One of the arbitrary systematizations of the order of formation of bonds restricts operations at first to the numbers 1 to 10, then to numbers under 100, then to numbers under 1000, then to numbers under 10,000. Apart from the avoidance of unreal and pedantic problems in applied arithmetic to which work with large numbers in low grades does somewhat predispose a teacher, there is little merit in this restriction of the order of formation of bonds. Its demerits are many. For example, when the pupil is learning to 'carry' in addition he can be given better practice by soon including tasks with sums above 100, and can get a valuable sense of the general use of the process by being given a few examples with three- and four-place numbers to be added. The same holds for subtraction. Indeed, there is something to be said in favor of using six- or seven-place numbers in subtraction, enforcing the 'borrowing' process by having it done again and again in the same example, and putting it under control by having the decision between 'borrowing' and 'not borrowing' made again and again in the same example. When the multiplication tables are learned the most important use for them is not in tedious reviews or trivial problems with answers under 100, but in regular 'short' multiplication of two- and three- and even four-place numbers. Just as the addition combinations function mainly in the higher-decade modifications of them, so the multiplication combinations function chiefly in the cases where the bond has to operate while the added tasks of keeping one's place, adding what has been carried, writing down the right figure in the right place, and holding the right number for later addition, are also taken care of. It seems best to introduce such short multiplication as soon as the × 5s, × 2s, × 3s, and × 4s are learned and to put the × 6s, × 7s, and the rest to work in such short multiplication as soon as each is learned. Still surer is the need for four-, five-, and six-place numbers when two-place numbers are used in multiplying. When the process with a two-place multiplier is learned, multiplications by three-place numbers should soon follow. They are not more difficult then than later. On the contrary, if the pupil gets used to multiplying only as one does with two-place multipliers, he will suffer more by the resulting interference than he does from getting six- or seven-place answers whose meaning he cannot exactly realize. They teach the rationale and the manipulations of long multiplication with especial economy because the principles and the procedures are used two or three times over and the contrasts between the values which the partial products have in adding become three instead of one.