Good books that explain the Why element.

[Darth Frodo]

Good books that explain the "Why" element.

Hi there, I'm in a bit of a pickle. I'm currently in what I think is the equivilent to the American Senior year. I am kind of angry with my maths teacher. While she is good she "Teaches to the exam". It really nettles me.

Does anyone know any good "Freshman" textbooks that explain the concepts behind the maths as opposed to rote learning. (Topics include, Calculus, Trig, Algebra)

Does anyone else have any experience with teachers and forced rote learning? Does this typically continue to college?

 

[JHamm]

I think you will find that most teachers (at least in high schools) will just teach you how to do well in exams rather than how to understand and apply the subject to interesting or new problems. Most people view the outcome of school as the numbers on test sheets rather than as a chance to learn, sad but true.
If you're really interested in mathematics go find something you enjoy about it; give complex numbers a go, maybe look at a book on maths proofs and try to write some of your own or find some high school maths competitions and look at the previous papers.

 

[DivisionByZro]

You'll find that many undergraduate texts in math explain the "Why" of math. Indeed, if one of your books don't (when you're an undergrad), then it's probably a bad book. It would help to know exactly what you're interested in, as I'm sure a lot of people can recommend some very interesting books.

Here's one I really like: Mathematics: A Discrete Introduction, Scheinerman
It's great for self-studying; there are solution to odd problems, the exercises are challenging and fun, and there are end of chapter exams with complete solutions. Perfect! It's basically an introduction to proofs and to number theory/graph theory, and logic.

 

[Snicker]

Most of us has been bastardized at least a little from our own high school math education. These books may help undue them:

Algebra by Gelfand justifies everything, from showing that multiplication commutes (three times five gives the same answer as five times three) to proving that the quadratic mean.

Calculus Made Easy by Thompson. This book can be bought very cheap. It is probably in your library (my small library has a dozen of them). You can also get it online for free because the copyright is long expired for the earlier editions (which were the better ones). The book is not 'rigorous' but very intuitive. Nothing goes unjustified.

Elementary Real and Complex Analysis by Shilov. This can be bought for only about $20. Despite its title, I do believe it was meant as an introductory text of calculus. However, it is far more rigorous than any calculus texts available (even Apostol). It treats fundamental concepts with care. It even gives a gorgeous analytical definition of the trigonometric function. There is plenty of motivation for everything. I would definitely recommend this.

William Chen's lecture notes are also helpful. They may be a bit more standard than the above texts, but Chen never fails to leave out any wonderful insight.

None of these books are easy. They require a lot of effort. However, if you really want to have a deep understanding of basic mathematics, then here you go!