A Shift in Thinking About Mathematics Instruction (page 2)
Traditionally, the teacher taught the mathematics, the students practiced it for a while, and then they were expected to use the new skills or ideas in solving problems. This approach, strongly engrained in our culture, has rarely worked well. First, it assumes that all children at that time possess the ideas required (the blue dots) to make sense of the explanation in the manner the teacher thinks is best. This means that there is only one way for each student to "get it." It's the teacher's way or no way. However, it is unrealistic to expect the existence of a singular set of ideas across any typical class. Although a show-and-tell approach sometimes succeeds with some children, showing and telling depends on passive absorption of ideas and leaves most students believing that mathematics is mysterious and beyond understanding.
The second difficulty with the teach-then-solve paradigm is that problem solving is separated from the learning process. Children who have come to expect the teacher to tell them the rules are unlikely to solve problems for which solution methods have not been provided. By separating teaching from problem solving and struggling with ideas, learning mathematics is separated from doing mathematics. This simply does not make sense.
Effective lessons begin where the students are, not where teachers are. That is, teaching should begin with the ideas that children already have, the ideas they will use to create new ones. To engage students requires tasks or activities that are problem based and require thought. Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than elements that must be taught before problem solving (Hiebert et al., 1996, 1997). Furthermore, the process of solving problems is now completely interwoven with the learning; children are learning mathematics by doing mathematics!
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