67. Some teachers lay great stress on the explication, step by step, of the smaller and smallest components of the subject, and insist on a similar reproduction on the part of the pupils. Others prefer to teach by conversation, and allow themselves and their pupils great freedom of expression. Others, again, call especially for the leading thoughts, but demand that these be given with accuracy and precision, and in the prescribed order. Others, finally, are not satisfied until their pupils are self-actively exercising their minds in systematic thinking. Various methods of teaching may thus arise; it is not necessary, however, that one should be habitually employed to the exclusion of the rest. We may ask rather whether each does not contribute its share to a many-sided culture. In order that a multitude of facts may be apprehended, explications or analyses are needed to prevent confusion; but since a synthesis is equally essential, the latter process may be started by conversation, continued by lifting into prominence the cardinal thoughts, and completed by the methodical independent thinking of the pupil: _clearness_, _association_, _system_, _method_. In teaching we need to have (1) _clearness_ in the presentation of specific facts, or the elements of what is to be mastered; (2) _association_ of these facts with one another, and with other related facts formerly acquired, in order that assimilation, or apperception, may be adequately complete; (3) when sufficient facts have been clearly presented and sufficiently assimilated, they must be _systematically_ ordered, so that our knowledge will be more perfectly unified than it could be did we stop short of thorough classification, as in the study of botany, or of the perception of rules and principles, as in mathematics and grammar; (4) finally the facts, rules, principles, and classifications thus far assumed must be secured for all time by their efficient _methodical_ application in exercises that call forth the vigorous self-activity of the pupil. These four stages of teaching may be considered fundamental, though varying greatly according to the nature of the subject and the ability of the pupil. It is good exercise for a pupil to take long, rapid steps when able to do so; it is hopeless confusion to undertake them when they are too great or too rapid for his capacity. These four stages in methods of teaching conceived to be essential, form the nucleus of an interesting development in the Herbartian school, under the title of “The Formal [_i.e._ Essential] Steps of Instruction.” The leading ideas will be further described in a subsequent paragraph (70). 68. On closer inspection we find that instead of being mutually exclusive, these various modes of instruction are requisite, one by one, in the order given above, for every group, small or large, of subjects to be taught.