What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers’ Knowledge of the Roles of Examples in Proving

  • Orly Buchbinder
  • Iris Zodik
  • Gila Ron
  • Alice L. J. Cook
Chapter
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 8)

Abstract

There is a consensus among mathematics educators that in order to provide students with rich learning opportunities to engage with reasoning and proving, prospective teachers must develop a strong knowledge base of mathematics, pedagogy and student epistemology. In this chapter we report on the design of a technology-based task “What can you infer from this example?” that addressed the content and pedagogical knowledge of the status of examples in proving of pre-service teachers (PSTs). The task, originally designed and implemented with high-school students, was modified for PSTs and expanded to involve multiple components, including scenarios of non-descript cartoon characters to represent student data. The task was administered through LessonSketch, an online interactive digital platform, to 4 cohorts of PSTs in Israel and the US, across 4 semesters. In this chapter we focus on theoretical and empirical considerations that guided our task design to provide rich learning opportunities for PSTs to enhance their content and pedagogical knowledge of the interplay between examples and proving, and address some of the challenges involved in the task implementation. We discuss the crucial role of technology in supporting PST learning and provide an emergent framework for developing instructional tasks that foster PSTs’ engagement with proving.

Keywords

Teacher education Reasoning and proof Examples in proving Technology-based task Virtual learning environments 

References

  1. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London: Holder and Stoughton.Google Scholar
  2. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14–17, 20–22, 43–46.Google Scholar
  3. Barkai, R., Tsamir, P., Tirosh, D., & Dreyfus, T. (2002). Proving or refuting arithmetic claims: The case of elementary school teachers. In A. D. Cockburn & E. Nardi (Eds.), Proceeding of 26th Conference of PME (Vol. 2, pp. 57–64). Norwich, UK: PME.Google Scholar
  4. Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 126–154). PME: Prague, Czech Republic.Google Scholar
  5. Buchbinder, O. (2010). The role of examples in establishing the validity of universal and existential mathematical statements. Unpublished dissertation manuscript (in Hebrew). Technion, Haifa.Google Scholar
  6. Buchbinder, O., & Zaslavsky, O. (2009). A framework for understanding the status of examples in establishing the validity of mathematical statements. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 225–232). Thessaloniki, Greece: PME.Google Scholar
  7. Buchbinder, O., & Zaslavsky, O. (2011). Is this a coincidence? The role of examples in fostering a need for proof. ZDM—The International Journal of Mathematics Education, 43(2), 269–281.CrossRefGoogle Scholar
  8. Buchbinder, O., & Zaslavsky, O. (2013). A holistic approach for designing tasks that capture and enhance mathematical understanding of a particular topic: the case of the interplay between examples and proof. In C. Margolinas (Ed.), Proceedings of ICMI Study 22: Task Design in Mathematics Education Conference, Oxford, UK (Vol. 1, pp. 27–35).Google Scholar
  9. Chieu, V. M., Herbst, P., & Weiss, M. (2011). Effect of an animated classroom story embedded in online discussion on helping mathematics teachers learn to notice. Journal of the Learning Sciences, 20(4), 589–624. doi: 10.1080/10508406.2011.528324.CrossRefGoogle Scholar
  10. Clay, E., Silverman, J., & Fischer, D. J. (2012). Unpacking online asynchronous collaboration in mathematics teacher education. ZDM—The International Journal of Mathematics Education, 44(6), 761–773.CrossRefGoogle Scholar
  11. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.CrossRefGoogle Scholar
  12. Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. (2009). Teaching practice: A cross-professional perspective. Teachers College Record, 111(9), 2055–2100.Google Scholar
  13. Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44(1 & 2), 127–150.CrossRefGoogle Scholar
  14. Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). NCTM. Reston, VA.: Information Age Pub Inc.Google Scholar
  15. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.CrossRefGoogle Scholar
  16. Herbst, P., & Chazan, D. (2015). Using Multimedia Scenarios Delivered Online to Study Professional Knowledge Use in Practice. International Journal of Research and Method in Education, 38(3), 272–287.Google Scholar
  17. Herbst, P., Chazan, D., Chieu, V. M., Milewski, A., Kosko, K. W., & Aaron, W. R. (2016). Technology-mediated mathematics teacher development: research on digital pedagogies of practice. In: M. Niess, S. Driskell, & K. Hollebrands (Eds.), Handbook of research on transforming mathematics teacher education in the digital age (pp. 78–106). Hershey, PA: Information Science Reference.Google Scholar
  18. Herbst, P., Chazan, D., Chen, C., Chieu, V. M., & Weiss, M. (2011). Using comics-based representations of teaching, and technology, to bring practice to university “methods” courses. ZDM—The International Journal of Mathematics Education, 43(1), 91–104.CrossRefGoogle Scholar
  19. Herbst, P., Chieu, V. M., & Rougee, A. (2014). Approximating the practice of mathematics teaching: what learning can web-based, multimedia storyboarding software enable? Contemporary Issues in Technology and Teacher Education, 14(4). Retrieved from http://www.citejournal.org/vol14/iss4/mathematics/article1.cfm.
  20. Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.CrossRefGoogle Scholar
  21. Ko, Y. Y. (2010). Mathematics teachers’ conceptions of proof: implications for educational research. International Journal of Science and Mathematics Education, 8(6), 1109–1129.CrossRefGoogle Scholar
  22. Lakatos, I. (1976). Proofs and refutations: the logic of mathematical discovery. Cambridge: Cambridge University Press.Google Scholar
  23. Lesseig, K. (2012). Mathematical knowledge for teaching proof. Unpublished dissertation manuscript. Retrieved from: http://ir.library.oregonstate.edu/xmlui/bitstream/handle/1957/23465/LesseigKristinR2011.pdf?sequence=2.
  24. Leung, A., & Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education, an ICMI study 22 (pp. 191–225). Switzerland: Springer International Publishing.CrossRefGoogle Scholar
  25. Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 38(1), 135–161.CrossRefGoogle Scholar
  26. Oztok, M., & Brett, C. (2011). Social presence and online learning: A review of research. The Journal of Distance Education, 25(3), 1–10.Google Scholar
  27. Ponte, J. P., & Chapman, O. (2008). Preservice mathematics teachers’ knowledge and development. In L. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 223–261). New York: Routldge.Google Scholar
  28. Putnam, R. T., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning? Educational Researcher, 29(1), 4–15.CrossRefGoogle Scholar
  29. Rieber, L. P. (2005). Multimedia learning in games, simulations, and microworlds. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 549–567). New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  30. Rissland, E. L. (1978). Understanding understanding mathematics. Cognitive Science, 2(4), 361–383.CrossRefGoogle Scholar
  31. Ron, G. (1998). Counter-Examples in mathematics: how do students understand their role? Unpublished dissertation manuscript (in Hebrew). Haifa: Technion.Google Scholar
  32. Santagata, R., & Yeh, C. (2014). Learning to teach mathematics and to analyze teaching effectiveness: evidence from a video- and practice-based approach. Journal of Mathematics Teacher Education, 17(6), 491–514.CrossRefGoogle Scholar
  33. Selden, A., & Selden, J. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123–151.CrossRefGoogle Scholar
  34. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.CrossRefGoogle Scholar
  35. Smith, M. S. (2001). Practice-based professional development for teachers of mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  36. Steele, M. D., & Rogers, K. C. (2012). Relationships between mathematical knowledge for teaching and teaching practice: The case of proof. Journal of Mathematics Teacher Education, 15(2), 159–180.CrossRefGoogle Scholar
  37. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.CrossRefGoogle Scholar
  38. Stylianides, A. J. (2011). Towards a comprehensive knowledge package for teaching proof: A focus on the misconception that empirical arguments are proofs. Pythagoras, 32(1), Art. #14, 10 pages.Google Scholar
  39. Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307–332.CrossRefGoogle Scholar
  40. Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314–352.Google Scholar
  41. Van Dreil, J. H., & Berry, A. (2010). Pedagogical content knowledge. In P. Peterson, E. Baker, & B. McGaw (Eds.), International encyclopedia of education (3rd ed., Vol. 7, pp. 656–661). Oxford, UK: Elsevier.Google Scholar
  42. Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317–346.Google Scholar
  43. Zazkis, R., & Chernoff, E. J. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195–208.CrossRefGoogle Scholar
  44. Zazkis, R., Sinclair, N., & Liljedahl, P. (2013). Lesson play in mathematics education: a tool for research and professional development. New York, NY: Springer.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Orly Buchbinder
    • 1
  • Iris Zodik
    • 2
  • Gila Ron
    • 3
  • Alice L. J. Cook
    • 4
  1. 1.University on New HampshireDurhamUSA
  2. 2.Technion, Israel Institute of TechnologyHaifaIsrael
  3. 3.Ohalo CollegeKatzrinIsrael
  4. 4.University of MarylandCollege ParkUSA

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