Cognitive Units, Concept Images, and Cognitive Collages

Abstract

The fragmentation of strategies that distinguishes the more successful elementary grade students from those least successful has been documented previously. This study investigated whether this phenomenon of divergence and fragmentation of strategies would occur among undergraduate students enrolled in a remedial algebra course. Twenty-six undergraduate students enrolled in a remedial algebra course used a reform curriculum, with the concept of function as an organizing lens and graphing calculators. The other major focus of this study was to investigate whether students who are more successful construct, organize, and restructure knowledge in ways that are different qualitatively from the processes utilized by those who are least successful. It was assumed that, though these cognitive structures are not directly knowable, it would be possible to document the ways in which students construct knowledge and reorganize their existing cognitive structures.

In an effort to minimize the extent of researcher inferences concerning cognitive processes and to support the validity of the findings, several types of triangulation were used, including data, method, and theoretical triangulation. Profiles of the students characterized as most successful and least successful were developed. Analyses of the triangulated data revealed a divergence in performance and qualitatively different strategies used by students who were most successful compared with students who were least successful. Students' concept maps and schematic diagrams of those maps revealed that most successful students organized the bits and pieces of new knowledge into a basic cognitive structure that remained relatively stable over time. New knowledge was assimilated into or added onto this basic structure, which gradually increased in complexity and richness. Students who were least successful constructed cognitive structures which were subsequently replaced by new, differently organized structures which lacked complexity and essential linkages to other related concepts and procedures.

About the Author

Mercedes McGowen began her teaching career in 1979 teaching junior high school and then high school mathematics in Elgin, Illinois. Beginning in 1990 she taught mathematics at William Rainey Harper College in Palatine, Illinois. Her major teaching focus since 1991 has been the mathematical content preparation of teachers. She has been on numerous national, state, and MAA committees and task forces involving the mathematical education of teachers and the role of two-year colleges in this education. She has also been active in MAA committees on developmental mathematics and mathematics courses in the first two years. Dr. McGowen has written two textbooks, has published numerous papers and participated in many panels on mathematics education. She has won several awards for teaching excellence, and an NSF Two-Year College Award for Exemplary Program in Teacher Preparation.