The Teaching of Number
“X. Y. Z.” writes: “How soon do
you think I ought to begin to teach my little boy of three, who seems to be
quite ordinarily intelligent to count? I heard of a boy not much older who was doing
little sums with his governess. Do you think this is a good thing, and if so,
how should I teach him?”
The question of the age at which
to we should begin to introduce arithmetical ideas to young children is really
bound up with the question of how we
should do it. Everything does depend upon how we go about it.
In the
first place, all those who have been studying little children to see how they
do learn to use arithmetical notions are agreed that formal lessons are quite
useless in the early years. Indeed they are worse than useless, because they
teach the child that arithmetic is dull and dreary; and once that emotional
attitude towards the subject has been set up it is extremely difficult to
change. It has been found that
more children get held up in their understanding of number from emotional
reasons than happens with any other subject of the school curriculum.
Fear and boredom are the two biggest factors in backwardness in arithmetic
among school children. It is easy to make minor mistakes of presentation in
showing the child how to add or subtract or divide, but these do not matter in
comparison with the fundamental mistake of associating arithmetic with a notion
of duty, or with situations of boredom and anxiety. It is very difficult for
even the best teaching later on to break down such an attitude once it has been
built up. The very first essential is, therefore, to make sure that from the
very beginning our methods are such as to let the child feel the natural
interest of number relations between things, and to discover that their practical value in his own active
interests and pursuits.
Fortunately
this is also much the easiest way to go about the problem, if only the grown-up
can rid our minds of the notion that children ought to be able to count and add and divide when these processes
have some real meaning to him for the normal activities of his age. For this
reason, all the number work of children in the home, indeed, even in the early
years at school, say seven or eight years at the very least, ought to be able
to be incidental to practical pursuits. Miss Margaret Drummond has made a very
interesting comparative study of young children who had regular formal lessons
in counting and adding and subtracting at school, and of children who have
never been given any formal instruction in number but whose parents have
responded to the child’s natural interest in number using incidental
opportunities in games and handicrafts. She has found that the latter are quite
as well advanced as the former at the age of six or seven, and quite as ready
for the more formal work that is useful at this school age. Such games as,
“One, two, Buckle my Shoe,” building towers of two or more blocks and counting
the number of blocks that can be balanced on one another, playing games of
shopping or helping mother in her real shopping, spending pocket money, playing
card games and dominoes, reckoning up the days of the month on the calendar,
the number of days before and since important events like a birthday or
Christmas treat, laying the table with the correct number of plates and spoons
and forks, and so on, are but examples of the endless opportunities which
ordinary life provides for stimulating the child’s interest in number in a way
that he can understand and use for himself. Miss Drummond is definitely of the
opinion that to introduce formal work to early, so far from developing the child’s natural interest
actually inhibits it by creating the wrong attitude.
Other
workers in education who have been studying the development of older children
have come to the conclusion that the same applies even in the later school
ages. It is only when the various arithmetical processes are linked up with the
child’s interests in making and doing as well as in his concern to understand
the general world around him, shopping and journeying, etc., that the child can
come to any deep understanding of arithmetical notions and facility in using
them. One of the most interesting recent developments in the teaching of arithmetic is the fruit
work of the Decroly school on the continent. In these schools the child
is introduced to number through measuring and weighing. They measure and weigh
things they have collected themselves, chestnuts, acorns, apples, fir cones,
and beans, using these to weigh against each other and only later on coming to
the conventional units of weight and volume. For example, when they have been
growing beans in water and have noticed that as the roots and stalks sprout,
the water grows less each day because the plant drinks it, they measure how
much it has drunk in a certain length of time with a medicine glass that
records drops of water, and find that twenty-five drops is also juts a
thimbleful. Form this they go on to the ordinary measures of volume. A great
deal can be learnt of weight and measures in connection with cooking,
gardening, shopping, and so on. In general, it is found that arithmetical
teaching that is connected with real situations is far more valuable and secure
as a foundation. The various formal materials such as the Montessori apparatus,
is valuable as a means of clarifying the child’s ideas, but it is not a
substitute for the practical working out of number in real situations.
In number
teaching in general an important principle is to make haste slowly, and it is
beginning to be thought that many of the abstract processes that are now taught
to children of eight to ten years would be better postponed to twelve or
thirteen, when the child was really ready to grasp them for their own sake.
There is
thus no reason why you should not play simple number games with your little
boy, connected with his everyday life. But I should not yet, for some time try
to introduce any formal material or to make any demand that he ought to learn
to count and add.
Those of my
readers who are interested in this whole question would find a most valuable
discussion of all the important aspects of the problem in the January number of
The New Era, which is devoted to the
teaching of mathematics both in its preliminary and its advanced stages.
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