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explained and understood by a class, are not so simple, and will bring out good teaching; while at the same time, besides its importance in a mathematical aspect, it is impossible to deal satisfactorily with division of decimals, or indeed to obtain any grasp of decimal fractions, until the principles upon which the reasons for this rule depend are apprehended; whereas, when once they are grasped, division of decimals becomes as simple an affair as division of money or avoirdupois.

And let us suppose that the lesson is designed to last forty-five minutes-from 11.40 to 12.25 (see § 20). Let us further suppose that not only has the pupil-teacher prepared his lesson carefully, (see § 10, 21), and taken counsel from the master, if there was any point in respect of which he was not clear in his own mind, or not satisfied that he could make it clear to others, but also that he has taken the precaution to tell his class beforehand, that they are to have a lesson next time on this subject, and has required them to avoid waste of time by learning the mechanical rule in preparation. Let us, in short, suppose that the teacher has not neglected ordinary precautions for making the most of his three-quarters of an hour's teaching. Then the scheme of the lesson will be something of this kind :

(1.) Five minutes. Questions on back work. For example: On powers of numbers; on the true meaning of multiplication; on the true meaning of the point in decimal fractions, and its function, as the exponent of the power of ten which is implied in the unexpressed denominator; on addition and subtraction of decimals.

(2.) Fifteen minutes. The mechanical rule for multiplication of decimals repeated by two or three

boys in different parts of the class, as a sample how far the class has mastered the process mechanically; and a few short examples, such as can be done wholly, or almost wholly, in the head, worked for the same purpose.

(3.) Twenty minutes. The reasons for this rule developed and explained. Law, that the product of powers of a number is found by taking the sum of their exponents, stated, demonstrated, and illustrated. Easy examples given, and questions asked, on the application of the law to integral numbers. The law applied to the process of multiplication of decimals, and to the rule for that process. Rule thus shown to be a mechanical process of applying that law.

(4.) Five minutes. Recapitulation, with application of above theory to three or four short, easy, examples.

40. Difference between Lesson of New Work and Lesson of Practice.-A lesson of new work in arithmetic should, mutatis mutandis, be something of this kind; and it is clear that in inspecting such a lesson the inspector will have a very different work on hand from the inspection of a lesson of practice or of review. And it must be observed that I mean by a lesson in new work an absolutely first lesson in a rule or process. The inspector, when the notes of the intended lesson are put into his hand by the pupil-teacher, will be careful to ask how far the lesson is really on an absolutely new rule. All lessons, except the first, on a given rule, I call lessons of practice; and I distinguish them on the one hand from lessons of new work, and on the other from lessons of review, which deal with a wider range of recapitulation, and

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in a more miscellaneous manner. not a certain amount of theory to be inculcated in lessons of practice. The reasons for every process should be constantly kept before the children's minds whenever working arithmetic. But that practice, and not explanation of theory, is the main object of these lessons of practice.

41. Second Division: A Lesson of Practice.In inspecting a lesson of new work in arithmetic, the inspector will look more to the teacher (according to the hints given above, see § 38) than to the class. In inspecting a lesson of practice, he will almost look more to the class than to its teacher. Let us suppose that the lesson is the next lesson on the same subject after a lesson of new work. The great points now for him to regard are—

(a). Are copying and prompting absolutely unknown? And does the class work as if they were unknown? That is, is each member of the class, to the extent of his abilities, self-reliant?

(b). Does the teacher sort his class? That is to say, does he find out quickly and accurately which boys have taken a firm and clear hold of the instruction he gave them in the last lesson, which have but an infirm grasp of it, and which (if any) have failed altogether to comprehend it?

(c). Does he understand that he ought to administer a different treatment to these different sections of his class; to push on the first division, and give them more and harder examples, and to select some of the best of them in turn to explain and drive home the subject to the second division, while he himself draws the third division, or worst laggards, out upon the floor, and makes another effort with them by way of recapitulation?

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(d). Are his examples carefully prepared beforehand, and well chosen? Arithmetical examples should not be always short, otherwise scholars will fail to acquire the valuable habit of patient, and yet intelligent, labour, with the mind all the while fixed on a goal. Long examples, as well as short, should be sometimes given; but they should be the exception. For everything, except a trial of endurance and accuracy, short examples are much more valuable than long. And they should be so chosen as to illustrate as many varieties of practical difficulties as possible.

(e). Does he do as much work as possible orallyputting on the worst scholars in the simpler parts of the processes, and making the better scholars keep watch to correct them; encouraging the diligent and accurate by marks or placetaking, and endeavouring to make the whole lesson as lively and as interesting as possible?

(f). Does he discourage speed at the expense of accuracy, and of neat figures, while encouraging it, particularly by means of abbreviated processes, in the careful?

42. Third Division. A Lesson of Review.— The lesson of review in arithmetic combines the leading features of the other two lessons. In it the teacher has not only to pass over back-work, for the purpose of preventing its being forgotten by his scholars, but he has the equally important work to do of trying to connect the different rules together, so as to show their bearing on one another, and to give his scholars a connected view of the science. It is in this kind of lesson that the teacher takes problems which involve the use and application of several rules, and works them through with his

class. This kind of lesson is, of course, from its nature, largely catechetical, and is as good a test of the teacher's capacities as any. If the lesson which the fourth-year pupil-teacher in question is to give is a lesson of review, the inspector will look particularly to such points as the following, in addition to those which I have noted in regard to the lessons in practice and new work, so far as they are applicable (see §§ 38, 41):

(a). Are the examples which the teacher gives of such a kind as to draw out the intelligence of the children, and to make them think; are they, not mere mechanical applications of rules, but problems, requiring the combined application of several rules, such as are met with in every-day life?

(b). When the scholars are puzzled by an example, does he understand how to help them judiciously? In showing them, for instance, how to attack a problem, does he endeavour to show them some general principle, wherewith all similar problems may be attacked, and whereby they may be better able to help themselves next time they meet with such a problem; or does he only look to helping them over the present difficulty?

(c). Does he require a fair proportion of the work in such a lesson to be done orally, and reasons for all processes to be regularly and clearly stated?

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43. Time given to these Lessons by the Inspector. It is supposed that the arithmetic lesson is designed to last forty-five minutes; but that the inspector can only give it twenty minutes. He must, therefore, do, in the case of this arithmetic lesson and of the geography lesson which is to follow it, as it is proposed he should do in the case of the reading and writing lessons of the first

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