Some Cognitive Difficulties Related to the Representations of two Major Concepts of Set Theory

Abstract

The main focus of this paper is on the study of students' conceptual understanding of two major concepts of Set Theory -the concepts of inclusion and belonging.To do so, we analyze two experimental classroom episodes. Our analysis rests on the theoretical idea that, from an ontogenetic viewpoint, the cognitive activity of representation of mathematical objects draws its meaning from different semiotic systems framed by their own cultural context. Our results suggest that the successful accomplishment of knowledge attainment seems to be linked to the students' ability to suitably distinguish and coordinate the meanings and symbols of the various semiotic systems (e.g. verbal, diagrammatic and symbolic) that encompass their mathematical experience.

Mathematical Representation of Grade 7 Students in Set Theory Topics Through Problem-Based Learning

“…Signs and symbols have a wide role in the concept of set. Bagni (2006) explained the representation of set, as follow : (1) Verbal: the definitions of set, element, subset, union, etc, (2) Symbolic: the set symbol (capital letter), the set operation symbols, the brackets, etc, and (3) Visual: the presenting of set in Venn diagram. These set representations are known as mathematical representations.…”

Mathematics and positive sciences: a reflection following Heidegger

“…Heidegger's philosophical work is a challenge to both individualism and instrumentalism. I hope that these aspects of Heidegger's philosophy contribute to current discussions in our field-particularly to discussions about subjectivity and agency (e.g., Bagni, 2006;Brown, 2008;Presmeg & Radford, 2008;Radford, Schubring, & Seeger, 2008).…”

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“…Respecto a la faceta epistémica se encontró que la falta de conocimiento avanzado radica, como explican D'Amore y Fandiño (2005), en el hecho del desconocimiento del desarrollo histórico y epistemológico de los números primos, lo que implica no asumir posiciones comprometidas y significativas respecto a ellos. A su vez, la dificultad para establecer conexión con otros objetos matemáticos puede tener origen en una concepción de la matemática en la que el profesorado desconoce que los objetos matemáticos no son entes aislados; por el contrario, están en constante relación (Bagni, 2006;Bagni y D'Amore, 2005). En este caso, el desconocimiento de la conexión de los números primos con otros objetos genera una conciencia débil en el profesorado de lo que estos números ofrecen matemática y didácticamente (Grugnetti y Rogers, 2000).…”