Educational Studies in Mathematics

, Volume 93, Issue 2, pp 155–173

Bridging the gap between graphical arguments and verbal-symbolic proofs in a real analysis context

  • Dov Zazkis
  • Keith Weber
  • Juan Pablo Mejía-Ramos
Article

Abstract

We examine a commonly suggested proof construction strategy from the mathematics education literature—that students first produce a graphical argument and then work to construct a verbal-symbolic proof based on that graphical argument. The work of students who produce such graphical arguments when solving proof construction tasks was analyzed to distill three activities that contribute to students’ successful translation of graphical arguments into verbal-symbolic proofs. These activities are called elaborating, syntactifying, and rewarranting. We analyze how engaging in these activities relates to students’ success in proof construction tasks. Additionally, we discuss how each individual activity contributes to the translation of a graphical argument into a verbal-symbolic proof.

Keywords

Proof Graphical argumentation Toulmin scheme 

References

  1. Alcock, L. (2010). Interactions between teaching and research: Developing pedagogical content knowledge for real analysis. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics. Dordrecht, The Netherlands: Springer.Google Scholar
  2. Alcock, L., & Weber, K. (2010). Undergraduates’ example use in proof production: Purposes and effectiveness. Investigations in Mathematical Learning, 3(1), 1–22.Google Scholar
  3. Bartolini-Bussi, M., Boero, P., Ferri, F., Garuti, R., & Mariotti, M. (2007). Approaching and developing the culture of geometry theorems in school. In P. Boero (Ed.), Theorems in school: From history, epistemology, and cognition to classroom practice (pp. 211–217). Rotterdam: Sense Publishing.Google Scholar
  4. Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on the Teaching and Learning of Mathematical Proof. Retrieved from: http://www.lettredelapreuve.org/OldPreuve/Newsletter/990708Theme/990708ThemeUK.html. Accessed 09 Mar 2016.
  5. Boero, P., Garuti, R., & Mariotti, M. A. (1996). Some dynamic mental processes underlying producing and proving conjectures. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 121–128). Valencia, Spain: PME.Google Scholar
  6. Douek, N. (2009). Approaching proof in school: From guided conjecturing and proving to a story of proof construction. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Viliers (Eds.), Proceedings of the 19th ICMI Study Conference: proof and proving in mathematics education (Vol. 1, pp. 142–147). Taipei, Taiwan: Department of Mathematics, National Taiwan Normal University.Google Scholar
  7. Fitzpatrick, P. M. (2006). Advanced calculus (2nd ed.). Providence, RI: American Mathematical Society.Google Scholar
  8. Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulty of proof. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 345–352). Stellenbosh, South Africa: PME.Google Scholar
  9. Gibson, D. (1998). Students’ use of diagrams to develop proofs in an introductory real analysis. Research in Collegiate Mathematics Education, 2, 284–307.CrossRefGoogle Scholar
  10. Hart, E. (1994). A conceptual analysis of the proof writing performance of expert and novice students in elementary group theory. In J. Kaput & E. Dubinsky (Eds.), Research issues in mathematics learning: Preliminary analyses and results (pp. 49–62). Mathematical Association of America: Washington.Google Scholar
  11. Iannone, P. & Inglis, M. (2010). Self-efficacy and mathematical proof: Are undergraduates good at assessing their own proof production ability? In Proceedings of the 13th Conference for Research in Undergraduate Mathematics Education. Raleigh, North Carolina. http://sigmaa.maa.org/rume/crume2010/Archive/Iannone%20&%20Inglis.pdf. Accessed 09 Mar 2016.
  12. Inglis, M., & Mejia-Ramos, J. P. (2009). The effect of authority on the persuasiveness of mathematical arguments. Cognition and Instruction, 27, 25–50.CrossRefGoogle Scholar
  13. Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J. P., & Weber, K. (2016). Lectures in advanced mathematics: why students might not understand what the professor is trying to convey. Journal for Research in Mathematics Education, 47(2), 163–198.Google Scholar
  14. Mamona-Downs, J., & Downs, M. (2010). Necessary realignments from mental argumentation to proof presentation. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Azarello (Eds.), Proceedings of the 6th Conference for European Research in Mathematics Education (pp. 2336–2345). Lyon, France: INRP.Google Scholar
  15. Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Rotterdam: Sense Publishers.Google Scholar
  16. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249–266.CrossRefGoogle Scholar
  17. Pedemonte, B. (2001). Some cognitive aspects of the relationship between argumentation and proof in mathematics. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 33–40). Utrecht, Netherlands: PME.Google Scholar
  18. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.CrossRefGoogle Scholar
  19. Pedemonte, B. (2008). Argumentation and algebraic proof. ZDM, 40, 385–400.CrossRefGoogle Scholar
  20. Pedemonte, B., & Reid, D. (2011). The role of abduction in proving processes. Educational Studies in Mathematics, 76, 281–303.CrossRefGoogle Scholar
  21. Raman, M. (2003). Key ideas: what are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52, 319–325.CrossRefGoogle Scholar
  22. Raman, M. (2004). Epistemological messages conveyed by three high-school and college mathematics textbooks. The Journal of Mathematical Behavior, 23(4), 389–404.CrossRefGoogle Scholar
  23. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.CrossRefGoogle Scholar
  24. Samkoff, A., Lai, Y., & Weber, K. (2012). Mathematicians’ use of diagrams in proof construction. Journal for Research in Mathematics Education, 14, 49–67.CrossRefGoogle Scholar
  25. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic press.Google Scholar
  26. Schoenfeld, A. (1991). On mathematics as sense-making: an informal attack on the unfortunate divorce of formal and informal mathematics. In J. F. Voss, D. N. Perkins, & J. W. Segal (Eds.), Informal reasoning and education (pp. 311–344). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  27. Selden, A., & Selden, J. (2008). Overcoming students’ difficulties in learning to understand and construct proofs. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: research and teaching in undergraduate mathematics education (MAA Notes, Vol. 73, pp. 95–110). Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar
  28. Strauss, A., & Corbin, J. (1990). Basics of qualitative research: grounded theory procedures and techniques. London: Sage.Google Scholar
  29. Stylianides, A. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289–321.Google Scholar
  30. Toulmin, S. (2003). The uses of argument. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  31. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101–119.CrossRefGoogle Scholar
  32. Weber, K. (2002). Beyond proving and explaining: Proofs that justify the use of definitions and axiomatic structures and proofs that illustrate technique. For the Learning of Mathematics, 22(3), 14–17.Google Scholar
  33. Weber, K. (2004). Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course. The Journal of Mathematical Behavior, 23(2), 115–133.CrossRefGoogle Scholar
  34. Weber. (2005). Problem-solving, proving, and learning: The relationship between problem-solving processes and learning opportunities in proof construction. The Journal of Mathematical Behavior, 24(3/4), 351–360.CrossRefGoogle Scholar
  35. Weber, K. (2010). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12, 306–336.CrossRefGoogle Scholar
  36. Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.CrossRefGoogle Scholar
  37. Weber, K., & Alcock, L. (2009). Semantic and syntactic reasoning and proving in advanced mathematics classrooms. In D. Stylinaiou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 323–338). New York: Routledge.Google Scholar
  38. Zazkis, R., & Liljedahl, P. (2004). Understanding primes: The role of representation. Journal for Research in Mathematics Education, 34(3), 164–186.CrossRefGoogle Scholar
  39. Zhen, B., Weber, K., & Mejia-Ramos, J. P. (2016). Mathematics majors’ perceptions of the admissibility of graphical inferences in proofs. International Journal of Research in Undergraduate Mathematics Education, 2(1), 1–29.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Dov Zazkis
    • 1
  • Keith Weber
    • 2
  • Juan Pablo Mejía-Ramos
    • 2
  1. 1.Arizona State UniversityTempeUSA
  2. 2.Rutgers UniversityNew BrunswickUSA

Personalised recommendations