“In principle, extra ‘depth’ in some metaphorical sense is to be expected whenever the information for … two descriptions is differently collected or differently coded” (Gregory Bateson 1979, p. 79).
Abstract
By introducing the concept of the “interactional niche in the development of mathematical thinking” (NMT), a new approach, in which the cultural historic and the interactionistic strands of socio-constructivist theories merge, will be presented. This concept is to be understood as an attempt to develop a theory about the generation of mathematical thinking of children in the ages between three and ten. Two specific niches will be described that arise under the condition of children’s non-canonical solutions and corresponding improvisation by (nursery) teacher and children.
Zusammenfassung
Es wird der Begriff der „interaktionalen Nische mathematischer Denkentwicklung“ (NMD) eingeführt. Mit ihm werden der kulturhistorische und der interaktionistische Ansatz im Rahmen sozialkonstruktivistischer Theorie verbunden. Er stellt ein Grundkonzept für eine zu entwickelnde Theorie mathematischer Denkentwicklung für Kinder im Alter von 3 bis 10 Jahren dar. Es werden zwei spezifische Nischen beschrieben, die unter der Bedingung ungewöhnlicher Lösungen von Kindern und darauf bezogener Improvisationen von Erzieherin, Lehrerin und den Kindern rekonstruiert werden konnten.
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Notes
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 has been funded for 6 years, beginning in July 2008; http://www.idea-frankfurt.eu/homepage/idea-projects/projekt-erstmal.
In the English-speaking world, the word “kindergarten” usually means an institution in which children spend just one year before entering primary school from age 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 US, 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.
This scene is first mentioned in Krummheuer (2011a).
The chipmunk, a stuffed animal, is the mascot of the project erStMaL.
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. In the Appendix, I attach the original German transcript. The numbering of the transcript lines is identical.
This scene is first mentioned in Krummheuer (1997).
The translation into English does not transport the double meaning of this second sentence. It also could be translated as “Four friends play together”.
In the wording of Prediger et al. (2008), I could speak of a “networking strategy”: … “networking strategies are those connecting strategies that respect on the one hand the pluralism and/or modularity of autonomous theoretical approaches but are on the other hand concerned with reducing the unconnected multiplicity of theoretical approaches…” (p. 170).
Culture taken here either as a macro-sociological global precondition or as a micro-sociological phenomenon of locally stabilized and routinized procedures of meaning negotiation.
The table is published in Krummheuer Krummheuer 2011a, p. 65.
Thanks to one of the reviewers for this alternative interpretation.
The notion of “overhearer“ belongs to four concepts of the “recipient design” as outlined in Krummheuer (2011b). It stands for the status of a recipient that is not directly addressed by the priorly speaking or acting person but is tolerated as listener and observer. In the given scene, the children always address their utterances and actions to the adult person B and define by this the status of the remaining children as that of overhearers.
In Sect. 2, I mentioned Prediger et al. (2008) and their concept of “networking strategies”. My aim, of course, is to synthesize these different aspects within the proposed conceptual framework on an NMT. In words of Cobb (2007), one can speak of an attempt of “theorizing as bricolage” (p. 16) • a metaphorization that one can already find in Lévi Strauss (1966); see also Lawler (1980).
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Krummheuer, G. The “Non-canonical” Solution and the “Improvisation” as Conditions for Early Years Mathematics Learning Processes: The Concept of the “Interactional Niche in the Development of Mathematical Thinking“ (NMT). J Math Didakt 33, 317–338 (2012). https://doi.org/10.1007/s13138-012-0040-z
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DOI: https://doi.org/10.1007/s13138-012-0040-z
Keywords
- Early years mathematics learning
- Interactional niche in the development of mathematical thinking
- Socio-constructivism
- Interactionism