Yvonne Lai

picture taken in Colorado

Assistant Professor
Department of Mathematics
University of Nebraska-Lincoln

Email: yvonnexlai at unl dot edu
Office: 306 Avery Hall

Yvonne Lai's research aims to improve instruction of mathematical reasoning at all levels by identifying and preparing teachers in the mathematical knowledge for teaching (MKT) necessary for such instruction. In her work on MKT, she builds on the work of Ball and colleagues (e.g., Ball, Thames, and Phelps (2008)). She brings a mathematics background to her work in mathematics education, specializing in hyperbolic geometry and geometric group theory prior to her interest in mathematical knowledge for teaching and practices of proof and reasoning. Her current projects investigate the cognition of teaching mathematical claim making and proving in secondary mathematics; the role of mathematical knowledge across different time scales (with Ruth Heaton and Mary Alice Carlson); differences between secondary MKT and elementary MKT and implications for professional development, teacher education, and measuring teachers' knowledge (with Heather Howell and Geoffrey Phelps); and the impact of content rich mathematics preparation on teaching practice (in collaboration with Jim Lewis, Lorraine Males, Wendy Smith, and Steve Swidler).

picture of a murphy stove in the first apartment I rented when 
	     	I lived in Ann Arbor (each range folded into the wall)


Areas of interest:
Mathematical Knowledge for Teaching, Cognition of Mathematical Claim Making, Practices of Proof and Reasoning

Mathematical interests:
Hyperbolic Geometry, Geometric Group Theory


Past project affiliations:


An Effective Compactness Theorem for Coxeter Groups. Geometriae Dedicata, Vol. 145, Issue 1 (2010), pp. 195-217. DOI 10.1007/s10711-009-9416-8. (arXiv) (journal)
(supervised by Misha Kapovich)

looking through a crevice, on top of a hike at Yosemite


University of Nebraska-Lincoln

Fall 2015: Math 405: Discrete Mathematics for Secondary Teaching

The overarching goal of this course is to do mathematics that will help teachers experience mathematical practice. Mathematical contexts to experience practice include graph theory and group theory. The emphasis in graph theory is on Eulerian paths/circuits, colorability, and genus of graphs and surfaces. The emphasis in group theory will be on permutation groups, dihedral groups, and polyhedral groups.

Spring 2015, Spring 2014: Math 407/807: Algebra for Secondary Teaching

A course emphasizing connections between algebraic concepts. Special emphasis on conic sections and trigonometric functions and understanding why different characterizations (e.g., slices of cones, ``stretched'' circles, foci-directrix-eccentricity, Ax2+By2+Cxy+Dx+Ey+F=0) indeed characterize the same class of objects.

Fall 2014, Fall 2013: Math 408/808: Geometry for Secondary Teaching

Euclidean geometry from a transformation perspective, with particular attention to proof, proving, and reasoning within this perspective. Emphasizes group theoretic properties of transformations, applying transformation perspective to topics in secondary geometry and algebra curricula. Coordinating course design with Sherry West, current mathematics department chair of Lincoln Southeast of Lincoln Public Schools.

Canada/USA Mathcamp

Since 2000, I have worked at the Canada/USA Mathcamp. It's a wonderful place where graduate students and faculty teach high school students about mathematics that has been personally inspiring.

I have taught the Banach Tarski paradox, Dehn invariants, the Scissors Congruence Problem, Disrete Fourier Transforms, and various pieces of hyperbolic geometry and geometric group theory (such as a class on SL(2,Z).)

At Mathcamp 2010, Ilya Grigoriev and I organized a viewing of Not Knot interspersed with a Q & A about the math. The film is visually beautiful, but can be frustrating to watch without a friend with whom to delve into the mathematics. It was great fun to examine the film with enthusiastic students and a fellow geometer.

Algebra Project

Throughout the academic year 2009-2010, I visited the Ypsilanti cohort of the Algebra Project and participated in the weekend planning sessions for the class. Here are some mathematical and non-mathematical highlights of the year.

University of Michigan

Fall 2013: Education 413: Secondary Mathematics Methods

Course for prospective teachers focusing on pedagogy for lower secondary and upper middle school mathematics; concurrent enrollment with field placement and practicum. Worked with field instructors Annick Rougee and Rachel Snider in designing the course.[course information from 2013]

Winter 2009, 2010, 2011: Math 486:
Mathematics for Secondary Teaching

Mathematics underlying the secondary and middle school curriculum. Related curricular topics to aspects of group/ring/field theory and real analysis. [course information from 2011]

Fall 2008, 2009: Math 105: Data, Functions, and Graphs

Mathematics in preparation for calculus.
[current course website]

I have drawn pedagogical inspiration from:

  • Loren Larson, who showed me the importance of motivation, and who compassionately coached me through my very first mathematical presentation, at the 1996 Canada/USA Mathcamp.
  • Misha Kapovich, who impressed upon me the power of well-chosen examples.
  • The 2005-2006 and 2006-2007 TAC groups at UC Davis, for giving the space for graduate students passionate about their disciplines and teaching to debate and compare the teaching philosophies prevalent in mathematics, computer science, physics, biology, linguistics, and literature.
  • The Elementary Mathematics Laboratory, which gave insight into how children think about fractions, offers a public teaching experience, and introduces a broad spectrum of people who care about teaching and mathematics to each other.



  • MIME 2014 (co-organizer, with Bill McCallum and Roger Howe)
  • MIME 2013 (co-organizer, with Bill McCallum and Roger Howe)
  • MIME 2012 (co-moderator, with Bill McCallum and Roger Howe)
  • MIME 2011 (co-moderator, with Bill McCallum)
  • Elementary Mathematics Laboratory MKTT Workshop (Mathematical Knowledge for Teaching Teachers) 2011, 2010 (assisted workshop led by Hyman Bass)
  • Canada/USA Mathcamp 2011 (Academic Coordinator, assisted by Eric Wofsey)
    Canada/USA Mathcamp 2008 (Academic Co-Coordinator ...or, "Academic Ordinator", with Mike Shulman and Alfonso Gracia-Saz)
  • RUME 2010 Working Group on Proof Presentation (organizer, with Tim Fukawa-Connelly)

Woman teaching geometry
(Geometry, personified, as woman teaching geometric construction)

Talks/Travel/Slides and handouts 2013-2014

  • Waterloo, ON. October.
  • Teachers Math Circle. November 14, 2013.
  • PME-NA 2013, November 14-17, 2013 (North American Chapter of the International Group for the Psychology of Mathematics Education). Setting up and enacting approximations of mathematical tasks of teaching in a practice-focused curriculum. (Joint w/ Y. Kim.) Saturday, November 16, 2013, 9:20am-10:00am. Palmer House Hilton-Price. Chicago, IL.
  • JMM 2014, Jan 15-16, 2014. Clear instruction of mathematical practice: Preparing teachers to use rich and ordinary problems to teach Common Core standards for mathematical practice. Invited 20-minute presentation at special session, "The changing education of pre-service teachers in light of the Common Core". Baltimore, MD.
  • Ann Arbor, MI-Matawan, NJ. December/January.
  • Mathematicians and School Mathematics Education: A Pan-American Workshop, January 26-31, 2014. Banff International Research Station for Mathematical Innovation and Discovery (BIRS).
  • AMTE 2014, February 6-8, 2014 (Association of Mathematics Teacher Educators). Teaching Teachers to Recognize and Use the Mathematical Practices (Joint w/ J. Jacobs, D. Kennedy, & D. Sherman). Hyatt Regency Irvine, Irvine, CA.
  • RUME 2014, Feb 27-Mar 1, 2014. (Special Interest Group of the Mathematical Association of American on Research in Undergraduate Mathematics Education; Conference on Research in Undergraduate Mathematics Education). Five considerations for using tasks assessing mathematical knowledge for teaching as representations of practice for undergraduate preparation of secondary mathematics teachers. Joint with Heather Howell (Educational Testing Service). Contributed presentation, Session 23, 4:10pm-4:40pm, Grand Mesa B,Hyatt Regency Denver Tech Center Denver Hotel, Denver, CO.
  • MIME 2014, March 16-18, 2014 (Mathematicians in Mathematics Education). An Introduction to Mathematical Knowledge for Teaching. Workshop leader and co-organizer. College Station, TX.
  • AERA 2014, April 3-6, 2014. (American Educational Research Association). Tasks assessing mathematical knowledge for teaching as representations of practice for professional preparation. Joint with Heather Howell (Educational Testing Service). One of seven presentations during the contributed paper session on Research on Issues in Teacher Education and Teacher Practice, Sun, April 6, 8:15 to 10:15am, Pennsylvania Convention Center, 100 Level, 125. Philadelphia, PA.
  • NCSM & NCTM 2014, April 7-9 & 9-12, 2014 (National Council of Supervisors of Mathematics, National Council of Teachers of Mathematics). Using Rich Problems to Teach the Mathematical Practices. (Joint w/ J. Jacobs, D. Kennedy, & D. Sherman). NCSM presentation tentatively scheduled for April 7, 2014 09:30 AM. NCTM presentation TBA. New Orleans, LA.

shortbread cookie endowed with hyperbolic structure

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