Una investigación sobre los esquemas conceptuales del continuo. Ensayo de un cuestionario

Abstract

In order to survey how students conceive the idea of continuum as they begin their training on Calculus, we have established a questionnaire and an analytical approach to responses, fully rooted in networks. The theoretical framework is based on the Dreyfus and Vinner idea of cconcept image)), where we attune the expression eperception of a propertyn. We have tested the questionnaire on 74 students in the range of 16-17 years old, acquiring valuable information in return about their perception on non-discrete ordering. On the other hand, we have established a draft ordered system for models) of geometric continuum, verifying the fact that, for non-integers, the way they are written comes out to be the most representative property of the various sets of numbers. At the end, we put forth an open question about the coherence of the concept images of people with different degrees of expertise.La modelización matemática de la naturaleza construida íntimamente ligada a la invención y desarrollo del cálcua partir del Renacimiento estuvo, ya desde su inicio, lo infinitesimal. Esto es todavía asíen nuestros días y, en

Students’ understanding of the number line / Estudiantes pensando en la recta numérica

“…In the first, the task is alien to them and they fail to engage in the activity, and in the second, they are unable to detach themselves from the real object (the microscope) or the external representation of the line. Both are associated with a lower level of mathematical studies and correspond with those students who 'do not play the game' according to Romero (1996).…”

“…In the first way, we find the vision of drawn or physical matter line (17%) associated with a naïve realism (Romero, 1996), according to which the line does not represent the real numbers, but is an external physical object. This conception is associated with the youngest students.…”

“…Twelve percent responded 'it is a line' and offered some justifications, 20% responded 'it is a line' without making any remarks, and 15% referenced a discrete model of the line. Romero (1996) also used a questionnaire with open-ended questions to obtain information about how students conceived the idea of a continuum at the beginning of their training in calculation, pointing out that:…”

“…Our study modified the proposal from Romero (1996) by: (i) adding the word 'ideal', in order to invite the participant to appeal to his/her imagination and to be able to detach from the physical situation of the proposed context; (ii) the proposal of a non-mathematical context (use of a microscope) so that even those students with less study in mathematics could approach the task; and (iii) calling the line 'number line', implicitly providing the correspondence of (real) numbers with the points on the line. A previous task from the same questionnaire provided a model of the number line over a tenths grid (Figure 1).…”

“…Our study is based on previous studies that have also asked this question. Robinet (1986) and Romero (1996) used similar tasks to investigate the ideas of students in mathematics courses, between the ages of 16 and 18 years, on the nature of the number line. They initially found distinct mental models regarding this line.…”