I mentioned in my last post that I was pleased to see a description by Henri Picciotto for a new course designed to give students an overview of fundamental ideas in mathematics. I had some thoughts and opinions about the course as well as some other things that had been on my mind. Earlier today, he posted a response to my article, called "All of High School Math In One Year?". I strongly encourage everyone reading this to read his article as well, because he brings up many important points that are worth discussing. As I read it, I started to write a comment in response, but it got too long. Here, without 4096-character limits, is my response to his post. I hope we can continue this conversation, because it's valuable and has been helpful to me.
Thank you very much for weighing in on my proposal, as I did on yours; I am hopeful that with your experience in the field and your obvious appreciation for and understanding of the issues in math ed, we can come to agreement more frequently than we disagree. I am curious, though, about what you mean by "teach better, not less"; you have said that to do what you'd like to in your course would take considerable time, perhaps 2 years. Where will the time come from to teach all of those other topics that you'd like to include? To take material that (as I mentioned in my post) is difficult for even the best teachers to teach well and engagingly, and then to do it "better" even more so. Doesn't mean we shouldn't try, but I suspect your willingness to believe that math teachers are capable of this higher level of "betterness" is at least equal to my own willingness to believe that math teachers could assist in other disciplines.
Moving on to the issue of fluency as a requirement for effective use of electronic tools, I certainly don't disagree. It'd be pointless in the extreme to ask algebra 1 students to use Wolfram|Alpha to find an integral. Or would it? Would students who have a firm grasp of the concept of area be able to grasp the concept of a definite integral well enough to understand it as an area under a curve, and therefore attach meaning to the result of an integral, even if they haven't the slightest clue how it's being calculated? I really think we use tech way too early for stupid reasons (as a replacement for the necessary development of automaticity in our students' arithmetical skills, as well as stunting their estimation skills) and too late for equally stupid ones (not allowing students to explore genuine problems involving real-world numbers, not made-up ones from a worksheet).
I'd like to address the three additional reasons you list.
I believe that the biggest point of contention between us is the volume of (current-mathematics-curriculum-specific) material that we think students should have mastered upon leaving high school. I argue for a great deal less, while (again) I read your post as arguing for an amount comparable to current levels, but taught in a richer and more meaningful way. I certainly would love to say that all students must learn a large quantity of mathematics, but I still am awaiting evidence for that genuine need. "Preparedness for calculus", when most students do not take math at that level in college, is not a sufficient justification to design a curriculum around in my mind. Nor, for that matter, are college acceptance requirements. We might be able to design a required Kaplan-esque course that we could teach to all of our high school juniors (regardless of their own preparation), which would guarantee them 800 math SAT scores, but what would be the point? With enough time, drill and kill, and a willingness to let other things go, people can (alas) be trained to do almost anything. Finally, if we design a mathematics curriculum around what "society has deemed essential", then we should likely be designing it around financial literacy, probability and statistics, and not much else. I could see an argument for that, but I too would like our students to have a little more breadth.
Moving on to your second point, I certainly accept that the idea of math teachers in a support role would be practically impossible. But I still contend that after a minimal amount of algebra and geometry, the vast bulk of students have learned the math they will need in life, and that any additional mathematics they need can be learned in context. I'm aware of the history of such interdisciplinary approaches, and how unsuccessful they have been in general - within the framework of a traditional high school, with traditional disciplines, and traditional teachers, they seem to be impossible to pull off. What I'm coming around to is a more radical position, then: perhaps the framework of school itself, with subject-specific teachers and disciplines that are detached from each other, is the problem.
To your third point, I had said in my original proposal that the students who would be taking the 100-hour course would be ones who were already arithmetically fluent; that in and of itself makes the course unrealistic for most students entering 9th grade. So I completely support your approach (including the additional time required), which would not only reinforce the arithmetic skills the students already should have developed, but would provide them with a solid underpinning in abstract mathematics.
I think my issue remains with the final point you make, and where I suspect our fundamental disagreement lies: I really do believe that the math we need to teach to students is a very different beast from the math we want to teach to the future mathematicians of the world, all 1% or so of them. I believe we both agree that it is critical for all students, of whatever interest level or ability, to have meaningful experiences learning mathematics in high school, but I also believe our definitions of meaningful are quite different. I am not convinced, in the end, that requiring all students to master important (whatever that may mean) topics from algebra through precalculus is good or necessary, regardless of how thoughtful, student-centered, constructivist, or project-based the teaching itself is. I think we do a great deal more damage, societally, in terms of perception and attitude toward mathematics, by forcing all of our students through an ever-more-abstract sequence than we do good.
But perhaps I'm misreading your last point, in which case, I'll eagerly await your future post on the subject.
Thanks for taking my idea seriously, by the way. Obviously I'm still working out the details, but I appreciate your perspective!