The objective of this study was to characterise prospective kindergarten teachers' development of noticing children's thinking about length and its measurement. We used the concepts of instrumental genesis and learning trajectories to identify the ways in which prospective kindergarten teachers used a learning trajectory to learn to notice children's mathematical thinking. Following a teaching experiment, we identified three ways in which prospective kindergarten teachers used the learning trajectory to notice children's mathematical thinking. Two instrumented action schemes supported these ways of using the learning trajectory: a scheme taking into account the mathematics learning progression to interpret children's answers, and a scheme for proposing instructional tasks based on the interpretation of children's mathematical thinking. Approaching the development of noticing as an appropriation process of a learning trajectory helps us to understand prospective teachers' difficulties at endowing meaning to a learning trajectory's conceptual structure. We suggest that these ways of using learning trajectory knowledge to interpret children's mathematical thinking and to make instructional decisions can be understood as an instrumentation process which reveals how noticing skills develop.

El objetivo de este trabajo es caracterizar el uso de una trayectoria de aprendizaje de la longitud y su medida como instrumento conceptual para favorecer la adquisición de la competencia docente “mirar profesionalmente” el pensamiento matemático de los niños. Participaron 64 estudiantes de la carrera de maestro de educación infantil que siguieron un módulo de enseñanza. Este módulo estaba articulado en torno a una trayectoria de aprendizaje sobre la longitud y su medida, con objeto de que los futuros maestros la usaran como instrumento conceptual para describir e interpretar las respuestas de niños de educación infantil a situaciones de enseñanza-aprendizaje (primer esquema de acción instrumental) y proponer tareas en función de la comprensión inferida (segundo esquema de acción instrumental). Los resultados muestran que hay estudiantes que se preparan para ser maestros que no desarrollan ningún esquema de acción instrumental, y otros desarrollan el primer esquema y algunos, ambos; esto proporciona evidencias de la adquisición de la competencia arriba señalada.

No abstract

The goal of this study was to characterise how third-grade primary school students understand the concept of polygon as well as relationships between polygons. Following a teaching experiment, students answered a questionnaire on the recognition of polygons and relationships between polygons, providing information about how a polygon was understood as an example of a class. Dimensional deconstruction of shapes (Duval, 2017) and Statistical Implicative Analysis were used to analyse students' answers. The findings indicated that students' understanding of the polygon concept depended on how students recognised and modified the relevant attributes considered in the definition of polygon. These results suggest that a progressive complexity underlies the understanding of the concept of polygon. Evidence of this progressive complexity was found in the relationship between recognising a figure as a polygon and transforming a non-example of a polygon into a polygon. Furthermore, the ability to identify a polygon as an instance of a class depended on the attribute that defines the class. That is, the fact of identifying a class of polygons was linked to the non-relevant attribute considered. Instructional implications are finally drawn regarding the key role of dimensional deconstruction to understand polygons and relationships between polygons.