An investigation of freshmen an senior mathematics and teaching mathematics students' conceptions and practices regarding proof

Abstract

This study was conducted with 93 freshmen and 82 senior students from Mathematics, Primary Education and Secondary Education Teaching Mathematics Programs in Boğaziçi University in order to investigate their attitudes and beliefs regarding proof, the types of reasoning and proof methods they use while constructing proof, as well as their proof evaluation practices. Three instruments were developed for this purpose: Attitudes and Beliefs Scale (ABS), Proof Exam (PE) and Proof Evaluation Exam (PEE). Data were collected in 2009-2010 fall and spring semesters during lecture sessions with the presence of the researcher. Factor analysis on ABS yielded four components, labeled background, attitude, self efficacy and beliefs. Students' responses to PE were categorized with respect to the types of proof they attempted and also scored according to their reasoning styles, ranging from experimental-inductive using numerical examples to formal deductive using symbolic mathematical language. PEE responses were categorized by examining whether the students think the provided argument proves the statement is true for all cases, true for some cases or does not prove the statement, and scored according to their accuracy of evaluating the given arguments. Results of related descriptive statistics regarding data collected by these three instruments are also reported. Comparisons of the results with respect to grade (freshmen and seniors) and department (Mathematics, Secondary Education Teaching Mathematics, and Primary Education Teaching Mathematics Programs) are done using appropriate quantitative analysis methods. Results indicate statistically significant differences among senior and freshmen students' background, attitude, self efficacy and beliefs subscales, where freshmen have higher background scores than seniors but for the other three subscales seniors' scores are higher. Seniors also have statistically significant higher PE scores than freshmen. While there are no significant differences among freshmen with respect to department in neither ABS nor PE scores; senior Mathematics, Secondary Education Teaching Mathematics and Primary Education Teaching Mathematics students' mean scores differ significantly both in PE and PEE, as well as some subscales of ABS. It has been observed that freshmen students mostly rely on inductive reasoning while attempting to prove given mathematical statements. Seniors are generally aware of the necessity of generalizing their results and attempting to use procedures involving deductive reasoning. Nonetheless seniors still have difficulties in constructing and evaluating proofs. Significant differences observed between senior Mathematics and Secondary and Primary Education Teaching Mathematics students' proof construction and evaluation practices are in favor of Mathematics majors. Implications for teaching and further studies are discussed.