MIT PRIMES is a math research program for high school students.
Its recommended reading list is copied below. Some parents, teachers, and administrators may fear math acceleration because they don't know what comes after calculus. Well, the student can take algebra again
. (Abstract algebra is not the same as the algebra taken by high school students.) Students who are good at math but are more interested in other subjects could look at what math courses economics or physics majors (for example) take in college.
http://mit.edu/primes/reading.shtmlRecommended readings for PRIMES applicants and students
Problem-solving and Proof-writing
A Decade of the Berkeley Math Circle: The American Experience, edited by Zvezdelina Stankova and Tom Rike (American Mathematical Society, 2008).
Paul Zeitz, The Art and Craft of Problem Solving, 2nd ed. (Wiley, 2006).
Sandor Lehoczky and Richard Rusczyk, The Art of Problem Solving, Vol. 1: the Basics (AoPS, 2006); The Art of Problem Solving, Vol. 2: and Beyond (AoPS, 2006).
Linear Algebra
Gilbert Strang, Introduction to Linear Algebra, 4th ed. (Wellesley-Cambridge Press, 2009).
Video lectures by Prof. Gilbert Strang on MIT OpenCourseWare.
More advanced reading (abstract linear algebra):
Sheldon Axler, Linear Algebra Done Right, 2nd ed. (Springer, 1997).
Abstract Algebra
Algebra Fact Sheet
Joseph Gallian, Contemporary Abstract Algebra, 7th ed. (Brooks Cole, 2009).
More advanced reading:
Michael Artin, Algebra, 2nd ed. (Addison Wesley, 2010).
Number Theory
Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An Introduction to the Theory of Numbers, 5th ed. (Wiley Text Books, 1991).
Harold Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers, 7th ed. (Cambridge University Press, 1999).
Lecture notes and assignments for the course 18.781, Theory of Numbers, by Prof. Martin Olsson, on MIT OpenCourseWare.
Combinatorics
Miklos Bona, A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, 2nd ed. (World Scientific, 2006).
More advanced reading:
Richard P. Stanley, Enumerative Combinatorics, Vol. 1, 2nd ed. (Cambridge University Press, 2011).
Knot Theory
Colin Adams, The Knot Book (American Mathematical Society, 2004).
