The relationship between mathematical induction, proposition functions, and implication functions.

摘要

In this study, I explored the relationship between mathematical induction ability and proposition and implication functions through a mixed methods approach. Students from three universities (N = 78) and 6 classrooms completed a written assessment testing their conceptual and procedural capabilities with induction and functions. In addition, I interviewed a subgroup of 10 participants to add context and meaning to the assessment results. This research study was unique in that it provided numeric correlations among important variables. The correlation between induction ability and function ability was r = 0.47 (p 0.001). The general linear model Mathematical Induction = -0.300 + 0.122 ACT Math + 0.222 Function Ability was significant at p 0.05 and explained 28.3% of the variation in induction ability. In the written assessment, I asked participants to construct two induction proofs. Out of the 156 attempts, 57 attempts were successful (37%). During the interview analysis, I identified participant subgroups based on mathematical goals, background, and motivation. I argued that the characteristics of these subgroups related directly to their scores on the written assessment. In particular, students who perceived mathematical induction as useful to themselves in their future career put forth the energy required to learn induction, both procedurally and conceptually. Based on the results of this study, I recommended that students learn about proposition functions prior to studying induction. I also recommended that the amount of class time spent on the instruction of induction increase along with a continued focus on the conceptual elements of the proof technique.