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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 ...
34
Every student question should be treated like a gift. It gives insight into student's thinking. Even a disrepectful question is an opportunity for you to teach, except that in such a case you don't teach math. There is no such thing as a "only math" teacher. Every teacher teaches life just by standing in front of the students.
I would say it is actually ...
32
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 ...
32
The think-pair-share technique is an oldie but a goodie:
Pose a question
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)
Give students 2 minutes to exchange / compare solutions with a neighbor
Ask for volunteers to share results with ...
27
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?...
18
Once upon a time I was one of those brighter students. Two things helped.
One was the opportunity to help other people. I got started tutoring mathematics when people would come to me for help with their homework, and I learned that the effort to explain a concept helped me understand it better. A couple of times, I found that I didn't understand a ...
15
I took an excellent set of related rates problems from Jim Belk's webpage and turned it into a nice worksheet. I spun the problems this way, to motivate them:
We have now covered related rates. This means you now have an extra power that works in all your science classes. Anytime anyone gives you an equation -- a fact -- that is true about the world, you ...
14
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 ...
14
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 ...
12
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 ...
11
There's already very nice answer of Andrej Bauer, but I would like to view the question from a slightly different perspective. Perhaps one should not call questions silly, but there are questions which we wouldn't want to answer, the main reason usually being that it would not be the best response. To name a few concrete examples:
As a teacher we have a ...
11
I've had a remarkable (remarkable to me) success in a different environment on a different topic, so I am not sure this will translate. But I'll mention in anyway.
With one change, the same quiet "labs" turned into such lively conversations that I now have to raise my voice above the din to interrupt them with instructions. The change was from allowing ...
10
One recommendation that I have also given in response to similar questions, check out Jo Boaler's edX course: How to Learn Math for Parents and Teachers. It's currently in progress, but I'm sure it will run again. (I don't know if you can join while it's going on).
Also, the book Powerful Learning: What We Know About Teaching for Understanding has a chapter ...
10
I strongly recommend consulting with the department AND Course Instructors before trying new interactive protocols. In one of my first Teaching Assistant jobs, I tried adding a quotation or factoid at the beginning of the session for the students to look at (and think about) if they wanted. I did some other things too.
My instructor sent out a midterm ...
10
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 ...
10
You said that most of your students are first year college students. Such students often lack basic academic skills. Perhaps they were smart enough in high school to get away with not taking notes or studying, perhaps they had teachers that provided too much scaffolding (and never took it away), or perhaps the standards were just lower. There are a lot of ...
9
In any case, you should invite the student to a little chat at your office.
You should probably tell him about your concern and ask him if there is something you can do to please him and to give him input for his motivation. In worst case, the student appreciate your honesty and gets a compliment how good he is and that it is a good way to be enthusiastic.
...
9
Differentiation is one of the hardest parts of teaching, for sure. One thing that has helped me a lot is reframing my questions so that they are more "open."
For instance, instead of asking my students to factor a quadratic, I'll ask them "Say that I've got $x^2 + bx + 9$. What can $b$ be if this expression is factorable?"
What's great about more open ...
9
I will try, though I am a biologist.
The main difference between your question and the one linked in the comments seems to be that you want students to realize that articulating mathematics leads to better learning of mathematics. Not just being willing to answer questions, but talking.
I would recommend a technique that can be done with either clickers or ...
9
Start calling on students individually without waiting for any one to raise their hand. For a long time I never did this since I am conflict avoidant and since I didn’t want to put students on the spot, but then I witnessed another instructor masterfully call on every student throughout the course of one class period. The key point to add is that when a ...
8
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 ...
8
This is a good question, and I hope my answer is just the first among other (likely better!) ones. [I'll try not to rant, but I'm not making any promises.]
Let me start with a few general remarks.
This is a good question, and the fact that you are thinking about teaching in this way is (obviously) a very good sign. Keep reflecting, keep asking, and keep ...
7
Two suggestions:
The first is to try clickers. They work like the voting system in "who wants to be a millionaire?". You can activate students and foster discussions on questions. There is no need for clicker hardware as long as your students have internet access via tablet, smartphone, etc. see PINGO software.
The second suggestion is flipped classroom. ...
7
Talk to them, make them see that you know they know the answers, but that you want to ensure that other (slower) students think it over by themselves before giving the answer. This has done the trick the times I had such a problem. Make a point of asking others first (sometimes/most of the time), then give them the opportunity to speak up. Perhaps ask just ...
7
For off-topic questions or just questions that we can't devote the time to answer right away, I have a devoted 'question wall'. The students have access to post-its and use them to post any question to the wall -- silly, content-focused, it doesn't matter to me. The result is if a student interrupts with a silly question I can quickly reply "Put it on a post-...
7
In my experience, everyone participates in a class of 10. By the time you get to 20, there's maybe a couple of students in the back who aren't really engaged, and class of 30 has a noticeable contingent of disengaged students. I haven't taught any classes in the 40 - 70 range, but a class of 80 really feels like a large lecture, with only a few students in ...
7
I'm not a natural extrovert, but I've found that this is less about social skills than about establishing clear expectations. For example, if I put up a question on the projector that says "Discussion question: ..." and tell the class we're going to discuss it, there's no real uncertainty about what is expected. On the other hand, when I observe other ...
7
I've taught about 100 biology discussions and get good interaction evaluations; you can decide if this advice applies to you and calculus.
I am making some assumptions about your teaching environment:
You have 10 - 30 students in the room
Students had lecture elsewhere, so you don't need to lecture
Nobody is telling you what you have to do in discussion (e....
6
My mother tells an excellent story from when I was learning to read. She finds that I am reading very quickly considering my age. After asking me some questions, she discovers that I thought "reading" meant "looking at and understanding each word on the page." I was able to do that very quickly!
Your students sound like they are doing the same thing -- ...
6
In some of my graduate courses, my professors tried to get a 'post your homework solutions and other thoughts' page going. It didn't work, which is surprising considering we are graduate students massively interested in the topic. Here are my thoughts in no particular order:
Pros:
Encourages students to practice mathematical typesetting (latex, etc)
Helps ...
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