[5] The facts concerning the present inaccuracy of school work in arithmetic will be found on pages 102 to 105.
[6] McLellan and Ames, Public School Arithmetic [1900].
[7] These concern allowances for two errors occurring in the same example and for the same wrong answer being obtained in both original work and check work.
[8] The very early learning of 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 3 × 3, and perhaps a few more multiplications is not considered here. It is advisable. The treatment of 0 × 0, 0 × 1, 1 × 0, etc., is not considered here. It is probably best to defer the '× 0' bonds until after all the others are formed and are being used in short multiplication, and to form them in close connection with their use in short multiplication. The '0 ×' bonds may well be deferred until they are needed in 'long' multiplication, 0 × 0 coming last of all.
[9] See page 76.
[10] At the end of a volume or part, the count may be from as few as 5 or as many as 12 pages.
[11] Certain paragraphs in this and the following chapter are taken from the author's Educational Psychology, with slight modifications.
[12] It should be noted that just as concretes give rise to abstractions, so these in turn give rise to still more abstract abstractions. Thus fourness, fiveness, twentyness, and the like give rise to 'integral-number-ness.' Similarly just as individuals are grouped into general classes, so classes are grouped into still more general classes. Half, quarter, sixth, and tenth are general notions, but 'one ...th' is more general; and 'fraction' is still more general.
[13] They may, of course, also result in a fusion or an alternation of responses, but only rarely.
[14] The more gifted children may be put to work using the principle after the first minute or two.