The passage of some of the subjects of elementary mathematics from the colleges to the secondary schools raised two very fundamental questions. The first of these concerned mostly the secondary schools, since it involved an adaptation to the needs of younger students of the more or less crystallized textbook material which came to them from the colleges. The second of these questions affected the colleges only, since it involved the selection of proper material to base upon the foundations laid by the secondary schools. It is natural that the influence of the colleges should have been somewhat harmful with respect to the secondary schools, since the interests of the former seemed to be best met by restricting most of the energies of the secondary teachers of mathematics to the thorough drilling of their students in dexterous formal manipulations of algebraic symbols and the demonstration of fundamental abstract theorems of geometry.
=Relation of mathematics in secondary schools and college=
Students who come to college with a solid and broad foundation but without any knowledge of the superstructure can readily be inspired and enthused by the erection of a beautiful superstructure on a foundation laid mostly underground, with little direct evidence of its value or importance. The injustice and shortsightedness of the tendency to restrict the secondary schools to such foundation work would not have been so apparent if the majority of the secondary school students would have entered college. As a matter of fact it tended to bring secondary mathematics into disrepute and thus to threaten college mathematics at its very foundation. It is only in recent years that strong efforts have been made to correct this very serious mathematical situation.
Much progress has been made toward the saner view of letting secondary mathematics build its little structure into the air with some view to harmony and proportion, and of requiring college mathematics to build _on_ as well as _upon_ the work done by the secondary schools. The fruitful and vivifying notions of function, derivative, and group are slowly making their way into secondary mathematics, and the graphic methods have introduced some of the charms of analytic geometry into the same field.