SUGGESTIONS TO TEACHERS

ON THE Method of USING) THIS WORK.

FCR a course of mental arithmetic, adapted to the capacities of very young pupils, they may take the mental exercises in each rule, as far as the first example for the slate. This course is not meant to include any of the exercises styled "Questions on the foregoing."

This course embraces the whole of the first 20 pages, together with the arith metical tables, extending to the Appendix. The necessity of impressing these tables on the minds of pupils at an early age is sufficiently obvious. When the pupil is perfect master of this course, as will, most probably, be the case after one or two reviews, the teacher will find no difficulty in making him understand the operations by slate. He may then take the whole in course,

In every school, it would be well to institute classes; and as there are seldom any answers given to the metal questions, the pupils may be allowed to read in their turns the questions from the book; thus giving the teacher no further trouble than occasional corrections By this, the reader will perceive, that the work may be used to advantage in monitorial schools, as the former editions have been In large schools these corrections may be made by an advanced scholar, instead of the teacher. Whenever an advanced scholar takes up the book with a view of profiting from it, he should omit nothing as he pro gresses, but make it his practice to qualify himself to answer any question, in the mental exercises, rules, or respecting the reason of the operations.

Teachers will find it to be a useful occupation for their scholars, to assign them a morning lesson, to be recited as soon as they come into school With little exertion on the part of teachers, pupils in this way may be made assiduous and ambitious, very much to their advantage, and to the credit of their teachers.

The mental questions, under the head of "Questions on the foregoing," will, intelligently answered, furnish to committees an admirable test of the pupil's knowledge of this subject.

The Appendix is designed for those who have time and opportunity to devote to the study of the more abstruse parts of mathematics.

Note. Lost some may mistake the object of the figures in the parentheses, it may here be remarked, that these figures are separate answers, left without Masigning any value to them, reserving this particular for the discretion of the pupil, which he must necessarily exercise, in order to obtain the answer which follows, that being the aggregate of the whole.

The above directions are those which seem the best to the author; but as every intelligent teacher has a way of his own, which, though not intrinsically the best, is, perhaps, the best for him, the subject is respectfully submitted to his own choice.

Were a new work is offered to the public, especially on a subject abounding with treatises, like tuis, the Jaquiry is very naturally made, Does this work contain any thing now?" "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 now; and, as to the second, a candid public, after a careful exammation of its contents, and not till then, it is hoped, must decide. Another inquiry may still be made: "Is this edition different from the preceding? The answer is, Yes, in many respects. The present edition professes to be strictly on the Pestalozzian, or inductive plan of teaching. This, how-~ ever, 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 mors closely to the true spirit of the Testalozzian 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, out 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 boasked, "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 correspond with the occurrences in actual business. Under this head is reckoned the application of Ratio to practical purposes, Fellowship, &c.

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

4. The mode of giving the individual results without points then the aggregate of these results, with points, for an ansier by which the relative value of the whole is determined, thus tur nishing a complete test of the knowledge of the pupil. This in a characteristic difference between this and the former edition's, 5. 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 recognises 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 accom pany this work.

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

As, in this work, the common rules of arithmetic are retained, perhaps the teader is ready to propose a question frequently asked, "What is the use of so ni by rules? Why not proscribe them?" The reader must here be reminded, that these rules are taught differently, in this system, from the common method. The pupil is first to satisfy himself of the truth of several distinct mathemati cal principles. These deductions, or truths, are then generalized; that is, briefly summed in the form of a rule, which, for convenience' wake, is named. ls. there any impropriety in this? On the contrary, is there not a great convenience in it? Should the pupil be loft to form his own rules, it is more than probable he might mistake the most concise and practical one. Besides, different minds view things differently, 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 Proportion, and annost all the successive rules growing out of it. With more reason, they night condemn Long Division, and even Short Division; and, in fact, all the common and fundamental rules of arithmetic, except Addition; for these may all be traced to that. The only question then is, "To what extent shall we go: 359 To this it is replied, As far as convenience requires. As the Rule of Three is generally taught, it must be confessed, that alinost any thing else, provided the mind of the pupil be exercised, would be a good substitute. But when taught us it should be, and the scholar is led on in the same train of thought that origimated the rule, and thus affectually made to see, that it is simply a convenient method of arriving at the result of both Multiplication and Division combined, its necessity may be advocated with as much reason as any fundamental rule. As taught in this work, it actually saves more figures than Short, compared 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 be 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 ar 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 without 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 call 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 a captain may be dispensed with; for, although the general, by merely summon ing bis captain, may summon 100 men, still he might call on each separately, although not quite su 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 the doctrine of Proportion, Fellowship, &c., as taught in this work.

The APPENDIX contains many useful rules, although a knowledge of 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, some time or other, with these rules, has long since been settled; the only question is, with regard to the expediency of introducing them into our arithmetics, and not reserving them for our algebras. In reply to this, the writer would ask, whether it can be sup posed, the developement of these truths, by figures, will invigorate, strengthen, and expand the mind less than by letters? Is not a more extensive knowledge of the power of figures 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 system, containing the same principles, only for the sake of discovering them by letters? Position, perhaps, may be said to be entirely useless. The same may be 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 mind 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 favoured few, and, in the mean time, to divert the atten tion of the pupil to less useful subjects?

The blocks, illustrative of the rule for the Cube Boot, 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 product, in decimals, should be 093 than either factor; as, for instance, why the solid contents of a half au 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 Bolid inch.

In this work, the author has endeavoured to make every part conform to thie maxim, viz. THAT NAMES SHOULD SUCCEED IDEAS. Communicating knowledge is diametrically opposed to that which obtains, in This method of many places, at the present day. The former, by first giving ideas, allures the pupil into a luminous comprehension of the subject, while the latter astounds him, at first, with a pompous name, to which he seldom affixes 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 givon 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 teaching 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 to 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 ploasantly; others are restiff, and some fractious. This sentiment is fully sustained by an article in Miss Edgeworth's works, from which the following extract is made: "A child's seeming stupidity, in learning arithmetic, may, perhaps, be a proof of intelligence 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 master, 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 eyes, that he might kno❤ the way to them again."

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

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

"In the regular order of instruction, arithmetic ought to take procedence of geometry, as it has a more immediate connexion with it than some 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 "

January

1829

THE AUTHOR