The analogy I used to introduce this blog was that of a food-tv show host and it's audience members. Let me explain this a bit more...
The job of the food-tv host is to develop recipes and demonstrate the recipe to the audience in an easy to understand step-by step format. The audience remains passive, they watch the host prepare the meal, take notes (or get the recipe online) and they attempt to follow that prescribed recipe at home following each step. This is the traditional approach to math. The teacher acts as the food-tv hosts, prepares the lesson, demonstrates the algorithms and concepts on the board while students act as an audience by taking notes and watching. They then attempt to follow the prescribed concepts and algorithms to the mathematical problems. Imagine a food-tv show where the audience becomes the host! They work together and start developing recipes on their own, they experiment, try out new ingredients, figure out what works together and what doesn't. The host simply presents the ingredients, assists when necessary and asks questions about what the audience is making. That's what should be occuring in math classrooms, students should be developing concepts on their own, experimenting, testing strategies, working with manipulatives and figuring out what works and what doesn't work..
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