Advertisement

Aspects that Affect Whole Number Learning: Cultural Artefacts and Mathematical Tasks

  • Maria G. Bartolini BussiEmail author
  • Maitree InprasithaEmail author
  • Ferdinando Arzarello
  • Hyman Bass
  • Ulrich Kortenkamp
  • Silke Ladel
  • Caroline Lajoie
  • Yujing Ni
  • Thomas Rottmann
  • Veronica Sarungi
  • Sophie Soury-Lavergne
  • Jenny Young-Loveridge
Chapter
Part of the New ICMI Study Series book series (NISS)

Abstract

The core of this chapter is the notion of artefact, starting from the discussion of the meaning of the word in the literature and offering a gallery of cultural artefacts from the participants’ reports and the literature. The idea of artefacts is considered in a broad sense, to include also language and texts. The use of cultural artefacts as teaching aids is addressed. A special section is devoted to the artefacts (teaching aids) from technologies (including virtual manipulatives). The issue of tasks is simply skimmed, but it is not possible to discuss about artefacts without considering the way of using artefacts with suitable tasks. Some examples of tasks are reported to elaborate about aspects that may foster learning whole number arithmetic (WNA). Artefacts and tasks appear as an inseparable pair, to be considered within a cultural and institutional context. Some future challenges are outlined concerning the issue of teacher education, in order to cope with this complex map.

Supplementary material

Video 9.1

(MP4 39738 kb)

Video 9.2

(MP4 116183 kb)

Video 9.3

(MP4 50589 kb)

Video 9.4

(MP4 55655 kb)

Video 9.5

(MP4 52215 kb)

Video 9.6

(MP4 39120 kb)

Video 9.7

(MP4 30084 kb)

Video 9.8

(MP4 5265 kb)

References

  1. Aebli, H. (1976). Psychologische Didaktik-Didaktische Auswertungen der Psychologie von Jean Piaget. Stuttgart: Klett-Cotta.Google Scholar
  2. Australian Curriculum Assessment and Reporting Authority. (2013). The Australian curriculum: Mathematics v2.4. Retrieved March 17, 2013, http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10
  3. Baccaglini-Frank, A., & Maracci, M. (2015). Multi-touch technology and preschoolers’ development of number-sense. Digital Experiences in Mathematics Education, 1(1), 7–27.CrossRefGoogle Scholar
  4. Baccaglini-Frank, A., Hoyles, C., & Noss, R. (2014, September 21–26). From notable occurrences to situated abstractions: A window for analysing learners’ thinking-in-change in a microworld. In Proceedings of the 12th international conference of the mathematics education into the 21st century project: The future of mathematics education in a connected world. Montenegro.Google Scholar
  5. Bartolini Bussi, M. G. (2011). Artefacts and utilization schemes in mathematics teacher education: Place value in early childhood education. Journal of Mathematics Teacher Education, 14(2), 93–112.CrossRefGoogle Scholar
  6. Bartolini Bussi, M. G. (2013). Bambini che contano: A long-term program for preschool teachers’ development. In B. Ubuz, et al. (Eds.), CERME 8. Proceedings of the eight congress of the European Society of Research in Mathematics Education, (pp. 2088–2097). Ankara: Middle East Technical University.Google Scholar
  7. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, B. Sriraman, & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 746–783). New York: Routledge Taylor & Francis Group.Google Scholar
  8. Bartolini Bussi, M. G., & Martignone, F. (2014). Manipulatives in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 365–372). Dordrecht: Springer.Google Scholar
  9. Barwell, R., Clarkson, P., Halai, A., Kazima, M., Moschkovich, J., Planas, N., Setati, M., Valero, P., & Villavicencio Ubillús, M. (2015). Mathematics education and language diversity. The 21st ICMI study (New ICMI Studies Series). New York/Berlin: Springer.Google Scholar
  10. Bruner, J. S. (1973). The relevance of education. New York: Norton.Google Scholar
  11. Butterworth, B., Reeve, R., Reynolds, F., & Lloyd, D. (2008). Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences of the United States of America, 105(35), 13179–13184.CrossRefGoogle Scholar
  12. Cheng, Z. J. (2012). Teaching young children decomposition strategies to solve addition problems: An experimental study. The Journal of Mathematical Behavior, 31, 29–47.CrossRefGoogle Scholar
  13. Chevallard, Y., & Bosch, M. (2014). Didactic transposition in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 170–174). Dordrecht: Springer.Google Scholar
  14. Cole, M. (1996). Cultural psychology. A once and future discipline. Cambridge, MA: The Belknap Press.Google Scholar
  15. Conti, A. (1920). Aritmetica per la prima classe elementare, Firenze: Bemporad e Figlio. Retrieved from http://www.indire.it/archivi/dia
  16. D’Amore, B., Radford, L., & Bagni, G. (2016). Obstáculos epistemológicos y perspectiva socio-cultural de la matemática. In B. D’Amore & L. Radford (Eds.), Enseñanza y aprendizaje de las matemáticas: Problemas semióticos, epistemológicos y prácticos (pp. 167–194). Bogotá: Editorial Universidad Distrital Francisco José de Caldas.Google Scholar
  17. Dienes, Z. P. (1963). An experimental study of mathematics learning. London: Hutchinson.Google Scholar
  18. Dong-Joong, K., Ferrini-Mundy, J., & Sfard, A. (2012). How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity. International Journal of Educational Research, 51/52, 86–108.CrossRefGoogle Scholar
  19. Engeström, Y. (1987). Learning by expanding: An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit.Google Scholar
  20. English, L. (Ed.). (1997). Mathematical reasoning: Analogies, metaphors, and images. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  21. Fauvel, J., & van Maanen, J. (Eds.). (2000). History in mathematics education. The ICMI study. Dordrecht: Kluwer.Google Scholar
  22. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393–425.CrossRefGoogle Scholar
  23. Hoyles, C., & Lagrange, J. B. (Eds.). (2010). Mathematics education and technology-rethinking the terrain. The 17th ICMI study. New York: Springer.Google Scholar
  24. Inprasitha, M., & Isoda, M. (eds). (2010). Study with your friends: Mathematics for elementary school 1st grade. Khon Kaen: Klungnanawittaya. (In Thai).Google Scholar
  25. Inprasitha, M., & Jai-on, K. (2016). The potential of dbook Pro to teach whole number arithmetic. [Online]. Retrieved on March 16, 2016, from: http://www.crme.kku.ac.th/The potential of dbookPro Exemplar 3+2
  26. Isoda, M. (2012). Problem-solving approach to develop mathematical thinking. In Isoda, M. & Katagiri, S. (Eds.), Monographs on lesson study for teaching mathematics and sciences–Vol. 1: Mathematical thinking-how to develop it in the classroom. (pp. 1–28). Singapore: World Scientific.Google Scholar
  27. Jacobsen, L. E. (1983). Use of knotted string accounting records in old Hawaii and ancient China. The Accounting Historians Journal, 10(2), 53–62.CrossRefGoogle Scholar
  28. Karp, A., & Schubring, G. (Eds.). (2014). Handbook on the history of mathematics education. New York: Springer.Google Scholar
  29. Kaur, B. (2010). A study of mathematical task from three classrooms in Singapore. In Y. Shimizu, B. Kaur, R. Huang, & D. J. Clarke (Eds.), Mathematical tasks in classrooms around the world (pp. 15–33). Rotterdam: Sense Publishers.Google Scholar
  30. Kempinsky, H. (1921). So rechnen wir bis hundert und darüber hinaus. Eine Anleitung für den Rechenunterricht besonders des zweiten Schuljahres. Leipzig: Verlag der Dürr’schen Buchhandlung.Google Scholar
  31. Kordemsky, B. A. (1992). The Moscow puzzles: 359 mathematical recreations. New York: Dover.Google Scholar
  32. Lam, L. Y., & Ang, T. S. (2004). Fleeting footsteps: Tracing the conception of arithmetic and algebra in ancient China. Singapore: World Scientific.Google Scholar
  33. Meaney, T., Trinick, T., & Fairhall, U. (2012). Collaborating to meet language challenges in indigenous mathematics classrooms. New York: Springer.CrossRefGoogle Scholar
  34. Menninger, K. (1969). Number words and number symbols: A cultural history of numbers. Cambridge, MA: The MIT Press. (Translated from the German edition of 1958).Google Scholar
  35. Miller, J., & Warren, E. (2014). Exploring ESL students’ understanding of mathematics in the early years: Factors that make a difference. Mathematics Education Research Journal, 26(4), 791–810.CrossRefGoogle Scholar
  36. Ministerium für Schule und Weiterbildung des Landes Nordrhein-Westfalen (MSW NRW). (2008). Richtlinien und Lehrpläne für die Grundschule in Nordrhein-Westfalen. Mathematik. Frechen: Ritterbach.Google Scholar
  37. Ministry of Education Science and Technology (MOEST). (2002). Primary education syllabus: Volume two. Nairobi: Kenya Institute of Education.Google Scholar
  38. Monaghan, J., Trouche, L., & Borwein, J. (2016). Tools and mathematics. Instruments for learning. New York: Springer.Google Scholar
  39. Murayama, K., Pekrun, R., Lichtenfeld, S., & vom Hofe, R. (2013). Predicting long-term growth in students’ mathematics achievement: The unique contributions of motivation and cognitive strategies. Child Development, 84(4), 1475–1490.CrossRefGoogle Scholar
  40. Ni, Y. J., Chiu, M. M., & Cheng, Z. J. (2010). Chinese children learning mathematics: From home to school. In M. H. Bond (Ed.), The Oxford handbook of Chinese psychology (pp. 143–154). Oxford: Oxford University Press.Google Scholar
  41. Nührenbörger, M., & Steinbring, H. (2008). Manipulatives as tools in teacher education. In D. Tirosh & T. Wood (Eds.), Tools and processes in mathematics teacher education. Volume 2 of The international handbook of mathematics teacher education (pp. 157–182). Rotterdam: Sense Publishers.Google Scholar
  42. Rabardel, R. (1995). Les hommes et les technologies. Approche cognitive des instruments contemporains. Paris: Armand Colin.Google Scholar
  43. Radford, L. (1997). On psychology, historical epistemology and the teaching of mathematics: Towards a socio-cultural history of mathematics. For the Learning of Mathematics, 17(1), 26–33.Google Scholar
  44. Rieber, R. W., & Wollock, J. (1997). The instrumental method in psychology. In R. W. Rieber & J. Wollock (Eds.), The collected works of L. S. Vygotsky: Problems of the theory and history of psychology (pp. 85–89). Boston: Springer.CrossRefGoogle Scholar
  45. Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3(3), 145–150.CrossRefGoogle Scholar
  46. Schipper, W., Dröge, D., & Ebeling, A. (2000). Handbuch für den Mathematikunterricht, 4. Schuljahr. Hannover: Schroedel.Google Scholar
  47. Shimizu, S., & Watanabe, T. (2010). Principles and processes for publishing textbooks and alignment with standards: A case in Japan. Paper presented at the APEC conference on replicating exemplary practices in mathematics education 2010, Koh Samul: Suratthani.Google Scholar
  48. Sierpinska, A. (1996). Understanding in mathematics. London: Falmer Press.Google Scholar
  49. Sinclair, N., & Baccaglini-Frank, A. (2015). Digital technologies in the early primary school classroom. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education: Third edition (pp. 662–686). Taylor and Francis.Google Scholar
  50. Sinclair, N., & Pimm, D. (2014). Number’s subtle touch: Expanding finger gnosis in the era of multi-touch technologies. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the 38th conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 209–216), Vancouver BC: PME.Google Scholar
  51. Stacey, K., Helme, S., Shona Archer, S., & Caroline Condon, C. (2001). The effect of epistemic fidelity and accessibility on teaching with physical materials: A comparison of two models for teaching decimal numeration. Educational Studies in Mathematics, 47(2), 199–221.CrossRefGoogle Scholar
  52. Stetsenko, A. (2008). From relational ontology to transformative activist stance on development and learning: Expanding Vygotsky’s (CHAT) project. Cultural Studies of Science Education, 3(2), 471–491.CrossRefGoogle Scholar
  53. Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3(4), 268–275.Google Scholar
  54. Tahan, M. (1996). O omem que calculava. Rio de Janeiro: Grupo Editorial Record.Google Scholar
  55. The Council of Chief State School Officers (CCSSO). (2016). National Governors Association Center for Best Practices and Council of Chief State School Officers. Common core state standards for mathematics. Retrieved March 20, 2016. http://www.corestandards.org/Math/
  56. The Ministry of Education. (2011a). Syllabus for the teaching of primary mathematics of the nine-year compulsory education. 北京:北京师范大学出版社. Beijing: Beijing Normal University Press. (in Chinese).Google Scholar
  57. The Ministry of Education. (2011b). Guidelines of mathematics curriculum for 9-year compulsory education. <基础教育课程改革纲要>. 北京:北京师范大学出版社. Beijing: People’s Education Publishing. (in Chinese).Google Scholar
  58. Theodore, R., Trustin, K., Kiro, C., Gollop, M., Taumoepeau, M., Taylor, N., Chee, K. S., Hunter, J., & Poulton, R. (2015, November). Maori university graduates: Indigenous participation in higher education. Higher Education Research & Development, 2015, 1–15.  https://doi.org/10.1080/07294360.2015.1107883
  59. Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18(1), 37–54.CrossRefGoogle Scholar
  60. Watson, A., & Ohtani, M. (Eds.) (2015). Task design in mathematics education. The 22nd ICMI study. New York: Springer.Google Scholar
  61. Wartha, S., & Schulz, A. (2012). Rechenproblemen vorbeugen. Berlin: Cornelsen.Google Scholar
  62. Xie, X., & Carspecken, P. F. (2007). Philosophy, learning and the mathematics curriculum dialectical materialism and pragmatism related to Chinese and American mathematics curriculums. Rotterdam: Sense Publishers.Google Scholar
  63. Zaslavsky, C. (1973). Africa counts: Number and pattern in African cultures. Chicago: Lawrence Hill Books.Google Scholar
  64. Zhou, Z., & Peverly, S. (2005). Teaching addition and subtraction to first graders: A Chinese perspective. Psychology in the Schools, 42(3), 259–272.CrossRefGoogle Scholar

Cited papers from Sun, X., Kaur, B., & Novotna, J. (Eds.). (2015). Conference proceedings of the ICMI study 23: Primary mathematics study on whole numbers. Retrieved February 10, 2016, from www.umac.mo/fed/ICMI23/doc/Proceedings_ICMI_STUDY_23_final.pdf

  1. Bakker, M., van den Heuvel-Panhuizen, M., & Robitzsch, A. (2015). Learning multiplicative reasoning by playing computer games (pp. 282–289).Google Scholar
  2. Baldin, Y., Mandarino, M. C., Mattos, F. R., & Guimarães, L. C. (2015). A Brazilian project for teachers of primary education: Case of whole numbers (pp. 510–517).Google Scholar
  3. Ball, D. L., & Bass, H. (2015). Helping students learn to persevere with challenging mathematics (pp. 379–386).Google Scholar
  4. Bartolini Bussi, M. G. (2015). The number line: A “western” teaching aid (pp. 298–306).Google Scholar
  5. Cao, Y., Li, X., & Zou, H. (2015). Characteristics of multiplication teaching of whole numbers in china: The application of nine times table (pp. 423–430).Google Scholar
  6. González, S., & Caraballo, J. (2015). Native American cultures tradition to whole number arithmetic (pp. 92–98).Google Scholar
  7. Hodgson, B. R., & Lajoie, C. (2015). The preparation of teachers in arithmetic: A mathematical and didactical approach (pp. 307–314).Google Scholar
  8. Inprasitha, M. (2015). An open approach incorporating lesson study: An innovation for teaching whole number arithmetic (pp. 315–322).Google Scholar
  9. Ladel, S., & Kortenkamp, U. (2015). Development of conceptual understanding of place value (pp. 323–330).Google Scholar
  10. Mercier, A., & Quilio, S. (2015). The efficiency of primary level mathematics teaching in French-speaking countries: A synthesis (pp. 331–338).Google Scholar
  11. Ni, Y. J. (2015). How the Chinese methods produce arithmetic proficiency in children (pp. 341–342).Google Scholar
  12. Peter-Koop, A., Kollhoff, S., Gervasoni, A., & Parish, L. (2015). Comparing the development of Australian and German children’s whole number knowledge (pp. 346–353).Google Scholar
  13. Pimm, D., & Sinclair, N. (2015). The ordinal and the fractional: Some remarks on a trans-linguistic study (pp. 354–361).Google Scholar
  14. Roberts, N. (2015). Interpreting children’s representations of whole number additive relations in the early grades (pp. 243–250).Google Scholar
  15. Rottmann, T., & Peter-Koop, A. (2015). Difficulties with whole number learning and respective teaching strategies (pp. 362–370).Google Scholar
  16. Siu, M. K. (2015). Pedagogical lessons from Tongwen Suanzhi (同文算指) – Transmission of bisuan (筆算 written calculation) in China (pp. 132–139).Google Scholar
  17. Soury-Lavergne, S., & Maschietto, M. (2015). Number system and computation with a duo of artifacts: The pascaline and the e-pascaline (pp. 371–378).Google Scholar
  18. Sun, X. (2015). Chinese core tradition to whole number arithmetic (pp. 140–148).Google Scholar
  19. Young-Loveridge, J., & Bicknell, B. (2015). Using multiplication and division contexts to build place-value understanding (pp. 379–386).Google Scholar
  20. Zou, D. (2015). Whole number in ancient Chinese civilisation: A survey based on the system of counting-units and the expressions (pp. 157–164).Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of Education and HumanitiesUniversity of Modena and Reggio EmiliaModenaItaly
  2. 2.Khon Kaen UniversityKhon KaenThailand
  3. 3.University of TurinTurinItaly
  4. 4.University of MichiganAnn ArborUSA
  5. 5.Universität PotsdamPotsdamGermany
  6. 6.Universität des SaarlandesSaarbrückenGermany
  7. 7.Université du Québec à MontréalMontrealCanada
  8. 8.Chinese University of Hong KongHong KongChina
  9. 9.Bielefeld UniversityBielefeldGermany
  10. 10.Institute for Educational Development, The Aga Khan UniversityDar es SalaamTanzania
  11. 11.IFE ENS de LyonLyonFrance
  12. 12.University of WaikatoHamiltonNew Zealand

Personalised recommendations