Adventures in Mathematics

Andrew M.H. Alexander

**Mathematics is beautiful, and should be taught as such**. It should be taught as an abstract, axiomatic art, and not as a series of formulas to be memorized, or a tool to be used in solving engineering problems.

From 2009 to 2011, I taught at Veritas Preparatory Academy, a small charter school in the Southwest. Veritas is a school devoted to rigorous, academic inquiry, modeled loosely after St. John’s College. All students take the same courses, which include two years of physics and two years of calculus, as well as a daily, two-hour literature and philosophy seminar based on the Great Books.

I taught the 11th and 12th grade math course. My teaching philosophy has already been best explained by Paul Lockhart, though my letter to the editor of the

*New York Times*describes it succinctly. (My 2009 teaching statement is pretty good, too.) Below are PDFs of my notes and problem sets. (If you want the original LaTeX source code, or any of my tests and quizzes, send me an email and I’d be happy to pass them along. Or just visit my house and take a look)

Algebraic synæsthesia • Logs • Trig • Derivatives • Integrals • Diff. Eqs • Infinite Series • Logic/Set Theory/Abstract Algebra

- Syllabus
- Mathematical Biography
**Preliminaries:**- Basic exponentiation laws and proofs
- Basic algebra (solving equations and whatnot)

**Algebraic synæsthesia:**- Parent functions and their linear transformations
- Sketching polynomials
- Sketching rational functions (lots of fun!)
**Logarithms:**- Inverse functions (as a primer)
- Logarithms!
**Trigonometry:**- Radians
- The unit circle and exactly evaluating trig functions
- Helpful mnemonic
- The proof of the Pythagorean Theorem
- Trig identities
- Equations with trig functions
- Construction of the sum and difference identities, and exactly evaluating
*even more*trig functions using the same - The Law of Sines
- The super-Pythagorean Theorem (a/k/a Law of Cosines)
- Some word problems with trig
**Differential calculus:**- Basic motivation and the construction of Fermat’s difference quotient
- Intuitive slope sketching
- The derivative of x
^{n} - Reading: "Infinitesimally Yours", by Jim Holt, in the
*New York Review of Books*, 20 May 1999 - Limits, informally
- Limits, formally
- Differentiation laws
- Trig derivatives (statements, not proofs)
- Derivatives of logarithms and exponentials
- Derivatives paper
- Other miscellaneous fun:
- Implicit differentiation
- Related rates
- Optimization and related word problems
- Write Your Own Adventure

- Calculus presentations
- Some themes in Calc 11
- Mathematical biography, part II, and review sheet
**Integral calculus:**- Basic ideas of integration and the FTC
- Integrals as "net area"
- Antidifferentiation (instructions, problems)
- Areas under and between curves
- Solids of revolution
- Centroids
- Improper integrals
- Integrals paper

**Differential equations:**- Basic differential equations notes and problems (seperable and first-order linear D.E.s)
- Differential equations problem sets:

**Infinite series:**- Basic ideas of infinite sequences and series
- Taylor series
- The most important test I’ve ever written (a step-by-step walkthrough of the proof of Euler’s identity)

**Higher math:**- Logic and set theory: I used the first two chapters of Velleman’s
*How To Prove It*(Cambridge UP, 1994), which is an absolutely fantastic book that I cannot say enough good things about. - Then I developed some other important concepts on my own:
- We also did a quick discussion of Russell’s paradox and the idea of axiomatizing set theory. We read (among other things) chapter 6 ("Axiomatic Set Theory") of Mary Tiles’s excellent and (relatively) accessible
*Philosophy of Set Theory*(Blackwell, 1989; 2004 Dover reprint). - This naturally segues into Gödel’s incompletness theorems, which we didn’t discuss as much as I wanted to. In retrospect, we should have read Rebecca Goldstein’s
*Incompleteness*(W.W. Norton, 2006), which is well-written and very accessible. - Then we started working out of Maxfield and Maxfield’s
*Abstract Algebra and Solutions by Radicals*(1971; 1992 Dover reprint), which is another fantastic book. It develops Galois theory in a way that is easily acessible to bright high school students, with the ultimate goal of proving that polynomials of degree five or greater are not neccessarily factorable.

- Logic and set theory: I used the first two chapters of Velleman’s
- Mathematical summer reading
- Is there life after calculus?