maclean

I am interested in various areas of Geometry and Analysis, and my current research is on asymptotic homotopy and a \(\pi_1\) de Rham Theorem. I am also engaged in research focused on how students learn mathematics. 

I am interested in understanding how we understand mathematics and I am working on projects that make use of comparative judgment to study how students learn mathematics. With some wonderful colleagues (faculty and staff), I co-developed ComPAIR, a tool for implementing comparative judgment exercises. This is an open education resource. I am currently working with colleagues in the School of Mathematics and Statistics at the University of Melbourne to study how using comparative judgment exercises may affect students' learning. I am also undertaking a UBC BREB approved study on the use of a natural language AI in comparative judgement assessments in MATH 253 Multivariable Calculus.  

I am involved in writing children's stories -- Adventures of Small Number -- to encourage interest in mathematics. This is a project with Veselin Jungic at SFU and involves collaborations with others. Many of our co-authors re-tell these stories in their Indigenous languages. There are some wonderful animations done from the illustrations for these stories. In addition to versions in English and French, there are versions in Blackfoot, Cree, Squamish, and Halq'eméylem, Sliammon, Nisga'a, Heiltsuk, Hul'q'umi'num, and Huu-ay-aht/Nuu-chah-nulth. 

I am on the organzing committee for the upcoming conference in the series Indigenising University Mathematics (IUM), which is part of a unique and emerging international conference series that has grown from a grassroots desire to explore what it may mean to Indigenize mathematical practices in university spaces. Since its inception, the IUM project has been a collaboration between mathematicians and Indigenous knowledge holders and practitioners, carried out with deep care for the integrity of the knowledge systems involved.

Awards

• Canadian Mathematical Society Adrien Pouliot Award in Mathematics Education, 2015.
• Pacific Institute for the Mathematical Sciences (PIMS) Education Prize, 2012.
• Killam Teaching Prize, Faculty of Science, UBC, 2005.

Manuscripts

1. Subriemannian Geometry:

A. The Asymptotic Development of Paths on Nilmanifolds, preprint, 18 August 2026.

Some animations for the asymptotic development of paths: (1) an animation of the subsequence macroscopic paths result in Proposition 3.6 for Example 4.2 in this paper, (2) an animation showing the full limit failure for the circular arc from Example 4.2, which provides insight into Proposition 3.5, (3) an animation showing that different macroscopic subsequences may have different Carnot developments, which illustrates Remark 5.3, (4) an animated shape gallery to see the realization of macroscopic paths established by Proposition 3.6, (5) an animation comparing the convergence along a suitable subsequence and the failure to converge at scales between the subsequence values, and (6) an animation in the Engel Lie group showing the depth-3 area moment in Example 4.4.

Some animations for the period array from Section 6 and the development on the Heisenberg group and nilmanifold:  (1) an animation showing two horizontal paths in the Lie algebra with the same macroscopic path and same asymptotic homotopy, but different sub-diagnonal period arrays, (2) an animation of the development in the Heisenberg group, and (3) an animation of the development on the Heisenberg nilmanifold with a second version highlighting a nontrivial drift in both horizontal directions. 

A Reading Guide to the Chacon-Fomenko Lie Integral Papers in Advances in Mathematics 88(2), 1991.  The Chacon-Fomenko Lie integral plays a central role in defining the asymptotic development of paths.

B. The Pendulum, Heisenberg Development, and the Period Array,  note, 1 September 2026.  I explore relationships between the asymptotic development results reported in "The Asymptotic Development of Paths on Nilmanifolds" and classical approaches to the dynamics and mechanics of the familiar nonlinear pendulum. These include interpretations of the subleading period array in terms of the classical action and period, Helmholtz mechanical temperature and integrating factor, and an associated Lie symmetry. The circulating energy regime also illustrates why asymptotic development must sometimes be considered along subsequences rather than only through full limits.  

2. Comparative Judgement (research with Jennifer Palisse and Deborah King (University of Melbourne)):

A. Shaping Attentional Focus in Mathematics: How Comparative Judgement Tasks Influence Student Noticing, Meanjin Delta 2025 Proceedings: The 15th Southern Hemisphere Conference on the teaching and learning of undergraduate mathematics and statistics, 24–28 November 2025, Brisbane, Australia.

B. Exploring the role of internalised standards in comparative judgement in mathematics, Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education, vol. 2, pp. 155 -- 162, 2025, Sanitago Chile. 

C. Comparative judgement and its impact on the quality of students' written work in mathematics, Fifth conference of the International Network for Didactic Research in University Mathematics, Centre de Recerca Matematica [CRM], Jun 2024, Barcelona, Spain. 

D. Does comparative judgement reduce students' perceived cognitive load when evaluating mathematics solutions? Proceedings of the Australian Conference on Science and Mathematics Education (2023). 

E. Comparative judgement and the hierarchy of students' choice criteria, International Journal of Mathematical Education in Science and Technology, 53(1), 2022.

F. Comparative Judgement and Affect: A Case Study, Annual Meeting of the Mathematics Education Research Group of Australisia, Singapore, 2021.