I teach calculus to college students, and find it very difficult to get them to speak up in class when I ask questions, or when I'm trying to get a pulse for how much they understand. I think students are afraid of being wrong, or asking questions that seem too easy to other students.

However, I firmly believe that an interactive classroom is far more conducive to learning than one in which I simply download information onto a blackboard.

What are some techniques I can use that help foster a safe environment for students to speak up and interact with me, other students, and the material?

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    Start out asking very simple questions. Increase the level gradually. – Rasmus Mar 13 '14 at 20:39
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    @RasmusBentmann: Your suggestion is definitely a good one. However, one has to be careful though to not ask questions that are too simple. This often has the effect of causing students to not answer because they feel condescended, or if they answer, it will show they were eager to answer a simple question whose answer is obvious to everyone. – Jared Mar 13 '14 at 20:51
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    I see what you mean. Maybe it depends on how one asks. In a better scenario, the students might answer the simple question in a way that makes clear that the answer was obvious and that they are ready for a step forward. For what it's worth, I've had a TA who did it this way and I found it great. – Rasmus Mar 13 '14 at 21:12
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    +1 Not only do I love this question but I think it is important. Most math teachers struggle with this question, and this page could become a very valuable catalog. The answers below are great and I hope there continue to be more and more. – benblumsmith Mar 14 '14 at 20:43
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    When I visit other instructors' classrooms to evaluate their teaching, one of the most common things that drives me crazy is that they ask the class a question but then immediately answer it themselves. This clearly sends the message that the professor doesn't actually want students to participate. – Ben Crowell Mar 29 '14 at 22:24

13 Answers 13

This is a small tip based on the obvious idea that it needs to feel safe to answer questions.

Suppose you need to take the derivative of x sin x, but you want students to speak up about it in the flow of lecture. Here are three ways to do it:

  • "Now I need the derivative of x sin x. What should I do first?"
  • "Now I need the derivative of x sin x. Which rule should I use?"
  • "Now I need the derivative of x sin x. The answer is not 1 * cos x. Why not?"

The first question is too open-ended and scary for students to answer. If you ask questions of this type, you will only possibly get answers from the top of the class, and no one else gains anything from the process.

The second one is a lot better even though it is almost the same question, because it explicitly tells them the format of the answer. You might get some brave B students who will name the correct rule to answer this one, because it is unlikely for them to answer in the completely wrong format.

The third one is my favorite form of question: instead of asking students to venture forth and dare them to avoid the trap, you ask a question, point at the trap, and say "don't step on that trap." Everyone can gather around the trap and many will think "What? I would have stepped into that trap." More students gain from it and more students are likely to venture forth an answer.

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    I strongly agree that the third type of question is the most effective. – Santiago Canez Mar 14 '14 at 14:38
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    I love this suggestion. Actually, though, I strongly disagree that the second form of the question is better than the first. It encourages students to relate to the question as a matter of already remembering something, which will make them silent and disempowered whenever they don't. An additional virtue of the third question beyond what you mentioned in the post is that it encourages them to relate to the question as something they can reason about. (They are not required to already remember in order to participate.) – benblumsmith Mar 14 '14 at 20:31
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    I actually ask what is $x\sin x$ and this leads us to the product rule. – JP McCarthy Apr 9 '14 at 13:59

The think-pair-share technique is an oldie but a goodie:

  1. Pose a question
  2. Give students 1 minute to quietly think of and write down their answer (if you have a computer/projector setup you can use an onscreen timer to enforce the "1 minute" frame)
  3. Give students 2 minutes to exchange / compare solutions with a neighbor
  4. Ask for volunteers to share results with the whole class

Devoting a minimum of 3 minutes explicitly to think/pair communicates to your class that your question is not rhetorical and that you expect an answer. The "pair" stage discussion provides them with an opportunity to pre-screen their answer so that they will (one hopes) feel safer about speaking up.

Of course the downside of this strategy is that it takes more time. If what you are hoping for is a quick check-in with your students to punctuate your presentation without interrupting the flow, this may not serve that purpose. But if your goal is to actually get students to stop, think, work, and talk to you (and each other) this may help.

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    This technique can be enhanced by the use of a classroom voting system ("clickers" or something fancier; there are free options available online). Having to vote for an answer is extra pressure on the students to actually think about the question and come up with an answer, and you can also display graphs to show the class, or just see yourself, how well people are understanding and whether the peer discussion helps. – Mike Shulman Mar 13 '14 at 20:49
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    I do often find that 1 minute for an answer is not long enough, at least if the question I asked is nontrivial and/or requires some computation. – Mike Shulman Mar 13 '14 at 20:50
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    Another option in step 4 is to actually call on students by name. I was very reluctant to do this, but now that I'm trying it, I find that it's not as bad as I feared. – Mike Shulman Mar 13 '14 at 20:51
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    @Mike I'd love to read a top-level answer from you about calling on students by name in a math course, and how you prepared them for it / some early pitfalls you experienced. – Chris Cunningham Mar 13 '14 at 21:03
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    @ChrisCunningham, I wouldn't be a good person to write that answer, since this is the first semester I've tried it. Maybe someone with more experience can share their advice. – Mike Shulman Mar 13 '14 at 21:36

One important point to make is that you should ensure that interaction is part of the culture of your lectures. It isn't enough to pose a question now and again and expect them to suddenly leap in to action to answer it. So you need to be asking questions consistently through the course.

The next point I'd like to make is that it will take time for the students to figure out that that is what is expected and to participate (and even longer to enjoy participating). This has two practical consequences. The first is that you should expect - and know how to deal with - an initial desire on the part of (most) students to not participate. The second is that you should think about participation on a continuous scale not as an on/off state, so in the early days ask questions that can be answered almost without thinking and make it easier for the students to answer than not answer. Then once they know what to expect, you can get them to do the more thoughtful questions.

A danger that is easy to fall in to with interactive environments is to focus your teaching on those who are interacting. This is natural, but can back-fire as all the others feel even less involved than they would normally. If the majority will not start the course feeling like they should interact, the initial interactors are out-liers and you shouldn't be shaping the course by their needs.

At the university level, I would start by telling them explicitly what is expected of them, and why. They are legally adults now and need to start taking responsibility for their own learning. Then every time I asked them a question, I would remind them of why I'm asking them.

On to practicalities. I'm currently using the "voting" method of asking questions. I'm teaching a course to non-mathematicians which is very much a techniques course. So the "I probably didn't understand the question" barrier to answering is quite high. Add in the fact that I'm not teaching in their native language, and the chance of someone answering a complicated question is quite small. So I ask simple questions and I get them to vote on them. Here's what is characteristic about my style (in reference to the points above):

  1. The questions are really easy. Quite frequently, the answer is really obvious, not just mathematically but from the very fact that I'm asking the question.

  2. I make it clear that guessing is absolutely fine. I frequently use the example of doing the exam: a guessed answer gives you a chance at some points, no answer gives you no chance. (I keep quiet about Oscar Wilde's quote about keeping one's mouth shut.)

  3. I don't take "no" for an answer. I mean, "no participation". After not many have voted the first time around, I berate them and ask again. Then I'll ask "Who didn't vote?", and "Who didn't say that they didn't vote (but didn't vote)?". The idea being that they're going to have to put their arm up anyway so they might as well put it up at the start and get it over and done with.

  4. I don't use technology. I want a phone-free environment in my lectures, and getting out phones or clickers every time is too much of a hassle. In addition, I'm often spontaneous about when I ask these questions and don't want to be taking time.

  5. I use the answer they give. If it's right, I'll make a joke about "We live in a democracy so let's go with the popular vote.". If it's wrong (or if a sizeable minority give the wrong answer), I'll take the time to explain why.

Lastly, let me share a different technique that I used quite successfully with a class of more mathematically minded students (so who could be expected to answer more complicated questions, though they were still actually quite straightforward). I would pick a starting point in the lecture hall and then simply go along the row every time I had a question (picking up where I left off next lecture). If someone passed, that was fine and I'd ask the next person. This once worked spectacularly well when someone had fallen asleep: I started the "answer snake" a few people before him in the row, and his neighbour realised what was going to happen so nudged him awake.

Safety: When a wrong answer is given, if you can figure out what would have made it right, you help the student feel safer. Teacher: 2*3 is...? Student: 5 Teacher: oh, I bet you're thinking about 2+3. In calculus, teacher: integral of sinx? Student: cosx. Teacher: If I were asking the derivative, you'd be exactly right. What's the derivative of your answers? (Now they get to tell me about the missed negative sign.)

I call on students equally with index cards. After years of doing it, I began to feel it was too ... cold? ... rigid? I stopped. But this winter I was re-reading something I've written about supporting women in math. Female students are less likely to raise their hands, and then they miss out. (Read Failing at Fairness, mostly about K12, to learn more about the scope of the problems female students face.) So I went back to my 3x5 cards. And my calc I class with 45 students is benefiting tremendously. It's at 8am, but is a very lively class. Shy students are willing to contribute when called on. (I used to say it was ok to pass. Too many students would pass. I did not do that this semester. It's working fine.)

Other ways to get interaction: If you have issues with clickers, you can do a low-tech alternative. I ask for students to give me thumbs up-down-or sideways many times during each class, to get feedback on how well they feel they understand something. It's very helpful.

If the answer is a small whole number, I ask them to indicate with fingers raised. (In pre-calc: What's the degree of this function?)

Some things can be shown with the body (and the movement helps students internalize the ideas). Show me the long-term behavior of this function with your arms. (y=x^5 has left arm down, right arm up.)

I use small groups whenever the task can best be done that way, and often have them pair up to compare answers.

When presenting some example, I like to let them cast votes on the correct answer (e.g. for choosing methods of integration, or for the question how many solutions a given linear system has, after reducing it to row echelon form). I offer them 3 possible answers (sometimes including "Who doesn't want to vote on this?"). This gives me a quick way to see how well the students are following, and it provides some start of a discussion if the vote wasn't unanimous, because then, you can ask "Why do you think it's A and not B?"

Also, what was kind of hard to learn, was to allow the students some time to think of the answer, and make a deliberate pause after the questions. Silcence in the classroom can be hard to endure, but I think creating a gap where students have time to think and answer is also important.

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    Definitely agree that silence is essential. – Loop Space Mar 14 '14 at 8:33

There are a lot of good answers already. That said, we have just scratched the surface. Shifting the environment of math classrooms from one in which students attempt (usually only semi-successfully) to passively absorb, to one in which students actually think, is a profound project and I think our profession is only partly underway with it. I would like to add just a few specific techniques and a general comment, and the caveat that there is a ton of research and writing on this subject already out there, and a ton of research and writing that has not been done yet and needs to be.

Specific techniques:

  • Asking not just questions but asking for students to summarize ideas. This is a technique I learned recently from some master teachers and it has transformed my practice. I blogged about it here.

  • Think-pair-share has been mentioned above by mweiss but I want to second it as an absolutely indispensable tool. Students engage much more readily when they have an opportunity to take some time to put their thoughts together and to attempt to articulate them in the comparative safety of a one-on-one interaction with a classmate rather than in the full glare of the room's complete attention.

  • Ask for questions: Ask, "what could a person ask about this?" Or, "what might bother somebody about this?" I often call on a student wearing a befuddled expression and ask, "what's on your mind?"

  • Wait time: Several people have mentioned the importance of tolerating silence after asking a question, to give students a chance to think. I'd add that, if you going to call on a raised hand, it is important to often not call on the first hand that goes up.

  • Dipsticking was mentioned by Sue VanHattum. I used to use thumbs-up, thumbs-sideways, thumbs-down as a quick gauge of the class' sense of its understanding, and I think it's great. More recently (since I started with students summarizing, see above), I'm more likely to say, "raise your hand if you think you understand, not necessarily if you buy it, but just if you understand, the last thing that was said." If less than 3/4 of the hands go up, I ask somebody who raised their hand to summarize the idea for the people who didn't.

  • Explicitly encouraging a culture of risk-taking. Praise all risks. When a student asks an honest question, thank them, no matter how obvious the answer seems to you or anyone else. If a student's wrong answer, confusion, question, or expression of doubt leads to the clarification of some point, call everyone's attention to this fact. Train students to see the value when they and other people are willing to ask their questions and express their thinking, by pointing out this value when it happens.

General comment:

Every single one of the techniques mentioned above slows down the coverage of material. In my experience, they all, always, slow it down more than I think they will. This makes them hard to implement consistently because of the pressure to get certain topics "done" and/or my enthusiasm to reach a certain goal I was excited about.

Be that as it may, when I do implement them consistently, I am always left with the strong conviction that the extra time was worth it. Often, for example, when dipsticking, or asking for student summaries, I find out that a point I thought I had taught well actually missed everybody. The time spent in discovering this and doing something about it was obviously worth it, because if I had pushed ahead, I would have been pretending the students had learned something they really hadn't.

I say this to encourage stick-to-it-iveness in broadening your repertoire of techniques to encourage participation, and to be ready to face down the costs to speed of coverage. It's worth it.

Disclaimer: I teach at university, it is much different than teaching younger students (e.g. I wouldn't really know how to handle them well).

The approach that works best for me is to force them to answer and to do it so frequently as to make it natural for them to ask questions, while, at the same time, letting them know it's alright to make mistakes.

To given an example, I often ask questions like "what technique would you want to use" which, sometimes, I make deliberately hard. Then, after a bad answer I respond "unfortunately, this approach will not work in this setting, however this is a perfectly reasonable answer" and then explain why the idea could work (i.e. intuitions, sometimes even explaining similar problems where it does work), and finally I would demonstrate why our current task is different and why that particular method fails.

There are particular personalities such that making them ask even a silly question helps. I remember one session where I made the whole class ask clownish things which, then, greatly eased the tension (it transformed into a sort of joke-questions contest) and activated even the most stubborn part of the group (and we did lose a lot of time, but I wouldn't say it was wasted).

After a two-three classes it becomes a standard thing and I don't even need to pick them. If someone is behind, it is often enough to apply a very gentle pressure (like a slightly longer eye-contact) to activate him/her. During the semester (i.e. when I earned enough respect) I sometimes taunt/tease them, even to the point of encouraging disobedience (tricking them into independent thoughts), e.g. proposing some bad solutions (sometimes blatantly bad, e.g. for comical effect, sometimes subtly wrong, to guard them against bad intuitions) or saying "no" to perfectly valid inferences (e.g. to emphasize which part of the process was the key part "oh, now I understand", and perhaps later to recall some previously solved problems "it is the same we did last week" and extract similar patterns, perhaps alike in some unobvious way; or to make the solution more formal and show them some traps which could be overlooked and later lead to incorrect reasoning).

As with any approach, there are disadvantages. There was one group which learned that saying "I don't know" is ok too fast and I was basically helpless - the contact was poor, and as things I prepare depend on class participation, they learned less and less, and that, in turn, made them even more remote. I was unable to break from the evil cycle until the very end.

Another thing is that these people start asking many (too many) questions in classes taught by others which makes them (the teachers) angry at you (e.g. it disrupts their flow).

Also, it works better with mature people, i.a. ones which are conscious that it is in their interest to learn, rather than just earn grades. One time such a situation turned itself into really dissonant tones: I scorned a "let-me-show-off" question of a good grade-earner and praised a bit naive, but honest question from person who usually doesn't know what's happening. There were some resentful looks and I had to give the class a much harder challenge so that it would be difficult even for the top part, so that the aforementioned person could reclaim his status, but in the process had to ask honest questions to complete the task.

Perhaps this is not the best way, but it works for me (among others, it matches my style of teaching).

I hope this helps $\ddot\smile$

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    I'd love to hear more about scorning the "let me show off" question. It seems like it would be tricky to do this while making it clear that others in the class should feel comfortable asking whatever question they like. – Chris Cunningham Mar 14 '14 at 12:41
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    @ChrisCunningham I don't remember my exact response, but it was kind of easy - mathematicians are usually bright people and even slight suggestions come across, in such an environment even a small explicit rebuke comes off as severe. It was something along the lines of "next time, to make even stronger impression on your friends, I would suggest ending your pronouncement with appropriate bibliography". Not very elegant, but did the job wonderfully. – dtldarek Mar 14 '14 at 18:43

In teaching calculus, I like to have something like 5% of the course grade be based on "Participation". On the first day of class, when going through the syllabus, I go over what I mean by 'participating' (it includes, for example, asking questions as opposed to only answering any questions I happen to pose to the class, coming to office hours, etc.) and in the first few days of class, when the students are still somewhat uncomfortable about speaking up, I sometimes reference their Participation Grade. I've also found that the more I meet with students in office hour and they get more comfortable interacting with me, the more likely they are to speak up in class. Almost all reasonable participation also gets a fair amount of positive reinforcement from me ('That's exactly right!' for correct things, or 'That's a good idea, let's figure out how this would work out...' for not-so-correct things.)

Don't tell my students, but almost everyone gets a near-perfect/perfect participation grade (this having been my secret plan all along), unless they never show up to class/some other relatively drastic thing .

I agree that grades should not be the primary reason that students are interested in doing well in any given course, but since they are something that people care about, why not put it to use.

(I imagine that this practice does not work very well with large classes where it isn't possible for everyone to participate in any real sense and it is presumably hard to keep track of the folks that do participate.)

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    One challenge I've found in enticing students with participation points is that some students become inclined to over-participate (dominate question/answer/discussion time). Then, I have to find a way to encourage such students to participate in the appropriate amount, without making them feel rejected. – Jared Mar 15 '14 at 17:49

This will, as others have pointed out, depend on the level. If the students are old enough, I would suggest having students present at the blackboard/whiteboard often. Make it part of your routine that you you call on people to go to the board and answer questions. Doing this regularly will create an atmosphere where speaking in class is "just what we do on a daily basis".

Also, just call on people to answer questions. This can be difficult in the beginning and some students will not like it. But if it becomes part of your routine, then students will get used to it and it can completely change the environment in the classroom (for the better).

And, as others have suggested, make sure all this is done in a way that doesn't make the students afraid. The classroom, of course, needs to be a safe place. You don't want to humiliate or embarrass students in any way.

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    Students are old enough to present at the board from the time they start school. In a supportive environment, this can add much to their learning. – Sue VanHattum Mar 15 '14 at 15:53
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    @SueVanHattum: I completely agree. The challenge for many teachers is that they don't have influence on what happens before the student joins their class. – Thomas Mar 15 '14 at 16:56

Interaction is fundamentally incompatible with lecture. So, the solution I adopted many years ago was to stop lecturing. But, even so, interaction is hard to get. So, I sort of not lecture and, bit by bit, as the semester goes on, I succeed in cajoling a number of students into interacting.

More precisely: the students are to read, according to a calendar, the chapter before each class and do a homework on that chapter. In class, we discuss the questions being brought up by the students---more often than not about the homework itself, and then the students submit (or not) their homework. The main advantage, at least as far as I am concerned, is that this allows me to spend more time on the aspects which, from experience, I think are important---for whatever reason---without feeling the need to "cover" the whole chapter.

P.S. I should probably mention that I write the texts and the homework that go with them.

Here 4 of 10 suggestions by Steven G. Krantz in an appendix entitled some suggestions for encouraging class participation of his book How to teach Mathematics

  • Get students to go to the blackboard.
  • Have students prepare oral reports or mini-lectures.
  • Have guest instructors.
  • Intentionally make mistakes in order that students can fix it.
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    This makes a lot more sense than groupwork and voting. – Rusty Core Nov 9 at 1:24

Establish that participation is compatible with being "cool"

I think one aspect that has not been metioned yet, is that there is often social pressure to not participiate too much in the class in order to not to be seen as a "uncool math geek".

This might sound strange to many math educators, because you are surrounded by people who are fascinated by math and you are one yourself. You enjoy passionate discussions about it. You can freely share that passion with your peers.

The "general population" often still has a stereotypical, negative, view of math professors teachers or in simply people who are good at it. If you really like math and talk to your "normal" friends about how facinating certain math concepts are, often you will get blank stares and people will change the subject, especially during the school years. Thus many of us learned to surpess this passion and not showing it publicly.

When they reach college, many will avoid participating too much, to avoid being seen like that. They don't want their "cool" friends (who are sitting next to them) to think: "Oh look, he answers again. I bet, all he does in his free time is read math books. He probalby has no friends and is a virgin.".

So when the teacher poses a question, they will not say anything although they could answer it.

What imho can alleviate this, is saying something like "Come on people, I know you are all tired from last night's partying, but let's see if we can solve this together." or "Come on people, I know this is hard but let's crack this one together". [There are probably better examples]. This gives students an excuse and kind of implies that they are one of those tired partying people and therefore not a "nerd" but someone with a "social life". As soon as some of the "cool kids" start participating it becomes "socially acceptable" and the general participation will be much higher.

Please don't think: "Well, if they think this way, I don't care if they learn something or not." Imho thats how social groups and social pressure works and educators should be aware of it.

So, don't think students are only afraid to be embarassed by being wrong, they are also afraid of being seen as the "uncool math nerd that has no friends". Change that!

  • I used a lot of quotes. I hope you undestand that I made some exagerations to get the point accross and that I don't share the mentioned views. – problemofficer Nov 8 at 20:45

Consider using the advice given by Catherine Attard in her 'Framework for Engagement with Mathematics'

Engagement with Mathematics: What Does It Mean and What Does It Look Like? Attard, Catherine

Australian Primary Mathematics Classroom, v17 n1 p9-13 2012

When discussing issues surrounding mathematics education, the topic of student engagement (or lack of) often dominates conversations. The low levels of engagement with mathematics experienced by students during the middle years have been of some concern to Australian mathematics educators and stakeholders in recent decades. Lowered engagement with mathematics has the potential to affect communities beyond the need to fill occupations that require the use of high level mathematics. It can also limit one's capacity to understand life experiences through a mathematical perspective. This article explores the concept of engagement against the backdrop of a recent longitudinal study into the influences on student engagement during the middle years of schooling, provides some insight into students' perceptions of engaging mathematics lessons and introduces a "framework for engagement with mathematics" that could be used to inform planning.

Full-text at https://eric.ed.gov/?id=EJ978128

a related article is at the following link: http://archive.uuworld.org/2003/03/interpretation.html

It might also be worth taking note of Steve Sigur’s remark that he preferred teaching high school students because they were still inquisitive, whereas college students have to be continually dragged along. Here is a link to the Youtube interview in which he makes that remark: https://www.youtube.com/watch?v=h256d1sfPwk

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    Please include a short summary of the paper you link to, so that your answer will still be useful even if the link expires. – Joel Reyes Noche Nov 9 at 4:00
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    I cannot see how the link to mathematical theology relates to your answer. – Joel Reyes Noche Nov 10 at 9:40
  • Such a perspective can support the notion of an interactive classroom, by giving the students a wider range of possible response. Remember, it’s dead-weight inertia that is the barrier that the teacher is trying to overcome. Once things get rolling – even if initially in the opposite direction (like a train that first moves a bit backwards before pulling out of the station, due to the great difference between static friction and rolling friction), the teacher can then direct the discussion appropriately. – EulerSpoiler Nov 10 at 16:12

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