Abstract
Abstract This article aims at presenting the results of a historical-epistemological study conducted to identify criteria for designing tasks that promote the understanding of the limit notion on a real variable function. As a theoretical framework, we used the Onto-Semiotic Approach (OSA) to mathematical knowledge and instruction, to identify the regulatory elements of mathematical practices developed throughout history, and that gave way to the emergence, evolution, and formalization of limit. As a result, we present a proposal of criteria that summarizes fundamental epistemic aspects, which could be considered when designing tasks that allow the promotion of each of the six meanings identified for the limit notion. The criteria presented allow us to highlight not only the mathematical complexity underlying the study of limit on a real variable function but also the richness of meanings that could be developed to help understand this notion.
Highlights
The notion of limit is the basis of infinitesimal calculus, as it is a fundamental concept for the comprehension and development of other concepts, such as continuity, derivative, integration, and series (ELIA et al, 2009; PARAMESWARAN, 2007)
In the last few years, multiple studies have examined the complexity of teaching and learning limits on a real variable function. These papers inform difficulties classified in three categories: a) epistemological obstacles (CORNU, 1991; ARTIGUE, 1995; SIERPINSKA, 1985; TALL; SCHWARZENBERGER, 1978); b) cognitive difficulties regarding the concept of infinity and the complexity of the formal concept ε, δ (BARAHMAND, 2017; MAMONA-DOWNS, 2001; BLÁSQUEZ et al, 2006); and c) didactic challenges concerning limits teaching (CAGLAYAN, 2015; MONAGHAN,1991; FUENTE; ARMENTEROS; FONT, 2012)
We have presented the historical-epistemological journey for the understanding of the notion of limit of a real variable function, analyzing from the practices that originated it to those that allowed its formalization
Summary
The notion of limit is the basis of infinitesimal calculus, as it is a fundamental concept for the comprehension and development of other concepts, such as continuity, derivative, integration, and series (ELIA et al, 2009; PARAMESWARAN, 2007). Research has evidenced difficulties regarding the tasks that teachers use to promote the learning of limits, such as the lack of meaningful activities and the emphasis on algebraic techniques and procedures (HEINE, 1988; ARTIGUE, 1995; KOIRALA, 1997) that negatively affect the development of students’ deep notion comprehension. In this regard, Koirala (1997) emphasizes that teachers teach rules, and students apply them not understanding what they are doing. Soler de Dios (2014) designed tasks using fractals to foster intuitive limits learning through their geometric representation
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