Semiotic Conflicts in the Learning of Proportionality: Analysis of a Teaching Experience in Primary Education

Abstract

In this paper, we analyse the results of an experience designed and implemented to evaluate the epistemic and cognitive conflicts identified in the study of proportionality, carried out by a group of 21 students in their last year of primary education. Initially the students exhibit difficulties to recognize situations where proportionality can be applied, to distinguish magnitudes, quantities and proportionality tables, and to identify procedures to solve proportionality problems other than the rule of three. Nevertheless, in the group discussion session, oriented to develop the dialogue and collective justification of the assessment tasks solutions, they are able to justify the methods employed, linking the rule of three with the equivalence of fractions and the proportionality relation.

Playing Mathematics in Early Childhood Based on Semiotics

“…At this stage, the activities carried out by the research subjects were the five subjects associated the signs with the knowledge they previously had and looked for all signs according to the objects found. A person can have different interpretations related to images, this depends on how the person interprets the image image (Burgos & Godino, 2020;Gravells, 2017;Kralemann, et al, 2013).The five subjects associated the number of sweeties found with a numeric symbol representing the number of sweeties. From the observations, subject 5 experienced confusion in looking for the symbol number 9.…”

Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis

“…Research about solving arithmetic word problems has shown that students use improper applications of linearity, referred to as the illusion of linearity (Van Dooren et al, 2005). Burgos and Godino (2020) studied written solutions of missing value tasks, followed by interviews with Spanish students (aged 11-12). The authors found that students struggled to identify proportional situations as well as to discern the different quantities in the proportional relationship.…”

“…Furthermore, textbooks provide limited support for students' development of mathematical structures (Vergnaud, 1983) necessary for the learning of proportional reasoning (Shield & Dole, 2013). For example, there are limited opportunities for students to distinguish proportional from non-proportional situations in textbook tasks (Burgos & Godino, 2020), and the intuitive approach to proportionality is also not recognized in the tasks presented in the textbooks. This article reports on an empirical study on how students (aged 12-13) in Sweden reason when they are confronted with two pattern tasks during one lesson, where the focus of the analysis is on indications of proportional reasoning as the learning processes occur.…”

Aplicação da proporcionalidade direta à resolução de uma situação do quotidiano por futuros professores dos primeiros anos

“…O conceito de proporcionalidade é um conceito complexo que requer a mobilização de vários outros conceitos, designadamente os conceitos de razão, proporção e função. Estudos realizados antes mostram que os alunos, em que se incluem também futuros professores dos primeiros anos escolares, têm dificuldades nos conceitos de razão, proporção e proporcionalidade (e.g., Berenson, Oldham, Price & Leite, 2013;Beswick, 2011;Burgos & Godino, 2020a;Fernandes, Barros & Gonçalves, 2019, 2020Fernandes & Leite, 2015;Livy & Vale;2011;Singh, 2000), dificuldades essas relativas à definição, à representação e à aplicação destes conceitos à resolução de situações-problema.…”