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

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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.