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It is believed that much of this guess-work in " figuring,” and its concomitant habits of listlessness and vacuity of mind, have arisen from the use, at first, of abstract numbers and intricate questions, requiring combinations above the capacity of children. Taking his slate and pencil, the pupil sits down to the solution of his problem, but soon finds himself involved in an impenetrable maze. He anxiously asks for light, and is directed to learn the rule." He does it to the letter, but his mind is still in the dark. By puzzling and repeated trials, he perhaps finds that certain multiplications and divisions produce the answer in the book ; but as to the reasons of the process, he is totally ignorant. To require a pupil to learn and understand the rule, before he is permitted to see its principles illustrated by simple practical examples, places him in the condition of the boy, whose mother charged him never to go into the water till he had learned to swim.
These embarrassments are believed to be unnecessary, and are attempted to be removed in the following manner :
1. The examples at the commencement of each rule are all practical, and are adapted to illustrate the particular principle under consideration. Every teacher can bear testimony, that children reason upon practical questions with far greater facility and accuracy
than they do upon abstract numbers.
62. The numbers contained in the examples are at first small, so that the learner can solve the question mentally, and understand the reason of each step in the operation.
3. As the pupil becomes familiar with the more simple combinations, the numbers gradually increase, till the slate becomes necessary for the solution, and its proper use is then explained.
4. Frequent mental exercises are interwoven with exercises upon the slate, for the purpose of strengthening the habit of analyzing and reasoning, and thus enable the learner to comprehend and solve the more intricate problems.
5. In the arrangement of subjects it has been a cardinal point to follow the natural order of the science. No principle is used in the explanation of another, until it
has itself been demonstrated or explained. Common fractions, therefore, are placed immediately after division, for two reasons. First, they arise from division, and are in fact unexecuted division. Second, in Reduction, Compound Addition, &c., it is frequently necessary to use fractions; consequently fractions must be understood, before it is possible to understand the Compound rules.
For the same reason, Federal Money, which is based upon the decimal notation, is placed after Docimal Fractions. Interest, Insurance, Commission, Stocks, Duties, &c., are also placed after Percentage, upon whose principles they are based.
6. In preparing the Tables of Weights and Measures, particular pains have been taken to ascertain those that are in present use in our country, and to give the legal standard of each, as adopted by the General Government.* It is well known that a great difference of weights and measures formerly existed in different parts of the country. More than ten years have elapsed since the Government wisely undertook to remedy these evils, by adopting uniform standards for the custom-houses and other purposes; and yet not a single author of arithmetic, so far as we know, has given these standards to the public.
7. The subject of Analysis is deemed so essential to a thorough knowledge of arithmetic and to business calculations, that a whole section is devoted to its development and application. The principles of Cancelation have been illustrated, and its most important applications pointed out, in their proper places. The Square and
* In the year 1836, Congress directed the Secretary of the Treasury to cause to be delivered to the Governor of each State in the Union, or to such person as he should appoint, a complete set of all the Weights and Measures adopted as standards, for the use of the States respectively; to the end that a uniform standard of Weights and Measures may be established throughout the United States. Most of the States have already received them; and may we not hope that every member of this great Union will promptly and cordially unite in the accomplishment of an object so conducive both to individual and public good ?
Cube Roots are illustrated by geometrical figures and cubical blocks.
Such is a brief outline of the present work. It is not designed to be a book of puzzles, or mathematical anomalies; but to present the elements of practical arithmetic, in a lucid and systematic manner. It embraces, in a word, all the principles and rules which the business man ever has occasion to use, and is particularly adapted to precede the study of Algebra and the higher branches of mathematics.
With what success the plan has been executed, remains for teachers and practical educators to decide. If it should be found to shorten the road to a thorough knowledge of arithmetic in any degree, its highest aims will be accomplished.
MODE OF TEACHING ARITHMETIC.
I. QUALIFICATIONS.—The chief qualifications requisite in teaching Arithmetic, as well as other branches, are the following:
1. A thorough knowledge of the subject. 2. A love for the employment. 3. An aptitude to teach. These are indispensable to success.
II. CLASSIFICATION.- Arithmetic, as well as reading, grammar, &c., should be taught in classes.
1. This method saves much time, and thus enables the teacher to devote more attention to oral illustrations.
2. The action of mind upon mind, is a powerful stimulant to exertion, and can not fail to create a zest for the study.
3. The mode of analyzing and reasoning of one scholar, will often suggest new ideas to the others in the class.
4. In the classification, those should be put together who possess as nearly equal capacities and attainments as possible.. If any of the class learn quicker than others, they should be allowed to take up an extra study, or be furnished with additional examples to solve, so that the whole class may advance together.
5. The number in a class, if practicable, should not be less than six, nor over twelve or fifteen. If the number is less, the recitation is apt to be deficient in animation ; if greater, the turn to recite does not come round sufficiently often to keep up the interest.
III. APPARATUS.— The Blackboard and Numerical Frame are as indispensable to the teacher, as tables and cutlery are to the housekeeper. Not a recitation passes without use for the blackboard. If a principle is to be demonstrated or an operation explained, it should be done upon the blackboard, so that all may see and understand it at once.
To illustrate the increase of numbers, the process of adding, subtracting, multiplying, dividing, &c., the Numerical Frame furnishes one of the most simple and convenient methods ever invented.*
IV. RECITATIONS.—The first object in a recitation, is to secure the attention of the class. This is done chiefly by throwing life, and variety into the exercise. Children loathe dullness, while animation and variety are their delight.
2. The teacher should not be too much confined to his text-book, nor depend upon it wholly for illustrations.
* Every one who ciphers, will of course have a slate. Indeed, it is desirable that every scholar in school, even to the very youngest, should be furnished with a small slate, so that when the little fellows have learned their lessons, they may busy themselves in writing and drawing various familiar objects. Idleness in school is the parent of mischief, and employment the best antidote against disobedience.
Geometrical diagrams and solids are also highly useful in illustrating many points in arithmetic, and no school should be without them.
3. Every example should be analyzeds the “ why and wherefore" of every step in ihe solution should be required, till each member of the class becomes perfectly familiar with the process of reasoning and analysis.
4. To ascertain whether each pupil has the right answer to all the examples, it is an excellent method to name a question, then call upon some one to give the answer, and before deciding whether it is right or wrong, ask how many in the class agree with it. The answer they give by raising their hand, will show at once how many are right. The explanation of the process may now be made.
Another method is to let the class exchange slates with each other, and when an answer is decided to be right or wrong, let every one mark it accordingly. After the slates are returned to their owners, each one will correct his errors.
V. THOROUGHNESS.—The motto of every teacher should be thoroughness. Without it, the great ends of the study are defeated.
1. In securing this object, much advantage is derived from frequent reviews.
2. Not a recitation should pass without practical exercises upon the blackboard or slates, besides the lesson assigned.
3. After the class have solved the examples under a rule, each one should be required to give an accurate account of its principles with the reason for each step, either in his own language or that of the author.
4. Mental Exercises in arithmetic, either by classes or the whole school together, are exceedingly useful in making ready and accurate arithmeticians, and should be frequently practised.
VI. SELF-RELIANCE. :-The habit of self-reliance in study, is confessedly invaluable. Its power is proverbial, I had almost said, omnipotent. “ Where there is a will, there is a way.”
1. To acquire this habit, the pupil, like a child learning to walk, must be taught to depend upon himself. Hence,
2. When assistance in solving an example is required, it should be given indirectly; not by taking the slate and performing the example for him, but by explaining the meaning of the question, or illustrating the principle on which the operation depends, by supposing a more familiar case. Thus the pupil will be able to solve the question himself, and his eye will sparkle with the consciousness of victory.
3. He must learn to perform examples independent of the answer, without seeing or knowing what it is. Without this attainment the pupil receivesbut little or no discipline from the study, and is unfit to be trusted with business calculations. What though he comes to the recitation with an occasional wrong answer; it were better to solve one question understandingly and alone, than to copy a score of answers from the book. What would the study of mental arithmetic be worth, if the pupil had the answers before him? What is a young man good for in the counting-room, who has never learned to perform arithmetical operations alone, but is obliged to look to the answer to know what figure to place in the quotient, or what number to place for the third term in proportion, as is too often the case in school ciphering?