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The mode of teaching arithmetic, and the text-books, used for the purpose, in a great portion of our country, are radically defective. Much of arithmetic is practised at school, but little is learned.
The scholar is put to ciphering without adequate mental preparation, and is referred to the direction of rules, whose phraseology and principles are to a learner equally obscure. By a tedious course of practice, perhaps he acquires a certain mechanical dexterity in performing operations; but no sooner does he enter upon the business of life, than he abandons the rules of his book, and, in his own way, learns so much of arithmetic as his occupation requires.
Whether the following treatise is calculated to afford any remedy for the defects I have alluded to, others will decide. I shall spare myself the task of a prefatory detail of what “the author conceives” to be its advantages, and will only add, that the design and execution of the work, have cost me much time and labor.
F. EMERSON. Boston, January, 1832.
NOTE TO TEACHERS. It will be most advantageous for young scholars, to go through with all the Oral Arithmetic before they enter upon the Written Arithme ic. Older scholars, however, after performing the exercises in the first chapter Oral Arithmetic, may pass immediately to the exercises in the first chapter of Written Arithmetic: and after concluding this chapter, may take up the two second chapters in the same order; and thus proceed through the book.
Much time has been wasted in some of our schools, by the practice of teaching individually, instead of teaching in classes. If this practice has been owing in any degree to the arrangement of text-books, it is hoped the present arrangement will afford a remedy. There can be no more objection to a distinct classification of a school for the purpose of teaching arithmetic, than there is to a like classification for the purpose of teaching orthography: and the advantages of class-instruction in the former branch, are as great as those in the latter.
The examples contained in the first six chapters, do not require the use of the slate. The answers, with the process of obtaining them, and the reasons which justify the process, are to be given orally. For example, the following question may be supposed to give rise to the subjoined exercise.
Example. A trader purchased 9 barrels of flour, at 7 dollars a barrel, and sold the whole for 68 dollars. What did he gain in the trade? Pupil. He gained five dollars. Teacher. How do you perceive it?' Pupil. If one barrel cost seven dollars, nine barrels must have cost nine times seven dollars, which is sixty-three dollars. He must have gained the difference between sixty-three dollars and sixty-eight dollars. 63 from 68 leaves 5.'
Learners should not be confined to any form of expression in solutions their reasoning should be their own. By a little practice, they will acquire an astonishing acuteness of apprehension, and facility of expression.
Section 1. When we have a large number of articles to count, such as quills, nuts, cents, &c., we may, if we please, count them by tens. Let us suppose we have a quantity of cents before us, and proceed to count them as follows.
We first count out ten cents, and lay them in a pile. We then count out ten more, and lay them in another pile; then ten more for another pile; and thus we continue to count out ten at a time, until we have counted ten piles. We put these ten piles together, and they make a large pile containing One Hundred cents.
Again we count out ten cents at a time, until we have counted ten small piles, as before. We
We put these together, and they make a large pile containing one hundred, like the hundred we first counted. We have now counted two hundred cents, and they lie in two large piles.
Having learned what is meant by two hundreds, we proceed to count out one hundred cents more; and after placing them by the side of the two hundreds, the three piles make three hundreds. Four large piles will be four hundreds; five piles will be five hundreds; six piles will be six hundreds; seven piles will be seven hundreds; eight piles
be eight hundreds; nine piles will be nine hundreds; and when we have counted out ten of these piles, we put the whole together. They make a pile still larger, and the number of cents contained in it is One Thousand.
Examine the arrangement of dots enclosed in the lines below, and find how many there are in each enclosure. Observe, that the figures standing over the several enclosures, represent the number of dots contained therein.
Example 1. Which of these numbers is the greatest, One, or Ten, or One Hundred, or One Thousand ?
2. How many ones are there in a ten?
3. How many tens are there in a hundred ?
5. Ten ones make what number? Ten tens make what number? Ten hundreds make what number?
6. What figures stand to represent the number ten? 7. What figures stand to represent one hundred ? 8. What figures stand to represent one thousand ?
SECTION 2. If one hundred scholars were in school, and one scholar more should come in, the number of scholars would then be one hundred and one; and would be expressed in figures thus ;-101. Again, if you had one hundred books, and you should buy two books more, you would then have one hundred and two books, and their number would be expressed in figures thus; -102.
In Part First, you learned to read figures expressing all numbers, from One to One Hundred. You will now see, in the following columns, how the figures stand to express numbers, from One hundred, to Two hundred.
100 One hundred,
120 one hund. and twenty, 101 one hund. and one,
121 one hund. and twenty-one, 102 one hund. and two,
122 one hund. and twenty-two, 103 one hund. and three, 123 one hund. and twenty-three, 104 one hund. and four, 105 one hund. and fivé, 130 One hund. and thirty. 106 one hund. and six, 107 one hund. and seven,
140 One hund, and forty. 108 one hund. and eight, 109 one hund. and nine, 150 One hund. and fifty. 110 one hund. and ten, 111 one hund. and eleven, 160 One hund. and sixty. 112 one hund. and twelve, 113 one hund. and thirteen, 170 One hund. and seventy. 114 one hund. and fourteen, 115 one hund. and fifteen, 180 One hund. and eighty. 116 one hund. and sixteen, 117 one hund. and seventeen, 190 One hund. and ninety. 118 one hund, and eighteen, 119 one hund. and nineteen, 200 Two hundred.
Edward's mother gave him one hundred walnuts, his sister gave him sixty, and his brother gave him eight; making together, one hundred and sixty-eight. Being required to tell what figures would express the number of his walnuts, Edward looked over the columns of figures on the last page, and discovered, (as you may), that 1 means one hundred, whenever two figures are standing at the right hand of it; and, that 6 means sixty, whenever one figure is standing at the right hand of it. He therefore said, "1, 6, 8, are the figures."
1. How many tens does the figure 6 represent, when there is one figure standing at the right of it?
2. What are 6 tens usually called, in reading numbers ?
3. How many tens does the figure 4 represent, when there is one other figure standing at the right of it?
4. What are 4 tens usually called, in reading numbers ?
5. What number does the figure 1 represent, when there is one other figure standing at the right of it?
6. What number does the figure 1 represent, when there are two other figures standing at the right of it?
7. What are 1 hundred and 5 tens usually called ? 8. What are 1 hundred and 9 tens usually called ? 9. What are 1 hundred and 3 ones usually called ? 10. What are 1 hundred and 8 ones usually called ? 11. What are 8 tens and 2 ones usually called ?
12. What are 1 hundred, and 7 tens, and 5 ones usually called, in reading numbers ?
Note to Teachers. Require the learners to read the numbers expressed in the following columns, without recourse to the preceding columns. 109 172 104 168
113 127 190 110 140
147 145 121 132 122
169 163 143 155 195
183 181 165 176 177
103 119 187 198 159
125 136 154 186 131
The comparisons on the next page will show you, that all the hundreds are expressed in the same manner that one hundred is expressed.