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46

It depends on how much time I can afford to spend on the problem. I check, as you did, whether I see a mistake. If not, I try to explain the math in a different way: a different conceptual approach, which is good for everyone anyway; perhaps replace the variables with numerical values, if appropriate; or replace a general function $f$ by a specific ...


44

In my view, mathematics education is not about the authority of the teacher, it's about developing the students as their own mathematical authorities. Ultimately, the authority in mathematics is that any one of us can use what we have learned, even via different methods, to verify what other people have done. In mathematics classrooms, we do not learn only ...


38

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 ...


32

Whether one can or can not double the brilliant in real life has nothing to do with the Banach-Tarski paradox. Various mathematical objects are models of various aspects of our universe. $\mathbb R^3$ is typically taken as a model of three-dimensional space. But, it is only a model. Some things that are true in the model are false in space and some that are ...


31

Edit (Nov 2015): I feel it would be disingenuous not to mention that my views on this matter have evolved since posting my original answer, which remains un-edited, below. I suppose the provided answer could be contextualized specifically in the setting of a mathematics course for undergraduate mathematics majors that is primarily lecture-based. If you are a ...


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?...


26

The Moore Method is alive and well, and so are a great many variants. These days the community is more likely to use the term Inquiry-Based Learning (IBL), because the Moore Method can be seen as a restrictive set of practices, and people use the same underlying ideas in a lot of different class structures. If you want to learn more and meet people who are ...


24

A blanket problem I've observed over-and-over is that the deficits that scuttle calculus students are even more fundamental than what is discussed in (typical) pre-calculus courses. Specifically, kids, as well as adults returning to school, often cannot do middle-school (pre-?) algebra. Either they get stuck and confused, or immediately and frequently commit ...


22

The Moore method is used at the University of Chicago in some sections of "Honors Calculus", which is really an introductory real analysis course for top incoming freshmen. I assisted with it a couple of times and taught it on my own once. It absolutely depends on having a well-constructed sequence of notes to use; we started with a truly excellent set of ...


21

I have had this problem before with students who always think they're right. If a student continues to insist you made a mistake, when you know that you haven't, then tell the student to hold the thought and ask them to discuss it with you after class. Once the class is over, write the original problem on the board and ask the student to solve it. If ...


20

I agree with @kathleen, that being a young (and female) instructor can lead some students to be less respectful. In this case, we can use that to our advantage... Long ago, when I was less secure as a teacher, I would be so embarrassed and bothered by making a mistake that it would decrease my ability to teach well for the rest of that class period. I knew ...


19

You should avoid a "I am smarter than you are" war with the student, the teacher must be above that. (You're already avoiding it, I am saying so for the benefit of the Internet.) In Japan you could just smile mysteriously, but I suspect we're talking about the Western world. The first and most important thing to resolve is this: Is your student right? If ...


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 ...


18

Does this student really already know everything? I ask because of an experience I had many years ago, teaching a "semi-honors" second-semester calculus class in the first (i.e., fall) semester. So the students were there because they had already done some calculus in high school and had done well on a placement test. The first topic of the semester was ...


17

This explanation from http://www.kuro5hin.org/story/2003/5/23/134430/275 might be useful: One important difference between S and a real, physical sphere is that S is infinitely divisible. Mathematically speaking, S contains an infinite number of points. This is not true of a physical sphere, as there are a finite number of atoms in any given physical ...


17

I don't think it's fair to tell the students to not read ahead. In fact, I'd encourage it. What you should make clear is that the material in the current section takes precedence, and if somebody else has a question on the current material you'll sadly have to help those people before the questions on the next section. If a weaker student is raising their ...


16

To approach this situation I would recommend you to imagine the same situation but that it is not kissing that is happening, but something else, say, two students were playing cards; so something that would also not cause any noise, but prevent the two from paying attention, and might also be disruptive to others in the class. Perhaps, in this case you ...


16

I took a number theory class at University of Cincinnati that was taught using a modified Moore method. The class size was pretty small, and credit-wise it was an "upper level mathematics elective" (i.e. not a core class), so our professor had some room to be experimental. Like with the Moore method, the students presented all course material. The ...


16

Well, at least in cases 1 and 3, I usually say (in a serious tone): "Well, I did it on purpose, to see if anybody is actually paying attention to what I'm saying". It usually takes the students about 3 times (sometimes over a couple of weeks) until they realize it's a joke. Then, I usually explain to them that it's normal, and in the process of learning ...


15

In some cases (e.g., during an open-book exam), you can opt for no-questions policy. However, I think that case-by-case is the best option. Each time a student asks a question, what I ask myself is (with some stretch, you could call this a “general rule”), $$\text{Would the answer give the student unfair advantage?}$$ Such an approach might mean that ...


14

There are a few reasonable approaches, and they vary mostly in (a) intrusiveness towards the students and (b) effectiveness Walk intently in the direction of the student, cough, and stare at them for a minute. From the perspective of the "potentially cheating student", this is quite effective. There's no way they'll pull any funny business while you're ...


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 ...


14

Since dtldarek's answer addresses well the issues of fairness to students, I'll mention another consideration. When writing exam questions, I try to make sure each question has a certain intent, that it probes the student's knowledge of a certain concept/definition/technique (or some combination). When a student asks a question during the exam, I have these ...


14

If you are worried about fairness, I'd offer the answer to any students' questions to the whole class, whether announced to the class verbally, or written down on the board.


14

Matter is no bounded set of points with a non-empty interior The sets in the Banach–Tarski theorem have to fulfill some requirements: They need to be bounded. They have to have a non-empty interior. They need to be, well, sets of points. In particular their elements are identical, i.e., do not contain any other information than position. No matter how you ...


14

Taking some extra care to verify the details of a confusing bit of material (preferably with class involvement, intuition-building examples, etc.) is usually an excellent use of class time. Offering to talk about the issue further after class or during office hours is also a reasonable tactic when time is short or when dealing with a particularly insistent ...


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