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."