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 and a more recent paper looking at the process of design and construction.
For most of my career, my focus has been on 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.
Well that all sounds good. Mathematicians, and indeed all of us, love to investigate, design structures and build things. But exactly what properties do an investigation or a design need to have to qualify them to be called mathematics? My answer to this is that they ought to belong to or be part of a sophisticated abstract structure. It is these structures: groups, dynamical systems, transformations, manifolds, that define the heart and soul of the subject. My current search is for curriculum units that embody this kind of sophistication. That's a tall order, but it's worth every penny that we put into it.Such units, when we find them, will be "big," so big that they will completely change the structure of the curriculum. A good example of such a unit is the grade 10 investigation of transformations that is currently our main project in Math9-12. We will be taking that into the classroom in Fall 2016.
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