Emerging Research-Empirical--Proving Styles in University Mathematics

这个实证研究项目正在开发一种工具来测量学生在构建证明时的推理风格，并将其应用于大规模的假设，即本科生在证明风格上是一致的，并且无论风格如何，都可以在数学上取得成功。将回答三个研究问题：1。数学专业本科学生对证明生成任务的推理在多大程度上表现出一致性？在试图构建一个给定命题的证明时，根据他们使用这些策略的程度，学生的分布是什么？大学生的推理方式与他们的一般数学能力、他们的一般思维方式和他们的学业成就之间有什么关系？这项工作是由罗格斯大学新不伦瑞克分校和蒙特克莱尔州立大学的一个研究小组进行的。个别学生的证明风格将通过分析他们对一个工具的反应来确定，该工具要求证明微积分和线性代数中的18个陈述，特别选择以区分证明风格。研究问题将通过对100名本科数学专业学生的一般推理风格测试和管理该工具来回答。参与者被录下来，并被要求在解决证明仪器上的一半问题时大声思考。回归分析用于研究证明风格，一般推理风格和数学SAT成绩之间的关系。聚类分析用于根据一般推理风格对学生进行排序，然后使用方差分析来查看聚类是否在证明风格上有所不同。拟议工作的最终产品将是检测证明方法的可靠和有效的工具。这种工具有可能帮助拓宽大学数学教师对学生对证明的理解的看法。这种拓宽的视角可能会导致证明教学方式的改变，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.