THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL NOTIONS IN
ARITHMETIC[11]
[11] Certain paragraphs in this and the following chapter are
taken from the author's _Educational Psychology_, with
slight modifications.
RESPONSES TO ELEMENTS AND CLASSES
The plate which you see, the egg before you at the breakfast table, and
this page are concrete things, but whiteness, whether of plate, egg, or
paper, is, we say, an abstract quality. To be able to think of whiteness
irrespective of any concrete white object is to be able to have an
abstract idea or notion of white; to be able to respond to whiteness,
irrespective of whether it is a part of china, eggshell, paper or
whatever object, is to be able to respond to the abstract element of
whiteness.
Learning arithmetic involves the formation of very many such ideas, the
acquisition of very many such powers of response to elements regardless
of the gross total situations in which they appear. To appreciate the
fiveness of five boys, five pencils, five inches, five rings of a bell;
to understand the division into eight equal parts of 40 cents, 32 feet,
64 minutes, or 16 ones; to respond correctly to the fraction relation in
2/3, 5/6, 3/4, 7/12, 1/8, or any other; to be sensitive to the common
element of 9 = 3 x 3, 16 = 4 x 4, 625 = 25 x 25, .04 = .2 x .2, 1/4 =
1/2 x 1/2,--these are obvious illustrations. All the numbers which the
pupil learns to understand and manipulate are in fact abstractions; all
the operations are abstractions; percent, discount, interest, height,
length, area, volume, are abstractions; sum, difference, product,
quotient, remainder, average, are facts that concern elements or aspects
which may appear with countless different concrete surroundings or
concomitants.
Towser is a particular dog; your house lot on Elm Street is a particular
rectangle; Mr. and Mrs. I.S. Peterson and their daughter Louise are a
particular family of three. In contrast to these particulars, we mean
by a dog, a rectangle, and a family of three, _any_ specimens of these
classes of facts. The idea of a dog, of rectangles in general, of any
family of three is a general notion, a concept or idea of a class or
species. The ability to respond to any dog, or rectangle, or family of
three, regardless of which particular one it may be, is the general
notion in action.
Learning arithmetic involves the formation of very many such general
notions, such powers of response to any member of a certain class. Thus
a hundred different sized lots may all be responded to as rectangles;
9/18, 12/27, 15/24, and 27/36 may all be responded to as members of the
class, 'both members divisible by 3.' The same fact may be responded to
in different ways according to the class to which it is assigned. Thus 4
in 3/4, 4/5, 45, 54, and 405 is classed respectively as 'a certain sized
part of unity,' 'a certain number of parts of the size shown by the 5,'
'a certain number of tens,' 'a certain number of ones,' and 'a certain
number of hundreds.' Each abstract quality may become the basis of a
class of facts. So fourness as a quality corresponds to the class
'things four in number or size'; the fractional quality or relation
corresponds to the class 'fractions.' The bonds formed with classes of
facts and with elements or features by which one whole class of facts is
distinguished from another, are in fact, a chief concern of arithmetical
learning.[12]
[12] It should be noted that just as concretes give rise to
abstractions, so these in turn give rise to still more
abstract abstractions. Thus fourness, fiveness, twentyness,
and the like give rise to 'integral-number-ness.' Similarly
just as individuals are grouped into general classes, so
classes are grouped into still more general classes.
Half, quarter, sixth, and tenth are general notions, but
'one ...th' is more general; and 'fraction' is still more
general.
FACILITATING THE ANALYSIS OF ELEMENTS
Abstractions and generalizations then depend upon analysis and upon
bonds formed with more or less subtle elements rather than with gross
total concrete situations. The process involved is most easily
understood by considering the means employed to facilitate it.
The first of these is having the learner respond to the total situations
containing the element in question with the attitude of piecemeal
examination, and with attentiveness to one element after another,
especially to so near an approximation to the element in question as he
can already select for attentive examination. This attentiveness to one
element after another serves to emphasize whatever appropriate minor
bonds from the element in question the learner already possesses. Thus,
in teaching children to respond to the 'fiveness' of various
collections, we show five boys or five girls or five pencils, and say,
"See how many boys are standing up. Is Jack the only boy that is
standing here? Are there more than two boys standing? Name the boys
while I point at them and count them. (Jack) is one, and (Fred) is one
more, and (Henry) is one more. Jack and Fred make (two) boys. Jack and
Fred and Henry make (three) boys." (And so on with the attentive
counting.) The mental set or attitude is directed toward favoring the
partial and predominant activity of 'how-many-ness' as far as may be;
and the useful bonds that the 'fiveness,' the 'one and one and one and
one and one-ness,' already have, are emphasized as far as may be.
The second of the means used to facilitate analysis is having the
learner respond to many situations each containing the element in
question (call it A), but with varying concomitants (call these V. C.)
his response being so directed as, so far as may be, to separate each
total response into an element bound to the A and an element bound to
the V. C.
Thus the child is led to associate the responses--'Five boys,' 'Five
girls,' 'Five pencils,' 'Five inches,' 'Five feet,' 'Five books,' 'He
walked five steps,' 'I hit my desk five times,' and the like--each with
its appropriate situation. The 'Five' element of the response is thus
bound over and over again to the 'fiveness' element of the situation,
the mental set being 'How many?,' but is bound only once to any one of
the concomitants. These concomitants are also such as have preferred
minor bonds of their own (the sight of a row of boys _per se_ tends
strongly to call up the 'Boys' element of the response). The other
elements of the responses (boys, girls, pencils, etc.) have each only a
slight connection with the 'fiveness' element of the situations. These
slight connections also in large part[13] counteract each other, leaving
the field clear for whatever uninhibited bond the 'fiveness' has.
[13] They may, of course, also result in a fusion or an alternation
of responses, but only rarely.
The third means used to facilitate analysis is having the learner
respond to situations which, pair by pair, present the element in a
certain context and present that same context with _the opposite of the
element in question_, or with something at least very unlike the
element. Thus, a child who is being taught to respond to 'one fifth' is
not only led to respond to 'one fifth of a cake,' 'one fifth of a pie,'
'one fifth of an apple,' 'one fifth of ten inches,' 'one fifth of an
army of twenty soldiers,' and the like; he is also led to respond to
each of these _in contrast with_ 'five cakes,' 'five pies,' 'five
apples,' 'five times ten inches,' 'five armies of twenty soldiers.'
Similarly the 'place values' of tenths, hundredths, and the rest are
taught by contrast with the tens, hundreds, and thousands.
These means utilize the laws of connection-forming to disengage a
response element from gross total responses and attach it to some
situation element. The forces of use, disuse, satisfaction, and
discomfort are so maneuvered that an element which never exists by
itself in nature can influence man almost as if it did so exist, bonds
being formed with it that act almost or quite irrespective of the gross
total situation in which it inheres. What happens can be most
conveniently put in a general statement by using symbols.
Denote by _a_ + _b_, _a_ + _g_, _a_ + _l_, _a_ + _q_, _a_ + _v_, and
_a_ + _B_ certain situations alike in the element _a_ and different in
all else. Suppose that, by original nature or training, a child responds
to these situations respectively by r_{1} + r_{2}, r_{1} + r_{7},
r_{1} + r_{12}, r_{1} + r_{17}, r_{1} + r_{22}, r_{1} + r_{27}. Suppose
that man's neurones are capable of such action that r_{1}, r_{2}, r_{7},
r_{12}, r_{22}, and r_{27}, can each be made singly.
Case I. Varying Concomitants
Suppose that _a_ + _b_, _a_ + _g_, _a_ + _l_, etc., occur once each.
We have _a_ + _b_ responded to by r_{1} + r_{2},
_a_ + _g_ " " r_{1} + r_{7},
_a_ + _l_ " " r_{1} + r_{12},
_a_ + _q_ " " r_{1} + r_{17},
_a_ + _v_ " " r_{1} + r_{22}, and
_a_ + _B_ " " r_{1} + r_{27}, as shown in
Scheme I.
Scheme I
_a_ _b_ _g_ _l_ _q_ _v_ _B_
r_{1} 6 1 1 1 1 1 1
r_{2} 1 1
r_{7} 1 1
r_{12} 1 1
r_{17} 1 1
r_{22} 1 1
r_{27} 1 1
_a_ is thus responded to by r_{1} (that is, connected with r_{1}) each
time, or six in all, but only once each with _b_, _g_, _l_, _q_, _v_,
and _B_. _b_, _g_, _l_, _q_, _v_, and _B_ are connected once each with
r_{1} and once respectively with r_{2}, r_{7}, r_{12}, etc. The bond
from _a_ to r_{1}, has had six times as much exercise as the bond from
_a_ to r_{2}, or from _a_ to r_{7}, etc. In any new gross situation, _a_
0, _a_ will be more predominant in determining response than it would
otherwise have been; and r_{1} will be more likely to be made than
r_{2}, r_{7}, r_{12}, etc., the other previous associates in the
response to a situation containing _a_. That is, the bond from the
element _a_ to the response r_{1} has been notably strengthened.
Case II. Contrasting Concomitants
Now suppose that _b_ and _g_ are very dissimilar elements (_e.g._, white
and black), that _l_ and _q_ are very dissimilar (_e.g._, long and
short), and that _v_ and _B_ are also very dissimilar. To be very
dissimilar means to be responded to very differently, so that r_{7}, the
response to _g_, will be very unlike r_{2}, the response to _b_. So
r_{7} may be thought of as r_{not 2} or r_{-2}. In the same way r_{12}
may be thought of as r_{not 12} or r_{-12}, and r_{27} may be called
r_{not 22} or r_{-22}.
Then, if the situations _a_ _b_, _a _g_, _a _l_, _a _q_, _a _v_, and
_a_ _B_ are responded to, each once, we have:--
_a_ + _b_ responded to by r_{1} + r_{2},
_a_ + _g_ " " r_{1} + r_{not 2},
_a_ + _l_ " " r_{1} + r_{12},
_a_ + _q_ " " r_{1} + r_{not 12},
_a_ + _v_ " " r_{1} + r_{22}, and
_a_ + _B_ " " r_{1} + r_{not 22}, as shown in Scheme II.
Scheme II
_a_ _b_ _g_ _l_ _q_ _v_ _B_
(opp. of _b_) (opp. of _l_) (opp. of _v_)
r_{1} 6 1 1 1 1 1 1
r_{not 1}
r_{2} 1 1
r_{not 2} 1 1
r_{12} 1 1
r_{not 12} 1 1
r_{22} 1 1
r_{not 22} 1 1
r_{1} is connected to _a_ by 6 repetitions. r_{2} and r_{not 2} are each
connected to _a_ by 1 repetition, but since they interfere, canceling
each other so to speak, the net result is for _a_ to have zero tendency
to call up r_{2} or r_{not 2}. r_{12} and r_{not 12} are each connected
to _a_ by 1 repetition, but they interfere with or cancel each other
with the net result that _a_ has zero tendency to call up r_{12} or
r_{not 12}. So with r_{22} and r_{not 22}. Here then the net result of
the six connections of _a_ _b_, _a_ _g_, _a_ _l_, _a_ _q_, _a_ _v_, and
_a_ _B_ is to connect _a_ with _r_, and with nothing else.
Case III. Contrasting Concomitants and Contrasting Element
Suppose now that the facts are as in Case II, but with the addition of
six experiences where a certain element which is the opposite of, or
very dissimilar to, _a_ is connected with the response r_{not 1}, or
r_{-1} which is opposite to, or very dissimilar to r_{1}. Call this
opposite of _a_, - _a_.
That is, we have not only
_a_ + _b_ responded to by r_{1} + r_{2},
_a_ + _g_ " " r_{1} + r_{not 2},
_a_ + _l_ " " r_{1} + r_{12},
_a_ + _q_ " " r_{1} + r_{not 12},
_a_ + _v_ " " r_{1} + r_{22}, and
_a_ + _B_ " " r_{1} + r_{not 22},
but also
- _a_ + _b_ responded to by r_{not 1} + r_{2},
- _a_ + _g_ " " r_{not 1} + r_{not 2},
- _a_ + _l_ " " r_{not 1} + r_{12},
- _a_ + _q_ " " r_{not 1} + r_{not 12},
- _a_ + _v_ " " r_{not 1} + r_{22}, and
- _a_ + _B_ " " r_{not 1} + r_{not 22}, as shown in
Scheme III.
Scheme III
_a_ opp. _b_ _g_ _l_ _q_ _v_ _B_
of _a_ (opp. of _b_) (opp. of _l_) (opp. of _v_)
r_{1} 6 1 1 1 1 1 1
r_{not 1} 6 1 1 1 1 1 1
r_{2} 1 1 2
r_{not 2} 1 1 2
r_{12} 1 1 2
r_{not 12} 1 1 2
r_{22} 1 1 2
r_{not 22} 1 1 2
In this series of twelve experiences _a_ connects with r_{1} six times
and the opposite of _a_ connects with r_{not 1} six times. _a_ connects
equally often with three pairs of mutual destructives r_{2} and
r_{not 2}, r_{12} and r_{not 12}, r_{22} and r_{not 22}, and so has zero
tendency to call them up. - _a_ has also zero tendency to call up any of
these responses except its opposite, r_{not 1}. _b_, _g_, _l_, _q_, _v_,
and _B_ are made to connect equally often with r_{1} and r_{not 1}. So,
of these elements, _a_ is the only one left with a tendency to call up
r_{1}.
Thus, by the mere action of frequency of connection, r_{1} is connected
with _a_; the bonds from _a_ to anything except r_{1} are being
counteracted, and the slight bonds from anything except _a_ to r_{1} are
being counteracted. The element _a_ becomes predominant in situations
containing it; and its bond toward r_{1} becomes relatively enormously
strengthened and freed from competition.
These three processes occur in a similar, but more complicated,
form if the situations _a_ + _b_, _a_ + _g_, etc., are replaced by
_a_ + _b_ + _c_ + _d_ + _e_ + _f_, _a_ + _g_ + _h_ + _i_ + _j_ + _k_,
etc., and the responses r_{1} + r_{2}, r_{1} + r_{7}, r_{1} + r_{12},
etc., are replaced by r_{1} + r_{2} + r_{3} + r_{4} + r_{5} + r_{6},
r_{1} + r_{7} + r_{8} + r_{9} + r_{10} + r_{11}, etc.--_provided the_
r_{1}, r_{2}, r_{3}, r_{4}, etc., _can be made singly_. In so far as any
one of the responses is necessarily co-active with any one of the others
(so that, for example, r_{13} always brings r_{26} with it and _vice
versa_), the exact relations of the numbers recorded in schemes like
schemes I, II, and III on pages 172 to 174 will change; but, unless
r_{1} has such an inevitable co-actor, the general results of schemes I,
II, and III will hold good. If r_{1} does have such an inseparable
co-actor, say r_{2}, then, of course, _a_ can never acquire bonds with
r_{1} alone, but everywhere that r_{1} or r_{2} appears in the preceding
schemes the other element must appear also. r_{1} r_{2} would then have
to be used as a unit in analysis.
The '_a_ + _b_,' '_a_ + _g_,' '_a_ + _l_,' ... '_a_ + _B_' situations
may occur unequal numbers of times, altering the exact numerical
relations of the connections formed and presented in schemes I, II,
and III; but the process in general remains the same.
So much for the effect of use and disuse in attaching appropriate
response elements to certain subtle elements of situations. There are
three main series of effects of satisfaction and discomfort. They
serve, first, to emphasize, from the start, the desired bonds leading to
the responses r_{1} + r_{2}, r_{1} + r_{7}, etc., to the total
situations, and to weed out the undesirable ones. They also act to
emphasize, in such comparisons and contrasts as have been described,
every action of the bond from _a_ to r_{1}; and to eliminate every
tendency of _a_ to connect with aught save r_{1}, and of aught save _a_
to connect with r_{1}. Their third service is to strengthen the bonds
produced of appropriate responses to _a_ wherever it occurs, whether or
not any formal comparisons and contrasts take place.
The process of learning to respond to the difference of pitch in tones
from whatever instrument, to the 'square-root-ness' of whatever number,
to triangularity in whatever size or combination of lines, to equality
of whatever pairs, or to honesty in whatever person or instance, is thus
a consequence of associative learning, requiring no other forces than
those of use, disuse, satisfaction, and discomfort. "What happens in
such cases is that the response, by being connected with many situations
alike in the presence of the element in question and different in other
respects, is bound firmly to that element and loosely to each of its
concomitants. Conversely any element is bound firmly to any one response
that is made to all situations containing it and very, very loosely to
each of those responses that are made to only a few of the situations
containing it. The element of triangularity, for example, is bound
firmly to the response of saying or thinking 'triangle' but only very
loosely to the response of saying or thinking white, red, blue, large,
small, iron, steel, wood, paper, and the like. A situation thus acquires
bonds not only with some response to it as a gross total, but also with
responses to any of its elements that have appeared in any other gross
totals. Appropriate response to an element regardless of its
concomitants is a necessary consequence of the laws of exercise and
effect if an animal learns to make that response to the gross total
situations that contain the element and not to make it to those that do
not. Such prepotent determination of the response by one or another
element of the situation is no transcendental mystery, but, given the
circumstances, a general rule of all learning." Such are at bottom only
extreme cases of the same learning as a cat exhibits that depresses a
platform in a certain box whether it faces north or south, whether the
temperature is 50 or 80 degrees, whether one or two persons are in
sight, whether she is exceedingly or moderately hungry, whether fish or
milk is outside the box. All learning is analytic, representing the
activity of elements within a total situation. In man, by virtue of
certain instincts and the course of his training, very subtle elements
of situations can so operate.
* * * * *
Learning by analysis does not often proceed in the carefully organized
way represented by the most ingenious marshaling of comparing and
contrasting activities. The associations with gross totals, whereby in
the end an element is elevated to independent power to determine
response, may come in a haphazard order over a long interval of time.
Thus a gifted three-year-old boy will have the response element of
'saying or thinking _two_,' bound to the 'two-ness' element of very many
situations in connection with the 'how-many' mental set; and he will
have made this analysis without any formal, systematic training. An
imperfect and inadequate analysis already made is indeed usually the
starting point for whatever systematic abstraction the schools direct.
Thus the kindergarten exercises in analyzing out number, color, size,
and shape commonly assume that 'one-ness' _versus_ 'more-than-one-ness,'
black and white, big and little, round and not round are, at least
vaguely, active as elements responded to in some independence of their
contexts. Moreover, the tests of actual trial and success in further
undirected exercises usually cooeperate to confirm and extend and refine
what the systematic drills have given. Thus the ordinary child in school
is left, by the drills on decimal notation, with only imperfect power
of response to the 'place-values.' He continues to learn to respond
properly to them by finding that 4 x 40 = 160, 4 x 400 = 1600,
800 - 80 = 720, 800 - 8 = 792, 800-800 = 0, 42 x 48 = 2016,
24 x 48 = 1152, and the like, are satisfying; while 4 x 40 = 16,
23 x 48 = 832, 800 - 8 = 0, and the like, are not. The process of
analysis is the same in such casual, unsystematized formation of
connections with elements as in the deliberately managed, piecemeal
inspection, comparison, and contrast described above.
SYSTEMATIC AND OPPORTUNISTIC STIMULI TO ANALYSIS
The arrangement of a pupil's experiences so as to direct his attention
to an element, vary its concomitants instructively, stimulate
comparison, and throw the element into relief by contrast may be by
fixed, formal, systematic exercises. Or it may be by much less formal
exercises, spread over a longer time, and done more or less incidentally
in other connections. We may call these two extremes the 'systematic'
and 'opportunistic,' since the chief feature of the former is that it
systematically provides experiences designed to build up the power of
correct response to the element, whereas the chief feature of the latter
is that it uses especially such opportunities as occur by reason of the
pupil's activities and interests.
Each method has its advantages and disadvantages. The systematic method
chooses experiences that are specially designed to stimulate the
analysis; it provides these at a certain fixed time so that they may
work together; it can then and there test the pupils to ascertain
whether they really have the power to respond to the element or aspect
or feature in question. Its disadvantages are, first, that many of the
pupils will feel no need for and attach no interest or motive to these
formal exercises; second, that some of the pupils may memorize the
answers as a verbal task instead of acquiring insight into the facts;
third, that the ability to respond to the element may remain restricted
to the special cases devised for the systematic training, and not be
available for the genuine uses of arithmetic.
The opportunistic method is strong just where the systematic is weak.
Since it seizes upon opportunities created by the pupil's abilities and
interests, it has the attitude of interest more often. Since it builds
up the experiences less formally and over a wider space of time, the
pupils are less likely to learn verbal answers. Since its material comes
more from the genuine uses of life, the power acquired is more likely to
be applicable to life.
Its disadvantage is that it is harder to manage. More thought and
experimentation are required to find the best experiences; greater care
is required to keep track of the development of an abstraction which is
taught not in two days, but over two months; and one may forget to test
the pupils at the end. In so far as the textbook and teacher are able to
overcome these disadvantages by ingenuity and care, the opportunistic
method is better.
ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS
We may expect much improvement in the formation of abstract and general
ideas in arithmetic from the application of three principles in addition
to those already described. They are: (1) Provide enough actual
experiences before asking the pupil to understand and use an abstract or
general idea. (2) Develop such ideas gradually, not attempting to give
complete and perfect ideas all at once. (3) Develop such ideas so far as
possible from experiences which will be valuable to the pupil in and of
themselves, quite apart from their merit as aids in developing the
abstraction or general notion. Consider these three principles in order.
Children, especially the less gifted intellectually, need more
experiences as a basis for and as applications of an arithmetical
abstraction or concept than are usually given them. For example, in
paving the way for the principle, "Any number times 0 equals 0," it is
not safe to say, "John worked 8 days for 0 minutes per day. How many
minutes did he work?" and "How much is 0 times 4 cents?" It will be
much better to spend ten or fifteen minutes as follows:[14] "What does
zero mean? (Not any. No.) How many feet are there in eight yards?
In 5 yards? In 3 yards? In 2 yards? In 1 yard? In 0 yard? How many
inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = .... qt.
5 pk. = .... qt. 0 pk. = .... qt. A boy receives 60 cents an hour
when he works. How much does he receive when he works 3 hr.? 8 hr.?
6 hr.? 0 hr.? A boy received 60 cents a day for 0 days. How much did he
receive? How much is 0 times $600? How much is 0 times $5000? How much
is 0 times a million dollars? 0 times any number equals....
232 (At the blackboard.) 0 time 232 equals what?
30 I write 0 under the 0.[15] 3 times 232 equals what?
----
6960 Continue at the blackboard with
734 321 312 41
20 40 30 60 etc."
--- --- --- --
[14] The more gifted children may be put to work using the principle
after the first minute or two.
[15] 232
30 If desired this form may be used, with the appropriate
--- difference in the form of the questions and statements.
000
696
----
6960
Pupils in the elementary school, except the most gifted, should not be
expected to gain mastery over such concepts as _common fraction_,
_decimal fraction_, _factor_, and _root_ quickly. They can learn a
definition quickly and learn to use it in very easy cases, where even a
vague and imperfect understanding of it will guide response correctly.
But complete and exact understanding commonly requires them to take,
not one intellectual step, but many; and mastery in use commonly comes
only as a slow growth. For example, suppose that pupils are taught
that .1, .2, .3, etc., mean 1/10, 2/10, 3/10, etc., that .01, .02, .03,
etc., mean 1/100, 2/100, 3/100, etc., that .001, .002, .003, etc., mean
1/1000, 2/1000, 3/1000, etc., and that .1, .02, .001, etc., are decimal
fractions. They may then respond correctly when asked to write a decimal
fraction, or to state which of these,--1/4, .4, 3/8, .07, .002,
5/6,--are common fractions and which are decimal fractions. They may
be able, though by no means all of them will be, to write decimal
fractions which equal 1/2 and 1/5, and the common fractions which
equal .1 and .09. Most of them will not, however, be able to respond
correctly to "Write a decimal mixed number"; or to state which of
these,--1/100, .4-1/2, .007/350, $.25,--are common fractions, and which
are decimals; or to write the decimal fractions which equal 3/4 and 1/3.
If now the teacher had given all at once the additional experiences
needed to provide the ability to handle these more intricate and subtle
features of decimal-fraction-ness, the result would have been confusion
for most pupils. The general meaning of .32, .14, .99, and the like
requires some understanding of .30, .10, .90, and .02, .04, .08; but it
is not desirable to disturb the child with .30 while he is trying to
master 2.3, 4.3, 6.3, and the like. Decimals in general require
connection with place value and the contrasts of .41 with 41, 410, 4.1,
and the like, but if the relation to place values in general is taught
in the same lesson with the relation to /10s, /100s, /1000s, the mind
will suffer from violent indigestion.
A wise pedagogy in fact will break up the process of learning the
meaning and use of decimal fractions into many teaching units, for
example, as follows:--
(1) Such familiarity with fractions with large denominators as is
desirable for pupils to have, as by an exercise in reducing to lowest
terms, 8/10, 36/64, 20/25, 18/24, 24/32, 21/30, 25/100, 40/100, and the
like. This is good as a review of cancellation, and as an extension of
the idea of a fraction.
(2) Objective work, showing 1/10 sq. ft., 1/50 sq. ft., 1/100 sq. ft.,
and 1/1000 sq. ft., and having these identified and the forms 1/10 sq.
ft., 1/100 sq. ft., and 1/1000 sq. ft. learned. Finding how many
feet = 1/10 mile and 1/100 mile.
(3) Familiarity with /100s and /1000s by reductions of 750/1000, 50/100,
etc., to lowest terms and by writing the missing numerators in
500/1000 = /100 = /10 and the like, and by finding 1/10, 1/100, and
1/1000 of 3000, 6000, 9000, etc.
(4) Writing 1/10 as .1 and 1/100 as .01, 11/100, 12/100, 13/100, etc.,
as .11, .12, .13. United States money is used as the introduction.
Application is made to miles.
(5) Mixed numbers with a first decimal place. The cyclometer or
speedometer. Adding numbers like 9.1, 14.7, 11.4, etc.
(6) Place value in general from thousands to hundredths.
(7) Review of (1) to (6).
(8) Tenths and hundredths of a mile, subtraction when both numbers
extend to hundredths, using a railroad table of distances.
(9) Thousandths. The names 'decimal fractions or decimals,' and 'decimal
mixed numbers or decimals.' Drill in reading any number to thousandths.
The work will continue with gradual extension and refinement of the
understanding of decimals by learning how to operate with them in
various ways.
Such may seem a slow progress, but in fact it is not, and many of these
exercises whereby the pupil acquires his mastery of decimals are useful
as organizations and applications of other arithmetical facts.
That, it will be remembered, was the third principle:--"Develop abstract
and general ideas by experiences which will be intrinsically valuable."
The reason is that, even with the best of teaching, some pupils will
not, within any reasonable limits of time expended, acquire ideas that
are fully complete, rigorous when they should be, flexible when they
should be, and absolutely exact. Many children (and adults, for that
matter) could not within any reasonable limits of time be so taught the
nature of a fraction that they could decide unerringly in original
exercises like:--
Is 2.75/25 a common fraction?
Is $.25 a decimal fraction?
Is one _x_th of _y_ a fraction?
Can the same words mean both a common fraction and a decimal fraction?
Express 1 as a common fraction.
Express 1 as a decimal fraction.
These same children can, however, be taught to operate correctly with
fractions in the ordinary uses thereof. And that is the chief value of
arithmetic to them. They should not be deprived of it because they
cannot master its subtler principles. So we seek to provide experiences
that will teach all pupils something of value, while stimulating in
those who have the ability the growth of abstract ideas and general
principles.
Finally, we should bear in mind that working with qualities and
relations that are only partly understood or even misunderstood does
under certain conditions give control over them. The general process of
analytic learning in life is to respond as well as one can; to get a
clearer idea thereby; to respond better the next time; and so on. For
instance, one gets some sort of notion of what 1/5 means; he then
answers such questions as 1/5 of 10 = ? 1/5 of 5 = ? 1/5 of 20 = ?;
by being told when he is right and when he is wrong, he gets from
these experiences a better idea of 1/5; again he does his best with
1/5 = _/10, 1/5 = _/15, etc., and as before refines and enlarges his
concept of 1/5. He adds 1/5 to 2/5, etc., 1/5 to 3/10, etc., 1/5 to 1/2,
etc., and thereby gains still further, and so on.
What begins as a blind habit of manipulation started by imitation may
thus grow into the power of correct response to the essential element.
The pupil who has at the start no notion at all of 'multiplying' may
learn what multiplying is by his experience that '4 6 multiplying
gives 24'; '3 9 multiplying gives 27,' etc. If the pupil keeps on doing
something with numbers and differentiates right results, he will often
reach in the end the abstractions which he is supposed to need in the
beginning. It may even be the case with some of the abstractions
required in arithmetic that elaborate provision for comprehension
beforehand is not so efficient as the same amount of energy devoted
partly to provision for analysis itself beforehand and partly to
practice in response to the element in question without full
comprehension.
It certainly is not the best psychology and not the best educational
theory to think that the pupil first masters a principle and then merely
applies it--first does some thinking and then computes by mere routine.
On the contrary, the applications should help to establish, extend, and
refine the principle--the work a pupil does with numbers should be a
main means of increasing his understanding of the principles of
arithmetic as a science.