Developing students’ ability to solve word problems through learning trajectory-based and variation task-informed instruction
Solving word problems is challenging in elementary schools, both for the teacher in teaching students to solve word problems and for the student in learning to solve them. This paper examines how the ideas of learning trajectory and variation pedagogy could be integrated as an instructional principle for teaching this content in the context of solving additive comparison problems. Based on research literature, a learning trajectory for solving additive comparison problems was identified. Informed by variation pedagogy and using a lesson study approach, a research team explored how to teach solving comparison word problems based on this learning trajectory. Data included lesson plans, videotaped research lessons, students’ pre- and post-tests, and students’ interviews. A fine-grained analysis of the data demonstrated that the lessons unfolded through exploration of a series of deliberate tasks along the learning trajectory, focusing on the structure of comparison problems and targeted at objects of learning. Purposefully constructed patterns of variation and invariance provided students with necessary conditions to discern and experience the objects of learning. Students were actively engaged in making sense of comparison problems and articulating their thinking using multiple representations. While the post-test and interview data show students’ understanding of key aspects of solving additive comparison problems to be at various levels, students’ gains in overall performance from pre- to post-test were statistically significant. Implications for teaching comparison word problems are discussed.
KeywordsAdditive comparison word problems Lesson study Variation theory Learning trajectory
We extend our thanks for the strong support from the participating school, Qiaotou no. 2 Elementary School of Yongjia County, City of Wenzhou, and especially to teaching research specialist Ms. Yuxiao Nan and mathematics teacher Jing Huang for their intellectual contribution to improving the design and teaching of the comparison word problem lessons.
- Baroody, A. J., & Purpura, D. J. (2017). Early number and operations: Whole numbers. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 308–354). Reston: National Council of Teachers of Mathematics.Google Scholar
- Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics, National Governors Association and the Council of Chief State School Officers, Washington, DC. Retrieved at: http://www.corestandards.org/Math/Practice. Accessed 9 Aug 2018.
- Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–275). New York: Macmillan Publishing.Google Scholar
- Fuson, K. C., & Murata, A. (2007). Integrating NRC principles and the NCTM process standards to form a class learning path model that individualizes within whole-class activities. National Council of Supervisors of Mathematics Journal of Mathematics Education Leadership, 10(1), 72–91.Google Scholar
- Fuson, K. C., Murata, A., & Abrahamson, D. (2014). Using learning path research to balance mathematics education: Teaching/learning for understanding and fluency. In R. Cohen, Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1036–1054). Oxford: Oxford University Press.Google Scholar
- Huang, R., Kimmins, D., Winters, J., Douglas, D., & Tessema, A. (2017). Teacher learning through perfecting a lesson through Chinese lesson study. Paper presented at PME-NA 39 in Indianapolis, Indiana, USA.Google Scholar
- Lesh, R., Post, T., & Behr, M. (1987). Representations and translation among representations in mathematical learning and problem solving. In C. Janvier (Ed.), Problem of representation in teaching and learning of mathematics learning (pp. 3–40). Hilldale: Erlbaum.Google Scholar
- Li, X. (1992). On integrity and development of children’s addition and subtraction cognitive structure (in Chinese). Psychological Development and Education, 8(1), 9–16.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(1), 7–22.Google Scholar
- Marton, F. (2015). Necessary conditions of learning. New York: Routledge.Google Scholar
- Ministry of Education, P. R. China (MoE). (2011). Mathematics curriculum standards for compulsory education (grades 1–9) (in Chinese). Beijing: Beijing Normal University Press.Google Scholar
- National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics (NCTM).Google Scholar
- Pang, M. F., Bao, J. S., & Ki, W. W. (2017). “Bianshi” and the variation theory of learning: Illustrating two frameworks of variation and invariance in the teaching of mathematics. In R. Huang & Y. Li (Eds.), Teaching and learning through variations (pp. 43–68). Rotterdam: Sense.CrossRefGoogle Scholar
- Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2016). Elementary and middle school mathematics: Teaching developmentally. Boston: Person Education Inc. (9 edition).Google Scholar
- Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557–628). Reston: NCTM.Google Scholar
- Wang, J. (2013). Mathematics education in China: Tradition and reality. Singapore: Galeasia Cengage Learning.Google Scholar