66 THE THIRD YEARBOOK 



possesses no significance to the pupil in his image of the quantity 

 involved, then neglect it. Fractions should be taught along with 

 whole numbers when they assist in determining an actual image — 

 not otherwise, any more than a useless or meaningless word. 

 Common and decimal fractions should be taught from the begin- 

 ning. The difference is merely one of written language. 



4. There must be careful selection of the processes by which the 

 comparisons are made. — This will be determined by the pupil's 

 ability to picture and use the different standards of comparison. 

 The earliest processes will be addition and subtraction ; then frac- 

 tions ; later ratio and percentage. 



5. The results obtained must be represented objectively until they 

 will be habitually imaged. — Thus, if it is found that a cubic foot 

 of soil contains three gallons of water, (a) the actual size of the 

 cubic foot must be shown (by a model or otherwise), and (b) the 

 water must be actually measured and put into some convenient 

 receptacle. In the same way, the units yard, foot, inch, acre, mile, 

 square mile, etc., etc., must be actually seen until they become a 

 part of the mental equipment. They should be ready for immediate 

 use, just as the color green or red is always ready when needed in 

 our thinking. Represent proportions by diagrams and models. 



6. Using the results obtained as data, a great culture- picture 

 must be constructed. — By calculation, the real magnitude of the 

 world-operations should be brought out with all possible distinct- 

 ness. For example, a certain series of rainstorms that crossed the 

 United States gave an average depth of five and one-half inches of 

 rainfall. This amounted to nearly three and one-half gallons of 

 water per square foot of surface. (The result should be shown 

 objectively, as suggested in 5.) When the amount of water is cal- 

 culated for the entire area, the mind is staggered at the quantity. 

 It would fill a hole having a cross-sectional area of twenty acres, 

 about two thousand miles deep. Athough the quantity of water is 

 so enormous, it fell as rain not only harmlessly, but in such a way 

 as to greatly revive and refresh the earth. There is no other means 

 of opening up these world-pictures to the pupil except through the 

 intelligent use of numbers. 



It is useless to merely enumerate topics in nature-study which 

 require number-work. The number-work is not required in fact, 

 unless the topics themselves are fully worked out. A random use 



