Primary school teacher’s noticing of students’ mathematical thinking in problem solving

Professional noticing of students’ mathematical thinking in problem solving involves the identification of noteworthy mathematical ideas of students’ mathematical thinking and its interpretation to make decisions in the teaching of mathematics. The goal of this study is to begin to characterize pre-service primary school teachers’ noticing of students’ mathematical thinking when students solve tasks that involve proportional and non-proportional reasoning. From the analysis of how pre-service primary school teachers notice students’ mathematical thinking, we have identified an initial framework with four levels of development. This framework indicates a possible trajectory in the development of primary teachers’ professional noticing.

An evolutive approach for the delineation of local labour markets

This paper presents a new approach to the delineation of local labour markets based on evolutionary computation. The main objective is the regionalisation of a given territory into functional regions based on commuting flows. According to the relevant literature, such regions are defined so that (a) their boundaries are rarely crossed in daily journeys to work, and (b) a high degree of intra-area movement exists. This proposal merges municipalities into functional regions by maximizing a fitness function that measures aggregate intra-region interaction under constraints of inter-region separation and minimum size. Real results are presented based on the latest database from the Census of Population in the Region of Valencia. Comparison between the results obtained through the official method which currently is most widely used (that of British Travel-to-Work Areas) and those from our approach is also presented, showing important improvements in terms of both the number of different market areas identified that meet the statistical criteria and the degree of aggregate intra-market interaction.

An application of fuzzy logic for hardware/software partitioning in embedded systems

Hardware/Software partitioning (HSP) is a key task for embedded system co-design. The main goal of this task is to decide which components of an application are to be executed in a general purpose processor (software) and which ones, on a specific hardware, taking into account a set of restrictions expressed by metrics. In last years, several approaches have been proposed for solving the HSP problem, directed by metaheuristic algorithms. However, due to diversity of models and metrics used, the choice of the best suited algorithm is an open problem yet. This article presents the results of applying a fuzzy approach to the HSP problem. This approach is more flexible than many others due to the fact that it is possible to accept quite good solutions or to reject other ones which do not seem good. In this work we compare six metaheuristic algorithms: Random Search, Tabu Search, Simulated Annealing, Hill Climbing, Genetic Algorithm and Evolutionary Strategy. The presented model is aimed to simultaneously minimize the hardware area and the execution time. The obtained results show that Restart Hill Climbing is the best performing algorithm in most cases.

A memetic algorithm for the delineation of local labour markets

Given a territory composed of basic geographical units, the delineation of local labour market areas (LLMAs) can be seen as a problem in which those units are grouped subject to multiple constraints. In previous research, standard genetic algorithms were not able to find valid solutions, and a specific evolutionary algorithm was developed. The inclusion of multiple ad hoc operators allowed the algorithm to find better solutions than those of a widely-used greedy method. However, the percentage of invalid solutions was still very high. In this paper we improve that evolutionary algorithm through the inclusion of (i) a reparation process, that allows every invalid individual to fulfil the constraints and contribute to the evolution, and (ii) a hillclimbing optimisation procedure for each generated individual by means of an appropriate reassignment of some of its constituent units. We compare the results of both techniques against the previous results and a greedy method.

El desarrollo de la competencia de estudiantes para profesor de matemáticas de educación secundaria en identificar la comprensión de la derivada en estudiantes de bachillerato

Esta investigación estudia el efecto de un módulo de enseñanza sobre la manera en la que estudiantes para profesor de matemáticas de educación secundaria identifican la comprensión de la derivada en estudiantes de Bachillerato. Los resultados indican que el desarrollo de esta competencia está vinculada a los elementos matemáticos de la noción de derivada que los estudiantes para profesor son capaces de considerar al identificar evidencias de la comprensión de la derivada en las respuestas de los estudiantes e interpretarlas.

Flexibilidad en la resolución de problemas de identificación de patrones lineales en estudiantes de educación secundaria

El objetivo de este trabajo es caracterizar la flexibilidad, entendida como habilidad para modificar la estrategia de resolución de un problema cuando se modifica la demanda de la tarea, de estudiantes de educación secundaria (12-16 años) en problemas de reconocimiento de patrones con varios apartados. Se utiliza una metodología de tipo cualitativo analizando las respuestas de los estudiantes en base a dos criterios: corrección de las respuestas y estrategias de resolución, y agrupando las que presentan características semejantes. Los resultados indican tres perfiles de estudiantes en relación a la flexibilidad en el uso de estrategias y el éxito alcanzado. El primero agrupa a los estudiantes que usan sólo la estrategia recursiva; la mayor parte de ellos se bloquea al aumentar la demanda cognitiva de la tarea; predominan los estudiantes de 12-13 años. El segundo perfil corresponde a los que cambian de una estrategia recursiva a una aproximación proporcional dando un resultado incorrecto; es más frecuente en los estudiantes de 13-14 años. Finalmente, el tercer perfil agrupa a los estudiantes que al aumentar la demanda cognitiva de la tarea cambian con éxito de una estrategia recursiva a una funcional; su frecuencia aumenta con la edad. Se concluye que la flexibilidad necesaria para identificar patrones cuando se incrementa la demanda de la tarea está relacionada con los conocimientos de los estudiantes y con el control y la regulación del proceso de resolución. Por otra parte, los estudiantes más jóvenes manifestaron menor grado de flexibilidad que los más mayores.