ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL
THE UTILIZATION OF INSTINCTIVE INTERESTS
The activities essential to acquiring ability in arithmetic can rely on
little in man's instinctive equipment beyond the purely intellectual
tendencies of curiosity and the satisfyingness of thought for thought's
sake, and the general enjoyment of success rather than failure in an
enterprise to which one sets oneself. It is only by a certain amount of
artifice that we can enlist other vehement inborn interests of childhood
in the service of arithmetical knowledge and skill. When this can be
done at no cost the gain is great. For example, marching in files of
two, in files of three, in files of four, etc., raising the arms once,
two times, three times, showing a foot, a yard, an inch with the hands,
and the like are admirable because learning the meanings of numbers thus
acquires some of the zest of the passion for physical action. Even in
late grades chances to make pictures showing the relations of fractional
parts, to cut strips, to fold paper, and the like will be useful.
Various social instincts can be utilized in matches after the pattern of
the spelling match, contests between rows, certain number games, and the
like. The scoring of both the play and the work of the classroom is a
useful field for control by the teacher of arithmetic.
Hunt ['12] has noted the more important games which have some
considerable amount of arithmetical training as a by-product and which
are more or less suitable for class use. Flynn ['12] has described
games, most of them for home use, which give very definite arithmetical
drill, though in many cases the drills are rather behind the needs of
children old enough to understand and like the game itself.
It is possible to utilize the interests in mystery, tricks, and puzzles
so as to arouse a certain form of respect for arithmetic and also to get
computational work done. I quote one simple case from Miss Selkin's
admirable collection ['12, p. 69 f.]:--
I. ADDITION
"We must admit that there is nothing particularly interesting in
a long column of numbers to be added. Let the teacher, however,
suggest that he can write the answer at sight, and the task will
assume a totally different aspect.
"A very simple number trick of this kind can be performed by
making use of the principle of complementary addition. The
arithmetical complement of a number with respect to a larger
number is the difference between these two numbers. Most
interesting results can be obtained by using complements with
respect to 9.
"The children may be called upon to suggest several numbers of
two, three, or more digits. Below these write an equal number of
addends and immediately announce the answer. The children,
impressed by this apparently rapid addition, will set to work
enthusiastically to test the results of this lightning
calculation.
"Example:-- 357 } 999
682 } A x 3
793 } ----
2997
642 }
317 } B
206 }
"Explanation:--The addends in group A are written down at
random or suggested by the class. Those in group B are their
complements. To write the first number in group B we look at the
first number in group A and, starting at the left write 6, the
complement of 3 with respect to 9; 4, the complement of 5; 2, the
complement of 7. The second and third addends in group B are
derived in the same way. Since we have three addends in each
group, the problem reduces itself to multiplying 999 by 3, or to
taking 3000 - 3. Any number of addends may be used and each addend
may consist of any number of digits."
Respect for arithmetic as a source of tricks and magic is very much less
important than respect for its everyday services; and computation to
test such tricks is likely to be undertaken zealously only by the abler
pupils. Consequently this source of interest should probably be used
only sparingly, and perhaps the teacher should give such exhibitions
only as a reward for efficiency in the regular work. For example, if the
work for a week is well done in four days the fifth day might be given
up to some semi-arithmetical entertainment, such as the demonstration of
an adding machine, the story of primitive methods of counting, team
races in computation, an exhibition of lightning calculation and
intellectual sleight-of-hand by the teacher, or the voluntary study of
arithmetical puzzles.
The interest in achievement, in success, mentioned above is stronger in
children than is often realized and makes advisable the systematic use
of the practice experiment as a method of teaching much of arithmetic.
Children who thus compete with their own past records, keeping an exact
score from week to week, make notable progress and enjoy hard work in
making it.
THE ORDER OF DEVELOPMENT OF ORIGINAL TENDENCIES
Negatively the difficulty of the work that pupils should be expected to
do is conditioned by the gradual maturing of their capacities. Other
things being equal, the common custom of reserving hard things for late
in the elementary school course is, of course, sound. It seems probable
that little is gained by using any of the child's time for arithmetic
before grade 2, though there are many arithmetical facts that he can
learn in grade 1. Postponement of systematic work in arithmetic to grade
3 or even grade 4 is allowable if better things are offered. With proper
textbooks and oral and written exercises, however, a child in grades 2
and 3 can spend time profitably on arithmetical work. When all children
can be held in school through the eighth grade it does not much matter
whether arithmetic is begun early or late. If, however, many children
are to leave in grades 5 and 6 as now, we may think it wise to provide
somehow that certain minima of arithmetical ability be given them.
There are, so far as is known, no special times and seasons at which the
human animal by inner growth is specially ripe for one or another
section or aspect of arithmetic, except in so far as the general inner
growth of intellectual powers makes the more abstruse and complex tasks
suitable to later and later years.
Indeed, very few of even the most enthusiastic devotees of the
recapitulation theory or culture-epoch theory have attempted to apply
either to the learning of arithmetic, and Branford is the only
mathematician, so far as I know, who has advocated such application,
even tempered by elaborate shiftings and reversals of the racial order.
He says:--
"Thus, for each age of the individual life--infancy, childhood,
school, college--may be selected from the racial history
the most appropriate form in which mathematical experience
can be assimilated. Thus the capacity of the infant and early
childhood is comparable with the capacity of animal consciousness
and primitive man. The mathematics suitable to later childhood
and boyhood (and, of course, girlhood) is comparable with Archaean
mathematics passing on through Greek and Hindu to mediaeval
European mathematics; while the student is become sufficiently
mature to begin the assimilation of modern and highly abstract
European thought. The filling in of details must necessarily
be left to the individual teacher, and also, within some such
broadly marked limits, the precise order of the marshalling of the
material for each age. For, though, on the whole, mathematical
development has gone forward, yet there have been lapses from
advances already made. Witness the practical world-loss of much
valuable Hindu thought, and, for long centuries, the neglect of
Greek thought: witness the world-loss of the invention by the
Babylonians of the Zero, until re-invented by the Hindus, passed
on by them to the Arabs, and by these to Europe.
"Moreover, many blunders and false starts and false principles
have marked the whole course of development. In a phrase, rivers
have their backwaters. But it is precisely the teacher's function
to avoid such racial mistakes, to take short cuts ultimately
discovered, and to guide the young along the road ultimately found
most accessible with such halts and retracings--returns up
side-cuts--as the mental peculiarities of the pupils demand.
"All this, the practical realization of the spirit of the principle,
is to be wisely left to the mathematical teacher, familiar with the
history of mathematical science and with the particular limitations
of his pupils and himself." ['08, p. 245.]
The latitude of modification suggested by Branford reduces the guidance
to be derived from racial history to almost _nil_. Also it is apparent
that the racial history in the case of arithmetical achievement is
entirely a matter of acquisition and social transmission. Man's original
nature is destitute of all arithmetical ideas. The human germs do not
know even that one and one make two!
INVENTORIES OF ARITHMETICAL KNOWLEDGE AND SKILL
A scientific plan for teaching arithmetic would begin with an exact
inventory of the knowledge and skill which the pupils already possessed.
Our ordinary notions of what a child knows at entrance to grade 1, or
grade 2, or grade 3, and of what a first-grade child or second-grade
child can do, are not adequate. If they were, we should not find
reputable textbooks arranging to teach elaborately facts already
sufficiently well known to over three quarters of the pupils when they
enter school. Nor should we find other textbooks presupposing in their
first fifty pages a knowledge of words which not half of the children
can read even at the end of the 2 B grade.
We do find just such evidence that ordinary ideas about the abilities of
children at the beginning of systematic school training in arithmetic
may be in gross error. For example, a reputable and in many ways
admirable recent book has fourteen pages of exercises to teach the
meaning of two and the fact that one and one make two! As an example of
the reverse error, consider putting all these words in the first
twenty-five pages of a beginner's book:--_absentees, attendance, blanks,
continue, copy, during, examples, grouped, memorize, perfect, similar,
splints, therefore, total_!
Little, almost nothing, has been done toward providing an exact
inventory compared with what needs to be done. We may note here (1) the
facts relevant to arithmetic found by Stanley Hall, Hartmann, and others
in their general investigations of the knowledge possessed by children
at entrance to school, (2) the facts concerning the power of children to
perceive differences in length, area, size of collection, and
organization within a collection such as is shown in Fig. 24, and
certain facts and theories about early awareness of number.
In the Berlin inquiry of 1869, knowledge of the meaning of two, three,
and four appeared in 74, 74, and 73 percent of the children upon
entrance to school. Some of those recorded as ignorant probably really
knew, but failed to understand that they were expected to reply or were
shy. Only 85 percent were recorded as knowing their fathers' names.
Seven eighths as many children knew the meanings of two, three, and four
as knew their fathers' names. In a similar but more careful experiment
with Boston children in September, 1880, Stanley Hall found that 92
percent knew three, 83 percent knew four, and 71-1/2 percent knew five.
Three was known about as well as the color red; four was known about as
well as the color blue or yellow or green. Hartmann ['90] found that two
thirds of the children entering school in Annaberg could count from one
to ten. This is about as many as knew money, or the familiar objects of
the town, or could repeat words spoken to them.
[Illustration: FIG. 24.--Objective presentation.]
In the Stanford form of the Binet tests counting four pennies is given
as an ability of the typical four-year-old. Counting 13 pennies
correctly in at least one out of two trials, and knowing three of the
four coins,--penny, nickel, dime, and quarter,--are given as abilities
of the typical six-year-old.
THE PERCEPTION OF NUMBER AND QUANTITY
We know that educated adults can tell how many lines or dots, etc., they
see in a single glance (with an exposure too short for the eye to move)
up to four or more, according to the clearness of the objects and their
grouping. For example, Nanu ['04] reports that when a number of bright
circles on a dark background are shown to educated adults for only .033
second, ten can be counted when arranged to form a parallelogram, but
only five when arranged in a row. With certain groupings, of course,
their 'perception' involves much inference, even conscious addition and
multiplication. Similarly they can tell, up to twenty and beyond, the
number of taps, notes, or other sounds in a series too rapid for single
counting if the sounds are grouped in a convenient rhythm.
These abilities are, however, the product of a long and elaborate
learning, including the learning of arithmetic itself. Elementary
psychology and common experience teach us that the mere observation of
groups or quantities, no matter how clear their number quality appears
to the person who already knows the meanings of numbers, does not of
itself create the knowledge of the meanings of numbers in one who does
not. The experiments of Messenger ['03] and Burnett ['06] showed that
there is no direct intuitive apprehension even of two as distinct from
one. We have to _learn_ to feel the two touches or see the two dots or
lines as two.
We do not know by exact measurements the growth in children of this
ability to count or infer the number of elements in a collection seen or
series heard. Still less do we know what the growth would be without
the influence of school training in counting, grouping, adding, and
multiplying. Many textbooks and teachers seem to overestimate it
greatly. Not all educated adults can, apart from measurement, decide
with surety which of these lines is the longer, or which of these areas
is the larger, or whether this is a ninth or a tenth or an eleventh of a
circle.
[Illustration]
Children upon entering school have not been tested carefully in respect
to judgments of length and area, but we know from such studies as
Gilbert's ['94] that the difference required in their case is probably
over twice that required for children of 13 or 14. In judging weights,
for example, a difference of 6 is perceived as easily by children 13 to
15 years of age as a difference of 15 by six-year-olds.
A teacher who has adult powers of estimating length or area or weight
and who also knows already which of the two is longer or larger or
heavier, may use two lines to illustrate a difference which they really
hide from the child. It is unlikely, for example, that the first of
these lines ______________ ________________ would be recognized as
shorter than the second by every child in a fourth-grade class, and it
is extremely unlikely that it would be recognized as being 7/8 of the
length of the latter, rather than 3/4 of it or 5/6 of it or 9/10 of it
or 11/12 of it. If the two were shown to a second grade, with the
question, "The first line is 7. How long is the other line?" there would
be very many answers of 7 or 9; and these might be entirely correct
arithmetically, the pupils' errors being all due to their inability to
compare the lengths accurately.
_A_ ______________ ________________
______________ ________________
_B_ |______________| |________________|
_C_ |-|-|-|-|-|-|-|
|-|-|-|-|-|-|-|-|
__ __ __ __ __ __ __
_D_ |__|__|__|__|__|__|__|
__ __ __ __ __ __ __ __
|__|__|__|__|__|__|__|__|
_E_ .'\##|##/`. .'\##|##/`.
/###\#|#/ \ /###\#|#/###\
|-----------| |-----------|
\###/#|#\###/ \###/#|#\###/
`./##|##\.' `./##|##\.'
The quantities used should be such that their mere discrimination offers
no difficulty even to a child of blunted sense powers. If 7/8 and 1 are
to be compared, _A_ and _B_ are not allowable. _C_, _D_, and _E_ are
much better.
Teachers probably often underestimate or neglect the sensory
difficulties of the tasks they assign and of the material they use to
illustrate absolute and relative magnitudes. The result may be more
pernicious when the pupils answer correctly than when they fail. For
their correct answering may be due to their divination of what the
teacher wants; and they may call a thing an inch larger to suit her
which does not really seem larger to them at all. This, of course, is
utterly destructive of their respect for arithmetic as an exact and
matter-of-fact instrument. For example, if a teacher drew a series of
lines 20, 21, 22, 23, 24, and 25 inches long on the blackboard in this
form--____ ______ and asked, "This is 20 inches long, how long is
this?" she might, after some errors and correction thereof, finally
secure successful response to all the lines by all the children. But
their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be
actually damaged by the exercise.
THE EARLY AWARENESS OF NUMBER
There has been some disagreement concerning the origin of awareness of
number in the individual, in particular concerning the relative
importance of the perception of how-many-ness and that of how-much-ness,
of the perception of a defined aggregate and the perception of a defined
ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and
Decroly and Degand ['12].)
The chief facts of significance for practice seem to be these: (1)
Children with rare exceptions hear the names _one_, _two_, _three_,
_four_, _half_, _twice_, _two times_, _more_, _less_, _as many as_,
_again_, _first_, _second_, and _third_, long before they have analyzed
out the qualities and relations to which these words refer so as to feel
them at all clearly. (2) Their knowledge of the qualities and relations
is developed in the main in close association with the use of these
words to the child and by the child. (3) The ordinary experiences of the
first five years so develop in the child awareness of the 'how many
somethings' in various groups, of the relative magnitudes of two groups
or quantities of any sort, and of groups and magnitudes as related to
others in a series. For instance, if fairly gifted, a child comes, by
the age of five, to see that a row of four cakes is an aggregate of
four, seeing each cake as a part of the four and the four as the sum of
its parts, to know that two of them are as many as the other two, that
half of them would be two, and to think, when it is useful for him to do
so, of four as a step beyond three on the way to five, or to think of
hot as a step from warm on the way to very hot. The degree of
development of these abilities depends upon the activity of the law of
analysis in the individual and the character of his experiences.
(4) He gets certain bad habits of response from the ambiguity of common
usage of 2, 3, 4, etc., for second, third, fourth. Thus he sees or hears
his parents or older children or others count pennies or rolls or eggs
by saying one, two, three, four, and so on. He himself is perhaps misled
into so counting. Thus the names properly belonging to a series of
aggregations varying in amount come to be to him the names of the
positions of the parts in a counted whole. This happens especially with
numbers above 3 or 4, where the correct experience of the number as a
name for the group has rarely been present. This attaching to the
cardinal numbers above three or four the meanings of the ordinal numbers
seems to affect many children on entrance to school. The numbering of
pages in books, houses, streets, etc., and bad teaching of counting
often prolong this error.
(5) He also gets the habit, not necessarily bad, but often indirectly
so, of using many names such as eight, nine, ten, eleven, fifteen, a
hundred, a million, without any meaning.
(6) The experiences of half, twice, three times as many, three times as
long, etc., are rarer; even if they were not, they would still be less
easily productive of the analysis of the proper abstract element than
are the experiences of two, three, four, etc., in connection with
aggregates of things each of which is usually called one, such as boys,
girls, balls, apples. Experiences of the names, two, three, and four, in
connection with two twos, two threes, two fours, are very rare.
Hence, the names, two, three, etc., mean to these children in the main,
"one something and one something," "one something usually called one,
and one something usually called one, and another something usually
called one," and more rarely and imperfectly "two times anything,"
"three times anything," etc.
With respect to Mr. Phillips' emphasis of the importance of the
series-idea in children's minds, the matters of importance are: first,
that the knowledge of a series of number names in order is of very
little consequence to the teaching of arithmetic and of still less to
the origin of awareness of number. Second, the habit of applying this
series of words in counting in such a way that 8 is associated with the
eighth thing, 9 with the ninth thing, etc., is of consequence because it
does so much mischief. Third, the really valuable idea of the number
series, the idea of a series of groups or of magnitudes varying by
steps, is acquired later, as a result, not a cause, of awareness of
numbers.
With respect to the McLellan-Dewey doctrine, the ratio aspect of numbers
should be emphasized in schools, not because it is the main origin of
the child's awareness of number, but because it is _not_, and because
the ordinary practical issues of child life do _not_ adequately
stimulate its action. It also seems both more economical and more
scientific to introduce it through multiplication, division, and
fractions rather than to insist that 4 and 5 shall from the start mean 4
or 5 times anything that is called 1, for instance, that 8 inches shall
be called 4 two-inches, or 10 cents, 5 two-cents. If I interpret
Professor Dewey's writings correctly, he would agree that the use of
inch, foot, yard, pint, quart, ounce, pound, glassful, cupful, handful,
spoonful, cent, nickel, dime, and dollar gives a sufficient range of
units for the first two school years. Teaching the meanings of 1/2 of 4,
1/2 of 6, 1/2 of 8, 1/2 of 10, 1/2 of 20, 1/3 of 6, 1/3 of 9, 1/3 of 30,
1/4 of 8, two 2s, five 2s, and the like, in early grades, each in
connection with many different units of measure, provides a sufficient
assurance that numbers will connect with relationships as well as with
collections.