This paper reports on different underlying structures of the derivative schema of three undergraduate students that were considered to be at the trans level of development of the derivative schema (action-process-object-schema). The derivative schema is characterized in terms of the students' ability to explicitly transfer the relationship between a function and its first derivative to the derivative function and the second derivative. This conscious shift of properties of derivatives is differently manifested by the students in the trans level of development of the derivative schema and can be considered evidence of the different characteristics of the thematization of derivative schema. From here we suggest that there are different underlying structures in the constructed schema due to the consciousness in which students use the relations between a function and its derivative.

Abstract. This research study examines the development of the ability of pre-service teachers to notice signs of students' understanding of the derivative concept. It analyses pre-service teachers' interpretations of written solutions to problems involving the derivative concept before and after participating in a teacher training module. The results indicate that the development of this skill is linked to pre-service teachers' progressive understanding of the mathematical elements that students use to solve problems. We have used these results to make some suggestions for teacher training programmes.

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This study focuses on how prospective teachers learn about students' mathematical thinking when (i) anticipating secondary students' answers reflecting different characteristics of understanding and (ii) propose new activities in relation to the classification of quadrilaterals. The data were collected from forty-eight prospective secondary school teachers enrolled in an initial training programme. The results indicate three changes in how the prospective teachers anticipate secondary students' answers in relation to the role given to a perceptual or relational perspective of the classification of quadrilaterals. These changes are described considering how prospective teachers grasp the students' understanding of the inclusive relation among quadrilaterals as a conceptual advance. We argue that prospective teachers' learning was promoted after participating in a structured environment where they had the opportunity to discuss how to recognize the features of student's understanding.

The aim of this work was to identify and characterize the levels of development of derivative schema. In order to do so, a questionnaire to 103 university students with previous instruction in Differential Calculus was applied. The questionnaire was composed of three tasks. For the identification of the levels of development of schema and their subsequent characterization, we consider the framework proposed by the APOS theory. In particular, this framework was operationalized through the establishment of 27 variables that allowed for the breakdown of the resolution protocols from the questionnaire into discrete elements. In this way, we obtained a vector associated with each of these variables. The identification of students assigned to each level of development of schema was carried out by a cluster analysis. Subsequently, we performed a statistical analysis of frequencies and implicative, with the 27 variables, which allowed to characterize the levels of development identified.

We summarize results obtained by the Didactics of Mathematics research group at the University of Alicante on the competence of professional noticing. The research focused on three issues over the last years: (i) characterizing how the skills that make up professional noticing interrelate; (ii) characterizing different degrees of competence development; and (iii) identifying contexts that support this competence development. Main results are described along with future challenges.

Resumen Este estudio analiza la importancia del tratamiento de puntos de no-derivabilidad en la tematización del esquema de derivada. Para ello, se consideró el marco propuesto por la teoría APOE, a través del uso y conexiones lógicas que los estudiantes universitarios establecen entre los elementos matemáticos y los modos de representación cuando resuelven tareas sobre derivada. Se utilizaron dos instrumentos: el primero fue un cuestionario conformado por tres tareas propuestas en distintos modos de representación, en cuya resolución era necesaria la utilización de los elementos matemáticos constitutivos del concepto de derivada; este instrumento fue aplicado a los 40 estudiantes participantes del estudio. El segundo instrumento correspondió a una entrevista clínica que se centró en el análisis de las respuestas de los estudiantes en relación con el tratamiento que realizan en los puntos de no-derivabilidad en derivadas sucesivas. Esta entrevista clínica se aplicó a 5 de los 9 estudiantes clasificados en el nivel de desarrollo Trans-derivada. Los resultados del análisis constatan que la coherencia del esquema es fundamental para identificar un esquema tematizado. Además, se observó el importante papel desempeñado por el análisis de los puntos de no-derivabilidad en la conexión y el tránsito entre derivadas de diferentes órdenes, especialmente desde el modo de representación gráfico, lo cual puede ser un factor que favorece la tematización del esquema de derivada.