should be formed before, not during, the training in short division. They are admirable at that point as practice on the division tables; are of practical service in the making-change problems of the small purchase and the like; and simplify the otherwise intricate task of keeping one's place, choosing the quotient figure, multiplying by it, subtracting and holding in mind the new number to be divided, which is composed half of the remainder and half of a figure in the written dividend. This change of order is a good illustration of the nearly general rule that "_When the practice or review required to perfect or hold certain bonds can, by an inexpensive modification, be turned into a useful preparation for new bonds, that modification should be made._" The bonds involved in the four operations with United States money should be formed in grades 3 and 4 along with or very soon after the corresponding bonds with three-place and four-place integers. This statement would have seemed preposterous to the pedagogues of fifty years ago. "United States money," they would have said, "is an application of decimals. How can it be learned until the essentials of decimal fractions are known? How will the child understand when multiplying $.75 by 3 that 3 times 5 cents is 1 dime and 5 cents, or that 3 times 70 cents is 2 dollars and 1 dime? Why perplex the young pupils with the difficulties of placing the decimal point? Why disturb the learning of the four operations with integers by adding at each step a second 'procedure with United States money'?" The case illustrates very well the error of the older oversystematic treatment of the order of topics and the still more important error of confusing the logic of proof with the psychology of learning. To prove that 3 × $.75 = $2.25 to the satisfaction of certain arithmeticians, you may need to know the theory of decimal fractions; but to do such multiplication all a child needs is to do just what he has been doing with integers and then "Put a $ before the answer to show that it means dollars and cents, and put a decimal point in the answer to show which figures mean dollars and which figures mean cents." And this is general. The ability to operate with integers plus the two habits of prefixing $ and separating dollars from cents in the result will enable him to operate with United States money. Consequently good practice came to use United States money not as a consequence of decimal fractions, learned by their aid, but as an introduction to decimal fractions which aids the pupil to learn them. So it has gradually pushed work with United States money further and further back, though somewhat timidly.