The chords theorem recalled to life at the turn of the eighteenth century

This paper is a historical account of the chords theorem, for conic sections from Apollonius to Boscovich. We comment the most significant proofs and applications, focusing on Newton's solution of the Pappus four lines problem. Newton's geometrical achievements drew L'Hospital's attention to the chords theorem as a fundamental one, and led him to search for a simple and direct proof, that he finally obtained by the method of projection. Stirling gave a very elegant algebraic proof; then Boscovich succeeded in finding an almost immediate geometrical proof, and showed how to develop the elements of conic sections starting from this theorem.     

Towards theoretical foundations of mathematics education

Although research in mathematics education developed in the last decades as a vivid scientific field, the nature and the place of theories in the field are still under discussion. H.-G. Steiner contributed to this debate in several ways. He not only intervened in the scientific debate but also played an active role in organizing the discipline and the scientific discussion at the international level. This paper attempts to give a synthetic view of the evolution of mathematics education with respect to theory by focusing on the French situation of research in mathematics education and paying a particular attention to the role and contribution of H.-G. Steiner.     

On the theoretical, conceptual, and philosophical foundations for research in mathematics education

The current infatuation in the U.S. with “what works” studies seems to leave education researchers with less latitude to conduct studies to advance theoretical and model-building goals and they are expected to adopt philosophical perspectives that often run counter to their own. Three basic questions are addressed in this article:What is the role of theory in education research? How does one's philosophical stance influence the sort of research one does? And,What should be the goals of mathematics education research? Special attention is paid to the importance of having a conceptual framework to guide one's research and to the value of acknowledging one's philosophical stance in considering what counts as evidence.     

Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course

The purpose of this study was to investigate pre-service teachers’ views about the history of mathematics course in which GeoGebra was used. The qualitative research design was used in this study. The participants of the study consisted of 23 pre-service mathematics teachers studying at a state university in Turkey. An open-ended questionnaire was used as a data collection tool. Qualitative data obtained from the pre-service teachers were analyzed by means of content analysis. As a result, it was determined that GeoGebra software was an effective tool in the learning and teaching of the history of mathematics.     

The rejected parts of Brouwer's dissertation on the foundations of mathematics

Brouwer launched his intuitionist attack on the formalistic trends in mathematics in his now famous dissertation “On the Foundations of Mathematics” in 1904. In the autumn of 1976, the author found what turned out to be the first version of Brouwer's dissertation. He also discovered that his supervisor disapproved and rejected what Brouwer considered to be the most important part of his dissertation. The extreme solipsistic views held by Brouwer in his later years are well known. The first version of the dissertation shows that these views were held before Brouwer began his intuitionist campaign and that they determined his philosophy of mathematics. The deleted parts not found in the final version of the dissertation are presented here in an English translation. In a brief introduction the author gives some of the historical background, based partly on the correspondence between Brouwer and his supervisor, Korteweg.