PREFACE.

THERE is a considerable space of time, even after children have learned to read and spell with some facility, and before they are capable of entering upon the study of the more difficult departments of science,when something adapted to their peculiar wants is needed to occupy their minds, and amuse them during such hours of the school as are not devoted to reading and spelling. If anything can be presented during this period which will attract the attention, and give the minds of children a moderate and healthy exercise, it will relieve them of many tedious hours, and lay a foundation for their future progress in any branch of science which they may afterwards pursue.

It is the design of this little work to furnish this desideratum. The first part contains several beautiful engravings, exhibiting scenes of juvenile sports and employments, which afford materials for questions, and also assist the young pupil in their solution.

The tables preceding the lessons present every combination of numbers that is employed in the formation or the solution of the practical examples that immediately follow them. These examples commence with the most simple combination of numbers, illustrated by visible objects, and gradually advance to the more difficult, as the pupil acquires the ability to master them.

A solution of the first question in each exercise, and in every new combination of numbers, is given, not only as a model form of solution for the pupil, but also to impress upon his mind the importance of analyzing every question, and of giving a reason for each step in the operation.

The lessons in Drawing and Writing are introduced for the purpose of furnishing an interesting and instructive pastime, during the leisure hours of the pupil.

With this explanation of the object and plan of the work, the author would express a hope that no teacher will permit his pupils to advance faster than they fully comprehend the subject. If such a course is pursued, it is believed that the work is so arranged, in the variety of the questions, and the frequency of the review, that it cannot be completed without imparting to the learner a pretty accurate and thorough knowledge of the first principles of arithmetic. Portland, Sept., 1849.

SUGGESTIONS TO TEACHERS.

WHILE children are attending to the introductory lessons in this work, the teacher should give familiar illustrations upon the black-board at every recitation.

The illustrations, by visible objects, in Addition, Subtraction, Multiplication, Division, and Fractions, should also be extended, if necessary, until each member of the class fully comprehends the process of performing examples according to the principles of each subject respectively.

The teacher should also require an analysis of each question, so far as time will allow. The pupils may perform the questions mentally, and when called upon by the teacher, should first give the answer, and then the analysis, or process of obtaining it.

MENTAL ARITHMETIC.

INTRODUCTION.

EVERY experienced teacher has, undoubtedly, often seen and felt more or less difficulty in teaching children the first principles of numbers.

This difficulty, it is believed, very commonly, if not always, arises from the confused and indefinite ideas in the minds of children in regard to the real and relative value of the different numbers, either when expressed by words or figures.

It is very important, therefore, that teachers take particular pains to impart clear and correct views on this point, at the very commencement of the pupil's course. They should endeavor to illustrate plainly, that all numbers are composed of single or individual things, or quantities, and that the figures used to represent them always express a certain number of single things, units or ones, respectively.

For the purpose of doing this in an intelligible manner, and at the same time making the subject interesting to young children, the teacher may draw some visible object on the blackboard, -a tree, for example, as represented on the following page, - and then ask the same or similar questions as are there introduced, varying and repeating the exercise until each member of the class fully comprehends what is meant by the names of the different numbers, one, two, three, &c., and clearly understands their relative value; that is, whether five means a greater or less number of things than six, ten a greater or less number of things than nine, &c.

6. You may now count, as I point to them, and tell me the number of apples on each branch of the tree, on the left hand side.

7. How many apples are there on the first branch? How many are there on the second branch? Does the number two here mean more or fewer apples than the number one? How many more? Then one and one more make how many?

8. How many apples are there on the third branch? Does the number three here mean more or fewer apples than the number two? How many more? Then two and one more make how many?

REMARK. The teacher may proceed in this manner with all the branches on the left hand side of the tree, and then, commencing with the tenth branch, ask similar questions in regard to the different branches on the right hand side; thus,

9. How many apples are there on the tenth branch, on the right hand side of the tree? How many are

there on the ninth branch? Does the number nine here mean a greater or less number of apples than the number ten? How many less, &c.?*

10. You may now count, commencing on the left hand side, and tell me the number of branches on both sides of the tree.

11. Now tell me how many apples there are on the first and second branches, on the left hand side of the tree, when counted together.

12. How many are there on the first, second and third? On the first, second, third and fourth? On the first, second, third, fourth and fifth, &c.?

*The teacher may here give the names of the nun bers employed when it is required to express more than ten, as eleven, twelve, &c., and also show the class how they are composed from these first ten numbers.