That said, for a rigorous proof-based approach to high school math, you may enjoy "Basic Mathematics" by Lang: https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/03879...
However, if you have a weak background in math and want to get up to speed before going into calculus and beyond, I have 2 suggestions.
1) Lial's Basic College Math is adequate and will get you up to speed.
2) Serge Lang's "Basic Mathematics" is great and will cover all you need to go into a rigorous theory based college math class.
 http://www.amazon.com/s?ie=UTF8&field-keywords=lials%20basic... The editions basically the same... pick the cheapest
which she has found very helpful.
As for mathematics, the subject I teach now, I have always cherished visual representations of mathematical concepts, for example those found in W. W. Sawyer's book Vision in Elementary Mathematics
But other mathematicians who taught higher mathematics, for example Serge Lang, recommended memorizing some patterns of multiplying polynomials by oral recitation, just like reciting a poem.
The acclaimed books on Calculus by Michael Spivak
and Tom Apostol
are acclaimed in large part because they use both well-chosen diagrams and meticulously rewritten words to deepen a student's acquaintance with calculus, related elementary calculus concepts to the more advanced concepts of real analysis.
Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma
demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.
Elementary school teachers having a poor grasp of mathematics and thus not helping their pupils prepare for more advanced study of mathematics continues to be an ongoing problem in the United States.
In light of recent HN threads about Khan Academy,
I wonder what Khan Academy users who also have read the submitted blog post by Cal Newport think about how well students using Khan Academy as a learning tool can follow Newport's advice to gain insight into a subject. Is Khan Academy enough, or does it need to be supplemented with something else?
I'll recommend a couple of books from that thread:
I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread.
Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart.
Get that right away.
Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer).
Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.
(Basic Mathematics is mostly high school level math, with a minimum of fuss and bother, and good exercises.)
(Numbers and Geometry is mostly undergraduate level math, with very good explanations and excellent exercises.)
Interpreting that as a request to name the textbooks I find useful, I'll do that here.
The Gelfand Correspondence Program series
Basic Mathematics by Serge Lang
The Art of Problem Solving expanded series
When a student has those materials well in hand, it is time to work on AMC and Olympiad style problem solving,
and also the best calculus textbooks, such as those by Spivak or Apostol.
By far the best initial reading text is
Let's Read: A Linguistic Approach
but there are many other good reading series, including
Teach Your Child to Read in Ten Minutes a Day
(I devote more time than that to reading instruction, typically, because I use multiple materials)
and quite a few others. There is more junk than good stuff among elementary reading materials, alas.
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