The goal of the study reported here is to gain a better understanding of the role of belief systems in the approach phase to mathematical problem solving. Two students of high academic performance were selected based on a previous exploratory study of 61 students 12-13 years old. In this study we identified different types of approaches to problems that determine the behavior of students in the problem-solving process. The research found two aspects that explain the students' approaches to problem solving: (1) the presence of a dualistic belief system originating in the student's school experience; and (2) motivation linked to beliefs regarding the difficulty of the task. Our results indicate that there is a complex relationship between students' belief systems and approaches to problem solving, if we consider a wide variety of beliefs about the nature of mathematics and problem solving and motivational beliefs, but that it is not possible to establish relationships of causality between specific beliefs and problem-solving activity (or vice versa).
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.
El objetivo de esta investigación es indagar en el conocimiento especializado de contenido matemático de estudiantes para maestro (EPM) sobre problemas de división-medida. Los EPM respondieron a dos cuestionarios en los que resolvieron dos problemas de división-medida e interpretaron respuestas dadas por alumnos de primaria. Un alto porcentaje de EPM resolvieron con éxito los dos problemas, pero pocos fueron capaces de interpretar de forma adecuada las respuestas de los alumnos de primaria cuando empleaban un procedimiento correcto alternativo a la división. Los resultados obtenidos ofrecen a los formadores referencias iniciales acerca de los conocimientos de los EPM sobre problemas de división-medida y sobre el uso que hacen de estos conocimientos en una de sus tareas profesionales: interpretar las respuestas de los alumnos.
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.
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