A teacher who has adult powers of estimating length or area or weight and who also knows already which of the two is longer or larger or heavier, may use two lines to illustrate a difference which they really hide from the child. It is unlikely, for example, that the first of these lines ______________ ________________ would be recognized as shorter than the second by every child in a fourth-grade class, and it is extremely unlikely that it would be recognized as being 7/8 of the length of the latter, rather than 3/4 of it or 5/6 of it or 9/10 of it or 11/12 of it. If the two were shown to a second grade, with the question, "The first line is 7. How long is the other line?" there would be very many answers of 7 or 9; and these might be entirely correct arithmetically, the pupils' errors being all due to their inability to compare the lengths accurately. _A_ ______________ ________________ ______________ ________________ _B_ |______________| |________________| _C_ |-|-|-|-|-|-|-| |-|-|-|-|-|-|-|-| __ __ __ __ __ __ __ _D_ |__|__|__|__|__|__|__| __ __ __ __ __ __ __ __ |__|__|__|__|__|__|__|__| _E_ .'\##|##/`. .'\##|##/`. /###\#|#/ \ /###\#|#/###\ |-----------| |-----------| \###/#|#\###/ \###/#|#\###/ `./##|##\.' `./##|##\.' The quantities used should be such that their mere discrimination offers no difficulty even to a child of blunted sense powers. If 7/8 and 1 are to be compared, _A_ and _B_ are not allowable. _C_, _D_, and _E_ are much better. Teachers probably often underestimate or neglect the sensory difficulties of the tasks they assign and of the material they use to illustrate absolute and relative magnitudes. The result may be more pernicious when the pupils answer correctly than when they fail. For their correct answering may be due to their divination of what the teacher wants; and they may call a thing an inch larger to suit her which does not really seem larger to them at all. This, of course, is utterly destructive of their respect for arithmetic as an exact and matter-of-fact instrument. For example, if a teacher drew a series of lines 20, 21, 22, 23, 24, and 25 inches long on the blackboard in this form--____ ______ and asked, "This is 20 inches long, how long is this?" she might, after some errors and correction thereof, finally secure successful response to all the lines by all the children. But their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be actually damaged by the exercise. THE EARLY AWARENESS OF NUMBER There has been some disagreement concerning the origin of awareness of number in the individual, in particular concerning the relative importance of the perception of how-many-ness and that of how-much-ness, of the perception of a defined aggregate and the perception of a defined ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and Decroly and Degand ['12].)