to devise a course in what is called in the recent Report of the Girls' Schools Committee of the Mathematical Association the "Arithmetic of Citizenship." This course is as applicable to boys' schools as to girls'. The main items in such a course are: (1) Local Rates -their relation to rent, analysis of rates; (2) County Finance sources of revenue, items of expenditure, loans; (3) Capital and Industry limited liability companies, value of investments, banking, building societies, land tenure, annuities; (4) National Finance; (5) Insurance-life, fire and marine, National Health -conditions that determine premiums.

There is not likely to be much difference of opinion either as to the interest attaching to such a course or as to the value of the training to be derived from it. The point to be settled-and it can only be settled by experience is, how far the subject-matter is within the compass of the ordinary adolescent boy or girl. The competent teacher will modify it in accordance with his pupils' capacities. Whatever the outcome, there is no doubt that it represents a great advance upon the old Arithmetic book.

It may, however, be as well to point out that even by the best teachers the capacity of the pupil for Mathematics is often misunderstood. Girls especially are frequently classed as unmathematical, withdrawn from lessons in Algebra and Geometry, and set to work instead at the traditional arithmetical sums. This practice is unfair to the girl. More often than not her alleged failure in Mathematics is merely failure to see sense in the subject through the mist of formalism by which it is obscured. To her1 Algebra is abhorrent

1 The reference here, of course, is not to all girls, but only to the girl commonly classed as unmathematical. Many girls not prac

because so deadly mechanical; Geometry, because progress is retarded at every step by the demand for proof of the obvious. The Arithmetic to which she is condemned has the advantage of having grown out of something concrete that she once understood, but it also soon becomes mechanical and meaningless. It is little other than a means of keeping her occupied. The only means of salvation in such instances is for the teacher to realize that, if the pupil has no interest in the subject, it is for the teacher to discover what interest the subject with saner treatment may have for the pupil. That interest will not be attained by narrowing, but by widening the field of study. There are all kinds of practical questions that can be dealt with direct without any formalism, either geometrical or algebraic-How to make a map, How to find the height of a hill, How to save calculation by use of tables, What speed is gained by the gear of a sewing machine or a bicycle-to mention only a few in addition to those specified in the "Arithmetic of Citizenship." When the girl might be tackling such questions as these, questions that call forth all her intellectual powers to their mastery, it is an injustice to undermine her confidence by stigmatizing her as unmathematical, and to throw her back on the sluggish performance of allotted tasks every one of which is a mere copy (digitis mutatis) of some one else's reasoning.

The criticism most often levelled at the "New Teaching" of Mathematics is that it lacks thoroughness as compared with the old. "If it were so, it were a grievous fault." It is possible that some

tically minded and not given to serious thinking are, unfortunately only too content to spend their time working mechanical sums. Such girls are not usually classed as unmathematical.

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reformers have been so intent on their reforms as to lose sight of the importance of maintaining the standard of strictness and precision inherited from the foregoing generation. It is possible even that the impatience characteristic of the reforming temperament is not always compatible with the prosaic demands. of the class-room, so that the most far-seeing and energetic reformers do not always make the best schoolmasters. If so, their failings ought not in justice to be charged against their creed. For there is nothing in the creed of the mathematical reformers which carries with it any lowering of standard in the pupil's work. The immediate aim of the teacher of Mathematics still remains, as it was a generation ago, to insist on clear thinking, followed or accompanied by accurate performance and expression. It is still his moral aim to get the best work out of the pupil. This object, he knows as well as the veriest martinet, is not to be attained by making the work soft or easy or amusing. But, unlike the mere martinet, he hopes to attain it by attaching to the work a human purpose that the boy or girl will appreciate, a purpose that acts as a stimulus to hard work in which the thinking shall be every bit as clear, and the working or drawing every bit as accurate, as under the regime of pure mental discipline.

For so far as the ordinary daily methods of teaching are concerned there is little change called for or brought about. It is still the pupil's work, not the teacher's, that counts. The change is in the type of work set and in the attitude of teacher and pupils towards it. If the work lacks life, it is the teacher's business to give it life. Mathematics is no longer merely the working through a set book with the

teacher as taskmaster. Still the tasks are there, and it is for the teacher to see that they are approached and carried out in such a spirit as not to mar the intellectual pleasure and mental enrichment to be derived from the performance of them. It is as true now as it ever was, that the good teacher of Mathematics does not do all the talking: neither does his class do all the working.

There is, however, one considerable change in the type of work set which the abandonment of formalism for scientific treatment necessarily carries with it. The standard type of mathematical exercise involves a hypothesis and requires a conclusion to be reached. In geometrical exercises it was customary that the conclusion should be stated as well as the hypothesis. For instance, "If adjacent angles are bisected, prove that the bisectors are at right angles to one another." This type of exercise is still useful, and any reform that banished the intellectual joy to be got out of riders would be pernicious. That, however, is only half the truth. There is a still greater intellectual joy to be derived from discovery, and it is a mistake to take the wind out of the sails of the would-be discoverer by always stating the conclusion to be reached. For this reason it is often of advantage to suppress the conclusion in stating the problem, putting an exercise like the above in the form, "If adjacent angles are bisected, do you notice anything else about the figure? If so, can you give a reason for what you find?" The simplest of riders is here chosen for the purpose of illustration. With slightly more complex figures the hunt for propositions can be carried much further and provides as much mental exhilaration as the acceptance of the intellectual chal

lenge conveyed in the old-fashioned and, let us hope, still vigorous rider.

On the same principle, that of encouraging the spirit of research or intellectual conquest, it is often advisable, paradoxical as it may seem, to suppress the hypothesis. A common kind of exercise in Trigonometry, for instance, takes the form of giving the distance between two points on a level plain and the angle of elevation of the top of a mountain from each, from which data the height of the mountain is to be found. In such an exercise the difficulties of the concrete problem are nearly all removed in the enunciation. The real difficulty consists in discovering a way to treat the problem by mathematical methods. It would be far better from the point of view of general education if the problem were set in the form, "How are we to find the height of an inaccessible peak?" A higher order of sagacity is required to discover what measurements are necessary than to work out the result when the measurements are given. Giving the measurements in the first instance tends, if anything, to suppress the development of this higher order of sagacity and to produce an unpractical type of mind in which concrete difficulties are not appreciated at their true value.

One further criticism of the "New Teaching" of Mathematics may be dealt with in conclusion. It is sometimes alleged that while the newer methods may be for the benefit of the intellectual democracy, they militate against the production of first-class mathematicians. Now, it is fairly safe to say that a really clever pupil is proof against most pedagogic contrivances, just as a healthy man is proof against all that a doctor can do for him. It is very unlikely that