This chronicle begins with a journey through the Balkans. It draws on literary admiration, on careful observation of cultural dynamics, especially what is perceived in Bulgaria. The nervous system of the chronicle concentrates the deep respect and enthusiasm for the author Elias Canetti, born in Ruse, Bulgaria and Nobel Prize winner for Literature. The chronicle also traces the author's diaspora, between cultures and languages, and, in addition, in a particular way, explores as a corollary the Bulgarian way of looking at things, which is direct and very human, without being uncomfortable or intrusive. In parallel, the chronicle reflects on the complex ethnic identities in the Balkan region, with their sensitive and unsettling histories, as well as, a very important part, the idiomatic and linguistic nuances. The apocryphal interview of the Mexican writer José María Pérez Gay with Elías Canetti is added.

Este artículo estudia los sentidos que futuras profesoras de matemática atribuyen a la validación antes y después de participar en un escenario de modelización matemática. La investigación es cualitativa, interpretativa y de diseño. El estudio revela que las futuras profesoras amplían y transforman sus sentidos sobre la validación, pasando de una concepción centrada en la demostración axiomático-deductiva a una comprensión que integra diversas formas de validación, entendidas como parte de un proceso social y complejo. Se enfatiza la importancia de abordar la validación como contenido de enseñanza y como medio para validar en la formación inicial docente.

[Objective] This paper aims to describe textbook tasks from the perspective of content analysis to demonstrate the relationship between the book and the curriculum in Costa Rica. [Methodology] This study is limited to the publishers Grupo Nación and Editorial Santillana; therefore, 88 tasks were selected from books from these publishers. A qualitative paradigm is used in combination with a descriptive approach. The bibliographic review conducted helps define the analysis categories that guide the data collection and the subsequent analysis. [Results] Some of the results helped to contrast the textbook proposal with the Ministry’s proposal. Both proposals strongly agree on the content of patterns. However, there were discrepancies in the use of mathematical concepts according to the modes of interpretation. The study revealed the absence of tasks enabling the creation or duplication of patterns or math tasks related to the application of mathematics in social or scientific environments. [Conclusions] This type of research highlights the importance of teacher training by analyzing tasks in response to current demands, having an impact both on teaching planning and on the skills that must be taught to students to achieve quality mathematics education. It is essential to consider, in teaching planning, those tasks that, according to the results, are absent from the textbook.

[Objective] This paper aims to describe textbook tasks from the perspective of content analysis to demonstrate the relationship between the book and the curriculum in Costa Rica. [Methodology] This study is limited to the publishers Grupo Naci & oacute;n and Editorial Santillana; therefore, 88 tasks were selected from books from these publishers. A qualitative paradigm is used in combination with a descriptive approach. The bibliographic review conducted helps define the analysis categories that guide the data collection and the subsequent analysis. [Results] Some of the results helped to contrast the textbook proposal with the Ministry's proposal. Both proposals strongly agree on the content of patterns. However, there were discrepancies in the use of mathematical concepts according to the modes of interpretation.The study revealed the absence of tasks enabling the creation or duplication of patterns or math tasks related to the application of mathematics in social or scientific environments. [Conclusions] This type of research highlights the importance of teacher training by analyzing tasks in response to current demands, having an impact both on teaching planning and on the skills that must be taught to students to achieve quality mathematics education. It is essential to consider, in teaching planning, those tasks that, according to the results, are absentfrom the textbook.

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.

This paper is part of broader research being conducted in the area of algebraic thinking in primary education. Our general research objective was to identify and describe generalization of a 2nd grade student (aged 7–8). Specifically, we focused on the transition from arithmetic to algebraic generalization. The notion of structure and its continuity in the generalization process are important for this transition. We are presenting a case study with a semi-structured interview where we proposed a task of contextualized generalization involving the function y = x + 3. Special attention was given to the structures evidenced and the type of generalization expressed by the student in the process. We noted that the student identified the correct structure for the task during the interview and that he evidenced a factual type of algebraic generalization. Due to the student’s identification of the appropriate structure and the application of it to other different particular cases, we have observed a transition from arithmetic thinking to algebraic thinking.

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 adopted a functional perspective on algebra. Our focus was on tables and how they were first used by second-grade elementary school students (7- and 8-year-olds) when working with functions. This qualitative, exploratory, and descriptive study consisted of five classroom sessions and two semi-structured interviews, one group and one individual, two weeks apart, focused on tasks involving different functions and uses of tables, with and without labels for headings. In this study, we ask the following research questions: How do children organise values in tables with or without (a priori) labels for headings? What are the regularities (structures) identified by students? Our data revealed that students were able to organise the values of variables by listing them in columns and labelling the headings (i.e., identifying the variables involved). The ways in which children organised the data in tables enabled us to identify the structures they identified as regularities between the variables involved in the functions. More structures were correctly identified in the second interview compared to the first.

Este trabajo es parte de una investigación más amplia que se desarrolla en el ámbito del pensamiento algebraico de estudiantes de educación primaria en España. Nos centramos aquí en identificar las estructuras (regularidades identificadas) y las representaciones que aparecen cuando los estudiantes trabajan con tareas de generalización. Involucramos funciones lineales en un estudio de caso dentro del experimento de enseñanza con tres estudiantes de 2º de educación primaria (7-8 años). Destacamos que el número de estructuras y la forma de generalizar la estructura dependen de las tareas planteadas en cada caso, según la función implicada. Las generalizaciones de todos los estudiantes se han representado mediante representaciones verbales y/o numéricas.

Este libro es un homenaje a los profesores Pablo Flores e Isidoro Segovia, del Departamento de Didáctica de la Matemática de la Universidad de Granada. Su tesón, su compromiso con la calidad y la innovación en la docencia superior y en la investigación en Educación Matemática, y, sobre todo, su empatía y cordialidad con todos sus compañeros durante su etapa de gestión, han sido un ejemplo para todos nosotros. Más aún, su contribución a la organización y la gestión de la Facultad de Ciencias de la Educación ha sido una constante en sus vidas profesionales. Y a todo esto se le añade una sostenida experiencia en centros educativos, en aulas de Educación Primaria y Secundaria, en los centros de profesorado y en la formación de profesores de matemáticas en ejercicio. Todas esas actividades están representadas de una forma u otra en este libro, cuyos capítulos abordan las líneas de trabajo de estos dos profesores. Pablo Flores ha desarrollado su actividad universitaria en torno a la formación de profesores, a los componentes de su conocimiento y desarrollo profesional y a sus sistemas de creencias. La de Isidoro Segovia se ha centrado en los procesos de estimación y cálculo, la resolución e invención de problemas, la noción de currículo y la formación de profesores. En este libro participan 44 autores de 15 instituciones educativas de España, Chile, Colombia y Costa Rica. El objetivo de esta obra es dedicarles un merecido reconocimiento por su trabajo desde el cariño, la admiración y el respeto más profundos.

This work is part of a broader investigation that is being developed in the field of algebraic thinking of elementary school students in Spain. We focus here on identifying the structures (identified regularities) and representations that appear during the generalization process of some students when students work with generalization tasks. For this purpose, we implemented generalization tasks involving linear functions in a case study within a teaching experiment with three students in the 2nd year of primary school (7-8 years old). We emphasize that the number of structures and the way of generalizing the structure depend on the tasks posed in each case. The generalizations of all students have been represented by verbal and/or numerical representations.

El objetivo de este estudio ha sido identificar y comparar las estructuras que evidencian los estudiantes de educación primaria en las formas directa e inversa de una función, tanto para el trabajo con casos particulares como en la generalización desde un enfoque funcional del early algebra. El estudio que se lleva a cabo es de tipo cualitativo y de carácter exploratorio y descriptivo. Se diseña una tarea contextualizada que involucra la función lineal y=x+4, en sus formas directa e inversa. Los seis estudiantes participantes de este estudio de educación primaria (7-8 años) trabajaron la tarea durante entrevistas semiestructuradas que se desarrollaron en el curso académico 2017/2018. Los estudiantes proceden de un colegio de Granada (España). Se describen las estructuras evidenciadas en ambas formas de la función y tanto en el trabajo con casos particulares como cuando se les pregunta por el caso general. Los seis estudiantes identificaron estructuras adecuadas de la forma directa de la función en al menos una ocasión durante la entrevista. En la forma inversa se observaron estructuras también adecuadas, pero hubo estudiantes que no respondieron o a los que no se les hicieron preguntas de esta parte. La mayoría de las estructuras generalizadas se evidenciaron al preguntarles explícitamente por la generalización tanto en la forma directa como en la forma inversa de la función. Al trabajar con la relación entre dos variables se identificaron diferencias entre las estructuras identificadas por el estudiantado en ambas formas de una función: directa e inversa. La mayor parte de las estructuras identificadas fueron adecuadas al problema y esto anima al trabajo con ambas formas de las funciones lineales en educación primaria.

This study is part of a broader study on algebraic reasoning in elementary education. The research objective of the present survey, namely to describe generalization among second grade (7- to 8-year-old) students, was pursued through semi-structured interviews with six children in connection with a contextualized generalization task involving the function y = x + 3. Particular attention was paid to the structures recognized and the type of generalization expressed by these students as they reasoned. In all six, we observed three phases of inductive reasoning: (a) abductive, (b) inductive and (c) generalization. The students correctly recognized the structure at least once during the interview and expressed generalization in three ways.

This study aimed to identify and compare the structures evidenced by primary school students in direct and inverse forms of a function, both working with particular cases and generalizing from an early algebra functional approach. The study was qualitative, exploratory, and descriptive. A contextualized task was designed involving the linear function y=x+4 in direct and inverse variations. Six 2nd graders (7-8 years of age) from a school in Granada (Spain) participated in the study performing a designed task during semi-structured interviews conducted in the 2017-2018 school year. We described the structures evidenced in both variations with particular cases and the general case. All six students identified adequate structures in the direct variation of the function at least once during the interview. Adequate structures were also observed in the inverse variation. However, some students did not respond to this section or were not asked these questions. The majority of the structures that students generalized were produced when explicitly asked for generalization, in both direct and inverse variations of the function. When using the relationship between two variables, differences were found between structures identified by students in both direct and inverse variations. Most of the structures identified were adequate for the problem, which encourages work with both variations of linear functions in primary education.