21st CENTURY

classes in the form of a Socratic dialogue with his students.
Typically, Steiner would present a problem in construction
to his class; for example, how to map one circle onto
another, so that for each point of one circle, there could
be found a definite corresponding point of the other. The
entire attention of the class would be directed toward generating
the solution. Steiner would intervene only to direct
a process of education through problem solving, largely
carried out by the students themselves. The outcome was
a process that made the most ordinary student capable of
thinking with scientific rigor.
Geometry and Socratic Dialogue
The primary result of instruction was not that the
students mastered individual problems, but rather that
in solving problems, they began thinking in terms of
relationships instead of fixed objects. Steiner emphasized
that the relationships discovered to exist between
geometrical constructions were primary to the individual
constructions themselves. In order to encourage the
students to make full use of their imaginations, Steiner
"used no figures in his classes," Felix Klein reported.
"The active thinking of the listener was supposed to
generate such a clear picture in his imagination that no
material image would be needed."
Already while a student at Yverdun, Steiner had sought
to advance the method of teaching geometry as well as
the subject matter itself. He described the early evolution
of his thinking in applications for financial support he made
later in 1826 and 1827:
The method brought into practice at the Pestalozzi
Institute to treat the mathematical truths as subjects of
free reflection (Nachdenken), caused me, as a student
there, to search in place of the propositions put forward
in the instruction, for other, where possible more
profound grounds than those which my teachers at
that time advanced, and in which I was often so successful
that the teachers preferred my proofs to their
own. . . . My many-sided occupation with mathematical
instruction, combined with my own intense scientific
efforts, gave my activity in this domain a direction
which began early to depart from the usual one.
As a student, what was thrust upon me, after I studied
several textbooks of geometry, was the arbitrary
nature of the order that, from the want of substantive
connection, arose from relating individual propositions
as such: I found somewhat arbitrarily, somewhat
empirically, that the necessity of science is proven from
its substantive content, and—according to a feeling
mysteriously enlivening me—that the entire manifold
of material must follow from its general unity and must
be exhausted accordingly.
It was clear to me, that so long as the synthetic method
was sought in this external, arbitrary connection of
individual propositions, we mislead the student in the
presentation that the individual propositions as such
were the object of science, and to him the perception
of its general synthetic unity is so obscured, that he
learns to comprehend its evidence always only as the
21st CENTURY November-December 1988 51