Educational Studies in Mathematics

, Volume 93, Issue 3, pp 293–313 | Cite as

Coding teaching for simultaneity and connections

Examining teachers’ part-whole additive relations instruction
Article

Abstract

In this article, we present a coding framework based on simultaneity and connections. The coding focuses on microlevel attention to three aspects of simultaneity and connections: between representations, within examples, and between examples. Criteria for coding that we viewed as mathematically important within part-whole additive relations instruction were developed. Teachers’ use of multiple representations within an example, attention to part-whole relations within examples, and relations between multiple examples were identified, with teachers’ linking actions in speech or gestures pointing to connections between these. In this article, the coding framework is detailed and exemplified in the context of a structural approach to part-whole teaching in six South African grade 3 lessons. The coding framework enabled us to see fine-grained differences in teachers’ handling of part-whole relations related to simultaneity of, and connections between, representations and examples as well as within examples. We went on to explore the associations between the simultaneity and connections seen through the coding framework in sections of teaching and students’ responses on worksheets following each teaching section.

Keywords

Simultaneity Connections Additive relations Coding framework Primary mathematics Variation theory South Africa 

Notes

Acknowledgments

This paper forms part of the work in progress within the Wits SA Numeracy Chair project, entitled the Wits Maths Connect—Primary project. It is generously funded by the FirstRand Foundation, Anglo American, Rand Merchant Bank, and the Department of Science and Technology and is administered by 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 content knowledge for teaching science and mathematics (pp. 132–146). London: Routledge.Google Scholar
  2. Alibali, M. W., Nathan, M. J., Wolgram, M. S., Church, R. B., Jacobs, S. A., Martinez, C. J., & Knuth, E. J. (2013). How teachers link ideas in mathematics instruction using speech and gesture: A corpus analysis. Cognition and Instruction, 32(1), 65–100. London: Routledge.CrossRefGoogle Scholar
  3. Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. Educational Studies in Mathematics, 70(2), 97–109.CrossRefGoogle Scholar
  4. Askew, M., Brown, M., Rhodes, V., Johnson, D. C., & Wiliam, D. (1997). Effective teachers of numeracy. Report of a study carried out for the Teacher Training Agency 1995–96 by the School of Education, King’s College London. London: Teacher Training Agency.Google Scholar
  5. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portmouth, NH: Heinemann.Google Scholar
  6. Clements, D. H. (1999). Playing math with young children. Curriculum Administrator, 35(4), 25–28.Google Scholar
  7. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258–277.CrossRefGoogle Scholar
  8. Doyle, W. (1983). Academic work. Review of Educational Research, 53, 159–199.CrossRefGoogle Scholar
  9. Flevares, L. M., & Perry, M. (2001). How many do you see? The use of nonspoken representations in first-grade mathematics lessons. Journal of Educational Psychology, 93, 330–345.CrossRefGoogle Scholar
  10. Fleisch, B. (2008). Primary education in crisis—Why South African schoolchildren underachieve in reading and mathematics. Cape Town: Juta & Co.Google Scholar
  11. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., et al. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study (NCES 2003–013). U.S. Department of Education. Washington, DC: National Center for Education Statistics.Google Scholar
  12. Kullberg, A., Runesson, U., & Mårtensson, P. (2014). Different possibilities to learn from the same task. PNA, 8(4), 139–150.Google Scholar
  13. Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in teaching and learning of mathematics (pp. 33–40). Hilldale, NJ: Erlbaum.Google Scholar
  14. McNeill, D. (1992). Hand and mind: What gesture reveals about thought. Chicago, IL: University of Chicago Press.Google Scholar
  15. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Erlbaum.Google Scholar
  16. Marton, F. (2015). Necessary condition of learning. London: Routledge.Google Scholar
  17. Marton, F., Runesson, U., & Tsui, A. B. (2004). The space of learning. In F. Marton & A. B. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3–40). Mahwah, N.J: Erlbaum.Google Scholar
  18. Marton, F., & Pang, M. F. (2006). On some necessary conditions for learning. The Journal of the Learning Sciences, 15(2), 193–220.CrossRefGoogle Scholar
  19. Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10–32.CrossRefGoogle Scholar
  20. Neuman, D. (1987). The origin of arithmetic skills. A phenomenographic approach. Göteborg: Acta Universitatis Gothoburgensis.Google Scholar
  21. Neuman, D. (1999). Early learning and awareness of division: A phenomenographic approach. Educational Studies in Mathematics, 40, 101–128.CrossRefGoogle Scholar
  22. Neuman, D. (2013). Att ändra arbetssätt och kultur inom den inledande aritmetikundervisningen [Changing approach and culture within the introduction of arithmetic; in Swedish]. Nordic Studies in Mathematics Education, 18(2), 3–46.Google Scholar
  23. Rowland, T. (2008). The purpose, design, and use of examples in teaching of elementary mathematics. Educational Studies in Mathematics, 69, 143–163.CrossRefGoogle Scholar
  24. Sun, X. (2011). ‘Variation problems’ and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educational Studies in Mathematics, 76, 65–85.CrossRefGoogle Scholar
  25. Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83, 323–340.CrossRefGoogle Scholar
  26. Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19(1), 19–43.CrossRefGoogle Scholar
  27. Schollar, E. (2008). Final Report: The primary mathematics research project 2004–2007—Towards evidence-based educational development in South Africa. Johannesburg: Eric Schollar & Associates.Google Scholar
  28. Venkat, H., & Adler, J. (2012). Coherence and connections in teachers’ mathematical discourses in instruction. Pythagoras, 33(3), 25–32.CrossRefGoogle Scholar
  29. Venkat, H., & Naidoo, D. (2012). Analyzing coherence for conceptual learning in a grade 2 numeracy lesson. Education as Change, 16(1), 21–33.CrossRefGoogle Scholar
  30. Venkat, H., Ekdahl, A-L., & Runesson, U. (2014). Connections and simultaneity: Analysing South African G3 Part-part-whole teaching. In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 5, pp. 337–344). Vancouver, Canada: PME.Google Scholar
  31. Watson, A., & Mason, J. (2006a). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.CrossRefGoogle Scholar
  32. Watson, A., & Mason, J. (2006b). Variation and mathematical structure. Mathematics Teaching, 194, 3–5.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Anna-Lena Ekdahl
    • 1
    • 2
  • Hamsa Venkat
    • 1
    • 2
  • Ulla Runesson
    • 1
    • 2
  1. 1.Jönköping UniversityJönköpingSweden
  2. 2.Univeristy of WitwatersrandJohannesburgSouth Africa

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