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The utility, and even necessity, of a work of this description, will scarcely be questioned by those who have had any experience in teaching Arithmetic. Most young persons. after having been persuaded, again and again, to review a long arithtu, *cal process, feel, or affect to feel, certain that they have performed it correctly, although the result, by the book, is erroneous. They then apply to their instructer ; and, unless he points out their mistake, or performs the operation for them, they become discouraged, think it useless 'to try' lunger, and the foundation for a habit of idleness is thus imperceptibly established. Now, in a large school, it is always inconvenient, and soinetinies impossible, for the instructer to devote the time necessary to overlook or perform a very simple, much more a complex, question in Arithmetic. This is at once obviated by having at hand a Key, to 'which reference can be easily and speedily made. The time of the teacher will thus be saved, and the pupil will not have his ardor damped by being told that his sum is wrong,' without learning where or how.

“ This work is not designed for, and can scarcely become, a help to laziness: its object is to lighten the burden of teachers, and facilitate the progress of scholars. To promote both of these important purposes n is now presented to the public.

January, 1834."

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Vhen a new work is offered to the public, especially on a subject a runding with treatises like this, the inquiry is very naturally made,

does this work contain any thing new?" "Are there not a hundred others as good as this?” To the first inquiry it is replied, that there are many things which are believed to be new; and, as to the second, a candid public, after a careful examination of its contents, and not till then, it is hoped, must decide. Another inquiry may still be inade : “Is this edition different from the preceding?"" The answer is, Yes, in many jespects. The present edition professes to be strictly on the Pestalozzian, or inductive, plan of teaching. This, however, is not claimed as a novelty. In this respect, it resembles many other systems. The novelty of this work will be found to consist in adhering more closely to the true spirit of the Pestalozzian plan; consequently, in differing from other systems, it differs less from the Pestalozzian. This similarity will now be shown.

1. The Pestalozzian professes to unite a complete system of Mental with Written Arithmetic. So does this.

2. That rejects no rules, but simply illustrates them by mental questions. So does this.

3. That commences with examples for children as simple as this, is as extensive, and ends with questions adapted to minds as mature.

Here it may be asked, “In what respect, then, is this different from that?” To this question it is answered, In the execution of our common plan.

The following are a few of the prominent characteristics of this work, in which it is thought to differ from all others.

1. The interrogative system is generally adopted throughout this work.

2. The common rules of Arithmetic are exhibited so as to cor respond with the occurrences in actual business. Under this head is reckoned the application of Ratio to practical purposes, Felloroship, &c.

3. There is a constant recapitulation of the subject attended to, styled “ Questions on the foregging."

4. The mode of giving the individual results without points, then the aggregate of these results, with points, for an answer by which the relative value of the whole is determined, thus furnishing a complete test of the knowledge of the pupil. This


is a characteristic difference between this and the former editions.

5. A new rule for calculating interest for days with months.

6. The mode of introducing and conducting the subject of Proportion.

7. The adoption of the Federal Coin, to the exclusion of Sterling Money, except by itself.

8. The Arithmetical Tables are practically illustrated, previously and subsequently to their insertion.

9. As this mode of teaching recognizes no authority but that of reason, it was found necessary to illustrate the rule for the extraction of the Cube Root, by means of blocks, which accompany this work.

These are some of the predominant traits of this work. Others might be mentioned, but, by the examination of these, the reader will be qualified to decide on their comparative value.

As, in this work, the common rules nf Arithmetic are retained, perhaps the reader is ready to propose a question frequently asked, “What is the use of so many rules?" “Why not proscribe them?” The reader must nere be reminded, that these rules are taught differently, in this system, from the common method. The pupil is first to satisfy hinself of the truth of several distinct mathematic principles. These deductions, or truths, are then generalized ; that is, briefly summed in the form of a rule, which, for convenience' sake, is named. Is there any impropriety in this ? On the contrary, is there not a great convenience in it? Should the pupil be left to form his own rules, it is more than probable he might mistake the most concise and practical one. Besides, different minds view things different. ly, and draw different conclusions. Is there no benefit, then, in helping the pupil to the most concise and practical method of solving the various problems incident to a business life?

Some have even gone so far as to condemn the Rule of Three, or Propor tion, and almost all the successive rules growing out of it. With more reason, they might condemn Long Division, and even Short Division; and, m fact, all the common and fundamental rules of Arithmetic, except Addi tion; for these may all be traced to that. The only question, then, is, “TO what extent shall we go?To this it is replied, As far as convenience requires. As the Rule of Three is generally taught, it must be confessed, that almost any thing else, provided the mind of the pupil be exercised, would be a good substitute. But when taught as it should be, and the scholar is led on in the same train of thought that originated the rule, and thus effectually made to see, that it is simply a convenient method of arriving at the result of both Multiplication and Division combined, its no cessity may be advocated with as much reason as any fundamental rula As taught in this work, it actually saves more figures than Short, compas ed with Long Division. Here, then, on the ground of convenience, it would be reasonable to infer, that its retention was more necessary than either. But, waiving its utility in this respect, there is another view to bę taken of this subject, and that not the least in importance, viz. the ideas of beauty arising from viewing the harmonious relations of numbers. Here is a delightful field for an inquisitive mind. It here imbibes truths as lasting as life. When the utility and convenience of this rule are once conceded, all the other rules growing out of this will demand a place, and for the same reason.

It may, perhaps, be asked by many, “Why not take the principle with out the name ?" To this it is again replied, Convenience forbids. The name, the pupil will see, is only an aggregate term, given to a process imbodying several distinct principles. And is there no convenience in this: Shall the pupil, when in actual business, be obliged to cry off his mind from all other pursuits, to trace a train of deductions arising from abstract reasoning, when his attention is most needed on other subjects? With as much propriety the name of captain may be dispensed with ; for, although the general, by merely summoning his captain, may summon 100 men, still he might call on each separately, although not quite so conveniently With these remarks, the subject will be dismissed, merely adding, by way of request, that the reader will defer his decision till he has examined his doctrine of Proportion, Fellowship, &c., as taught in this work.

The APPENDIX contains many useful rules, although a knowledge or these is not absolutely essential to the more common purposes of life. Under this head are reckoned Alligation, Roots, Progression, Permutation, Annuities, &c. The propriety of scholars becoming acquainted, sonie time or other, with these rules, has long since been settled; the only ques. tion is, with regard to the expediency of introducing them into our Arith metics, and not reserving them for our Algebras. In reply to this, the Writer would ask, whether it can be supposed, the developement of the truths, by figures, will invigorate, strengthen, and expand the mind less than by letters. Is not a more extensive knowledge of the power of fiy. ures desirable, aside from the improvement of the mind, and the practical utility which these rules afford? Besides, there always will, in some nook or other, spring up some poor boy of mathematical genius, who will be desirous of extending his researches to more abstruse subjects. Must he, as well as all others, be taxed with an additional expense to procure a systern, containing the same principles, only for the sake of discovering them bý letters?

Position, perhaps, may be said to be entirely useless. The same may love said of the doctrine of Equations by Algebra. If the former be taught ra tionally, what great superiority can be claimed for the one over the other? Is it not obvious, then, that it is as beneficial to the pupil to discipline his inind by the acquisition of useful and practical knowledge, which may be in the possession of almost every learner, as to reserve this interesting portion of Mathematics for a favored few, and, in the mean time, to diveni the attention of the pupil to less useful subjects?

The blocks, illustrative of the rule for the Cube Root, will satisfactorily account for many results in other rules; as, for instance, in Decimals, Mensuration, &c.; which the pupil, by any other means, might fail to perceive. By observing these, he will see the reason why his produci, in decimals, should be less than either factor; as, for instance, why the solid contents of a half an inch cube should be less than half as much as an inch cube. In this case, the factors are each half an inch, but the solid contents are much less than half a solid inch,

In this work, the author has endeavored to make every part conform to this maxim, viz. THAT NAMES SHOULD SUCCEED IDEAS.

This method of communicating knowledge is diametrically opposed to that which obtains, in many places, at the present day. The former, by first giving ideas, allures the pupil into a luminous comprehension of the subject, while the Jatter astounds him, at first, with a pompous name, lo which he seldom af fixes any definite ideas, and it is exceedingly problematical whether he ever will. In addition to this is the fact, that, by the last-mentioned method, when the name is given and the process shown, not a single reason of any operation is adduced; but the pupil is dogmatically told he must proceed thus and so, and he will come out so and so. This mode of teach ing is very much as if a merchant of this city should direct his clerk, without intrusting him with any business, first to go to South Boston, then to the state-house, afterwards io the market, and then to return, leaving him to surmise, if he can, the cause of all this peregrination. Many are fools enough to take this jaunt pleasantly; others are restiff, and some fractious. This sentiment is fully sustained by an article in Miss Edgor Wurth's works, from which the following extract is made : “A child's seeming stupidity, in learning arithmetic, may, perhaps, be a proof of in

telligence and good sense. It is easy to make a boy, who does not reason, repeat, by rote, any technical rules, which a common writing master, with magisterial solemnity, may lay down for him; but a child who reasons will not be thus easily managed; he stops, frowns, hesitates, questions his mas ter, is wretched and refractory, until he can discover why he is to proceed in such and such a manner; he is not content with seeing his preceptor make figures and lines on the slate, and perform wondrous operations with the self-complacent dexterity of a conjurer; he is not content to be led to the treasures of science blindfold; he would tear the bandage from his eges, that he might know the way to them again.”

In confirmation of the preceding remarks, and as fully expressive of tho author's views on this subject, the following quotation is taken from the preface to Pestalozzi's system.

“ The PestalozZIAN plan of teaching ARITHMETIC, as one of the great branches of the mathematics, when communicated to children upon tho principles detailed in the following pages, needs not fear a comparison with her more favored sister, GEOMETRY, either in precision of ideas, in dlearness and certainty of demonstration, in practical utility, or in the subfime deductions of the most interesting truths.

"In the regular order of instruction, arithmetic ought to take precedence of geometry, as it has a more immediate connection with it than soine are willing to admit. It is the science which the mind makes use of in measuring all things that are capable of augmentation or diminution ; and, when rationally taught, affords to the youthful mind the most advantageous exercise of its reasoning powers, and that for which the human intellect becomes early ripe, while the more advanced parts of it may try the energies of the most vigorous and matured understanding."

THE AUTHOR January, 1829

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