# Scholarship

My scholarly interests lie in two main areas: first in the intersections of commutative algebra, algebraic geometry, and graph theory; and second in scholarly teaching, including the use of mastery assessments, inquiry-based learning, and the interaction between mathematics, play, and the Christian faith.

### Mathematics

My dissertation research was in the exploration of the so-called containment problem. In its most general form, one may ask, given a homogeneous ideal I, for which m and r is the symbolic power I^(m) contained in the ordinary power I^r? In addition to exploring this question in my dissertation in the context of algebraic geometry, I explored a related question in the context of edge ideals I(G). In that case, certain invariants of I are shown to be related to invariants of the underlying (simple) graph G.

### Scholarly teaching

I am also very interested in the use of research-based methods for teaching and assessment. As I describe elsewhere, I am a proponent of active learning (thought of broadly) in all my courses, and I am also interested in the use of low-stakes, mastery-based assessments tied to clear standards for the purposes of encouraging mastery of fundamental skills and the development of a growth mindset in mathematics.

I compiled some WeBWorK-based preview activities for an elementary linear algebra course.

### Play

A new project (and one that I am keen to explore on this site) is an investigation of the interplay between mathematics, play, and the Christian faith. My teaching style is inquiry-oriented, and emphasizes problem solving; many of these problems can be thought of as puzzles, and thus hopefully introduce students to the playful side of mathematics. I aim to read and write on the literature linking these three areas.

## Publications

• Delivering Pre-Class Activities via WeBWorK to Impact Student Learning, Submitted July 2019 (with V. Zonnefeld)
• Comparing Powers of Edge Ideals, J. Algebra and its Applications, 2019, https://doi.org/10.1142/S0219498819501846 (with T. Kamp and J. Vander Woude)
• Mastery-based testing in undergraduate mathematics, PRIMUS, 29:5, 441-460, 2019, https://doi.org/10.1080/10511970.2018.1488317 (with J.B. Collins, A. Harsy, J. Hart, K. Haymaker, A. Hoofnagle, J. Stewart Kelly, A. Mohr, J. OShaughnessy)
• Start a Math Teacher Circle: Connect K-12 Teachers with Engaging, Approachable, and Meaningful Mathematical Problems, The 21st Proceedings of the Association of Christians in the Mathematical Sciences. (with T. Clark, A. Harsy, D. Klanderman, M. Maxwell, S. Robbert)
• The Waldschmidt constant for squarefree monomial ideals, J. Algbr Comb, 2016, http://dx.doi.org/10.1007/s10801-016-0693-7 (with C. Bocci, S. Cooper, E. Guardo, B. Harbourne, U. Nagel, A. Seceleanu, A. Van Tuyl, and T. Vu)
• On the Fattening of Lines in $\mathbf{P}^3$, Journal of Pure and Applied Algebra, 219(4):1055--1061, 2015, http://dx.doi.org/10.1016/j.jpaa.2014.05.033
• Containment problem for points on a reducible conic in $\mathbf{P}^2$, Journal of Algebra, 394(0):120--138, 2013, http://dx.doi.org/10.1016/j.jalgebra.2013.06.032 (with A. Denkert)

## Talks/Presentations

• Delivering Pre-Class Activities Online to Impact Student Learning, MAA Contributed Poster Session; MathFest 2019; Cincinnati, OH, August 2019
• Promoting Inquiry via Recreational Problems in a Liberal Arts Math Course, MAA Themed Contributed Paper Session on Inquiry-Based Learning; MathFest 2018; Denver, CO, August 2018
• A first experience in a flipped Calculus II course, MAA Themed Contributed Paper Session on Flipped Classes: Implementation and Evaluation; Joint Mathematics Meetings; San Diego, CA, January 2018
• Cut Your Cake (And Eat It, Too!), MAA Themed Contributed Paper Session on My Favorite Math Circle Problem''; MathFest 2017; Chicago, IL, July 2017
• An Overview of Specifications Grading, 21st ACMS Biennial Conference; Charleston Southern University, Charleston, SC, May 2017
• My Favorite Math Circle Problem: Cut Your Cake (And Eat It, Too!), 21st ACMS Biennial Conference; Charleston Southern University, Charleston, SC, May 2017
• Specifications Grading in a First Course in Abstract Algebra,
MAA Themed Contributed Paper Session on Teaching Abstract Algebra: Topics and Techniques''; Joint Mathematics Meetings 2017; Atlanta, GA, January 2017
• Improving proof-writing with reading guides,
MAA Themed Contributed Paper Session on Encouraging Early Career Teaching Innovation; MathFest 2016; Columbus, OH, August 2016
• Implementing Specifications Grading in a Linear Algebra course,
MAA Session on Assessing Student Learning: Alternative Approaches; Joint Mathematics Meetings 2016; Seattle, WA, January 2016
• The importance of $\alpha$,
MAA Invited Paper Session on Concrete Computations in Algebra and Algebraic Geometry; MathFest 2015; Washington, D.C., August 2015
• Symbolic Powers of Ideals: Problems and Progress (15 minutes),
20th ACMS Biennial Conference, Redeemer University College, Ancaster, Ontario, Canada, May 2015
• Symbolic Powers of Ideals: Problems and Progress (50 minutes),
Mathematics Seminar; University of South Dakota, March 2015
• On the Fattening of Lines in $\mathbf{P}^3$,
Oberwolfach Mini-Workshop 1508a, Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems; Mathematisches Forschungsinstitut Oberwolfach; Oberwolfach, Germany, February 2015
• On an Application of Bezout's Theorem,
Colloquium; Calvin College, February 2014
• Ideals of almost collinear points in $\mathbb{P}^2$,
Special session on Interactions Between Algebraic Geometry and Commutative Algebra; Summer 2012 Meetings of the Canadian Mathematical Society; Regina, Saskatchewan, June 2012
• Containment problem for ideals of points on a reducible conic,
Special session in Algebraic Geometry and Graded Commutative Algebra; Fall 2011 AMS Central sectional meeting; Lincoln, NE, October 2011
• Results on the containment problem of ideals of fat points,
Algebraic Geometry Seminar; University of Nebraska-Lincoln, April 2010
• What is Algebraic Geometry? (50 minutes),
Geometry and Physics on Graphs REU; Canisius College; Buffalo, NY, July 2009
• A Fifteen-Minute Survey of Nonunique Factorization,
IMMERSE at the University of Nebraska-Lincoln, July 2008