Maria del Mar Liñán García

This article aims to identify elements of the specialized knowledge mobilized by future teachers of Early Childhood Education in initial training on algebraic thinking when they develop a professional formative classification task. The selection was made because we believe that classification is a learning topic in Early Childhood Education that can foster logical-mathematical thinking, which in turn allows us to reflect on and develop algebraic thinking by considering the relationships that are evident in the classification process. Therefore, the theoretical framework we used for this analysis is Mathematics Teacher’s Specialized Knowledge (MTSK). The methodology used was a Teaching Experiment, applied in two initial training classes for early childhood education teachers at the University of Seville. Our results suggest the possibility of advancing discussions about the knowledge early childhood education teachers need regarding the specificities of classification.

This article aims to identify elements of the specialized knowledge mobilized by future teachers of Early Childhood Education in initial training on algebraic thinking when they develop a professional formative classification task. The selection was made because we believe that classification is a learning topic in Early Childhood Education that can foster logical-mathematical thinking, which in turn allows us to reflect on and develop algebraic thinking by considering the relationships that are evident in the classification process. Therefore, the theoretical framework we used for this analysis is Mathematics Teacher’s Specialized Knowledge (MTSK). The methodology used was a Teaching Experiment, applied in two initial training classes for early childhood education teachers at the University of Seville. Our results suggest the possibility of advancing discussions about the knowledge early childhood education teachers need regarding the specificities of classification.

La investigación, la práctica profesional y la formación inicial de maestros deben estar estrechamente conectadas. Proponemos el uso de tareas formativas para desarrollar el conocimiento especializado del profesor de matemáticas sobre geometría, que diseñamos desde situaciones reales de aula y validamos desde la investigación con un experimento de enseñanza. El modelo analítico del conocimiento especializado del profesor de matemáticas permite identificar situaciones de la práctica real y analizar el conocimiento movilizado en un aula de formación inicial al resolver dichas tareas. La reflexión que conlleva esta resolución hace que los estudiantes para maestro establezcan relaciones entre elementos del conocimiento especializado del profesor, generando su comprensión integral de la práctica.

Deepening the specialised knowledge of mathematics teachers is necessary for teaching, and it is one of our concerns as researchers; accordingly, in this study we analyze the knowledge that a teacher mobilizes when giving a first lesson on geometry to a class of fifth year of primary education at a Spanish school. We provide this analysis with a complementary perspective that leads us to broaden our focus of attention to other aspects of classroom management, thus identifying the specialised knowledge evoked in the researcher by the opportunities observed in the classroom. We take opportunities to mean classroom situations that occur as a result of the teacher’s management for which the researcher considers that an alternative management of such situations would be supported by relevant specialised knowledge for the teaching of the given content. Considering the possible alternative management of specific classroom situations makes this specialised knowledge emerge, which, together with that mobilized by the teacher, generates specialised knowledge useful for guiding the content of the training of primary education teachers.

Resumen: En este trabajo reflexionamos sobre el conocimiento profesional del profesor de Educación Infantil en relación con la resta. La atención al conocimiento deseable de este profesional es relativamente reciente, existiendo escasos estudios que aborden este tema. Partimos de la consideración de que este conocimiento es especializado; diferente al del profesor de Primaria o Secundaria; y su caracterización debe realizarse desde un enfoque centrado en la propia matemática. Esto nos lleva a considerar el modelo del Conocimiento Especializado del Profesor de Matemáticas (Mathematics Teachers’ Specialised Knowledge, MTSK) como herramienta teórica y analítica idónea para ofrecer una propuesta de elementos de conocimiento especializado que, a la luz de nuestra experiencia docente e investigadora, consideramos que deseables para este profesional en relación con la resta, incluyendo aspectos de conocimiento del contenido (Mathematical Knowledge, MK) y de conocimiento didáctico del contenido (Pedagogical Content Knowledge, PCK). La reflexión sobre esta propuesta nos lleva a destacar tres elementos principales que parecen caracterizadores de la naturaleza de este conocimiento: su densidad y cohesión, la profundidad de los conocimientos matemáticos implicados y su repercusión para construir un adecuado PCK; y la relevancia del conocimiento de las fases que los niños siguen en su proceso de comprensión del número. Se aportan sugerencias para la formación inicial y continua de estos profesionales.Palabras clave: Conocimiento especializado del profesor de matemáticas. MTSK. Profesor de Educación Infantil. Resta. Conocimiento matemático del contenido y conocimiento didáctico del contenido.KINDERGARTEN TEACHERS’ SPECIALISED KNOWLEDGE ON SUBTRACTION In this paper we reflect upon the early childhood teachers’ professional knowledge in relation to subtraction. Paying attention to this professional desirable knowledge is relatively recent, and thus there are few studies that address it. We assume as a starting point the consideration that this knowledge is specialized; differs from that of primary or secondary teachers’; and its characterization must be carried out from an approach centered on mathematics itself. This leads us to consider the model of Mathematics Teachers’ Specialised Knowledge (MTSK) as a suitable theoretical and analytical tool to offer a proposal of specialized knowledge elements that, in light of our teaching and research experience, we considered desirable for the work of teaching of these professionals in relation to subtraction. In this proposal we include aspects of Mathematical Knowledge (MK) and of Pedagogical Content knowledge (PCK). In the proposal we highlight three main elements that seem to characterize the nature of this knowledge: its density and cohesion; the depth of the mathematical knowledge involved and its impact to build an adequate PCK; and the relevance of knowledge of the phases that children follow in their process of understanding the number. Some recommendations for early years teachers’ education are provided.Keywords: Mathematics Teachers’ Specialised Knowledge. MTSK. Early childhood teacher. Subtraction. Mathematical knowledge. Pedagogical content knowledge.CONHECIMENTO ESPECIALIZADO DO PROFESSOR DA EDUCAÇÃO INFANTIL NO ÂMBITO DO TEMA DA SUBTRAÇÃO Resumo: Neste trabalho, refletimos sobre o conhecimento profissional do professor de Educação Infantil em relação à subtração. Considerar como foco de atenção o conhecimento desejável destes professores é algo relativamente recente, existindo poucos estudos que abordam esta questão. Partimos da consideração de que esse conhecimento é especializado; diferente do conhecimento do professor das etapas escolares seguintes; e sua caracterização deve ser realizada a partir de uma abordagem centrada na própria matemática. Nesse sentido consideramos a conceitualização do Mathematics Teachers’ Specialized Knowledge (MTSK) como uma ferramenta teórica e analítica para conceitualizar uma proposta de elementos constituintes do conhecimento especializado ideal do professor no âmbito que, à luz de nossa experiência de ensino e pesquisa, consideramos desejável para este profissional em relação à subtração. Nesta proposta incluímos aspectos do conhecimento do conteúdo e do conhecimento pedagógico do conteúdo (PCK). Nesta proposta destacamos três elementos principais que parecem caracterizar a natureza desse conhecimento: sua densidade e coesão; a profundidade do conhecimento matemático envolvido e seu impacto para construir um PCK adequado; e a relevância do conhecimento do professor das fases que as crianças seguem no seu processo de compreensão do número em etapas educativas posteriores. São fornecidas sugestões para a formação inicial e contínua de professores.Palavras chave: Conhecimento Especializado em Professores de Matemática. MTSK. Professor de Educação Infantil. Subtração. Conhecimento matemático do conteúdo e conhecimento pedagógico do conteúdo.

El profesor desempeña un papel crucial en la articulación de los Espacios de Trabajo Matemático. El modelo de conocimiento del profesor que aquí se presenta (denominado Mathematics Teacher’s Specialised Knowledge) propone una estructuración de dicho conocimiento basada en la noción de ‘especialización’. Esta estructuración, particularmente la consideración de sus componentes (subdominios), permite comprender el conocimiento que el profesor pone en juego en la transición entre los distintos Espacios (idóneo, personal y de referencia) de Trabajo Matemático, así como las relaciones entre estos.