Today’s post represents a further zooming out, to discuss my thoughts about how to organize content across courses. Of course, many if not most teachers are not consulted on such questions, so the audience for this post may be small. Still, I will hit on some ideas which may be useful to all.
Patrick Honner tweets: “To me, the problem with (advanced) HS trigonometry is that the course it usually appears in is typically a themeless hodgepodge.” Michael Pershan responds on Google+: “I used to make arguments in favor of themed courses, but teaching elementary school math has changed my mind. After all, there’s no real theme of 4th Grade math. We study geometry, statistics, fractions, whole number arithmetic and graphs, and the kids are better off for the hodgepodge.”
As I see it, this is not a matter of principle, but rather something that should be discussed in a case-by-case basis, taking into account the specifics of one’s school, such as school culture, schedule, and student preparation. Still, I’ll try to address the underlying issues.
Foundational courses, in general, need to be hodgepodge, to ensure the necessary breadth that students need further down the line. For example, you should not fail to teach the distributive law or the Pythagorean theorem because they do not fit into cleverly conceived themes. Moreover, in an ideal world, school mathematics is not just meant to just prepare students for calculus, as other directions should be available towards the end of high school, so it is unconscionable to merely work back from what they will need for the AP test. A premature adoption of themed courses (e.g. functions in Algebra 2) can be costly in its impact on breadth.
This should not be interpreted as a “mile-wide, inch-deep” forced march through too many topics. As I wrote before, breadth should not trump or undermine depth. A limited number of carefully chosen topics should be addressed in depth. Moreover, within each hodgepodge course, we should pay attention to connections between topics, such as for example between factoring and equation solving, slope and the tangent ratio, linear functions and arithmetic sequences, etc.
Let me be more specific with the example of trigonometry. At the school where I worked until my retirement, we found that our one-term trig course was spectacularly ineffective: while we enjoyed teaching it, and students enjoyed taking it, they retained next to nothing from it. This is not because of the lack of an overall narrative. Quite the opposite: the course was highly coherent. (It was based on the excellent Chakerian, Stein, and Crabill Trigonometry: A Guided Inquiry.) The problem was that students were not accustomed to thinking in this domain and nothing stuck.
So we distributed the content across three courses: right triangle trig in Math 2 (a course consisting mostly of geometry); law of sines, law of cosines, and intro to the unit circle in Math 3 (our version of Algebra 2); and trig functions in our themed, one-term precalculus course, Functions. One benefit of this largely hodgepodge approach is that it allowed us to extend student exposure to this important topic beyond a single term, in fact, beyond a single year. (See this post for the general argument about extending exposure.) In each round you can go in depth into that year’s topic, and review the previous years’ work as needed. This ended up working really well.
For us, it was after Math 3 that themed classes made sense. They allowed students to choose courses that appealed to them, and they allowed teachers to design courses for depth. We tried to create a spectrum of highly focused classes, some more accessible, some more challenging. Most of our post-Math 3 classes just take on two to four topics, and go deep. This is a strength of Calculus, as a course, which we tried to give to our other electives.
Here is a map of our courses, and a link to their descriptions, a few years ago:
(Note that the program is not tracked: no “honors” or “remedial” classes. But that’s a topic for another post.)