Evidence and argument in a proof based teaching theory

Abstract

In this article we outline the role evidence and argument plays in the construction of a framing theory for Proof Based Teaching of basic operations on natural numbers and integers, which uses tiles to physically represent numbers. We adopt Mariotti’s characterization of a mathematical theorem as a triple of statement, proof and theory, and elaborate a theory in which the statement “The product of two negative integers is a positive integer” can be proved. This theory is described in terms of a ‘toolbox’ of accepted statements, and acceptable forms of argumentation and expression. We discuss what counts as mathematical evidence in this theory and how that evidence is used in mathematical arguments that support the theory.

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Notes

  1. 1.

    This notation has been adapted from Ball (1993) as an alternative way to represent negative integers. Disadvantages with the standard notation (-1) are well documented in the literature.

  2. 2.

    This group of teachers learned ITT as described in this article. They referred to the grey tiles we use here to represent negative integers as ‘tilde unit tiles’. They also used this name later, when teaching.

  3. 3.

    These teachers took part in a training course in which they sometimes used oblong tiles to represent arbitrary numbers.

References

  1. Balacheff, N. (2008). The role of the researcher’s epistemology in mathematics education: An essay on the case of proof. ZDM, 40, 501–512. https://doi.org/10.1007/s11858-008-0103-2.

    Article  Google Scholar 

  2. Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 373–397.

    Article  Google Scholar 

  3. Bennett, A., Burton, L., & Nelson, T. (2010). Mathematics for elementary teachers: An activity approach. Boston: McGraw-Hill Science.

    Google Scholar 

  4. Biehler, R., & Kempen, L. (2019). Fostering first-year pre-service teachers’ proof competencies. ZDM, 51(4), 000–000.

    Google Scholar 

  5. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge: Belkapp.

    Google Scholar 

  6. Cid, E. (2015). Obstáculos epistemológicos en la enseñanza de los números negativos. (Doctoral dissertation). Universidad de Zaragoza, España. http://www.atd-tad.org/documentos/obstaculos-epistemologicos-en-la-ensenanza-de-los-numeros-negativos-tesis-doctoral/. Accessed 6 Jan 2019.

  7. Dienes, Z. P., & Golding, E. W. (1971). Approach to modern mathematics. New York: Herder and Herder.

    Google Scholar 

  8. Dietiker, L., Kysh, J., Sallee, T., & Hoey, B. (2010). Making connections: Foundations for algebra, course 1. Sacramento: CPM Educational Program.

    Google Scholar 

  9. Duval, R. (1995). Sémiosis et pensée: Registres sémiotiques et apprentissages intellectuels [Semiosis and human thought. Semiotic registers and intellectual learning]. Berne: Peter Lang.

    Google Scholar 

  10. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.

    Google Scholar 

  11. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Kufstein: Reidel.

    Google Scholar 

  12. Hefendehl-Hebeker, L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs. For the Learning of Mathematics, 11(1), 26–32.

    Google Scholar 

  13. Jahnke, H. N., & Wambach, R. (2013). Understanding what a proof is: A classroom-based approach. ZDM—The International Journal on Mathematics Education. 45(3), 469–482. https://doi.org/10.1007/s11858-013-0502-x.

    Article  Google Scholar 

  14. Joseph, G. G. (1991). The crest of the peacock: Non-European roots of mathematics. Princeton: Princeton University Press.

    Google Scholar 

  15. Keirinkan (2013). Gateway to the future: Math 1. Osaka: Keirinkan.

    Google Scholar 

  16. Kempen, L. (2018). How do pre-service teachers rate the conviction, verification and explanatory power of different kinds of proofs? In A. J. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving (pp. 225–237). Cham: Springer.

    Google Scholar 

  17. Mariotti, M. A. (2010). Proofs, semiotics and artefacts of information technologies. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 169–188). Dordrecht: Springer.

    Google Scholar 

  18. Martinez, A. A. (2006). Negative math: How mathematical rules can be positively bent; an easy introduction to the study of developing algebraic rules to describe relations among things. Princeton: Princeton Univ. Press.

    Google Scholar 

  19. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–289.

    Article  Google Scholar 

  20. Pino-Fan, L., Guzmán, I., Duval, R., & Font, V. (2015). The theory of registers of semiotic representation and the onto-semiotic approach to mathematical cognition and instruction: Linking looks for the study of mathematical understanding. In K. Beswick, T. Muir, & J. Wells (Eds.), Proceedings of the 39th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 33–40). Hobart, Australia: PME.

    Google Scholar 

  21. Reid, D., & Knipping, C. (2010). Proof in mathematics education: Research, learning and teaching. Rotterdam: Sense.

    Google Scholar 

  22. Reid, D., & Vallejo Vargas, E. A. (2017). Proof-based teaching as a basis for understanding why. In T. Dooley, & G. Gueudet, (Eds.), Proceedings of the tenth congress of the european society for research in mathematics education (pp. 235–242). Dublin: DCU Institute of Education and ERME.

    Google Scholar 

  23. Reid, D. A., & Vallejo Vargas, E. A. (2018). When is a generic argument a proof? In A. J. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving (pp. 239–251). Cham: Springer.

    Google Scholar 

  24. Semadeni, Z. (1984). A principle of concretization permanence for the formation of arithmetical concepts. Educational Studies in Mathematics, 15, 379–395.

    Article  Google Scholar 

  25. Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

    Google Scholar 

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Acknowledgements

Supported by research funds from the Bundesministerium für Bildung und Forschung (German Federal Ministry of Education and Research), Grant number 16SV7550K, through the grant program “Erfahrbares Lernen” (Experienceable learning). See https://www.technik-zum-menschen-bringen.de/projekte/mal.

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Reid, D.A., Vallejo Vargas, E.A. Evidence and argument in a proof based teaching theory. ZDM Mathematics Education 51, 807–823 (2019). https://doi.org/10.1007/s11858-019-01027-x

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Keywords

  • Evidence
  • Arguments
  • Integers
  • Theory
  • Algebra tiles
  • Proof based teaching