Two such means have been suggested in other connections. The first is the extension of training in checking and verifying work so that the pupil may work to a standard of approximately 100% success, and may know how nearly he is attaining it. The second is the use of standardized practice material and tests, whereby the pupil may measure himself against his own past, and have a clear, vivid, and trustworthy idea of just how much better or faster he can do the same tasks than he could do a month or a year ago, and of just how much harder things he can do now than then. Another means of stimulating the essential interest in quantitative thinking itself is the arrangement of the work so that real arithmetical thinking is encouraged more than mere imitation and assiduity. This means the avoidance of long series of applied problems all of one type to be solved in the same way, the avoidance of miscellaneous series and review series which are almost verbatim repetitions of past problems, and in general the avoidance of excessive repetition of any one problem-situation. Stimulation to real arithmetical thinking is weak when a whole day's problem work requires no choice of methods, or when a review simply repeats without any step of organization or progress, or when a pupil meets a situation (say the 'buy _x_ things at _y_ per thing, how much pay' situation) for the five-hundredth time. Another matter worthy of attention in this connection is the unwise tendency to omit or present in diluted form some of the topics that appeal most to real intellectual interests, just because they are hard. The best illustration, perhaps, is the problem of ratio or "How many times as large (long, heavy, expensive, etc.) as _x_ is _y_?" Mastery of the 'times as' relation is hard to acquire, but it is well worth acquiring, not only because of its strong intellectual appeal, but also because of its prime importance in the applications of arithmetic to science. In the older arithmetics it was confused by pedantries and verbal difficulties and penalized by unreal problems about fractions of men doing parts of a job in strange and devious times. Freed from these, it should be reinstated, beginning as early as grade 5 with such simple exercises as those shown below and progressing to the problems of food values, nutritive ratios, gears, speeds, and the like in grade 8. John is 4 years old. Fred is 6 years old. Mary is 8 years old. Nell is 10 years old. Alice is 12 years old. Bert is 15 years old. Who is twice as old as John? Who is half as old as Alice? Who is three times as old as John? Who is one and one half times as old as Nell? Who is two thirds as old as Fred? etc., etc., etc. Alice is .... times as old as John. John is .... as old as Mary. Fred is .... times as old as John. Alice is .... times as old as Fred. Fred is .... as old as Mary. etc., etc., etc.