:

question as to the fictitious part of pictorial representation, we mean when it is designed to teach children something by it if it is merely intended to amuse, we have no objection to all the absurdities and humours which we see in some of the common caricatures. The more absurd and laughable such things are, the better. But when pictures are systematically presented to children with the professed view of inculcating facts, (which it must be remembered will often incidentally inculcate opinions also,) we cannot be too careful to let our facts be true in all cases where particular truth can be attained; and in all other cases, we should give to our pictures at least that general truth and that reasonable probability which will bear the test of future examination, when the child is grown up into a man, with the recollections in his head which it is the professed design of the scheme to make permanent.*

*

*It is designed by this Society to publish a series of prints illustrative of English History. Most of them will be historical scenes, represented, we believe, as far as is practicable, with regard to the conditions which have been here laid down, as necessary to the usefulness of the scheme. Some of the prints will contain views of edifices and places of historical celebrity, together with correct views of armour, dress, and other things that illustrate the subject.

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ON TEACHING ARITHMETIC.

By A. DE MORGAN.

(From the Quarterly Journal of Education, No. IX.)

We may say of instructors, that each individual ought to have either his own system or no system at all. And this refers not only to those who live by tuition, but to all parents and guardians; and is said, not in ridicule of the various plans which appear every day, but from a conviction that the manner and degree in which the intellects of children develope themselves are so various, that few general rules are applicable: whence he must cease to be the slave of a system, and become its master, who would undertake the management of an infant inind. Very few can place themselves in such a position, as must be evident to any one who has had to instruct a class of boys who have left the nursery and the preparatory school to enter upon subjects which need a little previously-acquired power of thought. They then begin their education, as their parents think, who little guess that the most important part of it is already past, and that they themselves have incurred a greater responsibility than they can ever afterwards lay upon the shoulders of another. If any man who only knew the real meaning of the word education, were told that the rising generation of the richer class was mostly educated at Oxford and Cambridge, he would very much over-estimate the quantity of bread and milk consumed in those ancient institutions.

We have thought that some reflections on the method of teaching the simplest of all the sciences might be useful to parents, were it only that we might convince them of the difficulty of their undertaking, as well as of ts necessity. Most of the juvenile treatises on this subject rather tell what to do than how to do it; it is only of very late years that such works have appeared as the Lessons on Number, reviewed in the Third Number of this Journal, in which the process is explained, as well as the result of it. We shall, in succeeding articles, make some observations on other branches of mathematics; but for the present we shall confine ourselves to the elementary parts of arithmetic only.

It is a very common notion that this subject is easy; that is, a child is called stupid* who does not receive his first notions of number with facility. This, we are convinced, is a mistake. Were it otherwise, savage nations would acquire a numeration, and a power of using it, at least proportional to their actual wants; which is not the case. Is the mind, by nature, nearer the use of its powers than the body? If not, let parents consider how many efforts are unsuccessfully made before a single articulate sound is produced, and how imperfectly it is done after all; and let them extend the same indulgence, and, if they will, the same admiration, to the rude essays of the thinking faculty, which they are so ready to bestow upon those of the speaking power. Un

* Stupid is the word employed by teachers when children do not learn, or, which is the same thing, when teachers do not know how to teach. We have seen often enough in older children, the sluggishness of intellect which may be called stupidity: but we uever saw a stupid infant. And hence it suggested itself, that possibly something might have intervened; and by some concatenation of ideas, the Eton Grammar suggested itself.

fortunately the two cases are not equally interesting. The first attempts of the infant in arms to pronounce “papa” and “mamma," though as much like one language as another, are received with exultation as the promise of a future Demosthenes; but the subsequent discoveries of the little arithmetician, such as that six and four make thirteen, eight, seven, anything but ten, far from giving visions of the Lucasian or Savilian chairs, are considered tiresome, and are frequently rewarded by charges of stupidity or inattention. In the first case, the child is teaching himself by imitation, and always succeeds; in the second, it is the parent who instructs, and who does not always either succeed or deserve to succeed. Irritated or wearied by this failure, little manifestations of temper often take the place of the gentle tone with which the lesson commenced, by which the child, whose perception of such a change is very acute, is thoroughly cowed and discouraged, and left to believe that the fault was his own, when it really was that of his instructor.

It is not at all unusual to begin by making the infant repeat the words one, two, three, &c., in succession, and this is called teaching him to count. Of course this has no more to do with the matter than it would have if one, two, three, &c., were abandoned, and chair, sofa, table, &c., substituted in their place. To some others a little modification of the above process appears preferable ; they point their fingers to one object after another, pronouncing in succession the same words. Thus the child attaches to one, two, three, &c., the notion which properly belongs to first, second, third, &c. We have seen. án instance in which a child, on being asked the meaning of three, showed that finger which had usually been

the third in his reckoning. In most cases the abstract numbers only are used, and for the most part in connexion with one particular species of object; thus the learner, hearing the numerals, only or mostly in connexion with counters or his fingers, is led to imagine that what he is doing has some connexion with these things which it has not with others. The numbering is also carried too high, even up to thousands or millions, which is injudicious, as it gives the child names only, and not ideas. We should propose the following method:-Let a number of objects of several sorts be procured, say counters, marbles, and beans, and let the numerals never be used except in connexion with one of these; thus the child should never be allowed to say the word "five," except as a part of one of the phrases, "five marbles," "five counters," or "five beans." Let the different collections of each of these be ranged in rows, from one up to five, and let the child proceed through each set separately, beginning with the lowest, and being made to pronounce "one," " two,” and "three," in connexion with the name of the objects. So far he may be supposed to know the names of the different collections in the same way as he knows the meaning of the word "table." In this little range let him be restricted, until he can count or reckon everything which can be counted or reckoned by such numnbers. And in counting the collection of "three counters," let him use the words "first counter," second counter," and "third counter" instead of "one," " "two," and "three," beginning the reckoning first with one, then with another of the set. Let every possible question in addition or subtraction be proposed, in which no number or result above three is introduced, and let them be palpably solved by means of all the different objects.

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