Page:Popular Science Monthly Volume 85.djvu/464

460 it soon appeared desirable to include all mathematical instruction in the scope of its investigation.

Sub-commissions were appointed in various countries. The American sub-commission is composed of D. E. Smith, Columbia University; W. F. Osgood, Harvard University, and J. W. A. Young, Chicago University. Under the general direction of these sub-commissions a vast amount of material relating to the mathematical instruction has been collected and published. In our own country this material was published by the U. S. Bureau of Education in the form of thirteen reports. Some of the other countries have not yet completed their work, but about one hundred and sixty such reports have already been published in the twenty-six countries which have joined in this vast undertaking.

In addition to securing these valuable reports the central commission has arranged international meetings for the discussion of some of the fundamental questions relating to mathematical instruction. Such a meeting was held in Paris, France, in April of the present year, and the two subjects under consideration were: (1) The results obtained by the introduction of differential calculus in the advanced classes of the secondary schools, and (2) the place and the role of mathematics in higher technical education.

Some of the leading French mathematicians (including Appell,Darboux, Borel and d'Ocogne) took an active part in the deliberations. Professor Borel emphasized the fact that mathematics is not composed of a linear sequence of theorems such that each depends upon the preceding one. If this were the case, the only possible changes in methods of instruction would relate to what theorems could be omitted in this sequence or what theorems could be substituted for others. On the contrary, the number of different routes leading from first principles to an advanced mathematical proposition is often exceedingly large, and hence arises the possibility of employing widely different methods to achieve the same general results.

In other words, mathematics is a network formed by intersecting thought roads and the chief aim of the International Commission on the Teaching of Mathematics is to secure extensive information as regards the choice of roads in various nations. The Italian member of the central committee, G. Castelnouovo of Rome, stated explicitly in his address during the recent conference at Paris, that the commission did not aim to bring about any great reforms, but aimed to gather facts as regards existing conditions in order that the various nations might be enabled to profit by the experiences of other nations in instituting their own reforms.

In describing mathematics as a network of a certain type of thought-roads, it is not implied that thought is conveyed along these roads as the products of a country are conveyed on a railroad train. On the