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.
Overall this project achieved two main objectives: 1) To create theoretical and methodological means for understanding and interpreting student learning along the informal-formal continuum using and coordinating between four different lenses on student learning, using prior work and a new context of linear algebra as paradigmatic examples. 2) To conduct design research in linear algebra, leading to both practical outcomes for practitioners and more general analyses of student learning and means to support student learning. Overall we made very good progress on both of these goals and we have set the foundation for continued work on these objectives and related extensions. The foundation for this continued work consists of both the accumulated collection of publications (7 published journal articles, 5 under review, 5 near completion, 4 book chapters, and 41 conference proceedings) but also, more importantly, in the extent to which we have and continue to build capacity in the field. These capacity building efforts include mentoring and training of doctoral students who are now taking leadership positions in the field and continuing the work started in this project, supporting and mentoring undergraduates as they gain experience doing research and look forward to graduate school, building and sustaining national and international collaborations that extend the project’s objectives, and outreach activities to the broader mathematical community. Our work has also led to connections with undergraduate chemistry and physics education researchers. We have also fostered international collaborations with researchers in Israel and Mexico. Publications and presentations have reached basic researchers and undergraduate mathematics instructors. Our work has offered researchers new ways to understand and analyze student learning and produced instructional materials that actively engage students in creating powerful mathematics.
