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[Abhijith]

Abhijith

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This is an essay written by GH Hardy in 1940. Hardy was a mathematician of great repute and a mentor of Srinivasa Ramanujan.
In this essay the author discusses about the "aesthetics of mathematics".
It is a perspective of mathematics from a pure mathematician as against that of a appied mathematician.He tries to justify the existance of a tribe which pursues mathematics for its own sake.

A pdf version of the essay is available here www.maths.bris.ac.uk/~hb0262/Hardy.pdf.

[sri]

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Thanks Abhijith. I've read this book and would recommend it myself!

Er.. just wondering whether this is somehow related to the "abstract versus concrete" debate we had earlier..

Anyway, the question there was not about whether abstract mathematicians should exist. Of course they should exist; there is no question there I would say.. The question is more about which technique is more effective in teaching mathematics to students. My contention is that a purely abstract start for mathematics education may end up with the student making all the wrong or unintended interpretations for the abstractions (like matrices being an abstraction for parallel computation), causing a lot of confusion.

Keep posting. We enjoy your ideas!

Cheers
-Sri

[Abhijith]

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Though my attention to A Mathematician's Apology was drawn by the debate we had, the earlier post was not "Abhijith's Apology for the debate" .(The word apology used in the same context as used in the book  ).

 I agree with you and many others that examples are necessary in teaching mathematics or any other subject.But if you look carefully at the arguments posted by many in that debate , many almost have made an implicit assumption that a subject exists because of the examples.I seek to challenge that assumption.

My only grouse is that we in India rarely learn the underlying philosophy of the subjects.This leads to a kind of shallowness (if I may use the word) in thought thereby restricting us in doing good research.We fail to ask the really BIG questions that matter becuase we rarely understand the philosophy.I am not sure which is the best way to bring that in to our education system.
Any ideas?

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My only grouse is that we in India rarely learn the underlying philosophy of the subjects.This leads to a kind of shallowness (if I may use the word) in thought thereby restricting us in doing good research.We fail to ask the really BIG questions that matter becuase we rarely understand the philosophy.I am not sure which is the best way to bring that in to our education system.
Any ideas?

Interestingly, I have the exact same grouse, but come to almost exactly the opposite conclusion . What I see the reason for this shallowness is the "bookish" nature of our education system based on rote learning, memorization and symbolic knowledge without relevance to reality. We have an over-reliance on labels and notations than the real meaning behind what is being taught.

Reminds me of how our teacher taught us Karnaugh maps in engineering: "So why can't we pair two 1s on a diagonal to reduce them? Because there is no rule that says that you can pair two 1s on a diagonal.." Or like the case when I ask a doubt to our statistics teacher (in engineering again), she thinks for a while and says, "Don't worry, they won't ask such questions in the exam."

And the other day a student was asking me how to find "NP hard" problems in real life!! Just about everything we do from bus scheduling, time table planning, recognizing objects, driving, etc. are all NP hard problems. Somehow we don't see that at all.

Cheers
Sri

[Abhijith]

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When it comes to rote learning memorization etc I agree that we are the champs  . For example I shall recreate a conversation I had with one of my professor during my Bachelors.His name is Dr Shankar.
Dr Shankar : Abhijith, I was wondering what is multiplication? I just dont understand it.
Abhijith : Sir!! Of course it is repeated addition.
Dr Shankar : Are you sure?
Abhijith : Yes of course.
Dr Shankar : Then could you get me -2*-2 =4 through repeated addition?
Abhijith :    .

This he used to clearly establish in my mind that multiplication is in  infact a transformation between two entities.This would at the end of the day prevent one from wondering why matrix multiplication is called what it is?

So we need to explain THE CONCEPT before going into the examples.
Though one can show umpteen number of examples in explaining a concept , we need to be careful in choosing the appropriate ones. Also , we need to be careful that we are at the end of the day capable of modeling other situations with the tools learnt and not restrict the usage to the example taught.

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Agree with you there! This concrete example conversation showed me why abstract learning is important!!

PS: I am enjoying this debate and I value all the opinions very highly.. hope nobody is getting unduly offended out there..

[sids]

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It seems like the debate has just moved over to this thread . I am also quite enjoying this!

Abhijith, that example is an excellent illustration of the importance of understanding the "right" abstract concept (I've applauded you for it ).

I think that it is usually more straightforward to come up with examples that can be used for teaching at the beginning levels of mathematics than it is at the higher levels. In higher mathematics, examples are more useful for illustration of the concept and for establishing a connection to the real world -- it is simply infeasible to teach with examples here. In this context if the student does not understand the abstract concepts behind the basic mathematics that s/he has learned, it becomes increasingly difficult to grasp the higher level concepts. This is one of the greatest difficulties I personally have to grapple with when trying to understand higher mathematics.

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sids..

I don't know whether you got my (ironical) point about the concrete example (of the conversation with the teacher) being the crucial point to learn why abstract learning is important.. 

[Abhijith]

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I enjoy these discussions too. I got the pun too ).
I thank octave for creating a platform for such discussions. Missed it big time after leaving academia.

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Let's have more of these debates.. reminds me of my grad student days and of course of GEB.. 

(Goedel, Escher, Bach)

[Abhijith]

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Upanishads and Bhagvad geeta have tried explaining difficult philosophical concepts through conversations.I feel books on similar lines explaining scientific/mathematical concepts at high school level might add value and go a long way in improving analytical thinking.

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For most such questions, the right answer usually will be, "the right balance."

Yes it is possible to not see the underlying idea, if we stress too much on the concrete. Similarly, it is easy to make wrong interpretations if we stay purely abstract.

In our society specifically, I think we have too much of symbolic learning (let me not call it abstract) with hardly any association to reality. That is exemplified by the mugging up and over emphasis on examinations. This poses a great hindrance to first principles thinking, problem solving and inventing anything new. Our learning system strives to make students into database engines, who can efficiently answer queries; but not necessarily think on their own or be imaginative.

Is it any surprise that we are a nation of copy cats? Just see our media. Be they commercial movies or television programs (Indian Idol, KBC, etc.) all of them are total copies of programs from other places. The same thing is true with students. One of the first things students do nowadays when I give them a problem is to do a google search based on keywords that I have used, rather than think on their own about what the problem is about in the first place.

-Sri

[aditya]

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[I]

Should a student learn that a negative times another negative is positive without knowing why? Indian education boards might think so . Anyway I found this nice page http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/negative.html explaining why it is repeated addition. A I love the second answer in that page. If only everything we were taught was like this .

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That was a good one! Looks like osl is churning out argumentative monsters.. Anyway it is good if it can help provoke thinking.

The concept of a negative number is fairly easy to concretize. Other mathematical concepts like complex arithmetic and linear algebra are more difficult. (For instance why is e^it = sin t + i cos t ?) Fortunately, concretization of higher mathematical concepts simply means drawing a connection to more classical mathematical concepts -- not necessarily to real-world day-to-day experiences. If matrices are explained with a vector theoretic analogy, it makes it much more easier, even though vectors themselves are abstract mathematical concepts.

And yes, we don't generally see teachers deliberating on the meaning of concepts like that in the web page. It is hardly surprising because we as a population also believe that people take up academics only if they fail to get a "real" job and other such nonsense. Schooling is seen as something that is best finished and pushed out of the way as soon as possible. Anyway, let me not get started on that one.. 

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Should a student learn that a negative times another negative is positive without knowing why? Indian education boards might think so . Anyway I found this nice page http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/negative.html explaining why it is repeated addition. A love the second answer in that page. If only everything we were taught was like this .

Good one! Although I like the second explanation, I find the fourth (and last) explanation most interesting and convincing (conclusively). I guess it is a matter of personal perception as to who connects better to which kind of explanation.