En este artículo nos proponemos responder a qué conocimientos algebraicos se evidencian cuando un grupo de estudiantes de 9-10 años inventan sentencias numéricas e historias matemáticas, y qué aportes ofrecen estas tareas al desarrollo del pensamiento algebraico desde las dimensiones estructural y analítica. Analizamos las respuestas de estudiantes de estas edades, prestando atención a las operaciones y propiedades involucradas, los significados del signo igual y cómo se refieren y razonan sobre cantidades desconocidas. Los resultados muestran que el alumnado inventa sentencias que involucran números y cantidades desconocidas, apoyándose en diferentes propiedades de las operaciones y evidenciando comprensiones relacionales del signo igual. Al inventar historias, relacionan las cantidades desconocidas con situaciones cotidianas y plantean historias coherentes con la ecuación dada. Discutimos el rol que tiene la invención de problemas en la construcción del pensamiento algebraico en los primeros cursos de la educación primaria.

In this article, we describe the design and implementation of a professional development course based on the video club methodology, involving 21 Chilean primary school teachers. We supported teachers in noticing students' algebraic thinking in order to make decisions based upon their professional knowledge. The objective was to make teachers pay attention to both the algebraic content and the practices evidenced in students' strategies. The video club consisted of seven sessions, and its design was adapted to the dynamics and characteristics of the participants. In each session, we selected videos that highlighted key aspects of algebra learning and teaching at these educational levels. We conclude that the video club was a useful tool for addressing students' thinking and fostering ongoing professional development.

In this article, we aim to address the following research questions: What algebraic understandings are evidenced when a group of 9-10-year-old students invent numerical sentences and mathematical stories? And what contributions do these tasks make to the development of algebraic thinking from both structural and analytical perspectives? We analyse students' responses by focusing on the operations and properties involved, the meanings attributed to the equal sign, and the ways in which students refer to and reason about unknown quantities. The results show that students invent sentences involving numbers and unknown quantities, drawing on different properties of operations and demonstrating relational understandings of the equal sign. When inventing stories, they relate unknown quantities to everyday situations and construct narratives that are coherent with the given equation. We discuss the role of problem invention in the construction of algebraic thinking in the early years of primary education.

In this study, we focus on addressing the research question: How do prospective elementary school teachers make decisions when considering the algebraic thinking of 9-years-old children? Specifically, we describe how 21 prospective primary teachers make decisions regarding the strategies employed by three children to solve the open equality 6+4=+5. The prospective teachers participated in a Video Club and we focused the first two sessions, during which the participants responded to two questions intended to guide their decisions, one being more general and the other more specific. We conducted a sequential content analysis of their written responses, following both a deductive and inductive approach. The main findings reveal that, despite the limited use of evidence, the elements considered by prospective teachers when making decisions align with specific aspects of children’s algebraic thinking and related research in this area, which were addressed during the Video Club sessions. Finally, we found that prospective teachers make decisions following two approaches: (a) arithmetic, which focuses on sequentially referencing the procedures children should follow to complete the blank space, and (b) relational, which is centered on how children relate operations to each other with the equal’s sign.

En esta investigación, exploramos cómo un grupo de 21 futuras maestras de primaria atiende, interpreta y decide basándose en el pensamiento algebraico evidenciado por niñas al resolver igualdades numéricas. Nos situamos en las perspectivas conceptuales del noticing y el MTSK para profundizar en el conocimiento profesional de las participantes. Las futuras maestras participaron en una asignatura sobre la enseñanza y el aprendizaje del álgebra escolar, la cual siguió la metodología del análisis de vídeos. Los principales resultados muestran que las participantes consideran los elementos algebraicos involucrados en la situación, aunque el uso de evidencia es limitado. Además, destacamos la movilización de diversos conocimientos matemáticos para la enseñanza del álgebra escolar.

Responding to students' mathematical thinking is an essential teaching practice that drives high-quality education. This article explores how 21 future primary school teachers interact with students' algebraic thinking when solving numerical equalities, within the context of video-based course. To do so, we aim to address two research questions: (a) How do future teachers attend to, interpret, and decide based on students' strategies? and (b) What specialized mathematical knowledge for teaching school algebra do future teachers mobilize when attending to, interpreting, and deciding based on students' strategies? Conceptually, we situate ourselves within the perspectives of noticing and the mathematics teacher's specialised knowledge (MTSK) model to investigate the knowledge future teachers demonstrate and use when considering children's algebraic thinking. Participants engaged in a video-based course focusing on algebraic teaching and observed a video illustrating the reasoning of three students solving 6+4= +5. We analyze how participants attend to, interpret, and decide based on children's responses, considering relevant algebraic elements, the use of evidence, and domains/subdomains of MTSK model. The primary findings indicate that participants consider various pertinent algebraic elements in children's responses, although the use of evidence is limited. Moreover, we identify different types of mathematical knowledge used during interactions with children's algebraic thinking. Specifically, the results demonstrate that in attending and interpreting, there is evidence of mathematical content knowledge (MK) and pedagogical content knowledge (PCK), while in decision-making, PCK predominates, specifically in knowledge of mathematics teaching.

El objetivo de este trabajo es describir una propuesta de enseñanza que promueva el pensamiento algebraico a través de la expresión y justificación de ideas matemáticas al resolver tareas relacionadas con tres enfoques distintos del pensamiento algebraico. Diseñamos un experimento de enseñanza implementado durante la pandemia de la COVID-19 en Chile. Analizamos las discusiones orales y las producciones escritas de niños de cuarto de primaria (9-10 años). Los resultados muestran que los niños expresaron y justificaron ideas algebraicas cada vez más sofisticadas. Es decir, adoptaron paulatinamente un lenguaje matemático más preciso y abstracto. Concluimos que esta modalidad de trabajo, en la cual se destaca el carácter algebraico de la aritmética a través de diversas instancias de discusión, es un aporte para los docentes, al guiarlos en abordar los desafíos de enseñanza actuales.

center dot The aim of this paper is to describe a teaching proposal that promotes algebraic think-ing through the expression and justification of mathematical ideas when solving tasks related to three different approaches to algebraic thinking. We designed a classroom teaching experiment implemented during the COVID pandemic in Chile. We analyze the oral discussions and the written productions of children in fourth grade (9-10 years old). The results show that the children expressed and justified increasingly sophisticated algebraic ideas. That is, they gradually adopted a more precise and abstract mathematical language. We conclude that this modality of work, in which the algebraic character of arithmetic is highlighted through various instances of discussion, is a contribution for teachers, by guiding them in addressing current teaching challenges.

This study describes how 24 third graders (8–9 years old) relate and represent the relationships between variables when working with a functional thinking problem. This aspect contributes to providing insights about how elementary school students attend properties and relationships between covarying quantities rather than isolated computations. From a functional approach to early algebra, we describe written students' answers when working with a problem that involves a function, which includes questions for specific values and to generalize. Design research guidelines, specifically those set out for Classroom Teaching Experiment were followed. This study addresses the fourth and last Classroom Teaching Experiment session, which involved a function of the type y = ax + b and students had not previously worked it. Students primarily evidenced correspondence relationship, using natural language and numerical representation to express this functional relationship. Our findings let us to state that (a) although students were not used to working with these types of problems, eleven of them go beyond arithmetic computations, finding relationships that relate the variables; and (b) three students generalized using natural language as a useful vehicle, while there are other students who perceived the same regularity for different specific values but they are unable to represent generalization clearly.

In this study, we analyze how 9–10-year-old pupils work with equations, a central aspect of algebraic thinking in early grades and a cornerstone for more formal learning of algebra. Specifically, we seek: (a) to describe the main characteristics of the tasks that support algebraic thinking through a translation process from arithmetic word problems to algebraic language and vice versa, and (b) to identify how pupils refer to indeterminate quantities in these contexts and what meaning they give to them. The analysis focuses on the semantic congruence of the expressions proposed by them and on the dialogue they held during the translation process. We analyzed the oral discussion in the pools and the written responses to the problem that pupils posed. The results show that arithmetic word problems allow the indeterminate to become an object of thought for pupils, who represent it in multiple ways and refer to it when proposing equations that represent the structure of each problem. Another finding highlights that reflection on the interpretation of the equations supports the identification of two meanings associated with indeterminate quantities, namely, unknown and variable.

En este artículo presentamos ideas que podrían ayudar a profesores y profesoras a fomentar el pensamiento algebraico de sus estudiantes de Educación Primaria (6-12 años). Si bien diferentes países incluyen la idea de álgebra (y pensamiento algebraico) en sus normativas curriculares, todavía faltan elementos que ayuden a ver el álgebra más allá de letras y números. Los objetivos de este artículo son: (a) caracterizar elementos conceptuales y teóricos involucrados en el pensamiento algebraico, y (b) reflexionar qué decisiones pedagógicas podrían fomentar el pensamiento algebraico de estudiantes de Educación Primaria. Para abordar ambos objetivos, presentamos las respuestas de algunos estudiantes que participaron en diferentes sesiones de un taller de matemáticas, durante el verano de 2021. En concreto, presentamos tres problemas que trabajaron los estudiantes y exponemos algunas de sus respuestas (orales y escritas). A partir de dichas respuestas, resaltamos oportunidades para desarrollar el pensamiento algebraico en los diferentes cursos de la Educación Primaria.

Diversos reportes de investigación han mostrado cómo niños de las primeras edades construyen ideas matemáticas generales, aspecto central del pensamiento algebraico. Sin embargo, los estudios que abordan cómo estudiantes de estas edades justifican esas ideas generales son escasos y esto constituye la originalidad y contribución de esta investigación. Considerando el rol predominante de las representaciones manipulativas en las experiencias matemáticas de niños de las primeras edades, la pregunta de investigación que busco responder es: ¿cómo las representaciones manipulativas apoyan la construcción y justificación de ideas matemáticas generales al resolver problemas que tienen un carácter algebraico? Para aproximarme a dicha pregunta, realicé una entrevista individual semiestructurada a un niño de 5 años, en la cual le presenté dos problemas diferentes que buscaban que el participante atendiera a la relación entre una determinada cantidad de perros y la cantidad de colas/ojos respectiva. Los resultados muestran que a pesar de no emplear y/o referirse al material manipulativo para construir reglas generales y/o justificar, el participante encontró relaciones generales entre las variables involucradas en los problemas. Asimismo, sus justificaciones se basan en las reglas generales que él construyó. Finalmente, el participante extiende dichas reglas generales hasta el punto de interactuar con cantidades indeterminadas.

Uno de los desafios que nos deja la pandemia producida por la COVID-19 tiene relacion con identificar cuales son los aprendizajes matematicos esenciales que debieramos transmitir a nuestros y nuestras estudiantes. A partir de lo anterior, surge como un imperativo promover diferentes tipos de matematico desde las primeras edades. Es asi, como el pensamiento algebraico cobra relevancia pues permite que estudiantes encuentren reglas generales y favorece que estos seleccionen informacion matematica relevante; adapten, ajusten y reorganicen experiencias previas; pongan atencion a ideas, capacidades y propiedades involucradas en diferentes situaciones; mejoren su comprension y herramientas para resolver problemas, entre otros. En esta conferencia presentare ideas provenientes de la investigacion que permiten situar y entender que el algebra va mas alla que el uso de letras y numeros.

We describe 24 third (8–9 years old) and 24 fifth (10–11 years old) graders’ generalization working with the same problem involving a function. Generalizing and representing functional relationships are considered key elements in a functional approach to early algebra. Focusing on functional relationships can provide insights into how students work with two or more covarying quantities rather than isolated computations, and focusing on representations can help to identify the type of representations useful to them. The goals of this study are to (1) describe the functional relationships evidenced in students’ responses and (2) describe the representations that the students use. In addressing these research objectives, we describe student responses drawn from a Classroom Teaching Experiment in each grade. We analyzed students’ written responses to different questions designed to generalize the relationships in a problem that involves the function y = 2x + 6. Our findings illustrate that 11 third graders and 19 fifth graders provide evidence of functional relationships in their responses. Three third graders and all fifth graders generalized the relationship. We conclude that these differences may be due to the students’ previous classroom mathematical experiences, since students in higher grades would be more likely to focus on the relationships between variables, whereas third graders would focus on the details of arithmetic computations. In addition, we find that natural language is the main vehicle used to generalize in both grades. Unlike third graders, fifth graders perceive general rules from the numerical calculation and express these generalizations even when not explicitly requested to do so.

This paper describes the differences in the types of representations used by eight third-grade (8 to 9-years-old) and eight fifth-grade (10 to 11-years-old) students when working with problems that involve different linear functions. We present an analysis of students’ written and oral answers during a Classroom Teaching Experiment (CTE) and semi-structured interviews from a functional approach to early algebra. The study examines how students’ representations varied when working with different types of linear functions (y=x+a; y=ax; y=ax+b), when solving for specific values, and when generalizing. The findings show that students in both grades primarily used the representation present in the problem. The type of linear function involved appears to have had no effect on either group’s use of one representation or another.

This article discusses evidence of 24 fifth graders’ (10-11 year olds’) ability to generalize when solving a problem which involves a linear function. Analyzed in the context of the functional approach of early algebra, the findings show that 3 students generalized both when solving specific instances and when asked to provide the general formula; while 15 students generalized only when asked to define the general formula. The results are described in terms of the functional relationship identified, the types of representation used to express them and the type of questions in which students generalized their answers. Most of the pupils who generalized did so based on the correspondence between pairs of values in the function at issue.Generalización de estudiantes de quinto de primaria desde un enfoque funcionalEn este artículo presentamos evidencias de generalización de 24 estudiantes de quinto de primaria (10-11 años) al resolver un problema que involucra una función lineal. Desde el enfoque funcional del early algebra, los hallazgos muestran que 3 estudiantes generalizaron al trabajar con casos particulares y cuando se les pide expresar la regla general; mientras que 16 estudiantes solo lo hicieron cuando les pedimos expresar la regla general. Describimos los resultados en términos de la relación funcional identificada, los tipos de representaciones que emplearon para expresar dichas relaciones y el tipo de pregunta en la cual los estudiantes generalizaron. La mayoría de los estudiantes que generalizaron establecieron una relación de correspondencia entre los pares de valores de la función.Handle: http://hdl.handle.net/10481/50159Doi: https://doi.org/10.30827/pna.v12i3.6643Scopus record and citations

This paper focuses on functional thinking as an approximation to algebraic thinking in third-year primary-school students. It describes a study with a class group of 24 Spanish pupils displaying functional thinking to solve a contextualised problem, identifying the type of functional relationships distinguished by these students and the ability to generalise observed in some of them. It contains an analysis of the information collected in one questionnaire, which is part of a teaching experiment. The students distinguished two types of functional relationships, correspondence and covariation, predominantly the former. Three students generalised as well.

Existen varias aproximaciones a la definicion de pensamiento funcional (e.g., Blanton y Kaput, 2011; Canadas y Molina, 2016; Canadas, Brizuela y Blanton, 2016). Estas definiciones, que adquieren sentido en los estudiantes de edades tempranas, son empleadas en este taller para analizar las respuestas de los estudiantes y caracterizar tareas que promuevan este tipo de pensamiento. El objetivo de este taller es suministrar ideas y estrategias docentes para que el maestro seleccione y analice las respuestas de sus estudiantes y evalue el pensamiento funcional que evidencian. Este trabajo sera utilizado para informar el diseno de tareas que favorezcan el desarrollo de este tipo de pensamiento en los estudiantes de primaria. El taller se organiza en tres momentos: (a) discusion grupal, la cual busca que los docentes reflexionen sobre sus ideas iniciales alusivas al pensamiento funcional y el uso de las funciones en Educacion Primaria; (b) sintesis teorica, en la cual presentamos una sintesis de los principales aportes provenientes de la investigacion en edades tempranas sobre pensamiento funcional; (c) analizar en pequenos grupos, respuestas de estudiantes a tareas que involucran pensamiento funcional.