"When the sun is exactly opposite to us, in the south, at the highest point to which he rises, what o'clock is it?"
"When the sun is exactly opposite to us, can he be opposite to the Rocky Mountains?"
"Does he get opposite to the Rocky Mountains, before, or after, he is opposite to us?"
"When he is opposite to the Rocky Mountains, what o'clock is it there?"
"Is it twelve o'clock here, then, before, or after it is twelve o'clock there?"
"Suppose the river Mississippi is fifteen degrees from us, how long is it twelve o'clock here, before it is twelve o'clock there?"
"When it is twelve o'clock here then, what time will it be there?"
Some will probably answer "one," and some "eleven." If so, the step is too long, and may be subdivided thus:
"When it is noon here, is the sun going towards the Mississippi, or has he passed it?"
"Then has noon gone by, at that river, or has it not yet come?"
"Then will it be one hour before, or one hour after noon?"
"Then will it be eleven, or one?"
Such minuteness and simplicity would, in ordinary cases, not be necessary. I go into it here, merely to show, how, by simply subdividing the steps, a subject ordinarily perplexing, may be made plain. The reader will observe that in the above, there are no explanations by the teacher, there are not even leading questions; that is, there are no questions whose form suggests the answers desired. The pupil goes on from step to step, simply because he has but one short step to take at a time.
"Can it be noon, then," continues the teacher, "here and at a place fifteen degrees west of us, at the same time?"
"Can it be noon here, and at a place ten miles west of us, at the same time?"
It is unnecessary to continue the illustration, for it will be very evident to every reader, that by going forward in this way, the whole subject may be laid out before the pupils, so that they shall perfectly understand it. They can, by a series of questions like the above, be led to see by their own reasoning, that time, as denoted by the clock, must differ in every two places, not upon the same meridian, and that the difference must be exactly proportional to the difference of longitude. So that a watch, which is right in one place, cannot, strictly speaking, be right in any other place, east or west of the first: and that, if the time of day, at two places, can be compared, either by taking a chronometer from one to another, or by observing some celestial phenomenon, like the eclipses of Jupiter's satellites, and ascertaining precisely the time of their occurrence, according to the reckoning at both; the distances east or west, by degrees, may be determined. The reader will observe, too, that the method by which this explanation is made, is strictly in accordance with the principle I am illustrating,-which is by simply dividing the process into short steps. There is no ingenious reasoning on the part of the teacher, no happy illustrations; no apparatus, no diagrams. It is a pure process of mathematical reasoning, made clear and easy by simple analysis.
In applying this method, however, the teacher should be very careful not to subdivide too much. It is best that the pupils should walk as fast as they can. The object of the teacher should be to smooth the path, not much more than barely enough to enable the pupil to go on. He should not endeavor to make it very easy.
(2.) Truths must not only be taught to the pupils, but they must be fixed, and made familiar. This is a point which seems to be very generally overlooked.
"Can you say the Multiplication Table?" said a teacher, to a boy, who was standing before him, in his class.
"Yes sir."
"Well, I should like to have you say the line beginning nine times one."
The boy repeated it slowly, but correctly.
"Now I should like to have you try again, and I will, at the same time, say another line, to see if I can put you out."
The boy looked surprised. The idea of his teacher's trying to perplex and embarrass him, was entirely new.
"You must not be afraid," said the teacher; "you will undoubtedly not succeed in getting through, but you will not be to blame for the failure. I only try it, as a sort of intellectual experiment."
The boy accordingly began again, but was soon completely confused by the teacher's accompaniment; he stopped in the middle of his line saying,
"I could say it, only you put me out."
"Well, now try to say the Alphabet, and let me see if I can put you out there."
As might have been expected the teacher failed. The boy went regularly onward to the end.
"You see now," said the teacher to the class which had witnessed the experiment, "that this boy knows his Alphabet, in a different sense, from that in which he knows his Multiplication table. In the latter, his knowledge is only imperfectly his own; he can make use of it only under favorable circumstances. In the former it is entirely his own; circumstances have no control over him."
A child has a lesson in Latin Grammar to recite. She hesitates and stammers, miscalls the cases, and then corrects herself, and if she gets through at last, she considers herself as having recited well; and very many teachers would consider it well too. If she hesitates a little longer than usual, in trying to summon to her recollection a particular word, she says, perhaps, "Don't tell me," and if she happens at last to guess right, she takes her book with a countenance beaming with satisfaction.
"Suppose you had the care of an infant school," might the instructer say to such a scholar; "and were endeavoring to teach a little child to count, and she should recite her lesson to you in this way; 'One, two, four, no, three;-one, two, three,--stop, don't tell me,-five-no four-four-, five,---I shall think in a minute,-six-is that right? five, six, &c.' Should you call that reciting well?"
Nothing is more common than for pupils to say, when they fail of reciting their lesson, that they could say it at their seats, but that they cannot now say it, before the class. When such a thing is said for the first time, it should not be severely reproved, because nine children in ten honestly think, that if the lesson was learned so that it could be recited any where, their duty is discharged. But it should be kindly, though distinctly explained to them, that, in the business of life, they must have their knowledge so much at command, that they can use it, at all times, and in all circumstances, or it will do them little good.
One of the most common causes of difficulty in pursuing mathematical studies, or studies of any kind, where the succeeding lessons depend upon those which precede, is the fact that the pupil, though he may understand what precedes, is not familiar with it. This is very strikingly the case with Geometry. The class study the definitions, and the teacher supposes they fully understand them; in fact, they do understand them, but the name and the thing are so feebly connected in their minds, that a direct effort, and a short pause, are necessary to recall the idea, when they hear or see the word. When they come on therefore to the demonstrations, which, in themselves, would be difficult enough, they have double duty to perform. The words used do not readily suggest the idea, and the connexion of the ideas requires careful study. Under this double burden, many a young geometrician sinks discouraged.
A class should go on slowly, and dwell on details, so long as to fix firmly, and make perfectly familiar, whatever they undertake to learn. In this manner, the knowledge they acquire will become their own. It will be incorporated, as it were, into their very minds, and they cannot afterwards be deprived of it.