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Leveraging variation of historical number systems to build understanding of the base-ten place-value system

  • Eva Thanheiser
  • Kathleen Melhuish
Original Article

Abstract

Prospective elementary school teachers (PTs) come to their mathematics courses fluent in using procedures for adding and subtracting multidigit whole numbers, but many are unaware of the essential features inherent in understanding the base-ten place-value system (i.e., grouping, place value, base). Understanding these features is crucial to understanding and teaching number and place value. The research aims of this paper are (1) to present a local instructional theory (LIT), designed to familiarize PTs with these features through comparison with historical number systems and (2) to present the effects of using the LIT in the PT classroom. A theory of learning (variation theory) is paired with a framework related to motivation (intellectual need) to illustrate the mutually supporting roles they may play in mathematical learning and task design. The LIT, a supporting task sequence, and the rationale for task design are shared. This theoretical contribution is then paired with evidence of PTs’ changing growth in their conceptions of whole number before and after courses leveraging this task sequence.

Keywords

Variation theory Intellectual need Whole number Prospective teachers Historical number systems Number and operation Place value Base ten 

References

  1. Adler, J. (2017). Foreword. In R. Huang & Y. Li (Eds.), Teaching and learning mathematics through variation: Confucian heritage meets western theories (pp. xi-xiv). Rotterdam: SensePublishers.Google Scholar
  2. Ball, D. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. (unpublished doctoral dissertation). Ann Arbor: Michigan State University.Google Scholar
  3. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (1999). How people learn. Washington, DC: National Academy Press.Google Scholar
  4. Bussey, T. J., Orgill, M., & Crippen, K. J. (2013). Variation theory: A theory of learning and a useful theoretical framework for chemical education research. Chemistry Education Research and Practice, 14(1), 9–22.  https://doi.org/10.1039/C2RP20145C.CrossRefGoogle Scholar
  5. Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., & Hiebert, J. (2017). Making classroom implementation an integral part of research. Journal for Research in Mathematics Education, 48(4), 342–347.CrossRefGoogle Scholar
  6. Chapman, O. (2007). Facilitating preservice teachers’ development of mathematics knowledge for teaching arithmetic operations. Journal of Mathematics Teacher Education, 10, 341–349.CrossRefGoogle Scholar
  7. Cobb, P., & Wheatley, G. (1988). Children’s initial understandings of ten. Focus on Learning Problems in Mathematics, 10(3), 1–28.Google Scholar
  8. Coles, A., & Brown, L. (2016). Task design for ways of working: Making distinctions in teaching and learning mathematics. Journal of Mathematics Teacher Education, 19(2), 149–168.  https://doi.org/10.1007/s10857-015-9337-4.CrossRefGoogle Scholar
  9. Crespo, S., & Nicol, C. (2006). Challenging preservice teachers’ mathematical understanding: The case of division by zero. School Science and Mathematics, 106(2), 84–97.CrossRefGoogle Scholar
  10. Fuson, K. C. (1990). A forum for researchers: Issues in place-value and multidigit addition and subtraction learning and teaching. Journal for Research in Mathematics Education, 21, 273–280.CrossRefGoogle Scholar
  11. Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I.,.. . Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28, 130–162.CrossRefGoogle Scholar
  12. Graeber, A. O. (1999). Forms of knowing mathematics: What preservice teachers should learn. Educational Studies in Mathematics, 38(3), 189–208.  https://doi.org/10.1023/A:1003624216201.CrossRefGoogle Scholar
  13. Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning: An International Journal, 6(2), 105–128.CrossRefGoogle Scholar
  14. Gravemeijer, K., & Cobb, P. (2006). Design research from the learning design perspective. In J. Van den Akker, K. Gravemerijer, S. McKenney & N. Nieveen (Eds.), Educational design research (pp. 17–55). London: Routledge.Google Scholar
  15. Gu, F., Huang, R., & Gu, L. (2017). Theory and development of teaching through variation in mathematics in china. In R. Haung & Y. Li (Eds.), Teaching and learning mathematics through variation (pp. 13–41). Rotterdam: Sense Publishers.CrossRefGoogle Scholar
  16. Harel, G. (1998). Two dual assertions: The first on learning and the second on teaching (or vice versa). The American Mathematical Monthly, 105(6), 497–507.  https://doi.org/10.2307/2589401.CrossRefGoogle Scholar
  17. Harel, G. (2013). Intellectual need. In K. R. Leatham (Ed.), Vital directions for mathematics education research (pp. 119–151). New York: Springer.CrossRefGoogle Scholar
  18. Harkness, S. S., & Thomas, J. (2008). Reflections on “multiplication as original sin”: The implications of using a case to help preservice teachers understand invented algorithms. Journal of Mathematical Behavior, 27(2), 128–137.CrossRefGoogle Scholar
  19. Hart, J. (2010). Contextualized motivation theory (cmt): Intellectual passion, mathematical need, social responsibility, and personal agency in learning mathematics. (Ph.D.). Provo: Brigham Young University.Google Scholar
  20. Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14, 251–283.CrossRefGoogle Scholar
  21. Holmqvist, M., Gustavsson, L., & Wernberg, A. (2008). Variation theory: An organizing principle to guide design research in education. In A. E. Kelly, R. Lesh & J. Y. Baek (Eds.), Handbook of Design Research Methods in Education (pp. 111–130). New York: Routledge.Google Scholar
  22. Kaasila, R., Pehkonen, E., & Hellinen, A. (2010). Finnish pre-service teachers’ and upper secondary students’ understanding of division and reasoning strategies used. Educational Studies in Mathematics, 73(3), 247–261.CrossRefGoogle Scholar
  23. Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education, 1(2), 75–86.CrossRefGoogle Scholar
  24. Khoury, H. A., & Zazkis, R. (1994). On fractions and non-standard representations: Pre-service teachers’ concepts. Educational Studies in Mathematics, 27(2), 191–204.CrossRefGoogle Scholar
  25. Kontorovich, I. (2015). Why do experts pose problems for mathematics competitions? In C. Bernack-Schüler, R. Erens, T. Leuders, & A. Eichler (Eds.), Views and beliefs in mathematics education: Results of the 19th mavi conference (pp. 171–181). Wiesbaden: Springer Fachmedien Wiesbaden.CrossRefGoogle Scholar
  26. Kontorovich, I., & Zazkis, R. (2016). Turn vs. Shape: Teachers cope with incompatible perspectives on angle. Educational Studies in Mathematics, 93, 1–21.CrossRefGoogle Scholar
  27. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in china and the united states. Mahwah: Erlbaum.Google Scholar
  28. Marton, F., & Booth, S. A. (1997). Learning and awareness. Mahwah: Lawrence Erlbuam Associates, Inc.Google Scholar
  29. 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: Erlbaum.Google Scholar
  30. Marton, F., Tsui, A. B., Chik, P. P., Ko, P. Y., & Lo, M. L. (2004). Classroom discourse and the space of learning. Mahwah: Lawrence Erlbaum Associates, Inc.Google Scholar
  31. May, L. J. (1970). Teaching mathematics in the elementary grades. New York: The Free Press.Google Scholar
  32. Menon, R. (2009). Preservice teachers’ subject matter knowledge of mathematics. The International Journal for Mathematics Teaching and Learning. Retrieved from http://www.cimt.org.uk/journal/menon.pdf. Accessed 8 Aug 2018.
  33. National Research Council (Ed.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
  34. Overbay, S. R., & Brod, M. J. (2007). Magic with mayan math. Teaching Mathematics in the Middle School, 12, 340–347.Google Scholar
  35. Pang, M. F., Bao, J., & Ki, W. W. (2017). ‘Bianshi’ and the variation theory of learning. In R. Huang & Y. Li (Eds.), Teaching and learning mathematics through variation: Confucian heritage meets western theories (pp. 43–67). Rotterdam: Sense Publishers.CrossRefGoogle Scholar
  36. Ross, S. (2001). Pre-service elementary teachers and place value: Written assessment using a digit-correspondence task. In R. Speiser, C. A. Maher, & C. N. Walter (Eds.), Proceedings of the twenty-third annual meeting of the north american chapter of the international group for the psychology of mathematics education (Vol. 2, pp. 897–906). Snowbird: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  37. Runesson, U. (2006). What is it possible to learn? On variation as a necessary condition for learning. Scandinavian Journal of Educational Research, 50(4), 397–410.CrossRefGoogle Scholar
  38. Ryve, A., Larsson, M., & Nilsson, P. (2013). Analyzing content and participation in classroom discourse: Dimensions of variation, mediating tools, and conceptual accountability. Scandinavian Journal of Educational Research, 57(1), 101–114.CrossRefGoogle Scholar
  39. Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475–5223.CrossRefGoogle Scholar
  40. Simon, M. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 24(3), 233–254.CrossRefGoogle Scholar
  41. Simon, M., & Blume, G. W. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 13(2), 183–197.CrossRefGoogle Scholar
  42. Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New York: Springer.CrossRefGoogle Scholar
  43. Stylianides, A. J., & Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72(2), 237–253.CrossRefGoogle Scholar
  44. Thanheiser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40(3), 251–281.Google Scholar
  45. Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: Refining a framework. Educational Studies in Mathematics, 75(3), 241–251.  https://doi.org/10.1007/s10649-010-9252-7.CrossRefGoogle Scholar
  46. Thanheiser, E. (2014). Developing prospective teachers’ conceptions with well-designed tasks: Explaining successes and analyzing conceptual difficulties. Journal of Mathematics Teacher Education, 18(2), 141–172.  https://doi.org/10.1007/s10857-014-9272-9.CrossRefGoogle Scholar
  47. Thanheiser, E. (2018). Brief report: The effects of preservice elementary teachers’ accurate self-assessments in the context whole number. Journal for Research in Mathematics Education, 49, 39–56.CrossRefGoogle Scholar
  48. Thanheiser, E., Philipp, R., Fasteen, J., Strand, K., & Mills, B. (2013). Preservice-teacher interviews: A tool for motivating mathematics learning. Mathematics Teacher Educator, 1(2), 137–147.  https://doi.org/10.5951/mathteaceduc.1.2.0137.CrossRefGoogle Scholar
  49. The Design-Based Research Collective. (2003). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5–8.CrossRefGoogle Scholar
  50. Valeras, M., & Becker, J. (1997). Children’s developing understanding of place value: Semiotic aspects. Cognition and Instruction, 15, 265–286.CrossRefGoogle Scholar
  51. Woleck, K. R. (2003). Tricky triangles: A tale of one, two, three researchers. Teaching Children Mathematics, 10(1), 40.Google Scholar
  52. Zazkis, R., & Campbell, S. (1996). Divisibility and multiplicative structure of natural numbers: Preservice teachers’ understanding. Journal for Research in Mathematics Education, 27, 540–563.CrossRefGoogle Scholar
  53. Zazkis, R., & Khoury, H. A. (1993). Place value and rational number representations: Problem solving in the unfamiliar domain of non-decimals. Focus on Learning Problems in Mathematics, 15(1), 38–51.Google Scholar

Copyright information

© FIZ Karlsruhe 2018

Authors and Affiliations

  1. 1.Portland State UniversityPortlandUSA
  2. 2.Texas State UniversitySan MarcosUSA

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