of visual impressions as vivid as possible. The breaking up of a whole into parts really precedes in facility the additioning of parts into a whole, for the reason that the power of destruction in a child obviously precedes the power of construction. Froebel's fifth gift of cubical blocks has its first application on this fact, since the entire mass forming a cube may be broken up into twenty-seven smaller cubes. When we reached the number twenty-seven, I told the child it was the smallest cube that existed. But she having a year previously, when only four years old, learned to handle these same cubes, corrected my error, and demonstrated triumphantly that eight blocks would make a still smaller cube. The incident shows the tenacity of ideas once implanted in the right way and at the right time.
It is much more difficult to teach a child to subtract than to add, a fact upon which Warren Colburn sagaciously comments. In the discussion of practical problems, a hitch often occurs in the child's mind which may be quite unsuspected by the teacher. Thus, if Henry and Arthur go to buy a ball which costs sixteen cents, and one boy had six cents and the other seven, I found the child unable to solve the problem as to how many more cents were needed, because, as she said, she could not take thirteen from sixteen, since the very trouble was that the boys did not have sixteen cents. It was necessary to use sticks, and with the distinct formal agreement that those of one color should be known to represent an imaginary number, those of another color the number of actual things manipulated. But what a stride for a young child's mind to make, into a sphere neither real nor imaginary, but where the existent and the non-existent are indissolubly associated in an ordinary practical affair of every-day life!
From the beginning the decimal system imposed itself spontaneously upon the child's mind, on account of the facility of visibly recognizing groups of five and ten sticks, and of verbally recognizing their successive additions. In this way the multiplication-table the famous despair of little Marjorie Fleming—was mastered with great ease by this far less gifted child. Every one remembers the fierce vehemence of Pet Marjorie's protest, "But 7 times 9 is devilish, and what Nature itself can't endure!" It is so, if presented as an isolated fact. The child I taught, however, discovered of herself that the successive addition of tens was as easy as that of ones. After that, when she came to add (or multiply by) nines, she would say, first add ten, then say, and nine was one less. If it were eight, it was two less, etc. After a fortnight of these exercises, she was asked one day out of study hours what was the sum of 14 and 19, and answered immediately 33. Upon being asked to explain the process, she said, "10 and 19 makes 29, then I must add 4 more, and 1 and 29 are 30, and 3 more are 33." When three decimals were reached, a somewhat laborious exercise was performed. Thus, to operate with 138, the number 100 was constructed out of ten packages of purple sticks, each package
