Emerging and developing multiplicative structure in students' visuospatial representations: Four key configuration types

Abstract : Visuospatial representations of quantities and their relations are widely used to support the understanding of basic arithmetic, including multiplicative relationships. These include drawn imagery and concrete manipulatives. This paper defines four particular configurations of nonstandard representation according to the spatial organization of their visual elements. These are: unit containers, unit arrays, array-container blends, and number containers, all of which have been observed to support developing multiplicative thinking, allowing low-attaining students to work with the equal-groups structures of natural number multiplication-and division-based tasks. Student-created examples are discussed, and pedagogical and diagnostic implications considered. In their early encounters with quantitative relationships, children become aware of concepts such as conservation of number, counting, etc., through interactions with collections of objects. For example, addition as the joining of collections and subtraction as removing a subset of objects from a collection-in which the ordering of individual objects is unimportant-can be considered conceptual 'grounding metaphors' (Lakoff & Núñez, 2000). Various models of children's arithmetical problem-solving development indicate a broadly similar progression from early concrete/enactive-based reasoning, to imagic/iconic, to abstract/symbolic reasoning (e.g. Bruner, 1974; Piaget, 1952). Within this broad outline, the actual external representations of learners' thinking during problem-solving include many possible sub-varieties (e.g., sets of actual objects, pictures of objects, tally marks in different configurations, dot arrays, etc.), and many possible categorizations of these for analytical purposes. The construction of appropriate analytical frameworks is necessary for the discerning of inter-individual differences and intra-individual trajectories (Meira, 1995; Voutsina, 2012). This is particularly the case when studying atypically-developing learners (Fletcher et al., 1998). This aim of this paper is to share one aspect from the qualitative analytical framework for student-and co-created visuospatial data used in Finesilver (2014), delineating four particular types of visuospatial representation and demonstrating their use with selected examples. The project took an essentially grounded analytical approach, and so whilst this paper does not report results as such, a sample of research data is included with brief description of the process. Theoretical background To understand multiplication and division represents a significant qualitative change in learners' thinking compared to understanding addition and subtraction (Nunes & Bryant, 1996). These authors, amongst others, have recommended a replications model of multiplication, which is highly relevant both to counting-based strategies and to unitary drawn or modelled representations of multiplicative relationships. A central concept for considering this particular aspect of representation is spatial structuring: Thematic Working Group 24 Proceedings of CERME10 3920
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Carla Finesilver. Emerging and developing multiplicative structure in students' visuospatial representations: Four key configuration types. CERME 10, Feb 2017, Dublin, Ireland. ⟨hal-01950549⟩

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