The Mathematical Conference called by the Committee of Ten, U.S.A., writes as follows, and I quite agree with its view: “The conference recommends that the courses in arithmetic be abridged and enriched—abridged by omitting entirely those subjects which perplex and exhaust without affording any really valuable mental discipline, and enriched by a greater number of exercises in simple calculation and in the solution of concrete problems. Among the subjects which should be curtailed or omitted are compound proportion, cube root, abstract mensuration and the greater part of commercial arithmetic. Percentage should be reduced, and the needs of practical life—profit and loss, bank discount, compound interest, with such complications as result from fractional periods of time—are useless and undesirable. The metric system should be taught in application to actual measurements, and the weights and measures handled.
“Among the branches of this subject which it is proposed to omit are some which have survived from an epoch when more advanced mathematics was scarcely known in our schools, e.g., cube root, duodecimals; so far as any useful principles are embodied in them, they belong to algebra, and can be taught by algebraic methods with such facility, that there is no longer any sound reason for retaining them in the arithmetical course.”
I do not insist on algebra for all; it gives the same sort of mental discipline that arithmetic does, and so, educationally, is not of special value. Geometry, on the other hand, gives a different kind of training—opens a different set of ideas. Many girls, therefore, do not learn algebra, especially those who come late with no clear ideas about arithmetic. Those who have been taught arithmetic well from the beginning can be led on to use algebraic symbols and letters very early.
As soon as a pupil has gone through the course I have recommended, she is ready to take up algebra in a systematic way—I shall suppose she has already been familiarised with the use of letters as general symbols.
MATHEMATICS.
By Dorothea Beale.
How and when can we best introduce mathematical teaching? We have to do at present in girls’ schools with many who have come to the age, say of fifteen or sixteen, with no mathematical teaching except a very slight knowledge of arithmetical processes. For these it seems to me more important to give the mental training afforded by some initiation into geometrical ideas and methods, than to teach algebra.
Beginnings in the kindergarten.For the children (and they are happily a rapidly increasing number) who have had good teaching in the kindergarten, one may frame a course more approaching the ideal. Children can be quite early familiarised with geometrical forms and figures, and learn some of their simpler properties in connection with the drawing and modelling lessons.
Practical geometry.The Conference on Mathematics, called by the Committee of Ten, U.S.A., recommends that children from the age of ten should have some systematic instruction in concrete or experimental geometry. “The mere facts of plane and solid geometry should be taught, not as an exercise in logical deduction and exact demonstration, but in as concrete and objective a form as possible; the simple properties of similar plane figures and solids should not be proved, but illustrated and confirmed by cutting up and rearranging drawings and models. The course should include the careful construction of plane figures by the eye and by the help of instruments, the indirect measurements of heights and distances by the aid of figures drawn to scale, and elementary mensuration plane and solid.”
A small book by Paul Bert, First Elements of Experimental Geometry (Cassell), is very suggestive, and would throw much interest into the subject. Spencer’s Constructive Geometry may be referred to, but it is not altogether satisfactory. A useful and practical book is Geometry for Kindergarten Students, by Pullar (Sonnenschein).