ON TEACHING GRAMMAR IN NATIONAL SCHOOLS. REV. SIR,-The teaching of grammar, as of some other subjects, in schools for the poor, may be the occasion of many questions; among which, that of "what good will the knowledge of them do children in their station of life?" is one of the most prominent. If this question be taken in the sense which appears at first sight, I believe that few will maintain that it does them much good; for how few of them there are, who, from the language they use the moment they are out of school, appear to know that they can speak improperly. No; the most erroneous expressions and amusing provincialisms are used by them with utter thoughtlessness; and should no remark be made, they never think that there is any error in what they say. Even if they knew that there was, they would not speak properly amidst those whose pleasure it would be to submit them to ridicule on that account. Should this be disputed, I will venture to affirm, that not one in fifty of this class of children, who have been best educated, recollect any amount of this and like subjects when they are grown up; or who, by their language, could be discovered ever to have known them. No; their language does not depend upon what is taught them, but upon the class of people with whom it falls to their lot to spend their life: or why do children who move in a higher rank speak so properly, comparatively speaking, before they know what grammar means? Thus, if the apparent usefulness-producible results were all we trusted to, we might give it up at once; and not that alone, for few there are, who, having whilst young, in the old phrase even, 'gone through the arithmetic, can, when twenty-one, work a sum in compound division, unless they have been in an occupation where arithmetic was required. I am far, Sir, from attempting to discourage the teaching of grammar and other of those higher subjects so seldom taught in national schools. On the contrary, I would encourage them by every means in my power. Not, however, as an expression above would seem to insinuate, after "the old method," but one in which the children would learn, not from memory alone, but the understanding also. I do not say that memory should not be used; for whether the instruction be given from books, or altogether orally, the pupils must be so much acquainted with the subject, as to have it in some measure impressed on their memory, before the understanding can be brought sufficiently to bear upon it. The fact is, the chief thing for which I would teach these subjects, is the cultivation of the understanding; and for this purpose I do not think that the children can ever be taught too much, even if it be granted that it will afterwards be forgotten. How frequently must every one, who takes an interest in the management of schools, have been surprised at the senseless answers given to questions asked of those whose understandings have not been cultivated? How often when what do you mean by the communion of saints?" has been asked, the answer has been "the forgiveness of sins," &c., &c. This should not be: children should be taught to think; and what can At the very do this to a greater degree than teaching them grammar? outset, as there are but eight parts of speech in the English language, and as the same words may belong to several of these, it must necessarily require judgment to point out to which they belong. Then again, to distinguish the gender, case, mood, person, &c., and to exemplify each rule in syntax, do, in my opinion, greatly exercise the understanding, nor can these be done without calling it into action; and I venture to assert, that the boy or girl who is taught to parse well, will show this improvement of the intellectual faculties in other subjects, although they may never have been taught them as a lesson. If grammar is to be taught, or rather learnt from a book only, as was done when I was a boy, it is of little consequence what system be followed; whether Murray's, or M'Culloch's, the practical good derived from each will be about the same. If it is to be taught as it should be, and as, I am not afraid to say ere long in many places it must be, it is of great moment that some good method should be adopted, and some one book, if possible, prepared, as a text book for the teacher. I have taught grammar for a time, and with some success, and the children have never had a grammar in their hands; not that I think it wrong to allow them one, but because we have none in the school, and the children are many of them too poor to buy one. I cannot tell you whose system we follow, but I can state part of our plan, which is as follows:-at the commencement, and without much preparation, I tell them that there are eight parts of speech, and teach them their names; afterwards the definitions. The next step is to apply the knowledge gained, by requiring words in the different parts of speech from each child; and when this becomes moderately easy, we take up books, and pick out certain classes of words, or take each word successively; merely saying that such a word is such a part of speech. At the same time, I let nothing be said without a reason; for I should ask, "Why is 'John' a noun ?" and so on. This is much more difficult than without the book; for some of the words depending upon situation or termination, are not so easily distinguished; but they do not get so accustomed to a certain number of words, as to render the lessons monotonous and less useful. I then show them that there is a difference in nouns when used for one or more, explaining that this is number; and afterwards by degrees introduce proper nouns, cases, moods, &c.: so that a boy may be able to parse tolerably well, before he knows that there are regular and irregular verbs, &c. I will just add, that I teach no more than is essential, and for that reason use no more than three moods and two tenses; and leave out many useless distinctions in pronouns. I prefer noun to substantive, and adnoun to adjective, as I find it much easier, the boys knowing something of etymology; for having taught them the derivation of the names of the parts of speech, they are never at a loss for the definition, which would not be so readily given were those words retained. Rev. Sir, your humble servant, August, 1845. J. V. N. INSTRUCTIONS IN ARITHMETIC. (Continued from page 205.) Multiplication.* 40. How much will 5 yards of stuff cost, if each yard lings? * From (Lat.) multus, many, and plico, fold. cost 4 shil Now 5 yards are 5 times 1 yard; therefore 5 yards times 4 shillings. If, then, we call have a 4+ 4 + 4 + 4 + 4; will cost 5 the price of 5 yards, we shall but 5 = 1 + 1 + 1 + 1 + 1. Therefore, after effecting the addition of the 5 fours (which are 20), we find that x, or the price of 5 yards, is found when the price 4 shillings is known, in the same manner as 5 is formed from 1. Or, again, that we have repeated 5 times the price of 1 yard, or 4 shillings, in order to find the price of 5 yards. This process, which is, after all, only a species of addition in which the same number is added several times to itself, is called multiplication. It may, however, be considered under two points of view. 41. Thus, multiplication is an operation by which a number called the multiplicand, is repeated as many times as there are units in another number, called the multiplier. Or, multiplication is an operation by which a number is formed with a certain number called the multiplicand, in the same manner as another number called the multiplier is formed from unity. The result of multiplication is called the product,* and the multiplicand and the multiplier are called the factorst of the product. * From (Lat.) pro, forth, and duco, bring. + From (Lat.) facio, make or do. 42. In order to express, for instance, that 4 has to be repeated 5 times, we say, 4 multiplied by 5; and for this word " multiplied," in written arithmetic, is substituted the sign X, or sometimes a full stop or dot. Thus, 4 × 5, or 4.5 = 20, or (expressed in words), 4 multiplied by 5 equals 20, and sometimes, for the sake of brevity, for the words "multiplied by," we substitute the word "into;" thus we say, 4 into 5 equals 20.” 43. How many square yards of plank are required for a floor which is 8 yards in length and 5 in breadth? F D C Let A B C D represent the floor, in which the line AB, being divided into 8 equal parts, will represent 8 yards in length, and the line A D being divided into 5 such parts, will represent the breadth (or 5 yards). G A E Now it is evident, from the diagram anн nexed, that there will be 8 squares, repreB senting 8 square yards upon the line  B, and that the whole of the floor contains 5 such rows of 8 squares in each row, that is, 5 times 8, or 8 x 5. We may also say that there are 5 squares, representing 5 square yards upon the line A D, and that the figure ABCD is formed of 8 rows of 5 of such square; so that we shall likewise be able to state that the surface A B C D = = 5 x 8. Therefore a rectangular* surface, such as A B C D, contains as many square units as there are units in the product of two numbers representing the length of the sides A B and A D. ; * From (Lat.) rectus, right, and angulus, an angle. And hence we see that 8 × 5 = 5 x 8 for both products represent the same figure; and since the same argument will apply in all cases of a similar kind, we deduce the general inference, that in the multiplication of two factors, the result is the same whatever may be the order in which the two numbers are placed, or the multiplication is effected. 44. A wall 5 yards in length, 3 in height, and 2 in thickness, is to be built; how many cubic yards will it contain? D A F B E If we note the number of yards con- G may consider the solid contents un- it is found to contain either 3 rows of 5, or 5 or twice 5 rows of 3, or 2, or 2 rows of 3 Again, the face BCE G is formed of 3 rows of cubic feet, and the whole wall of 5 times 3 rows of 2, or or of 5 times 2 rows of 3, or 3 × 2 × 5. In a word, the face CEFD contains 2 rows of 5, or 5 rows of 2, and the whole wall 3 times 2 rows of 5 (i. e.) or 3 times 5 rows of 2, (i. e.) or, as the wall will contain 30 cubic yards. 5 x 2 x 3 ; 2 x 5 x 3; 2 × 5 × 3 = 10 × 3 = 30, 45. Hence it appears, that a rectangular parallelopiped* such as A E, contains as many cubic units as there are units in the product of the numbers which represent its length, breadth, and height. * From (Gr.) para, by the side of, allelon, one another, epi upon, and pedon, the ground. The six products which have been found to represent the same figures A E, are all equal the one to the other, that is to say, AE 5.3.2 = 3.5.2 2.3.5 = 3,2.5 = 5.2.3 = 2.5.3. And as the same process of reasoning is applicable to any like case and to any similar figures representing it, we may lay down the general principle, that in every case where there are three factors, the order in which the multiplication is effected does not affect the product (or result). It may likewise be observed, that in the above examples with 3 fac tors, we have six different combinations, and that it is not possible to have more than six. For when there are 2 factors, 5 and 3, there can only be 2 combinations, viz. 5.3 and 3.5, whereas when a third factor, 2, is added, we may place it either after or before, or between the other 2 factors, and this principle, which is as useful as it is important, may be shown to extend itself to any number of factors whatever. * For instance, let it be asked what is the product of three numbers, 5.9.2? Then, because 5 x 2 = 10; therefore 5.2.9 10.9 90. 46. The multiplication of a number by the product of two or more numbers, can be effected either by multiplying by the product, or by multiplying by each of the factors from which the product resulted. Thus, in multiplying 3 by 10, I may adopt either of the three following modes, and say 3 × 10 = 3.2.5 = 3.5.2, 10 being equal to 5 × 2 = 2 x 5; for, let the figure A B C D be formed of C10 squares, each of which contains three square yards, (or feet, or inches, &c.), and we have either 10 times 3, or 3 x 10; but A B C D is formed of twice 5 squares (containing 3, &c.), or of 5 times two squares (containing 3, &c.) D A 3 3333 33333 B Therefore ABCD = 3 x 10 = 3.5.2 = 3.2.5. It is also evident, that, in the expression 3 × 10 we may substitute in the place of the product 10, its factors 5.2, or 2.5. 47. The above principle is also clearly illustrated by combining the process of addition with that of multiplication; this will be seen in the following table, which indeed explains itself: = = *For let a and b be two factors, and P their product, the type or formula of multiplication will then be a x b = P, or a.b P, or in fact a b P; for it has been agreed by mathematicians, that the multiplication of quantities expressed by letters shall be signified by the union of these letters without any intervening sign. Now we have shown that a b = bd, and that a bc ac b = bac = bca = cab = c ba; and to extend this to any number, a b c d e f, it will be sufficient to prove that any two may be transposed; for then it will be easy to repeat the same process with each factor, ex. gr., To show that the factor d and e may change places; let us call the product of the preceding factors a b c d = P, or let a b c d = P; but (by No. 46), Pde = Ped, therefore abcde abc ed. Hence, also, a b c d e f |