MULTI-TEMPORAL COUNTING STRUCTURES. INDEXED NATURAL NUMBERS IN LIGHT OF COGNITIVE ARITHMETIC (Multi-temporalne struktury obliczeniowe. Indeksowane liczby naturalne w swietle arytmetyki kognitywnej)
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The paper presents a new model (conception) of the structure of basic arithmetical representations encoded in minds which enable them to solve simple story-tasks. According to the dominating paradigm the process of acquiring basic counting abilities culminates in encoding the exact number line in mind. This linear number representation enables the mind to solve simple story-tasks which do not require any mathematical mastery knowledge comprising laws, definitions and theorems. The controversy over the origin of this representational structure is the main subject of the recent debate. Some researchers try to show that the process of encoding the exact number line stems from transformations of the approximate number line (the mental number line) whereas others model this process as being dependent on the linguistic and logical resources of mind. In the paper the dominating approach is rejected in favor of a new paradigm of comprehending the structure of the basic mature arithmetic representation. The new paradigm assumes that the first, pre-school stage of developing arithmetical capacities is completed when a child acquires a cluster of exact number lines. Hence, the basic arithmetical structure enabling children to solve simple mathematical story-tasks cannot be a semantic model of Peano's arithmetic. If it was the case, then seven-years-old children engaged in solving simple story-tasks would have to use unconsciously very sophisticated set-theoretic tools. It is rather impossible because children provide solutions to these tasks in the very short time (sometimes in seconds) whereas the use of the set theoretic representations in processing input data given in the contents of the tasks would result in prolongation of the time needful for computing the outputs. The formal model of the cluster of number lines requires constructing a formal arithmetical theory which is called the 'arithmetic of indexed natural numbers'. The theory is a generalization of the standard arithmetic of natural numbers. In light of the proposed model, the verbal number line does not function as a tool for counting cardinalities of sets. Its main role is enabling the mind to construct categorial number lines belonging to the cluster-structures processed in the course of solving story-tasks. Unlike the classical model, the presented model explains children's abilities in solving tasks without reference to the tacit set theoretic knowledge encoded in children's minds.
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