About twenty years ago, along with Anita Wah, I wrote "A New Algebra", a paper for the Journal of Mathematical Behavior (JMB). It was an attempt to address what I saw as a crucial issue in secondary math education: on the one hand, the traditional Algebra 1 course was the gateway to any further work in math or science, and thus to college admission; but on the other hand, it was terribly ineffective. I was not the only one with this point of view, of course. In fact, the idea of "algebra for all" was taking off at the time, and by now is widely accepted.
The paper proposed that a new algebra course was needed, one that would aim for conceptual understanding, not merely mechanical facility. It would rest on two foundations: themes, and tools. Themes were interesting contexts for algebra, such as geometric questions, "real world" problems, or really anything to make the x's and y's come alive. This went along with a shift in emphasis towards functions and modeling, and to some extent away from strictly formal symbol manipulation. Such a shift was facilitated in part by then-nascent graphing technology. Again, I was not the only proponent of this shift, and the NSF funded a number of curriculum development projects that incorporated this kind of thinking. If the Common Core State Standards do take root, they may finally bring this idea to a much broader audience.
The other foundation of a new algebra was to be tools, specifically learning tools -- electronic, manipulative, and conceptual. While electronic tools have had a substantial impact, other tools (such as manipulatives, or function diagrams) have a much more limited audience. This is in part because of the widespread misconception that there is no need to worry about pedagogy at the secondary level: if the teacher knows the stuff, they can just pass it on to the students. Another reason might be that in our culture non-electronic tools do not generate the sort of fascination, fetishism, or profits as calculators and computers.
Even though a tool-rich pedagogy is still a bit of a niche phenomenon, I stand by what I wrote in 1993. The article made its point by taking an algebraic tour across a geometric landscape (the theme of area), using a variety of tools as vehicles. I believe it still reads well. Check it out! The article expresses the philosophy behind the textbook Algebra: Themes, Tools, Concepts, which is now available free on my Web site.
right above the abstract.) Davis was the editor of the JMB, and an early inspiration for me, as I had used some of his curriculum ideas (from the Madison Project at Syracuse University) during my years as an elementary school teacher. Thus I was quite pleased at his endorsement of our thinking.
PS: I recently returned to the topic of a tool-rich pedagogy in this article.