The intuition of infinity

Fischbein, Efraim; Tirosh, Dina; Hess, Peter

doi:10.1007/bf00311173

Cited by 142 publications

(105 citation statements)

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“…Dos visiones finitistas muy arraigadas en los estudiantes son: concebir el infinito como un número muy grande (Fischbein, Tirosh y Hess, 1979;Monaghan, 2001;Montoro, 2005; Juan, Montoro y Scheuer, 2012) y extender la propiedad de los conjuntos finitos el todo es mayor que la parte a los conjuntos infinitos, concepción que suele denominarse de inclusión (Fischbein et al, 1979;Fischbein;1987;Waldegg, 1993b;Falk, 1994).…”

Section: Las Concepciones De Los Estudiantes Acerca Del Infinitounclassified

“…Entre los estudiantes que aceptan la posibilidad de colecciones infinitas se ha identificado la concepción de aplanamiento donde se considera única la cantidad infinita (Arrigo y D´Amore, 2004;Fischbein et al, 1979;1987) y, muy minoritariamente, la concepción de infinito cardinal, sólo presente en universitarios (Moreno-Armella y Waldegg, 1991, Montoro, 2005.…”

Section: Las Concepciones De Los Estudiantes Acerca Del Infinitounclassified

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¿Cuán abundantes son los conjuntos de números? Estudiantes comparando infinitos

Montoro

Scheuer

Echeverría

2016

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146Educación MatEMática, vol. 28, núM. 3, diciEMbrE dE 2016 más compleja, la infinitista, sólo fue explicitada por estudiantes universitarios, según dos enfoques: pensar la cardinalidad de los conjuntos infinitos como una única cantidad infinita, o concebir distintos cardinales infinitos, este último expresado sólo por estudiantes avanzados de matemática.Palabras clave: conjuntos numéricos, infinito cardinal, educación secundaria, estudiantes universitarios.Abstract: We studied how students with different mathematical background conceive infinite cardinality of number sets. We analyzed a task in which high school and college students were requested to compare infinite number sets. Students were classified according to their ideas on infinity. Using this classification together with the students' level of math education, we performed a correspondence factorial analysis. A gradient was found in the depth of students' ideas. At one end we found what we called horror infiniti, based students propensity avoid infinity and instead construe it as something undefined. These views were associated with students with less mathematical education. In an intermediate zone, the finitist conception was placed. It was the most frequent way of thinking among the participants of the study, with three versions: tacitly infinitist, explicitly finitist, or taking the integers as model of inclusion. At the other end, the infinitist conception was placed. It was present among students with college mathematical education, according to two types: thinking of the cardinality of number sets as a unique infinite quantity; or conceiving different infinite cardinals. The latter was found only in advanced mathematics students.

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Section: Las Concepciones De Los Estudiantes Acerca Del Infinitounclassified

¿Cuán abundantes son los conjuntos de números? Estudiantes comparando infinitos

Montoro

Scheuer

Echeverría

2016

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“…Using the Cantorian hypothesis of continuum: According to Fischbein et al (1979) the world of ,... , ,…”

Section: The Hypothesis Of Continuummentioning

confidence: 99%

Mathematical, Philosophical and Semantic Considerations on Infinity (I): General Concepts

2015

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In the Reality we know, we cannot say if something is infinite whether we are doing Physics, Biology, Sociology or Economics. This means we have to be careful using this concept. Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of (the mathematical symbol for the set of all integers)? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on another worldly absolute truth. Many mathematicians believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that our knowledge of the physical world is imperfect and falls short of what we can apprehend with mathematical thinking. The objective of this paper is to present an epistemological rather than an historical vision of the mathematical concept of infinity that examines the dialectic between the actual and potential infinity.

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“…Research shows that such concept images often persist long after formal ideas are introduced (Fischbein et al, 1979). Personal informal images may involve essentially contradictory features.…”

Section: Axioms/definitionsmentioning

confidence: 99%