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I was reading answers to this question today when I realized we never had this followup question:

How do you effectively call on students by name in a math class?

As a way to encourage class participation and class preparation, many of my colleagues in other disciplines swear by the practice of randomly selecting students to ask questions of them during class. In my mind, the obvious issues are:

  • The practice is stressful for students (but wait, this is the primary alleged benefit of the practice).
  • It is different-levels-of-stressful for different students or groups of students.
  • It needs to be okay for a student to say they do not know something, but it can't always be okay, or you are defeating the purpose.

Do any of you regularly ask questions of non-volunteering students in your math classes in a way that avoids the pitfalls above? I'm personally interested in undergraduates but would be interested in answers from other perspectives as well.

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    $\begingroup$ Personally, I probably wouldn't have bothered getting a math degree if I constantly had professors annoy me by asking me to answer questions (in non-seminar classes). The idea that this is a good thing is just another example of the tyranny of the extroverted. Just because you find extroversion natural doesn't mean you have to inflict it on others. $\endgroup$ – John Coleman May 1 '17 at 18:50
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    $\begingroup$ the tyranny of the extroverted Oh, please. I'm an extreme introvert, but if someone calls on me to answer a question, I'll give it my best shot and not take it as a personal attack. $\endgroup$ – Ben Crowell May 1 '17 at 23:52
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    $\begingroup$ @BenCrowell The fact that I would find it annoying doesn't mean that I would take it as a personal attack. I do think that extroverts are over-represented in education departments and that as a consequence many of the ideas that they advance involve forced participation at the expense of contemplative learning. $\endgroup$ – John Coleman May 2 '17 at 0:39
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    $\begingroup$ @JohnColeman I'm a hard core introvert and I always hated being called on, but I'll not soon forget my advanced calc teacher calling on me for an example of a subset of the Reals with a specific character, accepting my answer and then asking someone else for a different kind of subset, at which time I interjected that my example worked for both questions, and he met my eyes and said "Oh yeah" in a funny way. That made me so happy that I did go for my math degree. Education must involve discomfort of some sort, otherwise it would be unnecessary. $\endgroup$ – Todd Wilcox May 2 '17 at 1:56
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    $\begingroup$ @ToddWilcox Yes -- that sort of thing happens a lot in pre-college education. Like many people, I detested school before college but loved college. Many reformers want to make college more like high school. I have no problem with well-motivated student-teacher interaction, including occasions where the teacher asks challenging questions (especially in upper-level courses with small class size), but the idea of randomly calling on students for the sole purpose of forced class participation rubs me the wrong way. As a professor I won't do it since (for me) it violates the golden rule. $\endgroup$ – John Coleman May 2 '17 at 19:39
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This isn't much of an answer, but I think it goes a long ways toward what to think about.

Know your class really well.

Different questions will work for different students. Your third point is a good one, and so you need to know the class well enough to know what sort of question will work for each student. This doesn't obtain in a huge class, though there one might be able to get away with this technique for other reasons. A corollary:

Have a backup plan.

There always needs to be something you can use to extricate yourself from a bad situation or a truly confused student. That could be to validate the student's feeling (see e.g. today's blog post by Allison Henrich which has a number of very useful points), or something else - not just to be able to retract the question. But there has to be a way to deal with the issue of when despite your best preparation you misread a student.

My own personal preference is to only ask very softball questions of completely random people, or questions whose answers are literally before their eyes. But I've seen it done effectively to every single student, given the right personality and expectations for the class. Also, in a less consumer-centric educational context than the USA (or in law school, I hear), this strategy might be perfectly appropriate all the time.

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    $\begingroup$ As part of backup plan, I try to have a second easier question that helps us to answer the first one. That way every student I ask answers some question even if it isn't the one I first asked. $\endgroup$ – MathIsKey May 1 '17 at 19:22
  • $\begingroup$ @kcrisman, I think I'd like to accept this answer, but I wonder if you could elaborate a little on why you linked us Paul Sally in a follow-up comment first? $\endgroup$ – Chris Cunningham May 4 '17 at 11:31
  • $\begingroup$ You certainly don't have to accept this answer - I tend to see questions like this as having many legitimate answers, so that the SX model isn't quite as appropriate (community wiki, but not EVERY question can be that). But it is very gratifying to hear you find it useful. $\endgroup$ – kcrisman May 4 '17 at 21:22
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    $\begingroup$ I linked to the late "Mr. Sally" because he is the person I have seen (and whom others reported) can do this effectively - namely, calling on every single student randomly. He said a class wasn't successful in this sense if he didn't call on every student twice! (Given a certain size class, of course.) But not everyone can imitate this, and you also have to have full rapport - that they really believe in you. This situation will definitely not always obtain. $\endgroup$ – kcrisman May 4 '17 at 21:25
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I emphasize from the beginning of the year that mistake-making is to be expected; the specific language that I use can be found in MESE 11794. I believe that by ensuring this is a norm from the beginning of the academic school year, one is better positioned to call on students who are not volunteering, which is something that I do frequently. I feel very strongly that much of the stress, or anxiety, of being cold-called can be averted by engaging in this practice deliberately and often, and weaving unsureness, or outright errors, into something productive as pertains to the topic at hand.

I do not, however, call on students "randomly": I know that some teachers use various methods to do this, e.g., spinning a name wheel or drawing popsicle sticks with student names out of a cup (possibly without replacement so that different people are called) or just circling around the room with one's questions. In my own estimation, part of the instructor's teaching expertise should be reflected in their choices around which student is called on, and when. Moreover, I believe that participation -- like mathematical tasks -- can be "low floor high ceiling" in the sense that, the initial student voicing can be very simple, and follow ups can be scaffolded towards the very difficult.

As an example of a low floor participatory task, I might say something like:

Chris, can you read problem 3 on page 11?

[Chris reads problem]

And do you, or does anyone else, have any ideas about how we can get started on this?

I would probably stay with the reader for a moment, to give them a chance to say something on their mind; otherwise, I will call on others who are volunteering, but I might do something slightly subtle and re-summarize at some point by using the reader's name. In the above example, this could mean, e.g., after Alice chimes in with an idea about how to proceed, I might say:

Okay, so Chris asked [problem 3], and Alice had an idea about how we could broach this. She suggested we [suggestion]. Does anyone else want to weigh in, either by building on what's been said, or by asking for something to be clarified further?

I will not make a claim that this is paradigm-shifting in its efficacy, but I believe that keeping track of names and who did what -- whether it is reading a problem aloud, or suggesting a (possibly incorrect) method -- is a good way to maintain a sense of participation by multiple individuals.

Although describing in full how I try to manage classroom discourse would make for a far too lengthy post, there is more about productive talk moves here, which I generally co-sign.

Succinctly:

  1. Establish a norm from the beginning of the year that making mistakes is expected and valued.

  2. Establish a norm from the beginning of the year that you will call on students who have not volunteered.

  3. Recognize that there are many forms of participation -- ranging from having one student asking another to clarify their reasoning, to giving a full and complete explanation for a problem, to simply reading aloud from a handout -- and value each of these forms, repeating contributions with attribution as you (the teacher) talk through the class' interactions around a problem/lesson.

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  • $\begingroup$ I'm curious about the context where you use this. In all classes I teach, attempting to have a majority of students follow the ideas well enough in order to do a problem in real time, would probably make me teach 10% of the syllabus. And, in most classes I teach, there are students who cannot answer the most basic questions because they carry a long string of "barely passing" their math classes with no understanding whatsoever. $\endgroup$ – Martin Argerami May 2 '17 at 11:13
  • $\begingroup$ @MartinArgerami I use problem sequences in a collaborative setting to teach (mostly) at the secondary level. We do not move as quickly as a lecture-based course could (although I question the depth of long-term understanding blitz courses impart, having taken some number in my past life as an undergrad...) nor as slowly as (e.g.) a Moore Method class would. $\endgroup$ – Benjamin Dickman May 2 '17 at 21:58
  • $\begingroup$ THe productive talk moves link is broken. $\endgroup$ – Sue VanHattum Nov 21 '17 at 18:35
  • $\begingroup$ @SueVanHattum Thanks; I will search for a working replacement! $\endgroup$ – Benjamin Dickman Nov 21 '17 at 18:59
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1. Be aware of your goals for calling upon students.

As teachers, it is vitally important that we are goal oriented. As such, we should establish goals not only for learning outcomes but for classroom management and maintaining an atmosphere that is beneficial to learning. Whether or not the method for calling on students is effective depends on whether it effectively supports your teaching goals. Do you want to make sure the students are paying attention? Do you want to generate formative discussion? Do you want to assess understanding? Set appropriate goals, then gear classroom questioning toward accomplishing them. Periodically, evaluate how effectively the method supports your goals, then adapt the method if necessary to better suit your needs.

2. Make question-and-answer a class activity, not an individual one.

Asking students questions, when done incorrectly, can actually be like giving the class permission to check out. Avoid creating situations that only require one person to think. If I ask, "Johnny, what is the square root of 9?" anyone not named Johnny knows they can relax and they miss out on the practice or the thinking that the question was meant to provoke. Instead, direct questions to the whole class. I often phrase questions similar to the following manner, "Here is a question that I'd like you to answer: what is the square root of 9? I'll give you ten seconds." This way, students know what the question is and how much time they have to think about it. If I ask, "Okay class, what is the value of $x$ in this diagram? I'll give you 40 seconds." Students know that they have enough time to think carefully, work out the answer on paper, and perhaps even consult a classmate- and they don't expect that they should know the answer immediately.

Clearly, it's important for this method that students are not shouting out answers when they arrive at them, but waiting until the time expires with their hand raised or their pencil down, and that the teacher actually gives enough time to think about the question before calling on a student volunteer or by name. This way the student who takes longer to think or has to write intermediate steps has time to arrive at an answer and is able to meaningfully participate. This makes a huge difference in student confidence when answering questions in front of the class.

3. Ask questions with open answers

Don't ask questions that allow for guessing, like "yes or no" questions, like "is this a right triangle?", or even worse, "This is a right triangle, isn't it?" Even if the student answers correctly, it's not as helpful for determining if they understand or are paying attention than an open ended question that must be answered in a complete sentence.

4. Follow through by asking about reasoning

Regardless of whether the answer is correct, ask the student how they arrived at their answer. This will allow other students who arrived at the same answer to understand common errors, which develops an atmosphere where it's okay to give a wrong answer, or allow students who are making errors to observe how to get to the right answer. I'll often ask for a second student to try to explain the reasoning of the first in the event that the first answer is correct.

5. Make student selection fair.

Unlike some comments on the OP, I am not of the opinion that asking open questions is unfair to introverts. On the other hand, allowing students to volunteer answers, and never calling upon students, will destroy a learning atmosphere. Students will learn that they simply don't need to try to think of an answer if they don't feel like it or they find it difficult. Then it's tempting to call on students as a disciplinary action- to ask them when you know they haven't paid attention. This is detrimental to a positive learning atmosphere, causes disdain for the teacher, and hinders learning by wasting time with poor guesses and "uh, can you repeat the question?". Relying on frequent volunteers can result in students feeling that the teacher favors students who volunteer regularly or that the students who do not volunteer are inferior to students who are able to answer- "She just get's it, because she's a math wiz. I just can't do that". In the event that a regular hand-raiser does get an answer incorrect, the atmosphere is not one that encourages or accepts incorrect answers.

In my classroom, each student has a number on their assigned desk. After I ask the question and time has expired or I see that the class has finished reasoning, I roll a large die and call on the student that it picks. This keeps the selection above criticism of unfairness and the distribution of students called upon relatively uniform. I often allow students to roll the die, which adds excitement and some humor. One can find twenty and thirty sided dice from a number of retailers.

6. Get multiple answers before revealing the correct one.

When a student answers a question, before discussing the answer, I often ask another student or two for their response. When the students have multiple answers, I get excited- it's a perfect opportunity to have them demonstrate their work or reasoning on the whiteboard. I keep extra credit tickets that I offer to students who go to the board, regardless of whether or not they are correct, so students are usually excited for the opportunity to get a bonus point on a quiz or test. This helps create a classroom atmosphere where giving the wrong answer is okay, and where being wrong can actually benefit the student directly.

Overall, find a question method that works for you, works for your students, and allows your students the greatest advantage for learning. More discussion of these ideas in addition to a wealth of helpful suggestions about classroom management can be found in David R. Johnson's book, Every Minute Counts: Making Your Math Class Work.

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If the question really is "how (best) to call on students?", then I'd say I don't know. If the question is "do I call on students?", then I'd say "no, I do not". I do not do so in undergrad or grad classes, required or elective. I do encourage questions, even if frivolous or humorous, and usually there are several students (whether academically strong or not...) who, by personality, are happy to ask the question that everyone else is thinking but too inhibited to ask.

That is, I tend to structure "lectures" not really as lectures, but as me-doing-examples (which, on one hand, are of the sort the students know they will be "graded on", but/and, also, are what I consider to be important scientifically/intellectually). Students are encouraged to ask clarification questions, background questions, please-repeat-what-you-just-said questions, or small-detail questions. Anything.

The most that I'd inclined to do is, when I hear a sharp inhalation, or see a perplexed facial expression, that I might ask a student whether they were unhappy with something...

I do think that asking non-volunteer students to ask questions is rather aggressive, and cultivates an adversarial/game-ish atmosphere, if only due to the naive perceptions of young students (sub-25?), regardless of the good intentions of the teacher.

For that matter, it is a non-trivial issue about whether one can easily, extemporaneously, formulate a coherent statement about a thing. Since I generally can do so, it took me a long time to understand that incapacity in this regard is not a sign of stupidity or lack of understanding or earnestness.

And, last, whether or not we declare that students "should not feel embarrassed/humiliated", immature people do manage to feel humiliated for reasons that don't seem reasonable to older people, etc. And, then, will be resentful, and distracted. The idea that they "should" behave otherwise is as impractical and untenable as the idea that they "should" have a different attitude in their approach to mathematics, etc.

In terms of "trust", the only time I ask students questions, expecting or hoping to be answered in real time, is with my research students in rehearsals of their presentations. This presumes many, many months of acquaintance, and a negotiated "trust" situation, and is absolutely not "public". And it is understood that "failure" is not literal failure, but is educational. But, even then, people need some reflection to understand that is indeed "how it could be". And usually needs substantial prior one-on-one discussion about "what the game is", and that we are not adversaries, etc. Even so, this does not seem to work well with immature grad students.

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  • $\begingroup$ Women will participate less, if not encouraged to do so. You might wish to read Failing at Fairness. $\endgroup$ – Sue VanHattum Nov 21 '17 at 18:33
  • $\begingroup$ @SueVanHattum, yes, indeed, by this point in my life I am very well acquainted with various such dynamics, as well as how they change depending on the demographics, and so on. It did take me a long time to catch on, though. $\endgroup$ – paul garrett Nov 21 '17 at 20:19
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As a teacher, I tend to avoid this because even when I ask generic questions to the whole class the discomfort of the students is often patent. With mathematics, also, you are frequently trying to get the students to understand a new idea, and it usually takes time and practice to get it; which means that being partially lost is a common ocurrence.

As a student, I never obtained any benefit from a teacher following this practice. In highschool, it made me really uncomfortable and it made me rebel and not study. In university, if a teacher had done this, I would have simply stopped attending class.

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    $\begingroup$ I guess I'm worried about your answer since it seems to imply we should give up entirely on class participation to make the students feel more comfortable. I'm very interested in carefully calibrating the level of discomfort in the room, but I'm not sure it is necessarily a good idea to set it to zero. $\endgroup$ – Chris Cunningham May 2 '17 at 10:33
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    $\begingroup$ I don't have a good answer to that. I feel strongly that educators put too much value on pedagogy, as opposed to deep knowledge from the teacher and personal effort from the students. $\endgroup$ – Martin Argerami May 2 '17 at 11:02
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    $\begingroup$ I don't think we disagree here, but I do think my primary purpose for thinking about pedagogy is to use it as a tool to induce more personal effort from the students. $\endgroup$ – Chris Cunningham May 2 '17 at 14:54
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    $\begingroup$ @MartinArgerami Considering that pedagogy means "the method and practice of teaching", it's probably a good thing that teachers put a high value on it. $\endgroup$ – Andrew May 3 '17 at 1:16
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    $\begingroup$ @Andrew: not if, as it is more and more common, they know how to teach but not what they teach. $\endgroup$ – Martin Argerami Jun 15 '17 at 13:51
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I have the opportunity to teach both high school students and undergraduate students in College Algebra and College Trigonometry. In my high school classes, I call on each student at least once per class. This has always been a practice of mine as it encourages participation and engagement. I strongly disagree with calling on students in a college class because it is not my job for students to be engaged in a college class. Students should be self motivated in college or they won't succeed. If they can't pay attention to a lecture, then they probably won't and shouldn't be successful in college. Self motivation has to be a factor to success or a college degree will continue to become more and more watered down.

To answer your question: An effective way to call on students is, as mentioned above, to know your students. Do not start immediately randomly calling on students. Only call on students after you've gotten a few formal assessments graded. Then, you know Johnny struggles, so you throw him your softball questions. And, you know Sally is doing very well. So, you may throw her some questions that require critical thinking.

Be sure to have follow up questions that could lead a student to an answer. This is important because some students will sit silently or say they don't know until you call on someone else. Guide the student to what you're looking for by asking simpler follow up questions.

Affirm students whether they are correct or incorrect. Using words like "No" will make students more anxious and less likely to be willing to participate in the future. Instead, you might say "You're on the right track... Let me ask you this follow up question....".

While I don't agree with the practice of calling on students in a college math class, these are a few approaches you may use.

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    $\begingroup$ +1; this is a great answer with a lot of detail. It looks like you recently joined the site and your posts are all this well-written; welcome! $\endgroup$ – Chris Cunningham May 4 '17 at 14:44
  • $\begingroup$ Not calling on students may be detrimental to female students, and for different reasons, to students of color. Building a sense of community, and making sure all students feel like they are a welcome part of that community, is important to broadening participation in mathematics. $\endgroup$ – Sue VanHattum Nov 21 '17 at 18:38
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I was routinely called on in high school and college. I think it is a fine practice. I would not obsess too much on how to do it perfectly or some big fear od doing it imperfectly.

Still remember snapping awake at the Naval Academy when the Indian ODEs prof asked me some ODE needing an integrating factor to solve it. I somehow woke instantly from total sleep, saw the board and spit out the answer xsqetothex or whatever it was. Not too simple though. He was amazed and commented how he always wished he could do that as a student.

Keep 'em awake, keep 'em guessing. Don't baby the ball!

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