Collaborative Research: Investigating Issues of the Individual and the Collective along a Continuum between Informal and Formal Reasoning

在美国，本科工程、科学和数学专业的学生从几门微积分课程开始他们的大学数学训练，然后转向微分方程和线性代数等课程。数学专业的学生和辅修学生也可以学习实分析或抽象代数。学生们经常发现从微积分课程到更正式的、基于证明的数学课程的过渡特别具有挑战性，而且往往是进一步学业成功的绊脚石。大二和大三的课程，如微分方程、线性代数、几何，以及介绍集合论和逻辑的课程，构成了有可能促进这种转变的核心课程。该项目的主要目标是在非正式和正式数学推理之间的连续体方面对理论和方法做出贡献。特别是，pi将开发理论手段来解释向正式的、基于证明的数学课程的过渡。他们通过使用四种不同的观点来分析个人和集体知识增长的本质。方法论产品将包括数据收集和数据分析的策略，允许洞察学生在四个不同理论视角内和之间的学习。这项工作的数学背景将主要是线性代数，并从我们之前在微分方程，几何和集合论方面的工作中获得见解。

Undergraduate engineering, science and mathematics majors in the United States begin their university mathematics training with several calculus courses, but then move on to such courses as differential equations and linear algebra. Mathematics majors and minors may also study real analysis or abstract algebra. Students often find the transition from taking calculus courses to taking more formal, proof-based mathematics courses particularly challenging, and often a stumbling block to further academic success. Sophomore and junior level courses such as differential equations, linear algebra, geometry, and courses introducing set theory and logic constitute a core collection of courses that have the potential to facilitate this transition. The main goals of this project are to make contributions to theory and methodology in terms of the continuum between informal and formal mathematical reasoning. In particular, the PIs will develop theoretical means for interpreting the transition to formal, proof-based mathematics courses. They do so by using four different perspectives on the nature of the individual and collective growth of knowledge. The methodological products will include strategies for data collection and data analysis that allow for insights into student learning within and between each of the four different theoretical perspectives. The mathematical context for this work will primarily be linear algebra, with insights drawn from our prior work in differential equations, geometry, and set theory.