Multiply 123456789
by 123456789
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Now, when you bring the example to one of the classes, address the pupils as follows:
"I have contrived for you a very difficult sum. It is the most difficult one that can be made, with the number of figures contained in it, and I do not think that any of you can do it, but you may try. I shall not be surprised if every answer should contain mistakes."
To the other class, say as follows:-
"I have prepared an example for you, which I wish you to be very careful to perform correctly. It is a little longer than those you have had heretofore, but it is to be performed upon the same principles, and you can all do it correctly, if you really try."
Now under such circumstances the first class will go to their seats with ardor and alacrity; determined to show you that they can do work, even if it is difficult. And if they succeed, they come to the class the next day, with pride and pleasure. They have accomplished something, which you admit it was not easy to accomplish. On the other hand, the second class will go to their seats, with murmuring looks and words; and with a hearty dislike of the task you have assigned them. They know that they have something to do, which, however easy it may be to the teacher, is really difficult for them, and they have to be perplexed and wearied with the work, without having at last, even the little satisfaction of knowing that the teacher appreciates the difficulties with which they had to contend.
2. We now come to consider the subject of rendering assistance to the pupil, which is one of the most important and delicate parts of a teacher's work. The great difference, which exists among teachers, in regard to the skill they possess in this part of their duty, is so striking that it is very often noticed by others; and perhaps skill here is of more avail, in deciding the question of success or failure, than any thing besides. The first great principle, is, however, simple and effectual.
(1.) Divide and subdivide a difficult process, until your steps are so short, that the pupil can easily take them.
Most teachers forget the difference between the pupil's capacity and their own, and they pass rapidly forward, through a difficult train of thought, in their own ordinary gait, their unfortunate followers vainly trying to keep up with them. The case is precisely analagous to that of the father, who walks with the step of a man, while his little son is by his side, wearying and exhausting himself, with fruitless efforts to reach his feet as far, and to move them as rapidly, as a full gown man.
But to show what I mean by subdividing a difficult process, so as to make each step simple, I will take a case which may serve as an example. I will suppose that the teacher of a common school, undertakes to show his boys, who, we will suppose, are acquainted with nothing but elementary arithmetic, how longitude is ascertained, by means of the eclipses of Jupiter's satellites; not a very simple question, (as it would, at first view, strike one,) but still one which, like all others, may be, merely by the power of the subdivision alluded to, easily explained. I will suppose that the subject has come up at a general exercise,-perhaps the question was asked in writing, by one of the older boys. I will present the explanation, chiefly in the form of question and answer, that it may be seen, that the steps are so short, that the boys may take them themselves.
"Which way," asks the teacher, "are the Rocky Mountains from us?"
"West," answer two or three of the boys.
In such cases as this, it is very desirable that the answers should be general, so that throughout the school, there should be a spirited interest in the questions and replies. This will never be the case, if a small number of the boys only take part in the answers; and many teachers complain, that, when they try this experiment, they can seldom induce many of the pupils to take a part.
The reason ordinarily is, that they say that any of the boys may answer, instead of that all of them may. The boys do not get the idea that it is wished that an universal reply should come from all parts of the room in which every one's voice should be heard. If the answers were feeble, in the instance we are supposing, the teacher would perhaps say;
"I only heard one or two answers: do not more of you know where the Rocky Mountains are? Will you all think, and answer together? Which way are they from us?"
"West," answer a large number of boys.
"You do not answer fully enough yet; I do not think more than forty answered, and there are about sixty here. I should like to have every one in the room answer, and all precisely together."
He then repeats the question, and obtains a full response. A similar effort will always succeed.
"Now, does the sun, in going round the earth, pass over the Rocky Mountains, or over us, first?"
To this question, the teacher hears a confused answer. Some do not reply; some say, "Over the Rocky Mountains;" others, "Over us;" and others still, "The sun does not move at all."
"It is true that the sun, strictly speaking, does not move; the earth turns round, presenting the various countries, in succession, to the sun, but the effect is precisely the same as it would be, if the sun moved, and accordingly I use that language. Now, how long does it take the sun to pass round the earth?"
"Twenty four hours."
"Does he go towards the west, or towards the east, from us?"
"Towards the west."
But it is not necessary to give the replies; the questions alone will be sufficient. The reader will observe that they inevitably lead the pupil, by short and simple steps, to a clear understanding of the point to be explained.
"Will the sun go towards, or from, the Rocky Mountains, after leaving us?"
"How long did you say it takes the sun to go round the globe, and come to us again?"
"How long to go half round?" "Quarter round?"
"How long will it take him to go to the Rocky Mountains?"
No answer.
"You cannot tell. It would depend upon the distance. Suppose then the Rocky Mountains were half round the globe, how long would it take the sun to go to them?" "Suppose they were quarter round?"
"The whole distance is divided into portions called degrees; 360 in all. How many will the sun pass, in going half round? In going quarter round?"
"Ninety degrees then make one quarter of the circumference of the globe. This you have already said will take six hours. In one hour then, how many degrees will the sun pass over?"
Perhaps no answer. If so, the teacher will subdivide the question, on the principle we are explaining, so as to make the steps such that the pupils can take them.
"How many degrees will the sun pass over in three hours?"
"Forty-five."
"How large a part of that, then, will he pass, in one hour?"
"One third of it."
"And what is one third of forty-five?"
The boys would readily answer fifteen, and the teacher would then dwell for a moment, on the general truth, thus deduced, that the sun, in passing round the earth, passes over fifteen degrees every hour.
"Suppose then it takes the sun one hour to go from us to the river Mississippi, how many degrees west of us, would the river be?"
Having thus familiarized the pupils to the fact, that the motion of the sun is a proper measure of the difference of longitude between two places, the teacher must dismiss the subject, for a day, and when the next opportunity of bringing it forward occurs, he would perhaps take up the subject of the sun's motion as a measure of time.
"Is the sun ever exactly over our heads?"
"Is he ever exactly south of us?"
"When he is exactly south of us, or in other words, exactly opposite to us, in his course round the earth, he is said to be in our meridian. For the word meridian means a line drawn exactly north or south from any place."
There is no limit to the simplicity which may be imparted, even to the most difficult subjects, by subdividing the steps. This point for instance, the meaning of meridian, may be the subject, if it were necessary, of many questions, which would render it simple to the youngest child. The teacher may point to the various articles in the room, or buildings, or other objects without, and ask if they are or are not in his meridian. But to proceed: