Counting in everyday life: Discrimination and enumeration

Nan, Yun; Knösche, Thomas R.; Luo, Yuejia

doi:10.1016/j.neuropsychologia.2005.10.020

Cited by 31 publications

(45 citation statements)

References 42 publications

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“…In particular, children who are magnitude aware have moved beyond counting as "a memorized list and a mechanical routine, without attaching any sense of numerical magnitudes to the words" (Lipton and Spelke 2005, p. 979). Moreover, magnitude awareness has been shown to be a predictor, irrespective of ability or age, of more general mathematical achievement (Aunio and Niemivirta 2010;De Smedt et al 2013;Desoete et al 2012;Holloway and Ansari 2009;Nan et al 2006;Stock et al 2010). …”

Section: Quantity Discriminationmentioning

confidence: 99%

Identifying Opportunities for Grade One Children to Acquire Foundational Number Sense: Developing a Framework for Cross Cultural Classroom Analyses

Andrews

Sayers

2014

Early Childhood Educ J

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It is known that an appropriately developed foundational number sense (FONS), or the ability to operate flexibly with number and quantity, is a powerful predictor of young children's later mathematical achievement. However, until now not only has FONS been definitionally elusive but instruments for identifying opportunities for children to acquire its various components have been missing from the classroom observation tools available. In this paper, drawing on a constant comparison analysis of appropriate literature, we outline the development of an eight dimensional FONS framework. We then show, by applying this framework to three culturally diverse European grade one lessons, one English, one Hungarian and one Swedish, that it is both straightforwardly operationalised and amenable to cross cultural analyses of classroom practice. Some implications are discussed.

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Section: Quantity Discriminationmentioning

confidence: 99%

Identifying Opportunities for Grade One Children to Acquire Foundational Number Sense: Developing a Framework for Cross Cultural Classroom Analyses

Andrews

Sayers

2014

Early Childhood Educ J

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“…This is not to say that subitizing and counting are unrelated; in fact, according to Trick and Pylyshyn (1994), subitizing is part of the groupand-add counting process. There are brain areas that are specific to subitizing or counting and others that the two processes share (e.g., Nan, Knösche, & Luo, 2006). Nonetheless, counting requires some operations that subitizing does not; as a result, there are dissociations, cases where item heterogeneity was not significantly weaker at 1 item (where there was no actual heterogeneity in the display) than it was at 2 and 3 items (where there was).…”

Section: Implications For the Study Of Enumerationmentioning

confidence: 99%

More than superstition: Differential effects of featural heterogeneity and change on subitizing and counting

Trick

2008

Perception & Psychophysics

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It has long been known that when people enumerate visual-spatial items, such as dots laid out in different locations all at one time, and they are instructed to indicate the number of items as fast and as accurately as they can, there are different patterns of performance with fewer than four or five dots than with more (e.g., Jevons, 1871). With small numbers of dots, enumeration is effortless, accurate, and fast, but with larger numbers, enumeration is effortful, relatively error-prone, and slow: For example, response time (RT) slope 40-100 msec/item for 1-3 items as opposed to 200-350 msec/item for more. Subitizing is the name given to the rapid, effortless form of enumeration that occurs for small numbers of dots, and the subitizing range is the range of numbers over which subitizing is observed (Kaufman, Lord, Reese, & Volkmann, 1949). There are individual differences in the maximal number of items that can be subitized, but for most adults the subitizing range is between 1-3 and 1-5 items (Akin & Chase, 1978;Jensen, Reese, & Reese, 1950;Mandler & Shebo, 1982;Trick & Pylyshyn, 1993). Counting is the name given to the more laborious and time consuming form of enumeration used with larger numbers of dots; the counting range is the range of numerosities over which counting is observed. There has been controversy about whether subitizing and counting are different processes or the same process applied to different numbers of items; in fact, some have gone so far as to say that subitizing is "mere superstition" and not a real phenomenon (Balakrishnan & Ashby, 1992;Cordes & Gelman, 2005). However, if there is no such thing as subitizing, if the enumeration of small and large numbers is accomplished by exactly the same mechanism, there would be no reason to expect a manipulation to have opposite effects on enumeration in the subitizing and counting ranges. These studies investigate the effects of featural heterogeneity and change on the enumeration of 1-8 items as it is influenced by attentional selection. The results have ramifications for enumeration literature and the research on selective attention.The debate about subitizing and counting is complicated by the fact that there are not just two proposed enumeration processes-but three. For example, Smitsman (1982) discusses two types of enumeration that calculate precise number (subitizing and counting), and a third that delivers approximate number (estimation). The empirical markers that distinguish among the three types of enumeration involve a combination of processing speed and accuracy. Subitizing and estimation are fast, although the increase in the amount of time for each additional item (the RT slope) is different for subitizing and estimation, and subitizing is extremely accurate, whereas estimation is extremely inaccurate (the exact number is typically wrong, though the estimate may be in the general vicinity). Similarly, both subitizing and counting are accurate (though subitizing is more accurate) but the RT slope is much smaller for subitizing than ...

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“…Magnitude-aware children have moved beyond counting as "a memorized list and a mechanical routine, without attaching any sense of numerical magnitudes to the words" (Lipton and Spelke, 2005, p. 979). Moreover, magnitude awareness has been shown to be a predictor, independently of ability or age, of more general mathematical achievement (Aunio and Niemivirta, 2010;De Smedt et al, 2009, 2013Desoete et al, 2012;Holloway and Ansari, 2009;Nan et al 2006;Stock et al 2010;). …”

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confidence: 99%

The Role of Conceptual Subitising in the Development of Foundational Number Sense

Sayers

Andrews

Boistrup

2016

Mathematics Education in the Early Years

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INTRODUCTIONOver the last fifteen years since the publication of Clements' (1999) well-known paper, various scholars, particularly in the United states, have been encouraging teachers to attend to the development of young learners' conceptual subitising (See, for example, Clements & Sarama, 2009;Conderman et al., 2014); where conceptual subitising is the ability to recognise quickly and without counting relatively large numerosities by partitioning these large groups into smaller groups that can be individually subitised (Clements & Sarama, 2007;Geary, 2011). Various claims, which we discuss below, have been made with respect to the efficacy of conceptual subitising-focused instruction. In a related vein, our own recent work has focused on a conceptualisation of foundational number sense (FoNS), which we describe as those number-related competences expected of a typical first grade student that require instruction (Back et al., 2014;Andrews & Sayers, 2014a). FoNS is characterised by eight components, which we describe below. The purpose of this paper, drawing on excerpts from grade one lessons taught by a case study teacher in each of Hungary and Sweden, is to examine the extent to which conceptual subitising-focused activities have the propensity to facilitate students' acquisition of the various FoNS components and, in so doing, examine the warrant for their claimed efficacy. WHAT IS SUBITISING?Subitising refers to being instantly and automatically able to recognise small numerosities without having to count (Clements, 1999;Jung et al., 2013;Moeller et al., 2009;Clements & Sarama, 2009). Children as young as three are typically able to subitise numerosities up to three (Fuson, 1988, Moeller et al., 2009), while most adults are able instantly to recognise without counting the numerosity represented by the dots on the face of a die (Jung et al. 2013). This process, innate to all humans, is typically known as perceptual subitising (Gelman & Tucker, 1975) and forms an element of the preverbal number sense we describe below. In short, perceptual subitising is recognizing a numerosity without using other mathematical processes (Clement, 1999). Conceptual subitisingHowever, a second form of subitising, conceptual subitising (Clements, 1999), which is not unrelated to FoNS, has been shown to have considerable implications for teaching and learning. Conceptual subitising relates to how an individual identifies "a whole quantity as the result of recognizing smaller quantities... that make up the whole" (Conderman et al., 2014, p.29). More generally, it can be summarised as the systematic management of perceptually subitised numerosities to facilitate the management of larger numerosities (Obersteiner et al., 2013). For example, when a child is confronted by two dice, one showing three and another showing four, each is perceptually subitised before any sense of seven can emerge.Subitising can be construed as having a synonymity with the spatial structuring of numbers (Battista et al., 1998). In this case, the ability to ...

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Counting in everyday life: Discrimination and enumeration

Cited by 31 publications

References 42 publications

Identifying Opportunities for Grade One Children to Acquire Foundational Number Sense: Developing a Framework for Cross Cultural Classroom Analyses

Identifying Opportunities for Grade One Children to Acquire Foundational Number Sense: Developing a Framework for Cross Cultural Classroom Analyses

More than superstition: Differential effects of featural heterogeneity and change on subitizing and counting

The Role of Conceptual Subitising in the Development of Foundational Number Sense

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