Salvador Llinares

Secondary School Mathematics Teacher's Professional Knowledge: A case from the teaching of the concept of function

This study examines the relationships between a secondary Mathematics teachers pedagogical content knowledge and the dilemmas posed when teaching the concept of function. Alternative tools and procedures for analysing these relationships are used. The results illustrate the flexibility of her knowledge use in Mathematics teaching and how it is related to her knowledge of the pupils as Mathematics learners. In addition, it highlights the central role played by the teachers knowledge about pupil comprehension in the content and organisation of her teaching agenda. The difference that the teacher establishes between exercise and problem shows her struggle with the two-fold understanding-memorisation of procedures as the content of a teaching dilemma. The results obtained suggest the need for secondary Mathematics teacher education programmes to adopt approaches that take into account the central role played by teacher knowledge about Mathematics learning, and the dilemmas generated when teachers have to reconcile objectives that are apparently contradictory during their teaching practice.

Characteristics of Pre-Service Primary School Teachers’ Configural Reasoning

The goal of this study is to identify the characteristics of pre-service primary teachers’ configural reasoning, understood as the relationships between concepts and figures set to solve geometrical proof problems. Ninety-seven primary teachers were asked to solve two geometrical proof problems in which a geometrical figure was provided. The results suggest the existence of two levels of the pre-service primary teachers’ discursive organization. The first, when pre-service primary teachers link the geometrical facts to the figure by discursive apprehensions, and the second, when several geometrical facts are related by logical chains to infer new information. The identification of a relevant configuration and the way in which geometrical facts are logically organized from discursive apprehensions are key factors in the shift between these moments. We contend that these factors help to explain how the image component stimulates thought and how conceptual constraints control the formal rigor of the process.

De la estructura aditiva a la multiplicativa : efecto de dos variables en el desarrollo del razonamiento proporcional

Resumen Este estudio analiza la influencia de las relaciones multiplicativas entre las cantidades y su naturaleza discreta o continua en el desarrollo de la capacidad de los estudiantes de educación primaria de diferenciar situaciones aditivas y proporcionales. Los resultados han mostrado que la relación multiplicativa entre las cantidades influye significativamente en cómo los estudiantes de educación primaria identifican ambas situaciones. En cambio, la naturaleza continua o discreta de las cantidades sólo influye en cómo se resuelven los problemas proporcionales. Además, el análisis implicativo de los niveles de éxito en los problemas refleja la independencia en la manera en la que los estudiantes resuelven las situaciones aditivas y proporcionales. Estos resultados subrayan la complementariedad de la comprensión de las relaciones proporcionales y aditivas en el desarrollo de la conceptualización de la idea de razón como una nueva unidad.

Algunos indicadores del desarrollo del esquema de derivada de una función

The aim of this study is to identify some indicators of post-secondary students’ development of derivative schema. In order to identify these indicators we use intra, inter and trans levels of development of a schema provided by Piaget and Garcia. We argue that students’ flexible use of the logical equivalence between mathematical elements when solving a problem is an indicator of the development of the derivative schema. This characterization helps to explain the transition between the levels inter and trans of the development of derivative schema.

Prospective Mathematics Teachers’ Perspectives on Technology

This chapter examined prospective secondary mathematics teachers’ perspectives on the role that technological resources play in supporting students’ learning. In particular, we study prospective teachers’ pedagogical reasoning in order to understand their decisions about the use of technology and their effects on students’ mathematics learning. We analysed prospective secondary teachers’ lesson plans on teaching mathematics through problem solving by integrating technology. Prospective secondary mathematics teachers’ perspectives on the use of technology for supporting students’ mathematical learning varied in two dimensions: (i) how technological resources are used, and (ii) what mathematical activity that prospective teachers should present to support students’ learning. These dimensions are related to the idea of instrumental integration that is used to describe how teachers organize the conditions for instrumental genesis. We identified three ways of integrating technological resources.

Digital Technology in Mathematics Education: Research over the Last Decade

In this survey paper we focus on identifying recent advances in research on digital technology in the field of mathematics education. We have used Internet search engines with keywords related to mathematics education and digital technology and have reviewed some of the main international journals. We identify five sub-areas of research, important trends of development, and illustrate them using case studies: mobile technologies, massive open online courses (MOOCs), digital libraries and designing learning objects, collaborative learning using digital technology, and teacher training using blended learning. These exemplary case studies may help the reader to understand how recent developments in this area of research have evolved in the last few years. We conclude the report discussing some of the implications that these digital technologies may have for mathematics education research and practice as well as making some recommendations for future research in this area.

Comprensión de las figuras geometrícas en niños de 6-9 años

Esta investigacion ha recibido el apoyo de los Proyectos I+D+i, EDU2014-54526-R del Ministerio de Economia y Competitividad, Gobierno de Espana.

El desarrollo de la competencia docente "mirar profesionalmente" la enseñanza-aprendizaje de las matemáticas

Este articulo se inserta en la linea de reflexion en educacion matematica centrada en caracterizar y desarrollar la competencia docente “mirar profesionalmente” la ensenanza-aprendizaje de las matematicas como uma componente de la practica profesional del profesor de matematicas. Un ejemplo procedente de un curso de formacion inicial de profesores nos permite caracterizar su significado. En este trabajo se argumenta que el desarrollo de esta competencia docente se apoya en la relacion dialectica entre el conocimiento de matematicas para la ensenanza (MKT) y el discurso generado en la resolucion de tareas profesionales. Finalmente se describen algunas caracteristicas de los entornos de aprendizaje en los programas de formacion de profesores dirigidos a fomentar el desarrollo de esta relacion dialectica.This report provides some reflections on the “professional noticing” skill as a component of professional practice of the mathematics teacher. We provide an example from a pre-service mathematics teachers program in order to characterize the aforementioned skill. We claim that the development of professional noticing is supported by the dialectic relation between mathematics knowledge for teaching (MKT) and the discourse generated in the resolution of professional tasks. Finally, some aspects of the learning environments in teacher education programs are described in order to support the relation between MKT and the resolution of professional tasks.

O desenvolvimento da competência docente de "olhar profissionalmente" o ensino-aprendizagem das matemáticas

Este articulo se inserta en la linea de reflexion en educacion matematica centrada en caracterizar y desarrollar la competencia docente “mirar profesionalmente” la ensenanza-aprendizaje de las matematicas como uma componente de la practica profesional del profesor de matematicas. Un ejemplo procedente de un curso de formacion inicial de profesores nos permite caracterizar su significado. En este trabajo se argumenta que el desarrollo de esta competencia docente se apoya en la relacion dialectica entre el conocimiento de matematicas para la ensenanza (MKT) y el discurso generado en la resolucion de tareas profesionales. Finalmente se describen algunas caracteristicas de los entornos de aprendizaje en los programas de formacion de profesores dirigidos a fomentar el desarrollo de esta relacion dialectica.This report provides some reflections on the “professional noticing” skill as a component of professional practice of the mathematics teacher. We provide an example from a pre-service mathematics teachers program in order to characterize the aforementioned skill. We claim that the development of professional noticing is supported by the dialectic relation between mathematics knowledge for teaching (MKT) and the discourse generated in the resolution of professional tasks. Finally, some aspects of the learning environments in teacher education programs are described in order to support the relation between MKT and the resolution of professional tasks.

The development of the "professional noticing" skill in teaching-learning mathematics

Este articulo se inserta en la linea de reflexion en educacion matematica centrada en caracterizar y desarrollar la competencia docente “mirar profesionalmente” la ensenanza-aprendizaje de las matematicas como uma componente de la practica profesional del profesor de matematicas. Un ejemplo procedente de un curso de formacion inicial de profesores nos permite caracterizar su significado. En este trabajo se argumenta que el desarrollo de esta competencia docente se apoya en la relacion dialectica entre el conocimiento de matematicas para la ensenanza (MKT) y el discurso generado en la resolucion de tareas profesionales. Finalmente se describen algunas caracteristicas de los entornos de aprendizaje en los programas de formacion de profesores dirigidos a fomentar el desarrollo de esta relacion dialectica.This report provides some reflections on the “professional noticing” skill as a component of professional practice of the mathematics teacher. We provide an example from a pre-service mathematics teachers program in order to characterize the aforementioned skill. We claim that the development of professional noticing is supported by the dialectic relation between mathematics knowledge for teaching (MKT) and the discourse generated in the resolution of professional tasks. Finally, some aspects of the learning environments in teacher education programs are described in order to support the relation between MKT and the resolution of professional tasks.