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.

In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers' belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I argue that the prospective teachers need a context allowing them to look at the topics they will teach in a different manner. This context may be provided by the history of mathematics. In this paper I shall discuss how history affected the construction of teaching sequences on algebra during the activities of the 'laboratory of mathematics education' carried out in a 2 year education program for prospective teachers. The conditions of the experiment, notably the fact that our prospective teachers had not had specific preparation in the history of mathematics, allow us to outline opportunities and caveats of the use of history in teacher education.Key words history of mathematics . mathematics teacher education . cognitive root . evolutionary mode . situated modeThe difference between us and the pupils entrusted to our care lies only in this, that we have traveled a longer tract of the parabola of life. If the pupils do not understand, the fault is with the teacher who does not know how to explain. Nor does it do any good to load the responsibility onto the lower schools. We must take the pupils as they are, and recall what they have forgotten, or studied under a different nomenclature. If the teacher torments his students, and instead of winning their love, he arouses hatred against himself and the science he teaches, not only will his teaching be negative, but the necessity of living with so many little enemies will be a continual torment for him. Each makes his own fortune, good or bad. Who causes his own troubles, cries alone. So said Jove, as Homer reported (Odyssey, I, 34). With these principles, dear reader and colleague, may you live happily.

It is widely recognized that purely cognitive behavior is extremely rare in performing mathematical activity: other factors, such as the affective ones, play a crucial role. In light of this observation, we present a reflection on the presence of affective and cognitive factors in the process of proving. Proof is considered as a special case of problem solving and the proving process is studied adopting a perspective according to which both affective and cognitive factors influence it. To carry out our study, we set up a framework where theoretical tools coming from research on problem solving, proof and affect are present. The study is performed within a university course in mathematics education, where students were given a statement in elementary number theory to be proved and were asked to write down their proving process and the thoughts that accompanied this process. We scrutinize the written protocols of two unsuccessful students, with the aim of disentangling the intertwining between affect and cognition. In particular, we seize the moments in which beliefs about self and beliefs about mathematical activity shape the performance of our students.

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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 Mathématique 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.