A study of the representational use of aggregates in the pedagogic elaboration of addition and subtraction in the Department of Basic Education Grades 1 to 3 Numeracy workbooks, prescribed for use in state funded South African schools

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2023

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Cognitive science demonstrates that a sensitivity to aggregates (groups, collections, classes, categories) forms part of the biologically endowed human (core domain) capacity for dealing with quantity, along with an ability to compute using aggregates, both approximately and exactly. Core domain computations using aggregates serve as a basis for the growth of noncore mathematical computations and principles, following exposure to number enculturation and the counting algorithm, both of which are enhanced by the growth of linguistic competence. The study focuses on the pedagogic use of the class of small, discrete aggregates in the teaching and learning of natural number addition and subtraction across the Foundation Phase of schooling. The central concern is the computational processes that use discrete aggregates, and operations over such aggregates. The six 2021 Department of Basic Education numeracy workbooks (Mathematics in English) for Grades One to Three, prescribed for use in state-funded SA schools, constitute the archive of information from which the data is produced for the study. The study adopts a computational analytic approach conditioned by the proposition that all thought is computational, entailing the use of operations over domains of objects that serve as arguments (inputs) and values (outputs). A mathematised notion of representation—as a structure-preserving mapping—comprised the chief analytical resource for describing computations related across representing and represented computational structures. The analysis, firstly, proceeds descriptively. The unit of analysis is a Task, made up of Subtasks containing Exercises, so that the analysis of a Task proceeds by way of an analysis of its Exercises. Only Tasks employing discrete aggregates for the purposes of teaching addition and subtraction are analysed to reveal the representations used by identifying the computational structures and the relations between such structures. Typically, the representations used in Tasks entail mappings from operations over discrete aggregates to operations over the natural numbers. As a further means of gauging the extent of the range of mappings/operations and structures identified across the workbooks, the descriptive data is extended by the use of quantitative databases, summarising and totalling all identified mappings/operations and structures. The study found that: (1) operations over discrete aggregates are used extensively as a ground for addition, subtraction, natural number order relations, and number partitions, including the use of iii partitions for teaching place value in the base-ten natural number system; (2) counting is the primary computational resource for relating operations over discrete aggregates to operations over the natural numbers; (3) addition and subtraction are often derived from operations over discrete aggregates in a manner that privileges a unary rather than binary form; (4) the treatment of discrete aggregates, together with the use of partitioning, suggests that aggregates are conceived of in a manner that has more affinity with fusions than with sets; (5) the general semantic basis for addition, subtraction and partitioning appears to be the universal cognitive operation referred to as merge (and its derivatives, unmerge and purge) as used by the human conceptual-intentional system.
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