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Theory of Learning
As math is traditionally taught in our schools, students memorize procedures for solving problems. Who cares if our students memorize procedures? But even this fails -- there is no understanding of math for them to build on, so they forget the procedures. (It is very difficult to remember a procedure you do not understand.)
Essays on Theory:
Teaching MathThe straightforward method of teaching mental models, which is ideally suited for math, is simply to give students problems to solve. Do NOT first tell your students how to solve the problem. They develop mental models in the process of solving the problem; you short-circuit that process if you first tell them how to solve the problem.
I call this method the "Problems-First" method of teaching math. Yes, when I teach math, the first thing I do is give the students problems to solve. If a student can't solve the problem, I usually take it away and find an easier problem. My students usually make good progress, learn math and thinking skills, and enjoy math.
The DetailsI am in the second year of working out the details. In addition to teaching strategies, I have been developing my own problems. These problems allow incremental learning through problem solving. They also teach some of the basic concepts of mathematics. When you are teaching mental models instead of memory, concepts rather than facts, your curriculum goals come out different.