The quotes are from this post on his blog.
I am curious about what you mean by "teach better, not less"I have discovered a truly remarkable approach to teaching which this post is too small to contain. See for example:
For a Tool-Rich Pedagogy
And feel free to follow the links therein.
Of course, I am not the only one to have developed approaches that are more effective than the traditional listen-to-me-and-practice standard. Anyone who wants to get better at this can find many ideas on how to do it, but for most of us changing our practice is difficult given inertia, school culture, and so on. Teacher collaboration is almost certainly the most powerful engine for such change.
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?At my school, we teach almost all the topics in that course by distributing them across three years. That leaves enough time during those three years to teach enough (not all!) of the traditional core sequence. Enough to prepare students for the wide range of electives we offer. Most (not all) of the students choose to do another year of math, or more, beyond our three required years -- that's a sign that our system works reasonably well.
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.Teaching better is far more difficult than assisting in other disciplines. The latter would be easy, as it is likely that the needed math would be fairly limited in any one course, and there would almost certainly be an attitude that deep understanding is not needed -- it would be sufficient to learn whatever techniques are needed for that course. (Unless the change you propose includes increasing the amount of time those other disciplines have, so as to include time for teaching the math well.)
But you are absolutely right: teaching better is difficult, for many reasons, including the ones I listed in my last post. It is not likely to happen on the sort of scale that is needed, at least not until this country changes its priorities. The way teachers are treated by society, the assumptions of the class system in education, the standardized test craze are just three among the obstacles to an overall systemic change.
Thus my own choice is to prioritize working to improve teaching with teachers and math departments, one at a time. I have seen impressive changes happen at that scale. Alas, even at that scale, change doesn't happen quickly.
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.My objection to your one-year proposal is not primarily about the topics that must be omitted. Even three years means you have to pick and choose and leave a lot more things out than you can include, especially if you teach for understanding. The point is that the habits of mind that are required to do math with understanding are hard to pick up, but valuable. It does take time.
Perhaps I can make my point better using writing as an example. You could say that given how much writing most people need to do, they already know the basics they will need by the time they hit high school. But we correctly insist that they keep struggling with writing year after year. Some of the students hate that, some of them love it. Some of it is taught poorly, some well. But it is a very hard skill that is hugely important. We don't know in advance who will become a journalist, or a playwright, but most professions require some writing, and being an educated citizen who can for example express opinions on a blog requires those skills. So we spend a lot of time and energy on writing in school, as we should.
Likewise, adults do not need to understand electricity. All they need to learn is how to flick a switch. They don't need to understand biology. When is the last time you have used what you know about cells? They don't need to understand difficult works of literature. They can just watch TV. And so on.
Education is not about learning any particular content because you will supposedly need it. It's about learning how to learn and how to think. By far the most applicable academic discipline is reading and writing. Math comes next, as it applies to considerably more endeavors than (say) history. Speaking of history: Stanford has a great Web site for history teachers, called Reading Like a Historian. Not many students will grow up to be historians. But if they learn to read like a historian, they will be so much better thinkers and citizens! The same applies to any discipline. Thinking like a mathematician! Experimenting like a scientist! Painting like an artist! That's what education is about, not the pointless accumulation of topics deemed useful by someone.
after a minimal amount of algebra and geometry, the vast bulk of students have learned the math they will need in lifeI'll go further: for many people, they have learned the math they will need in life before they even start algebra. Likewise for all the disciplines beyond reading, 'riting, and 'rithmetic -- not just math!
Some reasons to teach math:
- First and foremost, the habits of mind (see this article by Cuoco, Goldenberg, and Mark)
- The fact that it is one of the great achievements of the human spirit, and underlies everything we know about the natural world. I think of math as one of the humanities.
- The fact that we don't know in advance who will need more math because they want to become an engineer, a scientist, a statistician, a computer programmer, etc. If we make those decisions too early, we are arrogantly freezing a lot of people out of those careers and perpetuating the current hierarchies.
perhaps the framework of school itself, with subject-specific teachers and disciplines that are detached from each other, is the problem.You are of course not the first to suggest this. For example, I believe that is in part the thinking behind the Essential Schools movement. Perhaps one of those schools would be the right professional home for you.
As for me, I believe in disciplinary education, as I mentioned above. In any case, I have chosen to work in existing schools, and I've had the good fortune to have colleagues who share my vision, and an enlightened school administration.
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.There is no doubt that future mathematicians need more stimulation and more formalism than everyone else, and perhaps exposure to more topics. Some of that can happen in math clubs and math teams, and through differentiation within math class.
But everyone needs to learn math in a way that is closer to actual math than the cookbook approach that dominates in US schools. Look at the list of habits of mind in the article I cited above, or quicker, check out this blog post by Avery Pickford. Those are habits that are at the same time closer to what mathematicians do, and more useful to the general public, than the way math is traditionally taught in this country. The topics we choose don't matter nearly as much as the way we teach them.
Since the kind of teaching I'd like to see is the exception rather than the rule, would it be better to limit the damage by teaching less math? I don't think so, for the reasons I outlined above and in my previous post.
Teaching better, building a math department where there is good teaching, supporting colleagues at other schools as best I can -- those are things I can do. My impact is limited, but hey, it's better than giving up.