Diversity of student-built ideas about real numbers, irrational numbers, order and density

We present the analysis of the responses of high school and university students to four tasks that investigate how they understand what a number is in general and an irrational number, the order, the density and the supreme of an interval, in real numbers. We find a depth gradient in these ideas fro...

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Autor principal: Montoro, Virginia
Formato: Artículo revista
Lenguaje:Español
Publicado: Unión Matemática Argentina - Facultad de Matemática, Astronomía, Física y Computación 2022
Materias:
Número irracional
Número real
Concepciones numéricas
irrational number
numerical conceptions
real number
Acceso en línea:https://revistas.unc.edu.ar/index.php/REM/article/view/32442
Aporte de:
Revista de Educación Matemática de Universidad Nacional de Córdoba
id I10-R366-article-32442
record_format ojs
spelling I10-R366-article-324422022-05-17T18:50:50Z Diversity of student-built ideas about real numbers, irrational numbers, order and density Diversidad de ideas construidas por estudiantes sobre los números reales, los números irracionales, el orden y la densidad Montoro, Virginia Número irracional Número real Concepciones numéricas irrational number numerical conceptions real number We present the analysis of the responses of high school and university students to four tasks that investigate how they understand what a number is in general and an irrational number, the order, the density and the supreme of an interval, in real numbers. We find a depth gradient in these ideas from (i) a view of the integers as a model of number, distancing, or insecurity in front of these aspects of R, mainly in students with less studies in mathematics. In an intermediate zone, the (ii) conception of real numbers is identified with finite decimals and an explicit discretely, mainly in secondary and high school students (iii) a view in which the real numbers are identified with the rational numbers and as infinite potentially dense. Present mainly in first-year students of scientific careers. In the other extreme (iv) mainly advanced students of Mathematics, who understand the order, density, and property of the supreme in the real numbers. We show that to encourage students to appropriate the real number, teachers must anticipate in the last years of high school and the first years of university to work on these complex notions, to facilitate the transition from school mathematics to advanced mathematics. Presentamos el análisis de respuestas, de estudiantes de secundaria y de universidad a cuatro tareas que indagan cómo comprenden qué es un número en general y en particular un número irracional, el orden, la densidad y el supremo de un intervalo, en los números reales. Encontramos un gradiente de profundidad en sus ideas desde (i) una visión de los enteros como modelo de número, ajenidad o inseguridad frente a estos aspectos de R, principalmente en estudiantes con menor estudio de matemática.  En una zona intermedia la (ii) concepción de los reales identificados con los decimales finitos y de una discretitud explícita, principalmente en estudiantes de secundaria y (iii) una visión en la cual se identifican a los reales con los racionales y como infinitos-potencialmente densos; presente principalmente en ingresantes a las carreras científicas. Por último y principalmente estudiantes avanzados de Matemática, que (iv) comprenden el orden, la densidad y propiedad del supremo en los reales. Mostramos que para promover que los/las estudiantes se apropien del número real, la enseñanza debe prever para los últimos años de secundaria y primeros de universidad trabajar sobre estas complejas nociones, de modo de facilitar el pasaje de una matemática escolar a una matemática avanzada. Unión Matemática Argentina - Facultad de Matemática, Astronomía, Física y Computación 2022-04-29 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Artículo evaluado por pares application/pdf https://revistas.unc.edu.ar/index.php/REM/article/view/32442 10.33044/revem.32442 Revista de Educación Matemática; Vol. 37 Núm. 1 (2022); 61-92 1852-2890 0326-8780 spa https://revistas.unc.edu.ar/index.php/REM/article/view/32442/37592 Derechos de autor 2022 Virginia Montoro https://creativecommons.org/licenses/by-sa/4.0/
institution Universidad Nacional de Córdoba
institution_str I-10
repository_str R-366
container_title_str Revista de Educación Matemática
language Español
format Artículo revista
topic Número irracional
Número real
Concepciones numéricas
irrational number
numerical conceptions
real number
spellingShingle Número irracional
Número real
Concepciones numéricas
irrational number
numerical conceptions
real number
Montoro, Virginia
Diversity of student-built ideas about real numbers, irrational numbers, order and density
topic_facet Número irracional
Número real
Concepciones numéricas
irrational number
numerical conceptions
real number
author Montoro, Virginia
author_facet Montoro, Virginia
author_sort Montoro, Virginia
title Diversity of student-built ideas about real numbers, irrational numbers, order and density
title_short Diversity of student-built ideas about real numbers, irrational numbers, order and density
title_full Diversity of student-built ideas about real numbers, irrational numbers, order and density
title_fullStr Diversity of student-built ideas about real numbers, irrational numbers, order and density
title_full_unstemmed Diversity of student-built ideas about real numbers, irrational numbers, order and density
title_sort diversity of student-built ideas about real numbers, irrational numbers, order and density
description We present the analysis of the responses of high school and university students to four tasks that investigate how they understand what a number is in general and an irrational number, the order, the density and the supreme of an interval, in real numbers. We find a depth gradient in these ideas from (i) a view of the integers as a model of number, distancing, or insecurity in front of these aspects of R, mainly in students with less studies in mathematics. In an intermediate zone, the (ii) conception of real numbers is identified with finite decimals and an explicit discretely, mainly in secondary and high school students (iii) a view in which the real numbers are identified with the rational numbers and as infinite potentially dense. Present mainly in first-year students of scientific careers. In the other extreme (iv) mainly advanced students of Mathematics, who understand the order, density, and property of the supreme in the real numbers. We show that to encourage students to appropriate the real number, teachers must anticipate in the last years of high school and the first years of university to work on these complex notions, to facilitate the transition from school mathematics to advanced mathematics.
publisher Unión Matemática Argentina - Facultad de Matemática, Astronomía, Física y Computación
publishDate 2022
url https://revistas.unc.edu.ar/index.php/REM/article/view/32442
work_keys_str_mv AT montorovirginia diversityofstudentbuiltideasaboutrealnumbersirrationalnumbersorderanddensity
AT montorovirginia diversidaddeideasconstruidasporestudiantessobrelosnumerosrealeslosnumerosirracionaleselordenyladensidad
first_indexed 2024-09-03T22:36:48Z
last_indexed 2024-09-03T22:36:48Z
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