by their foliage. This is object teaching, of whose usefulness there can be no question. Could my own children exchange the misconceptions and crude notions formed while studying the "First Steps" and "Primary" of a popular geographical series, for knowledge like this, I should have an excellent reason for congratulating them on their good fortune.

T. W. H.

ELEMENTARY ARITHMETIC.

MR. EDITOR: At your suggestion I send you my views in regard to the proper mode of teaching elementary arithmetic. They are those which I have been accustomed to present when the subject of arithmetic has been assigned to me at Teachers' Institutes. The method here described has seemed to me to be natural and philosophical, and therefore worthy of presentation to teachers. For I think we may lay it down as a maxim in education, that a method which can be shown to be strictly in conformity with the order of nature, will be successful in practice.

But we must be careful not to confound two very different things. A philosophical method of teaching arithmetic is one thing; teaching the science of arithmetic is quite another thing. Every step in all elementary instruction should be taken in its proper order not too soon, not too late. The scanty results from much of the teaching in our schools, is in consequence of the disregard of this principle. The particular instruction imparted on a given day may be well enough in itself, but it is unseasonable. It is not adapted to the pupil's present intellectual condition. An explanation of a principle may be excellent for one class, and yet worthless for another.

In all elementary instruction I attach great importance to this matter of a natural order. One step paves the way for a second, and that second for a third, and so on. In this way progress is easy and rapid. The teacher may err by going too rapidly, or by attempting to exhaust one subject before proceeding to another. The latter errar is perhaps the more common at present. A teacher would have his class know everything pertaining to addition before doing anything with subtraction. This does not seem to me to be the wisest method. Algebra and geometry precede trigonometry; but shall we require the pupil to know all that is knowable of those two branches before entering upon the

study of this? If so, his school days would be over before he would be ready to begin.

Thoroughness is indispensable, but the most thorough teachers are not those who would exhaust subjects in elementary instruction. My impression is, that much time is lost in our lower schools because the teachers think that thoroughness requires them to elaborate every point, and subject each to microscopical examination. They are afraid of the charge of "mechanical" processes and "teaching by rote," and so they weary themselves and their pupils by long and repeated explanations, which, for all purposes of instruction at the time they are often given, are simply valueless.

What is the first thing to be done in teaching arithmetic? Teach the pupil to count objects. Generally, this has been done before the child begins to attend school. Suppose he can count ten. Then addition commences, the unit being the additive number. The figures should also be taught, so that written and mental arithmetic may be attended to simultaneously. At first, the exercises are threefold-adding objects, as, three apples and one apple are four apples; adding numbers, without slate or blackboard, as, three and one are four; and adding with the written characters on the board; as, 3+1 = 4, or. The child is supposed to have a slate and pencil when he begins to attend school, and to learn to make figures, as well as to read them, from the first.

I would say nothing of notation or numeration, and give no definitions of terms in a formal way, but aim merely to have the pupil comprehend the processes. And I would use as few words as possible in these exercises, especially those which require the board. Thus the teacher writes 2+1= 4+1 6+1=

3+1 =

3 5

1 1

, upon the board, then points to the example and requires the pupil to give the answer. Or he writes etc., then, as he points, the pupil answers 4, 6, 9, etc.

=

3,

Then we come to two as the additive number. With the aid of objects, perhaps, the pupil may form a table; as, 1+ 2 2 + 2 = 4, 3 + 2 = 5, etc., to 9 + 2 11. But whether he does or not, he is to learn such a table till he can answer any question any of the three forms given above. And of these three forms, the written one is the most satisfactory, as testing the pupil's knowledge. Many examples should therefore be given on the board, involving one and two as additive numbers, but no higher

in

2

1 1
7

8

number. Thus, etc. etc. The teacher points, and the

pupil answers, 4, 6, 8, 8, 10, 10, etc. Or, examples are written with more than two numbers; thus,

1

2

2

should be taught from the beginning to give results. He should not say, "four and two are six, and one is seven, and two are nine, and one is ten;" but, "four, six, seven, nine, ten." The process is, of course, mental, and the mind can work much more rapidly than the tongue. The latter will have enough to do to give results without delaying to record the processes.

Rapidity and accuracy are to be sought continually. The pupil is to be drilled on the various combinations that can be made with one and two as the additive numbers, until he can give the results as promptly and as accurately as the teacher himself. Having mastered these combinations, he takes three as the additive number, and follows the same method. Each day he learns a little, and is drilled on that little and also on what he had previously learned. This is what I call thoroughness in teaching elementary arithmetic. The teacher attempts but little each day, but that little is supposed to be just what the pupil needs, and is to be so learned that he will never forget it. You may call these processes mechanical, if you please, and affirm that this is rote teaching; to me it seems to be the way to lay the foundation on which to build arithmetic and all the branches of mathematics.

It is a narrow field which we are thus cultivating, I admit. There is a vast amount of knowledge pertaining to addition, of which not a word is supposed to have been said to the pupil as yet. He knows nothing of units, tens, hundreds, etc. He has never heard the words Arabic, Roman, literal, verbal, as applied to numbers. Carrying is Greek to him. He knows that a certain character represents eight, because you have told him so. So also of numbers represented by two figures. He has been taught to tell the number of his lesson in his Primer or First Reader, and of the page on which it is found. Perhaps this is learning by rote; but, with my present views, I recommend it unhesitatingly. By and by the time will come for explanations, but it has not come yet.

The knowledge thus far is mainly that of the tables. The drilling consists in using that knowledge in performing examples. It is of prime importance that no example should ever be given containing a combination which the pupil has not previously

learned. Thus, working examples is easy from the beginning. It is like giving him sentences to read containing no words which he can not call at sight. He is thus always on solid ground. He steps off with freedom and confidence. The boy can read his little sentence as perfectly as his teacher. He can work his little example as rapidly and accurately.

Having given many examples with one, two, and three as additive numbers, we proceed to four. Let the teacher exercise his judgment as to the mode in which the table shall be formed, whether by himself or by the pupil. The main thing is that the pupil is to learn it-learn it absolutely, so that the instant nine and four are given to him to be added, he can give the result. Whether the example is presented through the ear or the eye, the answer is to be given instantly.

So we go on till we have used all the numbers as far as nine as additive numbers. Thus far, the examples have all been so framed as to require no carrying. They have been composed of numbers in single rows or columns, horizontal or vertical. We have had 2+1 + 3 + 4 + 1 = 11. And

3

1

22

Of course, the pupils meanwhile have extended their counting range, and have become familiar with reading numbers with two figures, and perhaps with three. After being able to add by twos, threes, and fives, the pupil may be drilled in counting in that way. Thus, 2, 4, 6, 8, 10, 12, etc. So with the odd numbers, 1, 3, 5, 7, 9, etc. So, 5, 10, 15, 20, etc.

=

Now, we take examples with two or more columns. All possible combinations have been learned already, as tables, from 1+1 2 to 99 18. And in the exercises, the pupil has had 15 +8, and 25 +8, as well as 5+ 8. been such as to involve no carrying. given. We take the example,

8

15

But all his examples have
This method is now to be

22

15

The sum of the first column is written in full. Then under the second column, the sum of that is written. Adding the 3 to the 1 under which it stands, we have 45. In various ways the teacher can show the pupil that this is accurate, without saying anything of tens and units. After a few examples written out in full in this way, the teacher shows

the pupil that it is just as well to add the figure 1 of the 15 to the second column without the trouble of writing it down. Now we may practice on larger numbers, and also give the usual method of proof by adding downwards.

If all the steps have been taken in the order indicated, and the teacher has been faithful, addition, as a method, has now been learned, and the pupil is ready to proceed to subtraction. The range of his knowledge is narrow, indeed, but he is master of it. He knows nothing of the science of arithmetic, but he does know how to work examples in addition, and this knowledge will be to him of incalculable value.

Marietta College, May 13, 1867.

I. W. A.

READING.

Some years ago it was my good fortune to be thrown for a short time into the company of a son of Prof. Wm. Russel, the elocutionist, himself an elocutionist of rare powers. It is hardly necessary to say that it was a delight to listen to his readings, recitations, and instructions. One evening, just before the hour for "lecture," he was approached by a young man who handed him a short selection, made from the current literature of the day, with the request that he would render it that evening. After reading it over, Mr. R. returned it, saying to the young man: "It is a fine selection, my friend, and does credit to your taste, but as I have not time to study it, I can not undertake to read it. I should neither do justice to the selection nor to myself." Here, thought I, is the key to good reading. Study insures success in this department as well as in all others.

This is the question, then, which I wish briefly to discuss: What should constitute the study of a reading lesson, and how should the recitation be conducted so as to secure it?

First, as to the study. This, it will at once be admitted, must vary with the advancement of the pupils. Beginners can not be expected to study with the same scope as those who have several years the start; yet in all cases the work should begin at the same point, and should be extended in exact accordance with the capacity of the class. But here as elsewhere whatever is required in study, must first be definitely assigned. Setting aside all that class of teachers who practically require no study, but