This paper presents the Mathematics Teacher’s Specialised Knowledge (MTSK) model. It acknowledges earlier contributions to understanding and structuring teachers’ knowledge, in particular, the special debt owed to Shulman’s notion of pedagogical content knowledge and to Ball and collaborators’ Mathematical Knowledge for Teaching (MKT), influential for the specialised nature of one of its sub-domains. The authors’ research with teachers has led them to explore the characteristics of MKT and to refine the descriptors relating to its sub-domains, a task which has underlined the difficulty involved in unambiguously delimiting the boundaries which separate these. As a result, and taking into consideration a broader view of the specialised nature of the teacher’s mathematical knowledge, the authors propose a framework which, whilst respecting the major domains of Content Knowledge and Pedagogical Content Knowledge, regards the specialisation in respect of mathematical knowledge as a property which is inherent to the model and extends across all sub-domains.

: This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the sense of an adaptation of the knowledge in question. To promote a more holistic understanding, an alternative metaphor is offered: preparing mathematics for teaching as ecological engineering. By using the ecological engineering metaphor, the preparation of mathematics for teaching is presented as a two-sided process that involves both the adaptation of knowledge and the modification of its environment.

In this article, we analyse a lesson on Thales’s theorem in a Chilean secondary school classroom through the combination of two theories: Mathematics Teachers’ Specialised Knowledge (ThMTSK) and Mathematical Working Spaces (ThMWS). Both theories, first separately and then in relation to one another, are used to analyse two tasks proposed by the teacher in the classroom following a cross-methodology for networking of theories. Through a single case study research design, a content analysis of the transcript of the video recording of the lesson was conducted. The joint analysis of this lesson allows us to better understand the mathematical work taking place in the classroom. In particular, the results show the scope of each model and their complementarity through the detection of meeting points and blind spots, through the role of proof, representations, and the change between geometrical and numerical work in teaching Thales’s theorem. This allows for a deeper understanding of a teacher’s practice and teaching. Ultimately, relationships between the theoretical elements of both theories are established to show their complementarity. We conclude that networking between theories can contribute to the development of these theories by raising questions that involve examining their foundations and assumptions in greater depth.

Desde la educación matemática, la formación inicial del maestro debe promover la adquisición de competencias en la enseñanza y aprendizaje de las matemáticas, asumiendo adquiridas, al menos en gran parte, las relativas a la disciplina. Este objetivo se ve mermado por las numerosas y graves lagunas en conocimientos puramente matemáticos de los estudiantes para maestro que acceden a ella, que les impide implicarse en razonamientos didácticos específicos. En este artículo describimos las competencias numéricas deseables en los estudiantes para Maestro y, a través de un estudio exploratorio con alumnos de la Universidad de Huelva, indagamos sobre el grado de adquisición de algunas de ellas.

Mostramos un experimento de enseñanza en el que el análisis de vídeo supone el detonante para la construcción de elementos de conocimiento profesional, estructurados bajo la óptica del modelo de Conocimiento Especializado del Profesor de Matemáticas. Presentamos los resultados del análisis de la implementación de una tarea en un grupo completo de estudiantes para maestro, derivada de un vídeo en el que un maestro de 5º de Educación Primaria construye una definición de polígono. La evaluación de los logros se hará sobre la base de los elementos de conocimiento especializado identificados en una primera implementación de la tarea con tres estudiantes seleccionados ad hoc, en el contexto de un experimento de enseñanza uno a uno. Estos referentes contemplan tanto conocimiento matemático como didáctico del contenido, en particular conocimiento sobre la definición de polígono y sus elementos, la práctica matemática de definir, recursos y ejemplos para definir polígono, dificultades ligadas a dicho contenido, y su presencia en el currículo de primaria. Finalizaremos con reflexiones sobre cómo la tarea permite construir los elementos de conocimiento especializado pretendidos, así como con líneas de posibles futuros desarrollos de esta investigación.