PERSPICUOUS COMMUNICATION.

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school or college. When there is any doubt, we may settle the matter by leaving it out.

A very searching historical enquiry into modern events brings out such a variety of opinions in practical politics and still more in religion, as to make an obstacle to the introduction of the subject into the higher schools and colleges. This difficulty is felt in Germany, where professors are more outspoken than in England; it also occurs in connection with the Irish Roman Catholics in the Queen's Colleges. A history of the Reformation could scarcely be thorough, if it offended neither Protestants nor Roman Catholics; a history of the first centuries of the Christian era, if it dissatisfied nobody, would be worthless to everybody.

SCIENCE.

The Methods of teaching Science are as extensive and various as the field itself. They involve, in the highest degree, all the devices for the perspicuous communication of knowledge, as well as the more special devices of imparting generalized or abstract notions and truths. The teacher is usually supposed to have before him an exposition already shaped. He may of course modify the pre-existing exposition, and be a book to himself; but a convenient line may be drawn between the art of writing an expository work of science, and the art of bringing home the truths in actual teaching.

The methods of inculcating the abstract idea were incidentally sketched at an earlier stage. These methods are complete, as far as concerns the central fact of science -the generality or abstraction; only they do not include all that pertains to the teaching. Next to the

exposition of a single abstract notion or principle, is the setting forth of demonstrative reasoning in chains of abstractions; a process that has its own separate difficulties. The requirement here is not a new expository method, but the careful employment of the ordinary forms of perspicuous language; to which is to be added the making sure that the links of the chain are each made

secure.

Arithmetic.

The method for Arithmetical teaching is perhaps the best understood of any of the methods concerned with elementary studies. To illustrate Number by examples in the concrete, and to show the reasons of the rules by means of these examples, are the substance of the modern method, as opposed to the older practice of prescribing the tables and the rules, to be committed to memory, and carried into operation as the pupils best might.

Much is involved in the first attempts to work upon number. The distinction between one number and another is shown to the eye by concrete groups of various things; the identity of number appearing under disparity of materials and of grouping: ideas are thus acquired of unity, of two, three, &c., up to ten in a row. Difference or contrast is made use of, as well as agreement; five is placed by the side of four and of six.

the outset small tangible objects are used--balls, pebbles, coins, apples; then larger objects, as chairs, and pictures on a wall. Finally, dots, or short lines, or some other plain marks, are the representative examples to be deposited in the mind as the nearest approach to the abstract idea.

FUNDAMENTAL OPERATIONS.

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The conception of Number is not complete till it carry with it the ideas of more and less, of adding and taking away, and of the converting of one number into another by these means. More and less stands contrasted with the fundamental notion of equals, which also comes to the front in the first manipulation with numbers. Sameness in difference is exemplified in the notion of each number, obtained by comparing the concrete examples; and equality first conceived by coincidence of lengths, is transferred to number, by numerical coincidence in differently arranged groups; as when nine is set forth in one row, and also in three rows.

On the basis of the preliminary exercises with numbers in the concrete, the decimal system is reached, and with it the methods of adding and subtracting; all which can be made quite intelligible and rational, as the precursor of the exercises to be worked. The sums of the simple numbers, having first been exemplified, have to be committed to memory; and this is the commencement of the business of computation, and of all the severe part of the subject. It being the essence of the abstracting operations to enable us to leap to conclusions, without going through all the intermediate steps, the memory has to receive with firmness and precision all that is included in the addition and multiplication tables; and the test of aptitude for the subject is the readiness to come under this discipline. It is a kind of memory that in all probability depends on a certain maturity or advancement of the brain; so that no amount of concrete illustration will force it on before its time. On the old system, the pupil commenced arithmetic when able to imbibe the tables and to work sums without any preliminary explanations

U

of number; the ability arising in due course by the growth of the brain, and not depending on any aid from the teacher.

I am not aware of any special device for lightening this part of the process of arithmetical training. The general arts of teaching are of avail here as elsewherethe apportioning of the lessons in suitable amount, the graduated exercises, the unbroken application, the patience and encouragement of the teacher. It is plain, however, that the multiplication table is a grand effort of the special memory for symbols and their combinations, and the labour is not to be extenuated in any way. The associations must be formed so as to operate automatically, that is, without thinking, enquiring, or reasoning; and for this we must trust to the unaided adhesiveness due to mechanical iteration. It is not unimportant to have gone so far into the rationale of the process as to be able to work out any one product deductively; and it might be a certain relief in the work of committing the table, to select a few of the products for determination by manipulating the factors-four sixes equal two tens and four, seven twelves equal eight tens and four. Another collateral exercise would be to call attention to each column as a steady addition of one number twice six, three times six, and so on; which is the point where adding passes into multiplying. These explanations are useful in themselves, as contributing to the science of the subject; and they are a slight aid to the memory; we are not so apt to forget that four times six is twenty-four after having formed the twenty-four from the sixes. Still, I apprehend that the cementing of the requisite associations of the one hundred and forty

FRACTIONS AND THE RULE OF THREE. 291

four products must be mainly an affair of symbolical memory, the result of immense iteration, and not to be entered on until a suitable age.

While this complete and self-sufficing association is the groundwork of the process of multiplication, which enters into all the higher operations, there are various points in the actual exercises, where the intelligent conception of numbers comes in aid; as in the placing of the multiplier below the multiplicand, and the arrangement of the lines of the successive products. For these matters, a knowledge of the reasons is very serviceable. The same applies to Fractions; in them the reasons assist the mind in observing the rules, which are not so easily held in the unmeaning shape as are the addition and multiplication tables. Still more does the knowledge of reasons apply to the Rule of Three, which can hardly be applied under any mode of stating it that does not assign the explanation. Hence, this is justly counted the pons asinorum of Arithmetic; it is the place where mere rote acquirement is sure to break down. So long as the questions are given in a regular form, the unmeaning rule may be enough; but as against distorted arrangements, it is powerless. In the usual applications to computing Interest, the hackneyed rule suffices only for the easiest cases.

It is thus apparent that, while many of the links of arithmetical operations are blind unmeaning symbolical associations, which are possible at a certain age, which may be called the dawning of the Age of Abstract Reason, because it is the epoch when the mind can betake itself to symbolical and representative signs, and think and operate through their instrumentality,—yet there runs through the subject a necessity of perceiving