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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – CAPES This research is part of the field of training mathematics teachers to teach vector geometry (vector, properties and justifications), having the Anthropological Theory of Didactics (TAD) as a theoretical framework, and students of the Master's Degree in Education as research participants. Mathematics of Peru. The research aims to investigate how a device of Study and Research Pathways for Teacher Education (PEP-FP) can contribute, in particular, for this group of teachers in training to have a critical view, to question, analyze, design and experiment teaching processes with respect to the vector mathematical object to introduce vector geometry. The PEP-FP device shows the transition between monumentalism and visits of works to questioning the world. For which it has a qualitative research methodology based on Didactic Engineering, which has four stages: the first stage is the preliminary analysis in which the praxeological analysis and the study of the three dimensions of the didactic problem (epistemological, economic and and also here the construction of the epistemological reference model was made considering the praxeology that has as its reason for being the vector mathematical object, the second stage is the design and analysis a priori, in which the proposal of the PEP- FP, the third is the in vivo analysis considering the fundamental dialectics and the fourth step is the a posteriori analysis. As a result, we have that the PEP-FP contributed to the formation of this group of master's students who, at the end, proposed new generating questions to introduce them to the study of vector geometry, in addition to the methodology for the study of the mathematical object. The research methodology used in this work allowed the conditions for a connection between the Epistemological Reference Model and the PEP – FP and brought contributions to the Current Epistemological Model for teaching the introduction of vector geometry in teacher education Esta pesquisa se insere no campo da formação de professores de Matemática para o ensino da geometria vetorial (vetor, propriedades e justificações), tendo como quadro teórico a Teoria Antropológica do Didático e, como participantes da pesquisa, estudantes de mestrado de Ensino de Matemática no Peru. A pesquisa tem como objetivo investigar como um dispositivo de Percurso de Estudo e Pesquisa para Formação de Professores (PEP-FP) pode contribuir, em particular para esse grupo de professores em formação, para que tenham uma visão crítica, para que possam questionar, analisar, desenhar e experimentar processos de ensino com respeito ao objeto matemático vetor para assim poderem introduzir a geometria vetorial. O dispositivo PEP-FP mostra a transição do monumentalismo e das visitas das obras para o questionamento do mundo. E, para tanto, usa como metodologia de pesquisa do tipo qualitativa baseada na Engenharia Didática, e que possui quatro etapas: a primeira etapa é a análise preliminar, que é praxeológica e o estudo das três dimensões do problema didático (epistemológica, econômica e ecológica) e também aqui se fez a construção do modelo epistemológico de referência considerando as praxeologias que têm como razão de ser o objeto matemático vetor; a segunda etapa é o desenho e análise a priori, em que se planeja a proposta do dispositivo de PEPFP; a terceira é a análise in vivo considerando as dialéticas fundamentais e a quarta etapa é a análise a posteriori. Como resultado temos que o PEP-FP contribuiu na formação deste grupo de estudantes de mestrado que, ao final, propuseram novas questões geratrizes para introduzir o estudo da geometria vetorial, além da metodologia para o estudo do objeto matemático. A metodologia de pesquisa utilizada neste trabalho permitiu criar as condições para uma conexão entre o Modelo Epistemológico de Referência e o PEP-FP e trouxe contribuições para o Modelo Epistemológico Vigente para o ensino da introdução da geometria vetorial na formação de professores

Este artículo se encuentra disponible en la siguiente URL: https://www.mdpi.com/2227-7390/9/17/2117 Este artículo pertenece al número especial "Research on Teaching and Learning Mathematics in Early Years and Teacher Training". Early algebra proposes to incorporate algebra in primary school from the first years of schooling. The success of this incorporation depends, to a large extent, on the training of teachers, so the objective of this article is to study whether the degree of algebraic knowledge of student teachers at the beginning of their training. To conduct this, 106 preservice teachers were given a questionnaire. This survey is based on two daily life situations. They had to propose tasks to develop algebraic reasoning in primary school students. Most of the participants designed tasks in which they assigned specific values to the indeterminate ones and solved them arithmetically. In this way, they transformed open situations and numerous opportunities to promote algebraic thinking in students through the generalization and representation of relationships and functions into closed single-solution problems that do not promote algebraic thinking. We can see from the results that the participants’ algebraic knowledge is insufficient. Therefore, it is necessary to include in their training process the programs and experiences that will allow them to design tasks in order to detect and promote algebraic thinking in their future students. Sequences of tasks are presented to develop both situations by generalizing and representing relationships and functions, which can serve as a starting point for future training programs and experiences.

El objetivo de esta investigación es estudiar la relación entre el conocimiento matemático y la competencia docente “mirar profesionalmente el pensamiento matemático de los estudiantes” en el contexto de la generalización de patrones. Para ello se pidió a 40 estudiantes para maestros (EPM) que resolvieran un problema de generalización de patrones y que describieran e interpretaran las soluciones de tres alumnos de Educación Primaria al mismo problema. La resolución del problema permitió determinar el grado de conocimiento de los EPM y la capacidad para interpretar la comprensión de los alumnos a partir de la identificación de los elementos matemáticos significativos en las respuestas de los alumnos permitió determinar el grado de competencia. Del análisis de las destrezas identificar e interpretar se generaron descriptores para caracterizar cuatro perfiles en el desarrollo de la competencia, que posteriormente se refinaron incorporando descriptores del conocimiento. La investigación evidenció que los EPM con nivel bajo de conocimiento, y algunos con un nivel suficiente, no eran capaces de interpretar la comprensión de los alumnos de Educación Primaria, por lo que, aunque el conocimiento matemático del contenido es necesario para tener una mirada profesional, ese conocimiento no garantiza la competencia docente. Los materiales utilizados y la trayectoria inferida pueden servir de referencia para elaborar módulos de enseñanza sobre la competencia “mirar profesionalmente el pensamiento matemático de los estudiantes” en el contexto de la generalización de patrones. The aim of this research is to study the relationship between mathematical knowledge and teacher competence of professional noticing of children’s mathematical thinking in the context of generalization of patterns. For this purpose, 40 prospective teachers (PPT) were asked to solve a problem of generalization of patterns and to describe and interpret the answers of three elementary students to the same problem. The resolution of the problem allowed to determinate the degree of knowledge of the PPT and the ability to interpret the students’ comprehension from the identification of the significant mathematical elements in the students’ answers allowed to determine the degree of competence. From the analysis of the skills to identify and interpret were generated descriptors to characterize four profiles in the development of the competence, which were later refined by incorporating knowledge descriptors. The research evidenced that the PPT with low level of knowledge, and some with a sufficient level of knowledge, were not able to interpret the comprehension of the elementary students, therefore, although the mathematical knowledge of the content is necessary to have a professional noticing, this knowledge does not guarantee the teaching competence. The materials used and the inferred trajectory can be used as a reference to compose teaching modules about the competence of professional noticing of children’s mathematical thinking in the context of generalization of patterns. Esta investigación ha recibido el apoyo en parte del Proyecto I+D+i, EDU2014-54526-R y de EDU2017-87411-R, financiados por el Ministerio de Economía y Competitividad (MINECO), Gobierno de España.

Este artículo se encuentra disponible en la página web de la revista en la siguiente URL: https://revistas.ucm.es/index.php/RCED/article/view/55070/4564456548257 El objetivo de esta investigación es estudiar la relación entre el conocimiento matemático y la competencia docente “mirar profesionalmente el pensamiento matemático de los estudiantes” en el contexto de la generalización de patrones. Para ello se pidió a 40 estudiantes para maestros (EPM) que resolvieran un problema de generalización de patrones y que escribieran e interpretaran las soluciones de tres alumnos de Educación Primaria al mismo problema. La resolución del problema permitió determinar el grado de conocimiento de los EPM y la capacidad para interpretar la comprensión de los alumnos a partir de la identificación de los elementos matemáticos significativos en las respuestas de los alumnos permitió determinar el grado de competencia. Del análisis de las destrezas identificar e interpretar se generaron descriptores para caracterizar cuatro perfiles en el desarrollo de la competencia, que posteriormente se refinaron incorporando descriptores del conocimiento. La investigación evidenció que los EPM con nivel bajo de conocimiento, y algunos con un nivel suficiente, no eran capaces de interpretar la comprensión de los alumnos de Educación Primaria, por lo que, aunque el conocimiento matemático del contenido es necesario para tener una mirada profesional, ese conocimiento no garantiza la competencia docente. Los materiales utilizados y la trayectoria inferida pueden servir de referencia para elaborar módulos de enseñanza sobre la competencia “mirar profesionalmente el pensamiento matemático de los estudiantes” en el contexto de la generalización de patrones. / The aim of this research is to study the relationship between mathematical knowledge and teacher competence of professional noticing of children’s mathematical thinking in the context of generalization of patterns. For this purpose, 40 prospective teachers (PPT) were asked to solve a problem of generalization of patterns and to describe and interpret the answers of three elementary students to the same problem. The resolution of the problem allowed to determinate the degree of knowledge of the PPT and the ability to interpret the students’ comprehension from the identification of the significant mathematical elements in the students’ answers allowed to determine the degree of competence. From the analysis of the skills to identify and interpret were generated descriptors to characterize four profiles in the development of the competence, which were later refined by incorporating knowledge descriptors. The research evidenced that the PPT with low level of knowledge, and some with a sufficient level of knowledge, were not able to interpret the comprehension of the elementary students, therefore, although the mathematical knowledge of the content is necessary to have a professional noticing, this knowledge does not guarantee the teaching competence. The materials used and the inferred trajectory can be used as a reference to compose teaching modules about the competence of professional noticing of children’s mathematical thinking in the context of generalization of patterns.

Este artículo se encuentra disponible en la página web de la revista en la siguiente URL: https://www.revista.uclm.es/index.php/ensayos/article/view/1591/pdf El objetivo de esta investigación es estudiar cómo estudiantes para maestro (EPM) proponen acciones para mejorar el proceso de enseñanza‐aprendizaje de alumnos de Primaria a partir del análisis de sus respuestas a un problema de generalización de patrones. Las acciones propuestas por los EPM se clasificaron en tres categorías: actitudinales, genéricas y concretas. Los resultados muestran que algunos EPM se limitan a hacer comentarios sobre la actitud de los alumnos, que los EPM proponen más acciones a los alumnos que tienen dificultades en la resolución de los problemas y que pocos EPM son capaces de plantear acciones de forma concreta especificando una tarea. Como conclusión se reconoce que uno de los aspectos a mejorar en la formación y desarrollo profesional de maestros es la capacidad para proponer acciones adecuadas para mejorar la enseñanza‐aprendizaje de los alumnos. / The aim of this research is to study how Prospective Primary Teachers (PPT) propose actions to improve the teaching‐learning process of Primary students through the analysis of real answers to patterns generalization problems. The actions proposed by the PPT were classified into three categories: attitudinals, generics and concretes. In the results it was observed that some PPTs are limited to commenting on the attitude of the students, that the PPTs propose more actions to the students that have difficulties in the resolution of the problems and that few PPTs are able to plant actions in a concretes way specifying a task. In conclusion, it is recognized that one of the aspects to improve in the training and professional development of teachers is the ability to propose appropriate actions to improve the teaching‐learning of the students.