Research

My work centers on three broad themes: 

1. Using graph theory and combinatorics to analyze and improve the design of mathematical tasks; 

2. Developing quantitative combinatorial methods for examining the structure of mathematics textbooks and curricula.

3. Studying how students learn, persist, and engage with challenging mathematics, particularly in calculus; 

Across these areas, I am interested in how mathematical theory can be used to better understand teaching, learning, and the organization of mathematical knowledge.

My interests in discrete mathematics center on graph theory and combinatorics, particularly the ways these fields can be used to model structure, relationships, and organization. I study problems involving graph decompositions, combinatorial designs, and tree-based representations, with applications ranging from mathematical task design to the analysis of textbooks and curricula. I am especially interested in research that connects abstract mathematical theory with practical questions in teaching, learning, and educational design.

Recent Publications

• Gilroy, H. (2025). On the Existence of Balanced Chain Rule Task Sets. Journal of Combinatorial Mathematics and Combinatorial Computing, 127, 125-146. https://doi.org/10.61091/jcmcc127-09. 

• Bezdek, A., Gilroy, H., Henderschedt, O., & Lakhani, A. (2023). On Conway’s Brussels Sprouts. Studia Scientiarum Mathematicarum Hungarica: Combinatorics, Geometry, and Topology, 60(1), 76-90. https://doi.org/10.1556/012.2023.01535.

My work in mathematical task analysis and design examines how individual problems and collections of problems are structured. Using tools from graph theory and combinatorics, I study how mathematical features, procedures, and concepts are distributed across task sets. This allows me to identify repetition, imbalance, and gaps in the opportunities students receive to engage with important ideas. I also use these analyses to develop principled approaches for designing assignments and assessments that provide greater variety, balance, and mathematical coherence.

Recent Publications

• Gilroy, H. (2026, In Press). Measuring the Cognitive Orientation of Calculus Textbooks and Their Tasks. In V. Martinez-Luaces (Ed.), New Trends in Teaching and Learning of Calculus. Springer. (Book Chapter)

• Gilroy, H. & Harris, I. (2025). On doing and undoing in general and applied precalculus-calculus tracks. In L. Branchetti, S. Erduran, F. Feudel, A. Gonzalez-Martin, & O. Levrini (Eds.), The Learning and Teaching of Calculus Across Disciplines 2 – Proceedings of the Third Calculus Conference (pp. 77-81). Milan, Italy.

My work in mathematical textbook and curriculum analysis examines how mathematical ideas are organized, connected, and emphasized across instructional materials. I use graph-theoretic models, tree-based representations, and quantitative comparison methods to study the structure and sequencing of topics, examples, and tasks. This approach makes it possible to identify patterns of emphasis, gaps in coverage, and differences among textbooks or curricula that may not be apparent through traditional qualitative review. Ultimately, I am interested in how the organization of instructional materials shapes the mathematical experiences available to students.

Recent Publications

• Gilroy, H., Lanius, M., & Grate, S. (2025). Graph-theoretic reflection to foster alignment in coordinated courses. International Journal of Mathematical Education in Science and Technology, 1-18. https://doi.org/10.1080/0020739X.2025.2553315.

• Gilroy, H., & Lanius, M. (2023). On motivation and narrative in discipline-specific calculus texts. In T. Dreyfus, A. S. Gonzalez-Martin, E. Nardi, J. Monaghan, & P. W. Thompson (Eds.), The Learning and Teaching of Calculus Across Disciplines – Proceedings of the Second Calculus Conference (pp. 105-108). MatRIC. Bergen, Norway. https://matriccalcconf2.sciencesconf.org/

My work on student learning, persistence, and engagement examines how undergraduate students interact with challenging mathematics, particularly in calculus. I study factors such as homework behaviors, productive struggle, mathematical mindset, and students’ responses to difficulty to better understand why some students persist while others disengage. I am especially interested in how instructional practices, assessment design, and learning environments can support deeper engagement, strengthen students’ confidence, and improve their ability to succeed in advanced mathematics.

Recent Publications

• Gilroy, H. & Hensley, D. (2026). Calculus Students’ Mindsets are Not So Simplex. In A. P. Adiredja, B. P. Katz, K. Melhuish, & K. Gallagher (Eds.), Proceedings of the 28th Conference on Research in Undergraduate Mathematics Education (pp.1438-1439). Alexandria, VA.

• Hensley, D. & Gilroy, H. (2025). An Extension of Dweck’s Mindset Theory Within Calculus I. In R. Martinez, A. McCloskey, X. Yao, & R. M. Zbiek (Eds.), Proceedings of the 47th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1665-1669). Pennsylvania State University. https://doi.org/10.51272/pmena.47.2025.

• Hensley, D. & Gilroy, H. (2025). “I’VE ALREADY DONE THIS”: How prior exposure affects calculus students’ mindsets during a semester of regular reflections. In S. Cook, B. Katz, & K. Melhuish (Eds.), Proceedings of the 27th Annual Conference on Research in Undergraduate Mathematics Education (pp. 1496-1497). Alexandria, VA.