Mathematics, Writing, and Rhetoric: Deep Thinking in First-Year Learning Communities

Through the process of combining two seemingly unlikely bedfellows, mathematics and composition, two instructors explain how rhetoric connects the art of writing and the art of doing mathematics in an inquiry-based learning community. Combining these two courses in a learning community enables students and instructors to practice the deep thinking valued by each instructor and by a traditional liberal arts education while challenging both our and our students’ individual, disciplinary, and rhetorical conventions and beliefs. Using student writing from our course, our assignments from mathematics and composition, and survey evaluation results, we demonstrate how engaging in inquiry-based education provides unconventional (and conventional) learning opportunities for both students and instructors. Furthermore, through our discussions of the four iterations of our Learning Community, we examine some ways interdisciplinary learning challenges structural, individual, and disciplinary expectations, conventions, and learning

Two examples of ungrading in higher education from the United States and from Germany

In this paper the authors discuss their experiences with ungrading at a small public university in the U.S. as well as a large public university in Germany. The courses described are Calculus 1, Mathematics for Liberal Arts, and courses for pre-service secondary teachers of mathematics. We outline and compare our approaches, discuss student performance and feedback and present some interesting patterns relating to gender. A shortened and revised version of this paper appeared in PRIMUS 33 (2023), no. 9, 1035-1054, DOI: 10.1080/10511970.2023.2229819.Comment: 23 page

Supporting Teaching and Learning Reform in College Mathematics: Finding Value in Communities of Practice

Improving college STEM (science, technology, engineering, mathematics) student learning outcomes is an ongoing area of focus in Institutions of Higher Education (IHE). This reform includes challenging, changing, and adapting both teaching practices and the learning environment. Communities of practice (CoPs) can support faculty in making these shifts; however, creating large-scale instructional changes in STEM education requires a more careful look at the existing systems and structures in place. In this paper, we investigate a network of regional CoPs composed mainly of mathematics faculty from IHE focused on teaching with inquiry methods. Understanding what faculty need and value to support their instructional changes is important as CoPs and other mechanisms are put in place to increase student success. In this qualitative study, we use the value framework developed by Wenger et al. (2011) to dissect the variety of ways faculty engage and find value in their CoP participation. Faculty participants expressed that CoP participation created unique layers of value in helping them to identify resources to support teaching with inquiry especially during a pandemic, shift their beliefs about teaching, and engage with a network of peers about mathematics and teaching. Findings from this study, conducted during the COVID-19 global pandemic, provide preliminary insights for STEM stakeholders interested in large-scale, ongoing instructional reform to improve student learning outcomes and for networks interested in collectively supporting CoPs with ongoing rather than finite goals

Combinatorial aspects of toric varieties

According to Batyrev the Mori cone of a smooth, complete and projective toric variety can be generated by primitive relations. A primitive relation comes from a primitive collection, which is a set of 1-dimensional cones of the fan Σ such that the whole collection {ρ1, ..., ρk} does not generate a cone in Σ, but every subset does. To prove Batyrev\u27s smooth case you can relate wall collections and primitive collections. I generalize Batyrev\u27s statement to the non-complete, non-smooth but simplicial case and to the non-simplicial case. Lawrence toric varieties arise as GIT-quotients. Hausel and Sturmfels showed that the cohomology of Lawrence toric varieties is independent of the GIT parameter. I will give a different proof for this result. Moreover, I will show that a natural way of generalizing these varieties does not have independent cohomology anymore by presenting some counter-examples

PRIMITIVE COLLECTIONS AND TORIC VARIETIES

This paper studies Batyrev’s notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with an open problem