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Entered, according to Act of Congress, in the year 1849, by
TEMPERANCE C. COLBURN, Widow of Warren Colburn,
THs character of Colburn's First Lossons is too widely and thoroughly known to make it necessary to give, in this edition, any extended statement of its principles and method. Ideas which were now at the first publicadon of this work have now, through the great change” that has taken place in elementary instruction in Arithmetic, through its Influenco, beoome the common possession of all intelligent teachers.
The careful revision of the work which has now been made has suggestod very few points in which any change seemed to be required. It has been thought that a more easy and gradual introduction would render the work more useful to the most youthful beginners.*
The use of the book with beginners demands of the teacher considerable labor in the way of proposing original questions, and devising modes of Wustration; and a short course of Introductory Lessons is prefixed, which the teacher may use as materials and hints in the first steps of the study.
* In the city of Lowell, where this book has been used from its first pubLcation, the School Committee passed & vote in December, 1846, excluding all other Arithmetics in their Primary Schools ; thus showing, in the opinion of intelligent men who acted upon their ozperlenco, that Colburn', Hirst Lessons is suficiently casy for the most javenile scholar
Tas first instructions given to the child in Arithmetic are usually given on the supposition that the child is already able to count. This indeed seems a sufficiently low requisition; and if children were taught to count ut home in a proper manner, they would have this power in & sufficient degree when they enter the primary school. But it will be found on trial that most children, when they begin to go to school, do not know well how to count. This may be proved by requiring them to count 20 beans or Kornels of corn. Few of them will do it without mistake. The difficulty is they have been taught to repeat the numerical names, one, two, three, in order, without attaching ideas to them. They learn to count without connting things. This point then calls for the teacher's first attention to lead the child to apprehend the meaning of each numerical word by using it in connection with objects.
The kind of objects to be employed.ps counters should of course be similar, as marks on the blackboard, bea is, pieces of wood, or of cork, or the balls in & numeration frame. Provided they are similar, and large enough In be seen without effort by all the clase, it is of little consequence what they are; the simpler the better, and those which the teacher devises or makes will, other things being equal, be best of all.* Not more than ten should be used or exhibited to the children in the first few lessona.
Let the class have their attention called to the teacher; and when he lays down a counter, when all can see it, let them say one ; let the teacher lay down another, and the class say two, and so on up to ten. If any of the olas become inattentive, let the teacher stop at once; and, sfter the attenton is fully centred on him, let him begin again.
* The Numeration Frame should have ten balls on & bar. Three bars will be sufficient for all the necessary illustrations. It is sometimes proposed to employ a Frame with only nine bails on a bar: the use of such a Irame, however, would be a great error in the First Lessons of Mental Arithmetis. The Frame with nine balls is designed to illustrate the idea of local valno in the decimal notation, and has as many balls as there are signifi. cant ilgares. Bat Mental Arithmetic begins with the numerical words, and requires for its Wustration on a Frame as many balls as there are simple namerical worde. These are the first ten, those above being compound. Lloven is formerl of two obsolete words, signifying, one and ton; w twelve is a compound of two words, signifying two and ters. The names above theco, thtrteen, fourteen, &c., sufficiently indicate of themselves the simple words of wblob they are formed.
After going through this addition a few times in this form, it may be varied thus. The teacher laying down the counters, one by one, as before, the class may be led to say, one and one are two, two and one are three three and one are four, &o.
The above mode of adding may be shortened by leading the class to say as follows: One and one are two, and one are three, and one are four, &c.
At any time the word designating the counter may be used along with the number, as beans, balls, pieces, marks, or books, as the case may be.
At times it will be well to give some fictitious designation to the counters, such as the teacher, or still better such as some one of the class, may choose, calling them men, sheep, horses, &c.
Next to Addition, as illustrated above, should come Subtraction. Bay Ing counted ten, let the teacher take away one, and the class be made to say, one from ten leaves nine one from nine leaves eight, &o. In Subtraction the same variations may be introduced as in Addition. No further Illustrations of this operation need be given, as the teacher's discretion wil supply all that is necessary.
In connection with these exercises, let the pupil be taught to repeat in re versed order the numerical words they have employed, countieg from on ap to ten, and then in reverse order from ten to one.
It is not to be supposed that the whole of the foregoing lesson can b. learned at one exercise. It is only a small part of it that children will al first have suficient power of attention to go over with profit. The same romark may be made respecting the following Introductory Lessons.
Lot the teacher call the attention of the class, and require them to count, and then lay down, one by one, a small number of counters, gay, for ex. ample, five; then let him separate them into two parts, as one and four, thus,
one and four are five," and require the class to say the same. Then let him divide the number into different parts, et two and three, thror and two, four and one, Giao and one and three, &c., roquiring the class with each division to name the parts and make the addi. tion. Let them always begin at the left end of the line of counters as they face them. Having exhausted the combinations of ive, let the number six be taken, giving combinations like the following:
It may be found that a lower number than five should be made the first stop in this exercise.
After the combinations of six have been exhausted, the number seven may be taken, and then succeseively, eight, nine, and ten.
As a part of this Lesson, each question in addition should be converted into a question in subtraction ; thus, five and three are eight; then, having put the two parts together which make eight, remove the three, and lead ihe class to say,
three from eight leaves five.")" The following exercise is important in this connection. Let the teacher select some number, and give one part of it, and require the class as quick 20 possible to name the complementary part. Thus let six be the number, the exerciso will be as follows. Teacher. “Now attend, six is the number I am going to name one part of it; when you hear me name it, do you al bame the other pert as quick as you can; now be ready ; five.". Class a Onc.'' Teacher:
« Forer.” Olass : « Thoo." - Teacher: " Three.” Okey : "Throo."Teacher: “Ona.” Class : " Fime,” &c.
This exercise should not be pressed too fast, but carried on gradually ed the pupil's strength of mind will allow. Special pains should be taken that the number ten be perfectly mastered in this form of combining its parts. This will give the pupi the most important aid in all his calculations to larger numbers.
Yor & number of days after beginning the above exercises, the child should not have the book at all in his hands. If the child has the book in his
possession, it will be well for the teacher to take it for a few days, and lot the pupil employ himself at his seat in writing on a slate, or with other books. In this way the child has awakened within him the idea of calcalation in numbers, without having become wearied with the reading of what excites no interest. After a few days, however, the book may be put into the pupil's hands, and he may be directed to get a lesson in Section I. In the meanwhile the Introductory Lessons should be continued, and form a part of each day'g exercise till they are finished. In this way, the pupil, in stadying his first lesson from the book, will already have learned the use of counters, and will naturally resort to them at his seat, using beans or marks on his slate for this purpose. It will be far better for him to come to the use of counters in this natural way, than to be enjoined to use them before he has been interested in witnessing their application.
The pupil, in the preceding lessons, has become acquainted with all the numbers as far as ten, regarding them either as units, or as grouped into parts of a larger whole. The next step is to carry him through the numbers from ten to twenty.
First let the class count with the objects before them from one up to twenty; then, removing all but ten, let the ten be grouped in a pile; or, if they are marks on the board, let them be enclosed by & line drawn around them, and begin to count upward from ten. “ One and ten ere eleven; two and ton are twelve ; three and ten are thirteen;"bere pause, and examine the composition of the word, thirteen-three ten, or three and ten. Show bow the three is spelt in thirteen, and also how the ten is spelt. Then proceed," four and ten are fourteen, examining the word as in the former case ; “ five and ten are fifteen ; six and ten are," - perhaps some one in the class will now be able to give the compound word; then go on, “Sevon and ten, eight and ten, nine and ten, ten and ton."
When they can give the compound words readily from the simple ones, then give them the compound word, and let the class separate it into its two component words ; thus : Teacher: "Seventeen." Class:
“ Seven and ten, &c. Thus far let the teacher be careful to present the name of the smaller of the two numbers first, for that is the order in which the compound word presents them; let the teacher say four and ten, and not ten and four. After the class have caught the analogy between the simple words and the compounds which they form, so that one instantly suggests to them the other, then the order of the words may be changed, and the ten pat first. The caution here suggested may seem to some unnecessary bat & careful observation of the mental habits of children will not fall, I think, to show its importance.
In the analysis of the compound words from ten to twenty, eloven and twelve should be omitted till the last; for, as the simple words of which they are formed are disguised or obsolete, they tend to obscure rather than olacidate the subject to the mind of a child. Having obtained the idea through the other words in the series, he may take the statement respecting these on trust
Having counted twenty, and grouped the number in two tons, let the Glass count ten more, making in all thirty, or three tens. Keeping the tens separate, let the class count ten more, making forty, or four tens. Let the class then answer such questions as the following: - Twenty are how many bons? Thirty are how many tens? Forty are how many tens? Four tens are what number? Three tens are what number! Two tens are what number?
After this, they may proceed with the higher multiples of ton, Afty, staty, soventy, eighty, ninety, a hundred.
Through the whole of this exercise, each multiple of ten should be presented in groups of ten, so as to aid the idea by the visible representation.
The pupils should be led to see the significancy of each numerical name; that thirty-seven, for example, means three tens and seven ; fifty-six means ive tens and six.
In this way the pupils may be led to anderstand the Decimal Ratio at this early stage, and no further trouble need be taken in that direction. When, in a later stage of study, he comes to the Decimal notation in written Arithmetic, be will find it only a natural mode of expressing ideas already rondored familiar in practice.
Lot the teacher stand at the board, and call the attendon of the class to what he shall writo; then, making two marks, ask, "How many marks on the board ?" When the class havo Answered, let the teacher write two more, and ask, “How many now?” and so on to the number of twelve or more. Then take a writing book or sheet of paper, and covering all but two of the marks, let the glass repeat the same process while the teacher removes the book, so as to bring two more into view at each remove ; the numbers read by the class being two, four, six, eight, ten, &c. Thon let the process be reversed, subtracting two successively, which gives, boginning with sixteen, the following, -sixteen, fourteen, twelve, ten, &c. Again the teacher may say to the class,
"'When I 'made those marks, how many did I make at a time?” Class : “Two.” — Teacher: “Did I make two more than once ?" Class : “Yes, sir, & good many times.rs Then the teacher, covering up all but two: “Now loos, how many times two are there? "'Class : “Once." Teacher: “Once two are how many ?" Then, after the class have answered, showing two more, “How many times two do you see?" “ Twice two are how many ?" Then go on be the same way with three twos, four twos, &c., to the end.
At this point the pupils may be taught the distinction between even and odd Dumbers, and be trained to repeat; rapidly the oven numbers, from two up to twenty.
The pupils may derive important aid in adding and multiplying, by group Ing tho numerical names with the voice, in something like the following manner. Teacher: “Listen now to me; one, two- three, four - five, six, How many twou did I count?” Class : “Three twob.” Teacher: “ Court three twos just as I did.” Than let the teacher nak, “Three men ton &NO