Not to put too fine a point on it, I think we've blown it in the math education business. We have many superb teachers, and the subject itself is nothing short of spectacular, but somehow we fail to reach most of the students we teach.
I'm not the only one who thinks that, of course. It's quite a common perception. But what surprises me is that so little is done to put things right. Actually we discuss the problem a lot and I have been part of innumerable such discussions (Dialogue, Small napkins). But now, after all this, I've come to feel that we've focused too much on how to teach and not enough on what to teach. We talk about big classes and small, and investigations and drill, and group work and lectures, and formulas and manipulatives... and all these have their place but ultimately they simply generate heat--and the clouds of war (math war) roll in.
We need now to focus on the what, the material that we put before our students. I believe that if we can get that right, the rest will follow; but if we can't, the math-ed problem will always be with us.
What do I mean by this? Suppose your task is to teach an introductory calculus course. What should you teach? Well that's a silly question--I teach calculus! What's calculus? Here, you say patiently, taking a calculus text-book down from the shelf and opening it to the table of contents, We start with this, and then we do this, and then... So my point is that maybe there's a much better way.
Over the years I've written lots of stuff about this, slowly trying to converge on something that's at least right for me. Here's a recent paper which explores the metaphor of math teacher as artist.
My inspiration comes from many sources. Foremost of these is Alfred North Whitehead whose Aims of Education (Macmillan 1929) remains the definitive 20th century work on education. The central essay in this work sets forth his three stages of learning and I regard these as fundamental to any process of curriculum design. Recently I found this remarkable essay online. Whitehead says it all, in a direct and elegant manner, and it is a sobering thought that in 80 years so little has changed for the better.
My main interest is in the development of what can be called investigative problems--problems that reach out and grab hold of you, that suggest conjectures you can make and test, that engender discussion and argument among colleagues, that will yield to careful thought and persistence, perhaps with a guiding hand, encouraging mastery of some new skills along the way, and that at the end will display for you a canvas of beauty and power.
That's a tall order, but such problems are all around us and the tragedy is that there are so few of them in the school and early university curricula. One reason for this is that most curricula have very narrow technical objective, and typically too many of these to allow much in the way of intersting side-trips. I have fought against this type of curriculum structure for years and will continue to do so, but in spite of that, over the past decade, I have focused my efforts on problems which will fit standard curricula, introductory calculus and linear algebra, and senior high school mathematics.
In particular, I have recently been working with the Ontario Ministry to develop investigative modeling problems for the new grade 12 courses in Advanced Functions and Calculus and Vectors.
Publications and Lectures