• INTELLECTUAL ARITHMETIC, UPON THE INDUCTIVE METHOD instruction. BY WARREN COLBURN, A. M. STEREOTYPED AT THE BOSTON TYPE AND STEREOTYPE FOUNDIY. HALLOWELL, ME. : BOSTON: 1836. DISTRICT OF MASSACHUSETTS, TO WIT: "District Clerk's Office. BE it remembered, that on the twenty-third day of March, A. D. 1826, in the fiftieth year of tho Independence of the United States of America, Cummings, Hilliard, and Company, of the said District, have deposited in this office the title of a book, the right whereof thoy claim as proprietors, in the words following, to wit: “ Intellectual Arithmetic, upon the Inductive Method of Instruction. By Warren Colburn, A. M." In conformity to the act of the Congress of the United States, entitled, " An Act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned :" and also to an act, entitled, “ An Act, supplement ary to an act, entitled, An Act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies during the times therein mentioned; and extending the benefits thereof to the arts of designing, engraving, and etching historical, and other JOHN W. DAVIS, prints." RECOMMENDATIONS. Sir, Boston, 15 November, 1821. I have made use of the Arithmetic and Tables, which you sometime since prepared, on the system of Pestalozzi; and have been much grau ified with the improved edition of it, which you have shown me. I am satisfied, from experiment, that it is the most effectual and interesting mode of teaching the science of numbers with which I am acquainted. Respectfully, HENRY COLMAN. Having been made acquainted with Mr. Colburn's treatise on Arithmetic, and having attended an examination of his scholars, who had been taught according to this system, I am well satisfied that it is the most easy, simple, and natural way of introducing young persons to the first principles in the science of numbers. The method here proposed is the fruit of much study and reflection. The author has had considerable experience as a teacher, added to a strong interest in the subject, and a thorough knowledge not only of this but of inany of the higher branches of mathematics. This little work is therefore earnestly recommended to of those who are employed in this branch of early instruction, with the belief that it only requires a fair trial in order to be fully approve od and adopted. J. FARRAR, Prof Math, Harvard University. Cambridge, Nov. 16 1821. HARVARD LIBRARY ALLA PREFACE. As soon as a child begins to use his senses, nature continually presents to his eyes a variety of objects; and one of the first properties which he discovers, is the relation of number. He intuitively fixes upon unity as a measure, and from this ne forms the idea of more and less; which is the idea of quantity. The names of a few of the first numbers are usually learned very early; and children frequently learn to count as far as a hundred before they learn their letters. As soon as children have the idea of more and less, and the names of a few of the first numbers, they are able to make smal] calculations. And this we see them do every day about their playthings, and about the little affairs which they are called upon to attend to. The idea of more and less implies addition; hence they will often perform these operations without any previous instruction. If, for example, one child has three ap. oles, and another five, they will readily tell how many they both have ; and how many one has more than the other. Ifa child be requested to bring three apples for each person in the room, he will calculate very readily how many to bring, if the number does not exceed those he has learnt. Again, if a child be requested to divide a number of apples among a certain number of persons, he will contrive a way to do it, and will tell how many each must have. The method which children take to do these things, though always correct, is not always the most expeditious. The fondness which children usually manifest for these exercises, and the facility with which they perform them, seem to indicate that the science of numbers, to a certain extent, should be among the first lessons taught to them.* To succeed in this, however, it is necessary rather to fur. nish occasions for them to exercise their own skill in performing examples, than to give them rules. They should be allowed to pursue their own method first, and then they should be made to observe and explain it; and, if it was not * See on this subject two essays, entitled Juvenile Studies, in the Prize Book of the Latin school, Nos. I. and II., published by Cum. mings & Hilliard, 1820 and 1821. the best, some improvement should be suggested. By fol lowing this mode, and making the examples gradually increase in difficulty; experience propes, that, at an early age, children may be taught a great variety of the most useful combinations of numbers. Few exercises strengthen and mature the mind so much as arithmetical calculations, if the examples are made sufficiently simple to be understood by the pupil; because a regular, though simple process of reasoning, is requisite to perform them, and the results are attended with certainty. The idea of number is first acquired by observing sensible objects. Having observed that this quality is common to all things with which we are acquainted, we obtain an abstract idea of rumber. We first make calculations about sensible objects; and we soon observe, that the same calculations will apply to things very dissimilar; and, finally, that they may be made without reference to any particular things. Hence from particulars we establish general principles, which serve as the basis of our reasonings, and enable us to proceed, step by step, from the most simple to the more complex operations. It appears, therefore, that mathematical reasoning proceeds as much upon the principle of analytic induction, as that of any other science. Examples of any kind upon abstract numbers, are of very little use, until the learner has discovered the principle from practical examples. They are more difficult in themselves, for the learner does not see their use; and therefore does not so readily understand the question. But questions of a practical kind, if judiciously chosen, show at once what the combination is, and what is to be effected by it. Hence the pupil will much more readily discover the means by which the result is to be obtained. The mind is also greatly assisted in the operations by reference to sensible objects. When the pupil learns a new combination by means of abstract examples, it very seldom happens that he understands practical examples more easily for it, because he does not discover the connexion until he has performed several practical examples, and begins to generalize them. After the pupil comprehends an operation, abstract examples are useful to exercise him, and make him familiar with it. And they serve better to fix the principle, because they teach the learner to generalize. From the above observations, and from his own experience, the author has been induced to publish this treatise ; in which he has pursued the following plan, which seemed to him tho most agreeable to the natural progress of the mind. GENERAL VIEW OF THE PLAN. Every combination commences with practical examples Care has been taken to select such as will aptly illustrate the combination, and assist the imagination of the pupil in per. forming it. In most instances, immediately after the practical, abstract examples are placed, containing the same numbers and the same operations, that the pupil may the more easily observe the connexion. The instructer should be careful to make the pupil observe the connexion. After these are a few abstract examples, and then practical questions again. The numbers are small, and the questions so simple, that almost any child of five or six years old is capable of under standing more than half the book, and those of seven or eight years old can understand the whole of it. The examples are to be performed in the mind, or by means of sensible objects, such as beans, nuts, &c. or by means of the plate at the end of the book. The pupil should first perform the examples in his own way, and then be made to observe and tell how he did them, and why he did them * It is reinarkable, that a child, although he is able to perform a variety of examples which involve addition, subtraction, multiplication and division, recognises no operation but addition. Indeed, if we analyze these operations where we perform them in our minds, we shall find that they all reduce themselves to addition. They are only different ways of applying the same principle. And it is only when we use an artificial meihod of performing ihem, that they take a different form. If the following questions were proposed to a child, his answers would he, in substance, like those annexed to the questions. How much is five less than eight? Ans. Three. Why? Because five and three are eight. What is the difference between five and eight? Ans. Three. Why? Because five and three are eight. If you divide eight into two paris, such that one of the parts may be five, what will the other be? Ans. Three. Why? Because five and three are eight. . How much must you give for four apples at two cents apiece ? Ans. Eight cents. Why? Because two and two are four, and two are six and two are eight. How many apples, at two cents apiece, can you buy for eight cents ? Ans. Four. Why? Because two and iwo are four, and two are six, and two are eight. We shall be further convinced of this is we observe that the same rable serves for addition and subtraction; and another table which is |