Honors
For Honors in this class, I encourage you to explore: what mathematics are you curious to learn more about? Rather than having a fixed, standard option, I want students to consider their own interests and then propose an area of study that suits them. This can be an intimidating process, so here are some examples you might consider:
Would like to learn Calculus at the highest level? Curious about how our Calculus treatment is different from, say, Harvard Honors? The book "Calculus" by Michael Spivak is the BEST introduction to higher-level mathematics that is out there. Every page is filled with thought-provoking insights that can change the way you look at the mathematics you've learned (that was definitely my experience when I read it for the first time as a first-year college student). But it is not easy! Expect to have major challenges with some of the exercises I ask you to attempt! But the end is so worth it - this is a view of mathematics that few high schooler get to experience.
Want to go back an understand some of the mathematics you've learned previously, but at a higher level? The book "Measurement" by Paul Lockhart is absolutely beautiful - he's an extremist in his views on what mathematics really is; prepared to challenge your beliefs about the nature of mathematics education as you attempt to solve some extremely interesting problems in Geometry and Calculus! Lockhart's treatment of some topics (especially 3-dimensional Geometry) is the best I've seen, and informs my teaching in Geometry to this day.
For another Geometry option: want to see a completely, completely different side of Geometry? The book "Shape" by Jordan Ellenberg is so far removed from Geometry class you might have a hard time accepting that this is Geometry - but it is, as practiced by mathematicians today. THis book helps you get a sense of how much more there is to the world of mathematics than you have learned to date.
Interested in how to "think like a mathematician" usefully in the real world? Nate Silver's book "The Signal and the Noise", which is required Honors reading for Statistics, is a wealth of knowledge on when mathematics is a useful tool in the real world, when it isn't, and how to tell the difference.
You will find many, many different opinions on the book "Godel, Escher, Bach: an Eternal Golden Braid" (some hate it, some love it), but to me it is the most powerful, thought-inspiring book I've ever read. It is NOT an easy read (I've only gone through it once with a student for Honors, and then it was a one-on-one class), but it is one of the sacred texts of Artifical Intelligence despite being mostly about music, mathematics, and poetry. It's a true classic.
Interested in mathematical finance? I love the book "The Money Formula" by Paul Wilmott and David Orrell. They have their strong opinions on finance (let's just say their politics sound pretty close to Occupy Wall Street), but they are true experts in finance, and actually understand the mathematics underpinning hideously complicated modern algorithms and understand the absurdities that ensue from essentially having robots running the economy.
If you're just interested in something more "standard", you are welcome to practice and learn the concepts in Chapters 12 and 13 of your textbook, and complete the assigned problems there. These chapters introduce you to vector calculus, an essential multivariable calculus topic we won't touch on in this class.
By no means is this list exhaustive! And by no means do you have to read a book! It could be a series of videos, or an online course, or...
How to do Honors: Agree with me on a subject area, book, or course of study. Send me an email telling me that you are pursuing Honors, and what your overall plan of study will be.
Example: "Hi Peter, this is Jim and I want to do Honors in Calc 2. We talked about me reading the book Measurement. Every progress report I plan to have read more of the book and have attempted at least 3 interesting exercises in the book. I'll include in the progress report which exercises I attempted, and either what solutions I came up with or, if I got stuck, what my overall approaches in the exercises were".
Every 2-3 weeks a progress report on what you've accomplished in your studies needs to be submitted to me. There is no fixed goal except developing your mind, but it should be clear from every progress report that you've put in substantial time and concentrated thought (at least 2-3 hours) that past couple weeks developing your knowledge in your chosen area.