334 THE DISCOVERY OF THE CHILD It is very interesting to study the expressions on the faces of the possessors of the zero; the individual differences which appear form almost a revelation of the character of each individual. Some remain impassive, proudly trying to hide the inward pain of dis- illusion; others show by their momentary gestures that they are disappointed; others cannot hide the smile raised by the curious situation which will awaken the curiosity of their companions; some follow all the movements of their companions up till the end of the exercise with plainly shown feelings of longing, almost of envy; others show instant resignation. Their attitude when they confess to zero is also interesting. When one asks, during the inspection: " And you, have you taken nothing? " " I have the zero," " It is zero," " I had zero," These •are the common replies in spoken language, but the expressive ,gesturess the tone of the voice, express very diverse feelings. Rare are those who boldly seem to attribute the explanation to some * extraordinary fact; most are irritated or resigned. Some lessons on behaviour have to be given. " Take care, it is difficult to keep the secret of the zero; zero eludes you; be very watchful not to let it be known that you have nothing." After a while, pride and dignity prevail, and the little ones become accus- tomed to receiving the zero and the small numbers without disturbing themselves, satisfied if they no longer show the least trace of the feelings which at first overpowered them. ADDITION AND SUBSTRUCTION FROM ONE TO TWENTY MULTIPLICATION AND DIVISION The material which I use for teaching operations in arithmetic Is the same as that already used for numeration, that is, rods .graduated in length which already embody the first ideas of the •decimal system. The rods, as has been said, are called by the name of the number which they represent—one, two, three, etc. They are .arranged according to length, that is to say in numerical order.