618 THE POPULAR SCIENCE MONTHLY. 



containing ten sticks. These packages were placed in a row ; un- 

 derneath was a second row, containing, to represent the number 30, 

 three packages of yellow sticks, each containing ten ; finally, a third 

 row of eight units was made with green sticks in a single series. In 

 this exercise the sticks were all of the same size ; in another, later, a 

 hundred was represented by a single long stick, usually purple, a ten 

 by a yellow stick next in size, a unit by a stick still smaller and green. 

 Thus the original and clumsier representation was condensed by the 

 substitution of an expressive sign for the literal numbers, and as soon 

 as the sticks became used as signs, and not as the objects really to 

 be counted, the mutual relation of their respective sizes also ceased 

 to be literally exact, and became merely schematic. Thus was gradu- 

 ally managed a transition to the use of pure written signs or symbols. 

 The transition initiated and enlarged the condensation of Roman into 

 Arabic numerals. Knowledge of the process of subtraction, especially 

 in three and more decimals, was essentially facilitated by this device 

 with sticks, and the terrible difficulty of borrowing ten quite over- 

 come. Thus, if the number 288 were to be taken from 362, the larger 

 number would be represented by three long purple sticks, six shorter 

 yellow sticks, and two green sticks, the shortest of all. These colors 

 were always selected because harmonizing so well with each other. 

 Then, similarly, the 288 was represented by two purple, eight yellow, 

 and eight green sticks. It was easily recognized by the child, that 

 one of the yellow sticks could be removed from the ten sections of 

 the 362, and ten green sticks substituted, bringing the entire number 

 of units up to twelve, from which the eight of the lower figures could 

 be taken. It was also obvious that, when one yellow stick had been 

 taken away, only seven remained. There was no need, therefore, to 

 employ the usual confusing statement that a ten must be borrowed 

 from the upper figures, and later restored to a different place in the 

 lower. 



The study of abstract numbers, with Colburn's arithmetic, was 

 begun when the child was five and a half years. At the end of a year 

 she had thoroughly mastered the first four rules, including both 

 "short" and "long" division, and was considerably advanced in the 

 study of fractions, proper and improper. 



The last study entered upon during this year was that of natural 

 objects, and, for obvious reasons, plants were chosen for this purpose. 

 I suppose that most persons seriously interested in education are ac- 

 quainted with Miss Youmans's admirable little " First Lessons in Bot- 

 any," and the plea she makes for this science as a typical means of 

 training the observing powers of children. According to her plan, the 

 first object studied is the leaf and the pupil is taught at once, not 

 only to draw the leaf, but to fill out a schedule of description of it. 

 Much may be said in favor of this method, which proceeds from the 

 simple to the complex form, but it is by no means the only possible 



