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 after you have formed them all may be an extremely difficult order in which to form them. The criticism of other orders as 'scrappy,' or 'unsystematic,' valid enough if the course of study is thought of as an object of contemplation, may be foolish if the course of study is regarded as a working instrument for furthering arithmetical learning.
We must remember that all our systematizing and labeling is largely without meaning to the pupils. They cannot at any point appreciate the system as a progression from that point toward this and that, since they have no knowledge of the 'this or that.' They do not as a rule think of their work in grade 4 as an outcome of their work in grade 3 with extensions of a to a_1, and additions of b_2 and b_3 to b and b_1, and refinements of c and d by c_4 and d_5. They could give only the vaguest account of what they did in grade 3, much less of why it should have been done then. They are not much disturbed by a lack of so-called 'system' and 'logical' progression for the same reason that they are not much helped by their presence. What they need and can use is a _dynamically_ effective system or order, one that they can learn easily and retain long by, regardless of how it would look in a museum of arithmetical systems. Unless their actual arithmetical habits are usefully related it does no good to see the so-called logical relations; and if their habits are usefully related, it does not very much matter whether or not they do see these; finally, they can be brought to see them best by first acquiring the right habits in a dynamically effective order.
DECREASING INTERFERENCE AND INCREASING FACILITATION
Psychology offers no single, easy, royal road to discovering this dynamically best order. It can only survey the bonds, think what each demands as prerequisite and offers as future help, recommend certain orders for trial, and measure the efficiency of each order as a means of attaining the ends desired. The ingenious thought and careful experimentation of many able workers will be required for many years to come.