Research & Selected Writing

Guiding documents

Conference Board of the Mathematical Sciences Task Force Consensus Statement on Modernizing Mathematics in Grades 9-14. Transmitted to the Mathematical Sciences Education Board of the National Academies in 2025. (I was a co-lead of the Content section, and also contributed to the structure of the document as a whole.)

Guiding Principles for the CUPM Curriculum Guide’s Treatment of Teacher Education. This document, released in 2023, stakes out design principles for future program area reports on teacher education in the Curriculum Guides of the Mathematical Association of America’s Committee on the Undergraduate Program in Mathematics.

Reference list of guiding documents for the mathematical education of teachers. This reference is the Appendix of the Guiding Principles document above. Documents published by the Conference Board of the Mathematical Sciences (CBMS), including Mathematical Education of Teachers and Position Statements, are listed first. All other documents are listed in alphabetical order by common abbreviation for publisher.

National Council of Teachers of Mathematics. (2018). Catalyzing Change in High School Mathematics: Initiating Critical Conversations. Reston, VA: Author. (I led the writing of geometry section, and contributed to introductory chapters that frame this document.)

Lai, Y (2012). Teaching Undergraduate Mathematics. Critical Issues in Mathematics Education Series, Vol 5. Mathematical Sciences Research Institute, Berkeley, CA.

Essays (Selected)

See https://yvonnexlai.com/essays/ .

Teacher education

Lai, Y. (2025). Noticing mathematics education. Notices of the American Mathematical Society, 72(1), 49.

Lai, Y. & Enright, E. (2024). Decompressing mathematical definitions for communities of learners. For the Learning of Mathematics, 44(2), 22-24.

Boaler, J., Jones, L.L., Lai, Y., Nardo, J., & Schultens, J. (2024). Metacognition in the mathematics classroom. Notices of the American Mathematical Society, 71(1), 57-63.

Thanheiser, E., Bay-Williams, J., Velasco, R., Bondurant, L., & Lai, Y. (2024). Critical conversations of engaging with
the AMTE Standards. Journal for Theoretical & Marginal Mathematics Education, 3(1), 1-23.

Lai, Y., Strayer, J.F., Lischka, A.E., Anhalt, C., Bonnesen, C., Casey, S., Kohler, B., & Adamoah, K. (2023). Enhancing prospective secondary teachers’ potential competence for enacting core teaching practices—through experiences in university mathematics and statistics courses. ZDM–Mathematics Education, 55(4), 867-881. https://doi.org/10.1007/s11858-023-01485-4

Lai, Y. & Ahrens, S (online 2022). Mathematicians’ perceived and envisioned obligations to secondary teachers and teaching. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-022-09567-y

Lai, Y. & Baldinger, E. (2021). Authority Relations and the Tertiary-to-Secondary (Dis)continuity. For the Learning of Mathematics, 41(2), 26-31.

Bondurant, L., Howell, H., Kwon, M., Wilkerson, C.L,., Lai, Y., & Sapkota, B. (2021). Preservice teacher learning of practice through simulated teaching experiences before, during, and after covid. In Olanoff, D., Johnson, K., Spitzer, S. (Eds.) Proceedings of the 43rd Annual Meeting of the North American Chapter of the Psychology of Mathematics Education (PMENA). Philadelphia, PA.

Czap, L.,Ahrens, S., Lai, Y., & Lischka, A. (2021). Shaping the professional growth of mathematics faculty who teach prospective secondary teachers. In Olanoff, D., Johnson, K., Spitzer, S. (Eds.) Proceedings of the 43rd Annual Meeting of the North American Chapter of the Psychology of Mathematics Education (PMENA), pp. 733-737. Philadelphia, PA.

Ahrens, S. & Lai, Y. (2020) How Mathematicians Attend to Learning Goals for Teachers. In Proceedings of the 2020 Conference on Research in Undergraduate Mathematics Education. A longer version of this report is now published in JMTE.

Lai, Y., Lischka, A.E., & Strayer, J.F. (2021). Is the Medium the Message? Video-Recording and Writing for Approximations of Practice. In symposium by Sapkota, B., Howell, H., Lai, Y., Lee, C. on Opportunities and Challenges Associated with Approximations of Practice in Teacher Education Programs. Paper resented at the 2021 Annual Conference of the American Educational Research Association (AERA). (Slides)

Baldinger, E., & Lai, Y. (2019). Pedagogical Context and Proof Validation: The Role of Positioning as a Teacher or Student. The Journal of Mathematical Behavior. 10.1016/j.jmathb.2019.03.005

Lai, Y. (2019). Accounting for mathematicians’ priorities in mathematics courses for secondary teachers. The Journal of Mathematical Behavior, 53, 164-178. https://www.sciencedirect.com/science/article/pii/S0732312317301852

Baldinger, E. & Lai, Y. (2018). Validating Proofs in Parallel Mathematical and Pedagogical Tasks. In Proceedings of the 2018 Conference on Research in Undergraduate Mathematics Education. (Winner of a 2018 Top Three Paper Award, SIGMAA on Research in Undergraduate Mathematics Education.) This paper has now been expanded into a JMB publication.

Lai, Y., Carlson, M. A., & Heaton, R. (2017). Giving reason and giving purpose. In Li, Y., Lewis, W. J., Madden, J. (Eds.), Mathematics Matters in Education: Essays in honor of Roger E. Howe) (pp. 149-171). New York, NY: Springer.

Lai, Y. (2008). Personal Report on ICME-11: Concerns about Connections. Prepared for NCTM/NSF committee.

Lai, Y., McCallum, W., Soto-Johnson, H. (2008). The role of universities in preparing teachers. Technical report on ICME-11 for NCTM/NSF committee.

Mathematical knowledge for teaching

Lai, Y., Wasserman, N., Strayer, J.F., Casey, S., Weber, K., & Fukawa-Connelly, T. (accepted). Making Advanced Mathematics Work in Secondary Teacher Education. In Benken, B. (Ed.), AMTE Professional Book Series, Volume 5: Reflection on Past, Present and Future: Paving the Way for the Future of Mathematics Teacher Education, Chapter 6.

Lai, Y., Lischka, A., Strayer, J.F., & Adamoah, K. (2023). Characterizing Prospective Secondary Teachers’ Foundation and Contingency Knowledge for Definitions of Transformations. Journal of Mathematical Behavior, 70, 101030. https://doi.org/10.1016/j.jmathb.2022.101030

Lai, Y., Lischka, A.E., & Strayer, J.F. (2022). Characterizing prospective secondary teachers’ foundation and contingency knowledge for definitions of transformations. In Proceedings of the 2021 Annual Conference of the North American Chapter of the Psychology of Mathematics Education (PMENA). This was published in an expanded form in Lai, Lischka, Strayer, & Adamoah (2023) in JMB.

Lai, Y. & Jacobson, E. (2018). Implications of pedagogical context for eliciting pedagogical content knowledge. For the Learning of Mathematics, 38(2), 28-33.

Lai, Y. & Patterson, C. (2017). Opportunities presented by mathematics textbooks for prospective teachers to learn to use mathematics in teaching. In Smith, W.M., Lawler, B.R., Bowers, J., & Augustyn, L. (Eds.), Proceedings of the Sixth Annual Math Teacher Education Partnership Conference, (pp. 142-147)

Hoover, M., Mosvold, R., Ball, D. & Lai, Y. (2016). Making progress on mathematical knowledge for teaching. The Mathematics Enthusiast, 13(1&2), 3-34.

Suh, H., Howell, H., Lai, Y. (2016). Secondary teachers confronting mathematical uncertainty: Reactions to a teacher assessment item on exponents. In Tim Fukawa-Connelly, Nicole Engelke Infante, Megan Wawro, & Stacy Brown (Eds.), Proceedings of the 19^th Annual Conference on Research in Undergraduate Education, pp. 228-242.

Howell, H., Lai, Y. & Suh, H. (2017). Questioning assumptions about the measurability of subdomains of mathematical knowledge for teaching. In Aaron Weinberg, Chris Rasmussen, Jeffrey Rabin, Megan Wawro, & Stacy Brown (Eds.), Proceedings of the 20th AnnualConference on Research in Undergraduate Education, pp. 413-427.

Hoover, M. & Lai, Y. (2016). Pedagogical context and its role in articulating mathematical knowledge for teaching. Paper presented at Working Group on Teacher Resources and Research Methods of the 2016 European Research in Mathematics Education Topic Study Conference.

Howell, H., Lai, Y., & Phelps, G. (2016). Conceptualizing mathematical knowledge for teaching at secondary level: Do primary models extend? In Topic Study Group 46: Knowledge in/for teaching mathematics at secondary level. 13th International Congress on Mathematics Education. Hamburg, Germany.

Howell, H., Lai, Y., & Phelps, G. (2015). Challenges in assessing mathematical knowledge for secondary teaching: Do elementary models extend? Paper presented at the 2015 Annual Meeting of the American Educational Research Association.

Lai, Y. & Howell, H. (2014). Tasks assessing mathematical knowledge for teaching as representations of mathematics teaching practice. Paper presented at the 2014 Annual Meeting of the American Educational Research Association.

Lai, Y., Jacobson, E., & Thames, M. (2013). The role of pedagogical context in items assessing mathematical knowledge for teaching. Paper presented at the 2013 Annual Meeting of the American Educational Research Association.

Lischka, A.E., Lai, Y., Strayer, J.F., & Anhalt, C.O. (2020). Developing mathematical knowledge in and for teaching in content courses. In Martin, W. G., Lischka, A., Smith, W. M. and Lawler, B. R. (Eds.). The Mathematics Teacher Education Partnership: The Power of a Networked Improvement Community to Transform Secondary Mathematics Teacher Preparation (Chapter 5). IAP Press.

Proof and proving

Lai, Y., Weber, K., & Mejia-Ramos, J.-P. (2012). Mathematicians’ perspectives on features of a good pedagogical proof. Cognition & Instruction 30(2), 146-169 https://doi.org/10.1080/07370008.2012.661814

Lai, Y. & Weber, K. (2014). Factors mathematicians consider when creating pedagogical proof. Educational Studies in Mathematics, 85(1), 93-108. https://www.jstor.org/stable/43589808

Samkoff, A., Lai, Y., & Weber, K. (2012). How mathematicians use diagrams to construct proofs. Research in Mathematics Education 14(1), 49-67. (DOI: 10.1080/14794802.2012.657438) (Winner of 2012 Janet Duffin Award for most outstanding contribution to journal) https://doi.org/10.1080/14794802.2012.657438

Lai, Y. & Weber, K. (2010). What makes good proof: Investigating mathematicians’ proof revisions. Proceedings of the 13th Conference on Research in Undergraduate Mathematics Education (CRUME). (Winner of the 2010 Best Paper Award, SIGMAA on Research in Undergraduate Mathematics Education.) This was expanded into Lai & Weber (2014) and then Lai, Weber, & Mejia-Ramos (2012)

Secondary Mathematics Policy

Burrill, G., Cohn, H., Sinha, D., Son, J.Y., Stevenson, K.F. (2023). Listening for common ground in high school and early collegiate mathematics. Notices of the American Mathematical Society, 70(5), 798-805.

Mathematics Graduate Student Teaching/Undergraduate Education

Miller, E.R., Uhing, K., Hass, M., Zigterman, R., Quigley, K., Lai, Y., & Wakefield, N. (2018). Mathematics Graduate Teaching Assistants’ Growth as Teachers: An Unexamined Practice. In Aaron Weinberg, Chris Rasmussen, Jeffrey Rabin, Megan Wawro, & Stacy Brown (Eds.), Proceedings of the 20th Annual Conference on Research in Undergraduate Education, pp. 185-198.

Lai, Y (2012). Teaching Undergraduate Mathematics. Critical Issues in Mathematics Education Series, Vol 5. Mathematical Sciences Research Institute, Berkeley, CA.

Geometry

St. Goar, J., Lai, Y. (2020). Defining Key Developmental Understandings in Congruence Proofs from a Transformation Approach. In Sacristán, A.I., Cortés-Zavala, J.C. & Ruiz-Arias, P.M. (Eds.). (2020). Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of
Mathematics Education, Mexico. Cinvestav / AMIUTEM / PME-NA. https:/doi.org/10.51272/pmena.42.2020

St. Goar, J., & Lai, Y. (2021). Designing Activities to Support Prospective High School Teachers’ Proofs of Congruence from a Transformation Perspective. PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies), 32(7), 827-842, DOI: 10.1080/10511970.2021.1940403

Lai Y., & Donsig A. (2018). Using Geometric Habits of Mind to Connect Geometry from a Transformation Perspective to Graph Transformations and Abstract Algebra. In Wasserman N. (Ed), Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers. In the Research in Mathematics Education series. New York, USA: Springer, pp. 263-289. https://link.springer.com/chapter/10.1007/978-3-319-99214-3_13

Lai, Y. (2010). An effective compactness theorem for Coxeter groups. Geometriae Dedicata 145(1), 195-217. (DOI: 10.1007/s10711-009-9416-8. arXiv:0902.2718.) Key words: Coxeter groups, hyperbolic geometry, geometric group theory. (This paper is based on my dissertation.)

Reichardt, B., Heilmann, C., Lai, Y., & Spielman, A. (2003). Proof of the Double Bubble Conjecture in R⁴ and certain higher dimensional cases. Pacific Journal of Mathematics 208, 347-366. http://dx.doi.org/10.2140/pjm.2003.208.347

Grants/Projects

College of Arts & Sciences Strategic Priorities Grant: Understanding the experiences of underrepresented students in 300-level mathematics courses. Bennett, A. (PI), Lai, Y. (co-PI). Grant period: 2022-2023.

NSF DRK-12: Investigating the Role of Collaboration on the Development of Student Ideas using a Learning Progression for the Function Concept. Graf, E. (Educational Testing Service) (PI), Crombie, W. (Algebra Project), DeLeon, C. (Young People’s Project), Eames, C. (Southern Illinois State University-Edwardsville), & Lai, Y. (co-PIs). Grant period: 2020-2025. Total $3,733,791.

NSF IUSE: Collaborative Research: Using Networked Improvement Communities to Scale Up Program Transformation for Secondary Mathematics Teacher Preparation (NIC-Transform Scale Up). Martin, G. (lead PI) & Strutchens, M. (Auburn University) (co-PI); Smith, W. (PI), Lischka, A. (Middle Tennessee State University), Lai, Y., & Lawler, B. (Kennesaw State University) (co-PIs). Grant period: 2022-2026. Total $3M distributed across institutions.

NSF IUSE: Collaborative Research: Mathematics of Doing, Understanding, Learning, and Educating for Secondary Schools (MODULES). Anhalt, C. (PI) & Aubrey, J. (co-PI) (University of Arizona), Casey, S.A. (PI) & Ross, A. (co-PI) (Eastern Michigan University), Gobstein, H. (PI) (Association of Public and Land-grant Universities), Kohler, B. (PI) (University of Utah), Lai, Y. (PI)*, Strayer, J. (PI) (Middle Tennessee State University). Grant period: 2017-2022. Total $1,999,344 distributed across #DUE-1726707, 1726804, 1726252, 1726723, 1726744, & 1726098. *Research Lead

NSF BCECR: Development of Research to Inform the Design and Improvement of Mathematics Education for Teachers. St. Goar, J. (PI), Lai, Y. (Senior Personnel). Grant period: 2020-2022. Total $50,661.

NSF INTERN Supplement to NebraskaNOYCE Phase II: SWIRLS: Scratch Work Illuminating Representations and Learning for Students in High-need Districts. Smith, W. (PI) & Lai, Y. (co-PI). Grant period: 2018-2019. Total $49, 986 (#DUE-1747937)

Nebraska Department of Education Math and Science Partnerships : Nebraska Partnership TEAMS: Teaching to Enhance Achievement in Math and Science. Smith, W. (PI), Arthurs, L., Heaton, R., Homp, M., Lai Y., Lewis, E., Males, L., Searls, M., Thomas, A., & Thomas, J. (co-PIs). Grant period: 2017-2018. Total
$1, 000, 000.

NSF Noyce: NebraskaNOYCE Phase II: Investigating the Impact in High-Need Districts. Smith, W. (PI), Lai, Y., Lewis, W.J., & Males, L. (co-PIs). Grant period: 2014-2017. Total $299, 878 (#DUE-1439867).

NSF EAGER: Building Understanding and Measurement of Secondary Mathematical Knowledge for Teaching (MKT). Howell, H. (PI), Lai, Y. (PI), & Phelps, G. (co-PI).. Grant period: 2014-2017. Total $279, 906.

NSF Infrastructure Grant (2011-2014). Savitt, D. (PI), Snyder, N., Duchin, M., & Lai, Y. (co-PIs). Canada/USA Mathcamp: Research in Pairs and Scholarships for Students (#DMS-1135049, $76,608.00)

NCTM/NSF International Travel Grant for ICME-11 (2008). Awarded to ∼30 mathematicians, educators, and graduate students nationwide for travel to the 11th International Congress on Mathematical Education in Monterrey, Mexico.