"This is essentially the counting period, and any words that can be arranged into a series furnish all that is necessary. Counting is fundamental, and counting that is spontaneous, free from sensible observation, and from the strain of reason. A study of these original methods shows that multiplication was developed out of counting, and not from addition as nearly all textbooks treat it. Multiplication is counting. When children count by 4's, etc., they accent the same as counting gymnastics or music. When a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations. I would emphasize again that during the counting period there is a somewhat spontaneous development of the number series-idea which Preyer has discussed in his Arithmogenesis; that an immense momentum is given by a systematic series of names; and that these names are generally first learned and applied to objects later. A lady teacher told me that the Superintendent did not wish the teachers to allow the children to count on their fingers, but she failed to see why counting with horse-chestnuts was any better. Her children could hardly avoid using their fingers in counting other objects yet they followed the series to 100 without hesitation or reference to their fingers. This spontaneous counting period, or naming and following the series, should precede its application to objects." [D.E. Phillips, '97, p. 238.] THE RATIO IDEA OVEREMPHASIZED [Illustration: FIG. 1.] "Ratios.--1. Select solids having the relation, or ratio, of _a_, _b_, _c_, _d_, _o_, _e_. 2. Name the solids, _a_, _b_, _c_, _d_, _o_, _e_. The means of expressing must be as freely supplied as the means of discovery. The pupil is not expected to invent terms. 3. Tell all you can about the relation of these units. 4. Unite units and tell what the sum equals. 5. Make statements like this: _o_ less _e_ equals _b_. 6. _c_ can be separated into how many _d_'s? into how many _b_'s? 7. _c_ can be separated into how many _b_'s? What is the name of the largest unit that can be found in both _c_ and _d_ an exact number of times? 8. Each of the other units equals what part of _c_? 9. If _b_ is 1, what is each of the other units? 10. If _a_ is 1, what is each of the other units? 11. If _b_ is 1, how many 1's are there in each of the other units? 12. If _d_ is 1, how many 1's and parts of 1 in each of the other units? 13. 2 is the relation of what units? 14. 3 is the relation of what units? 15. 1/2 is the relation of what units? 16. 2/3 is the relation of what units? 17. Which units have the relation 3/2? 18. Which unit is 3 times as large as 1/2 of _b_? 19. _c_ equals 6 times 1/3 of what unit? 20. 1/3 of what unit equals 1/6 of _c_? 21. What equals 1/2 of _c_? _d_ equals how many sixths of _c_? 22. _o_ equals 5 times 1/3 of what unit?