In the spring of 2012, my district embarked on reorganizing Algebra II to reflect demands for a state test based on the subject. Myself, with others, developed a trial framework that jumbled a lot of topics to better match what we thought the state wanted to emphasize (rearranging the textbook essentially). In the spring of 2013, my colleagues felt that while ok, the framework needed some adjustment. In summer 2013, Texas abandoned their plan for testing Algebra II. New freedom in hand, by the end of the summer I emerged with a radical rethink on Algebra II.
It's been six weeks. How is it going?
I stumbled into a great situation for this kind of thing. I have one section of Algebra II with 22 students. It's academic level with mostly 11th grade but I have 10th grade and 12th grade in there. About as random as any other Algebra II in my school. It also hit during the longest period of the day (~7 minutes more than the rest).
As planned, after some simple refreshers, we open with parent function matching:
Also as planned, after completing the activity I said we were going to tackle four of them at once: linear, quadratic, absolute value, and radical.
Over the course of a week or so, we tackled basic versions of these types. Quadratics were done in ax^2=b form, so that answers were always +/- sqrt(b). I stuck with square roots only (no cubics or higher yet) for the radical portion. I introduced inequalities with linear, we discussed the mechanics of < becoming > when necessary, and I was then able to kick off our discussion of say, absolute value, with inequalities and no one in the room batted an eye.
My lesson planning revolved around the curriculum document I developed. We tackled each function at a basic level, then spiraled back to it at an intermediate one.
The main goal of the project is to keep the kids primed to solve many types of equations simultaneously. Can they spot an absolute value? Can they surface that solution strategy quickly? Can they remember to give the correct number of answers?
Our first assessment proved that at a basic level, yes, all 22 seem to adapt to different types when they are mixed together. There was even a section on the first assessment where they had to tell me the difference, and most (65% or so) did this well. Some basic errors surfaced: forgetting to include +/- with a quadratic, subtracting a term that should be divided, and failure to invert a < > when dividing by a negative.
I incorporated that into follow up lessons, and the mistakes seem to be decreasing. An assessment they took yesterday (our third) had a section on mechanics where they had to explain what can go wrong in a given situation.
After rounds of problem solving, we took a couple days off to produce fact sheets about our four functions:
Initially I was going to have the students pick two to explain, but decided what the heck, and everyone had to explain all four. This took time, but was worth it and made a good break. Many struggled on tips for linear because "they're easy."
A few things need some work and I don't think we're out of the woods yet. My curriculum document is great, but I need to supplement this with more specific details about what a problem that meets "linear intermediate" looks like. Writing assessments is a lot of guesswork right now. I write them by jumbling together problems from the last few days of classwork. The curriculum document is also a bit ambitious. My version has a thousand hand-written notes on it about modifications I need to make.
Speaking of assessments, I don't like the way I've named the standards, but I'm going to stay the course for the semester. Essentially, basics got named Level 1 with A, B, and C subdivisions. Intermediate became Level 2 and things like Theory and Mechanics got their own sections. Having them self generate a topic list at the front of the notebook needs more supervision than I was thinking.
In about a week we will stop discrete problem solving and talk about graphs for a month (?). We'll hit all the major topics at once: f(x) notation, domain, range, transformations, and inequalities. Ideally we'll take equations from previous classwork and talk about what they would like when graphed. My goal here is to develop an understanding that "3x^2 - 7 = 10" also has a graphical answer (while simultaneously a transformation of x^2) and it should confirm for them my previous statements about why certain functions have a certain number of answers.
This is a very wild experiment but the kids are being great. I already love the density of problem solving they've done. In our previous framework, they'd get six solid weeks of equation solving in the whole year maybe. This is but one equation unit of many. They might know some actual algebra when this is all over. Any student who schedule changes into this class might have a tough time adjusting. On average I introduce a new skill layer for about 10 minutes and they work the rest of the time. They don't have homework (officially, though some take things home) and spend easily 30 minutes a day working with each other. Lately I've gotten the sense that they're ready for a change of pace. I'm ready to spend some time on other things as well.
In conclusion, the early clinical trials are very promising. Updates will continue. Materials to help you try this yourself will be available at the end of the semester. I'm still sort of winging it.