It is becoming more widely recognised each year, among leaders of educational thought, that in the field of educational science, where mere opinion holds sway, this opinion should be replaced by conclusions scientifically obtained from carefully collected data.

In the teaching of mathematics we have such questions as:

(i) What is the most economical length of drill periods in first-year algebra ?

(ii) What is the best method of procedure for acquiring skill in the mechanical work in algebra ?

(iii) To what extent does mechanical skill in algebra help the pupil intelligently to use algebraic processes, such as the equation, the graph and the formula ?

(iv) What proportion of pupils benefits by a certain type of course and up to a given standard ?

(v) The results of (iv) would be a factor in deciding whether all secondary school pupils should take mathematics throughout their course.

The determination of the group of partial fractions corresponding to the powers of a linear factor x+b in the denominator of a rational function f (x)/{(x+b)nφ(x)} is one of the comparatively rare algebraic problems whose theoretical solutions are feasible in practice. The numerator of (x+b)n−r is the coefficient of yr in the expansion of f(−b+y)/φ(−b+y), and the calculation of the group of numerators is effected by means of two Horner transformations followed by one division, which can of course be arranged synthetically; no operation need be taken further than the term in yn−1, whatever the degrees of f(x) and φ(x).