Transfer and Meta-Cognitive Intervention in Conceptually Non-Isomorphic Linear Algebra Problem Settings

This dissertation arose out of an awareness of difficulties undergraduate linear algebra students encounter when solving linear algebra problems from non-isomorphic settings, even when the problems could be solved with matrix representations and similar procedures as problems from a more familiar setting. This mixed-methods study utilized both traditional and actor-oriented transfer paradigms. From a traditional transfer perspective, two factorial experiments demonstrated that novice undergraduate linear algebra students experienced difficulty in solving problems from non-isomorphic settings, even though the problems admitted similar matrix representations and solution-interpretation procedures. Upon qualitative interviewing using problems similar to those from Experiments 1 and 2, from an actor-oriented transfer perspective, three case studies revealed a scalar-variable conflict phenomenon as a factor impeding success for the production of a correct matrix representation or interpretation of a matrix solution, in the context of unfamiliar settings. In addition, interview evidence suggested Harel's (1999) contextual conception contributed as an obstacle to actor-oriented transfer due to overly-practical reliance upon proto-typical examples from familiar settings. Demonstrating evidence of a general pattern of actor-oriented transfer as a cycle of practical-theoretical thought processes, subjects were seen to perform similar personal constructions between conceptually non-isomorphic settings with the aid of meta-cognitive interventions which seemed to ease transitions from practical to theoretical thinking. The evidence suggests subjects produced forward co-ordinations relating setting-specific contextual information from unfamiliar settings to contextual information from familiar settings in the actor-oriented transfer of a matrix representation for a novel problem setting. Likewise, in a process of actor-oriented transfer, reflective-like backward co-ordinations were seen to aid the transfer of solution interpretations from a familiar setting to correct contextual interpretations in unfamiliar non-isomorphic settings. Based on Experiments 1 and 2 and the interview findings, the Intentional Transfer Hypothesis was conjectured and a final treatment/non-treatment designed Experiment 3 verified the effectiveness of a meta-cognitive intervention for linear algebra problem solving in an unfamiliar, non-isomorphic setting. Additional theoretical implications, limitations of the study, and recommendations for future research are also discussed.