C. Learning to add a seen to a thought-of number. D. Learning to neglect an empty space in the columns. E. Learning to neglect 0s in the columns. F. Learning the application of the combinations to higher decades may for the less gifted pupils involve as much time and labor as learning all the original addition tables. And even for the most gifted child the formation of the connection '8 and 7 = 15' probably never quite insures the presence of the connections '38 and 7 = 45' and '18 + 7 = 25.' G. Learning to write the figure signifying units rather than the total sum of a column. In particular, learning to write 0 in the cases where the sum of the column is 10, 20, etc. Learning to 'carry' also involves in itself at least two distinct processes, by whatever way it is taught. We find evidence of such specialization of functions in the results with such tests as Woody's. For example, 2 + 5 + 1 = .... surely involves abilities in part different from 2 4 3 - because only 77 percent of children in grade 3 do the former correctly, whereas 95 percent of children in that grade do the latter correctly. In grade 2 the difference is even more marked. In the case of subtraction 4 4 - involves abilities different from those involved in 9 3, - being much less often solved correctly in grades 2 and 4. 6 0 - is much harder than either of the above. 43 1 21 2 33 13 is much harder than 35. -- -- It may be said that these differences in difficulty are due to different amounts of practice. This is probably not true, but if it were, it would not change the argument; if the two abilities were identical, the practice of one would improve the other equally. I shall not undertake here this task of listing and describing the elementary functions which constitute arithmetical learning, partly because what they are is not fully known, partly because in many cases a final ability may be constituted in several different ways whose descriptions become necessarily tedious, and partly because an adequate statement of what is known would far outrun the space limits of this chapter. Instead, I shall illustrate the results by some samples. KNOWLEDGE OF THE MEANING OF A FRACTION