Textbook recommendations for a high school math student

[AsherA123]

My brother is currently doing his AS/A-Levels (11th - 12th grade) for maths and further maths. However he is unsatisfied with his textbooks as they focus purely on memorisation and don't really explain the mathematics behind concepts. He wants textbooks that not only teach the content but also have detailed explanations and focus on the proofs instead of the memorisation of facts. He doesn't mind if the textbooks contain university content, he doesn't even mind if they are university textbooks! Here is all the content he needs to learn:

http://oi59.tinypic.com/xqc66f.jpg
http://oi61.tinypic.com/29erh9f.jpg
http://oi61.tinypic.com/k0g481.jpg

 

[IDValour]

For A-level mathematics one cannot go far wrong with the series of books by Bostock and Chandler, though they were written for a slightly older and tougher specification. If I remember correctly they contain most if not all of the content needed for current A-Level Maths plus extra. Other options include the series of books by Gelfand on mathematics, which whilst accessible by GCSE students are still pleasant reads even for those at the A-level standard. Allendoefer and Oakley also have a text called "Principles of Mathematics" which doesn't perfectly align with the A-level specification, but that which it does cover it does an excellent job on. There are a large number of pre-calculus books in the US which have a lot of overlap with the non-calculus part of maths A-level, however I am not too familiar with these, although I have heard both Sullivan and Axler have written reasonably good texts. My final suggestion that excludes Calculus will be Basic Mathematics by Lang, which is in my opinion the most stretching of the books I have listed and more akin to a University level course in "pre-calculus" topics. Beyond this I feel Spivak's calculus is an excellent book for stretching talented students whilst also blending rather well with some of the pure and further A-level modules. Be warned though, if it is his first exposure to rigorous University style maths it may prove discouragingly difficult. It is helpful to have a teacher on hand for guidance in some parts, and I would also suggest that he read up on writing and constructing proofs if he does not have prior familiarity with this, to this end both How To Prove It by Velleman and Naive Set Theory by Halmos are very useful. Working through Lang's Basic Mathematics would also be good preparation for a text like Spivak. I personally elected to work through another of Lang's books, A First Course in Calculus, before approaching Spivak as I feel it can help to have a handle on some of the material first, there is certainly no shame in doing so as Spivak's text has oft been called more of an introduction to analysis than calculus, thus that book is another recommendation from me. Hopefully that should give your brother a fair selection of texts to examine and pick from. ;)

My overall suggested list:

A-level syllabus: Bostock and Chandler.
Rigorous Pre-calculus: Basic Mathematics by Lang.
Calculus: A First Course in Calculus by Lang.
Proof Writing: How To Prove it by Velleman and Naive Set Theory by Halmos.
Rigorous Calculus (essentially analysis): Calculus by Spivak.

 

[AsherA123]

For A-level mathematics one cannot go far wrong with the series of books by Bostock and Chandler, though they were written for a slightly older and tougher specification. If I remember correctly they contain most if not all of the content needed for current A-Level Maths plus extra. Other options include the series of books by Gelfand on mathematics, which whilst accessible by GCSE students are still pleasant reads even for those at the A-level standard. Allendoefer and Oakley also have a text called "Principles of Mathematics" which doesn't perfectly align with the A-level specification, but that which it does cover it does an excellent job on. There are a large number of pre-calculus books in the US which have a lot of overlap with the non-calculus part of maths A-level, however I am not too familiar with these, although I have heard both Sullivan and Axler have written reasonably good texts. My final suggestion that excludes Calculus will be Basic Mathematics by Lang, which is in my opinion the most stretching of the books I have listed and more akin to a University level course in "pre-calculus" topics. Beyond this I feel Spivak's calculus is an excellent book for stretching talented students whilst also blending rather well with some of the pure and further A-level modules. Be warned though, if it is his first exposure to rigorous University style maths it may prove discouragingly difficult. It is helpful to have a teacher on hand for guidance in some parts, and I would also suggest that he read up on writing and constructing proofs if he does not have prior familiarity with this, to this end both How To Prove It by Velleman and Naive Set Theory by Halmos are very useful. Working through Lang's Basic Mathematics would also be good preparation for a text like Spivak. I personally elected to work through another of Lang's books, A First Course in Calculus, before approaching Spivak as I feel it can help to have a handle on some of the material first, there is certainly no shame in doing so as Spivak's text has oft been called more of an introduction to analysis than calculus, thus that book is another recommendation from me. Hopefully that should give your brother a fair selection of texts to examine and pick from. ;)

My overall suggested list:

A-level syllabus: Bostock and Chandler.
Rigorous Pre-calculus: Basic Mathematics by Lang.
Calculus: A First Course in Calculus by Lang.
Proof Writing: How To Prove it by Velleman and Naive Set Theory by Halmos.
Rigorous Calculus (essentially analysis): Calculus by Spivak.

Thank you for that detailed response! I'll pass on your suggestions to my brother and let you know how it goes! :)