as reminders.  The number groups are important; they form what Miss Punnett calls “a scheme” for those who have no great visualising power, and they combine the smallest groups into large ones.  It ought to be remembered that the repetition of a group is an easier thing to deal with than the combination of two groups, that is, six is a name for two threes and eight for two fours, but five and seven have not so definite a meaning.[33]

[Footnote 33:  This very morning a child cutting out brown paper pennies for a shop said, ’Look! there are two sixes; that would be a big number!’]

The Tillich bricks are good playthings, and so is cardboard money—­shillings, sixpences, threepences, pence and halfpence.

When the names have a meaning the children will want signs, i.e. figures.  Clock figures (Roman) can be used first as simplest, showing the closed fingers and the thumb for V; the only difficulty is IX.  The Arabic figures can be made by drawing round the number groups, or by laying out their shapes in little sticks. 5 and 8 show very plainly how to arrange five and eight sticks; for two and three they are placed horizontally, the curves merely joining the lines.

In teaching children to count, the decimal system should be kept well in mind, and the teacher should see that thirteen means three-ten, and that the children can touch the three and the ten as they speak the word.  Eleven and twelve ought to be called oneteen and twoteen, half in joke.  The idea of grouping should never be lost sight of, and larger numbers should at first be names for so many threes, fours, fives, etc.  In order to keep the meaning clear the children should say threety, fourty and fivety, but there should be no need to write these numbers.  The Kindergarten sticks tied in bundles of ten are quite convenient counting material when any counting is necessary.  Tram tickets and cigarette pictures can be used in the same way.

The decimal notation is a great thing to learn, how great any one will discover who will take the trouble to work a simple addition sum, involving hundreds, in Roman figures.  Children are always taught the number of the house they live in, which makes a starting-point.  If, for instance, 35 is compared with XXXV a meaning is given to the 3.

Many teachers make formal sums of numbers which could quite well be added without any writing at all.  By using any kind of material by which ten can be made plain as a higher unit—­bundles of sticks or tickets, Sonnenschein’s apparatus, Miss Punnett’s number scheme, or the new Montessori apparatus with its chains of beads:  the material used is of no great consequence—­children should be able to deal as easily with tens as with ones, and there is no need for little formal sums which have no meaning.

Everything in daily life should be used before formal work is attempted.  “Measure, reckon, weigh, compare,” said Rousseau.  Children love to measure, whether by lineal or liquid measure, or by learning to tell the time or to use a pair of scales.