# Implementing mathematics teaching that promotes students’ understanding through theory-driven lesson study

## Abstract

Lesson study (LS) has been practiced in China as an effective way to advance teachers’ professional development for decades. This study explores how LS improves teaching that promotes students’ understanding. A LS group including didacticians (practice-based teaching research specialist and University-based mathematics educators) and mathematics teachers in China explored and documented how teacher participants shifted their attention to students’ learning by incorporating two notions of teaching: learning trajectory (LT) and variation pedagogy (VP). The former describes conjectured routes of children’s thinking and learning with pertinent tasks to move towards the learning goals along the route, while the latter suggests strategies for using systematic tasks progressively. The concepts of LT and VP are used to guide planning, teaching, and debriefing throughout the LS process. Data consist of lesson plans, videotaped lessons, post-lesson discussions, post-lesson quizzes, and teachers’ reflection reports. This study reveals that by building on the learning trajectory and by strategically using variation tasks, the lesson has been improved in terms of students’ understanding, proficiency, and mathematical reasoning. In addition, the LT was refined through the LS. This study displays how theory-driven LS could help teachers improve their teaching and develop the linkage between theory and practice.

## Keywords

Lesson study Learning trajectory Variation pedagogy Theory-driven lesson study## Notes

### Acknowledgments

We thank anonymous reviewers for their invaluable feedback on the revisions of the paper. We appreciate Dr. Dovie Kimmins and Mr. James Willingham from Middle Tennessee State University for their contribution to the improvement of the article. Our thanks go to participating teachers and didacticians for their commitment to the Lesson Study and support of data collection.

## References

- Cai, J., & Wang, T. (2006). U.S. and Chinese teachers’ conception and construction of representations: a case of teaching ratio concept.
*International Journal of Science and Mathematics Education,**4*, 145–186.CrossRefGoogle Scholar - Cai, J., & Wang, T. (2010). Conception of effective mathematic teaching with a cultural context: perspectives of teachers from China and the United States.
*Journal of Mathematics Teacher Education,**13*, 265–287.CrossRefGoogle Scholar - Carpenter, T. C., Lindquist, M. M., Brown, C. A., Kouba, V. L., Silver, E. A., & Swafford, J. O. (1988). Results of the fourth NAEP assessment of mathematics: trends and conclusions.
*Arithmetic Teacher,**36*(4), 38–41.Google Scholar - Chen, X., & Yang, F. (2013). Chinese teachers’ reconstruction of the curriculum reform through lesson study.
*International Journal for Lesson and Learning Studies,**2*, 218–236.CrossRefGoogle Scholar - Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education.
*Mathematical Thinking and Learning,**6*(2), 81–89.CrossRefGoogle Scholar - Clements, D., Sarama, J., Spitler, M., Lange, A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: a large-scale cluster randomized trial.
*Journal for Research in Mathematics Education,**42*, 127–166.Google Scholar - Common Core State Standards Initiative (CCSSI) (2010).
*Common Core State Standards for Mathematics*. http://www.corestandards.org/Math/Practice. - Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: an effective way of mathematics teaching in China. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.),
*How Chinese learn mathematics: perspectives from insiders*(pp. 309–348). Singapore: World Scientific.CrossRefGoogle Scholar - Hart, L. C., Alston, A. S., & Murata, A. (2011).
*Lesson study research and practice in mathematics education: learning together*. New York: Springer.CrossRefGoogle Scholar - Huang, R., & Bao, J. (2006). Towards a model for teacher’s professional development in China: introducing keli.
*Journal of Mathematics Teacher Education,**9*, 279–298.CrossRefGoogle Scholar - Huang, R., & Cai, J. (2011). Pedagogical representations to teach linear relations in Chinese and U.S. classrooms: parallel or hierarchical?
*The Journal of Mathematical Behavior,**30*(2), 149–165.CrossRefGoogle Scholar - Huang, R., & Han, X. (2015). Developing mathematics teachers’ competence through parallel lesson study.
*International Journal for Lesson and Learning Studies,**4*(2), 100–117.CrossRefGoogle Scholar - Huang, R., Su, H., & Xu, S. (2014). Developing teachers’ and teaching researchers’ professional competence in mathematics through Chinese Lesson Study.
*ZDM*—*The International Journal on Mathematics Education*,*46*, 239–251.Google Scholar - Kieran, C., Krainer, K., & Shaughnessy, J. M. (2013). Linking research to practice: teachers as key stakeholders in mathematics education research. In M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.),
*Third international handbook of mathematics education*(pp. 361–392). New York: Springer.Google Scholar - Langley, G. J., Moen, R. D., Nolan, K. M., Nolan, T. W., Norman, C. L., & Provost, L. P. (2009).
*The improvement guide*. San Francisco: Jossey-Bass.Google Scholar - Lee, C. K. E., & Lo, M. L. (2013). The role of lesson study in facilitating curriculum reform.
*International Journal For Lesson And Learning Studies,**2*, 200–206.CrossRefGoogle Scholar - Lewis, C. C. (2015). What is improvement sciences? Do we need it in education?
*Educational Researcher,**44*(1), 54–61.CrossRefGoogle Scholar - Lewis, C. C., Perry, R., & Murata, A. (2006). How should research contribute to instructional improvement? The case of lesson study.
*Educational Researcher,**35*(3), 3–14.CrossRefGoogle Scholar - Li, Y. (2008). What do students need to learn about division of fractions?
*Mathematics Teaching in the Middle School,**13*, 546–552.Google Scholar - Lo, M. L., & Marton, F. (2012). Toward a science of the art of teaching: using variation theory as a guiding principle of pedagogical design.
*International Journal for Lesson and Learning Studies,**1*, 7–22.Google Scholar - Maloney, A. P., Confrey, J., & Nguyen, K. H. (2014).
*Learning over time: learning trajectories in mathematics*. Charlotte: Informational age publishing.Google Scholar - Marton, F., & Tsui, A. B. M. (with Chik, P. P. M., Ko, P. Y., Lo, M. L., Mok, I. A. C., Ng, F. P., Pang, M.F., et al.) (Eds.). (2004).
*Classroom discourse and the space of learning.*Mahwah: Lawrence Erlbaum.Google Scholar - Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning.
*The Journal of the Learning Science,**15*, 193–220.CrossRefGoogle Scholar - Ministry of Education, P. R. China. (2011).
*Mathematics curriculum standards for compulsory education (grades 1–9)*. Beijing: Beijing Normal University Press.Google Scholar - Murata, A. (2011). Introduction: conceptual overview of lesson study. In C. L., Hart, A. S., Alston, & A. Murata (Eds.),
*Lesson study research and practice in mathematics education: learning together*(pp. 1–12). New York: Springer.Google Scholar - National Council of Teachers of Mathematics. (2014).
*Principles to action: ensuring mathematical success for all*. Reston: NCTM.Google Scholar - Ott, J. M., Snook, D. L., & Gibson, D. L. (1991). Understanding partitive division of fractions.
*The Arithmetic Teacher,**39*, 7–11.Google Scholar - Patton, M. Q. (2002).
*Qualitative research & evaluation methods*(3rd ed.). Thousand Oaks: Sage Publications.Google Scholar - Silvia, E. M. (1983). A look at division with fractions.
*The Arithmetic Teacher,**30*, 38–41.Google Scholar - Simon, M. A. (1995). Prospective elementary teachers’ knowledge of division.
*Journal for Research in Mathematics Education,**24*, 233–254.CrossRefGoogle Scholar - Sowder, J., Sowder, L., & Nickerson, S. (2010).
*Reconceptualizing mathematics for elementary school teachers*. New York: W.H. Freeman & Company.Google Scholar - Stein, M., Engle, R., Smith, M., & Hughes, E. (2008). Orchestrating productive mathematical discussions: five practices for helping teachers move beyond show and tell.
*Mathematical Thinking and Learning,**10*, 313–340.CrossRefGoogle Scholar - Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: an analysis of the relationship between teaching and learning in a reform mathematics project.
*Educational Research and Evaluation,**2*, 50–80.CrossRefGoogle Scholar - Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction: toward a theory of teaching.
*Educational Researcher,**41*, 147–156.CrossRefGoogle Scholar - Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: the case of division of fractions.
*Journal for Research in Mathematics Education,**31*, 5–25.CrossRefGoogle Scholar - Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: using variation to structure sense-making.
*Mathematical Thinking and Learning,**8*(2), 91–111.CrossRefGoogle Scholar - Woodbury, S., & Gess-Newsome, J. (2002). Overcoming the paradox of change without difference: a model of change in the arena of fundamental school reform.
*Educational Policy,**16*, 763–782.CrossRefGoogle Scholar - Yang, Y., & Ricks, T. E. (2013). Chinese lesson study: developing classroom instruction through collaborations in school-based teaching research group activities. In Y. Li & R. Huang (Eds.),
*How Chinese teach mathematics and improve teaching*(pp. 51–65). New York: Routledge.Google Scholar