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other branches have been gaining ground, and been better and better taught, arithmetick has lost, what other branches have gained ; and instead of being best and most successfully taught, as its importance demands, it has been the worst, and most carelessly taught. · No adequate reason can be assigned for the declining interest of arithmetick in our schools, for the last twenty years, but the vast disparity in improvements in the books on this, and other subjects. Some variety exists in the great abundance of elementary arithmeticks, but the same general principle of communicating knowledge pervades them all. This principle is wrong. It is wrong, first, because it does not give the best knowledge of the subject; and it is wrong, secondly, because it does not afford the best discipline to the mind. These are the only purposes, for which an elementary book is studied; and a failure in both or either of these points, is capital, and fatal to the branch to be taught. The systems have been formed, no doubt, by good mathematicians, but the object of a school book, as has been before observed, is not to reduce the science to the fewest general principles, and state those principles, as a philosopher would arrange and state them for his own .convenience. Adaptation to the mind, which is acquiring the science, must be ever kept in view, by the writer of a school book, which is destined to answer the only purposes for which it is written.
The plan of all arithmeticks, till quite lately, ha's been, to state the principle or rule to be taught in the most concise manner possible, and then arrange under it, examples of its application. This is called the synthetick, in contradistinction to the analytic method, which begins with examples, and at length arrives at a rule. Now the first part of the process by synthesis, cannot be said to give the best, if it can be said to give any knowledge of the subject. For, what is a learner wiser after he has committed the general principle or rule to his memory ? And it is impossible for him to do any thing more, without presupposing in him some knowledge of the subject. This operation in itself can, certainly, give him no knowledge ; because it is an abstract principle, stated in terms, of which he has probably never heard. And if he has no ideas attached to the principal terms, of which the sentence is composed, he cannot understand the relation of the ideas, intended to be expressed in the sentence.
Should the learner, therefore, after committing a rule to his memory, be able to solve a question under it, the operation must be merely mechanical. He begins as the rule directs, and when he has read or said a sentence, he puts his finger upon the place, lest he should do the same thing again, and conforms literally with his direction. This done, he proceeds to read another sentence, and in like manner to comply with its direction, and at length out comes the answer. If any pupil is able to do better, than I have described, it is not because the rule, he has committed, has made him able. He has not been called upon, in this process, to exercise any discrimination, judgment, or reasoning. It would be difficult, in fact, to tell by what powers of the mind he has done it. So that, as a discipline to his mind, he has derived none, or very little advantage. The powers of the mind are strengthened only by exereise. He has acquired no knowledge of the subject, except perhaps, a greater facility in the mechanical operation. He applies a rule with as little knowledge of the principles of the science, as the man has, who works in a chemical laboratory by receipts. He forms a compound of certain elements, as directed by his receipt, and obtains the desired result. But no one would call him a chemist. This process does not constitute, or impart a knowledge of that science. That is gained only by a minute analysis of the parts, which are to enter into the compound, and the examination of their affinities for each other.
When the pupil has been worried through his arithmetick; he is worried, because he cannot take pleasure in dwelling so long on what he does not in the least understand ; his mind is very little improved, for those faculties, which give the most decided character to a mind, have not been called into exercise. And he is hardly better prepared for the business of I can neither remember the rule, nor the application of it. But the parent is satisfied, because the child has been through the book, and can repeat all the rules it contains; and moreover, he can flourish in the application of any rule to the examples, which are put under it, and which his instructer has probably led him through again and again. The instructer is satisfied because the parent is ; and the pupil is doubly satisfied, on both accounts. But before any of the knowledge, which has been thus attained, can be very safely put into practice, it must be learned again, and rules for the individual must be arrived at, in the only legitimate method, viz : by induction of particular examples. In confirmation of this, if it needs confirmation, we need only refer to men of business.
Who, that is actually engaged in mercantile life, thinks of applying the dogmatical rule, he has learned at school ? In the frequent occasions the merchant has for arithmetical calculations, he examines the particular case, and makes a rule for himself. In this respect the man of business is a much better philosopher, than the student, who must hunt up an analogous case, and produce his rule from a book. In this manner, the rules of a man of business will be made to correct his knowledge, and put it in a form convenient to be remembered, and not by any means to give the knowledge, as the usual method seems to intimate. . All the evils, which result from a disgust of the study, from conveying inadequate ideas of the subject, and from paralyzing in a degree, the opening powers of the mind, are removed, when it is presented in the natural and most philosophical manner. There is nothing in it peculiarly difficult. On the contrary, when presented in a form adapted to the capacity of the learner, it has peculiar interest with most young minds; and is peculiarly calculated to call forth and strengthen their powers. On this point may be
cited the opinions of some of the most acute observ.ers of any age, of the phenomena of mind. “ Arithmetick,” says Locke, “is the easiest, and consequently the first sort of abstract reasoning, which the mind bears or accustoms itself to ; and is of so general use in all parts of life and business, that scarce any thing is to be done without it."* “ Would you have a man reason well,” says the same author, 66 you must use him to it betimes; exercise his mind in observing the connexion of ideas and following them in train. Nothing does this better than mathematicks; which, therefore, I think should be taught all those, who have time and opportunity, not so much to make them mathematicians, as to make them reasonable creatures.”+ " For," he says again, “ the business of education in respect of knowledge, is not to perfect a learner in all, or any one of the sciences, but to give his mind that freedom, that disposition, and those habits, that may enable him to attain any part of knowledge, he shall apply himself to, or stand in need of, in the future course of his life.” A word from Dr. Watts. "Converse much,” says he, in his work on the improvement of the mind, “ with those friends, and those books, and those parts of learning, where you meet with the * Treatise on Education. + Conduct of the Understanding,