Exploring some inattended affective factors in performing nonroutine mathematical tasks

The purpose of this study was to describe students’ attempts to solve nonroutine math problems and to explore possible correlates of their performance. The focus of this study was on inattended (i.e., intentionally avoided) dimensions that have been underrepresented in the literature, including attitudes, interests, values, aesthetics, metacognition, and representation. Both objective and subjective data— drawn from 13 separate sources—using quantitative and qualitative procedures, were analyzed. Fine-grained rubrics were developed and used to score student work on six nonroutine math problems. These objective data were complemented with students’ written “logs” of their work in real time, followed by semi-structured debriefing interviews after they had finished. Structured scales were used to document students’ math-related attitudes, career interests, and work-values, along with essays describing their long-term experience with math, in and out of school. Data was gathered on students’ math-aesthetics, including the features of “attractive” problems and their individual preferences for the different modes of instructional explanation. School records provided students’ demographic data and their scores on generic measures of aptitude and achievement. Students’ age, art discipline, attendance, sending school district, socioeconomic status (SES), and ethnicity were not found to be correlates for either students’ aptitude/achievement/experience measures or problem-solving ability. Girls significantly outperformed boys on ability/achievement/experience measures, but not on problem-solving measures. Individualized Education Plan (IEP) status was found to be a strong correlate of both aptitude/achievement/experience and problemsolving measures, with students without IEPs consistently scoring higher on all significant measures than students with IEPs. Overall, most math-related attitude variables had little effect on both aptitude/achievement/experience and problemsolving measures. However, there was strong evidence of students’ math-aesthetics in problem solving. Specifically, students appreciating more than one type of solution scored consistently higher in problem-solving measures and frequency of use of higher-order internal representations. A close relationship between metacognition, aesthetics, and representation was found, as well as a strong link between mathematics and language usage. The discovery of students’ use of higher-order internal representation during post-task video-taped interviews, undetected by paper-andpencil assessments, supported a conclusion that studies of problem-solving ability cannot be purely quantitative in method but must contain a qualitative component. ACKNOWLEDGEMENTS Above my desk hangs a framed picture of a mountain range, with each peak progressively increasing in height, left-to-right, beginning with one labeled “EDC 555” (the first course in the Joint URI/RIC Ph.D. in Education Program) and ending with the flag-raised-upon highest peak labeled “MOUNT NEVER-REST – DISSERTATION.” The title of the picture reads: CLIMB EVERY MOUNTAIN PHORD EVERY STREAM It was traditional for Dr. Louis J. Heifetz to present his graphical representation of the doctoral student’s journey – photocopied onto the frosting of a celebratory cake – to third-year cohort members at the end their last class of the threeyear EDP-641 (Field Research Seminar) course. The cake has long since been digested, but the picture remains in pristine condition – a constant reminder of the long journey I have traveled and the many to whom I owe thanks for their guidance and support. I begin with my Major Professor and mentor, Dr. Louis Heifetz. I learned quite a bit from my research of the literature – but I learned the most from the vast feedback received from Dr. Heifetz. Every comment, correction, and suggestion was always an improvement and kept me on that mountain path. The depth and breadth of his feedback required extensive hours, for which I owe him a tremendous amount of