Este trabajo forma parte de un proyecto de investigación sobre pensamiento algebraico en alumnos de educación infantil y primaria. El objetivo de investigación de este documento es analizar las relaciones entre variables, representaciones y estrategias que evidencian alumnos de infantil al trabajar con una tarea de generalización. Implementamos un experimento de enseñanza de 4 a 5 sesiones con alumnos de infantil. Analizamos el trabajo de una de las sesiones, donde se relacionaba el número de niños invitados a una fiesta de cumpleaños y el número de zumos necesarios, considerando que uno no toma ( ). La sesión se desarrolló en tres momentos: (a) introducción, (b) trabajo individual y (c) asamblea. En las producciones escritas de los alumnos predominó la representación pictórica e identificamos estrategias físicas y cognitivas. Hubo niños que generalizaron, indicando que “a este niño no le daremos zumo porque no le gusta”, para reflejar el “-1” implicado en la función.

This work is part of a research project on algebraic thinking in early childhood and primary education students. The research objective of this document is to analyze the relationships between variables, representations, and strategies that preschool students demonstrate when working with a generalization task. We implemented a teaching experiment of 4 to 5 sessions for preschool students. We analyzed the work in one of the sessions, where the number of children invited to a birthday party was related to the number of juices needed, considering that one child does not drink (f(n) = n-1). The session took place in three moments: (a) introduction, (b) individual work, and (c) assembly. In the students' written productions, pictorial representation predominated, and we identified physical and cognitive strategies. There were children who generalized, noting that "we will not give this child juice because he does not like it" to reflect the "-1" implied in the function.

This study is part of a broader research project on algebraic thinking in early childhood and elementary education (www.pensamientoalgebraico.en). [Objective] The objective of this work is to identify the change perceived by 3-year-old children when solving generalization tasks that involve qualitative or quantitative changes. [Methodology] Ateaching experiment consisting of 4 worksessions with a group of 25 3-year-old children from a public school in Granada was designed and implemented.The perception of qualitative and quantitative change was investigated.The first session involved color and size as attributes; the second session included the function f(n)=n+1. The focus was on identifying relationships between variables and on the way children expressed these relationships through the changes that affect these variables. [Results] The children identified the qualitative change by saying that the colored object that entered the rocket changed color and came out gray, for example, they used expressions such as "the color is gone," "it comes out black," or "it comes out gray." In the case of quantitative change, they indicated that the ice cream that was inserted always came out with one more scoop than those that entered the machine, using statements such as "it comes out bigger," "it comes out higher," or "many come out." [Conclusion] Based on the statements given bythe children, it can be concluded thatthey identified the changes in color, size and increase.

Early algebra research has focused primarily on documenting what students are able to do in tasks that involve functional thinking, in an attempt to refute preconceptions around young students' presumed inability to generalize. However, scant attention has been paid to the mistakes students make and difficulties they encounter in these tasks. This exploratory study analyzed eight 2nd-graders' (7- to 8-year-olds) responses, both appropriate and inappropriate, for individual interviews in which they were presented a contextualized task involving the function f(x) = x +5. We analyzed students' initial responses, the interviewer's interventions in face of inappropriate answers, and students' post-intervention responses. Specific types of pre- and post-intervention (appropriate and inappropriate) responses to generalization tasks in elementary mathematics were identified. All participants solved tasks involving algebraic notions involving functions. These 2nd-graders' work was significant considering the expectations of current curriculum frameworks. The types of responses identified contribute to the scholarly understanding of young students' functional thinking, being useful for research and teaching purposes.

Research on generalization in elementary school is a key topic in mathematics education. This study analyzes the generalization processes among fourth-grade students (9–10 years old) while working with functional relationships, focusing on the interplay between generalizations, justifications, and mediations during a lesson. Specifically, we identify and characterize the generalizations and justifications made by 22 students and describe the mediations carried out by the teacher-researcher throughout the class. Additionally, we examine the relationships between the levels of generalization, justifications, and mediations. Our findings reveal that students demonstrated varying levels of sophistication in their generalizations, which were consistently supported by justifications and mediations. Justifications played a crucial role in validating and explaining the students’ generalizations, while mediations facilitated their engagement with the task and supported the generalization process throughout the lesson. We conclude that justification and mediation are integral components of the generalization process, emphasizing its active and social nature. Our study underscores the importance of fostering classroom discussions and encouraging students to articulate their reasoning and conjectures, promoting deeper understanding and collaborative learning.

In this paper, we present a bibliometric analysis whose objective is to quantify and describe the scientific production of algebraic thinking in early childhood and primary education. We developed this analysis within the Scopus database, as it is one of the databases with the greatest coverage at the level of journals and volume of citations at the international level. We considered the initial open and final search ranges until December 2022. The results show a growing interest in the research community in Didactics of Mathematics in algebraic thinking at these educational levels. We identified authors who have produced a large amount of research on this topic, as well as groups of authors who have collaborated on different papers. In addition, key terms such as early algebra, functional thinking and generalization stand out as prominent topics in this area of study.

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.

En este trabajo presentamos un análisis bibliométrico cuyo objetivo es cuantificar y describir la producción científica sobre pensamiento algebraico en educación infantil y primaria. Desarrollamos este análisis dentro de la base de datos Scopus, por ser una de las de mayor cobertura a nivel de revistas y volumen de citación en el ámbito internacional. Consideramos los rangos de búsqueda inicial abierto y final hasta diciembre de 2022. Los resultados muestran un creciente interés de la comunidad investigadora en la Didáctica de la Matemática por el pensamiento algebraico en estos niveles educativos. Identificamos autores que han producido una gran cantidad de investigaciones en este tema, como a grupos de autores que han colaborado en distintos trabajos. Además de los términos clave como early algebra, functional thinking y generalization que destacan como temas prominentes en esta área de estudio.

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.

Given the growing emphasis on algebraic concepts in primary education curricula and the importance to investigate how students with Autism Spectrum Disorder (ASD) approach mathematical learning, the purpose of this study is to evaluate the effectiveness of an instruction based on multiple representations in promoting early algebraic thinking in students with ASD. Three students with ASD, aged 7, 7, and 9 years, enrolled in the same general education school in Spain participated in the study. The research followed a single-subject, multiple probe across participants design. The results of the study showed that the participants improved their performance on generalization tasks in functional contexts when using multiple representations. These improvements were maintained over time and transferred to everyday situations involving regularities. Social validity data collected from students' families and teachers indicated that both groups had positive perceptions of students' attitudes and motivation after the study. The implications of the results for promoting early algebraic thinking in students with ASD are discussed.

Given the relevance of graphs of functions, we consider their inclusion in primary education from the functional approach to early algebra. The purpose of this article is to shed some light on the students’ production and reading of graphs when they solved generalization problems from a functional thinking approach. We aim to explore how 3rd and 4th graders construct graphs associated to functions and what elements they use; and how they read function associated graphs and whether they connect pairs of values to see beyond the data. After four working sessions about functions, we designed and implemented individual interviews to 12 students. Through a qualitative analysis, we highlight that the students can read data in a graph on two different cognitive levels and also construct it from different elements of the graph initially provided. Regarding data reading, we evidence two levels: (a) literal reading of a given element in the graph, and (b) reading beyond the data. The construction of the graph is described with base on the axes, values and labels on the axes, scale of the axes, and construction techniques. We present examples of students’ work that evidence that graph construction varied depending on whether it was created from a blank sheet or it was necessary to provide help regarding the axes or the scale of the graph. We describe several techniques used by the students in the representation of data that yield non-canonical representations of a graph and that help glimpse how students are interpreting this representation.

Within the context of the functional approach inherent in early algebra education, this paper reports the structures recognized and the generalizations shown by second-grade students (aged 7-8 years). Semi-structured interviews were conducted with 4 students selected from a cohort participating in a previous classroom teaching experiment. These students were tasked with solving word problems that required understanding of both the direct and inverse forms of linear functions. This paper details the structures recognized by the students both when working with specific cases and when attempting generalization. All students successfully identified the structures involved in both the direct and inverse forms of the function. Differences in the generalized structures were observed depending on the function and its form. A greater number of students was able to generalize structures for the inverse form than for the direct form of the functions examined. This study is presented as part of a broader research project at XXXX.

The aim of this study was to determine how 5-year-old children identified the functional relationship of correspondence, and whether or not they generalized when working on a task that involved programmable robots. We conducted this study with 15 children (9 girls and 6 boys) in their last year of preschool education. The study was designed around a generalization task that involved the function f(n)=n+2. Our findings indicate that nine of the children identified the correspondence relationship and five children generalized, three of them correctly.

This work is part of a survey on functional thinking at an early age carried out in Spain (www.pensamientoalgebraico.es). We present part of the results obtained in a study carried out with first grade students (6-7 years). The general objective of this research is to describe the work of these students when solving a task that involves the function f(n)=3n. We focus on the representations, strategies, and structures they demonstrate. We highlight the formation of groups used by some students, the predominance of pictorial representation and the diversity of strategies in solving the task.

Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraictraining, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter.

This paper is part of a broader research initiative examining algebraic thinking. It presents a case study involving a 4 year-old female student, examining her functional perspective through individual work. Our interest lies in early childhood functional thinking. Specifically, our research objective is to describe how the child completed two tasks involving linear functions and their respective inverse functions. We posed questions related to specific near numbers, specific far numbers, and the generalizations themselves. We gathered data from her individual written work on the two tasks and from an individual interview. We analyzed the strategies she employed to generalize, the representations and generalizations she made, and how she established relationships between variables. We observed that she used pictorial and verbal representations to complete the proposed tasks and successfully achieved generalization. As a strategy, she counted drawings; however, when verbalizing her approach, she created groups of similar elements and innately distributed them into equal groups when working with the inverse function.

Este trabajo se enmarca en una investigación sobre pensamiento funcional en primeras edades realizada en España (www.pensamientoalgebraico.es). Presentamos parte de los resultados obtenidos en un estudio realizado con alumnado de primero de primaria (6-7 años). El objetivo general de esta investigación es describir el trabajo de estos alumnos al resolver una tarea que involucra a la función f(n)=3n. Nos centramos en las representaciones, las estrategias y las estructuras que evidencian. Destacamos la conformación de grupos utilizada por algunos alumnos, el predominio de la representación pictórica y la amplia gama de estrategias en la resolución de la tarea.