FIRST THOUGHTS
AND SOLVING PROBLEMS:
WHAT DO WE
LEARN FROM PH.D. MATHEMATICIANS
Jean McGivney-Burelle
University of Connecticut
Thomas C. DeFranco
University of Connecticut
Although the nature of expert mathematical
problem-solving performance has been documented, several issues need further
investigation. As a result, 15 Ph.D. mathematicians from research I
universities were asked to think aloud while solving four complex mathematics
problems and respond to questions regarding their solutions. Information
regarding participants’ “first thoughts” about a problem and control behavior
exhibited on the problem was organized in a contingency table and a chi-square
test was employed to analyze the data. Results indicated that: (1) in the
majority of problem-solving protocols, participants were unable to identify the
deep structure of the problems prior to solving them, and (2) there was no
significant relationship between identifying the deep structure of a problem
and exhibiting efficient control behavior. In addition, the 60 problem-solving
protocols were coded and a cross-case analysis of the protocols was employed.
General themes regarding efficient and inefficient control behavior with
respect to domain knowledge, problem-solving skills, and beliefs were
identified. Further analysis revealed that the use of appropriate heuristics
and efficient control behavior enabled participants to solve the problems in
spite of their inability to recognize the deep structure of a problem or
identify an approach to the problem prior to solving it. This study was, in
part, exploratory and more work needs to be done to understand control behavior
exhibited by experts in the midst of solving complex mathematics problems.