On recognizing and formulating mathematical problems

When mathematics is used to help people cope with real-life situations, a three-stage intellectual process is involved. First, a person becomes aware of a problem-situation which stimulates him to generate a problem-statement, a verbal story-problem. This may be in writing, expressed orally, or merely thought and evidenced by other behavior. Second, he transforms the verbal problem-statement into a mathematical formulation. Third, he analyzes this mathematically stated problem into subproblems to which the solution is more immediate.We propose an operational instrument for assessing how a person moves from the first to the second and third stage. This involves coding questions asked, actions taken, hesitancy and latency. Results of using this procedure in experimental situations were used to determine reliability. We also propose and compare methods for effecting and accelerating motion from the first to the third stage. These involve operant conditioning, need arousal cues, controlled verbal instructions. Experience with one such technique to teach problem-formulation in actual teaching practice is reported.The theoretical basis for these experiments is developed by showing how to program a computer to go through these stages for a reasonably large and interesting class of problem-situations to which mathematics is applicable.This research was partially supported by an Office of Education Grant OEG5-72-0050(509).