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will say, "There are seven." How many were there before t "Ten." How many have I taken away? "Three." Did these oblongs undergo any other change ?" You have moved that (pointing to it) nearer to the other." In order to vary these processes as much as possible, the chxldren should be desired to count the number of fingers on one or both hands, the number of buttons on their jackets or waisteoats, the number of chairs or forms in the room, the number of books placed on a table or book-shelf, or any other object that may be near or around them. By such exercises, the idea of number and the relative positions of objects would soon be indelibly impressed on their minds, and their attention fixed on the subject of instruction.

These exercises may be still farther varied, by drawing, on a large slate or board with chalk, lines, triangles, squares, circles, or other figures as under.

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Having chalked such figures as the above, the children may be taught to say, " One line, one triangle, one circle, one square—two lines, two triangles, two circles, two squares— three lines, three triangles, three circles, three squares," &c . which may be continued to twelve or twenty, or any other moderate number. They may be likewise taught to repeat the numbers either backwards or forwards, thus: "One triangle, two triangles, three triangles, four triangles"—" Four circles, three circles, two circles, one circle." The nature of the four fundamental rules of arithmetic may be explained in a similar manner. Drawing five squares or lines on the board, and afterwards adding three, it would be seen that the sum of 5 and 3 is eight. Drawing twelve circles, and then rubhing out or crossing three of them, it will be seen that if 3 be taken from 12, nine will remain. In like manner the operations of multiplication and division might be illustrated. But it would be needless to

dwell on such processes, as every intelligent parent and teacher can vary them to an indefinite extent, and render them subservient both to the amusement and the instruction of the young. From the want of such sensible representations of number, many young people have been left to the utmost confusion of thought in their first arithmetical processes, and even many expert calculators have remained through life ignorant of the raiionalx of the operations they were in the hahit of performing.

When the arithmetical pupil proceeds to the compound rules, as they are termed, cart should be taken to convey to his mind a well defined idea of the relative value of mcmry— the different measures of length, and their proportion to one another—the relative bulks or sizes of the measures of solidity and rapacity —angular measures, or the divisions of the circlesquare measure—and the measure of time. The value of money may be easily represented, by placing six penny pieces or twelve halfpennies in a row, and placing a sixpence opposite to them as the value in silver; by laying five shillings in a similar row, with a crown piece opposite; and twenty shillings, or four crowns, with a sovereign opposite as the value in gold; and so on, with regard to other species of money. To convey a clear idea of measures of length, in every school there should be accurate models or standards of an inch, a foot, a yard, and a pole. The relative proportions which these measures bear to each other should lie familiarly illustrated, and certain objects fixed upon, either in the school or the adjacent premises, such as the length of a table, the breadth of a walk, the extent of a bed of flowers, &c. by which the lengths and proportions of such measures may be indelibly imprinted on the mind. The number of yards or poles in a furlong or in a mile, and the exact extent of such lineal dimensions, may be ascertained by actual measurement, and then posts may be fixed at the extremities of the distance, to serve as a standard of such measures. The measures of surface may be represented by square boards, an inch, a foot, and a yard square. The extent of a perch or rod may be shown by marking a plot of that dimension in the school area or ganlen; and the superficies of an acre may be exhihited by setting oft" a square plot in an adjacent field, which shall contain the exact number of yards or links in that dimension, and marking its boundaries with posts, trenches, furrows, hedges, or §ther contrivances. Measures of capacity and solidity should be represented by models or standard measures. The gill, the pint, the quart, and the gallon, the peek and the bushel, should form a part of the fixrniturc of every school, in ordcr that their relative dimensions may be clearly perceived. The idea of a tolid fact may be represented by a box made exactly of that dimension; and the weights used in commerce may be exhibited both to tho eye and the sense of feeling, by having an ounce, a pound, a tlonc, and a hundred-weight, made of cast-iron, presented to view in their relative sizes, and by causing the pupil occasionally to lift them, and feel their relative weights. Where these weights and measures cannot be conveniently obtained, m general idea of their relative size may bo Imparted by means of figures, as under.

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dngular meature, or the divisions of the circle, might be represented by means of a very large circle, divided into degrees and minutes, formed on a thin deal board or pasteboard; and two indexes might lie made to revolve on its centre, for the purpose of exhibiting angles of different degrees of magnitude, and showing what is meant by the measurement of an angle by degrees and minutes. It mieht also be divided into twelve parts, to mark the signs or great divisions of the zodiac. From the want of exhibitions of this kind, and the necessary explanations, young persons generally entertain very confused conceptions on such subjects, and have no distinct ideas of the difference between minutes of time, and minutes of tpnet. In attempting to convey an idea-of the relative proportions of duration, we should begin by presenting a specific illustration of the unit of time, namely, the duration of a wrrW. This may be done by causing a pendulum of 39f inches in length to vibrate, and desiring the pupils to mark tho time whi^h intervenes between its passing from one side of the curve to the other, or by reminding them that the time in which we deliberately pronounce the word twenty-one, nearly corresponds to a second. The duration of a nimu/e may ho shown by causing the pendulum to vibrate 60 times, or by

counting delibeiately from twenty to eighty. The hours, half hours, and quarters, may bo illustrated by means of a common clock; and the pupils might occasionally be required to note the interval that elapses during the performance of any scholastic exercise. The idea of weeks, months, and years, might be conveyed by means of a largo circle or long stripe of pasteboard, which might be made either to run along one side of the school, or to go quite round it. The stripe or circle might be divided into 365 or 366 equal parts, and into 12 great divisions corresponding to the months, and 52 divisions corresponding to the number of weeks in a year. The months might be distinguished by being painted with different colours, and the termination of each week by . a black perpendicular line. This apparatus might be rendered uf use for familiarizing the young to the regular succession of the months and seasons; and for this purpose they might be requested, at least every week, to point out on the circle the particular month, week, or day, corresponding to the time when such exercises are given.

Such minute illustrations may, perhaps, appear to some as almost superfluous. But, in the instruction of the young, it may be laid down as a maxim, that we can never be too minute and specific in our explanations. We generally err on the opposite extreme, in being too vague and general in our instructions, taking for granted that the young have a clearer knowledge of first principles and fundamental facts than what they really possess. I have known schoolboys who had been long accustomed to calculations connected with the compound rules of arithmetic, who could not tell whether a pound, a stone, or a ton, was the heaviest weight—whether a gallon or a hogshead was the largest measure, or whether they were weights or measures of capacity— whether a square pole or a square acre waa the larger dimension, or whether a pole or a furlong was the greater measure of length. Confining their attention merely to the numbers contained in their tables of weights and measures, they multiply and divide according to the order of the numbers in these tables, without annexing to them any definite ideas: and hence it happens that they can form no estimate whether an arithmetical operation be nearly right or wrong, till they arc told tho answer which they ought to bring out Hence, likewise, it happens that, in the process of reduction, they so frequently invert the order of procedure, and treat tons as if they were ounces, and ounces as if they were tons. Such enors and misconceptions would generally be avoided were accurate ideas previously conveyed of the relative values, proportions, and capacities of the money, weights, and measures used in commerce.

Again, in many Floor and roof.

cases, arithmetic,
processes might be g
illustrated by dia- -'
grams, figures, and j
pictorial represen- i
tations. The fol-
lowing question is
stated in "Hamil-
ton's Arithmetic,"
as an exercise in
simple multiplica-
tion—" How many -
square feet in the jj
floor, roof, and 9
walls of a room. S
25 feet long, 18
broad, and 15 high?
It is impossible to
convey a clear idea
to an arithmetical
tyro, of the object
of such a question,
or of the process
by which the true
result may be olt-
tained, without fig- Breadth, 18.

ures and accompanying explanations. Yet no previous explanation is given in the book, of what is meant by the square of any dimension, or of the method by which it may be obtained. Figures, such as the foregoing, should accompany questions of this description.

The idea of superficial measure, and the 6

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be represented as above—6x6=36, and 9X4=36.

By such a representation it is at once seen what is meant by a square foot, and that the product of the length by the breadth of any dimension, or of the side of a square by itself, must necessarily give the number of square feet, yards, inches, &c., in the surface. It will also show that surfaces of very different shape*, or extent as to length or breadth, may contain the same superficial dimensions. In the same way we may illustrate the truth of such positions as the following:—That there are 144 inches in a square foot—9 square feet in a square yard—160 square poles in an acre— 640 square acres in a square mile—27 cuhical feet in a cuhical yard, &c . For example, the number of square feet in a square yard, or in two square yards, &c., may be represented in either of the following modes.

1 Square Yard.

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When the dimensions of the mason work of a house are required, the different parts of the building, which require separate calculations, as the side-walls, the end-walls, the gables, the chimney-stalks, if, should be separately delineated; and if such delineations are not found in the hooks where the questions arc stated, the pupil, before proceeding to his calculations, should bo desired to sketeh a plan of the several dimensions which require his attention, in order that he may have a clear conception of the operations before him. Such questions as the following should likewise be illustrated by diagram*. "Glasgow is 44 miles west from Edinburgh; Peebles is exactly south from Edinburgh, and 49 miles in a straight line from Glasgow. What is the distance between Edinburgh

GLASGOW. 41 Miles. FDJUst'RC.H.



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and Peebles V This question is taken from ■ Hamilton's Arithmetic," and is inserted as one of the exercises connected with the extraction of the Square Root; but no figure or explanation is given, excepting the following foot-note. "The square of the hypothenusc of a right-angled triangle, is equal to the sum of the squares of the oilier two sides." It should be represented as on page 92.

In a similar manner should many other examples connected with the extraction of roots be illustrated. The following question can scarcely be understood or performed, without an illustrative figure, and yet there is no figure given, nor hint suggested on the subject, in the book from which it is taken. "A ladder, 40 feet long, may be so placed as to reach a window 33 feet from the ground on one side of the street; and by only turning it over, without moving the foot out of its place, it will do the same by a window 21 feet high on the other side. Required the breadth of the street?" The following is the representation that should be given, which, with a knowledge of the geometrical proposition mentioned above, will enable an arithmetical tyro to perform the operation, and to perceive the reason of it.


By this figure the pupil will see that his calculations must have a respect to two rightangled triangles, of which he has two sides of each given to find the other sides, the sum of which will be the breadth of the street. The nature of fractions may be illustrated in a similar manner. As fractions are parts of a unit, the denominator of any fraction may be considered as the number of parts into which the unit is supposed to be divided. Tho following fractions, jj, J, J_, may therefore be represented by a delineation, as follows:

9 parts.

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By such delineations, the nature of a fraction, and the valut of it, may be rendered obvious to the eye of a pupil. A great many other questions and piocesses in arithmetic might, in this way, be rendered clear and interesting to the young practitioner in numbers; and where such sensible representations have a tendency to elucidate any process, they ought never to be omitted. In elementary books on arithmetic, such delineations and illustrations should frequently be given; and, where they arc omitted, the pupil should be induced to exert his own judgment and imagination, in order to delineate whatever process is susceptible of such tangible representations.

I shall only remark further, on this head, that the questions given as exercises in the several rules of arithmetic, should be all of a practical nature, or such as will generally occur in the octwil business of life—that tho suppositions stated in any question should all be consistent with real facts and occurrences— that facts in relation to commerce, geography, astronomy, natural philosophy, statisties, and other sciences, should be selected as exercises in the different rules, so that the pupil, while engaged in numerical calculations, may at the same time be increasing his stock of general knowledge—and that questions of a trivial nature, which are only intended to puzzle and perplex, without having any practical tendency, be altogether discarded. In many of our arithmetical books for the use of schools, questions and exercises, instead of being expressed in clear and definite terms, are frequently stated in such vague and indefinite language, that their object and meaning can scarcely be appreciated by the teacher, and far less by his pupils: and exercises are given which have a tendency only to puzzle and confound the learner, without being capable of being applied to any one useful object or operation. Such questions as the following may be reckoned among this class. "Suppose jE2 and f of J of a pound sterling will buy three yards and $ of | of a yard of cloth, how much will of J of a yard cost*" "The number of olars in a school was 80; there were onehalf more in the second form than in the first; the number in the third was $ of that in the second; and in the fourth, $ of the third. How many were there in each form V

In some late publications, such as " Butler'!



Arithmetical Exercises," and " Chalmers' Introduction to Arithmetic," a considerable variety of biographical, historical, scientific, and miscellaneous information is interspersed and connected with the different questions and exercises. If the facts and processes alluded to in such publications, were sometimes represented by accurate pictures and delineations, it would tend to give the young an interest in the subject of their calculations, and to convey to their minds clear ideas of objects and operations, which cannot be so easily imparted by mere verbal descriptions; and consequently, would be adding to their store of genial information. The expense of books constructed on this plan, ought to be no obstacle in the way of their publication, when we consider the vast importance of conveying well-defined conceptions to juvenile minds, and of rendering every scholastic exercise in which they engage interesting and delightful.

Section V.—Grammar.

Grammar, considered in its most extensive sense, being a branch of the philosophy of mind, the study of it requires a considerable degree of mental exertion; and is, therefore, in its more abstract and minute details, beyond the comprehension of mere children. Few things are more absurd and preposterous than the practice, so generally prevalent, of attempting to teach grammar to children of five or six years of age, by making them commit to memory its definitions and technical rules, which to them are nothing else than a collection of unmeaning sounds. In most instances they might as well be employed in repeating the names of the Greek characters, the jingles of the nursery, or a portion of the Turkish Alcoran. The following is the opinion of Lord Kaimes on this point:—" In teaching a language, it is the universal practice to begin with grammar, and to do every thing by rules. I affirm this to be a most preposterous method. Grammar is contrived for men, not for children. Its natural place is between language and logic: it ought to close lectures on the former; and to be the first lectures on the latter. It is a gross deception that a language cannot be taught without rules. A boy who is flogged into grammar rules, makes a shift to apply them; but he applies them by rote like a parrot Boys, for the knowledge they acquire of a language, arc not indebted to dry rules, but to practice and observation. To this day, I never think without shuddering, of Disputer's Grammar, which was my daily persecution during the most important period of my life. Deplorable it is that young creatures should be so punished, without being guilty of any fault, more than sufficient to produce a disgust at learning, instead of pro

moting it. Whence then this absurdity ,f persecuting boys with grammar rules V

In most of our plans of education, instead of smoothing the path to knowledge, we have been careful to throw numerous difficulties and obstacles in the way. Not many years ago, we had two characters for the letter s, one of them so like the letter f, that, in many cases, the difference could not be perceived. We had likewise comjxmrul Utters, such as ct, fl, fh, &c., joined together in such an awkward manner, that the young could not distinguish them as the same letters they had previously recognized m their separate state; so that, in addition to the ungracious task of learning the letters of the alphabet in their insulated state, under the terror of the lash, they had to acquire the names and figures of a new set of characters, before they could peruse the simplest lessons in their primers. Such characters, it is to be hoped, are now forever discarded. We have still, however, an absurd practice in our dictionaries and books of reference, which tends to perplex not only our tyros, but even our advanced students, when turning up such works—I mean the practice of confounding the letters I and J, and the letters U and V, which are as distinct from each other as a vowel is from a consonant; so that all the words beginning with J must be sought for under the letter I, and the words beginning with V, under the letter U, causing to every one a certain degree of trouble and perplexity, when searching for words beginning with any of these letters. Most of our school Dictionaries and Encyclopedias are still arranged on this absurd principle, which should now be universally discarded.

In the construction of our books of Grammar for the use of children,—instead of facilitating this study, we have done every thing to render it as dry and intricate as possible. We have definitions, general rules, exceptions to these rules, declensions and conjugations, profusely scattered throughout every part of these scholastic manuals, and a cart-load of syntactical rules and examples, all of which must of course be crammed, like a mass of rubbish, into the memories of the little urchins, although they should not attach a single correct idea to any portion of such scholastic exercises. Nothing can be more simple than the English verb, which, unlike the Greek and Latin verb, has only two or three varieties in its termination; yet, we perplex the learner with no less than tix different tentet—the present, the imperfect, the perfect, the pluperfect, the first future, and the future perfect,—while nature and common sense point out only tkres distinctions of time in which an action may be performed; namely, the past, the present, and the future, which of course are subject

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