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Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, i.e. professors lecture to students and sketch some proofs. Suffice it to say I'm not a fan of the current methods and I don't think it would be too far from the truth to say that I do all the actual learning outside the classroom. Has anyone tried anything different and seen any difference in student understanding and comprehension in graduate or undergraduate courses?

Some background motivation: I'm a TA and my current method of doing things is to just write some problems on the board and then go through their solutions. This is fine and it's what the students expect but sometimes I feel guilty because I'm just teaching them problem/solution patterns and reinforcing all the bad stereotypes about what mathematics is instead of showing them the underlying conceptual tapestry and helping them rethink their attitudes toward mathematics. It's kinda like the old saying “Give a man a fish; you have fed him for today. Teach a man to fish; and you have fed him for a lifetime”. So basically I throw a bunch of fish at the students hoping it will feed them for the semester.

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I highly recommmend the video to everyone. The irony here is that Mazur, who is arguing for the inefficacy of traditional "one-way" lecturing, is an absolutely mesmerizing lecturer. –  Pete L. Clark Dec 31 '09 at 23:37
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Well certainly he's an excellent speaker and that is one of the reasons he mentions that forces him to rethink his methods. Being a likeable lecturer does not mean your students are learning. –  davidk01 Jan 1 '10 at 0:56
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+1 for the link to the Mazur video. It deserves all the good reputation it has. I've had to spend many hours in teaching conferences being lectured on how not to lecture, and I wish they had all been half as interesting as watching Eric Mazur on a tiny screen... –  Thierry Zell Aug 10 '11 at 0:55
    
Mazur has a book, Peer Instruction: A User's Manual. IMO the book is less valuable for the specific technique it prescribes than for the summary of pedagogical research that it presents at the beginning. The research shows that traditional lecturing doesn't work in physics, and that the methods that do work are those that actively engage the students. There are many of these "active engagement" methods, and the research shows that they're all roughly equally successful. As a TA, unfortunately, David is in a weak position to institute a type of reform that is typically unpopular with students. –  Ben Crowell Oct 29 '12 at 15:39
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The topic you touch upon is vast, but I wanted to comment on this phrase: "problem/solution patterns which is very different from showing them the underlying conceptual tapestry".

If for some reason you have to use this format (department restrictions or whatnot) choosing your problems well will simulataneously introduce some of the conceptual tapestry. Rather than introducing a mathematical tool and then the problem that goes with it, you introduce the problem first (just out of range of the student ability) and bring it to the point where things get stuck, where something new is needed to go further. Then the motivation is clear for the new tool.

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Mazur is a fine speaker, but he fails to mention an important thing which importance is underestimated - teaching good students. Students who are smart, interested and willing to learn can be just as easily harmed by bad teaching (or helped by proper teaching, for that matter). Sadly, many people seem to think that teaching good students doesn't require much attention or conceptual thought, since they would learn the subject by themselves anyway. Quite the contrary, excellent students require more attention precisely because they are more capable, so there is more potential to be tapped by a good teacher.

Teaching gifted students is, IMHO, more difficult than teaching mediocre ones, but also more challenging, since it's difficult to make general statements and it's not obvious what the right approach is (in the case described by Mazur, we more or less know the "right" way of doing things - understanding instead of memorizing, thinking vs mindless application fo recipes etc.).

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Drawing such distinctions among the students misses the point about my question. The bell curve is everywhere so there will always be smart and not so smart students but the point is that certain methods reach more students and are more conducive to what the teacher is trying to accomplish. Also, I don't see what difficulties you mean when you mention gifted students. –  davidk01 Jan 1 '10 at 1:12
    
By the time you're teaching upper-level courses, the distribution is much more skewed toward the top in both motivation and intelligence. –  Harry Gindi Jan 1 '10 at 1:26
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@Harry: The problem for Mazur is that these are smart and academically proficient kids, but they're not future physicists, and they will be only too happy to go through the class without learning anything if given half the chance. But it's the instructor duty in that case to make sure they learn to appreciate the subject and its relevance to their future career. –  Thierry Zell Aug 10 '11 at 1:12
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Have you heard about this before: http://en.wikipedia.org/wiki/Moore_method ?

I think it's pretty different to the ordinary teaching method, and might be one possibility that differs significantly from the standard teaching method. Basically, it seems like the students are given the basic theorems, axioms and such, and then asked to prove these theorems for themselves given the axioms, and construct examples for these axioms to familiarize themselves with it. One thing I don't understand with this however, is how one manages to get through the course and cover a sufficient amount of content with this method (given that it naturally seems to take up a longer amount of time).

I think this could also apply to the way in which you might self-learn things from a book - perhaps instead of just reading the book, you could try proving the shorter theorems by yourself (instead of just reading their proofs), and get a sketch of how it's done for the longer theorems by a quick glance and fill in the details for yourself.

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I was taught point set topology using a modified version of the Moore method (we were allowed to discuss the problems and figure out solutions working together if we wished). It is true that you cover a little less material, but in my opinion you teach the students way more about mathematics. I have to say it was possibly the most rewarding class I have ever taken. My impression is that you have to be careful to select the material well. I've heard that it also works well for analysis. Point set topology is the perfect subject for this method because it is so "bottom up". –  Chris Schommer-Pries Dec 31 '09 at 12:31
    
There is a recent book from the MAA titled "The Moore method : a pathway to learner-centered instruction" that describes five teachers' variations on the Moore method. It may be helpful to look through it for ideas. –  Justin DeVries Dec 31 '09 at 17:01
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We were taught point-set topology and elementary group theory exactly in this way. I can't see learning them in any other way. It took a little while to understand the axiomatic approach, but I now understand it in any context instead of having to relearn it for every subject. –  Harry Gindi Dec 31 '09 at 20:28
    
What I've been told about the Moore method is that it is only intended for students with no prior background in the subject, which would never have worked for me, personally - I like to read ahead a lot and have, so far, never entered an undergraduate course with no prior background. Can anyone corroborate this? –  Qiaochu Yuan Jan 1 '10 at 1:50
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@Qiaochu: Moore used to demand from students that they swear to keep away from the library; at least one student who did not keep his word was kicked out of class permanently. But current practitioners of the method use a modified version, and I suspect (though I don't know for sure) that this is one of the conditions that gets most often relaxed. –  Thierry Zell Aug 10 '11 at 1:00
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I've seen Mazur speak on this and he says a fair number of his students don't like it. Some of their reasons for disliking it apply more to most other institutions than to Harvard. If they've always been taught that learning mathematics consists of memorizing algorithms to apply to assigned problems, they may regard a more intelligent approach as grounds for complaint, especially if the reason they're there is to get an "A+" in the course so that they can forget about it and get admitted to medical school (admission to which requires calculus or the like, not because students need to know that subject, but because they need to have demonstrated that the can succeed in courses that are challenging).

So there's some danger of being punished for doing the right thing. That doesn't mean you shouldn't do it, but it might affect the way in which you do it.

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You may also wish to look into the book How to Teach Mathematics by Steven G. Krantz. Perhaps Section 3.10 on the teaching reform and the references cited there could provide a reasonably good starting point for addressing your question.

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Most books by Krantz are very good but this is one I don't have in my personal library. I'll certainly look into it. –  davidk01 Jan 1 '10 at 1:24
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I just watched the video -- I didn't know of it and it is certainly very interesting and provides much food for thought.

From the perspective of someone who teaches, I could relate to parts of it: particularly the decreasing ability to understand student's questions. From the perspective of someone who was a student, though, I do remember learning during lectures: perhaps not in all courses, but certainly in some which I still remember fondly. These courses were usually lecture-based; although in one case the lecturer would ask questions to the students all the time: picking a starting person and then moving systematically along the audience. This used to instill the "fear of God" in some people, but it meant one had to be on top of the material. I enjoyed that and, in fact, it boosted my confidence.

Now to answer the question, in the University of Edinburgh (where I am based) we started a few years ago to teach some of the introductory courses incorporating some element of Peer Instruction. I personally have not taught introductory courses for a while, so I cannot say how this is panning out. The School of Physics (I'm in Maths) has been teaching the first-year introductory course using Peer Instruction for some time now and they seem to be very happy with the result.

I wonder whether some variant of this method can work for final-year courses, though.

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Reading things ahead of time in the upper-division courses is quite helpful but for some reason professors tend to be very secretive with their lecture notes and lesson plans so even if some students do try to be more on top of things it seems that the professors try to discourage it. It might not be their intention but it certainly feels that way. –  davidk01 Jan 1 '10 at 1:31
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Certainly not in my case. For some unknown reason in the UK it is not thought to be fair on the student to make them buy a book, so we are encouraged to produce our own lecture notes and, at least in Edinburgh, we tend to make them avaialble to the students ahead of the lectures. The first time one teaches a course there might be some delay in producing the notes, but the second time around the notes are available to the students in their entirety. How to motivate students to read the material before the lecture is another matter. That I'd like to learn how to do. –  José Figueroa-O'Farrill Jan 1 '10 at 4:04
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I've been to a couple of courses where the lecturers asked students to prove many fundamental theorems and examples as exercises (eg Ravi Vakil's algebraic geometry, you can see this in his online notes http://math.stanford.edu/~vakil/0708-216/ ). I know some students find it very frustrating, and for me this style is time-consuming and -inflexible (as in, if I don't try the questions straight after the lecture, I can't understand the next one), but I came away with a much deeper understanding than I usually get from courses. (I haven't heard of the Moore method before, but this sounds like something related.)

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This very stimulating presentation on teaching math effectively came out in May 2010 and addresses some of the issues presented in the other answers:

Dan Meyer: Math class needs a makeover

Although applied to high school math, at least some aspects of the technique could be incorporated, if only in a few sessions, into advanced classes. Comments to the video by educators and students provide some feedback on the technique.

Another potential method for revamping math classes for the 21'st century (maybe start viewing at time stamp 6:50):

Salman Khan: Let's use video to reinvent education

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You might like to see the argument in

What should be the context of an adequate specialist undergraduate education in mathematics?, Ronnie Brown and Tim Porter, The De Morgan Journal 2 no. 1, (2012) 411--67. http://education.lms.ac.uk/2012/04/r-brown-and-t-porter-what-should-be-the-context-of-an-adequate-specialist-undergraduate-education-in-mathematics/

and the comments that followed.

Here we argue against the concentration on content, without any discussion about mathematics. For more discussion on such teaching questions, see also my web page http://pages.bangor.ac.uk/~mas010/publar.html

Some of our students rated our course on "Mathematics in context" as the best of our courses. Students had to write a project under the terms of that title; we were totally surprised by the initiative of the students and the variety of topics chosen; some also said that the writing helped them to come to terms with their own attitude towards mathematics.

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I found this on a search for the "Moore Method", used for point-set topology at the University of Washington in the late 70's. Does anyone know where to find the (then mimeographed) hand-outs? As I recall, there were less than 20 pages of definitions, axioms, lemmas, and theorems. A beautiful thing. There is much on the Moore Method floating around, and yet this concrete demonstration is nowhere to be found. Help!

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Who was your instructor? The AIBL gathered a lot of IBL info over time - inquirybasedlearning.org/AIBL/Home.html –  François G. Dorais Aug 10 '11 at 1:02
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