23. 1/3 of what unit equals 1/5 of _o_? 24. 2/3 of _d_ equals what unit? _b_ equals how many thirds of _d_? 25. 2 is the ratio of _d_ to 1/3 of what unit? 3 is the ratio of _d_ to 1/2 of what unit? 26. _d_ equals 3/4 of what unit? 3/4 is the ratio of what units?" [Speer, '97, p. 9f.] THE RELATIONAL IDEA OVEREMPHASIZED An inspection of books of the eighties which followed the "Grube method" (for example, the _New Elementary Arithmetic_ by E.E. White ['83]) will show undue emphasis on the relational ideas. There will be over a hundred and fifty successive tasks all, or nearly all, on +7 and -7. There will be much written work of the sort shown below: _Add:_ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 2 -- -- -- which must have sorely tried the eyes of all concerned. Pupils are taught to "give the analysis and synthesis of each of the nine digits." Yet the author states that he does not carry the principle of the Grube method "to the extreme of useless repetition and mechanism." It should be obvious that all four meanings have claims upon the attention of the elementary school. Four is the thing between three and five in the number series; it is the name for a certain sized collection of discrete objects; it is also the name for a continuous magnitude equal to four units--for four quarts of milk in a gallon pail as truly as for four separate quart-pails of milk; it is also, if we know it well, the thing got by adding one to three or subtracting six from ten or taking two two's or half of eight. To know the meaning of a number means to know somewhat about it in all of these respects. The difficulty has been the narrow vision of the extremists. A child must not be left interminably counting; in fact the one-more-ness of the number series can almost be had as a by-product. A child must not be restricted to exercises with collections objectified as in Fig. 2 or stated in words as so many apples, oranges, hats, pens, etc., when work with measurement of continuous quantities with varying units--inches, feet, yards, glassfuls, pints, quarts, seconds, minutes, hours, and the like--is so easy and so significant. On the other hand, the elaboration of artificial problems with fictitious units of measure just to have relative magnitudes as in the exercises on page 5 is a wasteful sacrifice. Similarly, special drills emphasizing the fact that eighteen is eleven and seven, twelve and six, three less than twenty-one, and the like, are simply idolatrous; these facts about eighteen, so far as they are needed, are better learned in the course of actual column-addition and -subtraction.