ZDM

, Volume 48, Issue 4, pp 455–470 | Cite as

Teaching to add three-digit numbers in Hong Kong and Shanghai: illustration of differences in the systematic use of variation and invariance

  • Ming Fai Pang
  • Ference Marton
  • Jian-sheng Bao
  • Wing Wah Ki
Original Article

Abstract

In this study, the systematic use of variation and invariance in the teaching of mathematics is examined in accordance with two different but compatible explicit frameworks. We consider the following: (1) differences in tasks that follow on from each other can significantly change what can be learned; (2) Chinese teachers (and probably those in other high achieving countries in South-East Asia) pay considerable attention to the differences between tasks that follow on from each other; and (3) nonetheless, experienced teachers inspired by the two frameworks both of which pay attention to how tasks that follow on from each other can differ, while teaching the same mathematical topic, may still generate very different patterns of tasks and thereby make very different kinds of learning possible. Two experienced Hong Kong teachers devised and planned a series of lessons on the addition and subtraction of three-digit numbers, together with an expert on an explicit framework for systematic use of variation and invariance. Two Shanghai teachers did the same, together with an expert on a different, but compatible, framework. This paper focuses on the analysis of two lessons, one in Hong Kong and one in Shanghai, both being the first of their respective series. The video-recorded lessons were analysed with the aim of illustrating similarities and differences in terms of the systematic use of variation and invariance. These were not intended to be model lessons illustrating the respective frameworks of variation, but rather were examples of the ways teachers consciously and systematically make use of variation and invariance, as inspired by the two frameworks.

Keywords

Variation Invariance Addition Three-digit numbers Mathematics teaching 

References

  1. Bao, J., Huang, R., Yi, L., & Gu, L. (2003). Study of ‘Bianshi Jiaoxue’ [Teaching with variation]. Mathematics Teaching, 1–3.Google Scholar
  2. Fan, L., & Zhu, Y. (2004). How have Chinese students performed in mathematics? A perspective from large-scale international comparisons. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: perspectives from insiders (pp. 309–347). Singapore: World Scientific.CrossRefGoogle Scholar
  3. Gu, L. (1991). Xuehui Jiaoxue [Learning to teach]. Beijing: People’s Education Press.Google Scholar
  4. Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: perspectives from insiders (pp. 309–347). Singapore: World Scientific.CrossRefGoogle Scholar
  5. Häggström, J. (2008). Teaching systems of linear equations in Sweden and China: What is made possible to learn?. Göteborg: Acta Universitatis Gothoburgensis.Google Scholar
  6. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: the case of Mathematics (pp. 1–28). N J, Hillsdale: Lawrence Erlbaum.Google Scholar
  7. Jones, K., & Herbst, P. (2012). Proof, proving, and teacher-student interaction: Theories and contexts. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (the 19th ICMI Study) (pp. 261–277). New York: Springer.Google Scholar
  8. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah: Lawrence Erlbaum.Google Scholar
  9. Marton, F. (2015). Necessary conditions of learning. London: Routledge.Google Scholar
  10. Marton, F., & Tsui, A. B. M. with Chik, P. M., Ko, P. Y., Lo, M. L., Mok, I. A. C., Ng, F. P., Pang, M. F., Pong, W. Y., & Runesson, U. (2004). Classroom discourse and the space of learning. Mahwah, NJ: Lawrence Erlbaum AssociatesGoogle Scholar
  11. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  12. Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.CrossRefGoogle Scholar
  13. Marton, F., & Pang, M. F. (2008). The idea of phenomenography and the pedagogy for conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (pp. 533–559). London: Routledge.Google Scholar
  14. Marton, F., & Pang, M. F. (2013). Meanings are acquired from experiencing differences against a background of sameness, rather than from experiencing sameness against a background of difference: putting a conjecture to the test by embedding it in a pedagogical tool. Frontier Learning Research, 1(1), 24–41.Google Scholar
  15. Marton, F., Runesson, U., & Tsui, A. (2004). The space for learning. In F. Marton & A. Tsui (Eds.), Classroom discourse and the space for learning (pp. 3–40). Mahwah, NJ: Lawrence Erlbaum AssociatesGoogle Scholar
  16. Mason, J., & Johnston-Wilder, S. (2006). Designing and using mathematical tasks. London: QED Publishing.Google Scholar
  17. OECD (2013). PISA 2012 results: What students know and can doStudent performance in mathematics, reading and science (Vol.1). PISA, OECD Publishing.Google Scholar
  18. Pang, M. F. (2003). Two faces of variation—on continuity in the phenomenographic movement. Scandinavian Journal of Educational Research, 47(2), 145–156.CrossRefGoogle Scholar
  19. Pang, M. F., & Ki, W. W. (2016). Revisiting the idea of ‘critical aspects’. Scandinavian Journal of Educational Research. doi:10.1080/00313831.2015.1119724.Google Scholar
  20. Pang, M. F., & Marton, F. (2003). Beyond “lesson study”—Comparing two ways of facilitating the grasp of economic concepts. Instructional Science, 31, 175–194.CrossRefGoogle Scholar
  21. Pang, M. F., & Marton, F. (2013). Interaction between the learners’ initial grasp of the object of learning and the learning resource afforded. Instructional Science, 41, 1065–1082.CrossRefGoogle Scholar
  22. Ritchhart, R. (1999). Generative topics: building a curriculum around big ideas. Teaching Children Mathematics, 5, 462–468.Google Scholar
  23. Sun, X. (2011). ‘Variation problems’ and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educational Studies in Mathematics, 76(1), 65–85.CrossRefGoogle Scholar
  24. Vygotsky, L. S. (1986). Language and thought. Cambridge, MA: MIT Press.Google Scholar
  25. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  26. Wong, N. Y., Lam, C. C., Sun, X., Chan, & A. M. Y. (2009). From “exploring the middle zone” to “constructing a bridge”: Experimenting the spiral bianshi mathematics curriculum. International Journal of Science and Mathematics Education, 7, 363–382.Google Scholar
  27. Xie, H., Fan, Y., & Chu, H. (2016). A case study of three-digit number addition which is based on Variation Theory. Unpublished manuscript.Google Scholar

Copyright information

© FIZ Karlsruhe 2016

Authors and Affiliations

  • Ming Fai Pang
    • 1
  • Ference Marton
    • 2
  • Jian-sheng Bao
    • 3
  • Wing Wah Ki
    • 1
  1. 1.The University of Hong KongHong Kong SARChina
  2. 2.Gothenburg UniversityGothenburgSweden
  3. 3.East China Normal UniversityShanghaiChina

Personalised recommendations