Abstract
The literature in mathematics education identifies a traditional formal mechanistic-type paradigm in Integral Calculus teaching which is focused on the content to be taught but not on how to teach it. Resorting to the history of the genesis of knowledge makes it possible to identify variables in the mathematical content of the curriculum that have a positive influence on the appropriation of the notions and procedures of calculus, enabling a particularised way of teaching. Objective: The objective of this research was to characterise the anthology of the integral seen from the epistemic complexity that composes it based on historiography. Design: The modelling of epistemic complexity for the definite integral was considered, based on the theoretical construct “epistemic configuration”. Analysis and results: Formalising this complexity revealed logical keys and epistemological elements in the process of the theoretical constitution that reflected epistemological ruptures which, in the organisation of the information, gave rise to three periods for the integral. The characterisation of this complexity and the connection of its components were used to design a process of teaching the integral that was applied to three groups of university students. The implementation showed that a paradigm shift in the teaching process is possible, allowing students to develop mathematical competencies.
Highlights
It presents the results of this work in two senses: (1) The complexity of the integral, the object of this paper, and (2) we mention some details related to the characterisation of the complexity of the integral and its articulation when planning and implementing a sequence of tasks with university students, since the details and results of this characterisation can be found in [8]
This study allowed to identify the existence of certain limitations in each of the partial meanings put forward by the tertiary authors, given that they only focused their interest on the definite integral, leaving aside elements that became motivating situations for other more consistent “meanings” involving indefinite and improper integrals, here called secondary, and which are detailed in the three global epistemic configurations that we propose
We modelled the complexity of the integral in three global epistemic configurations: (1) origins of the integral (GEC1); (2) the operation of integration to support the nascent Integral Calculus (GEC2); (3) formalization of the Integral Calculus (GEC3)
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The onto-semiotic approach to mathematical cognition and instruction (hereafter, OSA [3]) was used as a theoretical support because it offers tools that enable the identification of the complexity of mathematical entities and the connection of the units in which this complexity erupts, following the model outlined in [4,5], through multiple meanings (partial meanings) described in terms of practices and epistemic configurations of the primary objects activated in these practices [6] This manuscript is focused on responding to the first objective, related to formalising the complexity of the integral, sharing the position put forward in [4,7] and considering that, in studies on the ontology of a mathematical object, logical keys and epistemological elements are evident in the process of theoretical constitution, which allow us to better understand the concept, and reveal characteristic aspects of the mathematical construction activity that must be taken into account for its comprehension. The Methodology section of this work describes how the proposal was developed, the results of which, due to their length and detail, can be consulted in [8,9]
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