The beginning of the higher mathematics may be taught children perfectly well under this method. Straight lines and angles are drawn, or constructed with little sticks, and named, and various figures thus formed. With blocks, the different geometrical figures are constructed and named--all being finished by the pupils themselves. On the blackboard certain lines are given, and with them "inventive drawing" goes on under the pupil's own suggestion. Weights and measures are learned by practical illustrations with real objects, and are thus not easily forgotten. Definition is very agreeably taught by the teacher's producing some object, say, an apple, and then making each scholar describe some quality of it, in taste, color, form, or material, and then write this word on the board. Very difficult adjectives, such as "opaque," or "pungent," or "translucent," or "aromatic," may thus be learned, besides all the simpler, and learned permanently. The old bugbear to children, spelling, is by no means so terrible under these methods. The teacher writes two initial consonants, say, "th," and each scholar makes a new word with them, and it is written on the board; or a terminal consonant is given, or certain combinations of letters are written down--say, _ough,_ in "though" and words of corresponding sound must be written underneath, or the different sounds of each vowel must be illustrated by the scholar, and the varying sounds of consonants, and so on endlessly--spelling becoming a perfect amusement, and, at the same time, training the pupil in many delicate shades of sound, and in analyzing and remembering words. Grammar is conveyed, not by that farce in teaching, and that cross to all children, grammatical rules, which are, in fact, the expressions of the final fruit of knowledge, but the teacher writes incorrectly on the blackboard, both in spelling, punctuation, and grammar, and the children must correct this; thus learning from the senses and usage, instead of from abstract rules. Reading is given as nearly as possible in conversational tones, and the old loud, mechanical sing-song is forbidden. The principle most insisted on in all this system is, that the child should teach himself as far as possible; that his faculties should do the work, and not the teacher's; and the dull and slow pupil is especially to be led on and encouraged. But, as might be supposed, the teacher's task, under the object method, is no sinecure. She can no longer slip along the groove of mechanical teaching. She must be wide-awake, inventive, constantly on the _qui vive_ to stir up her pupils' minds. The droning over lessons, and letting children repeat, parrot-like, long lists of words, is not for her. She must be always seeking out some new thing and making her pupils observe and think for themselves. Her duty is a hard one. But this is the only true teaching; and we trust that no Primary School in New York will be without a well-trained "Object-teacher."