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Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Here, student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different forms of attention. We analyse a number of actions observed in students in terms of forms of attention and shifts between them: in the short-term (in the moment), medium-term (over several tasks), and long-term (over a year). The main factors conditioning students´ attention and its movement are identified and some didactical consequences are proposed., This study was funded by a Spanish national project of Research, Development and Innovation, identified by the code SEJ2006-09056, financed by the Spanish Ministry of Sciences and Technology and FEDER funds.

La investigación que da origen a este artículo es un estudio exploratorio relacionado con la introducción temprana del pensamiento algebraico o álgebra en el currículum escolar (Early-Algebra). Se informa un análisis del trabajo realizado por 18 alumnos de entre ocho y nueve años, con igualdades numéricas. Las igualdades están basadas en propiedades aritméticas básicas y compuestas por números naturales y por las operaciones elementales de la estructura aditiva. Se analiza la evolución del significado del signo igual que manifestaron los alumnos, así como el uso de estrategias de resolución basadas en relaciones y propiedades aritméticas (pensamiento relacional)., The research repeported in this papeper is related to the early introduction of algebraic thinking or algebra in the school curriculum (Early-Algebra). It gathers the work of 18 eight and nine years old students in the context of number sentences. The sentences are based on basic arithmetic properties and composed of natural numbers and the basic operations of the additive structure. We study the students’ understanding of the equal sign and its evolution, as well as their use of solving strategies based on arithmetic relations and properties (relational thinking)., La investigación que aquí se presenta ha sido realizada dentro de un proyecto (SEJ2006-09056) financiado por el Plan Nacional de I + D + I del Ministerio de Educación y Ciencia y cofinanciado con fondos FEDER de la Comunidad Económica Europea.

Se describe una propuesta curricular basada en la integración de modos de pensamiento algebraicos en el currículo de la educación primaria, la cual está siendo objeto de numerosas investigaciones en la actualidad. En este contexto, partiendo del constructo pensamiento relacional, se presentan resultados de un experimento de enseñanza, basado en el trabajo con sentencias numéricas, que ejemplifica el potencial de dicha propuesta y permite evidenciar la capacidad de alumnos de tercer curso de educación primaria para trabajar en aritmética de un modo algebraico., We describe a curricular proposal, based on the integration of algebraic ways of thinking in the elementary curriculum, which is currently aim of numerous research studies. In this context and by using the construct relational thinking, we present some results from a teaching experiment based on working with number sentences. It exemplifies the potential of this proposal and allows us to evidence third grade students´ capacity to work algebraically in arithmetic., La investigación que aquí se presenta forma parte de una tesis doctoral (Molina, 2006) dirigida por la Dra. Encarnación Castro y el Dr. Enrique Castro, de la Universidad de Granada. Este trabajo se ha realizado dentro del proyecto “Representaciones, nuevas tecnologías y construcción de significados en Educación Matemática” (SEJ2006-09056) financiado por el Plan Nacional de I+D+I del Ministerio de Educación y Ciencia y cofinanciado con fondos FEDER de la Comunidad Económica Europea.

We use data from a teaching experiment with a group of eight years old students to explore the potential of examining number sentences to promote relational thinking. This type of thinking requires attention to mathematical structure through consideration of relationships between terms contained in the sentence and not just on computation and comparison of the numeric values of each side. We show that children came to “know-to act” in the context of written activities and orchestrated discussions about number sentences, overcoming some of their computational habits and developing new ways to see and more flexibly approach the sentences. The results help to advance the study of young students´ emergent algebraic modes of thinking., This study has been developed within a Spanish national project of Research, Development and Innovation, identified by the code SEJ2006-09056, financed by the Spanish Ministry of Sciences and Technology and FEDER funds.

We describe a teaching experiment about third grade students« understanding of the equal sign and their initial forays into analyzing expressions. We used true/false and open number sentences in forms unfamiliar to the students to cause students to reconsider their conceptions of the equal sign. Our results suggest a sequence of three stages in the evolution of students' understanding of the equal sign with students progressing from a procedural/computational perspective to an analytic perspective. Our data show that as students deepened their conceptions about the equal sign they began to analyze expressions in ways that promoted algebraic thinking. We describe the essential elements of instruction that advanced this learning.

En este artículo resumimos trabajos que abordan cuestiones relacionadas con el uso y desarrollo de pensamiento relacional en el contexto de la resolución de igualdades y sentencias numéricas. Nuestra intención es describir el estado de la cuestión e identificar líneas de investigación abiertas. Previamente detallamos el significado del término pensamiento relacional y señalamos otros términos más frecuentes en la literatura relacionados con este constructo., This paper summarizes studies which tackle questions related to the use and development of relational thinking in the context of solving number sentences. Our aim is to describe the state of the art and to identify open research lines. Previously, we detail the meaning of the term relational thinking and we list other terms which are more frequent in the literature and are related to this construct., Este trabajo ha sido desarrollado dentro del proyecto “Representaciones, nuevas tecnologías y construcción de significados en Educación Matemática” (SEJ2006-09056) financiado por el Plan Nacional de I+D+I del Ministerio de Educación y Ciencia y cofinanciado con fondos FEDER de la Comunidad Económica Europea.

En este documento presentamos algunos de los resultados de un estudio que aporta evidencias de la capacidad, de los alumnos de tercero de educación primaria, para desarrollar pensamiento relacional y comprensión del significado del signo igual “equivalencia numérica”, trabajando en un contexto de igualdades numéricas., In this document we present some of the results of a study which provides evidence of third grade students´ capacity to develop relational thinking and understanding of the meaning of the equal sign “numerical equivalence”, both in a context of number sentences., Este estudio ha sido desarrollado dentro de un proyecto del plan nacional de I+D+I financiado por el Ministerio de Ciencia y Tecnología, con número BSO2002-03035, y cofinanciado con fondos FEDER.