this practical kind, under stress of examination, even when it is on familiar lines: and there is great risk that those who have merely committed some facts to memory will appear to advantage in examination compared with those who have a much sounder working knowledge.

Experience of this kind suggests that too much importance must not be attached to examination results. If possible we must suit our methods of examination to the methods of teaching, though it is admitted that there may be serious difficulties in doing this. The difficulties may be surmountable; but even if they are not, it would seem to be undesirable to alter sound methods of teaching to suit examinations. And meanwhile we must be prepared for anomalies wherever examinations come into contact with the new methods. An instance may be given in illustration. To pass "Smalls" at Oxford (which has become a kind of entrance examination into the University) it is necessary for the candidate to show a slight acquaintance either with geometry or with algebra. Most boys of any ability qualify for this examination while at school, and it is only the weaker brethren who are left to face it on coming up to Oxford. Seeing that the new methods of geometrical teaching were designed to help the weaker brethren, it was expected that when they were introduced geometry would be preferred to algebra. But exactly the contrary happened; geometry was deserted and nearly all the candidates chose algebra. On making inquiry I found the reason to be that for the purposes of passing an examination it is undoubtedly easier to cram a dull boy with a few facts in the old way (which is still unreformed in algebra) than to teach him how to reproduce his knowledge of geometry. The difficulty does not arise in getting the knowledge into him, but in getting it out again in examination. Many of the boys in question formerly learnt their Euclid by heart, and this resource has been taken from them. They now learn a few rules in algebra by heart. Whether this does them much good is another question.

Relations of Teachers to one another.-With much more diffidence I remark on one other outcome of the modern methods of teaching. They have, if I am not mistaken, broken down, or at any rate are breaking down, the barriers between the teachers of Mathematics and of Science (especially Physics) which have seriously embarrassed education hitherto. In my early experience as inspector of some public schools in Mathematics on behalf of Oxford University, only half a dozen years ago, I could not but remark how little notice was taken by some of the teachers of what was being done at other schools, or even by their colleagues. I venture to think that a great improvement is being effected in this respect. In many schools a definite system of co-operation between the mathematical and scientific staffs has been already organised and doubtless others will adopt this plan in the future. The joint meetings of the

Mathematical Association and the Association of Science Masters will help to further this improved understanding.

Conclusion. In view of these considerations it seems reasonable to conclude that the methods of practical mathematical work recently adopted in our public schools have justified themselves hitherto. They have aroused the interest of the boys, have made learning easier, co-operation possible, and have stimulated imagination. More time is perhaps required, but there should not be any grave difficulty in assigning it if necessary. There are more serious difficulties in suiting examinations to the new methods, but these must be faced and not allowed to be a hindrance. There are also signs, which can scarcely be mistaken, that the new methods have been a welcome stimulus to the teachers as well.

H. H. TURNER.

PRACTICAL MATHEMATICS AT CLIFTON

COLLEGE.

As far back as 1875, Professor Worthington, then a master at Clifton College, organised a course of practical work in elementary mensuration and hydrostatics which ever since has formed an introduction to the practical science taught to all boys on the modern side and to some boys on the classical side of the school.

The methods of his text book, the first of its kind, have since been adopted throughout the educational world, and the development of his scheme at Clifton has been so successful that the new demand for what is called Practical Mathematics seems to be met quite adequately, in the lower part of the school at any rate, by what has been done here for many years.

Course of Practical Physics.

The time at present allotted to this work in the laboratory is as follows:

After this preliminary course the sets proceed to the practical work in Light, Magnetism and Electricity, &c.

The experiments are divided into Sets and now follow the order of Rintoul's "Introduction to Practical Physics," written by the present Head of the department at Clifton.

Reference to this book will show that sufficient explanation is given to enable the boy to perform an experiment without much assistance, and the educational value of the book lies in the fact that the boy is led to deduce the successive stages of the argument by a series of questions, the answers to which are to be written in such a way as to indicate the nature of the question which is being answered. For instance, the text book gives the following directions for finding, by the principle of Archimedes, the volume of any solid that sinks in water.

"Counterpoise your balance as before, and hang to the short pan a glass stopper. Weigh the stopper while hanging in air. Then immerse it in a beaker of water and weigh again while in water.

"What is the upthrust of the water on the stopper?

"What is the weight of water displaced by the stopper?

"What then is the volume of water displaced by the stopper, and therefore the volume of the stopper? Calculate the density of the stopper.

"In comparing this method of finding the volume of a solid with the method of displacement of water, which do you think capable of giving you the more accurate result?"

Perhaps the most valuable part of the work lies in the way in which the account of the experiment is written out, and it is regarded here as most important to take advantage of the opportunity thus provided for training the boy to express himself clearly and concisely. When satisfactory, both as regards accuracy and intelligibility, the boy's work is signed by the master, and the fact recorded on a board in the laboratory on which is fastened a tabulated list of the experiments of the Set.

It thus follows that the members of a given Form or Set are working quite independently of one another, and that each boy performs a logical sequence of experiments during his career at the school.

The system also allows a boy, who has for some reason dropped the subject for a time, to take it up again where he left off without interfering with the work of the particular block of the school of which he may then be a member.

As the Physical Laboratory is almost continuously occupied by Sets in Physics it is at present impossible to make arrangements for mathematical masters to take their Sets into the Laboratory to do practical work in combination with their theoretical work. Since, however, Mechanics is taught as a branch of experimental Science in the Physics course, most boys go through a course of Practical Mechanics even though they may not advance so far in their Mathematical Sets as to take it as a part of their mathematical work.

Practical Mathematics on the Military and Engineering Side.

At present, the Practical Mathematics now demanded in the Army Examinations is done in the Laboratory out of school hours, and is confined to practice in performing experiments similar to those set in previous examinations. It is hoped, however, that it may be possible shortly to provide accommodation for the Army Side so that their practical work may be done in close connection with their theoretical work.

On the Military Side, during the lessons in Mechanics, experiments are performed by the master in the presence of the class.

One room is provided with apparatus of a simple type sufficient for elementary experiments in Statics and Dynamics. The cost of the apparatus scarcely exceeded 21., and most of it is kept in the drawer of the table on which the experiments are performed. The most useful articles are the Rintoul dynamo

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meter (supplied by Messrs. Harris of Birmingham), and a board which can be clamped to the table and used for the demonstration of such theorems as the triangle of forces. When more elaborate apparatus is required such as Fletcher's trolley, it is generally possible to make arrangements for the Set to be taken in a Physics Lecture Room.

The class draws a diagram of the apparatus, the necessary measurements and readings are taken by the master and written down by the class who then make the calculations and deductions, after which a formal proof of the theorem is discussed. The advantage of introducing the apparatus into the Class Room was especially marked in the improvement which took place in the drawing of diagrams, and there is now a complete absence of the representation of bodies in equilibrium under impossible conditions.

Practical work to illustrate the application of the Calculus is only valuable when the boy is capable of working out theoretically the formula which is to be applied. There is very little educational value in an experiment which merely requires the substitution of numerical results in a formula which the boy is incapable of obtaining from his theoretical knowledge unless he is asked to make some deductions from the results which he has found. This seems to be a weak point in some of the questions set in the Army Examinations, but the Civil Service Commissioners are doing valuable work in insisting on practical work in the Competitive Examinations, and it is to be hoped that mathematical masters, whose training has been purely theoretical, will soon be brought to recognise the value of such practical work as part of a boy's mathematical education.

R. C. FAWDRY.