Sistemas de representación de números racionales positivos : Un estudio con maestros en formación

Prospective Teachers’ Difficulties in Integrating Technology Into Problem Solving and Teaching on Rational Numbers

“…Finally, concerning the use of the different interpretations of the rational number, we observed pedagogical-mathematical weaknesses in some couples that are restricted in their explanations to formal operations between fractions, neglecting the context offered by the problem. Gairín (2001) highlights that prospective teachers who show a lesser understanding of the different interpretations of the rational number design purely formal explanations, which may be related to the instruction received during their preuniversity mathematical education, guided by textbooks in which fundamentally procedural exercises predominate (Gairín & Muñoz, 2005;Martínez-Juste et al, 2017;Shield & Dole, 2013). All of the above shows a low development of our students' PCK component.…”

“…The development of the CK component of our prospective teachers is related to the mathematical errors they showed in the solution of the different items. The most common error is what we call "confusion of the part with the whole" (items a and c), possibly caused by one of the disadvantages of traditional teaching, generally limited to part-whole interpretation, in which the student does not need to recognise the unit because it coincides with the total quantity of magnitude (Gairín, 2001;Gairín & Muñoz, 2005). Simon et al (2018) also highlight this and other difficulties in some advanced concepts of fractions when teaching is limited exclusively to part-whole interpretation, and how they can be mitigated via sequences of tasks in interpretation as a measure, as Olive and Vomvoridi (2006) also assure.…”

“…Regarding the first, even some of those who solve the problem incorrectly make the effort to argue within a single interpretation, which would give them an advantage in elaborating future explanations. In contrast, other students who solve correctly reason, either by mixing ideas from various interpretations of the rational number or without using any interpretation, so the numbers lose their meaning, which could result in a didactic difficulty when giving their future students an explanation (Gairín, 2001). These considerations would go along the lines expressed by some authors (Ruíz de Gauna et al, 2013) who show the existence of a prospective teacher's profile of a mostly mathematical nature and another profile of a fundamentally pedagogical-didactic type.…”

“…However, instead of being of a geometric type, the content underlying our study is of an arithmetic nature, specifically on the rational number. We chose this mathematical object because it is important for the curriculum (Real Decreto 126/2014), and to know whether the teaching of this object to our preservice teachers when in elementary and high school might have limited them when making explanations from its different interpretations, as previous studies have already detected (Clarke et al, 2008;Escolano & Gairín, 2005;Freudenthal, 1983;Gairín, 2001;Martínez-Juste et al, 2017;Olive & Vomvoridi, 2006;Simon et al, 2018;Shield & Dole, 2013).…”

<p>Sobre las nociones de representación y comprensión en la investigación en educación matemática</p>

“…En la obra colectiva Didactics of Mathematics as a Scientific Discipline (Biehler, Scholtz, Strasser y Winkelmann, 1994), el concepto de representación se trabaja y emplea extensamente: "La representación de hechos y relaciones es un aspecto muy importante del aprendizaje y el pensamiento matemático, por ello los educadores matemáticos han estado fuertemente interesados en la investigación psicológica que contribuye a la comprensión de las representaciones. " (p. 19) Sin ánimo de ser exhaustivo, la noción de sistema de representación se encuentra explícitamente, en el seno del grupo de investigación Pensamiento Numérico, en las tesis de Encarnación Castro (1995), José L. González Marí (1995), Isabel Romero (1995), Eduardo Lacasta (1995), Francisco Fernández (1997, José M. Gairín (1998) y Francisco Ruiz (2000); también en Coriat y Scaglia (2000) hay un uso extenso de estas nociones. Castro y Castro (1997) hacen un análisis conceptual detallado de esta noción, que se ha utilizado en muchos otros trabajos e investigaciones.…”