There has been, especially in Germany, much argument concerning what sort of number-pictures (that is, arrangement of dots, lines, or the like, as shown in Fig. 60) is best for use in connection with the number names in the early years of the teaching of arithmetic. Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and others have measured the accuracy of children in estimating the number of dots in arrangements of one or more of these different types.[21] Many writers interpret a difference in favor of estimating, say, the square arrangements of Born or Lay as meaning that such is the best arrangement to use in teaching. The inference is, however, unjustified. That certain number-pictures are easier to estimate numerically does not necessarily mean that they are more instructive in learning. One set may be easier to estimate just because they are more familiar, having been oftener experienced. Even if the favored set was so after equal experience with all sets, accuracy of estimation would be a sign of superiority for use in instruction only if all other things were equal (or in favor of the arrangement in question). Obviously the way to decide which of these is best to use in teaching is by using them in teaching and measuring all relevant results, not by merely recording which of them are most accurately estimated in certain time exposures. [21] For an account in English of their main findings see Howell ['14], pp. 149-251. It may be noted that the Born, Lay, and Freeman pictures have claims for special consideration on grounds of probable instructiveness. Since they are also superior in the tests in respect to accuracy of estimate, choice should probably be made from these three by any teacher who wishes to connect one set of number-pictures systematically with the number names, as by drills with the blackboard or with cards. [Illustration: FIG. 60.--Various proposed arrangements of dots for use in teaching the meanings of the numbers 1 to 10.] Such drills are probably useful if undertaken with zeal, and if kept as supplementary to more realistic objective work with play money, children marching, material to be distributed, garden-plot lengths to be measured, and the like, and if so administered that the pupils soon get the generalized abstract meaning of the numbers freed from dependence on an inner picture of any sort. This freedom is so important that it may make the use of many types of number-pictures advisable rather than the use of the one which in and of itself is best.