Miguel R. Wilhelmi

Models for Statistical Pedagogical Knowledge

The education of statistics teachers should be based on adequate models for pedagogical knowledge that guide the teachers’ educators in implementing and assessing the training of teachers. In this chapter, some models that are relevant for mathematics and statistics are analysed, and a new framework that complements the previously described models is proposed. The different facets and levels that should be taken into account when educating mathematics and statistics teachers are highlighted. Some implications for the training of teachers are presented and a formative cycle directed to increase the teachers’ statistical and pedagogical knowledge simultaneously is briefly described.

Naturaleza del razonamiento algebraico elemental

The introduction of algebraic reasoning in primary education is a subject of interest for research and curricular innovation in mathematics education, which supposes an extended vision of the nature of school algebra. In this paper we propose a way to conceive of algebraic reasoning based on the types of mathematical objects and processes introduced in the onto-semiotic approach to mathematical knowledge. In particular, considering a mathematical practice as algebraic is based on the intervention of generalization and symbolization processes, along with other objects usually considered as algebraic, such as binary relations, operations, functions and structures. This way to conceive of elementary algebra is based on and compared to the characterizations given by other authors. We also propose a typology of algebraic configurations that allows defining degrees of algebrization of mathematical activity.

Analysis of Algebraic Reasoning and Its Different Levels in Primary and Secondary Education

An important objective in various curricular guidelines (e. g., NCTM, 2000) is the enhancement of algebraic reasoning since the first educational levels. This objective implies that we assume a new view of school algebra as being not limited to handling algebraic expressions (Aké, Godino, Gonzato & Wilhelmi, 2013; Godino, Ake, Gonzato & Wilhelmi, 2014). The effective implementation of this new conception of school algebra poses a challenge for the training of mathematics teachers, because few current training programs include the development of such new vision.

Evaluación de conocimientos didáctico - matemáticos sobre razonamiento algebraico elemental de futuros maestros

Diferentes estudios sugieren la necesidad de incorporar desde edades tempranas el razonamiento algebraico elemental, que en las propuestas curriculares clasicas se inicia en la Educacion Secundaria Obligatoria. En este trabajo se analizan los resultados de aplicar un cuestionario de evaluacion de conocimientos didactico - matematicos sobre razonamiento algebraico elemental a una muestra de estudiantes del Grado en Maestro en Educacion Primaria. El objetivo es la elaboracion de un diagnostico sobre la competencia algebraica elemental y su didactica de los futuros maestros, que permita enmarcar un programa formativo para estos, que garantice finalmente procesos de estudio efectivos en la educacion primaria. La muestra esta compuesta por 597 estudiantes de las Universidades de Granada, Jaen, Publica de Navarra, Santiago de Compostela en Espana y de Aveiro en Portugal. En estas universidades, los grupos se constituyen homogeneos ad extra y heterogeneos ad intro. Se propone pues la seleccion intencional de grupos completos en cada universidad. El cuestionario consta de 25 items que evaluan tanto conocimientos algebraicos como conocimientos sobre la ensenanza y aprendizaje del algebra en Educacion Primaria. El analisis cuantitativo de los resultados ha permitido explorar las caracteristicas psicometricas del instrumento (indices de dificultad, discriminacion, fiabilidad y validez). La comparacion de los programas de formacion en matematicas y su didactica entre las distintas universidades participantes revela el enfasis psicopedagogico del Plan de estudios vigente y muestra una formacion disciplinar deficiente, que, en particular, no incluye el bloque de razonamiento algebraico. Los resultados muestran un bajo nivel de conocimientos generalizado en las distintas componentes del conocimiento didactico - matematico, con diferencias significativas entre las universidades. Se concluye que es necesario revisar los programas de formacion y planificar el diseno de acciones formativas especificas sobre los contenidos algebraicos elementales, a fin de capacitar a los futuros maestros para que puedan promover en los alumnos de primaria el progresivo desarrollo del pensamiento algebraico.

Vectors, change of basis and matrix representation: onto-semiotic approach in the analysis of creating meaning

In a previous study, the onto-semiotic approach was employed to analyse the mathematical notion of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems in the context of multivariate calculus. This study approaches different coordinate systems through the process of change of basis, as developed in the context of linear algebra, as well as the similarity relationship between the matrices that represent the same linear transformation with respect to different bases.