life. Its
excellence does not consist in rules and illustrations by which examples
and problems are easily solved, but in leading the mind of the pupil
into natural and apparent processes of reasoning, by which he is enabled
to comprehend a proposition as an independent fact. Herein is a mental
discipline of great value, not only in the sciences, but in the daily
affairs of men of all classes and conditions. It is to be feared that
equally satisfactory results have not been attained in what is called
written arithmetic. This partial failure deserves consideration. The
first cause may be found in an erroneous opinion concerning the
difference between mental and written arithmetic. Written arithmetic is
mental arithmetic merely, with a record at given stages of the process
of what at that point is accomplished. But, as written arithmetic tends
to lessen the power of the pupil for the performance of those operations
that are purely mental, he should be subjected, each day, to a searching
and rapid drill in mental arithmetic also. This neglect on the part of
teachers explains the singular fact that pupils, well trained in mental
arithmetic, after attending to written arithmetic for three or six
months, appear to have lost rather than gained in their knowledge of the
science as a whole.

The second cause of failure may be found in the fact that rules,
processes and simple methods of solution, contained in the books, are
substituted for the power of comprehension by the pupil. He should be
trained to seize an example mentally, whether the slate is to be used or
not, and hold it until he can determine by what process the solution is
to be wrought. Nor is it a serious objection that he may not at first
avail himself of the easiest method. The difference between methods or
ways is altogether a subordinate consideration. There may be many ways
of reaching a truth, but no one of them is as important as the truth
itself. The text-books should contain all the facts needed for the
comprehension and the solution of the examples given; the teacher should
furnish explanations and other aids, as they are needed; but the
practice of adopting a process and following it to an apparently
satisfactory conclusion, without comprehending the problem itself, is a
serious educational evil, and it exerts a permanent pernicious
influence.

The remarks I have now made upon methods of teaching, which may seem to
have been offered in a spirit of seve