Programs What Do Math Majors Wish They Knew Before Graduation?

[espen180]

This way of teaching (called the Moore Method) is closer to the actual practice of mathematics. Once you have had a course based on this method, it is hard to go back to the traditional method where the instructor lectures for an hour and you just sit there and passively try understand it or are furiously scribbling notes so you can read them and understand later

But how much is covered? Wouldn't the pace suffer with this kind of method?

 

[dcpo]

This way of teaching (called the Moore Method) is closer to the actual practice of mathematics. Once you have had a course based on this method, it is hard to go back to the traditional method where the instructor lectures for an hour and you just sit there and passively try understand it or are furiously scribbling notes so you can read them and understand later

I'm a bit sceptical of this method as you describe it. I can't see how anything like the same amount of material could be covered, and the process doesn't sound very close to the practice of real maths, at least not as I have experienced it. Maths is either fully collaborative, with two or more people in a room working collectively on a problem, or independent, with most research involving a combination of the two. Real mathematicians will also use textbooks and papers to learn a subject, and only exceptional people can prove all the results contained in them themselves and still have time left over for creating original work. It's not so far from the standard model where students are asked to read things and then solve problems based on the material. The key difference is that real mathematicians have to work out the broader structure of the work themselves, formulating the problem clearly, deciding on relevant definitions etc. but the Moore method doesn't seem address this difference either.

 

[dkotschessaa]

I like the idea of this method, since I think pace is often substituted for understanding in most math classes. However I also agree with the above two posters. The compromise would be that I would like to see such classes offered as a supplement to regular classes - kind of like a "math lab" one credit course.

 

[Skrew]

The best math class in my experience is a class where there was no book - in fact, we were forbidden to consult any book on the subject. The instructor did not lecture, and students took minimal notes. First day in class, instructor comes in, writes a few definitions and proves a couple of simple theorems. Then he writes a theorem on the board with no proof. Asks someone to prove it. The student then goes to the board and works out a proof. The rest of the class can not help him or her but can criticize the proof and find the gaps. Instructor only gets involved to clear up misconceptions, point out holes in the arguments by questioning the students in a socratic manner. If the student can't solve it he is invited to try during the next class and the class moves on to another problem. The first student now has "homework"

As the class progresses, the theorems and problems becomes harder and more substantial until we were proving pretty advanced stuff. Students really enjoy this as they are "creating" mathematics. You don't really need to take notes until the whole class has agreed on the solution that was presented on the board. Sometimes it takes two classes to solve something. You can spend hours working on that one thing in order to present it at the next class.

This way of teaching (called the Moore Method) is closer to the actual practice of mathematics. Once you have had a course based on this method, it is hard to go back to the traditional method where the instructor lectures for an hour and you just sit there and passively try understand it or are furiously scribbling notes so you can read them and understand later

This sounds horrible. I can't say I would ever want to be in a class like this.

 

[jk]

But how much is covered? Wouldn't the pace suffer with this kind of method?

It starts out a little bit slow but the pace quickly picks up. You probably cover less material than a traditional class but you understand it much more deeply.

 

[Robert1986]

I have been enamoured with the Moore Method ever since I read Halmos' "Automathography". It seems like it would be very cool. Also, it sort of ties in very nicely with The Mathematician's Lament. (have you read this essay? its very good)

 

[jk]

I'm a bit sceptical of this method as you describe it.

That is a typical reaction from people who have not actually sat in a class (or taught a class) in this manner. But it has been used successfully in several places for decades now.

I can't see how anything like the same amount of material could be covered, and the process doesn't sound very close to the practice of real maths, at least not as I have experienced it. Maths is either fully collaborative, with two or more people in a room working collectively on a problem, or independent, with most research involving a combination of the two.

The process is actually VERY collaborative. When someone is on the board writing out their proof, the entire class is involved in helping them sharpen their argument by pointing out gaps, etc. It was one of the most participatory classes in math that I have ever seen.

Real mathematicians will also use textbooks and papers to learn a subject, and only exceptional people can prove all the results contained in them themselves and still have time left over for creating original work. It's not so far from the standard model where students are asked to read things and then solve problems based on the material. The key difference is that real mathematicians have to work out the broader structure of the work themselves, formulating the problem clearly, deciding on relevant definitions etc. but the Moore method doesn't seem address this difference either.

The typical math class involves assigned reading, lectures and homework. The first two tend to be somewhat passive activities because you are following someone's chain of reasoning. So you don't struggle as much to understand something. To me, an hour spent listening to someone lecture about the Heine-Borel theorem, for instance, is less useful than the same hour spent trying to prove it. You end up working much harder and understanding the topic more deeply, even if you don't succeed in proving the theorem, than you would have just following along in a lecture. It also stays with you longer.

 

[Robert1986]

I'm a bit sceptical of this method as you describe it. I can't see how anything like the same amount of material could be covered, and the process doesn't sound very close to the practice of real maths, at least not as I have experienced it. Maths is either fully collaborative, with two or more people in a room working collectively on a problem, or independent, with most research involving a combination of the two. Real mathematicians will also use textbooks and papers to learn a subject, and only exceptional people can prove all the results contained in them themselves and still have time left over for creating original work. It's not so far from the standard model where students are asked to read things and then solve problems based on the material. The key difference is that real mathematicians have to work out the broader structure of the work themselves, formulating the problem clearly, deciding on relevant definitions etc. but the Moore method doesn't seem address this difference either.

I haven't taken a Moore Method class, but (and someone correct me if I'm wrong) part of the idea (as I understand it) is that the instructor is there to sort of walk you through the stuff. So, its not a bunch of undergrads (or grads) in a room all re-discovering calculus. It is an instructor asking very well-placed questions and then usually forcing the students to work it out.

 

[jk]

I like the idea of this method, since I think pace is often substituted for understanding in most math classes. However I also agree with the above two posters. The compromise would be that I would like to see such classes offered as a supplement to regular classes - kind of like a "math lab" one credit course.

I think often times instructors try to cram as much information into a class as possible but that is counterproductive. I understand this if you're teaching a more practically oriented course like say, diff eq for engineers where it's plug and chug. But for courses intended for math majors, the goal should be to equip the student to be able to reason things out for him/her self and be able to follow the literature on her own. Learning has two sides, the presentation and absorption of knowledge. The lecture format accomplishes the first part very well but not necessarily the second, particularly when there is a lot of material.
I think combining the traditional approach with the Moore approach in the way you suggest would not work very well for two reasons: 1) It would be hard to split the material into parts that are proved by the instructor and other parts by the students. If the instructor is solving things for you in the "main" class, the motivation to do it yourself in the lab section is gone. 2) In a Moore Method class, you end up working much harder than in a regular class, at least in my experience. I don't know if a lot of students would have the time for both approaches in the same class.

 

[jk]

This sounds horrible. I can't say I would ever want to be in a class like this.

There are people who dislike this way. There were a couple of people in the class who just wanted things cut and dried: read section X of Chapter Y, do problems 3,4,5 and you're done. This way was too strange for them but I still think they benefited from the course.

 

[jk]

I have been enamoured with the Moore Method ever since I read Halmos' "Automathography". It seems like it would be very cool. Also, it sort of ties in very nicely with The Mathematician's Lament. (have you read this essay? its very good)

I have read the essay. He captures in a beautiful way everything that is wrong with the way mathematics is taught. I think everyone who has an interest in the teaching of mathematics should read it: http://www.maa.org/devlin/LockhartsLament.pdf

 

[Robert1986]

I have read the essay. He captures in a beautiful way everything that is wrong with the way mathematics is taught. I think everyone who has an interest in the teaching of mathematics should read it: http://www.maa.org/devlin/LockhartsLament.pdf

Not to mention, it is hilarious. Were you going "Yes this is exactly how my high school life was!" 'cause I was!

 

[jk]

Not to mention, it is hilarious. Were you going "Yes this is exactly how my high school life was!" 'cause I was!

Even some college classes were taught in a similar spirit

 

[noobilly]

The best math class in my experience is a class where there was no book - in fact, we were forbidden to consult any book on the subject. The instructor did not lecture, and students took minimal notes. First day in class, instructor comes in, writes a few definitions and proves a couple of simple theorems. Then he writes a theorem on the board with no proof. Asks someone to prove it. The student then goes to the board and works out a proof. The rest of the class can not help him or her but can criticize the proof and find the gaps. Instructor only gets involved to clear up misconceptions, point out holes in the arguments by questioning the students in a socratic manner. If the student can't solve it he is invited to try during the next class and the class moves on to another problem. The first student now has "homework"

As the class progresses, the theorems and problems becomes harder and more substantial until we were proving pretty advanced stuff. Students really enjoy this as they are "creating" mathematics. You don't really need to take notes until the whole class has agreed on the solution that was presented on the board. Sometimes it takes two classes to solve something. You can spend hours working on that one thing in order to present it at the next class.

This way of teaching (called the Moore Method) is closer to the actual practice of mathematics. Once you have had a course based on this method, it is hard to go back to the traditional method where the instructor lectures for an hour and you just sit there and passively try understand it or are furiously scribbling notes so you can read them and understand later

It sounds pretty interesting, though perhaps not very suitable as a Calculus 1 class for all freshmen, it could be a good optional class.

This style of teaching actually sounds more like a law class where one student defends his opinion from the attack of other students. I had a legal class something like that once and quite enjoyed it. It's just that one never hears about a math class being taught that way.

 

[QuarkCharmer]

The best math class in my experience is a class where there was no book - in fact, we were forbidden to consult any book on the subject. The instructor did not lecture, and students took minimal notes. First day in class, instructor comes in, writes a few definitions and proves a couple of simple theorems. Then he writes a theorem on the board with no proof. Asks someone to prove it. The student then goes to the board and works out a proof. The rest of the class can not help him or her but can criticize the proof and find the gaps. Instructor only gets involved to clear up misconceptions, point out holes in the arguments by questioning the students in a socratic manner. If the student can't solve it he is invited to try during the next class and the class moves on to another problem. The first student now has "homework"

As the class progresses, the theorems and problems becomes harder and more substantial until we were proving pretty advanced stuff. Students really enjoy this as they are "creating" mathematics. You don't really need to take notes until the whole class has agreed on the solution that was presented on the board. Sometimes it takes two classes to solve something. You can spend hours working on that one thing in order to present it at the next class.

This way of teaching (called the Moore Method) is closer to the actual practice of mathematics. Once you have had a course based on this method, it is hard to go back to the traditional method where the instructor lectures for an hour and you just sit there and passively try understand it or are furiously scribbling notes so you can read them and understand later

That sounds amazing.

 

[member 392791]

Not to mention, it is hilarious. Were you going "Yes this is exactly how my high school life was!" 'cause I was!

I was laughing when reading the Geometry section. The little table you made with the steps as a proof hahahah...ours was even worse, step 1 always had to be ''given''.

 

[Sankaku]

I have had a class that tried to use some Moore-type presentations for part of it. I admire the general idea, but it takes a capable professor to guide the discussion and keep things on track. I can see that it wouldn't be the right thing for some people's teaching styles - and traditional lectures done properly can be beautiful in their own right.

http://en.wikipedia.org/wiki/Moore_method

After the endless "technique-driven" classes in early undergrad, more of this kind of thing would be great (if done well). If you learn to think and really understand the core ideas, then you can easily learn anything that you didn't have time to cover, later.

It is better to learn less material really well, and gain real insight, while you have a mentor available. At some point you will end up having to do all your learning on your own, anyway.

 

[jk]

I haven't taken a Moore Method class, but (and someone correct me if I'm wrong) part of the idea (as I understand it) is that the instructor is there to sort of walk you through the stuff. So, its not a bunch of undergrads (or grads) in a room all re-discovering calculus. It is an instructor asking very well-placed questions and then usually forcing the students to work it out.

That is correct. Putting a bunch of students in a room with no ground rules and guidance will probably result in disaster. The instructor's role is very critical in that he or she has to break up the material into manageable chunks, correct misguided notions, gently hint at possible lines of attack (as a lost resort) etc...but this should be done in a very subtle way. The instructors should let the students struggle even if it looks like time is being wasted while the class seemingly goes nowhere. He should act as a referee and settle disagreements when someone is being unreasonable (It does happen that there is usually someone in the class who is stubborn and strong-willed and will not acknowledge errors).

 

[jk]

I have had a class that tried to use some Moore-type presentations for part of it. I admire the general idea, but it takes a capable professor to guide the discussion and keep things on track. I can see that it wouldn't be the right thing for some people's teaching styles - and traditional lectures done properly can be beautiful in their own right.

http://en.wikipedia.org/wiki/Moore_method

After the endless "technique-driven" classes in early undergrad, more of this kind of thing would be great (if done well). If you learn to think and really understand the core ideas, then you can easily learn anything that you didn't have time to cover, later.

It is better to learn less material really well, and gain real insight, while you have a mentor available. At some point you will end up having to do all your learning on your own, anyway.

I think it takes greater skill and patience to teach a Moore Method class than a traditional class. It must be very hard to sit back and watch someone come up with a convoluted argument when there is a three line proof that will do the job.

 

[dkotschessaa]

I think combining the traditional approach with the Moore approach in the way you suggest would not work very well for two reasons: 1) It would be hard to split the material into parts that are proved by the instructor and other parts by the students. If the instructor is solving things for you in the "main" class, the motivation to do it yourself in the lab section is gone. 2) In a Moore Method class, you end up working much harder than in a regular class, at least in my experience. I don't know if a lot of students would have the time for both approaches in the same class.

I don't know. I don't see it any differently then a physics class where you are given a lecture and then a separate lab period. The motivation may just be to get it over with, but you end up learning quite a bit. I've always learned more in labs than in lectures - usually against my will! And the work *is* much harder, because you have to think on your feet.

It's likely many students would not like this, especially if they have no real interest in math.

-Dave K

 

[jk]

I don't know. I don't see it any differently then a physics class where you are given a lecture and then a separate lab period. The motivation may just be to get it over with, but you end up learning quite a bit. I've always learned more in labs than in lectures - usually against my will! And the work *is* much harder, because you have to think on your feet.

It's likely many students would not like this, especially if they have no real interest in math.

-Dave K

I have had those lab periods too and it is very different. In a Moore class, you are essentially rederiving everything while in a lab you are basically solving problems. The fact that you have no crutches (no textbook, no direct help from the instructor or students) means that you have enormous pressure to work it out yourself. It also means an increase in confidence that what you thought were hard or impossible problems are now within your reach. Sometimes it takes 2 or 3 tries at the board to get it right so it might mean that you finish a proof days later. You don't necessarily have to "think on your feet" because some of the material takes time. You also learn to refine your arguments and present it in the clearest fashion because you have the time in between classes to work on it.
There are people who do not like it. Some prefer very structured classes and do not like the open ended nature of it. Some have no interest in working so hard...they just want to get their grade and be done with it. But if you have an interest in learning mathematics, you should give it a try if you have a chance.

 

[dkotschessaa]

Well, I'm not sure such a thing is happening where I am, but I'll keep an eye out.

I am very happy though that a professor has invited me to study with him during a summer session where he will be mostly bored teaching a precalculus summer course, but available all day, so he wants to do something. I am stoked.

 

[dkotschessaa]

Any more adds to the original post would be cool, if anybody has any ideas. I'd love to see this as the "corollary" to mathwonk's thread. (I dream big, you see).

It's kind of a "if I knew then what I know now" sort of thing.

-Dave K

 

[20Tauri]

I am very happy though that a professor has invited me to study with him during a summer session where he will be mostly bored teaching a precalculus summer course, but available all day, so he wants to do something. I am stoked.

That sounds fantastic. I wish I had professors who wanted me to spend the summer hanging out and doing math.

 

[dkotschessaa]

Question RE: Analysis

I know inevitably I will take Analysis, and it will likely be the most difficult course in my major. What are the things I can do now as I go along to make sure I'm prepared by the time I get there?

Is it the "abstract-ness" of the course that makes it difficult? Or is it not having a solid grasp of calculus? Fortunately we have a course devoted to abstract mathematics, which so far is my favorite course. I do think I am pretty inclined towards abstract math above "calculation" type courses, which is good news.

-Dave K

 

[homeomorphic]

Is it the "abstract-ness" of the course that makes it difficult? Or is it not having a solid grasp of calculus?

I think people who get that far usually have a sufficient grasp of calculus, and part of what you gain from the class will be a deeper mastery of it, so that's part of what the class is for, rather than something you have to have coming into it. I'm not sure it's the abstractness, although that is part of it. The difficulty is conceptual and in the fact that you have to do serious proofs.

 

[ZombieFeynman]

I took Abstract Algebra and Analysis using the moore method. The pace is slower, but i find it to be a far better method for upper divisional proof based classes.

 

[dkotschessaa]

I took Abstract Algebra and Analysis using the moore method. The pace is slower, but i find it to be a far better method for upper divisional proof based classes.

Cool, but I doubt we have anything like that. I'd love it though.

-Dave K

 

[dustbin]

I took Abstract Algebra and Analysis using the moore method. The pace is slower, but i find it to be a far better method for upper divisional proof based classes.

Can I ask where this was at?

 

[dkotschessaa]

Can I ask where this was at?

GooglePedia(A word I just made up) actually had an answer for this one:

http://en.wikipedia.org/wiki/Moore_method#Current_usage_of_the_Moore_Method

 

[dkotschessaa]

"When a flaw appeared in a 'proof' everyone would patiently wait for the student at the board to 'patch it up.' If he could not, he would sit down. Moore would then ask the next student to try or if he thought the difficulty encountered was sufficiently interesting, he would save that theorem until next time and go on to the next unproved theorem (starting again at the bottom of the class)." (Jones 1977)

It sounds kind of wonderful and terrifying at the same time.

-Dave K