Mathematical Cognition and Affect During Non-Routine Problem Solving

University Of Northern Colorado, Greeley CO

Investigators

Abstract

The objective of the proposed work is two?fold. At the level of the individual the objective is to advance understanding of the complexities of cognition and affect during problem?solving. The other objective is at the level of community: cognition?and?affect interaction in the context of collegiate mathematics classroom culture. In pursuing these objectives, the relationships among affect and cognition during non?routine problem solving should prove particularly important. The Proposal incorporates one of the most effective methods developed to date for research in collegiate mathematics education: research working groups involving graduate students as well as junior and senior researchers. The research methodology ensured by the make?up of the working groups is both broad: quantitative methods (e.g. survey), experimental methods (e.g. teaching experiment) and deep: qualitative methods (e.g. interview), experiential methods (e.g. case study). Investigating the relationships and causalities among what problem?solvers think of their feelings, attitudes, beliefs, and selves in mathematics and how those aspects of affect interact with cognitive efforts during problem solving would, of course, take more than the next year. What this Proposal provides for is the organizing of research working?groups around the objectives given above. One result of this cooperation will be the design, implementation, and completion of at least one pilot study. A second result will be the creation of a research design for continued work on both individual and community cognition?and?affect interaction in the various contexts of mathematics learning. At the individual?level, it is well known that significantly reducing the anxiety of a mathematically anxious student is challenging (at best, and often fails). What does show promise, however, is meta?affective self?control. Rather than insisting a student no longer feel anxious, an individuallevel pilot study would investigate ways to encourage an empowering meta?affect. Most work in this area has been at the school level. The proposed pilot?study would be at the collegiate level. A pilot study aimed at the community?level objective could offer a way to revitalize curricular reforms. Many curricular reforms in mathematics lose momentum because their "one?size fits all" approach fails to influence enough instructors. A community?level pilot study could investigate for which cognition?and?affect profiles an existing reform appears to be effective and then work through teaching experimentation and research to create several "sizes," i.e. tailor curricula for classroom and/or distance?learning formats based on the goals of the original reform efforts. The proposed avenue of research addresses one of the large issues confronting higher education today: distance learning. The format of an internet education site has, implicitly, assumptions about cognition?and?affect; i.e. a cognition?and?affect profile can be compiled from the forms and processes at a website. In many distance learning settings, the interaction of teachers and students does not disappear, but rather is made more complex by the mediation of the website (or other vehicle). This is especially so as distance learning in this country is shifting from a concern mainly with procedural knowledge to the teaching of conceptual knowledge (for example, in non?routine problem solving or theorem proving). As an area for research and development, this three?way relationship between students, teachers, and a mediating website has been little studied (in mathematics education) and is very promising. The proposed work would contribute to the foundation for future investigations of this topic.