AP PreCalculus?
I read a really interesting post by David Bressound on joining the advisory board for AP PreCalculus: link. The gist of which was despite misgivings he thought setting up the course would do more good than harm by providing an opportunity to standardize the curriculum. And in general I think this part of his analysis is probably true. Given an AP Pre-Calculus test, a lot of high schools that already teach AP Calculus are likely to switch over and replace their own course with one aligned to the AP standard. I’m much less sure if it will prove attractive to schools that don’t already do AP Calculus or AP Statistics as he asserts. Anecdotally watching trends around Seattle, it feels like dual enrollment classes at the community colleges are gaining traction for upper level high school math classes. For those out of the state this is partly because Running Start makes it particularly easy for students to access this option.
The meat of his article revolves around what should such a course focus on? First he rules out just copying college level paradigms which have not been particularly effective according to the data and are heavily focused on preparing for Calculus. There are a few points which immediately occur to me here. If you believe in multiple pathways and that Calculus is not the right capstone class for high school then reforming Pre-Calculus seems too narrow. By its very nature, precalc means to prepare students for the next year. Maybe its better to throw energy directly at discrete math, or data modelling or even reforming AP Statistics. Secondly, he discusses support classes that introduce precalculus concepts in tandem with a calculus class. While I share his feeling that the existence of college algebra and precalc classes are mostly an indictment of inadequate high school classes, I have trepidations about this model too. It consumes an entire class slot and I would like to see more data on long term outcomes. Would it make more sense to slow classes down rather than doing too separate ones at once? Why are students taking precalc/calc in this situation? Do they really need to be here or would they be better served by focusing on a different branch of mathematics?
But the main part that I’ve been considering is what would I do with the curriculum? To give some context, unlike middle school math which I’ve had the chance to watch multiple kids doing, for precalc I only really have my own son as a sample. But his experience this year drove home several shortcomings which I think are more general. To start with the in school class he took used “Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis” and occasional pieces from the ck12 site. Both of the resources were somewhere between boring to awful. The textbook spiraled so heavily that perhaps 2/3rds of the year were review of Algebra 2. There were units on linear functions, quadratics, polynomial, exponents, basic combinatorics, sequences etc. Its not just that this material didn’t extend anything from previous years. It also replicates my least favorite structure from Algebra 2: the “tour of functions”. This generally dull survey of function types without any direction or purpose doesn’t seem to stick well or we wouldn’t see so many students and classes repeating the process. Then the ck12 pages were often slip shod in quality with an overly mechanical/procedural focus that occasionally avoided discussing the reasons behind the procedure with the dreaded “The proof of this concept is beyond the scope of this page…“ All the time focused on review meant most of the new material was shoe horned into a few months at the end of the year. Inevitably time ran out before everything could be finished even within the limited scope the class had set.
Due to these shortcomings, I ended up home schooling the AoPS textbook through the whole year at the same time. What I liked about it was the focus on trigonometry, complex functions and linear algebra as well as the proof and problem based structure. As material these were fun to teach and compensated for a very thin treatment of the material in school. I particularly liked the sections on De Moivre’s theorem, Euler’s formula and the roots of unity. But there were several issues with the book that still don’t work well. For one, the chapters can drag and I saw his interest vary from point to point. There was a moment when it was clear we really had done enough complex transformation and the extension was not serving a good purpose. Secondly, while I had a lot of fun with linear algebra (about a half of the text) I don’t find it very well connected to the curriculum that follows. My concern remains that it will be forgotten before its reused multiple years later. I want a stronger connection in PreCalculus to Calculus without actually teaching derivatives, a trend I see more often in modern textbooks. As a result, I introduced several ideas from elsewhere that I thought were more narrative focused and focused on concepts that would recur in the next class.
- Using polynomial division to find tangent lines
- Newtons difference formula.
Ultimately that’s what I’m seeking out of precalculus. Discuss trigonometry which must be covered before Calculus and topics that directly relate to the notion of change, limits and series. Tie that all together by discussing a little more why this material was investigated in the first place. And have problems that reuse previous material like exponents without actually doing a unit on exponents. Most importantly, make a course that you want to take and is completely distinct from Algebra 2.
