The Kauffman bracket skein module of a twist knot exterior

We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other component is limited to the number of twists.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-6.abs.htm

Topological Interpretations of Lattice Gauge Field Theory

We construct lattice gauge field theory based on a quantum group on a lattice of dimension 1. Innovations include a coalgebra structure on the connections, and an investigation of connections that are not distinguishable by observables. We prove that when the quantum group is a deformation of a connected algebraic group (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of characters of the fundamental group of the lattice with respect to the corresponding algebraic group. Finally, we investigate lattice gauge field theory based on quantum SL(2,C), and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice.Comment: 35 pages, amslatex, epsfig, many figures; email addresses: [email protected], [email protected], [email protected]

Coherent Calculus Course Design: Creating Faculty Buy-In for Student Success

This paper recounts the process used and results achieved as first-semester Calculus at Boise State University was transformed over a period of approximately 16 months from a collection of independent, uncoordinated, personalized sections, into a single coherent multi-section course. During the process of this transformation, section size and the instructor pool remained relatively constant; however, profound changes were made across all sections in terms of pedagogy, homework, timing of course content, grade computation and exam content. The motivation for focusing on Calculus I arose from a five-year National Science Foundation Science Talent Expansion Program grant that was awarded in 2010 to a multi-disciplinary team that spanned engineering, mathematics and science. A major grant objective was to raise first-semester, full-time retention of students in STEM majors. The projects supported several yearlong faculty learning communities (FLCs) of about 10 instructors each. With significant involvement from mathematics faculty, the first two FLCs prepared the ground for pedagogical reform of calculus. In 2013-14, a final FLC was created with the express purpose of implementing consistent, student-learning focused strategies across several section of calculus. The specific approach used to design a coherent calculus course was tied to a decision made by the FLC to use identical homework assignments, with common due dates and times. The FLC structure facilitated buy-in and rapid communication and feedback between instructors, who as they came to agreement on the exact homework exercises, also came to agreement on learning goals and content for each individual lesson. Although there was no explicit attempt to have all instructors adopt the same pedagogy or classroom practices, because FLC discussions frequently turned to pedagogy, all members of the FLC chose to adopt a similar pedagogical approach which included devoting class time to solving problems, working in small groups, facilitated by the lead instructor and a learning assistant. In subsequent semesters, all calculus instructors have opted in to the common, coherent approach to the course (except for those teaching online or honors sections). Pass and withdrawal rates pre and post implementation reveal an increase in pass rate of 13.4% and a drop in withdrawal rate of 3.9% as a result of the project. Results from anonymous faculty surveys show that faculty in the project changed their teaching practices in Calculus, that they observed positive effects of this in their classrooms, that they took advantage of the FLC to learn from their colleagues and that their experiences with Calculus will have spillover impacts in their other classes. Results from student surveys show, among other things, that students were aware of the pedagogical difference in terms of their classroom experience, with some expressing discomfort in terms of working in groups to solve problems in class and not receiving a traditional lecture experience and others reporting group work as a valuable aspect

Longitudinal Success of Calculus I Reform

This paper describes the second year of an ongoing project to transform calculus instruction at Boise State University. Over the past several years, Calculus I has undergone a complete overhaul that has involved a movement from a collection of independent, uncoordinated, personalized, lecture-based sections, into a single coherent multi-section course with an activelearning pedagogical approach. The overhaul also significantly impacted the course content and learning objectives. The project is now in its fifth semester and has reached a steady state where the reformed practices are normative within the subset of instructors who might be called upon to teach Calculus I. Gains from the project include a rise in the pass rate in Calculus I, greater student engagement, greater instructor satisfaction, a general shift toward active learning pedagogies, and the emergence of a strong collaborative teaching community. Project leaders are seeking to expand these gains to other areas of the curriculum and to broaden the community of instructors who are fully accepting of the reforms. Common concerns expressed by faculty resistant to the overhaul include suspicion that pass rate gains might reflect grade inflation or weakened standards, and that altering the traditional content of Calculus I might leave students unprepared for Calculus II. External stakeholders also have a vested interest in ensuring students receive a solid preparation in Calculus I. In this paper we develop a response to ensure solid evidence of Calculus II readiness that we hope will be useful to change agents and campus leaders in many other settings. We address concerns about Calculus II readiness by conducting a natural experiment, tracking two cohorts of students through Calculus I and into Calculus II. The “treatment” cohort consists of students who reach Calculus II after passing the reformed Calculus I. The “control” cohort consists of students who reach Calculus II after passing non-reformed Calculus I at Boise State University. The experiment has no designed randomizing, but enrollment data shows that both cohorts spread out across all sections of Calculus II with apparent randomness. Our research question is: “Does the treatment cohort perform any worse than the control cohort in Calculus II?” Data on pass rates and grades in Calculus II will show that the answer is “No.

The Crux: Promoting Success in Calculus II

In the 2013-14 school year, Boise State University (BSU) launched a major overhaul of Calculus I. The details of the reform, described elsewhere, involved both pedagogical and curricular changes. In subsequent years, we developed several assessment tools to measure the effects of the project on students’ grades and retention. The toolkit includes: (1) pass rate and GPA in Calculus I, (2) longitudinal analysis of pass rates and GPA in subsequent courses, (3) impact of Calculus I on retention in STEM and retention at BSU, (4) all of the above comparing students in reformed Calculus vs traditional Calculus, (5) all of the above for underrepresented minorities, women, or other demographic subsets. While these tools were originally developed to study the Calculus I project, they are available for studying the effects of other courses on student academic performance and retention. In this paper, we briefly describe a rebuild of Calculus II, overhauled in the 2015-16 school year following the same general plan as was used for Calculus I. We then present the results of applying the full toolkit to the new Calculus II course. Pass rate and GPA improvements in Calculus II were evident immediately after scale up in the spring of 2016. Sufficient time has now passed so that we can apply the full set of assessment tools built for Calculus I to measure the effectiveness of the Calculus II transformation on academic performance in post-requisite coursework and on student retention in STEM