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Improving multiplicative reasoning in a context of low performance

  • Hamsa Venkat
  • Corin Mathews
Original Article

Abstract

In this paper we analyze the outcomes of a design experiment that sought to improve the multiplicative reasoning of 12–13 year-old learners across two schools in a South African context of low performance. Using a hybrid theoretical base in Realistic Mathematics Education, variation theory and analogical reasoning, a short-term intervention consisting of four lessons, designed with attunement to classroom culture and levels of learning, was implemented. Outcomes based on pre- and post-testing pointed to substantial gains in both schools, leading to interest in understanding the nature and extent of changes in models and calculation approaches in high performance and high gain item clusters. Increases in appropriate setting up of symbolic models of multiplicative situations and in more efficient calculation are discussed.

Keywords

Multiplicative reasoning South Africa Low attainment Design experiment RME 

Notes

Acknowledgements

This study is located within the South African Numeracy Chair project at the University of the Witwatersrand. It is generously supported by the FirstRand Foundation (with the RMB), Anglo American Chairman’s fund, the Department of Science and Technology and the National Research Foundation.

References

  1. Adler, J., & Venkat, H. (2014). Teachers’ mathematical discourse in instruction: Focus on examples and explanations. In H. Venkat, M. Rollnick, J. Loughran & M. Askew (Eds.), Exploring Mathematics and Science Teacher Knowledge: Windows into teacher thinking (pp. 132–146). Oxford: Routledge.Google Scholar
  2. Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics, 20, 367–385.CrossRefGoogle Scholar
  3. Askew, M. (2005a). Beam’s Big Book of Word Problems Year 3 and 4 (New edition). London: BEAM Education.Google Scholar
  4. Askew, M. (2005b). Beam’s Big Book of Word Problems Year 5 and 6 (New edition). London: BEAM Education.Google Scholar
  5. Askew, M., & Venkat, H. (2016). Developing South African primary learners’ multiplicative reasoning: The impact of a short teaching intervention. In: Presentation at the British Society for Research into Learning Mathematics Conference, Manchester Metropolitan University, February 27.Google Scholar
  6. Askew, M., & Venkat, H. (2018). Middle grade studentsperformance on arithmetic calculations presented as word problems or numeric problems. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.) Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (vol. 2), Umeä, Sweden, 3–8 July, pp. 75–83.Google Scholar
  7. Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. Educational Studies in Mathematics, 70(3), 217–241.CrossRefGoogle Scholar
  8. Burkhardt, H., & Schoenfeld, A. (2003). Improving educational research: Toward a more useful, more influential and better-funded enterprise. Educational Research, 32(9), 3–14.CrossRefGoogle Scholar
  9. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s Mathematics: Cognitively guided instruction. Portsmouth: Heinemann.Google Scholar
  10. Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1–5. Journal for Research in Mathematics Education, 27(1), 41–51.CrossRefGoogle Scholar
  11. Daroczy, G., Wolska, M., Meurers, W. D., & Nuerk, H.-D. (2015). Word problems: A review of linguistic and numerical factors contributing to their difficulty. Frontiers in Psychology, 6, 22–34.CrossRefGoogle Scholar
  12. DBE (2011). Curriculum and assessment policy statement (CAPS): foundation phase mathematics, grade R-3 Pretoria. Department for Basic Education.Google Scholar
  13. Dole, S., Clarke, D., Wright, T., & Hilton, G. (2012). Students’ proportional reasoning in mathematics and science. In T. Tso (Ed.), Proceedings of the 36th Conference of the International Group for the PME (vol. 2, pp. 195–202). Taipei: PME.Google Scholar
  14. English, L. D. (2012). Mathematical and analogical reasoning in early childhood. In L. D. English (Ed.), Mathematical and analogical reasoning of young learners (pp. 1–22). London: Routledge.Google Scholar
  15. Gravemeijer, K., Bruin-Muurling, G., Kraemer, J., & van Stiphout, I. (2016). Shortcomings of mathematics education reform in The Netherlands: A paradigm case? Mathematical Thinking and Learning, 18(1), 25–44.CrossRefGoogle Scholar
  16. Gravemeijer, K., & Terwel, J. (2000). Hans freudenthal: A mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777–796.CrossRefGoogle Scholar
  17. Graven, M. (2017). Blending elementary education research with development for equity: An ethical imperative enabling qualitatively richer work. Proceedings of the SEMT Conference. Prague. Aug 20–25, pp. 20–31.Google Scholar
  18. Graven, M., & Venkat, H. (2014). Primary Teachers’ Experiences Relating to the Administration Processes of High-stakes Testing: The Case of Mathematics Annual National Assessments. African Journal of Research in Mathematics, Science and Technology Education, 18(3), 299–310.CrossRefGoogle Scholar
  19. Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 276–295). New York: Macmillan.Google Scholar
  20. Hoadley, U. (2007). The reproduction of social class inequalities through mathematics pedagogies in South African primary schools. Journal of Curriculum Studies, 39(6), 679–706.CrossRefGoogle Scholar
  21. Kaur, B., & Dindyal, J. (2010). A prelude to mathematical applications and modelling in Singapore schools. In B. Kaur & J. Dinyal (Eds.) Mathematical applications and modelling: Yearbook 2010. Singapore: World Scientific Publishing Co.CrossRefGoogle Scholar
  22. Kieren, T. E. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 162–181). Reston: National Council of Teachers of Mathematics.Google Scholar
  23. Kouba, V. L. (1989). Children’s solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20, l47–l58.CrossRefGoogle Scholar
  24. Marton, F., & Booth, S. (1997). Learning and Awareness. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  25. Park, J., & Nunes, T. (2001). The development of the concept of multiplication. Cognitive Development, 16, 1–11.CrossRefGoogle Scholar
  26. Pritchett, L., & Beatty, A. (2012). The negative consequences of overambitious curricula in developing countries. Washington: Centre for Global Development.Google Scholar
  27. Reeves, C., & Muller, J. (2005). Picking up the pace: variation in the structure and organization of learning school mathematics. Journal of Education, 37, 103–130.Google Scholar
  28. Schollar, E. (2008). Final report: The primary mathematics research project 2004–2007—towards evidence-based educational development in South Africa. Johannesburg: Schollar & Associates.Google Scholar
  29. Schweisfurth, M. (2011). Learner-centred education in developing country contexts: From solution to problem? International Journal of Educational Development, 31, 425–432.CrossRefGoogle Scholar
  30. Skovsmose, O. (2011). An Invitation to Critical Mathematics Education. Rotterdam: Sense Publishers.CrossRefGoogle Scholar
  31. Sowder, J., Armstrong, B., Lamon, S., Simon, M., Sowder, L., & Thompson, A. (1998). Educating teachers to teach multiplicative structures in the middle grades. Journal of Mathematics Teacher Education, 1, 127–155.CrossRefGoogle Scholar
  32. Spaull, N., & Kotze, J. (2015). Starting behind and staying behind: The case of insurmountable learning deficits in South Africa. International Journal of Educational Development, 41, 13–24.CrossRefGoogle Scholar
  33. Streefland, L. (1985). Search for the roots of ratio: Some thoughts on the long term learning process (towards… a theory): Part II: The outline of the long term learning process. Educational Studies in Mathematics, 16, 75–94.CrossRefGoogle Scholar
  34. Thompson, I. (2008a). Deconstructing calculation methods, part 4: Multiplication. Mathematics Teaching incorporating Micromath, 206, 34–36.Google Scholar
  35. Thompson, I. (2008b). Deconstructing calculation methods, part 4: Division. Mathematics Teaching incorporating Micromath, 208, 6–8.Google Scholar
  36. Venkat, H., & Adler, J. (2012). Coherence and connections in teachers’ mathematical discourses in instruction. Pythagoras, 33(3), 25–32.CrossRefGoogle Scholar
  37. Venkat, H., & Spaull, N. (2015). What do we know about primary teachers’ mathematical content knowledge in South Africa? An analysis of SACMEQ 2007. International Journal of Educational Development, 41, 121–130.CrossRefGoogle Scholar
  38. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 128–175). London: Academic Press.Google Scholar

Copyright information

© FIZ Karlsruhe 2018

Authors and Affiliations

  1. 1.Wits School of EducationUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.School of Education and CommunicationJönköping UniversityJönköpingSweden

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