It is Week 3 of the New Blogger Initiative, school is about to start on Thursday, and I will begin teaching Algebra II soon after . Since it’s already something that’s been waking me up at night (once literally, often figuratively), hopefully trying to write out some unifying concepts of the swampland of more-advanced-but-still-high-school math counts as still doing work, right?

The big reason those courses feel like a hodgepodge of ideas is that…*they are!* All the higher degree polynomials, trigonometry, piecewise functions, conics, etc are thrown together into four semesters or so in order to prep students to take calculus and be able to differentiate and integrate as many well-behaved functions as possible. Not a single American friend of mine in college seemed to have the same sequencing of ideas as I in high school, while most international students wondered why we had spent so much time learning about cubics and trig identities instead of number theory, combinatorics, inequalities, and proof.

Another issue is that advanced high school mathematics, even more than Algebra I, is like an underwater mountain range in that on the surface it appears disconnected. Some of the deep ideas that connect the topics in the space between linear equations in two variables and calculus are very deep indeed (e.g. the Fundamental Theorem of Algebra, insolvability of the quintic).

All that said, here are three big themes that I want to emphasize in my Algebra II class, in other words, themes that are accessible to my students on an everyday basis, not deep, deep results that I hope to give them a glimpse at.

- Modifying the behavior of functions by changing different parameters. These classes can be a place for “mathematics as tinkering.”
- Detecting what answers make sense. I suppose there is some throwing out of extraneous solutions in Algebra I, but Algebra II is where you really need to have your guard up re: answers given you by formulas and algorithms.
- Translating between different written forms of a function and its graph. To me, this is the largest. Two expressions that are equivalent may reveal completely different information about a function. For example, it is hard to see a parabola’s axis of symmetry if the function is written . The process of first figuring out what quantity is important and then illuminating that quantity with algebraic techniques is probably more useful than any of those quantities themselves.

Now I suppose I have some talking points when we decide what to cut from Algebra II tomorrow!