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Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, and Analysis

Felix Klein
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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Felix Klein's famous lectures on Elementary Mathematics from an Advanced Standpoint have recently been reprinted by Dover, which should be cause for great celebration. The lectures come in two volumes. (The early reviews refer to a third volume, which, as far as I can tell, was never published.) The first volume is dedicated to Arithmetic, Algebra, and Analysis, and the second to Geometry. Both derive from lectures given by Klein to future school teachers.

Klein's major goal seems to have been to expose future teachers to a rigorous development of elementary mathematics in the hope that this would improve both their understanding of what they were to teach and the quality of their teaching. One early reviewer highlighted the main point: "The elements of mathematics, presented from an advanced perspective by a creative mathematician rather than a pedant, prove to be vastly more intelligible and fascinating than the orthodox presentation would permit one to suspect." (John M. Reiner, writing in Philosophy of Science in 1941 — isn't JSTOR wonderful?).

The two volumes are actually very different. The first, on arithmetic, algebra, and analysis combines an interest in the logical development of the subject with extensive discussion of pedagogy. Each section asks first "what is the state of our knowledge?" and then "what are the implications for teaching?". The geometry volume focuses rather on giving an overall "take" on geometry as a whole (Klein even uses the word "encyclopedic"), which of course reflects Klein's famous idea that the subject should be organized in terms of the groups of isometries attached to various geometries. In the German edition, this volume included, at the end, a discussion of pedagogy, but this is omitted from the translation.

These books were enormously influential in the United States and in the early days of the MAA (one of the translators of this edition was E. R. Hedrick, the first president of the Association). Klein's account of elementary mathematics is still worth reading and pondering.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College.
I. Calculating with Natural Numbers
II. The First Extension of the Notion of Number
III. Concerning Special Properties of Integers
IV. Complex Numbers
2. Algebra
I. Real Equations with Real Unknowns
II. Equations in the Field of Complex Quantities
3. Analysis
I. Logarithmic and Exponential Functions
II. The Goniometric Functions
III. Concerning Infinitesimal Calculus Proper