The variation within a single class for which a single teacher has to provide is great. Even when teaching is departmental and promotion is by subjects, and when also the school is a large one and classification within a grade is by ability--there may be a wide range for any given special component ability. Under ordinary circumstances the range is so great as to be one of the chief limiting conditions for the teaching of arithmetic. Many methods appropriate to the top quarter of the class will be almost useless for the bottom quarter, and _vice versa_. [Illustration: FIGS. 67 and 68.--The scores of ten 6 B classes in a 12-minute test in computation with integers (the Courtis Test 7). The score is the number of units done. Certain long tasks are counted as two units.] Figures 67 and 68 show the scores of ten classes taken at random from ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of computation done in 12 minutes. Observe the very wide variation present in the case of every class. The variation within a class would be somewhat reduced if each pupil were measured by his average in eight or ten such tests given on different days. If a rather generous allowance is made for this we still have a variation in speed as great as that shown in Fig. 69, as the fact to be expected for a class of thirty-two 6 B pupils. [Illustration: FIG. 69.--A conservative estimate of the amount of variation to be expected within a single class of 32 pupils in grade 6, in the number of units done in Courtis Test 7 when all chance variations are eliminated.] The variations within a class in respect to what processes are understood so as to be done with only occasional errors may be illustrated further as follows:--A teacher in grade 4 at or near the middle of the year in a city doing the customary work in arithmetic will probably find some pupil in her class who cannot do column addition even without carrying, or the easiest written subtraction (8 9 78) (5 3 or 37) (- - --), who does not know his multiplication tables or how to derive them, or understand the meanings of + - × and ÷, or have any useful ideas whatever about division. There will probably be some child in the class who can do such work as that shown below, and with very few errors. Add 3/8 + 5/8 + 7/8 + 1/8 2-1/2 1/6 + 3/8 6-3/8 3-3/4 ----- Subtract 10.00 4 yd. 1 ft. 6 in. 3.49 2 yd. 2 ft. 3 in. ----- ---------------------- Multiply 1-1/4 × 8 16 145 2-5/8 206 ------ --- _______ _____ Divide 2)13.50 25)9750