**Guiding documents**

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**

Lai, Y. (2021, Fall). *Mathematical Claims. *AWM Newsletter.

Lai, Y. (2021, October 17). *From Teaching Math to Teaching Students Math. *AMS Blog on Teaching and Learning Mathematics.

Lai, Y. (2021, Fall). *Please, Do Tell. *AWM Newsletter.

Lai, Y. (2020, November 30). *The Calculus of Context. *AMS Blog on Teaching and Learning Mathematics.

**Teacher education**

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

Bondurant, L., Howell, H., Kwon, M., Wilkerson, C.L,., Lai, Y., & Sapkota, B. (2022). Preservice teacher learning of practice through simulated teaching experiences before, during, and after covid. In Proceedings of the 2021 Annual Conference of the *North American Chapter of the Psychology of Mathematics Education (PMENA)*

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 has been provisionally accepted to JMTE.

Lai, Y., Lishcka, 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. Presented at the 2021 Annual Conference of the *American Educational Research Association (AERA)*.

Baldinger, E., & Lai, Y. (2019). Pedagogical Context and Proof Validation: The Role of Positioning as a Teacher or Student. *The Journal of Mathematical Behavio*r.

Lai, Y. (2019). Accounting for mathematicians’ priorities in mathematics courses for secondary teachers. *The Journal of Mathematical Behavior*, *53*, 164-178.

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*.)

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., 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. *AMTE Professional Book Series, Volume 5: Reflection on Past, Present and Future: Paving the Way for the Future of Mathematics Teacher Education*. In this chapter, we argue that it is essential to provide opportunities for prospective secondary mathematics teachers to connect advanced mathematics content to secondary *mathematics teaching practice *(in addition to connections only to secondary *mathematics*), if advanced mathematics courses are to be useful to these teachers — and that there is much room for designing these opportunities in anti-deficit ways that support equitable instructional practice.

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)*.

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.

Howell, H., Lai, Y. & Suh, H. (2017). Questioning assumptions about the measurability of MKT sub-domains. In Proceedings of the 2017 *Conference on Research in Undergraduate Mathematics Education*

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

Lai, Y. & Weber, K. (2014). Factors mathematicians consider when creating pedagogical proof. *Educational Studies in Mathematics*, 85(1), 93-108

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)

Lai, Y. & Weber, K. (2010). What makes good proof: Investigating mathematicians’ proof revisions. *Proceedings of the *13*th Conference on Research in Undergraduate Mathematics Education (CRUME)*. (Winner of the 2010 *Best Paper Award, SIGMAA on Research in Undergraduate Mathematics Education*.)

**Geometry**

St. Goar, J., Lai, Y. (2020). Defining Key Developmental Understandings in Congruence Proofs from a Transformation Approach. In Proceedings of the *42nd Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PMENA)*

St. Goar, J., & Lai, Y. (2021). Designing Activities to Support Prospective High School Teachers’ Proofs of Congruence from a Transformation Perspective. *Problems, Resources, and Issues in Mathematics Undergraduate Studies*

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.

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 R^{4} and certain higher dimensional cases*. Pacific Journal of Mathematics* 208, 347-366.

**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: 2020-2024. 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.