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and only considers those acquirements as lasting and reliable which were obtained by self-activity of the mind. It is evident, that in a book only outlines or sketches for lessons can be given ; a great many incidental questions and observations will suggest themselves during the course of a conversation; and a mother who has once begun to try these instructions, will soon find that she requires very few hints as to the order and general character of her instructions, and that she does best not to bind herself too strictly to the models here given.

The idea of publishing the present treatise gradually presented itself to me. Being engaged in schools, as well as in private families, in England ever since 1851, I had ample opportunity of observing the deficient knowledge and surety of my pupils in Arithmetic, owing chiefly to the want of a rational system in their first instruction. But, as Mathematics were not my particular department, I took no further notice of it until, from changed circumstances, I became more lively interested in the general proficiency of my pupils. I then undertook one special branch of that science-the fractions; and being myself, as well as my friends and pupils, well satisfied with the result of my teaching, wrote down a sketch of the method I followed, and published it in the JOURNAL OF EDUCATION. But even during those lessons in fractions, I was very often checked and unpleasantly disappointed by finding that the original foundations of my pupils' knowledge and skill in Arithmetic were so very loose and unsatisfactory; their science and ability were strictly limited by a certain number; the common operations of adding, subtracting, and multiplying, by mental Arithmetic, went up to a given number ; beyond that, pen and ink were absolutely necessary, and then even the slightest deviation from a mechanical rule put them out. Thus it often happened that an otherwise satisfactory demonstration or solution of a problem was impeded by such common-place difficulties. I now inquired more closely into the matter, examined the respective schoolbooks, attended lessons, and had conversations with masters and parents about the subject. The result of my studying the books generally used in schools, I have already given ; I did not find one intended for the teacher himself, containing an exposition of the method to be followed in teaching. As for the lessons, it may be easily inferred, from the want of proper guides for the masters, and the generally acknowledged deficiency of systematical training of teachers in public as well as private establishments, that they do not materially differ from the mere mechanical treatment of the subject in the respective books. What I heard from parents and tutors, only confirmed me in my idea that nothing more beneficial for this branch of education could be done than to write some explicit models of lessons, such as they ought to be given in schools or private families. It had been my particular good fortune to meet with many pupils who had received their first instruction from their own mother, and, as is always the case, turned out very intelligent pupils. The only reason why those mothers had entirely neglected Arithmetic was, that they considered the subject too dry for little children; they did not know themselves much about it, and had no idea how to give those lessons. The following pages are especially also intended for such mothers; if they only try, they will soon find that it is not so difficult after all, and their efforts will certainly have a most beneficial effect upon the intellectual development of their children.

First Notions. The Numbers from 1 to 10. Master. (Having placed the whole class before him in front of the black-board.) What do I hold here in my hand ?-Answer. A. pen. M. Say : That is a pen.

How many pens are there -A. One pen. M. Say again : That is one pen.

(We observe here in the beginning, that it is very important to insist always upon full sentences. Such complete answers not only prove that the question has been well understood, but also induce the pupil to think more distinctly, and express himself more correctly.]

M. Now say all together : That is one pen. (Taking another pen into his left hand :) How many pens have I here ?-A. That is also one pen.

M. Say so all together. Now listen to what I say, and we shall afterwards see who can repeat it (putting the two pens together): One pen and one pen are two pens. Who can say the same ? (Several children do so.)

M. Say so all together.

M. Who can lift up one finger ? Do all so. Henry, show me two pens—lift up two fingers. What is more, one shilling or two shillings ? -A. Two shillings are more than one shilling.

M. How much more -A. One shilling more. Or better : Two shillings are one shilling more than one shilling.

M. How many tables, fireplaces, &c. are in this room ?-A. In this room there is one table, &c.

M. (Naming an object which is twice in the room :) How many maps are in this room ?--A. There are two maps in this room.

M. Name other objects of which there two in this room.-A. We have here two black-boards, two windows, &c.

M. What have you once on your body ?-A. I have one head, one mouth, one forehead, &c.

M. What have you twice on your body ?--A. I have two eyes, two arms, two hands, &c.

M. What is only once in this town ?-A. In our town there is one market-place, one town-hall, one inayor, &c.

M. Now look here; I make one stroke on the black-board, now two more beneath the first. Take this piece of chalk and do the same; try to make your strokes straight, and equally long and thick.

[For a first regular lesson with little children the preceding exercises are sufficient. With all preparations, getting them in order, corrections, additional questions which may suggest themselves during its course, it will take nearly half an hour; and as uninterrupted attention is required, that time must not be exceeded. The next lesson begins with a repetition of the preceding one, particularly of its first questions. Then the master continues.]

M. How many pencils do I hold here ?—A. There are two pencils.

M. (Taking another pencil) How many pencils have I here ?-A. That is one pencil.

M. Now listen to me : Two pencils and one pencil are called three pencils. John, say the same ; now Henry ; now all together. Give me one pencil, now two, now three ; always say at the same time how many you give me. What is three times in this room ? Take the chalk and

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make one stroke on the black-board, now two underneath it, and after
that three more strokes. Now say with me (pointing to the strokes) :
This is one stroke, there are two strokes, there are three strokes. Now
backwards. Say the numbers without looking at the black-board, now
backwards.

(Most of these answers are to be given first by one child, and to be
repeated by all of them.

The object of all preceding questions and exercises was not so much to make the children acquainted with the numbers one, two, three, (which they probably very well knew before they entered the schoolroom), as to accustom them to pay attention, express distinctly their ideas, and learn a few expressions which are necessary for the following exercises. To obtain this important object we purposely dwelt long on the first numbers, which in themselves offer no great difficulty. The master now goes on in the same way, varying his questions and the objects he uses for illustrating the same ; but he must never omit first to show to the eye what he intends to teach, and frequently cause his pupils to show themselves, on material objects, what they are saying about numbers. This is also the best way to fix their attention and interest them in the lessons. To avoid repetition with too little variation for the reader,—which, however, with little children is necessary,—we now suppose our pupils taught in the manner given above the numbers as far as 7. The master then begins the next lesson.]

M. Charles, say the numbers from 1 to 7; Tom, say them back-
wards ; now, all together. Count how many books I have here.-
A. One, two, three, four, five,—there are five books.

M. How many fingers do I lift up ? Count them in the same way.
Now pay attention to how many strokes I make on the black-board.
Henry, how many brothers have you ? Name them.
fingers have
you on your

left-hand ? Tell me how many days there are
in a week.-A. A week has seven days.

M. Name them. Who can name them backwards ? Which is the first day of the week, the second, the last, the fifth ?

[Several of the latter questions are, of course, a kind of deviation from the exact subject of Arithmetic, but we insert them on purpose to show that they are not altogether objectionable. It has been already mentioned that in the course of a lesson many additional questions may suggest themselves, and any master or mother knows that from experience. If such slight deviations from our principal object do not occur too often, and are not too long, they prove very useful by bringing a little more variation and interest into an otherwise dry subject. Besides, we must not forget that at this period of education a rapid progress of our pupils in one particular branch is not so much our object as a general and harmonious development of all mental faculties, and it therefore matters little if our conversation turns for a moment from the chief object.]

M. Charles, say the numbers from 2 to 6, from 3 to 7 ; backwards, from 7 to 2, &c. Which is more,—3 or 4, 2 or 5, 7 or 6? Now answer this question: Which is less,—3 or 41–A. 3 is less than 4.

M. We are now going to learn a new number. Here I take seven pens,—one, two, three, &c. ; and here is another pen which I add to the

Seven and one are called eight. Repeat that, and several

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questions as above. I make here strokes on the black-board from one to eight; here dots from one to eight; and, lastly, little rounds from one to eight.

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Show me five points, three little rounds, seven strokes, &c. Go now to your place and try to make the same on your slate. What have you to do?-A. We have to make strokes on our slate from one to eight, then points from one to eight, then little rounds from one to eight.

[At another time the master or pupils make the same backwards, either on the black-board or on the slate. When the children have thus at last become fully acquainted with the numbers from 1 to 10, we begin another exercise, in order to make them write down what they have only been saying until now.]

M. Tell me once more how many are one pen and one pen, one stroke and one stroke -A. One stroke and one stroke are two strokes.

M. Let us try to write that down. Look at this sign (making a little cross +), I call this "and.” How do you call this sign? This other sign (making two horizontal strokes =) I call “are.” How do you call this sign ? Now look what I am writing here :

| + | = || What is this (pointing to the first stroke) ?-A. That is one stroke.

M. How do you call this sign (pointing to the cross) ?-A. I call this " and.”

And so on : that is one stroke; that sign I call “are ;” these are two strokes.

M. I am going to read the whole line to you: one stroke and one stroke are two strokes. John, repeat the same and point to each sign as you pronounce it. (Several children do so.)

M, Charles, come here and write it once more on the black-board. I rub it out,—who can write it now once more? (Several children will present themselves, and are admitted to try.)

M. Now let us go on. How many are two pens and one pen, two strokes and one stroke? We must write that down. Here I make two strokes ; what comes next ?-A. “And."

M. Show me how I write “and." Very well ; now I have written, two strokes and—what comes nextkA. One stroke.

M. Here it is. Read once more as far as we have written. (Thus the second line is gone through.)

M. Read the whole line again ; now all together. Repeat both lines. What do you think we are going to write now 1–4. Three strokes and one stroke are four strokes.

[In this manner the whole series is gone through as far as 10, in one or more lessons. The whole sum is then read over several times by single children and the whole class, and afterwards they have to write the same on their slate. As a repetition, they may write the same with dots or little rounds.]

Addition of 2. M. William, how many books have I here !—A. You have there three books.

M. And here -A. There are two books.

M. Very well ; now I will add these three books and the two books together. Look, first I add one book to the three books—how many are there now A. Three books and one book are four books.

M. Now I add the other book-how many are there ?-A. There are now five books.

M. How many did I add to the three I had first l-A. Two.

M. Now say after me : Three books and two books are five books. (Several children and the whole class say the same.)

M. How many books have I here ?—count them.-A. There are six books.

M. Let us add two to them. How many do I add first ?-A. First I add one book.

M. And how do you say in adding one book ?—A. Six books and one book are seven books.

M. Very well ; but I must add one more, and how do you say now ? -A. Seven books and one book are eight books.

M. How many books had I first ; and how many did I add to them ? Now say again after me : Six books and two books are eight books. Here I take five pencils and two pencils. Let me see which of you is able to add the two to the five and tell me how many they are together. (One or more children will, aided by a few suggestive words and signs of the master, begin: Five pencils and one pencil are six pencils; six pencils and one pencil are seven pencils.)

M. Now say also what you have found, beginning with "therefore.”A. Therefore five pencils and two pencils are seven pencils.

M. Who can repeat that who else? Now all together.

The master now makes them find out how many are 8 and 2, 2 and 2, at last 1 and 2. In the beginning, pens, books, or similar things, used to facilitate the exercise. After some practice these objects are removed, and, at last, even the words “ books," "pens,” &c. left out. Thus the children say : Seven and one are eight, eight and one are nine ; therefore, seven and two are nine. This gradual progress, from that which is visible to the abstract, is to be observed in all following exercises. Now the children are again led on to write what they have found out.

M. Say once more, how many are 1 and 2 ? Let us write that on the black-board. What do you say first -A. One.

M. Here I write one (making one stroke). What now k-1. And.

M. Do you remember how I write, and show it me ? Well, what comes after that? In this way the master goes on and writes : 1 + ll= lll. Read the whole line over. Now say once more, how

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