EXPERIENCES OF MATHEMATICAL TRUTHS
By means of toys, handwork and games, as well as various private
individual experiments, a child touches on most sides of mathematics in
the nursery class. In experimenting with bricks he must of necessity
have considered relative size, balance and adjustment, form and
symmetry; in fitting them back into their boxes some of the most
difficult problems of cubic content; in weighing out "pretence" sugar
and butter by means of sand and clay new problems are there for
consideration; in making a paper-house questions of measurement evolve.
This is all in the incidental play of the Nursery School, and yet we
might say that a child thus occupied is learning mathematics more than
anything else. Here, if he remained till six, he did a certain amount of
necessary counting, and he may have acquired skill in recognising
groups, he may have unconsciously and incidentally performed
achievements in the four rules, but never, of course, in any shortened
or technical form. Probably he knows some figures. It is best to give
these to a child when he asks for or needs them, as in the case of
records of games. On the other hand he may be content with strokes.
Various mathematical relationships are made clear in his games or trials
of strength, such as distance in relation to time or strength, weight in
relation to power and to balance, length and breadth in relation to
materials, value of material in relation to money or work. By means of
many of his toys the properties of solids have become working knowledge
to him. Here, then, is our starting-point for the transition period.
AFTER THE NURSERY STAGE
Undoubtedly the aim of the transition class is partly to continue by
means of games and dramatic play the kind of knowledge gained in the
Nursery School; but it has also the task of beginning to organise such
knowledge, as the grouping into tens and hundreds. This organisation of
raw material and the presenting of shortened processes, as occur in the
first four rules, forms the work also of the junior school. To give to a
child shortened processes which he would be very unlikely to discover in
less than a lifetime, is simply giving him the experience of the race,
as primitive man did to his son. But the important point is to decide
when a child's discovery should end and the teacher's demonstration
begin.
This is the period when we are accustomed to speak of beginning
"abstract" work; it is well to be clear what it means, and how it stands
related to a child's need for experience. When we leave the problems of
life, such as shopping, keeping records of games and making measurements
for construction; and when we begin to work with pure number, we are
said to be dealing with the abstract. Formerly dealing with pure number
was called "simple," and dealing with actual things, such as money and
measures, "compound," and they were taken in this order. But experience
has reversed the process, and a child comes to see the need of abstract
practice when he finds he is not quick enough or accurate enough, or his
setting out seems clumsy, in actual problems. This was discussed at
greater length in the chapter on Play.
For instance, he might set down the points of a game by strokes, each
line representing a different opponent:
John ||||||||||||||||
Henry |||||||||||
Tom |||
He will see how difficult it is to estimate at a glance the exact score,
and how easy it is to be inaccurate. It seems the moment to show him
that the idea of grouping or enclosing a certain number, and always
keeping to the same grouping, is helpful:
John |||||||||| |||||| = 1 ten and 6 singles.
Henry |||||||||| | = 1 ten and 1 single.
Tom ||| = 3 singles.
After doing this a good many times he could be told that this is a
universal method, and he would doubtless enjoy the purely puzzle
pleasure in working long sums to perfect practice. This pleasure is very
common in children at this stage, but too often it comes to them merely
through being shown the "trick" of carrying tens. They have reached a
purely abstract point, but they cannot get through it without some more
material help. The following is an example of the kind of help that can
be given in getting clear the concept of the ten grouping and the
processes it involves:
[Illustration: Board with hooks, in ranks of nine, and rings]
The whole apparatus is a rectangular piece of wood about 3/4 of an inch
thick, and about 3x1-1/2 feet of surface. It is painted white, and the
horizontal bars are green, so that the divisions may be apparent at a
distance; it has perpendicular divisions breaking it up into three
columns, each of which contains rows of nine small dresser hooks. It can
be hung on an easel or supported by its own hinge on a table. Each of
the divisions represents a numerical grouping, the one on the right is
for singles or units, the central one for tens, and the left side one
for hundreds: the counters used are button moulds, dipped in red ink,
with small loops of string to hang on the hooks: it is easily seen by a
child that, after nine is reached, the units can no longer remain in
their division or "house," but must be gathered together into a bunch
(fastened by a safety pin) and fixed on one of the hooks of the middle
division.
Sums of two or three lines can thus be set out on the horizontal bars,
and in processes of addition the answer can be on the bottom line. It is
very easy, by this concrete means, to see the process in subtraction,
and indeed the whole difficulty of dealing with ten is made concrete.
The whole of a sum can be gone through on this board with the
button-moulds, and on boards and chalk with figures, side by side, thus
interpreting symbol by material; but the whole process is abstract.
The piece of apparatus is less abstract only in degree than the figures
on the blackboard, because neither represents real life or its problems:
in abstract working we are merely going off at a side issue for the sake
of practice, to make us more competent to deal with the economic affairs
of life. There is a place for sticks and counters, and there is a place
for money and measures, but they are not the same: the former represents
the abstract and the latter the concrete problem if used as in real
life: the bridge between the abstract and the concrete is largely the
work of the transition class and junior school, in respect of the
foundations of arithmetic known as the first four rules.
Games of skill, very thorough shopping or keeping a bankbook, or selling
tickets for tram or train, represent the kind of everyday problem that
should be the centre of the arithmetic work at this transition stage;
and out of the necessities of these problems the abstract and
semi-abstract work should come, but it should _never_ precede the real
work. A real purpose should underlie it all, a purpose that is apparent
and stimulating enough to produce willing practice. A child will do much
to be a good shopkeeper, a good tram conductor, a good banker; he will
always play the game for all it is worth.