We math teachers generally know a lot about our subject, and we have perhaps a tendency to want to give all of this wonderful knowledge to our students as straightforwardly as possible. Thus, the "classroom lecture" is born.

Yet the measure of our success is not how much we know, nor even how much we tell our students-it is how much mathematical understanding they take away with them at the end of the day and the end of the year.

There has been a lot of research on this subject, but I am just talking about my own experience here: students learn best by doing, not by listening. So I do everything I can to give them opportunities to do.

An effective activity has several of these characteristics:

· It provides a way for the students to see-and touch-a physical example of a mathematical concept. 

· It requires the solution of a problem.

· It gives students a chance to make discoveries of their own.

· It involves a subject or theme that interests students and excites their imaginations.

· It presents an opportunity for students to exercise mathematical skills.

· It illustrates the connections between mathematics and other parts of the curriculum and/or between mathematics and life outside of the classroom.

· It requires students to think, to communicate, and to work together.

Fortunately, one of the characteristics not in the above list is "it has to be original." There are lots of wonderful sources for ideas, and more and more curriculum materials available that support this type of math instruction.

But even with those resources, teaching mathematics this way requires a lot of thought and preparation. You have to work out how to motivate the activity and establish a context for it. Often, you have to know exactly what question to ask, to help a student make the connection between what his hands are doing and what his mind should be thinking about. And it requires good management skills to allow students to be active and noisy while ensuring that they remain on task. 

But, to me, these things-preparation, motivation, creating contexts, asking questions, classroom management-are teaching. And when I do them well, I find that I am always more than happy with how much of my knowledge that my students end up with-and retain.

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