"I should like to ascertain," says the teacher, "how great is the difference of rapidity, with which different members of the class work in addition. I will give you another example, and then notice by my watch, the shortest and longest time required to do it."
The result of the experiment was, that some members of the class were two or three times as long in doing it, as others.
"Perhaps you think," said the teacher, "that this difference is altogether owing to difference of skill, but it is not. It is mainly owing to the different methods adopted by various individuals. I am going to describe some of these, and as I describe them, I wish you would notice them carefully, and tell me which you practice.
There are then three modes of adding up a column of figures, which I shall describe."
1. "I shall call the first counting. You take the first figure, and then add the next to it, by counting up regularly. There are three distinct ways of doing this.
(a.) Counting by your fingers. ("Yes sir.") You take the first figure,-suppose it is seven, and the one above it, eight. Now you recollect that to add eight, you must count all the fingers of one hand, and all but two again. So you say seven-eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen."
"Yes sir," "Yes sir," said the scholars.
(b.) "The next mode of counting is to do it mentally, without using your fingers at all, but as it is necessary for you to have some plan to secure your adding the right number, you divide the units into sets of two each. Thus you remember that eight consists of four twos, and you accordingly say, when adding eight to seven, seven;-eight, nine;-ten, eleven;-twelve, thirteen;" &c.
(c.) "The third mode is, to add by three, in the same way. You recollect that eight consists of two threes and a two; so you say, seven;-eight, nine, ten;-eleven, twelve, thirteen;-fourteen, fifteen."
The teacher here stops to ascertain how many of the class are accustomed to add in either of these modes. It is a majority.
2. "The next general method is calculating. That is, you do not unite one number to another by the dull and tedious method of applying the units, one by one, as in the ways described under the preceding head, but you come to a result more rapidly by some mode of calculating. These modes are several.
(a.) Doubling a number, and then adding or subtracting as the case may require. For instance in the example already specified; in order to add seven and eight, you say, "Twice seven are fourteen and one are fifteen;" ("Yes sir," "Yes sir,") or "Twice eight are sixteen, and taking one off, leaves fifteen. ("Yes sir.")
(b.) Another way of calculating is to skip about the column, adding those numbers which you can do most easily, and then bringing in the rest as you best can. Thus, if you see three eights in one column, you say three times eight are twenty-four, and then you try to bring in the other numbers. Often in such cases, you forget what you have added and what you have not, and get confused, ("Yes sir,") or you omit something in your work, and consequently it is incorrect.
(c.) If nines occur, you sometimes add ten, and then take off one, for it is very easy to add ten.
(d.) Another method of calculating, which is, however, not very common, is this. To take our old case, adding eight to seven, you take as much from the eight to add to the seven as will be sufficient to make ten, and then it will be easy to add the rest. Thus, you think in a minute, that three from the eight will make the seven a ten, and then there will be five more to add, which will make fifteen. If the next number was seven, you would say five of it will make twenty, and then there will be two left, which will make twenty-two. This mode, though it may seem more intricate than any of the others, is in fact more rapid than any of them, when one is little accustomed to it.
These are the four principal modes of calculating which occur to me. Pupils do not generally practice any one of them exclusively, but occasionally resort to each, according to the circumstances of the particular case."
The teacher here stopped to inquire how many of the class were accustomed to add by calculating in either of these ways; or in any simpler ways.
3. "There is one more mode which I shall describe: it is by Memory. Before I explain this mode I wish to ask you some questions which I should like to have you answer as quick as you can.
How much is four times five?-Four and five?
How much is seven times nine?-Seven and nine?
Eight times six?-Eight and six?
Nine times seven?-Nine and seven?"
After asking a few questions of this kind, it was perceived that the pupils could tell much more readily what was the result when the numbers were to be multiplied, then when they were to be added.
"The reason is," said the teacher, "because you committed the multiplication table to memory, and have not committed the addition table. Now many persons have committed the addition table, so that it is perfectly familiar to them, and when they see any two numbers, the amount which is produced when they are added together comes to mind in an instant. Adding in this way is the last of the three modes I was to describe.
Now of these three methods, the last is undoubtedly the best. If you once commit the addition table thoroughly, you have it fixed for life; whereas if you do not, you have to make the calculation over again every time, and thus lose a vast amount of labor. I have no doubt that there are some in this class who are in the habit of counting, who have ascertained that seven and eight for instance, make fifteen; by counting up from seven to fifteen, hundreds of times. Now how much better it would be, to spend a little time in fixing the fact in the mind once for all, and then when you come to the case, seven and eight are-say at once "Fifteen,"-instead of mumbling over and over again, hundreds of times, "Seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen."
The reason then, that some of the class add so slowly, is not probably because they want skill and rapidity of execution, but became they work to a great disadvantage, by working in the wrong way. I have often been surprised at the dexterity and speed with which some scholars can count with their fingers, when adding, and yet they could not get through the sum very, quick-at least they would have done it in half the time, if the same effort had been made in travelling on a shorter road. We will therefore study the addition table now, in the class, before we go on any farther."
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The foregoing narratives, it is hoped, may induce some of the readers of this book to keep journals of their own experiments, and of the incidents which may, from time to time come under their notice, illustrating the principles of education, or simply the characteristics and tendencies of the youthful mind. The business of teaching will excite interest and afford pleasure, just in proportion to the degree in which it is conducted by operations of mind upon mind, and the means of making it most fully so, are, careful practice, based upon, and regulated by, the results of careful observation. Every teacher then should make observations and experiments upon mind a part of his daily duty, and nothing will more facilitate this, than keeping a record of results. There can be no opportunity for studying human nature, more favorable than the teacher enjoys. The materials are all before him; his very business, from day to day, brings him to act directly upon them; and the study of the powers and tendencies of the human mind is not only the most interesting and the noblest that can engage human attention, but every step of progress he makes in it, imparts an interest and charm, to what would otherwise be a weary toil. It at once relieves his labors, while it doubles their efficiency and success.