21st CENTURY

INTRODUCTION
The great Swiss geometer Jacob Steiner (1796-1863) led
a revolution in the teaching of geometry and science
in Prussia early in the 19th century, as part of the
education reform movement initiated by J. G. Fichte, Baron
vom Stein, and Prussian Education Minister Wilhelm von
Humboldt. The reformers targeted the stuffy Prussian academic
establishment and overthrew its prevailing reactionary
teaching practices that emphasized memorization—
"drill and grill." Instead, the reformers promoted a program
for the many-sided development of the cognitive
powers of the student via the Socratic method of instruction.
The leading feature of the new approach was the
teaching of "synthetic geometry" as a language for concept
formation, distinct from verbal language per se. Synthetic
geometry is based on demonstration by construction rather
than deductive "proof."
With his fresh outlook, Steiner also produced a grand
synthesis of all the work of geometry and mathematics up
to his time, under a unified conception of human creative
mentation. The impact of his work swept across the European
continent like a tidal wave, reverberating in the labors
of Bernhard Riemann, Georg Cantor, Ludwig Prandtl, and
many others. When Steiner died, Ludwig Crelle's journal
for Pure andApplied Mathematics praised him as "the greatest
geometer of his time."
The Swiss Steiner came to prosper in Berlin only because
of the educational reform movement launched by Fichte
after Prussia's devastating defeat by Napoleon's armies in
1806. The reformers argued that the soldiers of the Prussian
army turned and ran when the fighting started because they
had no conception of what they were fighting for, and that
this was the underlying cause of the defeat. The reformers
went to the heart of the matter: It is necessary to teach our
citizens how to think. They called for introducing into Prussia
the Socratic teaching methods of the Swiss educational
reformer Johann Heinrich Pestalozzi in order to instill patriotism
in the citizens of the next generation.
Their reasoning was that if the student can be guided to
experience the creative power of his own mind, he will
appreciate the value of a republic and the need to defend
its freedom. Fichte went so far as to say that applying Pestalozzi's
pedagogical principles would "awaken the civic
and military spirit of the nation" and that it was "the only
possible means of saving German independence." All of
these goals were anathema to G.W.F. Hegel, who was entrusted
by the Vienna powers to use his influence in education
to suppress republicans across the continent. Pestalozzi's
followers became Hegel's chief opponents.
The Pestalozzi Method
Pestalozzi placed emphasis on bringing out the capabilities
of the individual student through instruction in synthetic
geometry. "All knowledge is to be earned, discovered,
produced by the student himself; the teacher is only
to guide the independently thinking student in the right
direction," wrote the mathematician and educator Felix Klein
50 November-December 1988 21st CENTURY
on the Pestalozzi method. To guide the students, the teacher
asks questions in order to promote their reflection on a
problem in precisely the same way that Socrates uses this
method in the Meno dialogue to lead a slave boy to discover
the solution of the Pythagorean theorem. Pestalozzi wrote:
Ignorance is better than knowledge that is but prejudice,
a glass through which to view the world. To
arrive at knowledge slowly, by one's own experience,
is better than to learn by rote, in a hurry, facts that
other people know, and then, glutted with words, to
lose one's own free, observant and inquisitive ability
to study. . . .
The higher purpose of education is to prepare the
individual to make free and self-reliant use of all the
faculties with which the Creator has endowed him, and
then to direct these faculties that they may perfect all
human life.
Fichte, Vom Stein, and Humboldt sent many of their followers
to Pestalozzi's institute in Yverdun, Switzerland, to
master his method. In addition, some of Pestalozzi's own
assistants emigrated to Prussia to join in the reform movement,
working at its centers in Berlin at the Cauer and
Plamann educational institutes and the Friedrichs Werder
trade school, all established by disciples of Pestalozzi and
Fichte.
Into the middle of this reform movement in Switzerland
and Prussia walked Jacob Steiner at age 18. The young Swiss
farmer had finally got the support he required for beginning
his formal education at Pestalozzi's Yverdun academy in
1814. Felix Eberty, whom Steiner later tutored at the Cauer
Institute, described how Steiner displayed his inherent aptitude
for thinking in the language of synthetic geometry
when he was a student with Pestalozzi:
That he already as a youth saw eight triangles,
where the teacher had asked for only one, is, so
to speak, a typical example of his method. His mind
possessed with respect to geometrical propositions,
one might say, a kaleidoscopic power. The simplest
one shaped itself before his inner eye into a manysided
harmonic vision. One hexagon became 15
hexagons, whose radiating rays met again at nodal
points, and these points connected by further lines,
form new figures. He could construct most of his
theorems only in the head, because no illustration
was able to adhere to its complexity.
After a year and a half of study, Steiner was appointed
by Pestalozzi to a teaching position at his institute. At this
time, Fichte's 10 followers who were to found the Cauer
Institute in 1817 were studying at Yverdun, and they may
have then recruited Steiner to join them in Berlin. In 1818,
Steiner left Yverdun for the University of Heidelberg to
familiarize himself with the formal branches of mathematics
in preparation to joining the reform movement in
Prussia.
As a teacher, Steiner did not lecture in the ordinary way.
In accordance with Pestalozzi's method, he conducted his