The third case concerns bonds whose importance for practical uses in life or as notable facilitators of other bonds is so great that they may profitably be brought to a greater strength than 199 correct out of 200 at a speed of 2 sec. or less, or be brought to that degree of strength very early. Examples of bonds of such special practical use are the subtractions from 10, 1/2 + 1/2, 1/2 + 1/4, 1/2 of 60, 1/4 of 60, and the fractional parts of 12 and of $1.00. Examples of notable facilitating bonds are ten 10s = 100, ten 100s = 1000, additions like 2 + 2, 3 + 3, and 4 + 4, and all the multiplication tables to 9 × 9.
In consideration of these three modifying cases or principles, a volume could well be written concerning just how much practice to give to each bond, in each of the types of complex situations where it has to operate. There is evidently need for much experimentation to expose the facts, and for much sagacity and inventiveness in making sure of effective learning without wasteful overlearning.
(1) The textbook or other instrument of instruction which is a teacher's general guide may give far too little practice on certain bonds.
(2) It may divide the practice given in ways that are apparently unjustifiable.
(3) The teacher needs therefore to know how much practice it does give, where to supplement it, and what to omit.
(4) The omissions, on grounds of apparent excess practice, should be made only after careful consideration of the third principle described above.
(5) The amount of practice should always be considered in the light of its interest and appeal to the pupil's tendency to work with full power and zeal. Mere repetition of bonds when the learner does not care whether he is improving is rarely justifiable on any grounds.
(6) Practice that is actually in excess is not a very grave defect if it is enjoyed and improves the pupil's attitude toward arithmetic. Not much time is lost; a hundred practices for each of a thousand bonds after mastery to 199 in 200 at 2 seconds will use up less than 60 hours, or 15 hours per year in grades 3 to 6.
(7) By the proper division of practice among bonds, the arrangement of learning so that each bond helps the others, the adroit shifting of practice of a bond to each new type of situation requiring it to operate under changed conditions, and the elimination of excess practice where nothing substantial is gained, notable improvements over the past hit-and-miss customs may be expected.