One of the most valuable learning experiences for students outside of the classroom is a research project involving an interest of the student. Research exposes students to cutting-edge technologies in their discipline, stimulates creativity and innovation, strengthens critical thinking skills, improves technical writing ability, and promotes collaboration. Part of my research vision is to enhance undergraduate learning through independent studies and honor’s theses, as well as through research that results in a student publication or a talk at a conference.
Specifications Grading
Over the past decade, a new paradigm of grading called specifications grading (also called specs grading) has become more popular at the college level. While others may use the term a little more broadly, I will use specs grading to refer to a grading system in which
• students are provided with a clear list of course learning objectives they must complete;
• final course grades are based primarily on how many objectives the student completes;
• students are provided ample opportunities to reassess mastery of any objective;
• only the best mark for each objective is used toward the course final grade.
There is much evidence that specs grading can increase student receptivity to meaningful feedback, enhance student motivation to learn from their mistakes, lower their stress level related to academic expectations, and strengthen their work ethic through the process of reassessments.
I have been using specs grading in my classes since 2016. I co-authored an article in the journal PRIMUS describing my implementation of specs grading to mathematics courses and how it promoted a growth mindset among our students. In that article, I describe a pass/fail grading system (i.e. no partial credit) to measure mastery of learning outcomes. I have since changed the appearance of the grading to involve points instead (and hence a form of partial credit). I have found my students are more receptive to the grading scheme when presented in the form of points rather than just which problems were right and wrong.
Here is a more in-depth description of the alternative grading schemes, standards-based grading, specs grading, and ungrading.
Active Pedagogical Research
Specs grading allows students multiple attempts to try problems that are each tied to a specific learning outcome. Over the years, I have been collecting anecdotal data of students' grades from my courses. I have observed that students who have personal issues that cause them to disconnect with my course and/or fall behind can be given the opportunity to catch up when those issues subside. Usually, I would reach out to any such students and invite them to my office hours, where we can go over material and decide a plan for when they can retake certain problems and what that student will do to prepare. In essence, specs grading can allow students to achieve passing grades in ways that would typically not be allowed using traditional grading, where exams are only given on pre-set dates and cannot be retaken for a higher grade.
Of particular interest is the effect specs grading may have on test anxiety and/or math anxiety. It is an unfortunate fact that more and more students are experiencing a form of anxiety related to their grades or performance. On my end-of-semester surveys, many students report feeling less stressed on assessments since there are multiple opportunities. To test this further, I used a test anxiety inventory (i.e., STAI-5, a validated measure of test anxiety) to measure any difference in a student’s anxiety at the beginning and end of my courses. The data was collected in the fall 2022 semester. There is not much research on the possible effects that alternative grading may have on student anxiety, though more studies are being performed. I plan to use this "quasi-experiment" to develop motivation for studying this effect on a grander scale.
I am also in the process of writing an updated article describing my current implementation of specs grading, and how I have created all the files I use to help organize the grades and to help students understand how the grading works.
Active Mathematical Research
Previously, I worked with Bojko Bakalov (my former Ph.D. advisor) from North Carolina State University on research that extended the work I did for my dissertation. This problem (on which I worked from 2012 to 2014) was a smaller piece to a larger puzzle that Bojko, Ivan Todorov, and Victor Kac (Bojko's former Ph.D. advisor) had previously been working on but could not finish. This project studied the representation theory of a fixed-point subalgebra (called an "orbifold") of a vertex algebra generated from an even lattice. The fixed-points correspond to an automorphism of the vertex algebra, stemming from an isometry of the lattice. I my dissertation, I had explicitly described the representations in the case of an order two isometry. In my work with Bojko, Ivan, and Victor Kac, we extended these explicit representations to automorphisms of prime order. This extension involved heavy use of Theta functions and what we called "modified characters". This project is also described in the "Publications" section of this portfolio.
More recently, I have been collaborating with a former student, Anthony Stefan (who is currently a graduate student at FIT), and Aaron Welters (who is known for his work in the theory of composites and also Anthony's Ph.D. advisor), and Ian Orzel (who recently graduated from FIT and is now a graduate student in computer science). This project involves the Bessmertnyi realization theory for rational functions of several complex variables, which is well known, but its general construction is still an open question. Another related problem we are working on concerns the determinantal representation of a polynomials. Given a polynomial P, a determinantal representation of P is a matrix M such that the determinant of M is the polynomial P. There are conditions one may put on the entries of M which result in more special or more general representations. While this problem of finding determinantal representations has been solved in fields of characteristic not equal to two, the problem is still open in characteristic two. In particular, we have already found an example polynomial that does not have such a representation in characteristic two (but it does in other characteristic). I will also describe this project in the "Publications" section of this portfolio.
More on my research agenda, and sources for the statements made within this section, can be found in my research statement.