Concepciones dinámicas y estáticas del infinito: procesos continuos y sus totalidades

Abstract

El estudio del infinito en matemática educativa desde una perspectiva cognitiva se ha centrado en analizar el infinito potencial y actual. A partir de la teoría APOE (Acción, Proceso, Objeto, Esquema), esta noción se ha explicado en términos de procesos iterativos infinitos y objetos trascendentes, respectivamente. En este artículo se presenta una descomposición genética, que sustenta el diseño de dos problemas relacionados con la recta tangente a una curva, en el que se toman en consideración los aspectos dinámicos y estáticos del infinito. A través de una entrevista realizada a un profesor universitario, se presenta evidencia de Totalidad, una posible nueva estructura que permite que el individuo conciba un Proceso como un todo, aunque no pueda aplicar Acciones sobre ella.

What’s new with APOS theory? A look into levels and Totality

“…This instrument was designed specifically in order to be able to differentiate between possible Totality and Object conceptions. Villabona et al (2022) report the case of one instructor, who during his interview showed evidence of a possible conception Totality.…”

“…They refer the matter for future research; that is to see if new studies point to its existence in the context of other concepts. Villabona et al (2022) point out the importance of finding evidence in the context of other mathematical concepts as well as in relation with contexts of infinity other than repeating decimals, in order for Totality to be accepted as a new stage in the theory.…”

“…On the one hand the instructor showed that he can conceive the tangent line as the totality of a process of approaching a point on the curve through secant lines: "when I have finished approaching totally, they will be the same" (referring to the secant and tangent lines, Villabona et al, 2022). But on the other hand, he has difficulty applying Actions on that totality: in another situation involving a circle, two AIEM (2022), 21, 9-21 parallel tangent lines and another line crossing them, he could not conceive of some of the angles that were formed by the figures becoming zero in a limiting case, in which a secant line gets closer to a tangent line by means of approaching some point on the circle to another point on the circle where a tangent line was drawn.…”

“…This is an indication that he has not yet constructed an Object conception, although he can conceive of the Process as a Totality. Villabona et al (2022) posit that if Totality were considered a new stage, the mechanism of encapsulation would correspond to the passage from Totality to Object, and that a new mechanism would be in place for transiting from Process to Totality, named completez in Spanish, which can be translated roughly as completeness. This term was first used by Roa Fuentes (2012) to denote a mechanism for construction of a transcendental Object structure in the context of mathematical infinity.…”

La concepción dinámica del límite de una función desde APOE y los registros semióticos

“…Es por esto por lo que, en el contexto del infinito, la estructura de Totalidad se ubica entre el Proceso y el Objeto. El mecanismo mediante el cual un individuo transita de la estructura Proceso del infinito a la Totalidad fue denominado completez por Roa-Fuentes (2012), y se usó para demostrar la Totalidad en Villabona et al (2022). El infinito está inmerso en el concepto de límite; para ver el Proceso infinito como terminado, el sujeto necesita considerar el límite como Objeto, como el Objeto que trasciende el Proceso (Villabona y Roa-Fuentes, 2016).…”