Some time ago, Mike Thayer posted a comparison of Algebra 1 and Geometry as they are experienced in the classroom. He concluded that since geometry is so much more real to students, and lends itself to interesting connections, perhaps it should be taught first.
In response, I suggested that moving specific topics up and down the high school sequence is a more flexible tool than moving entire courses, and I cited my experience at the Urban School, where we had some success with this approach:
- Math 1: basic intro to functions and graphs (linear, quadratic,
exponential), a manipulatives-based approach to symbol manipulation, and
"real world" (including geometric) contexts
- Math 2: largely geometry, some programming, some algebra (systems of equations, working with radicals), some basic trig
- Math 3: quadratics in depth, iterating linear functions, plus many standard Algebra 2 topics
Note that this is largely about postponing traditional end-of-the-book Algebra 1 topics until the students are more ready for them. (There is more info about that curriculum here.)
In response, Mike pointed out that such flexibility is not available to public school teachers, as the curriculum they are supposed to teach is mandated externally.
Since my career has been entirely in private schools, I know very little about the politics of how the public school curriculum is set. Nevertheless, I will make two suggestions.
The first is to use the coming of the Common Core State Standards as an opportunity to ask for a reevaluation of the sequencing of topics. The CCSS do not mandate any particular sequence in high school. They can be implemented through integrated courses, or in the traditional sequence, or even in some other sequence altogether. So for example, some version of the sequence suggested above would be legitimate.
Moreover, the CCSS require substantial shifts in emphasis. In algebra, modeling and families of functions take a bigger role, and manipulation of symbols and "simplifying" are demoted. In geometry, transformations become foundational, and congruence / similarity postulates follow. Those changes are tremendous opportunities to create a program that is at the same time more accessible, more interesting, and mathematically deeper. I hope that teachers who want these outcomes will take the lead in the coming curriculum discussions, because inertia reigns in the teaching of high school math, and there will be tremendous resistance.
My second thought is that even within a problematic mandated curriculum, there is always some wiggle room. One can and must prioritize, because superficial coverage of too many topics is in no one's interest. It does not even adequately prepare students for the standardized tests that are supposed to reflect that content. Much more effective is to choose what to emphasize, and give more time and multifaceted teaching to those topics. Yes, that means spending less time on other things, but on balance, that approach is sure to yield better results in understanding, motivation, and —very likely— test scores.
--Henri
"There is no one way"
Wednesday, May 29, 2013
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