Pupils in the elementary school, except the most gifted, should not be expected to gain mastery over such concepts as _common fraction_, _decimal fraction_, _factor_, and _root_ quickly. They can learn a definition quickly and learn to use it in very easy cases, where even a vague and imperfect understanding of it will guide response correctly. But complete and exact understanding commonly requires them to take, not one intellectual step, but many; and mastery in use commonly comes only as a slow growth. For example, suppose that pupils are taught that .1, .2, .3, etc., mean 1/10, 2/10, 3/10, etc., that .01, .02, .03, etc., mean 1/100, 2/100, 3/100, etc., that .001, .002, .003, etc., mean 1/1000, 2/1000, 3/1000, etc., and that .1, .02, .001, etc., are decimal fractions. They may then respond correctly when asked to write a decimal fraction, or to state which of these,--1/4, .4, 3/8, .07, .002, 5/6,--are common fractions and which are decimal fractions. They may be able, though by no means all of them will be, to write decimal fractions which equal 1/2 and 1/5, and the common fractions which equal .1 and .09. Most of them will not, however, be able to respond correctly to "Write a decimal mixed number"; or to state which of these,--1/100, .4-1/2, .007/350, $.25,--are common fractions, and which are decimals; or to write the decimal fractions which equal 3/4 and 1/3. If now the teacher had given all at once the additional experiences needed to provide the ability to handle these more intricate and subtle features of decimal-fraction-ness, the result would have been confusion for most pupils. The general meaning of .32, .14, .99, and the like requires some understanding of .30, .10, .90, and .02, .04, .08; but it is not desirable to disturb the child with .30 while he is trying to master 2.3, 4.3, 6.3, and the like. Decimals in general require connection with place value and the contrasts of .41 with 41, 410, 4.1, and the like, but if the relation to place values in general is taught in the same lesson with the relation to /10s, /100s, /1000s, the mind will suffer from violent indigestion. A wise pedagogy in fact will break up the process of learning the meaning and use of decimal fractions into many teaching units, for example, as follows:-- (1) Such familiarity with fractions with large denominators as is desirable for pupils to have, as by an exercise in reducing to lowest terms, 8/10, 36/64, 20/25, 18/24, 24/32, 21/30, 25/100, 40/100, and the like. This is good as a review of cancellation, and as an extension of the idea of a fraction. (2) Objective work, showing 1/10 sq. ft., 1/50 sq. ft., 1/100 sq. ft., and 1/1000 sq. ft., and having these identified and the forms 1/10 sq. ft., 1/100 sq. ft., and 1/1000 sq. ft. learned. Finding how many feet = 1/10 mile and 1/100 mile.