homeomorphic said:I think there's a place for less visual people in math, but there are large chunks of math that I don't see how anyone could make heads or tails of if they are geometrically challenged. Even if they can somehow manage, they'd be missing all the beauty of it. So, I don't think it's possible to be a broad or deep mathematician without being at least somewhat comfortable with visual arguments.
dkotschessaa said:What editor specifically?
Could you expand on this? I'm having trouble following what you're saying.ingenvector said:You do bring up a good point. Though I personally think it's largely useless, symbolic logic does not lend itself to visualisation readily unless the problem is already defined as a relational problem. It is also important to algorithms, though I don't think people usually visualise flow charts or trees beyond a certain size. What you are seeing in Set Theory I think is principally a similar syntax, but it doesn't use first-order or propositional logic and consigns itself to the more restrictive intuitionistic logic. It's what makes Set Theory so... intuitive.
dcpo said:Could you expand on this? I'm having trouble following what you're saying.
dcpo said:Could you expand on this? I'm having trouble following what you're saying.
Thanks. What do you mean by 'intuitionistic logic'? Do you mean it in the sense of Brouwer and Heyting?ingenvector said:Actually, I think I may know what you are asking after all. Let me backtrack since I was drunk with sleepiness when I wrote that post, as I am also now I suppose. But I digress...
In the context of visualisation, the most visual aspects of set theory are generally those which were first derived using intuitionist logic in naive set theory. Thus, the relations that the person was seeing in set theory that were visual were I think the intuitional syntactic structures of Set Theory which are also expressible in other logics. Naturally, nearly everything in mathematics is translatable, and first-order logic and the such is not exempt. I did not mean to imply that Set theory did not use first-order logic, indeed it can as in Zermelo–Fraenkel set theory, I merely meant that that was most likely not the visual component as this tends to delve into the equation classification over visualisation. I wrote rather sloppily, and I see now that it can be rather ambiguous and more open to interpretation than I had intended. I hope this answered your initial question.
Sankaku said:I agree heartily with this. However, I am biased since I am very visual, too.
Dave, don't give up on visualizing curves & surfaces in calculus just because it is hard to start with. It is worth the time you put in.
mathwonk said:Taking notes is maybe relevant or not depending on how the prof teaches. Some profs just come in and copy the same stuff onto the board that is in the book. In those classes there is no need to take notes at all. The right way to treat that situation is to read the book in advance, and make notes on things that are puzzling and then use class for repetition (it never hurts to hear something hard twice) and to ask questions.
In other classes the prof ignores the book, leaving it as auxiliary reading and presents a complete self contained course of his own devising on the board. If you don't take notes there you may not know what was in the course afterwards. All Harvard courses were like this in my day as an undergrad there. There was also a book that one was advised to read for an alternate version of the material.
dkotschessaa said:Man, so I just discovered the joys of unlined paper...
bpatrick said:Haha. I almost included that in my previous post when I was talking about rolling with the clipboard + recycled (printed on one side + discarded) paper combo.
I haven't used lined paper since 2003 and have no intention of going back.
I concur with what mathwonk said above, and why I hardly ever take notes in class. It's so much better to read ahead so you know what is going to happen in lecture, then listen closely during the lecture rather than mindlessly copying stuff down. Keep some paper around for working examples along with the class or taking note of something that the prof may do that is super profound or gives a different take on the material / technique / whatever.
In case you missed it in my previous post:
A Mathematician's Survival Guide:
Graduate School and Early Career Development
Steven G. Krantz
Publication Year: 2003
ISBN-10: 0-8218-3455-X
ISBN-13: 978-0-8218-3455-8
well worth the read if you're seriously thinking about a career in math.
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"mathwonk said:Taking notes is maybe relevant or not depending on how the prof teaches. Some profs just come in and copy the same stuff onto the board that is in the book. In those classes there is no need to take notes at all. The right way to treat that situation is to read the book in advance, and make notes on things that are puzzling and then use class for repetition (it never hurts to hear something hard twice) and to ask questions.
In other classes the prof ignores the book, leaving it as auxiliary reading and presents a complete self contained course of his own devising on the board. If you don't take notes there you may not know what was in the course afterwards. All Harvard courses were like this in my day as an undergrad there. There was also a book that one was advised to read for an alternate version of the material.
In such a situation, one cannot fully benefit from taking notes unless one goes over them faithfully every night after class, filling details, thinking through the arguments and making remarks about things to raise in class or in office hours later.
A very few professors write out a pristine textbook version of the material on the board word for word, and expect you to memorize it and regurgitate it back on the exam. I had one such professor. Although I learned very little math from him, I did learn how to get a guaranteed A.
...
jk said: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.jk said: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
jk said: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
espen180 said:But how much is covered? Wouldn't the pace suffer with this kind of method?
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.dcpo said:I'm a bit sceptical of this method as you describe it.
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.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 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.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.
dcpo said: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 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.dkotschessaa said: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.
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.Skrew said:This sounds horrible. I can't say I would ever want to be in a class like this.
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.pdfRobert1986 said: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 said: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 said:Not to mention, it is hilarious. Were you going "Yes this is exactly how my high school life was!" 'cause I was!
jk said: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
jk said: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
Robert1986 said:Not to mention, it is hilarious. Were you going "Yes this is exactly how my high school life was!" 'cause I was!
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).Robert1986 said: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.
Sankaku said: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 said: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.