To many observers, there is something very seriously wrong with school mathematics in general, and ninth-grade algebra in particular. There may be less agreement as to what the main deficiencies really are. One diagnosis says: school math has been set up—and evaluated—in terms of skills. There was also a heavy focus on notations. Both of these emphases are inappropriate nowadays, if they ever were appropriate.

More central to real mathematics are concepts, which receive little emphasis in most school curricula today. Almost completely ignored are the matters of habits, an orientation toward inquiry and exploration. a realistic assessment of what one can accomplish, and a good idea of how one may be able to get started on doing it. What is most essential to mathematics is almost completely absent from our efforts to help students. Instead, we focus on, and test, primarily skills.

Experienced teachers and shrewd observers are coming to see a possible way out of the usual sterile approaches, by changing our emphasis to problem solving. In its strongest form. this approach does not first study mathematics, then seek places to "apply" it. On the contrary. it starts with problems, and the concepts and techniques of mathematics emerge as one grapples with the process of inventing ways to deal with the problems.

But perhaps one needs to go even further, as Picciotto and Wah suggest. What is mathematics really about? It is about looking at a confusing world, feeling that there is no possibility of understanding it, then trying to think about it—and finding that, if you think about it in an appropriate way you can make sense of it. They do this, for example, when they ask students to study the area of triangles with vertices at (0,0), at (b,O), and at (x,y). There is complexity here—but, if we think about this appropriately, we can come to understand it.

They also provide students with the kinds of experiences that can give meaning to the symbols and operations and constraints of mathematics. in the sense of letting students build up, in their own minds, what have been called "tools to think with".

Central to mathematics is the activity of thinking. and Picciotto and Wah show us how to make this central to the teaching and learning of algebra. There are good ideas here—but, even more, there is an extremely valuable approach to the task to helping students learn. They show us not only clever things we can do in our next few classes—they show us how we can begin to reorient our entire course.

--Bob Davis