reviewing or testing progress in arithmetic. They should also be taught, when taking a new process in arithmetic with their classes, always to work through a certain number of examples, orally, with the children, on the black board, taking care to make those who are usually slow, inattentive, or inaccurate in arithmetic do the greater share of this work. It is marvellous what a reform is made in the arithmetic of a school when once steps have been taken to render copying impossible. Boys who have been inattentive, learn to attend; boys who have been in the habit of relying on others, get the habit of self-reliance, and find themselves so much happier and better that it becomes no very difficult matter, with a little care and judgment, to maintain that habit in them. And this change in their habits, as regards arithmetic, affects not only their progress in that subject, but improves their capacity and their work in all the subjects taught in the school. It is, therefore, as I have said, impossible to overrate the importance of preventing copying in arithmetic in a school, and the inspector will make a point of inquiring, in the course of his inspection, what methods are adopted in the school to secure this result.

35. Pupil-Teachers to be Questioned on Method. -He will also inquire, as part of his general inspection of the school, how far the multiplicationtable is learnt; whether it is the practice of the school to teach it up to twenty times instead of stopping, as is usually thought sufficient, at twelve times; and whether the knowledge of it is secured and rendered readily available by frequent and regular repetition of it, at least throughout all but the highest classes in the school. He will

inquire what system of mental arithmetic is in use in the school; and whether the pupil-teachers, or at any rate the seniors among them, are acquainted with handy methods of working rules by shortened processes. It will be found to have a great and most wholesome effect if the inspector, when holding the collective examination of pupil-teachers, will call up the third, fourth, and fifth year pupil-teachers, or some of them, and question them orally as to the methods adopted in their schools in respect of these and similar matters. By so doing the inspector will not only get some light to guide him in his forthcoming inspection, and will learn something of the intelligence of pupil-teachers, when taken out of the ordinary routine of their paper work; but he will also awaken in their minds a desire to study method, when they see what importance is attached by the inspector to all the processes which they pursue in the exercise of their profession. A very good opportunity may be found for doing this, at the collective examination, by the inspector, when he calls out the older pupil-teachers to work their Euclid orally on the black-board. I used, when acting as an inspector, always to give an oral examination in Euclid, as well as the written one, at my collective examinations; of course not allowing the same letters as are employed in the text-book to be affixed to their figures by the examinees. I found that this practice had a most salutary effect on the study of Euclid among the pupil-teachers, as the principal teachers of the school were generally present (as well as many managers and others), and were ashamed that their pupil-teachers should break down in so public a manner. I also found that I was very often able to use that opportunity to put one or

two unexpected questions to the senior pupil-teachers on method or other matters which I had reason to think were apt to be neglected in the schools.

36. Fractions to be Taught next after the Simple Rules.-The new Code does not require vulgar fractions to be taught below the sixth standard (see Appendix I.). The inspector cannot therefore, of course, insist on any instruction being given in fractions in elementary schools below that standard. But he can point out to teachers how defective and slipshod all teaching in arithmetic must be in which fractions are not introduced, and can encourage them to begin instruction in fractions as early as possible. When the Revised Code was first introduced, fractions did not form part of the standard examination at all; and many schools in which that subject had been regularly taught, gave it up and confined themselves to the standard course. I was so persuaded of the evil of this, that I issued a circular to the effect that old established boys' schools would still be examined in fractions, and that those in which instruction in this subject was maintained would have a better general report. My own conviction is that teachers will find that it answers, for the mere purposes of the Standard examination, to teach fractions to all their classes immediately after the first four simple rules; while I think there can be no question that the general effect of pursuing this course will be excellent. The teaching of arithmetic will thus become much more sound and intelligent, and can also be made much more interesting to the children. I do not believe that, in the long run, teachers would find they had lost any time, or any grant on arithmetic, by teaching vulgar fractions to their third and fourth standards, and decimal

fractions to their fifth standards; but rather that the time was, in every sense, well bestowed in securing to the children that their arithmetical training was really sound and scientific, which it never can be until they have learnt something of fractions. The gain, too, in the popularity of the schools among the intelligent artisans and other skilled labourers would be very great, and more would be done by such a course than by any thing else to remove one of the great scandals of our elementary schools, viz., that they turn out the mass of their scholars (who never reach the sixth standard) so deficient in their knowledge of arithmetic, that it is useless to attempt to give them any technical education until they have first gone through a course of improved arithmetic. I am quite aware, however, that while the standard course of arithmetic remains as it now is, and requires the teaching of the compound rules, of the weights and measures, and of practice and proportion, before vulgar and decimal fractions, the inspector can do nothing in this matter except exhort and encourage. Earnest exhortation and hearty encouragement will, however, do much more than people suppose.

37. Inspection of an Arithmetic Lesson. Three Divisions of the Subject.—With these general principles in view respecting the teaching of arithmetic, the inspector will proceed to criticize the arithmetic lesson which I have supposed is to be delivered before him by the fourth year pupil-teacher. And the first thing which he will have to consider, in directing his attention to this particular lesson is, with what part of the art of teaching arithmetic, is this lesson concerned? In teaching arithmetic, there are three essential parts-new work, practice,

and review. The inspector will have inquired during the above-mentioned interval (see § 25), or he will inquire before the lesson begins, with which of these three divisions the lesson of the fourth-year pupil-teacher is to be concerned, and will look carefully to see if the pupil-teacher understands the distinction of the provinces of these divisions.

38. First Division. A Lesson of New Work.— If the lesson is one of new work, that is in which the class first breaks ground on a new rule, the great points for the inspector to look to are

(a). Is the teacher thoroughly master of his subject? Does he treat of it mechanically, or does he seem saturated with it, so that he can put it in many various ways, and can illustrate it largely?

(b). Is he clear and logical in his treatment? Do the parts of his lesson lead up to one another, and to the conclusion, by well-arranged, clear, definite, and yet easy steps, so that each one suggests the preceding and the following?

(c). Does the lesson show thought and preparation? Does he simply adopt the line of any well-known good text-book, in his arrangement of the subject, his examples, and his reasons for the different processes, or has he so far thought over the matter, as to give it a turn of his own? No intelligent teacher, however young, can think over his work out of school, and by himself, without giving it some originality of aspect.

39. Illustration of an Arithmetic Lesson of New Work. For example, let us suppose it is a first lesson on multiplication of decimals. I select this subject because it is one in which the mechanical rule is exceedingly simple and easy, but in which the reasons for the rule, though perfectly capable of being