Advertisement

ZDM

, Volume 51, Issue 1, pp 69–80 | Cite as

Mathematicians and teachers sharing perspectives on teaching whole number arithmetic: boundary-crossing in professional development

  • Jason CooperEmail author
Original Article

Abstract

Teachers and mathematicians hold different perspectives on the teaching and learning of whole number arithmetic. Though these perspectives may be complementary, sharing them across communities is challenging. An unusual professional development course for primary school teachers, initiated and taught by research mathematicians, provided a setting for productive sharing of both mathematical and pedagogical perspectives. Drawing on the theory of commognition, I analyze two lesson segments, one, designed by one of the mathematician-instructors, included a discussion of an alternate method for performing vertical subtraction; the other, initiated by a 3rd grade teacher, is a discussion of an authentic classroom activity. Through these analyses, and drawing on the notion of boundary as sociocultural differences between communities, I reveal some mechanisms of the perspective-sharing that took place, and discuss what and how the parties learned from and with each other. I also highlight the role of a participant-observer researcher as a broker in this process, supporting events of boundary-crossing in which the parties came to explicate, and sometimes change, their own perspectives on teaching and learning mathematics with respect to the perspectives of others. I propose this PD setting as a model for sharing perspectives across these communities.

Keywords

Professional development Mathematicians Commognition Boundary-crossing 

Notes

Acknowledgements

This research was supported by the Israel Science Foundation (Grant no. 615/13). Sincere thanks are due to Abraham Arcavi for his guidance and advice.

References

  1. Akkerman, S. F., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research, 81(2), 132–169.CrossRefGoogle Scholar
  2. Bass, H. (2005). Mathematics, mathematicians, and mathematics education. Bulletin (New Series) of the American Mathematical Society, 42(4), 417–430.CrossRefGoogle Scholar
  3. Boland, R. J., & Tenkasi, R. V. (1995). Perspective making and perspective taking in communities of knowing. Organization Science, 6, 350–372.CrossRefGoogle Scholar
  4. Cooper, J. (2014). Mathematical discourse for teaching: A discursive framework for analyzing professional development. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36. 2 (pp. 337–344). Vancouver: PME.Google Scholar
  5. Cooper, J. (2015). Growth of mathematical knowledge for teaching—The case of long division. In CERME 9-ninth congress of the European society for research in mathematics education (pp. 2081–2088).Google Scholar
  6. Cooper, J. (2016). Mathematicians and primary school teachers learning from each other. Unpublished doctoral dissertation, Weizmann Institute of Science, Israel. Retrieved Dec. 15, 2017, from https://stwww1.weizmann.ac.il/wp-content/uploads/2016/09/Cooper-dissertation-finalAppendix.pdf.
  7. Cooper, J., & Arcavi, A. (2013). Mathematicians and elementary school mathematics teachers—Meetings and bridges. In Y. Li & J. N. Moschkovich (Eds.), Proficiency and beliefs in learning and teaching mathematics—Learning from Alan Schoenfeld and Günter Törner (pp. 179–200). Rotterdam: Sense.Google Scholar
  8. Cooper, J., & Karsenty, R. (2018). Can teachers and mathematicians communicate productively? The case of division with remainder. Journal of Mathematics Teacher Education. 21, 237–261.  https://doi.org/10.1007/s10857-016-9358-7.CrossRefGoogle Scholar
  9. Cooper, J., & Pinto, A. (2018). Jourdain and Dienes effects revisited—Playing Tic Tac Toe or learning non-Euclidean Geometry? 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, pp. 307–314). Umeå, Sweden: PME.Google Scholar
  10. Engeström, Y., Engeström, R., & Karkkainen, M. (1995). Polycontextuality and boundary crossing in expert cognition: Learning and problem solving in complex work activities. Learning and Instruction, 5, 319–336.CrossRefGoogle Scholar
  11. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.Google Scholar
  12. Fried, M. N. (2014). Mathematics and mathematics education: Searching for common ground. In M. N. Fried & T. Dreyfus (Eds.), Mathematics and mathematics education: Searching for common ground (pp. 3–22). New York: Springer (advances in mathematics education series).CrossRefGoogle Scholar
  13. Hill, H., Schilling, S., & Ball, D. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11–30.CrossRefGoogle Scholar
  14. Israeli Mathematicians. (2010). Math-Program-letter.pdf. Retrieved Dec. 15, 2017, from http://www.wisdom.weizmann.ac.il/~dnovikov/MathEd/Math-Program-letter.pdf.
  15. Klein, D., Askey, R., Milgram, R. J., Wu, H.-H., Scharlemann, M., & Tsang, B. (1999). An open letter to Richard Riley, United States Secretary of Education. Retrieved Dec. 15, 2017, from California State University Northridge. http://www.csun.edu/~vcmth00m/riley.html.
  16. Knowles, M. S. (1990). The adult learner: A neglected species. Houston: GulfGoogle Scholar
  17. Konkola, R., Tuomi-Grohn, T., Lambert, P., & Ludvigsen, S. (2007). Promoting learning and transfer between school and workplace. Journal of Education and Work, 20, 211–228.CrossRefGoogle Scholar
  18. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New Jersey: Routledge.Google Scholar
  19. Overview of ICMI. (2017). Retrieved Dec. 15, 2017, from International Commission on Mathematics Instruction. http://www.mathunion.org/icmi/icmi/overview-of-icmi/.
  20. Pedagogical Secretariat of the Israeli Ministry of Education. (2009, 6 24). A new mathematics curriculum for primary school for all sectors (in Hebrew). Retrieved Dec. 15, 2017, from http://cms.education.gov.il/EducationCMS/Units/Tochniyot_Limudim/Math_Yesodi/PDF/.
  21. Pinto, A., & Cooper, J. (2017). In the pursuit of relevance—Mathematicians designing tasks for elementary school teachers. International Journal of Research in Undergraduate Mathematics Education, 3(2), 311–337.CrossRefGoogle Scholar
  22. Ralston, A. (2004). Research mathematicians and mathematics education: A critique. Notices of the American Mathematical Society, 51(4), 403–411.Google Scholar
  23. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253–286.CrossRefGoogle Scholar
  24. Sfard, A. (1998). The many faces of mathematics: Do mathematicians and researchers in mathematics education speak about the same thing? In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 491–511). Dordrecht: Springer.Google Scholar
  25. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  26. Shklovskij, V. (1998). Art as technique. In J. Rivkin & M. Ryan (Eds.), Literary theory: An anthology. Malden: Blackwell Publishing Ltd.Google Scholar
  27. Star, S. L. (1989). The structure of ill-structured solutions: Boundary objects and heterogeneous distributed problem solving. In L. Gasser & M. Huhns (Eds.), Distributed artificial intelligence (pp. 37–54). San Mateo: Morgan Kaufmann.CrossRefGoogle Scholar
  28. Sun, X., Kaur, B., & Novotná, J. (2015). Discussion document. In X. Sun, B. Kaur, & J. Novotná (Eds.), Primary mathematics study on whole numbers: ICMI study 23 conference proceedings, June 3–7, 2015 in Macau, China (pp 625–641). Macao: University of Macau.Google Scholar
  29. van Zanten, M., & van den Heuvel-Panhuizen, M. (2014). Freedom of design: The multiple faces of subtraction in Dutch primary school textbooks. In Y. Li & G. Lappan (Eds.), Mathematics curriculum in school education (pp. 231–259). Dordrecht: Springer Netherlands.CrossRefGoogle Scholar
  30. Wu, H. (2011). The mis-education of mathematics teachers. Notices of the AMS, 58(03), 372–384.Google Scholar

Copyright information

© FIZ Karlsruhe 2018

Authors and Affiliations

  1. 1.Department of Science TeachingWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations