As regards the formal introduction of generalised arithmetic or algebra, one cannot lay down any limit of age, owing to the very untrained state in which girls come to secondary schools, but with children who have been taught thoroughly the principles of arithmetic up to fractions, it is easy to introduce literal symbols and so prepare the way: this should be done much earlier than is usual.
Children well taught in arithmetic might perhaps begin the subject formally about thirteen, and I think it well for the first term to drop arithmetic altogether, so as to get as much time as possible for overcoming the initial difficulties, and making use of the zeal which a new study gives; but of course every good teacher of arithmetic will train his pupils to use letters for numbers very much earlier. There is a good deal put into arithmetic books, which would be much better dealt with by algebraical methods, and should be postponed, e.g., involution and evolution, and much time should be saved by omitting long sets of examples on weights and measures, etc., and giving sums to be worked out mechanically with large numbers. As in arithmetic, it is extremely important to give an insight into the composition of quantities, so that de-composition may be easy, subsequent mechanical work in multiplication, division, involution, etc., minimised, and the pupil reach sooner the more attractive branches of the subject, and feel the power it gives.
Mixed mathematics.If children have acquired early a fair knowledge of geometry and algebra, they may, say at sixteen, be ready to pass on to those branches in which the alliance of the two is most intimate, and which are so closely correlated with all the teaching in mechanics and physics. It takes most girls some time to assimilate the ideas of the trigonometrical ratios, and it is fatal to hurry them.[24] Those who are able to proceed further, and enter upon the study of co-ordinate geometry, usually take great delight in it; and it is well, too, to lead them gradually on by some such books as Proctor’s Easy Lessons in the Differential Calculus, to form some idea of what a powerful instrument the Calculus is, before they actually make use of it or formally study it; it takes time for a new method to infiltrate the mind of an ordinary student.
[24] I may add that there is an interesting chapter in Herbart’s A B C of Sense-Perception, in which he works out trigonometrical ratios on the basis of his philosophical system: this chapter would interest those teaching mathematics.
Historical method.Finally, I would once more recommend that, whenever it is possible, pupils should be led along the path of discovery pursued by original investigators, both in physics and applied mathematics; I have found the interest of logarithms greatly increased by this method.[25]
[25] Professor Salford (Monographs on Education and Health) insists on the importance of teaching logarithms as a part of scientific arithmetic. “Often logarithms are first taught in connection with trigonometry, and the average pupil does not learn the difference between a logarithmic and a natural sine; there is no cure for this confusion but to teach logarithms where they belong and to apply them to purely arithmetical problems.” He advises the introduction of logarithms “as soon as the pupil has reached in algebra the proposition am × an = am × n, and he should be shown that the practical method of dealing with powers and roots is the logarithmic. Teachers will then abstain from annoying young pupils with difficult and needless problems solved in the antiquated manner; they will learn how to calculate a compound interest table, an excellent exercise in itself, as well as a labour-saving contrivance in arithmetic. The reason why logarithms are so little appreciated, is that teachers of arithmetic have not as a rule really learned their use; they go on wasting time in arbitrary exercises in evolution, interest, etc., done by tedious methods, and do not appreciate how instinctively the best calculators employ logarithms.”
Professor Lodge’s popular book, Pioneers of Science, is very much appreciated by the young, and I may quote à propos evidence given by Dr. Bryce of Glasgow before the Royal Commission of 1864:—
“Pure mathematics cultivates the power of deductive reasoning, and as soon as boys are capable of forming abstract ideas, and grasping general principles, as soon as they have got correct notions of numbers, and an accurate knowledge of the essential parts of arithmetic, and have made some progress in geometry, then natural philosophy may be advantageously taught. I speak on this matter from experience. My relative and colleague, who had charge of the mathematical department in the Belfast Academy, introduced natural philosophy as part of the work of all the mathematical classes. After these classes had gone a certain length in geography and algebra, he took up the elements of natural philosophy two days in the week, as part of the work of every mathematical class. He began with simple experiments, and according as the progress of the boys in Euclid and algebra admitted of it, more mathematical views of natural philosophy were introduced. The great advantage of the study of physical science is that, when properly taught, it interests boys in intellectual pursuits generally. For instance, Newton’s great discovery, the identity of the power which retains the moon in her orbit with terrestrial gravity, was being explained to a class of from twelve to eighteen boys. The teacher did not tell them the result; he enumerated the phenomena by which Newton arrived at it, taking care to present them in the order most likely to suggest it. As fact after fact was marshalled before them, they became eager and excited more and more, for they saw that something new and great was coming; and when at last the array of phenomena was complete, and the magnificent conclusion burst upon their sight, the whole class started from their seats with a scream of delight. They were conscious that they had gone through the very same mental operation, as that great man had gone through. The consciousness of fellowship with so great a mind was an elevating thing, and gave them a delight in intellectual pursuits. An unusual proportion of those boys who passed through the Belfast Academy during the twenty years that I was able to have natural and physical science taught on those principles, have, as men, been distinguished and successful; and they owe it, I am convinced, in a large degree to the taste for intellectual pursuits thus formed.”