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Mathematics learning and aesthetic production

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Abstract

Some teaching projects in which the learning of mathematics was combined with mainly theatrical productions are reported on. They are related and opposed to an approach of drama in education by Pesci and the proposals of Sinclair for mathematics teaching and beauty. The analysis is based on the distinction between aesthetics as related to beauty or as related to sensual perception. The usefulness of concepts of model and metaphor for the understanding of aesthetic representations of mathematical subject matter is examined. It is claimed that the Peircean concept of the interpretant contributes to a concise analytical approach. The pedagogical attitude is committed to a balanced relationship of scientific and aesthetic values.

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Notes

  1. “SAGR. How can it be true that the continuous can endlessly be divided into parts that again can be divided? SALV. Because this distinction of actual and potential in a certain manner makes seem feasible what in another manner would be impossible. But I shall adjust this discourse in making up another reckoning. To the question whether the parts of a limited continuum are finite or infinite, I give an answer that is the contrary of the one Sign. Simplicio gave before, and I say these parts are neither finite nor infinite. SIMP. I never could have answered like this, because I’d never thought that there could be an intermediate term between finite and infinite and so the distinction, or division, namely that something be either finite or infinite, would be incomplete and defective. SALV. It actually is, as it seems to me. …“(transl. HG).

  2. “Anna: Well, one could write down a term, I guess there is a certain regularity in there, i.e. at a certain integer the number of squares is smaller in a definite way, isn’t it? Amn’t I right? And once you have this term, you can name any number x, and if this arbitrary x is infinite, yes … (pause) … difficult then.” “Benno: Not for every integer there is an identical square, but all the same there remain infinitely many because as you can stretch the integers into the infinite, you could … hmm … the squares. But in this frame, there remain less and less of them; I really don’t know how to express … perhaps by a drawing at the blackboard. Anna: Yes, come on, do that for me with a drawing, and you just paint Infinity at the blackboard and then you show me the thing.”

  3. “Benno: Not for every integer there is an identical square, but all the same there remain infinitely many because as you can stretch the integers into the infinite, you could … hmm … the squares. But in this frame, there remain less and less of them; I really don’t know how to express … perhaps by a drawing at the blackboard. Anna: Yes, come on, do that for me with a drawing, and you just paint Infinity at the blackboard and then you show me the thing.”

  4. See, e.g. the famous “sucking stones sequence” in Beckett’s (1951) Molloy. A semiotic analysis of a narrative about counting is given in Gerstberger 2006b.

  5. In another seminar on aesthetic production and presentation, two participants who are already working as teachers contributed an alternative way of examining the pupils. Instead of an ordinary test they made the students produce a video clip, that was to convey essential parts of the subject matter. In the evaluation of the results, special emphasis was given to presentational competence, which was made operational by precise criteria (Crossley & Ricart Brede 2007).

  6. Pesci describes some theatre plays of mathematical significance in detail. The list could easily be enlarged. Proof by Auburn (2001) is a more recent example of a play often set on stage. I would especially recommend short sketches like L’addition by Prévert (1963).

  7. Cf. The German term Konkrete Poesie.

  8. In Hofstadter’s Gödel, Escher, Bach (1979), a balanced combination of mathematical logic, graphical art and music is suggested.

  9. The term dimension is used here in a non-technical way. Eventually we could subsume the above-mentioned “dimensions” under the concept of representational system. One has to be careful, however, because with the introduction of a technical term a decision or even a prejudice can be tacitly implied. Whether the abstract is represented by the concrete or vice versa, depends on context and interpretation. More generally, instead of representational systems O’ Halloran (2005) investigates the sign systems of mathematics in the framework of systemic functional linguistics.

  10. In the title “A beautiful mind” (Nasar 1998) an attribution of beauty to a mental entity might cause astonishment and curiosity.

  11. The term object is used here in distinction from the term sign—see Sect. 3.

  12. Pesci (2001) proposes and examines an autobiographic approach where metaphors are offered in order to reflect on the student’s experience with the teaching and learning of mathematics—see Sect. 3. A biographic approach together with historical elements was essential in a theatrical production by the drama educator Felicitas Miller and myself on the occasion of the Einstein year 2005 (“Alles relat… nein!”). In the case of this latter production, the students’ interests varied strongly between scientific ones on the one hand and biographic, historical, and ethical ones on the other.

  13. Inszenierungswert—the value of setting on stage—is added by Böhme to the economical concepts of exchange value and value in use.

  14. In a similar way, the MACAS-conferences (Beckmann et al. 2005; Sriraman et al. 2008) expose mathematics and its connections to the arts and sciences in an interdisciplinary effort. To add one more example, PM Heft 16/August 2007/48. Jg., is a topical edition of a mathematics education journal with the title Kunstvoller Mathematikunterricht (artful mathematics teaching). Most of the articles in this edition express the spirit of encouraging children to explore and to produce aesthetical and mathematical structures in a creative way and in a balanced appreciation of both domains.

  15. See Sect. 4.

  16. “Orthogonal” becomes an inner-mathematical metaphor if the term is used in a general, non-geometrical sense—the vanishing inner product of vectors.

  17. According to the definitions of model and metaphor given below, it may be doubted whether Lakoff & Nuñez describe metaphors or rather prototypical models.

  18. By the transition from the action—and the verb to collect—to the result—and the noun collection—a reification (Sfard 1994) or hypostatic abstraction (Peirce) is performed. Halliday & Mackensen (1999) introduced the concept of grammatical metaphor for phenomena like this. This concept is further developed in Gerstberger 2006a.

  19. A detailed definition of the technical term source is given below.

  20. Catachrestical transfer of (pictorial) signs is constitutive in the paintings of René Magritte (cf. Foucault 1973).

  21. The term congruent is due to Halliday (1994).

  22. It does, however, not contradict the definition if the source is more abstract than the target. One can also find a blend of abstract and concrete elements like in “quantum jump”.

  23. The very same expression can be metaphorical in several ways according to context and interpretation: “to head” can mean “to be on top”, “to lead” or “to go toward”. All these versions derive from different functions of the head as part of the human body.

  24. In the previous example “to head”, the relevant relationship can either be seen in the spatial one between upper and lower parts, or in the directional on, when the head goes “ahead”, or else in the structural between the centre of command and the rest of the system.

  25. Peirce himself explains that the spoken word can be considered an interpretant of the written word (cf. Seeger 2004).

  26. That would be a physical blend of the two domains.

  27. This interdisciplinary research project is entitled Aesthetic Production and Presentation in Learning and Education (APPLE).

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  1. University of Education, Weingarten, Germany

    Herbert Gerstberger

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Gerstberger, H. Mathematics learning and aesthetic production. ZDM Mathematics Education 41, 61–73 (2009). https://doi.org/10.1007/s11858-008-0144-6

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