Situated Counting
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Situated Counting Peter Gärdenfors 1,2 & Paula Quinon 3 # The Author(s) 2020
Abstract We present a model of how counting is learned based on the ability to perform a series of specific steps. The steps require conceptual knowledge of three components: numerosity as a property of collections; numerals; and one-to-one mappings between numerals and collections. We argue that establishing one-to-one mappings is the central feature of counting. In the literature, the so-called cardinality principle has been in focus when studying the development of counting. We submit that identifying the procedural ability to count with the cardinality principle is not sufficient, but only one of the several steps in the counting process. Moreover, we suggest that some of these steps may be facilitated by the external organization of the counting situation. Using the methods of situated cognition, we analyze how the balance between external and internal representations will imply different loads on the working memory and attention of the counting individual. This analysis will show that even if the counter can competently use the cardinality principle, counting will vary in difficulty depending on the physical properties of the elements of collection and on their special arrangement. The upshot is that situated factors will influence counting performance.
1 Introduction Most theories of the development of counting focus on the cognitive representations of the counting tasks (e.g., Schaeffer et al. 1974, Gelman and Gallistel 1978, Sarnecka and Carey 2008). In particular, the cardinality principle – defined as that the last number tag used in a count, represents the cardinality of the items counted – has received a lot of
* Paula Quinon [email protected] Peter Gärdenfors [email protected]
1
Department of Philosophy and Cognitive Science, Lund University, Lund, Sweden
2
Palaeo-Research Institute, University of Johannesburg, Johannesburg, South Africa
3
International Center for Formal Ontology, Faculty of Administration and Social Sciences, Warsaw University of Technology, Warsaw, Poland
P. Gärdenfors, P. Quinon
attention. The principle has been used as a benchmark to determine when a child knows how to count (e.g., Gelman and Gallistel 1978, Wynn 1992, Sarnecka and Carey 2008, Davidson et al. 2012). LeFevre et al. (2006) distinguish conceptual and procedural knowledge of counting and they study how the two types of knowledge depend on each other. In this paper, we follow their distinction, by first identifying three conceptual components and then analyzing the procedure of counting. Predecessors to this approach are Schaeffer et al. (1974), who wrote about the “action sequence” of counting, and Cavanagh and He (2011), who presented a series of steps involved in counting. We also adopt the perspective of situated cognition (Zhang and Norman 1994). From this perspective, counting is not studied anymore as an internal cognitive process, but it is always treated as inseparable from the spatial and physical propertie
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