Marta Menghini

From geometrical figures to definitional Rigour: Teachers’ analysis of teaching units mediated through van Hiele's theory

The present study reports on curriculum design implemented in eight parallel classes of high school, with students aged 14–15, by means of work-sheets dealing with properties of the principal plane geometric figures. Van Hiele’s theory was used to facilitate teachers’ discussion of student performances on work-sheets in order to enhance their didactical awareness. We hypothesize that van Hiele’s theory is a proper means to provide encouragement and support for teachers in rethinking their teaching goals and practices. This environment enables teachers to manage their students’ transition from the perception of a figure to its definition. In this paper, we will recount a teachers’ discussion as it unfolded in a school-university work group. The analysis of students’ work on work-sheets interweaves with an analysis of the teaching actions that foster the process of understanding definitions.RésuméLa présente étude rend compte d’un programme mis en pratique dans 8 classes parallèles de niveau secondaire, auprès d’étudiants de 14 et 15 ans, grâce à des fiches d’exercices traitant des propriétés qui caractérisent les principales figures géométriques planes. La théorie de van Hiele a été utilisée pour faciliter la discussion chez les enseignants quant au rendement des élèves dans ces fiches d’exercices, afin d’affiner leur niveau de conscience pédagogique. Nous posons comme hypothèse que la théorie de van Hiele constitue un instrument adéquat pour encourager et soutenir les enseignants qui souhaitent revoir leurs objectifs et leurs pratiques d’enseignement. Un tel environnement permet aux enseignants d’aider leurs élèves à faire la transition qui consiste à passer de la perception d’une figure à sa définition. Dans cet article, nous rapportons certaines discussions entre enseignants, telles qu’elles ont eu lieu au sein d’un groupe de travail mixte scolaire et universitaire. L’analyse du travail fait par les étudiants sur les fiches d’exercices est ensuite mise en rapport avec l’analyse des techniques d’enseignement servant à favoriser le processus de compréhension des définitions.

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.

Notes on the correspondence between Luigi Cremona and Max Noether

Abstract This paper contains an analysis of the scientific climate in which the Italian school of algebraic geometry came into being and developed in a direction different from that of the German school, even though both shared a common initial matrix. Reference is made to a number of letters written by Max Noether to Luigi Cremona.

History of Teaching Geometry

Since the beginning of the transmission of geometric knowledge, two aspects of geometry have been present: the abstract “speculative” and the practical. These two aspects correspond to an essential dialectic in geometry teaching, between a deductive/rational science and a practical/intuitive one. In the nineteenth and twentieth centuries, the stress on the second aspect led to experimental geometry. After the work of Descartes, another tension concerned the solution of problems: between “pure” methods and methods coming from algebra or analysis. Attempts to find a new language for school geometry reached their apex when geometry was substituted by linear algebra in the 1960s.

What Is and What Might Be the Legacy of Felix Klein

Felix Klein always emphasised the great importance of teaching at the university, and he strongly promoted the modernisation of mathematics in the classrooms. The three books “Elementary Mathematics from a higher (advanced) standpoint” from the beginning of the last century gave and still give a model for university lectures especially for student teachers. The “Merano Syllabus” (1905), essentially initiated and influenced by Felix Klein, pleaded for an orientation of mathematics education at the concept of function, an increased emphasis on spatial geometry and an introduction of calculus in high schools. The Thematic Afternoon “The legacy of Felix Klein” will give an overview about the ideas of Felix Klein, it will highlight some developments in university teaching and school mathematics related to Felix Klein in the last century, and it will especially be asked for the meaning, the importance and the legacy of Klein’s ideas nowadays and in the future in an international, worldwide context.

From Practical Geometry to the Laboratory Method: The Search for an Alternative to Euclid in the History of Teaching Geometry

This paper wants to show how practical geometry, created to give a concrete help to people involved in trade, in land-surveying and even in astronomy, underwent a transformation that underlined its didactical value and turned it first into a way of teaching via problem solving, and then into an experimental-intuitive teaching that could be an alternative to the deductive-rational teaching of geometry. This evolution will be highlighted using textbooks that proposed alternative presentations of geometry.