A Relational View of Mathematical Concepts

Coles, Alf

doi:10.1017/9781316471128.013

Cited by 12 publications

(9 citation statements)

References 6 publications

(4 reference statements)

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“…While this contrast between arithmetical problem-solving and structural re-arranging has been noted in the Davydov-linked literature (e.g., Polotskaia, 2017), the more salient contrast for us, and following Coles (2017), is between pedagogic approaches launching from a base in structure to those launching from a base in counting.…”

Section: Davydov's Approach To Early Number Teachingmentioning

confidence: 94%

“…Davydov's approach to early number offers attention to quantities in a relational sense rather than in a counting-based sense. Coles (2017), advocating for the relational sense, details the distinctions between these two approaches, and argues that the relational sense of quantity seen in Davydov's curriculum (and in Gattegno's (1974) work) provides a route into an awareness of number and number relationships as simultaneously material and abstract. However, curricular and pedagogical norms in South Africa present counting as the introductory route into number and into calculation, echoing Coles' broader (2017) sentiment that "the predominant narrative in schooling is a counting world" (p. 206).…”

Section: Introductionmentioning

confidence: 99%

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Shape-shifting Davydov’s ideas for early number learning in South Africa

2020

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In this paper, we share details of a South African early grades’ number intervention informed by aspects of Davydov’s writing on early number teaching and learning. A key part of Davydov’s approach to early number teaching involves starting with attention to relationships between quantities rather than with counting. The Structuring Number Starters (SNS) intervention focused—over a nine-year period—on supporting early grades’ students to move beyond the calculating-by-counting approaches that are prevalent in South Africa. In attending to this focus, the intervention shifted increasingly towards an emphasis on relationships between quantities, though not in the same format or task sequence as advocated by Davydov. The contextual and cultural features that led to our adaptations—or shape-shifting—are highlighted in this paper. We interrogate key aspects of Davydov’s approaches to early number teaching in relation to key features typical of South African classroom mathematics teaching in order to understand the evolution of the SNS initiative. Quasi-longitudinal interview-based assessment data available from a cross-attainment sample of students in 2011, 2014 and 2018 indicate shifts over time from calculating-by-counting to calculating-by-structuring. These outcomes point to successes with moves into increasingly structured ways of working with early number, but suggest also that these successes may be contingent on some fluency with forward and backward number word sequences. The outcomes suggest that it is feasible to explore interventions directing attention to early number structure from the outset in larger scale studies.

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Section: Davydov's Approach To Early Number Teachingmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Shape-shifting Davydov’s ideas for early number learning in South Africa

2020

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“…In this text, we will be mobilising the terms abstract and concrete but using them to distinguish different forms of relationship, or modes of engaging, with the world (Coles, 2017). In broad terms, we find it helpful to draw the distinction that we can attend to things (e.g., objects, concepts) and we can attend to relations between things (e.g., differences, changes).…”

Section: Concrete and Abstractmentioning

confidence: 99%

“…The key mathematical symbols are the numerals. Numerals are introduced, in both curricula, as relations between objects (Coles, 2017). Hence the concrete objects are used as a context in which to make meaningful a set of symbols, but where the key symbols are abstract from the start, denoting actions on, or relations between objects.…”

Section: Manipulatives and Symbolsmentioning

confidence: 99%

Re-thinking ‘Concrete to Abstract’ in Mathematics Education: Towards the Use of Symbolically Structured Environments

Coles

Sinclair

2019

Can. J. Sci. Math. Techn. Educ.

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In this article, we question the prevalent assumption that teaching and learning mathematics should always entail movement from the concrete to the abstract. Such a view leads to reported difficulties in students moving from manipulatives and models to more symbolic work, moves that many students never make, with all the implications this entails for life chances. We propose working in "symbolically structured environments" as an alternative way of characterising students' direct engagement with the abstract and exemplify two such environments, both of which involve early number learning. We additionally propose some roles for the teacher working in a symbolically structured environment.

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“…For example, in a remain static, coupled as it is with particular tasks and tools that enable children's learning. The ideas of Davydov in particular have received recent attention (Coles, 2017;Dougherty, 2008;Savard, 2017), yet have not gained widespread acceptance and have not challenged the narrative of the inevitability of children progressing through predictable hierarchies of achievement.…”

mentioning

confidence: 99%

Re-thinking ‘normal’ development in the early learning of number

Coles

Sinclair

2018

J. Numer. Cogn.

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In this article we suggest that, notwithstanding noted differences, one unmarked similarity across psychology and mathematics education is the continued dominance of the view that there is a 'normal' path of development. We focus particularly on the case of the early learning of number and point to evidence that puts into question the dominant narrative of how number sense develops through the concrete and the cardinal. Recent neuroscience findings have raised the potential significance of ordinal approaches to learning number, which in privileging the symbolic-and hence the abstract-reverse one aspect of the 'normal' development order. We draw on empirical evidence to suggest that what children can do, and in what order, is sensitive to, among other things, the curriculum approach-and also the tools they have at their disposition. We draw out implications from our work for curriculum organisation in the early years of schooling, to disrupt taken-forgranted paths.Keywords: early number, development, mathematics, teaching, technology The Developmental Narrative in Psychology and Learning NumberFollowing Piaget, it is perhaps an easy temptation to interpret sequences of child behaviour in terms of a developmental path both in general and, in our particular concern in this article, when considering the learning of number. Despite differences between the fields of psychology and mathematics education, we see one similarity (which perhaps goes unmarked) in the use of development or trajectory metaphors to explain behaviour. A good example of such logic is from the first volume of this journal: 'The results of the current study revealed a clear developmental pattern through which preschoolers traverse towards Arabic digit knowledge' (Knudsen et al., 2015, p. 21), a conclusion reached as a result of interpreting children's responses to a series of tasks. We can understand the interest that both cognitive scientists and mathematics educators might have in identifying developmental progressions, however, we would like to call into question the very notion of a 'natural' or 'normal' developmental sequence. We question, for example, whether tasks framed in a different manner might have 'revealed' different patterns of 'development' (Hughes & Donaldson, 1979). We also want to In the context of early childhood education, Fowler (2017) argues that tensions around the use of developmental theory arise in part from the different types of development in question. In the Piagetian approach, in which development drives learning, the focus is on development of "universals" such as object permanence and speech. In the Vygotskian approach, where learning drives development, the focus is on learning non-universals, such as reading and counting. In recommending that teachers adopt a multidimensional framework, in which they are aware of how the relationship varies between learning and development within the universal versus non-universal developmental sequence, Fowler does not disrupt the notion of a normal developmental path....

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A Relational View of Mathematical Concepts

Cited by 12 publications

References 6 publications

Shape-shifting Davydov’s ideas for early number learning in South Africa

Shape-shifting Davydov’s ideas for early number learning in South Africa

Re-thinking ‘Concrete to Abstract’ in Mathematics Education: Towards the Use of Symbolically Structured Environments

Re-thinking ‘normal’ development in the early learning of number

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