"(3) Decomposition in order. It may easily happen that the children will at once point out the decomposition of the number into component parts in order; even then the third exercise cannot be regarded as superfluous: Here we have formed four cubes of twos, of separate cubes, and of threes,--in what order had we best place the cubes on the board? With what shall the decomposition of the four cubes begin? With the decomposition into separate cubes. How are four cubes to be formed from separate cubes? We must take four times one cube. How are four cubes to be formed from twos, from a pair? We must take two twos,--twice two cubes, two pairs of cubes. How shall we afterward break up the four cubes? They can be formed of threes: for this purpose we take three and one, or one and three. The teacher explains to the pupils that the last decomposition, that is, 1 1 2, does not come under the accepted order, and is a modification of one of the first three." Why does Mr. Evtushévski not admit this last decomposition? Why must there be the order indicated by him? All that is a matter of mere arbitrariness and fancy. In reality, it is apparent to every thinking man that there is only one foundation for any composition and decomposition, and for the whole of mathematics. Here is the foundation: 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, and so forth,--precisely what the children learn at home, and what in common parlance is called counting to ten, to twenty, and so forth. This process is known to every pupil, and no matter what decomposition Mr. Evtushévski may make, it is to be explained from this one. A boy that can count to four, considers four as a whole, and so also three, and two, and one. Consequently, he knows that four was produced from the consecutive addition of one. Similarly he knows that four is produced by adding twice one to two, just as he knows twice one is two. What, then, are the children taught here? That which they know, or that process of counting which they must learn according to the teacher's fancy. The other day I happened to witness a lesson in mathematics according to Grube's method. The pupil was asked: "How much is 8 and 7?" He hastened to answer and said 16. His neighbour, too, was in a hurry and, without raising his left hand, said: "8 and 8 is 16, and one less is 15." The teacher sternly stopped him, and compelled the first boy to add one after one to 8, until he came to 15, though the boy knew long ago that he had made a blunder. In that school they had reached the number 15, but 16 was supposed to be unknown yet. I am afraid that many people, reading all these long refutals of the methods of object instruction and counting according to Grube, which I am making, will say: "What is there here to talk about? Is it not evident that it is all mere nonsense which it is not worth while to criticize? Why pick out the errors and blunders of a Bunákov and Evtushévski, and criticize what is beneath all criticism?"