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Component Fluency Hypothesis

Math problem solving procedures are important tools in a problem solver’s toolbox. Fluency at using those procedures frees up cognitive resources for problem solving. This is the component fluency hypothesis described by van Merriënboer in his book Training Complex Cognitive Skills and in an ETR&D article. These algorithmic skills are not everything though. Common taxonomies of knowledge such as those described in Jim Cangelosi’s book on Teaching Mathematics in Secondary and Middle School include facts, concepts, procedures, principles, problem solving and application.

Problem solving is what you do when you don’t know what to do. Problem solving requires recognizing and defining the problem, selecting an approach, breaking the problem down into sub-problems, selecting procedures for solving those sub-problems, executing those procedures, evaluating and diagnosing progress, recognizing when a solution is satisfactory, and interpreting results. Note, if practice makes perfect, we better give students opportunities practice in all of these aspects of problem solving, not just simple recall and algorithmic procedures.

Common wisdom says that we should wait until people have developed the basics before we ask them to solve problems and do higher level thinking. I reject that notion. Higher level thinking may not so much be “higher level” as it is “different level”. Kids at the youngest ages can and need to be given opportunities to engage in real problem solving. Maybe, part of why kids learn to hate math is because we spend so much time focusing on “repeat what I just said” and “do what I just did”, to the exclusion of authentic problem solving.

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