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“Memorable Diagonals”: Exploratory Problems as Propositions for Doing Mathematics

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Part of the book series: Research in Mathematics Education ((RME))

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Abstract

Learning by experience is of essential importance in everyday life and also in the academic field. Academically engaging with experience can be called induction or inductive reasoning, which Klauer explains as “recognising regularity or order in the only apparently non-ordered while giving awareness to disruptions or non-order in the only apparently ordered” (Klauer, 1991, p. 137; translated by T.F.). Even if it might seem surprising, induction according to Pólya plays an important role especially in mathematics:

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Notes

  1. 1.

    American philosopher and mathematician Charles Sanders Peirce distinguishes in this context between induction and abduction.

  2. 2.

    The pattern supposed by John cannot be reconstructed. For a 4 × 5 rectangle , there are seven points of intersection.

  3. 3.

    Indeed, Liam is not yet sure regarding the universality of the pattern assumed (cf. the following remarks).

  4. 4.

    And therefore visualized lighter in Fig. 3

References

  • Borel, A. (1984). Mathematik: Kunst und Wissenschaft (2nd ed.). München, Germany: Carl Friedrich von Siemens Stiftung.

    Google Scholar 

  • Chazan, D. (1990). Quasi-empirical views of mathematics and mathematics teaching. Interchange, 21(1), 14–23.

    Article  Google Scholar 

  • Fritzlar, T. (2010). “Investigations” und Explorationen in der Elementarmathematik. Der Mathematikunterricht, 56(3), 3–13.

    Google Scholar 

  • Fritzlar, T., Rodeck, K., & Käpnick, F. (2006). Mathe für kleine Asse. 5./6. Schuljahr. Berlin, Germany: Cornelsen.

    Google Scholar 

  • Hadamard, J. (1949). An essay on The Psychology of Invention in the mathematical field. New York: Dover.

    Google Scholar 

  • Heintz, B. (2000a). Die Innenwelt der Mathematik: Zur Kultur und Praxis einer beweisenden Disziplin. Wien, Austria: Springer.

    Book  Google Scholar 

  • Heintz, B. (2000b). “In der Mathematik ist ein Streit mit Sicherheit zu entscheiden”: Perspektiven einer Soziologie der Mathematik. Zeitschrift für Soziologie, 29(5), 339–360.

    Article  Google Scholar 

  • Klauer, K. J. (1991). Erziehung zum induktiven Denken: Neue Ansätze der Denkerziehung. Unterrichtswissenschaften, 19(2), 135–151.

    Google Scholar 

  • Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, England: Cambridge University Press.

    Book  Google Scholar 

  • Leuders, T., Naccarella, D., & Philipp, K. (2011). Experimentelles Denken—Vorgehensweisen beim innermathematischen Experimentieren. Journal für Mathematikdidaktik, 32(2), 205–231.

    Article  Google Scholar 

  • Philipp, K. (2013). Experimentelles Denken: Theoretische und empirische Konkretisierung einer mathematischen Kompetenz. Wiesbaden, Germany: Springer Spektrum.

    Book  Google Scholar 

  • Pólya, G. (1954a). Mathematics and plausible reasoning: Vol I: Induction and analogy in mathematics. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Pólya, G. (1954b). Mathematics and plausible reasoning: Vol. 2: Patterns of plausible inference. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Pólya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving (Vol. I). New York: Wiley.

    Google Scholar 

  • Pólya, G. (1971). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Putnam, H. (1979). What is mathematical truth? In H. Putnam (Ed.), Mathematics, matter and method (2nd ed., pp. 60–78). Cambridge, MA: Cambridge University Press.

    Chapter  Google Scholar 

  • Strauss, A. L., & Corbin, J. M. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Thousand Oaks, CA: Sage.

    Google Scholar 

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Authors and Affiliations

  1. University of Halle-Wittenberg, Franckeplatz 1, 06099, Halle, Germany

    Torsten Fritzlar

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  1. Torsten Fritzlar
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Correspondence to Torsten Fritzlar .

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Editors and Affiliations

  1. University of Chile, Santiago, Chile

    Patricio Felmer

  2. University of Helskini, Helsinki, Finland

    Erkki Pehkonen

  3. University of Georgia, Athens, USA

    Jeremy Kilpatrick

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Fritzlar, T. (2016). “Memorable Diagonals”: Exploratory Problems as Propositions for Doing Mathematics. In: Felmer, P., Pehkonen, E., Kilpatrick, J. (eds) Posing and Solving Mathematical Problems. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-28023-3_10

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  • DOI: https://doi.org/10.1007/978-3-319-28023-3_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28021-9

  • Online ISBN: 978-3-319-28023-3

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