Mr. Fearon (British Schools, Wales). The Arithmetic is the worst elementary subject in these schools, and yet it is the one in which comparative backwardness is least excusable in Wales, because the children's ignorance of the English language does not offer so much obstruction to their instruction in Arithmetic as to that in reading and spelling. The unsatisfactory results in this subject are partially due, no doubt, to defective powers of explanation on the part of the teachers, and partly to the intrinsic difficulty of the subject. But I believe a still greater part is due to the simple fact that the teachers do not take effective measures to prevent copying. It is of no use to lecture and caution the children against copying; to represent to them that the practice is dishonourable and injurious to their best interests. Except, perhaps, to a very few bright-minded and noble natures, all such talk is merely souring the mind. What pupils at Eton or Harrow, passmen at Oxford and Cambridge, candidates for Holy Orders, and examinees all over the world will do if they can, these Welsh children will not be likely to abstain from doing because they are told it is wrong. Unless a habit of not copying has been created in the minds of all the children, and is diligently fostered, it is useless to preach to them against the sin of copying. And how is such a habit to be created? Not by logical conviction, but by exercise. The teacher must make it constantly impossible for the children to copy. But he must remember the force of the word impossible. He must remember that a child when hard pressed is like the fox in the Second Irish Reading Book, "he has ten tricks, and ten times ten more than you can guess.' He can at a pinch see many yards, and one glance at a friend's slate will often undo the benefit of an hour's careful teaching, by vitiating the result and concealing from the teacher the fact that that child has not mastered the subject. Then when the Inspector comes, and takes measures which render it impossible for that child to copy, he breaks down in a manner which perfectly astonishes the teacher. The only way to secure the habit of not copying in a school is to begin by giving all the children different sums. This can easily be done by a little forethought and contrivance, if a sufficient number of examples be prepared beforehand. Then when a habit of self-reliance has been created by this method, the children may be taught in draughts, standing back to back.'

Mr. Fitch (British Schools, Yorkshire). Of the Arithmetic I have little to add to the testimony given in my former report. In many cases I observe great improvement in the methods of teaching, and in all the good schools it is the invariable practice to illustrate and explain every rule, not only before the working of each new group of sums is attempted, but also at intervals whenever a difficulty occurs. Nothing is more depressing to the intelligence of a child than the handling of figures to

which he attributes no meaning, and the employment of rules which he does not understand; yet there are still many schools in which sums are merely regarded as problems to be done, not as exercises to be understood. I know that the hardest of all teaching is teaching to think, and that there is always a temptation to avoid it, or accept any mechanical substitute for it. Yet if the effort is made it always repays the cost of making it. No child can learn the reason of the mysterious process in subtraction, commonly known as "borrowing 10," without receiving a lesson in logic, and without having a proportionately brighter intellect to carry to all his other lessons. But because it takes a little time and trouble to effect this it is rarely done. Teachers find it more easy to take for granted that if the process is used now it will be understood hereafter; or they content themselves with a jargon about "borrowing" and "carrying," which looks like explanation, but which is utterly without meaning. At any rate the number of schools in which any attempt is made to teach the simple truth, "that the difference between two unequal numbers remains the same if we add 10 to each of them," is very small. In course of a recent visit to one of the schools of the Christian Brothers, in France, I noticed that the boys were called up one after another to work a sum on the black board in the presence of a class, and to account for the reason of each part of the operation. The performance seemed to interest the boys, to quicken their attention, and to impart a good deal of intellectual animation to the whole treatment of the subject. Such exercises, though not rare in our best schools at home, might be increased in number with great advantage.'

Mr. Johnstone (S. Lancashire).- With regard to the Arithmetic some explanation requires to be made. One-fourth of the children are found to have failed in it, and this proportion is more than double that in reading and writing. Great allowance, however, has to be made in consequence of the enforcement of what has been in this district a novel requirement. Numeration was hardly taught in the schools two years ago and the children had been accustomed to have the figures set down for them upon the examination cards. When this system was put an end to, and the dictation of sums introduced, the consequences in each school were most fatal. Although the teachers had received due notice of the change that was to be made, their scholars were found in almost every instance unprepared, and the long list of failures was followed by corresponding reduction in the grants. I cannot say that these results were met in every case as they might, or perhaps ought to have been; I can hope, however, that the feeling excited was but temporary, and I can report that other results have since followed, which show a progressive improvement that is likely to be sustained. In another year the schools will have accustomed themselves

to the new teaching required, and to the new form of examination, and from what has already been done there is every reason to hope that this branch of instruction will no longer appear so far behindhand in the official return, and there will at least be the knowledge that it is founded upon a firmer basis.'

Mr. Kennedy (Lancashire and Isle of Man).- Fewest children are actually passed in Arithmetic, yet notwithstanding, to my mind, Arithmetic is the most satisfactory. I need not repeat again my explanations of this seeming paradox.'

Mr. Kerr (Scotland). The percentage of passes in Arithmetic is 76 per cent.'

Mr. King (Durham and Northumberland).—' Arithmetic is methodically taught, notation is improving, and as far as the standard requirements are concerned, the subject continues to promise well; some danger appears to me to exist of its being treated in too mechanical a way with the upper classes, so that unless questions are put exactly in the accustomed form, the children experience some difficulty in applying the rules with which they are really familiar.'

Mr. Moncrieff (Kent).-'I am not prepared to recommend any change in the Arithmetic, either of Standard II. or Standard III. The latter is decidedly improving, specially notation. For Standard IV. division, leaving long division (simple and compound) for the following year. Standard V. might be confined to a few of the easier tables of weights and measures, reserving the others for Standard VI. The step to this highest standard is really too short, but I should not like to increase its length, unless my subsequent suggestion were adopted as to the treatment of children who have failed. As long as it is absolutely necessary to carry every child from grade to grade in consecutive years, it is well to leave the last year's task so easy as to give time for a careful revision of the work of previous standards.'

Mr. Oakeley (British Schools, North of England).-' In Arithmetic boys' schools are very decidedly the best; and I cannot call to mind a single girls' school where the best scholars can work any ordinary sum in any common rule, which could be done in 20 or 30 boys' schools. I am always pleased when I find some knowledge of the principles, some ability to prove the rule; this, however, is very seldom the case.'

Mr. Robinson (Buckingham and Hertford). The pressure put upon the teacher to pass his scholars in the given subject, leads him to study what will conduce to that particular end, to shut the door to other considerations but the best means to secure the money payment which follows the examination of the children in standards; now the latter prescribes the limits for each successive grade or class; a couple of rules of simple

computation, or the four money rules as the case may be; to take the case of Standard III., into which most of the children pass, simple multiplication and simple short division only are expected to be known; but if the previous rules are well grounded, the bare working up these two rules need not occupy all the year, or bind down the performance to the exclusion of mental Arithmetic, or the advance upon higher rules, or the ensuring of ground already traversed. And it is found that such is the literal adherence to the limits of each standard, for fear of failure in the examination, that a slight deviation from the beaten track causes instant consternation; and the fortunes of those who are thus drilled up and down, over and over, in long acquired rules, is by no means more successful than that of others, who endeavour to instil life, and intelligence and diversity, into their teaching of this subject.'

Mr. Scoltock (British Schools, Midland Counties). In Arithmetic the teaching is too mechanical; straightforward questions are either set on the black board, or worked from books, consequently the simplest sum which needs any amount of thought is scarcely ever done. In Standard VI. bills of parcels are less frequently correctly done than long sums of practice.'

Mr. Scougal (North and North-East of Scotland).-'It is gratifying to see the important subject of Arithmetic in a higher position,' (passes being 79-81 per cent. in 1866-7 against 66-62 per cent. in 1864-5), evidently receiving more attention than formerly, and at an earlier stage of a child's school life. Still it deserves to be noticed that comparatively few attempt questions a little out of the beaten track, and requiring a little independent thought. I often give a question (optional) of this kind in Standards IV., V., VI., and in the majority of cases it is not attempted; and when attempted it is not unfrequently quite wrong. Such questions should be frequently given to the scholars, and would form, in my opinion, very good home exercises.'

Mr. Sewell (Lincoln and Notts.)-The exercises in Arithmetic are still less successful than those in reading and writing. The requirement of accurate notation as part of the exercise under the instructions of January, 1866, has increased the difficulty of it, but has undoubtedly increased the children's aptitude for arithmetical processes. It is, after all, a question whether this work is really inferior to, or rather that it appears to a disadvantage beside reading and writing; because the slightest mistake is fatal to the correctness of a sum, and leaves the Examiner no opportunity to exercise the charity which overlooks a misspelling or a fault of pronunciation.'

Mr. Temple (Cheshire).-'The Arithmetic is generally taught with greater accuracy than I should have thought possible before the days of

"paying by results," but there is still a want of variety in method, and too much slavery to black boards and cards.'

Mr. Warburton (Hampshire).—The instruction in Arithmetic, of which almost all of H.M. Inspectors have complained, continues to produce very uncertain and irregular results. A sufficiently large number of schools pass well in this subject to show that children can be taught to do the sums required with certainty, but in the great majority of my schools, and more especially in those under mistresses, the work does not appear to be worth very much. It is quite an exception when a scholar in the VI. Standard, a girl more especially, gets the right answer to two out of the three sums required to be set in practice and bills. In the latter kind of sums there is generally a considerable number of details, though of the simplest character, requiring absolute accuracy in the first four rules and reduction, but nothing beyond this. In the former rather more thought and contrivance is required, and an apparently trifling error in method will produce a result 10 or 100 too great or too small; but the children hardly ever seem to apply, before bringing up their answers, the test of common sense, which would probably convince them in a moment of the impossibility of the result obtained being the correct one.'

Mr. Watkins (Lincoln, Nottingham, York). In Arithmetic the comparison of the last two years is, in my district, more unfavourable than in writing. In 1865 the passes were 83 per cent., in 1867 75 per cent. This is a considerable falling off. Possibly-though this is an ungallant explanation-it may be owing to a cause I have already mentioned, the larger employment of female teachers in mixed schools. And I observe that the report of your Lordships' Committee bears me out in this view. "The schools under mistresses contribute more than their proportionate share of failures in this subject (Arithmetic)." My impression is that the chief failure is in the V. and VI. Standards, perhaps more in the former than in the latter. The step from the work of the Fourth to that of the Fifth Standard is not an easy one. The pounds, shillings, and pence of the Fourth Standard are familiar things, the perches and roods, the grains and pennyweights of the Fifth Standard are things known only, if at all, by name, and not by sight. They certainly ought, for the work of life, to be better known. And the "bills of parcels," which the Sixth Standard has to make out, should be more correctly worked, and in a much shorter time than that generally allowed to them. There is no doubt that Arithmetic is taught too much by rules and too little in its principles, mechanically and not intelligently.'

Mr. Wilkinson (Episcopal Church Schools, Scotland). The results of examination in Arithmetic continue to betray some deficiency in that subject of instruction. The failures in Arithmetic amount to about 17