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

在美国，工程、科学和数学专业的本科生开始大学数学训练时先学习几门微积分课程，然后再学习微分方程和线性代数等课程。数学专业和未成年人也可以学习真实的分析或抽象代数。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，并且往往是进一步学术成功的绊脚石。二年级和初级课程，如微分方程，线性代数，几何，以及介绍集合论和逻辑的课程，构成了一个核心集合的课程，有可能促进这种转变。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性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过渡到正式的，基于证明的数学课程。他们这样做是通过使用四个不同的角度对知识的个人和集体增长的性质。方法学产品将包括数据收集和数据分析的策略，这些策略允许深入了解学生在四个不同理论视角中的学习情况。这项工作的数学背景将主要是线性代数，从我们以前在微分方程，几何和集合论方面的工作中汲取见解。

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.