Professor LIANG, Biyao

Research Areas:

• One of my research lines is centred on students' and teachers' mathematical cognition regarding algebra, geometry, pre-calculus and STEM ideas. I investigate individuals' mathematical thinking through the lenses of quantitative reasoning, covariational reasoning, and computational thinking. 
• Another research line is focused on the area of (mathematics) teacher education and professional development. I study teachers' mathematical knowledge for/in teaching, and in particular, how teachers develop their mathematical knowledge and knowledge of students' mathematical thinking through listening, noticing, and decentering when interacting with students or analysing student thinking. Relatedly, I am interested in understanding what conditions teachers to learn from students and ways to support teachers as researchers.
• I am also interested in how students learn from each other mathematically through student-student interactions. 

Epistemological and Theoretical Orientations: radical constructivism, Piagetian learning theories

Qualitative Methodologies: constructivist teaching experiment, task-based clinical interview, video-stimulated recall interview

Teaching Areas: mathematics content, mathematics pedagogy, mathematics learning

The Chinese University of Hong Kong Research Fellowship Scheme (2021/22)

Graduate Education Advancement Board Fellowship (UGA; 2020)

100 Honor Graduates of South China Normal University (Class of 2015) 

Principal Investigator

1. Exploring the Application of Peer-assisted Learning in Hong Kong Mathematics Education (Jun 2025–Jun 2027). Seed Fund for PI Research – Basic Research (HK$73,840)
2. MathBuddy: Developing and Piloting an AI Companion Agent for Algebra Learning (Jan 2026–Dec 2026). Faculty Research Fund (HK$60,000)
3. Innovating Programming-enhanced Mathematics Curriculum Through Teacher-researcher Partnership: Design-based Research as a Pathway to Teachers’ Professional Learning (Jun 2024–Jun 2026). Seed Fund for PI Research – Basic Research (HK$93,040)
4. Investigating and Supporting Students’ Mathematical Modelling Through the Quantitative Reasoning Perspective (Apr 2025–Dec 2025). Faculty Research Fund (HK$60,000)
5. Investigating Interplays Between Computational Thinking and Mathematical Thinking (Jun 2023-Jun 2025). Seed Fund for Basic Research for New Staff (HK$150,000)
6. Cross-ethnic Differences in Parenting Styles and Their Influence on Students’ Mathematics Learning in China (Jan 2023–Jun 2025). Research Grants Council (Hong Kong), General Research Fund (HK$744,600)
7. Leveraging the Affordances of Programming Environments to Develop Conceptual Mathematics Curriculum (Jan 2024–Dec 2024). Faculty Research Fund (HK$60,000)
8. Learning to Conduct Constructivist Teaching Experiment: A Collaboration with Graduate Students, Pre-service Teachers, and In-service Teachers (Jan 2023- July 2024). Faculty’s Project-based Research Funding (HK$60,000) 

Co-Investigator

1. Investigating the Secondary–Tertiary Mathematics Transition in Hong Kong: Toward Local and Global Significance (2025-2027). Research Grants Council (Hong Kong), General Research Fund (HK$490,500). P.I.: Oi-Lam Ng
2. Collaborating with Generative Artificial Intelligence (AI) to Transform Learning and Teaching (2023-2026). Fund for Innovative Technology-in-Education (FITE), University Grants Committee (Hong Kong) (HK$250,000). P.I.: Chun Lai
3. Mathematical Problem Solving through Digital Making: Envisioning a Computationally Enhanced Mathematics Curriculum in Hong Kong’s Primary and Secondary Schools (2021-2023). Research Grants Council (Hong Kong), General Research Fund (HK$638,908). P.I.: Oi-Lam Ng

Other Collaborative Work

1. Supporting Hong Kong Ethnic Minority Learners’ Multimodal Mathematics Learning Through Responsive Teaching in Technology-Enhanced Environments (2021-2022). Research Grants Council (Hong Kong), General Research Fund . P.I.: Oi-Lam Ng
2. Advancing Secondary Mathematics Teachers’ Quantitative Reasoning (2015-2021). National Science Foundation(USA). P.I.: Kevin C. Moore 
3. Generalization Across Multiple Mathematical Areas (GAMMA) (2016-2017). National Science Foundation (USA). P.I. Amy Ellis
4. Diagnosing Chinese In-service Teachers’ Multiplicative Reasoning (2017-2020). National Science Foundation (USA). P.I. Andrew Izsak; Subproject Leader: Rui Kang
5. Eye-tracking Dynamic Geometry Proofs (2012-2013). Q-Center at Kansas State University. P.I. Andrew Bennett 

 Journal Articles

1. Liang, B., Wu, Y., & Liu, R. (under review). Third-order modeling: A constructivist approach to teacher knowing of students’ mathematical thinking. 
2. Liang, B. & Moore. K. C. (under review). Learning through learners: A framework for teachers’ productive struggles with student thinking.
3. Liang, B., Chen, Q., Zhang, Y., & Ng, O. (accepted). From output to understanding: How programming outputs mediate mathematics learning via perturbation. International Journal of Science and Mathematics Education.
4. Moore, K. C., Tasova, H. I., Stevens, I. E., & Liang, B. (in press). Competing meanings, perturbation, and engendering shifts in (prospective) teacher meanings. Frontiers in Education.
5. Zhang, Y., Ng, O., & Liang, B. (2026). From pre-ritual to exploration: Young learner’s gestural routine development in manipulative-based number discourse. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-025-10457-2
6. Ye, H., Liang, B., & Ng, O. (2025). A learner‐centred exploration of teachers’ solution pathways in K‐12 programming‐based mathematical problem‐solving. Journal of Computer Assisted Learning, 41(5). Article e70102. https://doi.org/10.1111/jcal.70102
7. Liang, B. (2025). Mental processes underlying mathematics teachers' learning from student thinking. Journal of Mathematics Teacher Education. 28(1), 7-32. https://doi.org/10.1007/s10857-023-09601-7 
8. Ye, H., Liang, B., Ng, O., & Chai, C. S. (2023). Integration of computational thinking in K-12 mathematics education: A systematic review on CT-based mathematics instruction and student learning. International Journal of STEM Education, 10, Article 3. https://doi.org/10.1186/s40594-023-00396-w
9. Liang, B., Ng, O., & Chan, Y. (2023). Seeing the continuity behind “double discontinuity”: Investigating Hong Kong prospective mathematics teachers’ secondary–tertiary transition. Educational Studies in Mathematics, 113, 107-124. https://doi.org/10.1007/s10649-022-10197-7
10. Ng, O., Liang, B., Chan, A., Ho, T. C., Lam, L. P., Law, M. H., Li, E. M., Lu, T. Y (2022). A collective reflection on the transition from secondary to university mathematics through the lens of the “double discontinuity” by Felix Klein. EduMath, 45. 
11. Liang, B. & Moore, K. C. (2021). Figurative and operative partitioning activity: Students’ meanings for amounts of change in covarying quantities. Mathematical Thinking and Learning, 23(4), 291-317.  https://doi.org/10.1080/10986065.2020.1789930
12. Liang, B., & Castillo-Garsow, C. (2020). Undergraduate students’ meanings for central angle and inscribed angle. The Mathematics Educator, 29(1), 53-84. https://openjournals.libs.uga.edu/tme/article/view/2093/2599. 
13. Moore, K. C., Stevens, I. E., Paoletti, T., Hobson, N. L. F., & Liang, B. (2019). Pre-service teachers’ figurative and operative graphing actions. The Journal of Mathematical Behavior, 56, Article 100692. https://doi.org/10.1016/j.jmathb.2019.01.008

Book Chapters

1. Moore, K. C., Stevens, I. E., & Liang, B. (under review). The teaching experiment: A method to operationalize an epistemology. How Theories Work in Mathematics Education: Foundation, Form, and Function.
2. Moore, K. C., Stevens, I., Tasova, H., & Liang, B. (2024). Operationalizing figurative and operative framings of thought. In P. C. Dawkins, A. J. Hackenberg, A. J. Norton (Eds), Piaget’s Genetic Epistemology in Mathematics Education (pp. 89-128). Springer. https://doi.org/10.1007/978-3-031-47386-9_4
3. Ng, O., Sinclair, N., Ferrara, F., & Liang, B. (2023). Transforming arithmetic through digital resources. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), Handbook of Digital Resources in Mathematics Education. Springer. https://doi.org/10.1007/978-3-030-95060-6_17-1
4. Ng, O., Liang, B., & Leung, A. (2023). Using first- and second-order models to characterise in-service teachers’ video-aided reflection on teaching and learning with 3D Pens. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era: International Research on Professional Learning and Practice (pp. 95-117). Springer. https://doi.org/10.1007/978-3-031-05254-5_4
5. Moore, K. C., Liang, B., Stevens, I. E., Tasova, H., & Paoletti, T. (2022). Abstracted quantitative structures: Using quantitative reasoning to define concept construction. In G. Karagöz Akar, İ.Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), Quantitative Reasoning in Mathematics and Science Education (pp. 35-69). Springer. https://doi.org/10.1007/978-3-031-14553-7_3

Selected Conference Proceedings

1. Liang, B. (2020). Theorizing teachers’ mathematical learning in the context of student-teacher interaction: A lens of decentering. In S. S. Karunakaran, Z. Reed, & A. Higgins (Eds.), Proceedings of the 23rd Annual Conference on Research in Undergraduate Mathematics Education (pp. 742-751). Boston, MA. http://sigmaa.maa.org/rume/RUME23.pdf

2. Liang, B., Ying, Y., & Moore, K. C. (2020). A conceptual analysis for optimizing two-variable functions in linear programming. In S. S. Karunakaran, Z. Reed, & A. Higgins (Eds.), Proceedings of the 23rd Annual Conference on Research in Undergraduate Mathematics Education (pp. 374-381). Boston, MA. http://sigmaa.maa.org/rume/RUME23.pdf

3. Liang, B. (2019). A radical constructivist model of teachers’ mathematical learning through student-teacher interaction. In S. Otten, A. G. Candela, Z. de Araujo, C. Haines, & C. Munter (Eds.), Proceedings of the 41st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1814-1819). St. Louis, MO. https://www.pmena.org/pmenaproceedings/PMENA%2041%202019%20Proceedings.pdf

4. Liang, B. (2019). Construction and application perspective: A review of research on teacher knowledge relevant to student-teacher interaction. In A. Weinberg, D. Moore-Russo, H. Soto & M. Wawro (Eds.), Proceedings of the Twenty-Second Annual Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education (pp. 35-43). Oklahoma City, OK. http://sigmaa.maa.org/rume/RUME22_Proceedings.pdf

5. Stevens, I. E., Paoletti, T., Moore, K. C., Liang, B. & Hardison, H. (2017). Principles for designing tasks that promote covariational reasoning. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro & S. Brown (Eds.), Proceedings of the Twentieth Annual Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education (pp. 928-936). San Diego, CA. http://sigmaa.maa.org/rume/RUME20.pdf 

• Journal Reviewer: Educational Studies in Mathematics; Journal of Mathematics Teacher Education; The Journal of Mathematical Behavior; Digital Experiences in Mathematics Education; Asia Pacific Education Review; The Mathematics Educator
• Member of the Organizing Committee of the Topic Study Group 5.3 (Cognition, learning sciences, and neurosciences in mathematics education) at the ICME15 (International Congress on Mathematics Education, Sydney, Australia, 2024).