or altering the figures when 'borrowing' in subtraction, and the like. Since it is undesirable that the pupil should regard the 'crutch' response as essential to the total procedure, or become so used to having it that he will be disturbed by its absence later, it is supposed that the bond between the situation and the crutch should not be fully formed. There is a better way out of the difficulty, in case crutches are used at all. This is to associate the crutch with a special 'set,' and its non-use with the general set which is to be the permanent one. For example, children may be taught from the start never to write the crutch sign or crutch figure unless the work is accompanied by "Write ... to help you to...." Write - to help you to Find the differences:-- remember that you must 39 67 78 56 45 subtract in this row. 23 44 36 26 24 -- -- -- -- -- Remember that you must Find the differences:-- subtract in this row. 85 27 96 38 78 63 14 51 45 32 -- -- -- -- -- The bond evoking the use of the crutch may then be formed thoroughly enough so that there is no hesitation, insecurity, or error, without interfering to any harmful extent with the more general bond from the situation to work without the crutch. THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS Another instructive case concerns the bonds between certain words and their meanings, and between certain situations of commerce, industry, or agriculture and useful facts about these situations. Illustrations of the former are the bonds between _cube root_, _hectare_, _brokerage_, _commission_, _indorsement_, _vertex_, _adjacent_, _nonagon_, _sector_, _draft_, _bill of exchange_, and their meanings. Illustrations of the latter are the bonds from "Money being lent 'with interest' at no specified rate, what rate is charged?" to "The legal rate of the state," from "$_X_ per M as a rate for lumber" to "Means $_X_ per thousand board feet, a board foot being 1 ft. by 1 ft. by 1 in." It is argued by many that such bonds are valuable for a short time; namely, while arithmetical procedures in connection with which they serve are learned, but that their value is only to serve as a means for learning these procedures and that thereafter they may be forgotten. "They are formed only as accessory means to certain more purely arithmetical knowledge or discipline; after this is acquired they may be forgotten. Everybody does in fact forget them, relearning them later if life requires." So runs the argument.