Juan D. Godino

Errors and difficulties in understanding elementary statistical concepts

This paper presents a survey of the reported research about students’ errors, difficulties and conceptions concerning elementary statistical concepts. Information related to the learning processes is essential to curricular design in this branch of mathematics. In particular, the identification of errors and difficulties which students display is needed in order to organize statistical training programmes and to prepare didactical situations which allow the students to overcome their cognitive obstacles. This paper does not attempt to report on probability concepts, an area which has received much attention, but concentrates on other statistical concepts, which have received little attention hitherto.

Clarifying the Meaning of Mathematical Objects as a Priority Area for Research in Mathematics Education

The specific aim of mathematics education as a research field is to study the factors that affect the teaching and learning of mathematics and to develop programs to improve the teaching of mathematics. In order to accomplish this aim mathematics education must consider the contributions of several disciplines: psychology, pedagogy, sociology, philosophy, etc. However, the use of these contributions in mathematics education must take into account and be based upon an analysis of the nature of mathematics and mathematical concepts, and their personal and cultural development. Such epistemological analysis is essential in mathematics education, for it would be very difficult to efficiently study the teaching and learning processes of undefined and vague objects.

Training Teachers to Teach Probability.

In this paper we analyze the reasons why the teaching of probability is difficult for mathematics teachers, describe the contents needed in the didactical preparation of teachers to teach probability and analyze some examples of activities to carry out this training. These activities take into account the experience at the University of Granada, in courses directed to primary and secondary school teachers as well as in an optional course on Didactics of Statistics, which is included in the Major in Statistical Sciences and Techniques course since 1996. The aim is encouraging other colleagues to organize similar courses at their universities, either as part of their official programs or in their postgraduate training.

INSTITUTIONAL AND PERSONAL MEANINGS OF MATHEMATICAL PROOF

Although studies on students’ difficulties in producing mathematical proofs have been carried out in different countries, few research workers have focussed their attention on the identification of mathematical proof schemes in university students. This information is potentially useful for secondary school teachers and university lecturers. In this article, we study mathematical proof schemes of students starting their studies at the University of Córdoba (Spain) and we relate these schemes to the meanings of mathematical proof in different institutional contexts: daily life, experimental sciences, professional mathematics, logic and foundations of mathematics. The main conclusion of our research is the difficulty of the deductive mathematical proof for these students. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty.

Modelo para el análisis didáctico en educación matemática

Resumen La finalidad de este artículo es presentar la viabilidad de un modelo teórico para el análisis de procesos de enseñanza y aprendizaje de las matemáticas. Dicho modelo contempla cinco niveles de análisis, los cuales son aplicados conjuntamente a un episodio de clase. Este modelo se ha elaborado para describir (¿qué ha ocurrido aquí?), explicar (¿por qué ha ocurrido?) y valorar (¿qué se podría mejorar?) procesos de instrucción en el aula de matemáticas. Nos basamos en una síntesis teórica de aspectos del enfoque ontosemiótico del conocimiento y la instrucción matemática, que venimos desarrollando desde hace una década. Aunque algunas partes del modelo son específicas de la actividad matemática, investigadores de otras áreas educativas pueden adaptarlas de modo que resulten eficaces en el análisis didáctico de otros tipos de prácticas escolares. El principal resultado esperado de la aplicación del modelo es llegar a una valoración fundamentada de la idoneidad didáctica de procesos de instrucción.

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.

Institutional and Personal Meaning of Mathematical Objects

The concept of meaning, which is frequently used informally in didactic research, is a central and controversial subject in philosophy, logic, semiotics and other sciences and technologies interfering in human cognition. The analysis of this concept from a didactic point of view could be useful for understanding the relationships between the different mathematical frameworks in mathematics education and could throw a new light on some research questions, particularly those referring to the assessment of knowledge. This paper tries to approach this analysis, and presents a pragmatic theory of the meaning of mathematical objects, by establishing a triple conditioning of this concept: institutional, personal and temporal. In addition the relationship between the proposed notion of meaning on the one hand and those of conception and „rapport à l’objet” on the other hand will be studied.ZusammenfassungDer Begriff Bedeutung, der in der didaktischen Forschung so oft in informeller Weise gebraucht wird, ist in der Philosophie, in der Logik, in der Semiotik sowie in anderen Wissenschaften und Bereichen, die sich mit dem menschlichen Denken beschäftigen, viel diskutiert und umstritten. Eine Analyse dieses Begriffs aus didaktischer Sicht könnte für das Verständnis der Beziehungen zwischen verschienen mathematischen Zusammenhaengen im Unterricht nützlich sein. Sie könnte neues Licht auf einige Forschungsfragen werfen, vor allem auf diejenigen, die sich mit der Leistungsfeststellung befassen. In diesem Beitrag soll eine solche Analyse versucht und eine pragmatische Theorie der Bedeutung mathematischer Objekte vorgestellt warden. Sie bringt zum Begriff Dedeutung ein Feld von drei Bedingungen zur Geltung: institutionelle, personale und temporal. Erörtert warden auch die Beziehungen zwischen dem vorgeschalegenen Wort „Bedeutung” und den Bezeichnungen „Begriff” bzw. „Konzept” und „rapport à l’object”.

Three Perspectives on the Issue of Theoretical Diversity

Since the birth of the didactics of mathematics in the 1970s, the research community has aimed to build theories that may be used as models for studying phenomena in the teaching and learning of mathematics, within a milieu designed for their teaching. The creativity of researchers giving birth to many theories has created certain problems in the community. Over the last decade, researchers have shared thoughts and views on the relationships between these theories. This chapter proposes three different perspectives on this topic. The first examines the richness of a multidimensional approach based on the mobilisation and networking of various well-identified theories, enabling a segmentation of reality that is well suited to the study of didactic phenomena. The second perspective defends a possible methodology for reducing theoretical diversity. The third perspective examines a social viewpoint of the multiplicity of theories in the didactics of mathematics and the search for connections.

Influência dos padrões de interação didática no desenvolvimento da aprendizagem Matemática: análise de uma atividade exploratório-investigativa sobre sequências

Abstract The aim of this paper is to explore the didactic configurations and, more specifically, thepatterns of interaction established between teachers and students during an exploratory-investigative mathematical activity. This activity was implemented in a seventh gradeclass (students approximately 12 years old) and aimed to contribute to understandingregarding the emergence of the patterns of interactions and their effects on thedevelopment of learning. We identified the didactic configurations using a theoretical-methodological model designed to describe and interpret the interactive processes inthe classroom. In conclusion, we emphasize the need for management between thedistinct configurations, and the consequent productive cooperation (synergy) betweenthe different interaction patterns, to stabilize the negotiation of meanings and enable thedevelopment of students’ learning. Keywords : Onto-semiotic Approach. Didactic Configurations. Patterns of interaction.Exploratory-investigative activities. Sequences.