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I read a great book a few years ago that gave itself this description:

For disciplines as diverse as literature, music and art, there is a tradition of examining masterpieces - the "great novels", the "great symphonies," the "great paintings" - as the fittest and most illuminating objects of study. Books are written and courses are taught on precisely these topics in order to acquaint us with some of the creative milestones of the discipline and with the men and women who produced them.

The present book offers an analogous approach to mathematics, where the creative unit is not the novel or symphony, but the theorem.

I found this to be a highly enjoyable way of learning math: first a problem is motivated, then it is solved. I just finished a course on algebra which was very well taught, and I learned a lot, but I often felt it wasn't well motivated - I spent lots of time learning about, say, rings, but no reasons as to why rings are useful and we should reason at that level instead of the more familiar algebra.

Anyway, this is a very long-winded introduction to my question: can anyone recommend books or (preferably) courses which follow the basic structure of: here's a problem, here's why it's important, now let's solve it?

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Well, I was going to recommend Journey through Genius, until I clicked on the link... –  Michael Lugo Aug 30 '11 at 2:36
    
I don't have a reference, but the Euler-Lagrance equations are well-motivated by optimization and extremization of functionals. It's how one derives the geodesic equation, which (sort of) describes how distance-minimizing paths in curved spaces look locally. –  anon Aug 30 '11 at 4:00
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3 Answers 3

You might like this one: Abel's Theorem through Problems and Solutions

This is based on a series of lectures given by V.I Arnold to High School students in the USSR.

By your criterion,

a. Here's a problem (theorem):

Is a generic algebraic equation of degree greater than 4 solvable by radicals?

b. Here's why its important

Its a lot of fun and involves interesting mathematics.

c. Now let's solve it..

Beginning from groups, to complex numbers (all the way to monodromy groups). The last section is titled Abel's theorem. I haven't read that far yet, but I so far it has been a very enjoyable experience.

The author was also a very respected mathematician.

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Galois theory is usually motivated by the question of when the roots of a polynomial can be written in terms of radicals. Though there are some detours in things like finite field theory. Is this the sort of thing that you're looking for?

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I've since found some other books that I thought I should include for others' reference:

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