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Professor LIANG, Biyao

Professor LIANG, Biyao


Assistant Professor

Academic Unit of Mathematics, Science, and Technology


BSc. (South China Normal University), PhD (University of Georgia)


[javascript protected email address]


(852) 2241 5085


Room 321, Runme Shaw Building

Research Expertise

  • Mathematics Education
  • Teacher Education and Development
  • Primary and Secondary Education
  • Learning Sciences
  • Collaborative Learning

Prospective PhD/ EdD/ MPhil Applications

I am available to supervise PhD/EdD/MPhil students and would welcome enquiries for supervision.

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. Investigating Interplays Between Computational Thinking and Mathematical Thinking (2023-2025). Seed Fund for Basic Research for New Staff (HK$150,000)
  2. Learning to Conduct Constructivist Teaching Experiment: A Collaboration with Graduate Students, Pre-service Teachers, and In-service Teachers (2023-2024). Faculty’s Project-based Research Funding (HK$60,000) 



  1. 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
  2. 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. (2023). Mental processes underlying mathematics teachers' learning from student thinking. Journal of Mathematics Teacher Education. Advance online publication. 
  2. 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.
  3. 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.
  4. 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
  5. 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.
  6. Liang, B., & Castillo-Garsow, C. (2020). Undergraduate students’ meanings for central angle and inscribed angle. The Mathematics Educator, 29(1), 53-84.
  7. 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.


Book Chapters


  1. 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.
  2. 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.
  3. 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.
  4. 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.


Selected Conference Proceedings


  1. Liang, B. (in press). Unfolding the cognitive processes underlying the interactions between knowledge of content and knowledge of student thinking. Paper submitted to the 15th International Congress on Mathematics Education (ICME). Sydney, Australia.

  2. Chen, Q., Liang, B., Zhang, Y. (in press). Characterizing the role of programming outputs in mediating students’ mathematical learning. Paper submitted to the 15th International Congress on Mathematics Education (ICME). Sydney, Australia.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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. 

  • 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).