THE MONTESSORI METHOD: CHAPTER
XIX

TEACHING OF NUMERATION;
INTRODUCTION TO ARITHMETIC

CHILDREN of three years already know
how to count as far as two or three when they enter our schools.
They therefore *very easily* learn numeration, which consists
*in counting objects.* A dozen different ways may serve toward
this end, and daily life presents may opportunities; when the
mother says, for instance, "There are two buttons missing
from your apron," or "We need three more plates at table."

One of the first means used by me,
is that of counting with money. I obtain *new *money, and
if it were possible I should have good reproductions made in cardboard.
I have seen such money used in a school for deficients in London.

The *making of change* is a
form of numeration so attractive as to hold the attention of the
child. I present the one, two, and four centime pieces and the
children, in this way learn to count to *ten.*

No form of instruction is more *practical*
than that tending to make children familiar with the coins in
common use, and no exercise is more useful than that of making
change. It is so closely related to daily life that it interests
all children intensely.

Having taught numeration in this
empiric mode, I pass to more methodical exercises, having as didactic
material *[Page
327]* one of the sets
of blocks already used in the education of the senses; namely,
the series of ten rods heretofore used for the teaching of length.
The shortest of these rods corresponds to a decimetre, the longest
to a metre, while the intervening rods are divided into sections
a decimetre in length. The sections are painted alternately red
and blue.

Some day, when a child has arranged
the rods, placing them in order of length, we have him count the
red and blue signs, beginning with the smallest piece; that is,
one; one, two; one, two, three, etc., always going back to one
in the counting of each rod, and starting from the side A. We
then have him name the single rods from the shortest to the longest,
according to the total number of the sections which each contains,
touching the rods at the sides *[Page 328]*
B, on which side the stair ascends. This results in the same numeration
as when we counted the longest rod1, 2, 3, 4, 5, 6, 7, 8,
9, 10. Wishing to know the number of rods, we count them from
the side A and the same numeration results; 1, 2, 3, 4, 5, 6,
7, 8, 9, 10. This correspondence of the three sides of the triangle
causes the child to verify his knowledge and as the exercise interests
him he repeats it many times.

We now unite to the exercises in
*numeration* the earlier, sensory exercises in which the
child recognised the long and short rods. Having mixed the rods
upon a carpet, the directress selects one, and showing it to the
child, has him count the sections; for example, 5. She then asks
him to give her the one next in length. He selects it *by his
eye*, and the directress has him *verify* his choice by
*placing the two pieces side by side and by counting their sections.
*Such exercises may be repeated in great variety and through
them the child learns to assign a *particular name to each one
of the pieces in the long stair.* We may now call them piece
number one; piece number two, etc., and finally, for brevity,
may speak of them in the lessons as one, two, three, etc.

At this point, if the child already
knows how to write, we may present the figures cut in sandpaper
and mounted upon cards. In presenting these, the method is the
same used in teaching the letters. "This is one." "This
is two." "Give me one." "Give me two."
"What *number* is this?" The child traces the number
with his finger as he did the letters.

*Exercises with Numbers.* Association of the graphic sign with the
quantity. *[Page
329]*

I have designed two trays each divided into five little compartments. At the back of each compartment may be placed a card bearing a figure. The figures in the first tray should be 0, 1, 2, 3, 4, and in the second, 5, 6, 7, 8, 9.

The exercise is obvious; it consists in placing within the compartments a number of objects corresponding to the figure indicated upon the card at the back of the compartment. We give the children various objects in order to vary the lesson, but chiefly make use of large wooden pegs so shaped that they will not roll off the desk. We place a number of these before the child whose part is to arrange them in their places, one peg corresponding to the card marked one, etc. When he has finished he takes his tray to the directress that she may verify his work.

*The Lesson on Zero.* We wait until the child, pointing to the
compartment containing the card marked zero, asks, "And what
must I put in here?" We then reply, "Nothing; zero is
nothing." But often this is not enough. It is necessary to
make the child *feel *what we mean by *nothing. *To
this end we make use of little games which vastly entertain the
children. I stand among them, and turning to one of them who has
already used this material, I say, "Come, dear, come to me
*zero* times." The child almost always comes to me,
and then runs back to his place. "But, my boy, you came *one*
time, and I told you to come *zero *times." Then he
begins to wonder. "But what must I do, then?" "Nothing;
zero is nothing." "But how shall I do nothing?"
"Don't do anything. You must sit still. You must not come
at all, not any times. Zero times. No times at all." I repeat
these exercises until the children understand, and they are then
immensely amused at remaining quiet when I call to them to come
to me zero times, or to throw me zero kisses. *[Page 330]*
They themselves often cry out, "Zero is nothing! Zero is
nothing!"

When the children recognise the written figure, and when this figure signifies to them the numerical value, I give them the following exercise:

I cut the figures from old calendars
and mount them upon slips of paper which are then folded and dropped
into a box. The children draw out the slips, carry them still
folded, to their seats, where they look at them and refold them,
*conserving the secret. *Then, one by one, or in groups,
these children (who are naturally the oldest ones in the class)
go to the large table of the directress where groups of various
small objects have been placed. Each one selects the *quantity
*of objects corresponding to the number he has drawn. The number,
meanwhile, has been left *at the child's place*, a slip of
paper mysteriously folded. The child, therefore, must *remember
*his number not only during the movements which he makes in
coming and going, but while he collects his pieces, counting them
one by one. The directress may here make interesting individual
observations upon the number memory.

When the child has gathered up his objects he arranges them upon his own table, in columns of two, and if the number is uneven, he places the odd piece at the bottom and between the last two objects. The arrangement of the pieces is therefore as follows:

o o o o o o o o o o X XX XX XX XX XX XX XX XX XX X XX XX XX XX XX XX XX X XX XX XX XX XX X XX XX XX X XX

*[Page 331]* The crosses represent the objects,
while the circle stands for the folded slip containing the figure.
Having arranged his objects, the child awaits the verification.
The directress comes, opens the slip, reads the number, and counts
the pieces.

When we first played this game it
often happened that the children took *more objects *than
were called for upon the card, and this was not always because
they did not remember the number, but arose from a mania for the
having the greatest number of objects. A little of that instinctive
greediness, which is common to primitive and uncultured man. The
directress seeks to explain to the children that it is useless
to have all those things upon the desk, and that the point of
the game lies in taking the exact number of objects called for.

Little by little they enter into
this idea, but not so easily as one might suppose. It is a real
effort of self-denial which holds the child within the set limit,
and makes him take, for example, only two of the objects placed
at his disposal, while he sees others taking more. I therefore
consider this game more an exercise of will power than of numeration.
The child who has the *zero*, should not move from his place
when he sees all his companions rising and taking freely of the
objects which are inaccessible to him. Many times zero falls to
the lot of a child who knows how to count perfectly, and who would
experience great pleasure in accumulating and arranging a fine
group of objects in the proper order upon his table, and in awaiting
with security the teacher's verification.

It is most interesting to study the
expressions upon the faces of those who possess zero. The individual
differences which result are almost a revelation of the "character"
of each one. Some remain impassive, assuming a *[Page 332]*
bold front in order to hide the pain of the disappointment; others
show this disappointment by involuntary gestures. Still others
cannot hide the smile which is called forth by the singular situation
in which they find themselves, and which will make their friends
curious. There are little ones who follow every movement of their
companions with a look of desire, almost of envy, while others
show instant acceptance of the situation. No less interesting
are the expressions with which they confess to the holding of
the zero, when asked during the verification, "and you, you
haven't taken anything?" "I have zero." "It
is zero." These are the usual words, but the expressive face,
the tone of the voice, show widely varying sentiments. Rare, indeed,
are those who seem to give with pleasure the explanation of an
extraordinary fact. The greater number either look unhappy or
merely resigned.

We therefore give lessons upon the meaning of the game, saying, "It is hard to keep the zero secret. Fold the paper tightly and don't let it slip away. It is the most difficult of all." Indeed, after awhile, the very difficulty of remaining quiet appeals to the children and when they open the slip marked zero it can be seen that they are content to keep the secret.

The didactic material which we use for the teaching of the first arithmetical operations is the same already used for numeration; that is, the rods graduated as to length which, arranged on the scale of the metre, contain the first idea of the decimal system.

The rods, as I have said, have come
to be called by the numbers which they represent; one, two, three,
etc. They *[Page
333]* are arranged in
order of length, which is also in order of numeration.

The first exercise consists in trying to put the shorter pieces together in such a way as to form tens. The most simple way of doing this is to take successively the shortest rods, from one up, and place them at the end of the corresponding long rods from nine down. This may be accompanied by the commands, "Take one and add it to nine; take two and add it to eight; take three and add it to seven; take four and add it to six." In this way we make four rods equal to ten. There remains the five, but, turning this upon its head (in the long sense), it passes from one end of the ten to the other, and thus makes clear the fact that two times five makes ten.

These exercises are repeated and
little by little the child is taught the more technical language;
nine plus one equals ten, eight plus two equals ten, seven plus
three equals ten, six plus four equals ten, and for the five,
which remains, two times five equals ten. At last, if he can write,
we teach the signs *plus *and *equals *and *times.
*Then this is what we see in the neat note-books of our little
ones:

9+1=10

8+2=10

7+3=10

6+4=105x2=10

When all this is well learned and
has been put upon the paper with great pleasure by the children,
we call their attention to the work which is done when the pieces
grouped together to form tens are taken apart, and put back in
their original positions. From the ten last formed we take away
four and six remains; from the next we take away three and seven
remains; from the next, two and eight remains; from the last,
we take away one and nine *[Page
334]* remains. Speaking
of this properly we say, ten less four equals six; ten less three
equals seven; ten less two equals eight; ten less one equals nine.

In regard to the remaining five, it is the half of ten, and by cutting the long rod in two, that is dividing ten by two, we would have five; ten divided by two equals five. The written record of all this reads:

10-4=6

10-3=7

10-2=8

10-1=910 / 2=5

Once the children have mastered this exercise they multiply it spontaneously. Can we make three in two ways? We place the one after the two and then write, in order that we may remember what we have done, 2+1=3. Can we make two rods equal to number four? 3+1=4, and 4-3=1; 4-1=3. Rod number two in its relation to rod number four is treated as was five in relation to ten; that is, we turn it over and show that it is contained in four exactly two times: 4 / 2=2; 2x2=4. Another problem: let us see with how many rods we can play this same game. We can do it with three and six; and with four and eight; that is,

2x2=4 3x2=6 4x2=8 5x2=10 10 / 2=5 8 / 2=4 6 / 2=3 4 / 2=2

At this point we find that the cubes with which we played the number memory games are of help:

From this arrangement, one sees at
once which are the numbers which can be divided by twoall
those which have not an odd cube at the bottom. These are the
*even* numbers, because they can be arranged in pairs, two
by two; and the division by two is easy, all that is necessary
being to separate the two lines of twos that stand one under the
other. Counting the cubes of each file we have the quotient. To
recompose the primitive number we need only reassemble the two
files thus 2x3=6. All this is not difficult for children of five
years.

The repetition soon becomes monotonous, but the exercises may be most easily changed, taking again the set of long rods, and instead of placing rod number one after nine, place it after ten. In the same way, place two after nine, and three after eight. In this way we make rods of a greater length than ten; lengths which we must learn to name eleven, twelve, thirteen, etc., as far as twenty. The little cubes, too, may be used to fix these higher numbers

Having learned the operations through
ten, we proceed with no difficulty to twenty. The one difficulty
lies in the *decimal numbers *which require certain lessons.

The necessary didactic material consists
of a number of square cards upon which the figure ten is printed
in large type, and of other rectangular cards, half the size of
the square, and containing the single numbers from one to nine.
We place the numbers in a line; 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10.
Then, having no more numbers, we must begin over again and take
the 1 again. This 1 is like that section in the set of rods which,
in rod number 10, extends *[Page
335]* beyond nine. Counting
along *the stair *as far as nine, there remains this one
section which, as there are no more numbers, we again designate
as 1; but this is a higher 1 than the first, and to distinguish
it from the first we put near it a zero, a sign which means nothing.
Here then is 10. Covering the zero with the separate rectangular
number cards in the order of their succession we see formed: 11,
12, 13, 14, 15, 16, 17, 18, 19. These numbers are composed by
adding to rod number 10, first rod number 1, then 2, then 3, etc.,
until we finally add rod number 9 to rod number 10, thus obtaining
a very long rod, which, when its alternating red and blue sections
are counted, gives us nineteen.

The directress may then show to the child the cards, giving the number 16, and he may place rod 6 after rod 10. She then takes away the card bearing 6, and places over the zero the card bearing the figure 8, whereupon the child takes away rod 6 and replaces it with rod 8, thus making 18. Each of these acts may be recorded thus: 10+6=16; 10+8=18, etc. We proceed in the same way to subtraction.

When the number itself begins to have a clear meaning to the child, the combinations are made upon one long card, arranging the rectangular cards bearing the nine figures upon the two columns of numbers shown in the figures A and B.

Upon the card A we superimpose upon
the zero of the second 10, the rectangular card bearing the 1:
and under this the one bearing two, etc. Thus while the one of
the *[Page
337]* ten remains the
same the numbers to the right proceed from zero to nine, thus:

In card B the applications are more complex. The cards are superimposed in numerical progression by tens.

Almost all our children count to 100, a number which was given to them in response to the curiosity they showed in regard to learning it.

I do not believe that this phase of the teaching needs further illustrations. Each teacher may multiply the practical exercises in the arithmetical operations, using simple objects which the children can readily handle and divide.