I taught mathematics to pre-service teachers this summer and the mean age of the student in my class was 28. These students had trouble differentiating between the algorithm and the meaning of a computation. For example, I gave them several fraction addtion problems with the same error pattern where they had to reproduce the error pattern and then give the correct answers. The next question was "How would you explain to a student the reasoning behind the correct answer?" Five of the six students simply described the algorithm for addition of fractions. The other student used fraction bars to describe the process.
If these (older) students have trouble differentiating between an algorithm and the meaning of the operation, how do we expect grade school students to make this distinction?
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