ion
for too long a time. A well-known specialist says that such exercises
should not be allowed at first to take up more than a minute or two at
a time; then, that their duration should gradually extend to five and
ten minutes. The length of time which children closely and voluntarily
attend to an exercise is as follows: Children from five to seven
years, about fifteen minutes; from seven to ten years, twenty minutes;
from twelve to eighteen years, thirty minutes. A magnetic teacher can
obtain attention somewhat longer, but it will always be at the expense
of the succeeding lesson. "By teachers of high pretensions, lessons
are often carried on greatly and grievously in excess of the proper
limits; but when the results are examined they show that after a
certain time has been exceeded, everything forced upon the brain only
tends to drive out or to confuse what has been previously stored in
it."

We find, of course, that the mind can sustain more labor for a longer
time when all the faculties are employed than when a single faculty is
exerted, but the ambitious teacher needs to remind herself every day
that no error is more fatal than to overwork the brain of a young
child. Other errors may perhaps be corrected, but the effects of this
end only with life. To force upon him knowledge which is too advanced
for his present comprehension, or to demand from him greater
concentration, and for a longer period than he is physically fitted to
give, is to produce arrested development.[55]

[55] "Whoever sacrifices health to wisdom has generally
sacrificed wisdom, too." (Jean Paul.)

MATHEMATICAL FORMS.

We must beware of abstractions in these forms of knowledge, and let
the child see and build for himself, then lead him to express in
numbers what he has seen and built. He will not call it Arithmetic,
nor be troubled with any visions of mathematics as an abstract
science.[56]

[56] "Perceptions and recognitions which are with difficulty
gained from _words_ are easily gained from facts and deeds.
Through actual experience the child gains in a trice a total
concept, whereas the same concept expressed in words would be
only grasped in a partial manner. The rare merit, the
vivifying influence of this play-material is that, through
the representations it makes possible, concepts are
recognized at once in their wholeness and unity, whereas such
an idea of a whole can only very graduall