly, lets get an anonymous answer from everyone, with a digital voting system. There are lots available, some of them free. You ask the question, then all the students answer.

    A. If 95% of the students got the right answer, you can just confirm it and move on.

    B. If about 50% of the students got the right answer, ask them to discuss their answer with their neighbour. Don't say which answer was correct, just tell them to try an convince their neighbour of their answer. This is normally very effective, particularly for basic topics, the ones who got it right are quickly able to convince the friend next to them who got it wrong. Or when I try to justify my wrong answer to the person beside me, I'm very likely to notice that it was incorrect. Crucially, nobody is ashamed, as they are just chatting with the student sitting next to them, so they are open about their confusion. Ask them to vote again after 3 mins of discussion, and you can check it worked.

    C. If more like 25% of the student got it right, you need a quick review. Go thought it on the board, and then run another poll. Hopefully, you are able to explain well enough that you at least get the majority of the students understanding.

People will react with anger when they are ashamed, that's almost hard-coded into human psyche. People protect their ego more fiercely than just about anything else. So you are more likely to get a constructive reaction if you are careful not to make anyone lose face. But these two suggestions are ways to mitigate the shame associated with being wrong.

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    The prof who taught you, and all of us, should keep talk about sex out of the classroom. All that does is make women (who are already too often alienated from STEM communities) feel even more uncomfortable. May 16 at 1:59
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    @GregMartin I identify as a feminist, and most of my female friends seem to have a healthy interest in sex. Banging on about it all the time could be unpleasant (pun unintended), but do you have any evidence that the occasional jest is going to be make women feel uncomfortable? May 16 at 13:19
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    @GregMartin - very sexist to assume women would be uncomfortable with a remark like that!!
    – deep64blue
    May 16 at 15:19
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    @deep64blue Ok, I didn't really want to weigh in on this. In part because it was a real thing a prof said, and it was well judged for the time and place. But I'm not insensitive to the idea that sex related humour is socially complicated. It's not unreasonable of Greg to point out that this could be a problem, and if the joke had been told after a student's mistake I agree it would have been in poor taste, because then the prof is directing a sex related comment at a student.
    – Clumsy cat
    May 16 at 15:32
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    I had a math professor who would say in joking tone "ah, you are careless" when students made mistakes. Which you'd think would make people feel bad, but just by his tone of voice and manner made it clear enough that he was joking. And crucially, when he made a rare mistake, he'd say "I am careless!", with a similar same joking tone. He was well-respected by our class as a good instructor, and great at freehand diagrams on the blackboard. (He was Korean, and the style of respected and competent authority figure worked well for him, like a martial arts sensei, but not unapproachable.) May 17 at 1:59
34

Ask yes or no questions. Then ask why.

Instead of asking what shape a drawing is, ask the students if it’s a square. Then ask why it’s a square.

This lets students know what the right answer is. If they have trouble getting to “all sides are the same length and at 90 degree angles” then you know a review session is in order.

As a professor it’s difficult to understand what an easy undergrad question is. You’ve had 5+ years of extra education on top of likely being a good student.

Using yes/no questions at first limits the answer, and the why follow up ensures it wasn’t a lucky guess. If people take more than a minute to completely answer why time for a review session.

EDIT

A commenter pointed out that my original answer “all sides are the same length” isn’t enough. They sides also need to be at 90 degree angles.

Answering on the spot is hard. As I’ve shown it’s easy to leave out information that you know. It’s even harder when you’re being ask by an authority figure like a professor.

If a student had said that - follow with something like “that’s part of it, you’re almost there. What other thing has to be true about the lines.”

EDIT 2

And even that isn’t technically enough to correctly answer the why. Leaving the question as is to show just how difficult answering can be.

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    Especially the "ask why" is very important! I suspect you ask questions with the purpose of helping the students learn. Questions are a way for you to figure out the student's thought process. Without asking "How did you arrive at that answer?" you miss this opportunity and asking it was worthless.
    – Dirk
    May 15 at 9:05
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    @Stef you are correct - edited to include 90 degree angles. Also probably a good example of how even easy questions can be hard to answer on the spot like I tried to. May 15 at 19:45
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    Note this is also not the only way to characterise squares. "Diagonals are the same length and intersect in their middle at a right angle" works too. Also, combinations of your characterisation and mine can work too: "all sides are the same length and diagonals are the same length and intersect in their middle"; "diagonals intersect in their middle at a right angle and the sides make right angles"; etc
    – Stef
    May 15 at 19:47
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One key point to consider here that hasn't been brought up is that your job in the class room, as a professor is to teach the student who are actually in front of you, rather than the student you would like to be in front of you, or think should be in front of you.

Its no good saying that "they should have learned this material during the first few weeks of their first year." - this particular student clearly doesn't know this right at this moment, wheather they "should" or not. What you do with that information depends on whether they are alone in this deficit, or it is indicative of the class as a whole. If a substaintial fraction of the class doesn't remember that log(a*b) = log a + log b, then you need to teach them.

This circles back to asking why we ask questions in class. Asking questions doesn't, in it self, make teaching better. There are generally two reasons to ask a question:

  1. Guide the students to reflect or think about a particular point, that will hopefully lead them to coming to an important incite about the topic themselves. This is reffered to i pedagogy as "co-construction", as you are helping the students to construct the knoewlege for themselves.
  2. For a rapid, real time assessment of a students knowledge. This is only useful if you are going to do something with the result - tailor the class to how well the students are coping with the material. Asking about prerequisites can be useful, but only if you are prepared to divert the class to explain a prerequisite that appears to be missing in a significant section of the audience.
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There are several ways you could nudge your students to the correct domain, for example:

  • a : b <=> c : ? question formats, which they may be familiar with from some types of testing. This gives them two hints - the relationship between a and c, and the relationship between a and b.
    E.g. ▲ : triangle <=> ■ : ?
  • minimal pairs, examples where the only difference between two items is the one you are interested in. E.g. ▮ / ■
  • Chains where you start from a concept you know they grasp, perhaps because you've just refreshed it, and then move on by very small increments to the concept you want. You can combine this tip with the above - present the chain, then ask your students to identify what feature is introduced at each step.
    E.g. plane figure -> polygon -> quadrilateral -> parallelogram -> rectangle -> square

Other replies have comments have mentioned other useful techniques, like multiple choice, yes/no, etc.

However, I'm more worried about the fact that you don't really seem to consider the possibility that your questions may be genuinely hard to answer. I remember many teachers who would go through this process: think of a concept A that they want to assess -> think of a plausible question Q that A is a correct answer to -> get flustered when students answered with one of the many, many other possible answers to Q and be unable to give more guidance as to what it is they wanted, other than "try again" or "that's not right". These exams turned into a game of "read the teacher's mind" and were stressful and distracting from the subject.

Your example is obviously hyperbolic, so it's hard to assess this possibility. I'll point out, however, that you use "it is obviously a rectangle" as an example of an incorrect answer to "What is the shape of this ■ figure". Surely that figure is a rectangle? And a simple polygon and a trapezoid and a number of other correct things that are not what you had in mind when you asked the question. So I really suspect that this is what is going on. If you are willing to share a real example, people here will be able to give you pointers to making your questions less confusing and frustrating for you and your students.

Edit

The OP’s comments and edits make this answer completely obsolete - there isn’t, in fact, a domain/category problem at all, just “normal” wrong answers that need some teaching skill to be turned into a learning opportunity instead of a put down. anjama, Clumsy cat and Jessica have covered what I would say about this already.

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    These exams turned into a game of "read the teacher's mind" and were stressful and distracting from the subject. Quite likely that this is what OP perceives as students being "sensitive".
    – henning
    May 15 at 15:45
  • I really like the first point of your answer. I play a game with my students when I see them struggling... "Let's play the game of Complete this!" or I start with the sound of the first syllable of the word of the correct answer, and possibly tell them how many words the answer has... This may sound silly, but the idea is always to give them clues/guidance, even tough the answer should already be known by them. I would also add that we need patience, lots of that, and remember that we do not know the background of our students even though it may seem easy to guess. May 15 at 18:57
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    "▲ : triangle = ■ : ?" I would be soooooo confused by this. "triangle = ■"? What does that even mean? And then it ends with a question mark, so it's a question? The answer to that question is "no", I guess, since a triangle is usually not equal to ■? (I still upvoted your answer, because of the paragraph about reading the teacher's mind).
    – Stef
    May 15 at 19:00
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    @Stef, yes if you've never seen this notation before it is not particularly intuitive... It comes from fractions but in its loose usage it is read "▲ is to triangle as ■ is to what?" and it means "the relationship between ▲ and 'triangle' is the same as the relationship between ■ and the solution". For example: "foot" is to "shoe" as "hand" is to ? (solution: glove). This notation is commonly used in some types of standardised testing, so students may be familiar with it.
    – Ottie
    May 15 at 19:58
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    @Kimball: I agree that "=" is confusingly nonstandard, but I wish you had explicitly stated a standard notation for analogies, such as "colon notation" a : b :: c : d ( en.wikipedia.org/wiki/Analogy ).
    – David Cary
    May 16 at 20:24
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The question conflates a number of different things.

  • "How to deal with category mistakes and other ontological problems when teaching a concept?" (the example about failing to recognize a square as such)
  • "How to spare a student's feelings when telling them they are wrong?" (the point about being sensitive)
  • "What to do about students who falsely believe they are right?" (the point about insisting the square is peanut butter)
  • "How to help students make fewer mistakes when applying techniques they learned?" (the realistic example of incorrect log transformation)

You have of course added information to try and make the question more clear, but I think it's nevertheless confused things a bit, and it's better to answer these separately. Taken on its own, each sub-question has a simple and direct answer, and when individually understood, these answers make the overall issue plainer to see.

Mistakes in applying techniques

The time-tested solution is practice, practice, practice. If students make mistakes with algebraic operations that they already "know", first thing you want to make sure is that they're getting adequate practice via homework, problem sessions and in-class example problems/solutions. When solving non-trivial problems, one must not only apply techniques correctly, but also be able to select which techniques are appropriate. You can supplement this with trivial problems, such as asking them to rewrite "a=logx" as "x=e^a" and vice versa, or evaluate with certain numeric values of x or a.

It is also useful to teach fall-back techniques when their knowledge of the primary skill fails. For example, even if they don't remember whether the distributive property applies to logs, they can try to prove/disprove it in some quick way (such as: "log2(16*4)=log2(64)=6" clearly does not equal "log2(16) * log2(4) = 4*2 = 8"). Another option when failing to remember a rule is to try and remember its derivation instead.

Sparing a student's feelings

There's a lot to be said here but briefly, you want to create an environment where students feel safe in making mistakes and asking questions, while also having confidence in their ability to learn. You should set up the lesson plan so that everyone is always learning something, even if not everything, in that lecture. When explaining concepts and answering questions, try to identify what part of the explanation seems like a "leap" to the student, and break it down into simpler steps. Recall earlier lectures where students were not able to solve a problem which they now can, and point out that what seems intractable now will soon become soluble with some effort and practice.

Keep in mind also that not everyone will be a prodigy in every subject. You want to present the subject as a ladder of knowledge and techniques, where student will see that increasing commitment of effort will yield increasing mastery, and yet there are evenly spaced "exit points" where they can stop investing into the subject and still have some partial mastery to take away from it. This also helps them recognize the proverbial steps by which mountains are climbed.

Dealing with stubborn students

Assuming the student is acting in good faith, the problem here is typically your failure to establish rapport and authority in the class. Ideally, you want to establish and maintain as clear a picture as possible in the students' minds of what they are attempting to learn in a class and what the utility of this is. Even if you are not able to justify to them what good logarithms are in every day life, you can at least emphasize that they are a prerequisite of many other interesting topics. You can also look for examples of real-world problems where they apply (exponential population growth is often a good one).

When the a student claims that you are wrong, you should be prepared to justify your claim with various proofs and examples, as well as refuting the student's claim persuasively. This of course requires a pre-agreed upon standard of truth: Either the student body must implicitly believe certain criteria for accepting an argument as correct, or you must establish them from the first day of class.

Note that students claiming you are wrong are an excellent opportunity to (a) figure out what parts of material you failed to explain adequately and (b) teach students techniques for independently verifying their own work and catching their own mistakes.

Category and other ontological mistakes

What I mean by this is things like confusing a square with a rectangle.

This is a broad pedagogical topic but ultimately you want to avoid relying too much on the Socratic method if your rapport with the students is not excellent. Do not draw a square, ask what it is, and then get frustrated when they say "a drawing". Tell them the rule right away:

In this class, we are mainly interested in whether things are squares, triangles and circles. If they have four equal sides we will call them squares. There will not be trick questions about equilateral parallelograms or non-Euclidean planes so don't worry about that.

As the class goes on and students learn your style, you can ramp up the Socratic element, but you should always start out didactically.

You want to pick a closely related set of learning objectives, ideally in the same or similar ontological categories, and focus on those. Leave the philosophy to philosophy class. When teaching algebraic techniques, don't distract them with difficult ontological dilemmas. It's okay (and arguably useful) to point out ontological asides, but don't quiz them on these.

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    +1 for the "sensitivity/feeling-sparing" part, which I think most of the other answers have missed. You want to set up an environment where students feel safe being wrong. While it's easier/better to be right, being wrong out in the open is a way for students to learn. (I think another part of the conflation in the original question is "students don't know what they should/are underprepared", which is yet another can of worms ...)
    – Ben Bolker
    May 15 at 22:20
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    Very good answer. Separating out the issues really helps!
    – Vincent
    May 16 at 18:51
  • The last section introduces a really good point about teaching style: it needs to change based on the student's current level of knowledge in the particular area being taught. I've found From Novice to Expert, Patricia Benner's explication of the Dreyfus model of learning, to be very useful for this.
    – cjs
    May 18 at 5:50
  • Obviously I can't get into it much in a comment or two, but the flavour of it is: 1) Novice: tell them to do X, and exactly how they should do it. 2) Advanced beginner: give them a (simple) goal and let them determine how to reach it. 3) Competent: give them rules to determine the goal. 4) Proficient: give them (sometimes conflicting) guidelines from which they need to choose.
    – cjs
    May 18 at 5:50
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If you're asking them about something that they were supposed to learn a long time ago, it's possible that they forgot, or maybe they never learned it when they were supposed to. One thing you could say (if you have enough time) is:

"Before I get to today's topic, I want to review [topic from previous class], because it's essential for understanding today's topic. It's usually taught in [name of class], but I understand that you all have different backgrounds. Some of you may have learned it a long time ago and forgot, and I want to make sure we're all on the same page. You can also read up about it on your own by consulting [references]. [Insert review]"

Edit: Now that I saw your example, it's possible that your students didn't pay much attention to the properties of logarithms. One kind of sneaky way to get more correct responses is to teach something that uses that logarithm property at the beginning of class. Then when you get to the derivation and ask what happens, you're more likely to get correct responses.

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    +1. Currently as a computer science student I can say that there a lot of basics in math that are never covered but always assumed as already taught. I assume the same happens in other fields, for other topics, as well.
    – lucidbrot
    May 15 at 13:38
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    @lucidbrot: In undergrad physics, the physics classes actually know they'll be ahead of the math classes, and need to teach the practical how-to-use-it aspects of the math. This was a known thing that I think even got mentioned to the class by the professors in a couple of my 2nd year undergrad physics classes. (Math takes its time and cares about proving things. Physics needs to use that math to talk about actual physics of oscillators, surface integrals for fields such as electric and magnetic fields, and so on.) So that's the other side of the coin; common in many fields but not all. May 17 at 2:15
8

Are you clear about your aim when asking students such questions? I think that asking such questions when being more or less sure that students will give the correct answer is fake interaction, and students will realise this. What's the value of showing a square and students then saying "that's a square"?

Now your students tell you something else. I'd actually like that, because I like that students give me something that I didn't expect. It is information for you that students tell you something else than what you expect; but of course you have to make some sense of that information. The obvious (but not necessarily the only) reason would be that the basic material was not learnt as well as you hoped for, which of course you can take into account in further teaching (maybe there's more repetition required, maybe it is worthwhile to make more references to earlier material in the sense that some later learned things also agree with earlier concepts or provide examples/special cases for them and the like). You may also question your way of teaching the basic material, or you may want to find out (just to given an example for another possible issue) whether the life situation of many students prevents them from learning enough, or learning well enough.

A more positive spin on your communication with the students can be given by thanking them for contributing something to the class, and thanking them for showing to you how these concepts can also be perceived, even if wrong, because it means you've got to do something about their understanding.

Personally I tend to avoid questions of this kind (I may occasionally use them, see below) because I know that it is not nice to be told in front of an audience that you got something wrong. It may produce a bad feeling, at least if this happens in a bad way.

I rather like to ask questions that are meant to give me information about how the students think, generally trying to value what they contribute in other ways than just stating whether it's correct or wrong. I prefer to ask questions about which more than one opinion is possible, or questions that even though they seem simple are connected to some hidden difficulties. Sometimes I want students to get things wrong in order to prepare a next result that shows a concept from a different angle and will likely change their view on what was taught earlier. Sometimes I ask rather hard right/wrong questions and if there are different opinions, I ask the students to try to convince each other. I then also thank all involved for their contributions; those who were on the wrong side of the argument had the effect that more explanation was given which was apparently necessary, so they would have a positive impact on the teaching.

I know that some students are happy to show off their knowledge by answering a simple right/wrong question correctly, or even to test it by guessing an answer that may be wrong. Such questions can serve to make sure everyone operates on the same basis (or rather to push they group at least a bit in that direction), and may serve to bring attention to your teaching again when some students may have lost it. So I'm not saying they shouldn't be used, but my attitude when asking questions is always that I won't expect one particular answer (as then I'd be bored when getting it), but rather that this is, as much communication, a vehicle to find out something also for me, rather than another way to assess the students (for which there are exams).

6

In the example you've cited, it would be more educational to avoid saying "No - wrong answer. It's actually loga + logb."

All students - old as well as young - detest trip-up teaching. Especially in a class situation where the superficial view of the situation by other students is usually humorous.

Far better to say: "Why do you say that, now ?" in an even tone.

You could then take an example of log(to base 10) of 100 * 100 where the student's answer just happens to be right.

Then take log(to base 10) of 1000 * 1000 where the student's answer is incorrect. But don't say he/she is incorrect after you discover it, say "We are wrong here".

Then go into deducing the correct expression from particular situations, not algebraically. After you've got a seemingly right-for-all-cases expression, do a once-over with the formal derivation.

By now the student's involvement - be it right or wrong - is forgotten. The class is far too engrossed with your - seemingly - bumbling deductions . . .

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    Now I would be really interested to learn about the reason for the downvote. This is the first answer so far that suggests to actually explain to the students why "log(ab) = log(a)log(b)" is false while "log(ab) = log(a)+log(b)" is true (and moreover, to explain it in way which makes it easy for the students to understand). If one believes that learning something in class should somehow be related to understanding what you learn, then this seems to be the canonical course of action. May 16 at 15:58
  • (+1) Very good to work out a counterexample to the student's purported "rule". But I think I would skip the step of doing an example that just "happens" to be right (100*100) because that will tend to reinforce the wrong answer as well as waste time before getting to the point. May 17 at 18:29
  • 10 seconds, Ari. The 100 * 100 example might well be the one that students do in their head to "prove" it to themselves. Then go no further . . . Though I accept that no teacher can kid glove students indefinitely. Comes a time when they have too be left on their own, to sink or swim.
    – Trunk
    May 17 at 19:41
5

Don't tell people they are wrong. Tell them when they are right.

Let's consider your actual "log(a*b)" question because it's real.

When you ask the question remind them where they would have heard this - say "you probably remember this from your pre-calc class".

  • Let's say the first student answers "log(a) * log (b)". You say "OK, any other ideas?"
  • The second student says "multiply a by b and take the log", you say "yes, that works, but is there an alternative?".
  • Hopefully somebody eventually says "log(a) + log(b)" and you say "yes, that's right" and move on.

You haven't upset anyone by telling them they are wrong.

If there is pushback, and someone says "what about log(a) times log(b)" ask the class if they think that's right. If there is doubt say "OK, let's try it". Get out a calculator. Calculate log(2), log(3) and log(6) and see if log(6)=log(2) + log(3) or log(6)=log(2) * log(3). Or if that takes too much time ask them to look up the result in the pre-calc course notes.

The other possibility, since this isn't key information for your class, is to just go ahead and state the result without asking the question. Just say "..since log(a*b) = log(a)+log(b) we get the result..."

5

Don't ask questions about prerequisites, only ask questions about the ideas you're teaching.

Just assume the prerequisites are a given and puzzled students will just look it up later. In the unlikely event that a student asks a question about them, you give the (correct) answer and move on.

Asking questions is good, but you need to be prepared to dedicate some time to them. Questions about the core of your class ensure that if the answer is wrong it will actually be worth the time to revise that material.

There is also the issue that pointing out that the student misses some required knowledge could be taken as a suggestion that they are not good enough to be in your class. This is likely to hurt the student's feelings as well as those of some of the onlookers.

5

The only solution I see for this is to just lecture and not encourage participation in class

You would be doing them a disservice.

Now, what you can do depends on how the feedback they provide impacts your position on a professional level. It also depends on the culture of the country.

If it does impact your career then I would do what is best for me: get wonderful marks from the pupils. Yes, this is sad but if my company university sees this as a way to rank me then obviously I will comply.

Bad luck for the students.

If it does not impact your career then you can do real coaching in addition to teaching. You would be doing them a disservice by letting it go because they are adults who as supposed to behave like adults.

People may have all sorts of opinions about how to not bring discomfort to the youth but tomorrow they will be on the job market where they will get a kick in the butt without warning.

I would encourage them to answer, but also to review the basic information they will need for the course. Actually, with your experience, I would provide them what they need to know in math right at the start.


Despite not having a feedback system at the end of the year, I usually asked students to give me one. They had the opportunity to tell me who they were if they wanted to help me understand.

I was less interested in the great ones (though it is always nice) and was looking at the medium and bad ones - and trying to understand what went wrong.

So I find feedback a good thing when it is not a way to rank a teacher (which is the case in the US if I understand correctly).

1
  • If the college wants to be a diploma-mill and he stands in their way, then his career is going to end up as roadkill. How do you pay your mortgage or feed your kids without a career? May 17 at 17:01
4

I'd like to add a few comments to some of the already excellent answers posted here:

  1. Be careful about asking "recall this (potentially basic) fact" type questions. To see an extreme example, think about me posing the question "what is 17+19?" to my Calculus 1 students. They should of course be able to answer, but there is very little reward for answering correctly and considerable shame for answering incorrectly. I used to ask many more questions like this when I first started teaching because I felt they made the class more accessible, but over time I learned that they actually discouraged student engagement and I have tried to cut as many as possible out of my lectures.

  2. That said, there are certain basic facts that I would like to remind students about in my class. For example, it would be great if my Calculus 1 students all knew off the top of their head that cos(0)=1, but many have forgotten, never learned it well to begin with, etc. Rather than telling them this fact, calling on a particular student, or posing it as a question to the class at large, I have found success doing one of the following two things:

a. Assigning every student a partner (this happens every day in my class) and then having pairs discuss the answer with each other.

b. Posing the question to the class as a poll question and then collecting responses electronically--this forces students to think about it, but doesn't put any negative social pressure on making a mistake.

  1. All this considered, I still think that the best questions ask students to think, rather than recall. There are several good answers above discussing this, so I will simply ask any readers of my answer to refer to those.
3

Try rewarding good answers. One of my professors used to give treats (like mini chocolate bars) to students who ask good questions or answer well to his questions. This way

  1. You encourage interactions.
  2. You make a clear distinction between good / bad answers, but without making the bad ones looking bad.

He was always getting positive feedback in the student evaluation for this.

2
  1. First of all, determine whether the students are really dumb or intelligent.
  2. If they are intelligent, are they giving wrong answers because of lack of basic or per-requisite knowledge.
  3. If they are intelligent and have basic knowledge, then do they want to tease you or they are just careless.
  4. Also find out how they are doing in other courses.

All the methods told in other answers need intelligence, per-requisite knowledge and seriousness of the students.

Analyze the students as told in steps above and decide how to pursue.

EDIT

Within first week of classes, teacher knows about intelligence and seriousness of each student in his class and the students (even the dumb ones) know intelligence and knowledge level of the teacher. But if you still need to find more, then google the topic. I found some interesting websites.

4
  • 15
    How would you determine whether the students are really dumb or intelligent? May 15 at 16:33
  • @Peter Mortensen I have edited my answer.
    – imtaar
    May 16 at 7:55
  • 1
    Corollary: What if the students are dumb?
    – user253751
    May 16 at 13:13
  • 1
    @user253751 In some countries, there are Condition-Specific Schools for children with autism, ADHD, OCD, "language-based learning challenges," anxiety, cognitive disabilities, or mental health issues. Also there are Therapy-Specific Schools. But such schools are not found everywhere. If a child is not doing very well in school, try to find if he is good in some specific field like music, agriculture, smithy, computer graphics etc. Get him private tuition in that field.
    – imtaar
    May 16 at 13:33
1

I see a fair number of purely immediate/tactical answers and not so many structural/architectural ones, so I will try to share some structural things you can do that make this work.

Assumed:

  • they do not actually know the material that they should
  • they do not know that they do not know that material
  • they need to understand what they are taught in order to be able to perform acceptably as a professional
  • you have to teach them that

Modelling from Dr. Daniel Jankowski (Jaws**):
One of my professors was a bear. You get into his class and he has a pile of 3x5 cards, one for each student, and instead of asking at random he works through the pile. Everyone knows they will be called equally. He always asks hard/important questions, no fewer than 5 per class. Part of the rules is that if you don't know the answer, you say you do not know it, and he makes the mark, and goes to the next card. Another piece of this is that he forces everyone to sit forward. The center and front get the best value, so he calls the front-row the A-row, the second the B-row, and he forces the students to move to seats as far front as they can. Their convenient cliques are disrupted out the gate, and those cliques are the ones the students are most worried about looking dumb around, not the nebulous "class".

Anger is a secondary emotion:

Your brain is required for anger. Expections - Reality := Disappointment (anger). It does not exist without reason. The most important events in a trip, holiday, vacation, or such, the ones that form the kids opinion, and the latter ones in the series. They do not remember the events as a dispassionate objective observer, and this applies to how humans process emotional events like anger or shame over time.

So how do you exploit this background? Don't ask just one, ask a sequence. Make the last one be something you know they know. I like to ask in the sequence: medium-easy-hard-easy and let the brain work it out.

Finally I like to give them the tools to teach themselves. There is an old saying that the first job of a teacher is to put themselves out of a job. It is implicit to the current class of students. They do it by giving the students enough of the fundamentals to be capable, and then training the students to be able to teach themselves in the area.

They say that "eternal review is the price of knowledge" or "repetition is the mother of learning". I find things I was an ace at 12 years ago have rusted nearly to dust, but if I keep the nuggets/seeds from which that tree grew, I can re-grow in a hundredth the time that former and formerly rusted capability.

A fund way to do that is "prove it" or asking if someone else can prove or disprove the wrong answer. Knowing the few axioms that you can use to test your answer is useful for re-teaching yourself. It helps them exercise those seeds.

Best of luck. Also, academia can be hell, and it pays poorly. "Those who can, do; ..." The market is hungry for capable folks, now more than ever. A capable scientist is an asset. Bureau of Labor Statistics says a Bachelors Chemist has a median pay of $80,000 per year, not including benefits, stock, heath, vacation, nice equipment, and such.

** Jaws. His class was like fighting a bear. I spent twice or three times the effort to get by in that class than it took to ace higher level classes in the same chain. But like becoming a werewolf you are torn apart and the spirit of the wolf enters you, the spirit of that bear entered me. In latter classes with peers who took the same Course-number from other professors, I had super-powers. I could first-principles derive things they couldn't set up.
Bottom line: Fight the bear. Charge the dragon head-on, because you can't lose. Every single thing that most honest of enemies, that most noble of opponents, called truth, does it makes you more powerful than you could imagine.

0

One technique that I was taught for cultural contexts in which it's profoundly shameful to have your mistake called out, is to give the correct answer without calling attention to the mistake:

Teacher: What's another way to write log (a * b)?

Student: log(a) * log(b)

Teacher: Another way to write log (a * b) is log(a) + log(b).

The teacher is then reinforcing the correct information. If the student isn't paying attention to what you're saying, he'll miss the correction. But if he is, he'll appreciate not being called out. Students pay more attention to what the teacher says than what a classmate says (at least this is my experience), so they will hear the correct fact and can move forward.

(Of course, you can make a mental note to call students' attention to “an easy mistake to make” later in the lecture. I don't think it's less effective for being outside the context of someone actually making the mistake.)

2
  • 2
    Arguably worse than OP's more direct engagement. after being given a wrong answer.
    – Trunk
    May 16 at 11:18
  • 1
    Of course “arguably”—because it's entirely relative to the context and the goals of the instructor. If one doesn't care what gets back to one's supervisor, or about how the students feel, the best feedback might well be, “Maybe you shouldn't be in an upper-level chemistry class if you can't do basic algebra.”
    – adam.baker
    May 16 at 12:28
0

Try asking the other students for a second opinion. If one student gives a wrong answer, ask the class if anyone else has an idea. You're not directly telling the first student they were wrong (and you can also thank them for volunteering an answer) so they won't feel put down.

Alternatively, there's the "you're nearly there" or "you're on the right lines" response.

If you have time, get them to work out the correct answer as a team. "What's log(3)? (You can use your calculators)". "And what's log(4)? So what's log(12)?" Write the answers on the board, and someone will spot that it's the sum, not the product.

(And just to make it more fun, and more memorable, at the next class you can bring in your ancient slide rule and invite the students to play with it...)

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