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But in the First Part of this work, in the compass of twenty pages, the pupil gains all the fundamental principles of the science, which in the succeeding parts, are developed in more minute particulars.
3. To aid in the same object, and to secure other advantages, a new method of classification has been adopted. The chief benefit aimed at, by this new arrangement, is to simplify the science, by leading the pupils to understand, that all the various exercises of arithmetic are included under the same general principles of addition, subtraction, multiplication, division, and reduction.
The common method of teaching the simple rules first, and then of introducing vulgar and decimal fractions, has a tendency to render the science much more complicated and perplexing. Thus the pupil is first taught the process of simple addition. Then follows the exercises of the other simple rules, and by the time the pupil has somewhat forgotten the principles of simple addition, comes compound addition, which seems to the child as much on a new principle, as multiplication, or division. Then, after another interval, comes decimal addition, and then the addition of vulgar fractions.
But a child who is first taught the system of numeration, as including whole numbers and fractions, and the nature of each of these modes of expressing numbers, can immediately commence the operation of addition, in all its various particulars, and recognize the general principle, that runs through the whole, and at the same time the peculiarity which distinguishes each.
The difficulties arising from the common mode of arrangement, are particularly felt
, when the processes of multiplying and dividing by fractions, are introduced. In all previous operations, the pupil has found that multiplication increases a number, and division diminishes it. But when fractions are introduced, a new science seems to commence, in which multiplication lessens and division increases a number, and all heretofore learned, seems to be contradicted and undone.
But if, at the commencement of the science, the pupil understands the peculiar character of fractions, and then finds them arranged with whole numbers, so as to be able to compare and distinguish the same general principles of the various exercises, and at the same time the specific difference, the perplexity arising from an apparent multiplicity of operations, and their seemingly contradictory nature, is avoided.
When this plan was first attempted, some difficulty was felt from the necessity of the operation of division in the previous operation of multiplying by a fraction. But this difficulty has been obviated by having the First Part precede, in which mul
tiplication and division are explained, without entering minutely into the various processes of fractional multiplication.
It will be found that pupils, by learning the division table, and the first part, can perform all the exercises in simple and fractional multiplication, without any other knowledge of the rule of division. There are two or three exceptions, however, where some exercises for the slate are introduced, where the rule of division must be employed. These are intended for older pupils, who are supposed to understand the process of division, and are to be omitted by new beginners, until a review.
The writer has herself employed this method of classification, in teaching, and it has been used by the teachers in the institution under her care, for several years; and thus experience has enforced the conviction, which at first was the result of reasoning, that this mode of classification will better secure the benefits sought for, in all attempts at generalization. It certainly attains advantages, and avoids difficulties, much more than the common method. But should any teachers prefer the common method of classification and arrangement, the simple rules can be taken first, and directions are inserted in the work for this purpose.
4. Another difficulty experienced in using some of the most popular works of this kind, has arisen from the fact that the mental and written exercises have been entirely separate; in some cases being placed in different books. Thus the pupil, after completing an arithmetic designed for mental exercise alone, will often be found repeating exactly the same processes in written arithmetic, without recognizing the principles, which, in mental operations, have been constantly employed. To remedy this, both mental and written exercises are placed tog
her under every general rule.
5. Another defect in teaching this science, has arisen from a want of some method of stating and explaining the rationale, of each arithmetical process. In many arithmetics, a mechanical method is presented, of performing certain operations according to rule, without any exhibition of the reason for such operations. Thus, in subtraction, why one is carried, and ten borrowed; or, in multiplication, why the figures are placed in a certain method; or, in division, why multiplication and subtraction are performed, is never explained or illustrated. To a child, they are a sort of cabalistical process, which he finds will bring the right answer, and this is all he can know from any thing he gains from the book. To remedy this, in the following work, every rule is accompanied by a full explanation of the reason, for each process employed. In the mental operations also, a proper mode of stating each process is given. The definitions, rules, and explanations, will be found to be more simple and
concise than in many works of this kind, and perhaps may be considered as improvements.
As the writer has been in a situation, in which she has employed various teachers, of different qualifications, it has been one great aim to furnish a work, by which new and inexperienced teachers could avail themselves of the experience of others. This work has been maturing for several years, and the results of the experience of several able and ingenious teachers employed by the writer, have been added to her own. It is believed that any teachers with common talents and industry, can, with the aid of this work, do all for the pupils in this science, which needs to be done, in order to make them thorough and expert proficients.
For the purpose of perfecting such a work as this, and to make a fair trial of the several improvements contemplated, a small work on this plan was printed some years ago, for the use of the pupils of the writer. But as it was intended for an experiment, and was necessarily very imperfect and incomplete, it was never published. Yet, as in several cases, those who have been teachers and pupils in this institution, have introduced it into their schools, it
proper to add, that this work is very different from the former, being much easier, much more extensive and complete, and improved in several respects which it is unnecessary to mention.
The writer does not lay alaim to any great originality, in these various particulars, but has aimed to unite in one work the various excellencies, which might otherwise be found only in a great variety
Hartford Female Seminary, Jan. 1, 1832.
If any teacher prefers the common method of arrangement and classification, the simple rules can be taken together first, and directions are given in the work for this purpose.
It is very desirable that new beginners should review the first part, till it is very thoroughly understood. It will save much trouble to both teacher and pupil
. It will be found advantageous, to require older pupils to study the first part, previous to commencing the second ; for though some of the exercises are very simple, there are some important explanations and illustrations, not found in the second part.
It is very desirable that pupils should become thorough and expert in numeration, especially in decimal numeration, before taking the next lessons, and one or two reviews are recommended previous to proceeding.
If the pupil has never practised simple division, omit those exercises which require this rule, till a review. They will be found in smaller type, as are some other exercises, which though they do not demand a knowledge of the rule of division, are too difficult for young pupils.
The second part should be reviewed, before commencing the
If any teachers have a preference for the common method of classification, it is very easy to direct the pupils to learn the simple rules first. But every pupil will find it advantageous at least to review on the plan of arrangement adopted in this work.
When young beginners take the second part, it is recommended, that they take the easiest exercises, and reserve the more difficult, till a review.