Identifying mathematically gifted students is an important objective in mathematics education. To describe skills typical of these students, researchers pose problems in several mathematical domains whose solutions require using different mathematical capacities, such as visualization, generalization, proof, creativity, etc. This paper presents an analysis of the solutions to two problems by 75 students (aged 11–14), as part of the selection test for a workshop to stimulate mathematical talent. These problems require the use of the capacity for mathematical generalization of solution methods. We define a set of descriptors of such capacity, use them to analyze students’ solutions, and evaluate how well students with high capacity for generalization can be distinguished from average students. The results indicate that the two problems are suitable for identifying potential mathematically gifted students and several descriptors have high discriminatory power to identify students with high or low capacity for generalization.

As part of a doctoral research, whose objective is to analyze the learning of proof in the context of three-dimensional geometry through construction problems in a dynamic geometry environment, we want to study the support and impact of the teacher's interventions when he talks to the students about the proofs elaborated by them. In this document we illustrate the relevance of interventions aimed at deepening students' ideas and guiding their work when blockages occur. We highlight the importance of teacher's management and participation in the learning of elaboration of the proofs.

In mathematics education, technology offers many opportunities to enrich curricular contents. Plane symmetries is a topic often skipped by primary teachers. However, it is important and may be worked in attractive ways in dynamic geometry software environments. In any regular classroom there are students with different levels of mathematical attainment, some needing easy tasks while others, particularly mathematically-gifted students, need challenging problems. We present a teaching unit for plane symmetries, adequate for upper primary school grades, implemented in a fully interactive electronic book, with most activities solved in GeoGebra apps. The book allows student to choose which itinerary to follow and attention is paid to different levels of students' mathematical attainment. The research objective of the paper is to make a networked analysis of the structure and contents of the teaching unit based on the Van Hiele levels of mathematical reasoning and the levels of cognitive demand in mathematical problem solving. The analysis shows the interest of networking both theories, the suitability of the teaching unit, as the Van Hiele levels and the cognitive demand of the activities increases, and its usefulness to fit the needs of each student, from low attainers to mathematically-gifted students.

El software educativo de geometría 3D es una interesante ayuda para que los estudiantes representen y manipulen cuerpos espaciales. En este artículo presentamos una secuencia de actividades basada en la rotación de cubos, combinando cubos reales y representaciones en papel y en ordenador. Las actividades en ordenador se basan en diversos cubos creados en Adobe Flash y en GeoGebra. Un objetivo de esta investigación es observar si los estudiantes de distintos cursos de Educación Primaria, en particular aquellos con alta capacidad matemática, muestran diferencias en la presencia de sus habilidades de visualización y su talento matemático al utilizar uno u otro software. El análisis de los datos muestra además que las actividades planteadas con estas aplicaciones son útiles para que los estudiantes aprendan a visualizar los giros y desarrollen sus habilidades de visualización.

La geometría analítica 3-dimensional es importante en los cursos de cálculo multivariado, no obstante los estudiantes tienen dificultades para su aprendizaje. Presentamos resultados de una investigación sobre aprendizaje de las superficies cuádricas mediante una intervención didáctica basada en el software dinámico 3-dimensional educativo GAnalitica3D. Analizamos la influencia del carácter constructivista y colaborativo de la enseñanza y de las habilidades de visualización necesarias para interactuar con el software. La experimentación se realizó con estudiantes universitarios, que resolvían los problemas en pequeños grupos y después hacían puestas en común. Informamos sobre la evaluación de conocimientos previos realizada y mostramos ejemplos de resoluciones en las que se aprecian el progreso y las dificultades de los estudiantes. Concluimos que el software permite relacionar comprensivamente las representaciones algebraica y visual de las superficies cuádricas, lo cual facilita su aprendizaje.

Presentamos una experiencia dirigida a estudiantes de 1o y 2o de ESO, de un taller extraescolar, realizado de manera virtual, mediante videoconferencia y un canal de mensaje instantaneo (movil). En el taller se resolvian y discutian problemas, dirigidos principalmente a potenciar el aprendizaje de la demostracion matematica. Los estudiantes procedian de localidades y centros de estudio distintos y las sesiones las realizaban mediante un ordenador o tableta con conexion a Internet. La eleccion de este entorno esta motivada porque se trataba de estudiantes con nivel superior al medio en matematicas, con varios sujetos de alta capacidad, unicos de ese nivel en su aula. En el trabajo completo mostramos la organizacion del taller virtual, su potencial y algunos inconvenientes con los que nos hemos encontrado por el hecho de utilizar videoconferencia.

We analyze the solutions given by secondary school mathematically gifted students to a collaborative task designed to promote the development of students' competence of visualization. Each student was provided with two different orthogonal projections of a set of buildings made of cubes and other verbal data, and they were asked to place the buildings on a squared grid. We analyze students' use of visualization abilities and the complexity of their reasoning. Results show that there is a relation between the objective of students' actions and the kind of visualization abilities used, and, also, between students' strategies of solution and the cognitive demand necessary to fulfill them. Finally, we network both analyses to gain insight and look for global conclusions.

El objetivo de esta investigacion es identificar que habilidades de visualizacion ponen en juego estudiantes de edades tempranas al resolver problemas en un entorno tecnologico Bee-bot y si los estudiantes de acm manifiestan mejor uso de esas habilidades que sus companeros de capacidad media. Los problemas planteados consisten en planear recorridos del robot Bee-bot, el cual admite instrucciones de avance o retroceso de 1 paso y giros de 90o a derecha o izquierda. Los estudiantes usaban fichas con estas instrucciones para planificar la secuencia de pasos del recorrido. Despues programaban el robot con esa secuencia y finalmente observaban el recorrido hecho por el robot, lo cual les permitia evaluar si su secuencia era correcta o en que pasos fallaba. La toma de datos se realizo mediante grabacion en video de las sesiones experimentales.

Mathematics education researchers have focused on mathematically talented students’ behaviour. Three types of studies have been carried out most frequently: identification, description, and intervention. We present results related to identification.

En nuestra investigacion nos planteamos la forma de razonar de estudiantes de talento matematico y estudiantes con rendimiento academico alto, pero no de talento, a traves de un taller de 10 horas de duracion.

Mathematically gifted students require specific teaching methodologies to foster their interest in mathematics and their engagement in solving problems. Geometric pattern problems are an interesting context in which to introduce algebra to those students. We present the case of a nine-year-old student engaged in a teaching unit based on geometric pattern problems that was aimed at helping him start learning algebra, equations, and algebra word problems. To analyze and assess the cognitive effort the student made to solve the problems, we used a particularization to this context of the cognitive demand model. We analyzed answers typical of the different kinds of problems posed throughout the teaching unit, showing the student's learning trajectory and related characteristics of mathematical giftedness.

Some theoretical models emerged from specific mathematical contents or educational levels, so they are well adapted to the context where they were created but, sometimes, they do not fit well in a different context. Related to the issue of the need to go beyond a specific theory when researching a phenomenon, to solve the tension between home-grown needs and borrowed theories, we present the reasons for and steps of an adaptation we have had to do of the Cognitive Demand model so that it fits the requirements of our research questions, methods and data. We systematized the definition of the model stated by its authors, and completed this definition with some necessary statements. Then, we re-stated some characteristics of the model to avoid inconsistencies when using it. Finally, we particularized the general model to several specific mathematical topics.

Los estudiantes con alta capacidad matematica deben recibir atencion diferenciada en clase. Presentamos una propuesta metodologica para primaria y secundaria basada en la resolucion de problemas que son parte de la programacion ordinaria, dirigidos a todo el alumnado y que incluyen extensiones para que los estudiantes con mayor capacidad matematica puedan completar su formacion.