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 many 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 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.
[Illustration]
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 B, on which side the stair
ascends. This results in the same numeration as when we counted the
longest rod--1, 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.
THE NUMBERS AS REPRESENTED BY THE GRAPHIC SIGNS
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.
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. They themselves often cry out, "Zero is nothing! Zero is
nothing!"
EXERCISES FOR THE MEMORY OF NUMBERS
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
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 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.
ADDITION AND SUBTRACTION FROM ONE TO TWENTY: MULTIPLICATION AND DIVISION
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 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 5 x 2 = 10
7 + 3 = 10
6 + 4 = 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 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 = 5
10 - 2 = 8
10 - 1 = 9
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;
2 x 2 = 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,
2 x 2 = 4 3 x 2 = 6 4 x 2 = 8 5 x 2 = 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:
2 4 6 8 10
X X|X XX X|X XX X|X XX X|X XX X|X
| X X|X XX X|X XX X|X XX X|X
| | X X|X XX X|X XX X|X
| | | X X|X XX X|X
| | | | X X|X
From this arrangement, one sees at once which are the numbers which can
be divided by two--all 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 2 x 3 = 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.
LESSONS ON DECIMALS: ARITHMETICAL CALCULATIONS BEYOND TEN
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 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.
A B
+----+ +----+
| 10 | | 10 |
| 10 | | 20 |
| 10 | | 30 |
| 10 | | 40 |
| 10 | | 50 |
| 10 | | 60 |
| 10 | | 70 |
| 10 | | 80 |
| 10 | | 90 |
+----+ +----+
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 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.
+----+
| 10 |
| 11 |
| 12 |
| 13 |
| 14 |
| 15 |
| 16 |
| 17 |
| 18 |
| 19 |
| 20 |
+----+
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.