Fulvia Furinghetti

Rethinking Characterizations of Beliefs

In this chapter we consider beliefs and the related concepts of conceptions and knowledge. From a review of the literature in different fields we observe that there is a diversity of views and approaches in research on these subjects. We report on a small research project of our own attempting to clarify the understanding of beliefs among specialists in mathematics education. A panel of 18 mathematics educators participated in a panel that we termed “virtual”, since the participants communicated with us only by e-mail. We sent nine characterizations related to beliefs, selected from the literature, to the panelists, asked them to express their agreement or disagreement with the statements, and also asked each to give their own characterization of the term. The answers were analyzed, searching for the elements around which the concept of beliefs has developed along the years. We discuss issues on which there was agreement and disagreement and conjecture what lies behind the differences. As a final step we make some suggestions relating to characterization of the term belief and ways of dealing with it in future research.

The use of original sources in the mathematics classroom

The study of original sources is the most ambitious of ways in which history might be integrated into the teaching of mathematics, but also one of the most rewarding for students both at school and at teacher training institutions.

The political context

People have studied, learned and used mathematics for over four thousand years. Decisions on what is to be taught in schools, and how, are ultimately political, influenced by a number of factors including the experience of teachers, expectations of parents and employers, and the social context of debates about the curriculum. The ICMI study is posited on the experience of many mathematics teachers across the world that its history makes a difference: that having history of mathematics as a resource for the teacher is beneficial.

The history of mathematics as a coupling link between secondary and university teaching

During the years they spend in university, many mathematics students develop a very poor conception of mathematics and its teaching. This fact is bad in all cases, but even more in the case of those students who will be mathematics teachers in school. In this paper it is argued that the history of mathematics may be an efficient element to provide students with flexibility, open-mindedness and motivation towards mathematics. The theoretical background of this work relies both on recent research in mathematics education and on papers written by mathematicians of the past. Opinions are supported with examples. One example concerns a historical presentation of ‘definition’; it was developed with mathematics students who will become mathematics teachers. For students oriented to research or to applied mathematics, an example is presented to address the problem of the secondary-tertiary transition.

Teacher Education Around the World

EDITOR’S NOTE: The “Teacher Education Around the World” section will consist of descriptions and critical analyses of creative programs or models of educating teachers used in different parts of the world. Submissions should consist of not more than 10 pages, double spaced, including references and figures and should address the general intent of JMTE. The purpose of these short contributions will be to inform others of programs that address fundamental problems in mathematics teacher education or provide information about educating teachers that is generally not available elsewhere in the literature. Authors should clearly indicate that their paper is intended for the “Teacher Education Around the World” section of the journal. Submissions will be evaluated on the basis of their potential contribution to the field, the clarity of the program’s description including attention to the educational and cultural context in which the program exists, and the quality of the critical analysis. We hope authors will take advantage of this opportunity so that more information about mathematics teacher education around the world can be shared.

From Mathematics and Education, to Mathematics Education

This chapter takes a historical view of the development of mathematics education, from its initial status as a business mostly managed by mathematicians to the birth of mathematics education as a scientific field of research. The role of mathematical communication is analyzed through the growth of journals and research conferences. Actions of internationalization and cooperation in facing instructional and educational problems are illustrated with reference to the journal L’Enseignement Mathematique and to ICMI. Curricular and methodological reforms in the 20th century which generated changes in school mathematics are considered. Starting from the acknowledgement that research in mathematics education demands more than the traditional focus on discussing curricular options at distinct grade levels, we identified several specialized clusters, debating specific issues related to mathematics education at an international level. We grouped the clusters into three main areas: relationships with psychology, the study of social, cultural and political dimensions, and the relevance of a theory for mathematics education.

The computer in mathematics teaching: scenes from the classroom

In this paper we analyse, through a case-study approach, the role assigned by mathematics teachers to the use of educational software. We considered cases in which teachers autonomously chose and used the software. Our analysis was carried out by the direct observation of classroom activities. Two areas of particular note were those of how the computer was used, e.g., in the development of mathematical knowledge and/or for reinforcing ideas already taught, and the nature of the teacher-pupil classroom discourse. Results indicate that even experienced users make limited use of the exploratory potential of the technology, and the dynamics of communication (teachers, pupils, computer) is still very much that of a ‘teacher directed and led’ classroom.

Conceptions of Proof – In Research and Teaching

This chapter first analyses and compares mathematicians’ and mathematics educators’ different conceptualisations of proof and shows how these are formed by different professional backgrounds and research interests. This diversity of views makes it difficult to precisely explain what a proof is, especially to a novice at proving. In the second section, we examine teachers’, student teachers’ and pupils’ proof conceptions and beliefs as revealed by empirical research. We find that the teachers’ beliefs clearly revolve around the questions of what counts as proof in the classroom and whether the teaching of proof should focus on the product or on the process. The third section discusses which type of metaknowledge about proof educators should provide to teachers and thus to students, how they can do this and what the intrinsic difficulties of developing adequate metaknowledge are.

The construction of a didactic itinerary of calculus starting from students’ concept images (ages 16‐19)†

It is well known that calculus is a topic difficult to teach because of educational and epistemological reasons. The Italian programmes enforced by the Ministry of Education for the age range 16‐19 years contemplate calculus in many types of schools. These programmes are precise for the content, but do not suggest any methodology; so the freedom left to teachers mainly concerns this last point. We will describe the main features of a didactic itinerary which combines com‐pulsoriness for the contents and freedom for the methodology. The main points of this itinerary are as follows. (i) We approach calculus starting from the ‘pre‐knowledge’ of students and proposing problems set inside mathematics and outside mathematics (physics, philosophy). In this way we hope to provide students with motivation and to offer them stimuli for constructing concepts. (ii) Afterwards we attach the traditional programme and construct the concepts using the textbook mediated by worksheets we prepared. In these worksheets there...

Integration of computer science and mathematics in upper secondary school: reflections and realizations

Abstract In this paper we outline the main features of the first part of the project Integration of Computer Science and Mathematics (Integrazione di informatica e matematica). This part of the project is aimed at introducing basic elements of computer science in mathematics courses for students aged from 14 to 16. In particular, we discuss the didactic methodology and the computer science contents to be introduced in relation to the ages in question. As the connections between the two disciplines of mathematics and computer science are the problems underlying the proposed integration, we briefly analyse these connections and, in particular, fix our attention on the influence of the teaching of computer science on the teaching of mathematics.