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The first class is taught all about the numbers, genders, cases, persons, tenses, and moods. The etymology and meaning of every difficult word are explained, and pieces are parsed every day. Every difficult word in the reading lesson presents a subject for a question or two. I explain, where a word may be a verb, where it may be an adjective, and where it may be a noun. I have found it to be very useful to employ the first class boys, in parsing a verse (which is written on the slates), after the following example.
“The way was long, the wind was cold,
Was carried by an orphan boy.' “The" article, because it limits the signification of the noun “way.”
way" noun, the name of a thing. “ was” verb, it affirms something of the noun
way.” “Jong” adjective, it qualifies the noun “way.”
the" article, it limits the signification of the noun wind.” “ wind" ) noun, the name of a thing. And so on with the rest of the verse. After this manner the pupil is compelled to think, and to put down what he thinks. Another day he writes down the number, gender, case, person, &c., &c., and gives his reason for everything he asserts. This is a sure method, and so greatly does it contribute to correctness of judgment, that out of abundance of questions I put to my scholars, I seldom, when the question is rightly understood, get an improper answer.
The rules of syntax are learned, and occasionally one is written out legibly on the slate ; and bad sentences, illustrating the violation of the rule, are placed under. The rule is then driven into the mind, and the pupil is shown how to apply his knowledge. I think it to be a good plan, to teach what is right, by showing what is wrong. Sentences of false construction are placed before the boys, which with their knowledge of syntax they can readily correct. But question must follow question, and the teacher must employ all his art of explaining, illus. trating, and of fixing the attention. There is not so much skill required in teaching syntax : the difficulty is over when etymology is mastered. As to orthography and prosody, I do not trouble my head much about them. The former is best learned by giving the attention to words when reading, and by copying sentences.
Your obedient servant,
F. R. Smalley, near Derby.
AN OBJECTION TO TOO MUCH ORAL INSTRUCTION. Rev. Sir,--Since it is the general practice to teach English grammar in our national schools, if not with a view to impart a knowledge of our language, at least to afford a mind-strengthening course of study, the method to be preserved with respect to eliciting the greatest possible
benefit therefrom, may be deemed a subject worthy of some sideration. There is a great predilection evinced in our large schools for what may be called vivd voce teaching, more especially with regard to instruction in grammar, and although this mode of teaching may have its advantages, when the subject is in the hands of one who is acquainted with the capacities of children's minds, still there is a danger attending it which claims great attention from those who are anxious to see the studies of the school exert an influence in disciplining the mind. The danger I advert to, is that of rendering study easy and soon acquired, instead of allowing the mind to exercise itself in such an investigation of the subject as would lead to an invigorating exertion of the reasoning faculties. A child accustomed to have the whole of his studies simplified in such a manner as to render that acquisition an affair requiring very little mental labour, will be but ill-prepared in riper years to encounter difficulties connected with subjects requiring close attention, and could calculate upon very little success from his own unassisted exertions.
As far, therefore, as discipline of the mind is concerned, we cannot be too cautious as to how much we do for a child which he might do for himself
, thereby fostering a timid feeling for support and explanation with respect to matters not beyond the capacity of his mind to unravel and comprehend.
On this account, in teaching grammar, or any other subject requiring close application, I should prefer the use of books of a well-arranged and authorized stamp, with a view to the committal to memory of those parts which convey clear and comprehensive definitions. By such a method the ground work might be laid upon which vivd voce teaching could be usefully employed, more, however, in seeking to draw out what may have been acquired by strength of mind alone, than with an intention to simplify and explain.
In subjects of the highest importance, such as religious instruction, we all know the value of sound views committed to memory, and why should not the same system be pursued in matters of minor import, but, nevertheless, requiring that firm foundation which oral communication can never effect?
Your most obedient servant, Sept. 19, 1845.
INSTRUCTIONS IN ARITHMETIC.
* From (Lat.) di. Division.*
vido, divide. 49. What is the price of one yard of stuff, when four yards cost 20 shillings?
If I knew the price of one yard, by repeating this four times, or by multiplying it by 4, I should find the price of 4 yards 20s.
Let me then call the price of one yard X, and I shall have x x 4 = 20s.
By means of the table of products in section 48, I shall find 5 X 4 = 20; hence x will equal 5, that is, the single yard will cost 58.
This operation or process is called Division. Division, therefore, is an operation by which, when the product of two factors, and one of the two factors are given, we inquire what is the other factor ?
The given product is called the dividend, the given factor the divisor; whilst the factor sought is termed the quotient :*
* From (Lat.)
quoties, how thus, in the last question, 20 is the dividend, 4 is the divisor, and 5 is the quotient.
50. We wish to distribute 24 shillings equally among six poor persons, how many shillings should each of them receive ?
I might place these poor men in a row, as for instance the letters and give to each a shilling at a time, until the sum
1 1 was exhausted ; in which case it would be seen
1 that each person had received 4 shillings.
Or, again, I might reason thus: If I call wc the number of shillings which each person will have to receive, the six persons will receive 6 times x, or 6 x ; and as the sum to be divided among them is 24 shillings, we shall have 6 x = 24.
The question might likewise be answered according to the definition of division in the preceding paragraph, for the product, 24, is known, as likewise one of the factors, 6; we have therefore only to look for the other factor, or x, in the table of products (in section 48); or, again, in accordance with the words of the question itself, we might define division to be an operation by which we divide a number into as many equal parts as there are units in another number.
51. How many yards of stuff can I purchase for 20s. when one yard costs 5s. ?
I can buy as many yards as there are 5s. (which is the price of one yard) in 20s.; that is to say, the number of yards I can purchase will be equal to the number of times I can subtract 5s. from 20s. or 5 from 20. In taking away 5 once, I shall have
20 - 5 = 15 twice
15 5 = 10 three times
5 four times
5 - 5 = 0 Whence I perceive that 5 is contained exactly 4 times in 20; or, in other words, that I can purchase 4 yards of stuff for 20s. when 1 yard costs 4s.
Again, I can reason thus :—To have the price of a certain number of yards, I may repeat the price of one yard as many times as there are yards; that is, I might multiply the price of one yard by the number of yards.
Thus, if x represents the number of yards which we may buy with 20s., and if 1 yard costs 5s., we shall have 5 x x = 20; in which case the question would be answered as in section 49; for we have then to seek the factor x, knowing both what is the product, and what is the other factor, or 5.
Again, referring to the original question, we may say, that division is
likewise an operation by which we subtract a certain number from some other given number, as many times as we can, and the quotient shows how often the divisor is contained in the dividend.
52. Hence also we perceive, that these three views of the process of division are in fact one and the same ; for they at last resolve themselves into this, that division, whether by one or other of the methods proposed, brings us to the multiplication of two factors, when the product and one of the factors are known. By a diagram this may be made still more evident. Let the figure
ABCD, which represents 20, or the product of two factors, 5 and 4, be formed of four rows of 5 squares, or of 5 rows of 4 squares. If then the product, or 20, and the factor, 5, are known (or given), the figure A B C D is likewise given, as
also the side A B, which is the base to a number of similar squares ; and in this case the number of rows of 5 on A D, or 4, will be the other factor.
The figure ABCD, too, is naturally divided into 5 equal parts, or into 5 rows of 4.
Again, if we take away from A B C D a row of 5 squares as many times as we can, it will be seen that it is possible to do this 4 times, since there are 4 rows of 5 similar squares.
This rule, which we have just given, will solve any question in division.
53. Division is signified by two dots, :, or by a horizontal line between two dots, +, or by a horizontal line between the dividend and
20 the divisor, ; and for the sake of brevity, instead of saying 20 di
5 vided by 5, we may say 20 by 5.
54. If multiplication is the repeated addition of a given number a certain number of times, division is subtraction, in which we take away one number from another as many times as we can ; or, in other words, division is the converse of multiplication. For, ex. gr., to form the product 20, I add 5 + 5 + 5 + 5; and to divide 20 by 5, I take away 5 four times from 20, and thus by one process we destroy the effect of the other :
5 This principle may be expressed by saying, that to divide a product by one of its factors, is to suppress or cancel this factor; for, in dividing 20 or 5.4 by 4, the factor 4 is suppressed or cancelled, and we have for a quotient the other factor, 5; and, conversely, in order to multiply a quantity by a number which divides it, we have only to suppress or
20 cancel this divisor ; for x 4 = 20.
4 For 20 being both multiplied and divided by 4 in this case, the two operations are contrary the one to the other; and as by dividing 20 (or 5.4) by 4, we cancel the 4 which is one of the factors, so by multiplying
20 by 4, the factor 4 is once more brought back. Thus no effect takes
20.4 place, and the number 20 remains unchanged, or
4 55. Multiplication and division mutually assist each other.
Thus, if we to multiply 8 by 5, I observe that 10 = 5 x 2. If, then, I multiply 8 by 10, I shall obviously introduce the factor 2, which is in excess, or more than is required. It can, however, be readily suppressed or cancelled; for 8=4x2, and we shall have 8x5= 4 x 2 x 10
= 4 x 10 = 40; or, in other words, in the expression 2 4 x 2 x 10
the two twos, in the dividend and in the divisor, destroy 2 each other. Again, if it be required to divide 60 by 5, I might for the 5 substi
60 tute 10, and say, = 6, thus cancelling the 10, according to the
10 definition of division in section 54; but in cancelling 10, I have not only, as required by the original question, cancelled the factor 5, but likewise the factor 2, for 10 = 5 x 2. We must therefore bring back the 2 by multiplication, and in that case in figures the expression would be
10 56. Addition, subtraction, and multiplication may be viewed as complete operations, but this is not the case with division ; for let it be required to divide 27 by 6, and we perceive that there is no whole number which, multiplied by 6, will give 27; or, in other words, we cannot divide 27 into six equal parts without a remainder, or six is not contained an exact number of times in 27. In such a case, we seek then for a number which, multiplied by 6, will give a product which, without exceeding 27, may approach as near as possible. This will be 4; for 6 x 4 = 24, and 6 X 5 = 30.
Again, if we divided 27 into 6 equal parts, each part would contain 4 units, and there would be three units in excess, or as a remainder; and the same result would follow if we took away 6 from 27 as many times as we could do so; for, after having taken away 6 four times from 27, there will remain 3.0
In considering division as a separation into parts, we conceive that to part 27 into 6 equal parts, we must separate in the first instance 24; since to each part we must add the sixth part of 3. At present we would only indicate the division, 27 24 + 3 24 3
= 4 +
6 57. When a number is divisible exactly by another number, the one is said to be a multiple of the second, and the other is said to be a submultiple and a divisor of the other.
likewise say that the first number is divisible by the second; thus 24 is a multiple of the numbers