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The relationship between diagrammatic argumentation and narrative argumentation in the context of the development of mathematical thinking in the early years

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Abstract

This paper deals with one aspect of the endeavor to generate a theory of the development of mathematical thinking of children in the early years ages 3 to 10. By comparing two scenes, one from preschool and one from a first grade mathematics class, the relationship between diagrammatic and narrative argumentations among children and teachers is reconstructed and related to possible developmental trajectories of mathematical thinking. Theoretically, I attempt to implement these developmental paths in a concept of an “Interactional Niche in the Development of Mathematical Thinking.”

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Notes

  1. The empirical background is the project “early Steps in Mathematics Learning” (erStMaL) that is part of the research activities of the “Center for Individual Development and Adaptive Education” (IDeA) located in Frankfurt am Main, Germany. It is funded for 6 years, beginning in July 2008; http://www.idea-frankfurt.eu/homepage/idea-projects/projekt-erstmal.

  2. In the English-speaking world, the word “kindergarten” usually means an institution in which children spend just one year before entering primary school from ages 5 to 6. In Germany, the word “Kindergarten” designates the institution, which children visit from ages 3 to 6 before they start primary school. In the USA one might use the abbreviation “pre-k”. In its organization and administration, the German Kindergarten is separate from primary schools. In order to remind the reader that my data originate from a German institution, I consequently write this word with a capital letter, as one usually does for nouns in German. “Early Kindergarten age” addresses an age group between age 3 and 4. Usually the age of third graders is between 8.5 and 9.5.

  3. Surprisingly, Ernest (2010) does not mention this research tradition that usually is subsumed under the name “micro-sociology.” For its reception in mathematics education, see Bauersfeld, 1995; Krummheuer, 1995; Voigt, 1995.

  4. For more details, see Krummheuer, 2011a.

  5. Super and Harkness (1986) defined “their” development niche by the three components ”the physical and social settings in which the child lives,” “culturally regulated customs of child care and rearing” and “the psychology of the caretakers” (Super and Harkness 1986, p. 552). They conducted anthropological studies without focusing on the situational aspects of social interaction processes.

  6. Leeway taken here in the colloquial meaning of “room for freedom of action” (see Webster, 1983, p. 1034), originally the notion of “Partizipationsspielraum” in the book by Brandt (2004).

  7. The chipmunk, a stuffed animal, is the mascot of the project erStMaL.

  8. For the notion “didactical design pattern” see, for example, Acar Bayraktar, Hümmer, Huth, Münz, & Reiman, (2011, p. 17ff.) and below Section 4.

  9. Here starts the transcript that is printed in the Appendix. Numbers in angle brackets refer to the numbered lines in the transcript.

  10. The transcript is printed in the Appendix. Numbers in angle brackets refer to the numbered lines in the transcript.

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  1. Goethe University, Frankfurt am Main, Germany

    Götz Krummheuer

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Correspondence to Götz Krummheuer.

Appendix: the two transcripts

In the original German transcript, we do not use the standard interpunctuation, and denote speaking pauses, raisings of the pinch, and so forth. In the English translation, we do not use these paralinguistic notations. Word order and tone of voice differ too much between German and English.

Appendix: the two transcripts

figure b

The place value problem

figure c

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Krummheuer, G. The relationship between diagrammatic argumentation and narrative argumentation in the context of the development of mathematical thinking in the early years. Educ Stud Math 84, 249–265 (2013). https://doi.org/10.1007/s10649-013-9471-9

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