A big stumbling-block in the way of the child beginning Algebra is the transition from the conception of definite numbers in Arithmetic to that of indefinite quantities in Algebra, and the performance on these of the fundamental operations with facility and certainty. It is a truism that our algebraic teaching must grow out of our arithmetical: in its initial stages it may with advantage be based on some such method as the following. From a box of counters, or little wooden cubes, or in fact any handy articles, ask several pupils—better still, all the pupils, if the class be of convenient size—to pick out respectively five counters, eight counters, and so on. Their possessions can be exhibited on the black-board thus:—“A has five counters,” “B has eight counters,” etc. The counters being replaced, now ask others to take out some counters each. So long as the boys hold these in their closed hands no one can tell what quantity of counters any boy has; all that can be said is that each boy has some counters. Hence on the black-board is written:—“L has some counters,” “M has some counters,” and so on for each boy. Now the pupils have been accustomed in their arithmetical work to use various shorthand symbols in order to avoid writing out in full frequently occurring words or phrases. Thus, instead of the word “one” we use 1; the expression “multiplied by” is represented by ×, and so on. We therefore rub out the words “five” and “eight” that we wrote at first, and write in place of them their symbols “5” and “8.”