Higher Math for High School Students - A Book

When I was in high school, mathematics seemed to be the only thing that I could do marginally well. In fact, that went as far back as elementary school where upon completing my own daily problem sheet I would walk around the room and help all the other kids with theirs. What a weirdo, that guy.

Cameron Brown, a mathematics teacher at Bellarmine Prep and a good friend, told me during my junior year of a brand new summer mathematics program: the Summer Institute of Mathematics at the University of Washington or SIMUW. I completed the sheet of math problems that came with the application (which were really hard), sent it in, and waited for a response. For some insane reason they decided that I would be talented enough for their program and that summer I spent six weeks learning about research-level mathematics from a surprisingly elementary point of view.

We were exploring problems combinatorics, elliptic curves, abstract algebra (transformations of the plane to be exact), computational mathematics, and many other fascinating topics and were able to make discoveries without the need of a degree in the subject. For someone who was itching to learn something other than more calculus this was a real treat and a true eye-opener to a world of mathematics that couldn't really be taught in the high school classroom; probably due to the ever-increasing need to meet standardized test...standards. That situation is a complicated one that I shouldn't get into, but in my experience as a prospective mathematician I always wanted some high school-level resources to cultivate this interest.

Therefore, I decided to write a book for all of those students out there who are interested in mathematics and would like to learn about totally awesome topics in an interactive and exploratory way. Similar to SIMUW, this book isn't meant to be a substitute for any college-level courses. Rather, I want to expose students to some of the key concepts that underly some of the hot research topics in math and science.

Each "chapter" will focus on a topic: methods of proof, number theory, graph theory, numerical analysis, combinatorics, to name a few. Within each chapter, I see four main sections:

History: A brief discussion of the history of the chapter's topic including short bios on the field's most influential people.
Mathematics: Discussion and proofs of some of the key ideas and concepts of the topic. This section is not intended as a reference style textbook (state theorem, prove theorem, state theorem, prove theorem, ...) but more of a guided exploration with plenty of motivation that will lead the reader to derive the theorems for themselves.
Examples: Here will be some real-world situations that incorporate the mathematics discussed in the previous section. These are meant to be exciting and meaningful to a wide assortment of aspiring scientists. Some computational examples will make use of Sage and Matlab so that students can become exposed to some of the key mathematical tools used in academia and industry.
Exercises: The exercises will primarily be used as a continuation of the theory outlined in the "Mathematics" section. Doing exercises is essential to learning math and motivated students would appreciate and would hopefully be excited about deriving some key ideas themselves. Many proofs of important theorems can be divided into several, easier to solve exercises. (Dummit and Foote's Algebra does an excellent job of this.)

Since I'm a proud supporter of the open-source community, especially after my work on Sage, I plan on releasing this book free of charge and open source under some GNU public license.

If you have any suggestions for topics, structure, or even public licenses then please send me an email or, better yet, post a comment. I look forward to hearing your thoughts and opinions on this project.