and of the amount of practice it requires, may mislead the reader into thinking that these bonds and abilities are to be formed each by itself alone and kept so. They should rarely be formed so and never kept so. This we have indicated from time to time by references to the importance of forming a bond in the way in which it is to be used, to the action of bonds in changed situations, to facilitation of one bond by others, to the coöperation of abilities, and to their integration into a total arithmetical ability.

As a matter of fact, only a small part of drill work in arithmetic should be the formation of isolated bonds. (Even the very young pupil learning 5 and 3 are 8 should learn it with '5 and 510,' '5 and 2=7,' at the back of his mind, so to speak. Even so early, 5+3=8 should be part of an organized, coöperating system of bonds. Later 50+30=80) should become allied to it. (Each bond should be considered, not simply as a separate tool to be put in a compartment until needed, but also as an improvement of one total tool or machine, arithmetical ability.

There are differences of course. Knowledge of square root can be regarded somewhat as a separate tool to be sharpened, polished, and used by itself, whereas knowledge of the multiplication tables cannot. Yet even square root is probably best made more closely a part of the total ability, being taught as a special case of dividing where divisor is to be the same as quotient, the process being one of estimating and correcting.

(In general we do not wish the pupil to be a repository of separated abilities, each of which may operate only if you ask him the sort of questions which the teacher used to ask him, or otherwise indicate to him which particular arithmetical tool he is to use. Rather he is to be an effective organization of abilities,) coöperating in useful ways to meet

the quantitative problems life offers. He should not as a rule have to think in such fashion as: "Is this interest or discount? Is it simple interest or compound interest? What did I do in compound interest? How do I multiply by 2 percent?" The situation that calls up interest should also call up the kind of interest that is appropriate, and the technique of operating with percents should be so welded together with interest in his mind that the right coöperation will occur almost without supervision by him.

As each new ability is acquired, then, we seek to have it take its place as an improvement of a thinking being, as a coöperative member of a total organization, as a soldier fighting together with others, as an element in an educated personality. Such an organization of bonds will not form itself any more than any one bond will create itself. If the elements of arithmetical ability are to act together as a total organized unified force they must be made to act together in the course of learning. What we wish to have work together we must put together and give practice in teamwork.

We can do much to secure such coöperative action when and where and as it is needed by a very simple expedient; namely, to give practice with computation and problems such as life provides, instead of making up drills and problems merely to apply each fact or principle by itself. Though a pupil has solved scores of problems reading, "A triangle has a base of a feet and an altitude of b feet, what is its area?" he may still be practically helpless in finding the area of a triangular plot of ground; still more helpless in using the formula for a triangle which is one of two into which a trapezoid is divided. Though a pupil has learned to solve problems in trade discount, simple interest, compound interest, and bank discount one at a time, stated in a few

set forms, he may be practically helpless before the actual series of problems confronting him in starting in business, and may take money out of the savings bank when he ought to borrow on a time loan, or delay payment on his bills when by paying cash he could save money as well as improve his standing with the wholesaler.

Instead of making up problems to fit the abilities given by school instruction, we should preferably modify school instruction so that arithmetical abilities will be organized into an effective total ability to meet the problems that life will offer. Still more generally every bond formed should be formed with due consideration of every other bond that has been or will be formed; every ability should be practiced in the most effective possible relations with other abilities.)

CHAPTER VII

THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS

THE bonds to be formed having been chosen, the next step is to arrange for their most economical order of formation to arrange to have each help the others as much as possible to arrange for the maximum of facilitation and the minimum of inhibition.

The principle is obvious enough and would probably be admitted in theory by any intelligent teacher, but in practice we are still wedded to conventional usages which arose long before the psychology of arithmetic was studied. For example, we inherit the convention of studying addition of integers thoroughly, and then subtraction, and then multiplication, and then division, and many of us follow it though nobody has ever given a proof that this is the best order for arithmetical learning. We inherit also the opposite convention of studying in a so-called "spiral" plan, a little addition, subtraction, multiplication, and division, and then some more of each, and then some more, and many of us follow this custom, with an unreasoned faith that changing about from one process to another is per se helpful.

Such conventions are very strong, illustrating our common tendency to cherish most those customs which we cannot justify! The reductions of denominate numbers ascending and descending were, until recently, in most courses of study,

kept until grade 4 or grade 5 was reached, although this material is of far greater value for drills on the multiplication and division tables than the customary problems about apples, eggs, oranges, tablets, and penholders. By some historical accident or for good reasons the general treatment of denominate numbers was put late; by our naïve notions of order and system we felt that any use of denominate numbers before this time was heretical; we thus became blind to the advantages of quarts and pints for the tables of 2s; yards and feet for the tables of 3s; gallons and quarts for the tables of 4s; nickels and cents for the 5s; weeks and days for the 7s; pecks and quarts for the 8s; and square yards and square feet for the 9s. Problems like 5 yards = feet or

15 feet= yards have not only the advantages of brevity, clearness, practical use, real reference, and ready variation, but also the very great advantage that part of the data have to be thought of in a useful way instead of read off from the page. In life, when a person has twenty cents with which to buy tablets of a certain sort, he thinks of the price in making his purchase, asking it of the clerk only in case he does not know it, and in planning his purchases beforehand he thinks of prices as a rule. In spite of these and other advantages, not one textbook in ten up to 1900 made early use of these exercises with denominate numbers. So strong is mere use and wont.

Besides these conventional customs, there has been, in those responsible for arithmetical instruction, an admiration for an arrangement of topics that is easy for a person, after he knows the subject, to use in thinking of its constituent parts and their relations. Such arrangements are often called 'logical' arrangements of subject matter, though they are often far from logical in any useful sense. Now the easiest order in which to think of a hierarchy of habits