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These three volumes constitute the first complete English
translation of Felix Klein's seminal series "Elementarmathematik
vom hoheren Standpunkte aus". "Complete" has a twofold meaning
here: First, there now exists a translation of volume III into
English, while until today the only translation had been into
Chinese. Second, the English versions of volume I and II had
omitted several, even extended parts of the original, while we now
present a complete revised translation into modern English. The
volumes, first published between 1902 and 1908, are lecture notes
of courses that Klein offered to future mathematics teachers,
realizing a new form of teacher training that remained valid and
effective until today: Klein leads the students to gain a more
comprehensive and methodological point of view on school
mathematics. The volumes enable us to understand Klein's
far-reaching conception of elementarisation, of the "elementary
from a higher standpoint", in its implementation for school
mathematics. In Volume III, Klein explores the relationship between
precision and approximation mathematics. He crosses the various
fields of mathematics - from functions in one and two variables to
practical geometry to space curves and surfaces - underlining the
relation between the exactness of the idealised concepts and the
approximations to be considered in applications. Logical procedures
are confronted with the way in which concepts arise starting from
observations. It is a comparison between properties pertaining only
to the theoretical field of abstract mathematics and properties
that can be grasped by intuition. The final part, which concerns
gestalt relations of curves and surfaces, shows Klein to be the
master of the art of description of geometric forms.
This book seeks to explore the history of descriptive geometry in
relation to its circulation in the 19th century, which had been
favoured by the transfers of the model of the Ecole Polytechnique
to other countries. The book also covers the diffusion of its
teaching from higher instruction to technical and secondary
teaching. In relation to that, there is analysis of the role of the
institution - similar but definitely not identical in the different
countries - in the field under consideration. The book contains
chapters focused on different countries, areas, and institutions,
written by specialists of the history of the field. Insights on
descriptive geometry are provided in the context of the
mathematical aspect, the aspect of teaching in particular to
non-mathematicians, and the institutions themselves.
This book seeks to explore the history of descriptive geometry in
relation to its circulation in the 19th century, which had been
favoured by the transfers of the model of the Ecole Polytechnique
to other countries. The book also covers the diffusion of its
teaching from higher instruction to technical and secondary
teaching. In relation to that, there is analysis of the role of the
institution - similar but definitely not identical in the different
countries - in the field under consideration. The book contains
chapters focused on different countries, areas, and institutions,
written by specialists of the history of the field. Insights on
descriptive geometry are provided in the context of the
mathematical aspect, the aspect of teaching in particular to
non-mathematicians, and the institutions themselves.
This open access book provides an overview of Felix Klein's ideas,
highlighting developments in university teaching and school
mathematics related to Klein's thoughts, stemming from the last
century. It discusses the meaning, importance and the legacy of
Klein's ideas today and in the future, within an international,
global context. Presenting extended versions of the talks at the
Thematic Afternoon at ICME-13, the book shows that many of Klein's
ideas can be reinterpreted in the context of the current situation,
and offers tips and advice for dealing with current problems in
teacher education and teaching mathematics in secondary schools. It
proves that old ideas are timeless, but that it takes competent,
committed and assertive individuals to bring these ideas to life.
Throughout his professional life, Felix Klein emphasised the
importance of reflecting upon mathematics teaching and learning
from both a mathematical and a psychological or educational point
of view. He also strongly promoted the modernisation of mathematics
in the classroom, and developed ideas on university lectures for
student teachers, which he later consolidated at the beginning of
the last century in the three books on elementary mathematics from
a higher standpoint.
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