Emerging Research-Empirical--Proving Styles in University Mathematics

本实证研究项目旨在开发一种在构建证明时测量学生推理风格的工具，并将其用于大规模检验本科生证明风格一致且无论风格如何都能在数学上取得成功的假设。将回答三个研究问题：1.数学专业本科生在证明任务上的推理一致性在多大程度上？2.根据学生在试图构建给定陈述的证明时使用这些策略的程度，学生的分布情况如何？3.本科生的推理风格与他们的一般数学能力、一般思维方式和他们的学术成就之间有什么关系？这项研究是由罗格斯大学、新不伦瑞克大学和蒙特克莱尔州立大学的一个研究小组完成的。个别学生的证明风格将通过分析他们对一种工具的反应来确定，该工具要求从微积分和线性代数中特别选择18个陈述的证据来区分证明风格。这些研究问题将通过对100名数学专业本科生使用该工具和一般推理风格的测量来回答。参与者被录制下来，并被要求在解决校对工具上一半的问题时大声思考。采用回归分析的方法研究证明风格、一般推理风格与数学SAT成绩之间的关系。聚类分析被用来按照一般的推理风格对学生进行分类，然后使用方差分析来查看聚类是否在证明风格上存在差异，所提出的工作的最终结果将是检测证明方法的可靠和有效的工具。这样的工具有可能有助于拓宽大学数学教师如何看待学生对证据的理解。这种开阔的视角可能会导致教证明的方式发生变化，并使更多的大学生更容易接触到高等数学。

This empirical research project is developing an instrument to measure students' reasoning styles when constructing proofs and applying it to test on a large scale the hypotheses that undergraduate students are consistent in their proving style and can be successful in mathematics regardless of style. Three research questions will be answered: 1. To what extent do undergraduate mathematics majors show a consistency in their reasoning on proof production tasks?2. What is the distribution of students according to the extent to which they employ these strategies when attempting to construct a proof of a given statement?3. What is the relationship between undergraduates' reasoning styles and their general mathematical ability, their general thinking style, and their academic success?The work is being performed by a research team at Rutgers University New Brunswick and Montclair State University. The proving styles for the individual students will be identified from analysis of their responses to an instrument that calls for proofs of 18 statements from calculus and linear algebra specifically selected to distinguish proving styles. The research questions will be answered by administering the instrument along with a measure of general reasoning style to 100 undergraduate mathematics majors. Participants are videotaped and asked to think aloud when solving half of the items on the proof instrument. Regression analyses are used to study the relationship between performance on proving style, general reasoning style, and mathematics SAT scores. Cluster analyses are used to sort students by general reasoning styles and then an ANOVA is used to see if clusters vary in proving styles.The final product from the proposed work will be a reliable and valid instrument to detect approaches to proof. Such an instrument has the potential to help broaden how teachers of mathematics at the college level view students' understanding of proof. Such a broadened perspective could lead to a change in how proof is taught and make higher mathematics more accessible to a larger group of college students.