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knowing how to apply procedures is extremely important because, without fluency in calculations, it is hard to focus your attention on higher-level concepts.
They would, however, become more effective mathematicians if they learned to stop and think first about what would allow them to make the most progress.
Part of deciding which procedure to apply is giving yourself a moment to think about it before you leap in and do the first thing that comes to mind.
When you finish an exercise, instead of just moving on to the next one, stop and think about these questions: 1. Why did that procedure work? 2. What could be changed in the question so that it would still work? 3. What could be changed in the question so that it would not work? 4. Could I modify the procedure so that it would work for some of these cases?
Developing a proper understanding of why things work is generally harder and more time-consuming, but the resulting knowledge is easier to remember and more supportive of flexible and accurate reasoning.
