ABSTRACT
In 1908, Felix Klein not only became the founding president of the Commission internationale de l’enseignement mathématique (CIEM, anglicized as the International Commission on the Teaching of Mathematics) but also published the first volume of his groundbreaking Elementarmathematik vom höheren Standpunkte aus (Elementary Mathematics from a Higher Standpoint). In the introduction, Klein identifies a central problem in preparing teachers to teach mathematics: a double discontinuity that the prospective teacher encounters in going from school to university and then back to school to teach. School mathematics and university mathematics typically seem to have no connection. Klein’s course assumes that the prospective teachers are familiar with the main branches of mathematics, and he attempts to show how problems in those branches are connected and how they are related to the problems of school mathematics. Throughout his career, Klein saw school mathematics as demanding more dynamic teaching and consequently university mathematics as needing to help prospective teachers “stand above” their subject.

In print for a century, the volumes of Klein’s textbook have been used in countless courses for prospective and practicing teachers. They provide excellent early examples of what today is termed mathematical knowledge for teaching. Klein’s courses for teachers were part of his reform efforts to improve secondary mathematics by improving the preparation of teachers. Despite the many setbacks he encountered, no mathematician has had a more profound influence on mathematics education as a field of scholarship and practice.

In the talk, I will also discuss the work of two mathematicians whose contributions to mathematics education resemble those of Klein: George Pólya and Hans Freudenthal. I will discuss why higher is a better translation of höheren than advanced is and will end by noting some problems posed when considering mathematics education from a higher standpoint.