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with the most important arithmetical operations performed with whole:
The third part is mostly practical, and composed of such rules and
NOTE. In his pregress through the second part the pupil should be
numbers by fig. tables, &c. 16 questions
24 Sec. IX. Geometrical det.
That frequent exercises in mental computations, havea salutary influ. ence upon the mind, by inducing hahits of attention, by strengthening the memory, and by producing a promptness of recollection, is, at present, very generally admitted. And, that exercises of this kind sliould be more extensively introduced into our primary schools, is aeknowl. edged, and even urged, by our most experienced and successful teach. ers. The success, which has, in most cases, attended the introduction of Intellectual Arithmetic into schools, has been such, as would doubtless appear incredible to those unaccustomed to witness it; but expe. rience has shown that children may be made acquainted with the first principles of Arithmetic at as early an age as they can be taught thie Alphabet and its most simple combinations. We have seen, says Dr. Griscom, a class of girls, whose ages everage not more than nine years, by the force of memory, and a few plain rules, multiply seven or eight figures by an equal number, enumerate and announce accurately ihe product and then extract the square root of this large product, and state the root and the remainder, witbout varying a figure from the truth.
In the ordinary course of instruction, Arithmetic has been studied only by the boys; and by them it has usually been defered to the very last portion of their attendance at school. The consequence has been, that few bave become familiar with its first principles, before they have been obliged to quit school and enter upon the business of life. Commencing the study of Arithmetic at this advanced period, the scholar is sensible that he bas but little time to devote to it, but, being determined to cipher through his book, he applies bimself withi diligence, yet he hurries on from rule to rule with such rapidity, that he learns nothing as he ought. He may indeed reach the end and thus accomplish his principal purpose; but, of what he has gone over, 'scarcely a trace remains upon his mind. He has not even made him self thoroughly acquainted with the elements of the science, nor bas he made himself so familiar with the rules as to derive from them ang considerable advantage in the traasaction of business.
It is asserted with confidence, that children, after having learned to talk, cannot too soon be made acquainted with numbers, and exercise ed in mental computations. But great care should be taken that these exercises be adapted to the age and capacity of the child-that the questions proposed, be such as the child can fully comprehend. And as young children are incapable of the exercise of abstraction, the instructer will find it necessary to begin by employing sensible objects. These should be placed before the child, and the first questions proposed should relate to the objects themselves, and be solved by them. Questions may then be asked respecting things which are not present; and the child may soon be led to conceive the objects before him to represent men, cents, or any other things you please. In performing these exercises the child will at length discover that numbers are not inherent qualities of the objects them.selves, but that they merely de. note a succession of similar quantities, and may be applied as well to one kind of quantity as another. After this discovery the child will find but little difficulty in forming a conception of abstract numbers, that is, of numbers, or successions, without applying them to any spe. cific objects.
By repeating and varying these simple operations, children will soon become familiar with the fundamental principles of Arithmetic and their application to practical purposes. They will at the same time be acquiring habits of attention, and a promptness of computation, which will be of inestimable value to them in after life. And this may be done in our primary schools, as an amusement and relaxation to the scholars, without interrupting, in the least, their other pursuits. The proper place to commence these instructions is in our summer schools. These, it is true, are usually taught by females, many of whom have not bad the advantages of much arithmetical instruction. But this defect in their qualifications, is not owing to a want of capa. city to learn, but to a fault of the times when the study of Arithmetic was regarded as propier only for boys. But those times are passing by and with them this defect will vanish. A moderate share of attention to the subject would enable every young lady, who en. gages in teaching, to give instruction in the Intellectual Arithmetic contained in this work, and it is believed that tbey would find themselves amply repaid for this attention by the improvement of their owa minds. By beginning with children at the commencement of their going to school, every boy and girl of ordinary capacity may be made more thoroughly acquainted wiih the principles of Arithme. tic before they arrive at the age of ten years, than most of our scholars are on leaving school, after having plodded through all the rules of Arithmetic in the ordinary way. Some knowledge of Arithmetic is no less necessary to the female sex than to our own; and experience has proved, that, if the course, here recommended, be pursued, they will not be found less capable of proficiency in this science. It is hoped that our instrnuters, both male and female, will take this subject into consideration and unite their efforts in bringing about a reformasion so desirable in the course of arithmetical instruction.
1. In commencing a course of instruction in Intellectual Arithmetic with very young children, it should be the teacher's first object to learn them to coont. For this purpose beans, small blocks of wood, marks on a slate or paper, or some other sensible objects must be employed. It wonld perhaps be advisable to use no more than five counters at first, and in selecting these, care should be taken that they resemble each other as nearly as possible, that the child may not be led to suppose that the names used in counting denote a difference among the objects employed. Having called the little class around him, the instructer should begin by laying down one of the counters, which he has provided, and which we shall bere suppose to be beans, and saying, there is one, require the children to repeat after him, one. Then, putting down another, he should say, one and one are two. Another bean may then be laid down, and the children taught in like manner to count ihree; and so on to five. After the children have learned to count five with facility, five more beans may be taken and the chil. dren taught in the same way to count ten; after which they may be taught, by the help of the beans, to answer the following questions :
2. í. How many beans are one beans and two beans ? bean and one bean more?
15. How many beans are sevet 2. How many beans are two beans and two beans ? beans and one bean?
16. How many beans are eighi 3. How many beans are three beans and two beans ? beans and one bean ?
17. How many beans are two 4. How many beans are four beans and three beans ? beans and one bean ?
18. How many beans are three 5. How many beans are five beans and three beans? beans and one bean?
19. How many beans are four 6. How many beans are six beans and three beans ? beans and one bean ?
20. How many beans are five 7. How many beans are seven beans and three beans ? beans and one bean ?
21. How many beans are six 8. How many beans are eight beans and three beans ? beans and one bean ?
22. How many beans are seven 9. How many beans are nine beans and three beans ? beans and one bean ?
23. How many beans are two 13. How many beans are two beans and four beans ? beans and two beans?
24. How many beans are three 11. How many beans are three beans and four beans ? beans and two beans?
25. How many beans are four 12. How many beans are four beans and four beans ? beans and two heans?,
26. How many beans are five 13. How many beans are five beans and four beans ? beans and two beans?
27. How many bean
are six 14. How many beans are sixbeans and four beans ?
* This Section is designed for rery young children ; older ones may com. mence at Section II.
28. How many beans are two 38. How many cents are two beans and five beans ?
cents and two cents ? 29. How many beans are three 39. How many cents are three beans and five beans?
cents and tiro cents ? 30. How many beans are four 40. How many plumbs are three beans and five beans ?
plumbs and three plumbs? 31. How many beans are five 41. How many nuts are four beans and five beans?
nuts and three nuts ? 32. How many beans are two 42. How many feet has one beans and six beaus ?
horse ? 33. How many beans are three 43. How many feet have two beans and six beans?
horses ? 34. How many beans are four 44. How many hands have two beaus and six beans ?
boys ? 35. How many beans are two 45. How many hands have four beans and seven beans ?
boys? 36. How many beans are three 46. How many hands have five beans and seven bear's ?
boys ? 37. How many beans are two 47. How many legs are there to beans and seven beans ?
3. 1. How many beans are two times two beans ? times one bean?
17. How many beans are three 2. How many beans are three times three beans ? times one bean?
18. How many beans are four 3. How many beans are four times two beans ? times one bean?
19 How many beans are five 4. How many beans are fiveltimes two beans ? times one bean :
20. If I give two boystwo plumbs 5. How many beans are six times a piece, how many plurbs will one bean?
both have ? 6. How inany beans are seven 21. At one cent a piece how times one hean ?
much do four apples cost? 7. How many beans are eight) . 22 At two cents a piece how times one bean?
inuch do three pears cost ? 8. How many beans are nine 23. What animal bas as many times one bean ?
again feet as you have? and how 9. How many beans are ten many feet has it? times one bean?
24. How many eyes have two 10. How many beans are one boys? time one beap ?
25. How many ears have three 11. How many beans are one boys ? time two beans ?
26. How many eyes and ears 12. How many beans are two have two boys ? times tyo beans ?
27. How many gloves do two 13. How many beans are two pair of hands require ? times three beans ?
28. How many ear rings must I 14. How many beans are two get for three pair of ears? times four béans ?
29. I gave a boy three cents and 15. How many beans are two he gave me twice as many apples, tipies five beans?
how many did he give me? 16. How many beans are threel 30. What do three oranges cast