I taught a student last year who has since decided to home school. She was very bright and at 16 had already finished all of the math credits required by our district. This leaves two years for her to finish up her other requirements (english, social studies, science, etc), and as such, will basically not be doing any math for the next two years. She has come to me looking for opportunities for math practice/enrichment so that she will not get rusty over the next two years. Ideally these would be things that would be accessible to her on her own, as her parents are by no means mathematicians. My only idea was to participate in a local math circle. What other ways are there for home-schooled students to get enrichment in math?
The University of Waterloo (Canada's MIT, roughly speaking) keeps a large catalog of high quality math problems online at http://www.cemc.uwaterloo.ca/contests/past_contests.html. A strategy for a student in the situation you've described could be to download and attempt one contest every week. These aren't as curriculum focused as some other materials might be, but they'll definitely prevent any atrophy of mathematical thinking skills.
Similarly, they also publish a set of challenge problems on a weekly basis, which are available via email subscription or download at http://www.cemc.uwaterloo.ca/resources/potw.php.
A word of warning - these materials are quite challenging and can lead to frustration or burnout, especially in the case of a young student unaccustomed to failure and lacking immediate mathematical mentorship. The contests, though, have the nice feature that the difficulty ramps up in each individual one.
A bright high school student might do well to start by writing the grade nine contests, where she could treat the introductory questions as exercises, the intermediate ones as more substantive work, and the final sections as serious problems for deeper consideration. If it turns out that these contests are too easy or too difficult, then she can move up or down a grade.
edit: (Solutions are also hosted on the site, which I've got mixed feelings about. Sometimes it's nice to see people tortured with mathematics.)
Here are some resources:
- nrich They have a wide range of rich problem solving explorations for students in primary through secondary and some post secondary. They have student pages and teacher pages.
- NCTM illuminations Explore the interactives for grades 9-12. They have tools to explore fractals, a game to explore vector addition as a boat travels through a current, etc.
- Khan Academy This is a wonderful place to practice different skills so they don't become rusty.
- Stella's Problems Wonderful non-routine problems for all levels.
Many homeschoolers attend community college while of high school age. She could take whatever math class she wants at a community college.
The local math circle sounds like a good idea.
Girls' Angle posts interesting math problems.
She might like my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers.
In addition to the above recommendations, I'd add resources from the Art of Problem Solving.
(BTW, the answer to this question depends heavily on the homeschooler. I was homeschooled and found my way to mathematics through physics. I didn't like pure mathematics at all until I needed it for a better understanding of physics. Only much later did I develop a taste for pure mathematics. Somehow I ended up a pure mathematician so my perspective has changed. I figured I should share my memory of this…since I'm not sure how the old me would have responded to the material at AOPS above…)
Unfortunately, all too often what is done with bright students (especially middle school and beyond) is that their mathematical clocks are pushed - that is if they are in "grade x" they are shown the mathematics at grade (x+1) because they are already proficient with the mathematics at grade x. There are similar issues for the home schooled. My advice is to show students who are outpacing the mathematics at their grade level the books of Martin Gardner. These are available at many public libraries. Here is a partial list:
Gardner's books often show people a different vision of what mathematics is and can be. My only reservation about his work is that he does not do justice to using applications of mathematics as an entry point to creating excitement and interest in mathematics.