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Methods of Teaching Mathematics

Where is mathematics learning going today? There are many books nowadays, that emphasis conciseness and rigour over all else. The Rudin "series" is a perfect example. There is hardly any motivation, and emphasis is put on rigour, rather than intuition. I am sure that this could be argued away as being a valuable tool for training "real" mathematicians, but how effective is this approach, really? Has the use of such learning tools really produced a new generation of brilliant mathematicians? What was it that produced the previous generation of great mathematicians? How would the masters have responded to this change in the teaching of mathematics? I am not trying to make a point here, just curious as to how much success these approaches have achieved.

An interesting article to read if you have the time, by the esteemed V. I. Arnold http://pauli.uni-muenster.de/~munsteg/arnold.html

Seems to me as if the Soviets were on top of things with regards to teaching. (Kolmogorov and such)
 
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Well books that are concise can be good or they can be bad, it depends on how the material is presented and the wording that is used. I've read a theorom in one book and it's taken me a minute or two to understand it, then I go to another book and the author words it completely different and I fully understand it right away. Both books were concise, but one was just written better, or maybe it just happened that I understood one of the explanations better than the other. It's hard to say.

Generally though, I'd rather have a book that gets to the point, states a theorom or a definition, gives a proof of it, and then shows how to apply it to the problems. One gripe I have is alot of books don't provide motivation behind the proofs, this is something that I guess can't always be done maybe? I remember a certain proof of a method used to solve nonhomogenous differential equations where the author gave no motivation behind the proof. I asked my instructor and he said we just have to assume those conditions and it ends up working itself out heh.

As for teaching mathematics, I think teachers play a big role in this. A good teacher can make a big difference in someones view about mathematics in general. I remember taking a certain math course where I could not understand my teacher at all. He was a nice guy but I just could not follow his teaching style. I thought to myself what would have happened if this was the FIRST math class I had ever taken, how would I have viewed mathematics in general? I probably would have hated it.

If I don't like a book, I just go to another one. I still try to look for conciseness more than anything though. There is nothing worse than reading 3 paragraphs for an explanation of something that could be written in one or two lines followed by a short example. For example I have a book that takes 15 pages to talk about something and another takes 45 pages to discuss the same thing! Of course the one with 45 pages has more examples, but that does not mean it is better, it's actually alot worse, the examples are not well explained and the proofs are incredibly difficult to understand because no motivation is provided for them.

I think books that are rigorous are good because what you learn sticks with you because you had to spend more time learning/reading it. I personally prefer a well written, easy to read, concise book, over a very rigourous one. It just depends on the mood I am in.
 
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MalleusScientiarum

In my experience as a physics student, mathematicians don't teach intuition because trying to use pure intuition can create problems when you're dealing with a mathematical situation. It almost feels like mathematicians sit around coming up with really weird counterexamples to otherwise perfectly intuitive proofs just to spite me (at least that's how I feel). I have had the benefit of having a few very good math professors that have given me a sort of intuition about reading proofs, and translating proofs into actual situations.

One good thing to do is try to find good math textbooks. I personally think that the Bartle Elements of Real Analysis and the Brown & Churchill Complex Analysis books are very good at grounding some really odd theorems into reality. Books like that are rare; it's much easier to just write down a series of theorems, lemmas, postulates and proofs; but they are worth the search.
 

quasar987

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Eratosthenes said:
As for teaching mathematics, I think teachers play a big role in this. A good teacher can make a big difference in someones view about mathematics in general.
This is all the teachers are worth imo. I don't need them to tell me exactly what the book says and copy the proof of every theorem on the blackboard. I got that in the book, written better, and with no mistake every 2 lines. On the other hand, those bits of info that a teacher might leak from time to time are priceless. These are great because when a doctor talks about math, he's talking about it very generally, i.e. in terms of the big picture. And that's beautiful. It's something we the students can't do, and something most books don't do. A fabulous exemple of this, I think, is how mathwonk starts his book on linear algebra: "Linear algebra is the study of linear spaces and linear maps between them." Ask any student of a linear algebra class what linear algebra is about and he'll say something to the effect of "Well there's matrices, and those helps solve systems of equations. Then there's vectors spaces. Those are sets of objects satisfying specific axioms. Yeah that's it." I like to read mathwonk's posts (just to name one) in the "Tensors and differential geometry" section even though I have no idea what he's talking about; even though I'm not learning anything concrete, I feel like I'm learning something about the essence of mathematics itself.


Btw, my ideal view of a mathematics class would be a teacher just answering any questions the students might have about the stuff they read before the class.

And about books, I think both the "non-rigourous but intuitive" type and the "rigourous" type lack efficiency if used alone, but powerful when used together. I.e. use the rigorous book to see what the theorem IS, and use the intuitive book to see what that MEANS.
 

quasar987

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As for the physics teachers, what they must do to make their classes better (worthwhile!), is, ok, first explain the naive texbook exemples that we find in every textbooks, and then tell us how closely this naive exemple relates to real life phenomena, and exemples of when it happens.
 

MalleusScientiarum

I'm of the opinion that any physics undergraduate program with any research department attached to it should require its undergraduates to complete at least a year of actual research with a professor. That would take care of the "real life phenomena" and would also give undergrads a chance to see if they like what physics actually is.
 

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