What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's Elementary Mathematics from an Advanced Standpoint and the books of the Gelfand Correspondence School – school-level books with a university ethos.
[Math] Insightful books about elementary mathematics
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Have you taught this course before? After teaching it several times from Millman/Parker and other materials using Birkhoff's axioms, I suggest you consider using Euclid himself plus Hartshorne's guide, Geometry: Euclid and beyond, which uses a form of Hilbert's axioms.
The problem for me is that real numbers are much more sophisticated than Euclidean geometry, and the Birkhoff approach is thus a bit backwards except for experts like us who know what real numbers are.
When we covered as much of Millman/Parker as we could manage, the most enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.
If you like assuming that every line in the plane is really the real numbers R, what about going the rest of the way and assuming the plane itself is R^2? Then you can use matrices to define rigid motions and do a lot that connects up to their calculus courses.
Moise is more succinct than the 500 pages suggests as I recall, and is an excellent text from a mathematician's standpoint, but very forbidding probably from a student's. I noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.
The old SMSG books in the 1960's were based on Birkhoff's approach, but are not short. They are also available free on the web.
I just looked at the old SMSG book and found the following circular sort of discussion of real numbers: "if you fill in all those other non rational points on the line, you have the real numbers."
Clint McCrory spent several years developing his own course using Birkhoff's approach at UGA, and made it very successful. Here is a link to his course page. The students loved his class at least in its evolved form after a couple years. they especially appreciated the GSP segment at the start. Apparently many students had little geometric intuition and used that to acquire some. Clint apparently never found an appropriate book to use though.
http://www.math.uga.edu/~clint/2008/geomF08/home.htm
After teaching this course myself from Greenberg, Millman/Parker, Clemens, supplemented by Moise, and the original works of Saccheri, my own Birkhoff axioms, I finally found Euclid and Hartshorne to be my favorite, by a large margin.
But the beauty of the topic is that there is no perfect choice. You will likely enjoy the search for your favorite too. There is a reason however that Euclid has the longevity it has.
In a nutshell, there are two equivalent concepts, similarity and area, that are treated in opposite order in Euclid and Birkhoff. Euclid's theory of equal content, via equidecomposability, in his Book I, allows him to use area to prove the fundamental principle of similarity in Book VI. Birkhoff assumes similarity as an axiom, and area is relatively easy using similarity, e.g. similarity allows one to show that the formula A = (1/2)BH for area of a triangle is independent of choice of base. Euclid himself uses similarity to deduce a general Pythagorean theorem in Prop. VI.31, whose proof many people prefer as "simpler" than Euclid's own area based proof of Pythagoras in Prop.I.47. The problem is that there has, to my knowledge, never been a civilization in which similarity or proportionality developed before the idea of decomposing and reassembling figures. Briefly, congruence, on which equidecomposability is based, is more fundamental than similarity. Hence, although logically either concept can be used to deduce the other, it seems to me at least that the more primitive concept should be placed first in a course.
I believe there have been similar questions, but not one exactly of this flavor.
To answer your last question, it is true that you need to know many different areas of mathematics in order to delve deeply into algebraic geometry. On the other hand, to get a basic grounding in the field, one need only have a basic understanding of abstract algebra.
That being said, I will give my recommendations.
If you have already done complex variables, and I'm not sure that every student in your position will have completed this, I recommend Algebraic Curves and Riemann Surfaces by Rick Miranda. Although this book also develops a complex analytic point of view, it also develops the basics of the theory of algebraic curves, as well as eventually reaching the theory of sheaf cohomology. Multiple graduate students have informed me that this book helped them greatly when reading Hartshorne later on.
If you want a very elementary book, you should go with Miles Reid's Undergraduate Algebraic Geometry. This book, as its title indicates, has very few prerequisites and develops the necessary commutative algebra as it goes along. More advanced students may complain that this book does not get very far, but I think it may very well satisfy what you are looking for.
Another book you might want to check out is the book Algebraic Curves by William Fulton, which you can thankfully find online for free.
If you would not mind a computational approach, and furthermore a book which requires even fewer algebraic prerequisites than you seem to have, you might want to check out Ideals, Varieties, and Algorithms by Cox and O'Shea.
Thierry Zell's suggestion is also supposed to be good.
That being said, if you decide that you like algebraic geometry and decide to go more deeply into the subject, I highly recommend that you learn some commutative algebra (such as through Commutative Algebra by Atiyah and Macdonald). But for the moment, I think the above recommendations will suit you well.
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Best Answer
Geometry and the Imagination by Hilbert and Cohn-Vossen.