THE CONDITIONS OF LEARNING
We shall consider in this chapter the influence of time of day, size of
class, and amount of time devoted to arithmetic in the school program,
the hygiene of the eyes in arithmetical work, the use of concrete
objects, and the use of sounds, sights, and thoughts as situations and
of speech and writing and thought as responses.[17]
[17] Facts concerning the conditions of learning in general will
be found in the author's _Educational Psychology_, Vol. 2,
Chapter 8, or in the _Educational Psychology, Briefer Course_,
Chapter 15.
EXTERNAL CONDITIONS
Computation of one or another sort has been used by several
investigators as a test of efficiency at different times in the day.
When freed from the effects of practice on the one hand and lack of
interest due to repetition on the other, the results uniformly show an
increase in speed late in the school session with a falling off in
accuracy that about balances it.[18] There is no wisdom in putting
arithmetic early in the session because of its _difficulty_. Lively and
sociable exercises in mental arithmetic with oral answers in fact seem
to be admirably fitted for use late in the session. Except for the
general principles (1) of starting the day with work that will set a
good standard of cheerful, efficient production and (2) of getting the
least interesting features of the day's work done fairly early in the
day, psychology permits practical exigencies to rule the program, so far
as present knowledge extends. Adequate measurements of the effect of
time of day on _improvement_ have not been made, but there is no reason
to believe that any one time between 9 A.M. and 4 P.M. is appreciably
more favorable to arithmetical learning than to learning geography,
history, spelling, and the like.
[18] See Thorndike ['00], King ['07], and Heck ['13].
The influence of size of class upon progress in school studies is very
difficult to measure because (1) within the same city system the average
of the six (or more) sizes of class that a pupil has experienced will
tend to approximate closely to the corresponding average for any other
child; because further (2) there may be a tendency of supervisory
officers to assign more pupils to the better teachers; and because
(3) separate systems which differ in respect to size of class probably
differ in other respects also so that their differences in achievement
may be referable to totally different differences.
Elliott ['14] has made a beginning by noting size of class during the
year of test in connection with his own measures of the achievements of
seventeen hundred pupils, supplemented by records from over four hundred
other classes. As might be expected from the facts just stated, he finds
no appreciable difference between classes of different sizes within the
same school system, the effect of the few months in a small class being
swamped by the antecedents or concomitants thereof.
The effect of the amount of time devoted to arithmetic in the school
program has been studied extensively by Rice ['02 and '03] and Stone
['08].
Dr. Rice ['02] measured the arithmetical ability of some 6000 children
in 18 different schools in 7 different cities. The results of these
measurements are summarized in Table 10. This table "gives two averages
for each grade as well as for each school as a whole. Thus, the school
at the top shows averages of 80.0 and 83.1, and the one at the bottom,
25.3 and 31.5. The first represents the percentage of answers which were
absolutely correct; the second shows what per cent of the problems were
correct in principle, _i.e._ the average that would have been received
if no mechanical errors had been made."
The facts of Dr. Rice's table show that there is a positive relation
between the general standing of a school system in the tests and the
amount of time devoted to arithmetic by its program. The relation is
not close, however, being that expressed by a correlation coefficient
of .36-1/2. Within any one school system there is no relation between
the standing of a particular school and the amount of time devoted to
arithmetic in that school's program. It must be kept in mind that the
amount of time given in the school program may be counterbalanced by
emphasizing work at home and during study periods, or, on the other
hand, may be a symptom of correspondingly small or great emphasis on
arithmetic in work set for the study periods at home.
A still more elaborate investigation of this same topic was made by
Stone ['08]. I quote somewhat fully from it, since it is an instructive
sample of the sort of studies that will doubtless soon be made in the
case of every elementary school subject. He found that school systems
differed notably in the achievements made by their sixth-grade pupils in
his tests of computation (the so-called 'fundamentals') and of the
solution of verbally described problems (the so-called 'reasoning'). The
facts were as shown in Table 11.
TABLE 10
AVERAGES FOR INDIVIDUAL SCHOOLS IN ARITHMETIC
KEY A: CITY
B: SCHOOL
C: Result
D: Principle
E: Percent of Mechanical Errors
F: Minutes Daily
===========================================================
| |6TH YEAR |7TH YEAR |8TH YEAR |SCHOOL AVERAGE |
| |----+----+----+----+----+----+----+----+-----+-----
A | B | C | D | C | D | C | D | C | D | E | F
---+---+----+----+----+----+----+----+----+----+-----+-----
III| 1 |79.3|80.3|81.1|82.3|91.7|93.9|80.0|83.1| 3.7 | 53
I| 1 |80.4|81.5|64.2|67.2|80.9|82.8|76.6|80.3| 4.6 | 60
I| 2 |80.9|83.4|43.5|50.9|72.7|79.1|69.3|75.1| 7.7 | 25
I| 3 |72.2|74.0|63.5|66.2|74.5|76.6|67.8|72.2| 6.1 | 45
I| 4 |69.9|72.2|54.6|57.8|66.5|69.1|64.3|70.3| 8.5 | 45
II| 1 |71.2|75.3|33.6|35.7|36.8|40.0|60.2|64.8| 7.1 | 60
III| 2 |43.7|45.0|53.9|56.7|51.1|53.1|54.5|58.9| 7.4 | 60
IV| 1 |58.9|60.4|31.2|34.1|41.6|43.5|55.1|58.4| 5.6 | 60
IV| 2 |59.8|63.1| -- | -- |22.5|22.5|53.9|58.8| 8.3 | --
IV| 3 |54.9|58.1|35.2|38.6|43.5|45.0|51.5|57.6|10.5 | 60
IV| 4 |42.3|45.1|16.1|19.2|48.7|48.7|42.8|48.2|11.2 | --
V| 1 |44.1|48.7|29.2|32.5|51.1|58.3|45.9|51.3|10.5 | 40
VI| 1 |68.3|71.3|33.5|36.6|26.9|30.7|39.0|42.9| 9.0 | 33
VI| 2 |46.1|49.5|19.5|24.2|30.2|40.6|36.5|43.6|16.2 | 30
VI| 3 |34.5|36.4|30.5|35.1|23.3|24.1|36.0|42.5|15.2 | 48
VII| 1 |35.2|37.7|29.1|32.5|25.1|27.2|40.5|45.9|11.7 | 42
VII| 2 |35.2|38.7|15.0|16.4|19.6|21.2|36.5|40.6|10.1 | 75
VII| 3 |27.6|33.7| 8.9|10.1|11.3|11.3|25.3|31.5|19.6 | 45
===========================================================
High achievement by a system in computation went with high achievement
in solving the problems, the correlation being about .50; and the
system that scored high in addition or subtraction or multiplication or
division usually showed closely similar excellence in the other three,
the correlations being about .90.
TABLE 11
SCORES MADE BY THE SIXTH-GRADE PUPILS OF EACH OF TWENTY-SIX SCHOOL
SYSTEMS
=================================================
SYSTEM | SCORE IN TESTS WITH | SCORE IN TESTS IN
| PROBLEMS | COMPUTING
-------+---------------------+-------------------
23 | 356 | 1841
24 | 429 | 3513
17 | 444 | 3042
4 | 464 | 3563
25 | 464 | 2167
22 | 468 | 2311
16 | 469 | 3707
20 | 491 | 2168
18 | 509 | 3758
15 | 532 | 2779
3 | 533 | 2845
8 | 538 | 2747
6 | 550 | 3173
1 | 552 | 2935
10 | 601 | 2749
2 | 615 | 2958
21 | 627 | 2951
13 | 636 | 3049
14 | 661 | 3561
9 | 691 | 3404
7 | 734 | 3782
12 | 736 | 3410
11 | 759 | 3261
26 | 791 | 3682
19 | 848 | 4099
5 | 914 | 3569
=================================================
Of the conditions under which arithmetical learning took place, the one
most elaborately studied was the amount of time devoted to arithmetic.
On the basis of replies by principals of schools to certain questions,
he gave each of the twenty-six school systems a measure for the
probable time spent on arithmetic up through grade 6. Leaving home study
out of account, there seems to be little or no correlation between the
amount of time a system devotes to arithmetic and its score in
problem-solving, and not much more between time expenditure and score in
computation. With home study included there is little relation to the
achievement of the system in solving problems, but there is a clear
effect on achievement in computation. The facts as given by Stone are:--
TABLE 12
CORRELATION OF TIME EXPENDITURES WITH ABILITIES
Without Home Study { Reasoning and Time Expenditure -.01
{ Fundamentals and Time Expenditure .09
Including Home Study { Reasoning and Time Expenditure .13
{ Fundamentals and Time Expenditure .49
These correlations, it should be borne in mind, are for school systems,
not for individual pupils. It might be that, though the system which
devoted the most time to arithmetic did not show corresponding
superiority in the product over the system devoting only half as much
time, the pupils within the system did achieve in exact proportion to
the time they gave to study. Neither correlation would permit inference
concerning the effect of different amounts of time spent by the same
pupil.
Stone considered also the printed announcements of the courses of study
in arithmetic in these twenty-six systems. Nineteen judges rated these
announced courses of study for excellence according to the instructions
quoted below:--
CONCERNING THE RATING OF COURSES OF STUDY
Judges please read before scoring
I. Some Factors Determining Relative Excellence.
(N. B. The following enumeration is meant to be suggestive rather than
complete or exclusive. And each scorer is urged to rely primarily on his
own judgment.)
1. Helpfulness to the teacher in teaching the subject matter outlined.
2. Social value or concreteness of sources of problems.
3. The arrangement of subject matter.
4. The provision made for adequate drill.
5. A reasonable minimum requirement with suggestions for valuable
additional work.
6. The relative values of any predominating so-called methods--such as
Speer, Grube, etc.
7. The place of oral or so-called mental arithmetic.
8. The merit of textbook references.
II. Cautions and Directions.
(Judges please follow as implicitly as possible.)
1. Include references to textbooks as parts of the Course of Study.
This necessitates judging the parts of the texts referred to.
2. As far as possible become equally familiar with all courses before
scoring any.
3. When you are ready to begin to score, (1) arrange in serial
order according to excellence, (2) starting with the middle one
score it 50, then score above and below 50 according as courses
are better or poorer, indicating relative differences in
excellence by relative differences in scores, _i.e._ in so far
as you find that the courses differ by about equal steps, score
those better than the middle one 51, 52, etc., and those poorer
49, 48, etc., but if you find that the courses differ by
unequal steps show these inequalities by omitting numbers.
4. Write ratings on the slip of paper attached to each course.
The systems whose courses of study were thus rated highest did not
manifest any greater achievement in Stone's tests than the rest. The
thirteen with the most approved announcements of courses of study were
in fact a little inferior in achievement to the other thirteen, and the
correlation coefficients were slightly negative.
Stone also compared eighteen systems where there was supervision of the
work by superintendents or supervisors as well as by principals with
four systems where the principals and teachers had no such help. The
scores in his tests were very much lower in the four latter cities.
THE HYGIENE OF THE EYES IN ARITHMETIC
We have already noted that the task of reading and copying numbers is
one of the hardest that the eyes have to perform in the elementary
school, and that it should be alleviated by arranging much of the work
so that only answers need be written by the pupil. The figures to be
read and copied should obviously be in type of suitable size and style,
so arranged and spaced on the page or blackboard as to cause a minimum
of effort and strain.
[Illustration: FIG. 26.--Type too large.]
[Illustration: FIG. 27.--12-point, 11-point, and 10-point type.]
_Size._--Type may be too large as well as too small, though the latter
is the commoner error. If it is too large, as in Fig. 26, which is a
duplicate of type actually used in a form of practice pad, the eye has
to make too many fixations to take in a given content. All things
considered, 12-point type in grades 3 and 4, 11-point in grades 5 and 6,
and 10-point in grades 7 and 8 seem the most desirable sizes. These are
shown in Fig. 27. Too small type occurs oftenest in fractions and in the
dimension-numbers or scale numbers of drawings. Figures 28, 29, and 30
are samples from actual school practice. Samples of the desirable size
are shown in Figs. 31 and 32. The technique of modern typesetting makes
it very difficult and expensive to make fractions of the horizontal type
(1 3 5
- - -
4, 8, 6)
large enough without making the whole-number figures with which they
are mingled too large or giving an uncouth appearance to the total.
Consequently fractions somewhat smaller than are desirable may have
to be used occasionally in textbooks.[19] There is no valid excuse,
however, for the excessively small fractions which often are made in
blackboard work.
[19] A special type could be constructed that would use a large
type body, say 14 point, with integers in 10 or 12 point and
fractions much larger than now.
[Illustration: FIG. 28.--Type of measurements too small.
This is a picture of Mary's garden. How many feet is it
around the garden?]
[Illustration: FIG. 29.--Type too small.]
[Illustration: FIG. 30.--Numbers too small and badly designed.]
[Illustration: FIG. 31.--Figure 28 with suitable numbers.]
[Illustration: FIG. 32.--Figure 30 with suitable numbers.]
_Style._--The ordinary type forms often have 3 and 8 so made as to
require strain to distinguish them. 5 is sometimes easily confused with
3 and even with 8. 1, 4, and 7 may be less easily distinguishable than
is desirable. Figure 33 shows a specially good type in which each figure
is represented by its essential[20] features without any distracting
shading or knobs or turns. Figure 34 shows some of the types in common
use. There are no demonstrably great differences amongst these. In
fractions there is a notable gain from using the slant form (2/3, 3/4)
for exercises in addition and subtraction, and for almost all mixed
numbers. This appears clearly to the eye in the comparison of Fig. 35
below, where the same fractions all in 10-point type are displayed in
horizontal and in slant form. The figures in the slant form are in
general larger and the space between them and the fraction-line is
wider. Also the slant form makes it easier for the eye to examine the
denominators to see whether reductions are necessary. Except for a few
cases to show that the operations can be done just as truly with the
horizontal forms, the book and the blackboard should display mixed
numbers and fractions to be added or subtracted in the slant form. The
slant line should be at an angle of approximately 45 degrees. Pupils
should be taught to use this form in their own work of this sort.
[20] It will be still better if the 4 is replaced by an open-top 4.
When script figures are presented they should be of simple design,
showing clearly the essential features of the figure, the line being
everywhere of equal or nearly equal width (that is, without shading, and
without ornamentation or eccentricity of any sort). The opening of the 3
should be wide to prevent confusion with 8; the top of the 3 should be
curved to aid its differentiation from 5; the down stroke of the 9
should be almost or quite straight; the 1, 4, 7, and 9 should be clearly
distinguishable. There are many ways of distinguishing them clearly, the
best probably being to use the straight line for 1, the open 4 with
clear angularity, a wide top to the 7, and a clearly closed curve for
the top of the 9.
[Illustration: FIG. 33.--Block type; a very desirable type except
that it is somewhat too heavy.]
[Illustration: FIG. 34.--Common styles of printed numbers.]
[Illustration: FIG. 35.--Diagonal and horizontal fractions
compared.]
[Illustration: FIG. 36.--Good vertical spacing.]
[Illustration: FIG. 37.--Bad vertical spacing.]
[Illustration: FIGS. 38 (above) and 39 (below).--Good and bad
left-right spacing.]
The pupil's writing of figures should be clear. He will thereby be saved
eyestrain and errors in his school work as well as given a valuable
ability for life. Handwriting of figures is used enormously in spite of
the development of typewriters; illegible figures are commonly more
harmful than illegible letters or words, since the context far less
often tells what the figure is intended to be; the habit of making clear
figures is not so hard to acquire, since they are written unjoined and
require only the automatic action of ten minor acts of skill. The
schools have missed a great opportunity in this respect. Whereas the
hand writing of words is often better than it needs to be for life's
purposes, the writing of figures is usually much worse. The figures
presented in books on penmanship are also commonly bad, showing neglect
or misunderstanding of the matter on the part of leaders in penmanship.
_Spacing._--Spacing up and down the column is rarely too wide, but very
often too narrow. The specimens shown in Figs. 36 and 37 show good
practice contrasted with the common fault.
Spacing from right to left is generally fairly satisfactory in books,
though there is a bad tendency to adopt some one routine throughout and
so to miss chances to use reductions and increases of spacing so as to
help the eye and the mind in special cases. Specimens of good and bad
spacing are shown in Figs. 38 and 39. In the work of the pupils, the
spacing from right to left is often too narrow. This crowding of
letters, together with unevenness of spacing, adds notably to the task
of eye and mind.
_The composition or make-up of the page._--Other things being equal,
that arrangement of the page is best which helps a child most to keep
his place on a page and to find it after having looked away to work on
the paper on which he computes, or for other good reasons. A good page
and a bad page in this respect are shown in Figs. 40 and 41.
[Illustration: FIG. 40.--A page well made up to suit the action
of the eye.]
[Illustration: FIG. 41.--The same matter as in Fig. 40, much
less well made up.]
_Objective presentations._--Pictures, diagrams, maps, and other
presentations should not tax the eye unduly,
(_a_) by requiring too fine distinctions, or
(_b_) by inconvenient arrangement of the data, preventing easy
counting, measuring, comparison, or whatever the task is, or
(_c_) by putting too many facts in one picture so that the eye
and mind, when trying to make out any one, are confused by the
others.
Illustrations of bad practices in these respects are shown in Figs. 42
to 52. A few specimens of work well arranged for the eye are shown in
Figs. 53 to 56.
Good rules to remember are:--
Other things being equal, make distinctions by the clearest method, fit
material to the tendency of the eye to see an 'eyeful' at a time
(roughly 1-1/2 inch by 1/2 inch in a book; 1-1/2 ft. by 1/2 ft. on the
blackboard), and let one picture teach only one fact or relation, or
such facts and relations as do not interfere in perception.
The general conditions of seating, illumination, paper, and the like are
even more important when the eyes are used with numbers than when they
are used with words.
[Illustration: FIG. 42.--Try to count the rungs on the ladder,
or the shocks in the wagon.]
[Illustration: FIG. 43.--How many oars do you see? How many
birds? How many fish?]
[Illustration: FIG. 44.--Count the birds in each of the three
flocks of birds.]
[Illustration: FIG. 45.--Note the lack of clear division of the
hundreds. Consider the difficulty of counting one of these
columns of dots.]
[Illustration: FIG. 46.--What do you suppose these pictures are
intended to show?]
[Illustration: FIG. 47.--Would a beginner know that after
THIRTEEN he was to switch around and begin at the other end?
Could you read the SIX of TWENTY-SIX if you did not already know
what it ought to be? What meaning would all the brackets have
for a little child in grade 2? Does this picture illustrate or
obfuscate?]
[Illustration: FIG. 48.--How long did it take you to find out
what these pictures mean?]
[Illustration: FIG. 49.--Count the figures in the first row,
using your eyes alone; have some one make lines of 10, 11, 12,
13, and more repetitions of this figure spaced closely as here.
Count 20 or 30 such lines, using the eye unaided by fingers,
pencil, etc. ]
[Illustration: FIG. 50.--Can you answer the question without
measuring? Could a child of seven or eight?]
[Illustration: FIG. 51.--What are these drawings intended to
show? Why do they show the facts only obscurely and dubiously?]
[Illustration: FIG. 52.--What are these drawings intended to
show? What simple change would make them show the facts much
more clearly?]
[Illustration: FIG. 53.--Arranged in convenient "eye-fulls."]
[Illustration: FIG. 54.--Clear, simple, and easy of comparison.]
[Illustration: FIG. 55.--Clear, simple, and well spaced.]
[Illustration: FIG. 56.--Well arranged, though a little wider
spacing between the squares would make it even better.]
THE USE OF CONCRETE OBJECTS IN ARITHMETIC
We mean by concrete objects actual things, events, and relations
presented to sense, in contrast to words and numbers and symbols which
mean or stand for these objects or for more abstract qualities and
relations. Blocks, tooth-picks, coins, foot rules, squared paper, quart
measures, bank books, and checks are such concrete things. A foot rule
put successively along the three thirds of a yard rule, a bell rung five
times, and a pound weight balancing sixteen ounce weights are such
concrete events. A pint beside a quart, an inch beside a foot, an apple
shown cut in halves display such concrete relations to a pupil who is
attentive to the issue.
Concrete presentations are obviously useful in arithmetic to teach
meanings under the general law that a word or number or sign or symbol
acquires meaning by being connected with actual things, events,
qualities, and relations. We have also noted their usefulness as means
to verifying the results of thinking and computing, as when a pupil,
having solved, "How many badges each 5 inches long can be made from
3-1/3 yd. of ribbon?" by using 10 x 12/5, draws a line 3-1/3 yd. long
and divides it into 5-inch lengths.
Concrete experiences are useful whenever the meaning of a number, like 9
or 7/8 or .004, or of an operation, like multiplying or dividing or
cubing, or of some term, like rectangle or hypothenuse or discount, or
some procedure, like voting or insuring property against fire or
borrowing money from a bank, is absent or incomplete or faulty. Concrete
work thus is by no means confined to the primary grades but may be
appropriate at all stages when new facts, relations, and procedures are
to be taught.
How much concrete material shall be presented will depend upon the fact
or relation or procedure which is to be made intelligible, and the
ability and knowledge of the pupil. Thus 'one half' will in general
require less concrete illustration than 'five sixths'; and five sixths
will require less in the case of a bright child who already knows 2/3,
3/4, 3/8, 5/8, 7/8, 2/5, 3/5, and 4/5 than in the case of a dull child
or one who only knows 2/3 and 3/4. As a general rule the same topic will
require less concrete material the later it appears in the school
course. If the meanings of the numbers are taught in grade 2 instead of
grade 1, there will be less need of blocks, counters, splints, beans,
and the like. If 1-1/2 + 1/2 = 2 is taught early in grade 3, there will
be more gain from the use of 1-1/2 inches and 1/2 inch on the foot rule
than if the same relations were taught in connection with the general
addition of like fractions late in grade 4. Sometimes the understanding
can be had either by connecting the idea with the reality directly, or
by connecting the two indirectly _via_ some other idea. The amount of
concrete material to be used will depend on its relative advantage per
unit of time spent. Thus it might be more economical to connect 5/12,
7/12, and 11/12 with real meanings indirectly by calling up the
resemblance to the 2/3, 3/4, 3/8, 5/8, 7/8, 2/5, 3/5, 4/5, and 5/6
already studied, than by showing 5/12 of an apple, 7/12 of a yard, 11/12
of a foot, and the like.
In general the economical course is to test the understanding of the
matter from time to time, using more concrete material if it is needed,
but being careful to encourage pupils to proceed to the abstract ideas
and general principles as fast as they can. It is wearisome and
debauching to pupils' intellects for them to be put through elaborate
concrete experiences to get a meaning which they could have got
themselves by pure thought. We should also remember that the new idea,
say of the meaning of decimal fractions, will be improved and clarified
by using it (see page 183 f.), so that the attainment of a _perfect_
conception of decimal fractions before doing anything with them is
unnecessary and probably very wasteful.
A few illustrations may make these principles more instructive.
(_a_) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000,
need more concrete aids than are commonly given. Guessing contests about
the value in dollars of the school building and other buildings, the
area of the schoolroom floor and other surfaces in square inches, the
number of minutes in a week, and year, and the like, together with
proper computations and measurements, are very useful to reenforce the
concrete presentations and supply genuine problems in multiplication and
subtraction with large numbers.
(_b_) Numbers very much smaller than one, such as 1/32, 1/64, .04,
and .002, also need some concrete aids. A diagram like that of
Fig. 57 is useful.
(_c_) _Majority_ and _plurality_ should be understood by every citizen.
They can be understood without concrete aid, but an actual vote is well
worth while for the gain in vividness and surety.
[Illustration: FIG. 57.--Concrete aid to understanding fractions
with large denominators. A = 1/1000 sq. ft.; B = 1/100 sq. ft.;
C = 1/50 sq. ft.; D = 1/10 sq. ft.]
(_d_) Insurance against loss by fire can be taught by explanation and
analogy alone, but it will be economical to have some actual insuring
and payment of premiums and a genuine loss which is reimbursed.
(_e_) Four play banks in the corners of the room, receiving deposits,
cashing checks, and later discounting notes will give good educational
value for the time spent.
(_f_) Trade discount, on the contrary, hardly requires more concrete
illustration than is found in the very problems to which it is applied.
(_g_) The process of finding the number of square units in a rectangle
by multiplying with the appropriate numbers representing length and
width is probably rather hindered than helped by the ordinary objective
presentation as an introduction. The usual form of objective
introduction is as follows:--
[Illustration: FIG. 58.]
How long is this rectangle? How large is each square? How many
square inches are there in the top row? How many rows are
there? How many square inches are there in the whole rectangle?
Since there are three rows each containing 4 square inches, we
have 3 x 4 square inches = 12 square inches.
Draw a rectangle 7 inches long and 2 inches wide. If you divide
it into inch squares how many rows will there be? How many inch
squares will there be in each row? How many square inches are
there in the rectangle?
[Illustration: FIG. 59.]
It is better actually to hide the individual square units as in Fig. 59.
There are four reasons: (1) The concrete rows and columns rather
distract attention from the essential thing to be learned. This is not
that "_x_ rows one square wide, _y_ squares in a row will make _xy_
squares in all," but that "by using proper units and the proper
operation the area of any rectangle can be found from its length and
width." (2) Children have little difficulty in learning to multiply
rather than add, subtract, or divide when computing area. (3) The habit
so formed holds good for areas like 1-2/3 by 4-1/2, with fractional
dimensions, in which any effort to count up the areas of rows is very
troublesome and confusing. (4) The notion that a square inch is an area
1' by 1' rather than 1/2' by 2' or 1/3 in. by 3 in. or 1-1/2 in. by 2/3
in. is likely to be formed too emphatically if much time is spent upon
the sort of concrete presentation shown above. It is then better to use
concrete counting of rows of small areas as a means of _verification
after_ the procedure is learned, than as a means of deriving it.
There has been, especially in Germany, much argument concerning what
sort of number-pictures (that is, arrangement of dots, lines, or the
like, as shown in Fig. 60) is best for use in connection with the number
names in the early years of the teaching of arithmetic.
Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and
others have measured the accuracy of children in estimating the number
of dots in arrangements of one or more of these different types.[21]
Many writers interpret a difference in favor of estimating, say, the
square arrangements of Born or Lay as meaning that such is the best
arrangement to use in teaching. The inference is, however, unjustified.
That certain number-pictures are easier to estimate numerically does not
necessarily mean that they are more instructive in learning. One set may
be easier to estimate just because they are more familiar, having been
oftener experienced. Even if the favored set was so after equal
experience with all sets, accuracy of estimation would be a sign of
superiority for use in instruction only if all other things were equal
(or in favor of the arrangement in question). Obviously the way to
decide which of these is best to use in teaching is by using them in
teaching and measuring all relevant results, not by merely recording
which of them are most accurately estimated in certain time exposures.
[21] For an account in English of their main findings see
Howell ['14], pp. 149-251.
It may be noted that the Born, Lay, and Freeman pictures have claims for
special consideration on grounds of probable instructiveness. Since they
are also superior in the tests in respect to accuracy of estimate,
choice should probably be made from these three by any teacher who
wishes to connect one set of number-pictures systematically with the
number names, as by drills with the blackboard or with cards.
[Illustration: FIG. 60.--Various proposed arrangements of dots
for use in teaching the meanings of the numbers 1 to 10.]
Such drills are probably useful if undertaken with zeal, and if kept as
supplementary to more realistic objective work with play money, children
marching, material to be distributed, garden-plot lengths to be
measured, and the like, and if so administered that the pupils soon get
the generalized abstract meaning of the numbers freed from dependence on
an inner picture of any sort. This freedom is so important that it may
make the use of many types of number-pictures advisable rather than the
use of the one which in and of itself is best.
As Meumann says: "Perceptual reckoning can be overdone. It had its chief
significance for the surety and clearness of the first foundation of
arithmetical instruction. If, however, it is continued after the first
operations become familiar to the child, and extended to operations
which develop from these elementary ones, it necessarily works as a
retarding force and holds back the natural development of arithmetic.
This moves on to work with abstract number and with mechanical
association and reproduction." ['07, Vol. 2, p. 357.]
Such drills are commonly overdone by those who make use of them, being
given too often, and continued after their instructiveness has waned,
and used instead of more significant, interesting, and varied work in
counting and estimating and measuring real things. Consequently, there
is now rather a prejudice against them in our better schools. They
should probably be reinstated but to a moderate and judicious use.
ORAL, MENTAL, AND WRITTEN ARITHMETIC
There has been much dispute over the relative merits of oral and written
work in arithmetic--a question which is much confused by the different
meanings of 'oral' and 'written.' _Oral_ has meant (1) work where the
situations are presented orally and the pupil's final responses are
given orally, or (2) work where the situations are presented orally and
the pupils' final responses are written or partly written and partly
oral, or (3) work where the situations are presented in writing or print
and the final responses are oral. _Written_ has meant (1) work where the
situations are presented in writing or print and the final responses are
made in writing, or (2) work where also many of the intermediate
responses are written, or (3) work where the situations are presented
orally but the final responses and a large percentage of the
intermediate computational responses are written. There are other
meanings than these.
It is better to drop these very ambiguous terms and ask clearly what are
the merits and demerits, in the case of any specified arithmetical work,
of auditory and of visual presentation of the situations, and of saying
and of writing each specified step in the response.
The disputes over mental _versus_ written arithmetic are also confused
by ambiguities in the use of 'mental.' Mental has been used to mean
"done without pencil and paper" and also "done with few overt
responses, either written or spoken, between the setting of the task and
the announcement of the answer." In neither case is the word _mental_
specially appropriate as a description of the total fact. As before, we
should ask clearly, "What are the merits and demerits of making certain
specified intermediate responses in inner speech or imaged sounds or
visual images or imageless thought--that is, _without_ actual writing or
overt speech?"
It may be said at the outset that oral, written, and inner presentations
of initial situations, oral, written, and inner announcements of final
responses, and oral, written, and inner management of intermediate
processes have varying degrees of merit according to the particular
arithmetical exercise, pupil, and context. Devotion to oralness or
mentalness as such is simply fanatical. Various combinations, such as
the written presentation of the situation with inner management of the
intermediate responses and oral announcement of the final response have
their special merits for particular cases.
These merits the reader can evaluate for himself for any given sort of
work for a given class by considering: (1) The amount of practice
received by the class per hour spent; (2) the ease of correction of the
work; (3) the ease of understanding the tasks; (4) the prevention of
cheating; (5) the cheerfulness and sociability of the work; (6) the
freedom from eyestrain, and other less important desiderata.
It should be noted that the stock schemes A, B, C, and D below are only
a few of the many that are possible and that schemes E, F, G, and H have
special merits.
PRESENTATION OF MANAGEMENT OF ANNOUNCEMENT OF
INITIAL SITUATION INTERMEDIATE PROCESSES FINAL RESPONSE
A. Printed or written Written Written
B. " " Inner Oral by one pupil,
inner by the rest
C. Oral (by teacher) Written Written
D. " " Inner Oral by one pupil,
inner by the rest
E. As in A or C A mixture, the pupil As in A or B or H
writing what he needs
F. The real situation As in E As in A or B or H
itself, in part at
least
G. Both read by the pupil As in E As in A or B or H
and put orally by the
teacher
H. As in A or C or G As in E Written by all
pupils, announced
orally by one pupil
The common practice of either having no use made of pencil and paper or
having all computations and even much verbal analysis written out
elaborately for examination is unfavorable for learning. The demands
which life itself will make of arithmetical knowledge and skill will
range from tasks done with every percentage of written work from zero up
to the case where every main result obtained by thought is recorded for
later use by further thought. In school the best way is that which, for
the pupils in question, has the best total effect upon quality of
product, speed, and ease of production, reenforcement of training
already given, and preparation for training to be given. There is
nothing intellectually criminal about using a pencil as well as inner
thought; on the other hand there is no magical value in writing out for
the teacher's inspection figures that the pupil does not need in order
to attain, preserve, verify, or correct his result.
The common practice of having the final responses of all _easy_ tasks
given orally has no sure justification. On the contrary, the great
advantage of having all pupils really do the work should be secured in
the easy work more than anywhere else. If the time cost of copying the
figures is eliminated by the simple plan of having them printed, and if
the supervision cost of examining the papers is eliminated by having the
pupils correct each other's work in these easy tasks, written answers
are often superior to oral except for the elements of sociability and
'go' and freedom from eyestrain of the oral exercise. Such written work
provides the gifted pupils with from two to ten times as much practice
as they would get in an oral drill on the same material, supposing them
to give inner answers to every exercise done by the class as a whole; it
makes sure that the dull pupils who would rarely get an inner answer at
the rate demanded by the oral exercise, do as much as they are able to
do.
Two arguments often made for the oral statement of problems by the
teacher are that problems so put are better understood, especially in
the grades up through the fifth, and that the problems are more likely
to be genuine and related to the life the pupils know. When these
statements are true, the first is a still better argument for having the
pupils read the problems _aided by the teacher's oral statement of
them_. For the difficulty is largely that the pupils cannot read well
enough; and it is better to help them to surmount the difficulty rather
than simply evade it. The second is not an argument for oralness
_versus_ writtenness, but for good problems _versus_ bad; the teacher
who makes up such good problems should, in fact, take special care to
write them down for later use, which may be by voice or by the
blackboard or by printed sheet, as is best.
CHAPTER XIV
THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE
Dewey, and others following him, have emphasized the desirability of
having pupils do their work as active seekers, conscious of problems
whose solution satisfies some real need of their own natures. Other
things being equal, it is unwise, they argue, for pupils to be led along
blindfold as it were by the teacher and textbook, not knowing where they
are going or why they are going there. They ought rather to have some
living purpose, and be zealous for its attainment.
This doctrine is in general sound, as we shall see, but it is often
misused as a defense of practices which neglect the formation of
fundamental habits, or as a recommendation to practices which are quite
unworkable under ordinary classroom conditions. So it seems probable
that its nature and limitations are not thoroughly known, even to its
followers, and that a rather detailed treatment of it should be given
here.
ILLUSTRATIVE CASES
Consider first some cases where time spent in making pupils understand
the end to be attained before attacking the task by which it is
attained, or care about attaining the end (well or ill understood) is
well spent.
It is well for a pupil who has learned (1) the meanings of the numbers
one to ten, (2) how to count a collection of ten or less, and (3) how to
measure in inches a magnitude of ten, nine, eight inches, etc., to be
confronted with the problem of true adding without counting or
measuring, as in 'hidden' addition and measurement by inference. For
example, the teacher has three pencils counted and put under a book; has
two more counted and put under the book; and asks, "How many pencils are
there under the book?" Answers, when obtained, are verified or refuted
by actual counting and measuring.
The time here is well spent because the children can do the necessary
thinking if the tasks are well chosen; because they are thereby
prevented from beginning their study of addition by the bad habit of
pseudo-adding by looking at the two groups of objects and counting their
number instead of real adding, that is, thinking of the two numbers and
inferring their sum; and further, because facing the problem of adding
as a real problem is in the end more economical for learning arithmetic
and for intellectual training in general than being enticed into adding
by objective or other processes which conceal the difficulty while
helping the pupil to master it.
The manipulation of short multiplication may be introduced by
confronting the pupils with such problems as, "How to tell how many
Uneeda biscuit there are in four boxes, by opening only one box."
Correct solutions by addition should be accepted. Correct solutions by
multiplication, if any gifted children think of this way, should be
accepted, even if the children cannot justify their procedure.
(Inferring the manipulation from the place-values of numbers is beyond
all save the most gifted and probably beyond them.) Correct solution by
multiplication by some child who happens to have learned it elsewhere
should be accepted. Let the main proof of the trustworthiness of the
manipulation be by measurement and by addition. Proof by the stock
arguments from the place-values of numbers may also be used. If no child
hits on the manipulation in question, the problem of finding the length
_without_ adding may be set. If they still fail, the problem may be made
easier by being put as "4 times 22 gives the answer. Write down what you
think 4 times 22 will be." Other reductions of the difficulty of the
problem may be made, or the teacher may give the answer without very
great harm being done. The important requirement is that the pupils
should be aware of the problem and treat the manipulation as a solution
of it, not as a form of educational ceremonial which they learn to
satisfy the whims of parents and teachers. In the case of any particular
class a situation that is more appealing to the pupils' practical
interests than the situation used here can probably be devised.
The time spent in this way is well spent (1) because all but the very
dull pupils can solve the problem in some way, (2) because the
significance of the manipulation as an economy over addition is worth
bringing out, and (3) because there is no way of beginning training in
short multiplication that is much better.
In the same fashion multiplication by two-place numbers may be
introduced by confronting pupils with the problem of the number of
sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square
inches in 35 square feet, or the number of days in 32 years, or whatever
similar problem can be brought up so as to be felt as a problem.
Suppose that it is the 35 square feet. Solutions by (5 x 144) +
(30 x 144), however arranged, or by (10 x 144) + (10 x 144) +
(10 x 144) + (5 x 144), or by 3500 + (35 x 40) + (35 x 4), or by
7 x (5 x 144), however arranged, should all be listed for verification
or rejection. The pupils need not be required to justify their
procedures by a verbal statement. Answers like 432,720, or 720,432,
or 1152, or 4220, or 3220 should be listed for verification or
rejection. Verification may be by a mixture of short multiplication
and objective work, or by a mixture of short multiplication and
addition, or by addition abbreviated by taking ten 144s as 1440, or
even (for very stupid pupils) by the authority of the teacher. Or the
manipulation in cases like 53 x 9 or 84 x 7 may be verified by the
reverse short multiplication. The deductive proof of the correctness
of the manipulation may be given in whole or in part in connection
with exercises like
10 x 2 = 30 x 14 =
10 x 3 = 3 x 44 =
10 x 4 = 30 x 44 =
10 x 14 = 3 x 144 =
10 x 44 = 20 x 144 =
10 x 144 = 40 x 144 =
20 x 2 = 30 x 144 =
20 x 3 = 5 x 144 =
30 x 3 = 35 = 30 + ....
30 x 4 = 30 x 144 added to 5 x 144 =
3 x 14 =
Certain wrong answers may be shown to be wrong in many ways; _e.g._,
432,720 is too big, for 35 times a thousand square inches is only
35,000; 1152 is too small, for 35 times a hundred square inches would be
3500, or more than 1152.
The time spent in realizing the problem here is fairly well spent
because (1) any successful original manipulation in this case
represents an excellent exercise of thought, because (2) failures show
that it is useless to juggle the figures at random, and because (3) the
previous experience with short multiplication makes it possible for the
pupils to realize the problem in a very few minutes. It may, however, be
still better to give the pupils the right method just as soon as the
problem is realized, without having them spend more time in trying to
solve it. Thus:--
1 square foot has 144 square inches. How many square inches are there in
35 square feet (marked out in chalk on the floor as a piece 10 ft. x 3
ft. plus a piece 5 ft. x 1 ft.)?
1 yard = 36 inches. How many inches long is this wall (found by measure
to be 13 yards)?
Here is a quick way to find the answers:--
144
35
----
720
432
----
5040 sq. inches in 35 sq. ft.
36
13
---
108
36
---
468 inches in 13 yd.
Consider now the following introduction to dividing by a decimal:--
Dividing by a Decimal
1. How many minutes will it take a motorcycle, to go 12.675 miles
at the rate of .75 mi. per minute?
16.9
______
.75|12.675
7 5
---
5 17
4 50
----
675
675
---
2. Check by multiplying 16.9 by .75.
3. How do you know that the quotient cannot be as little as 1.69?
4. How do you know that the quotient cannot be as large as 169?
5. Find the quotient for 3.75 / 1.5.
6. Check your result by multiplying the quotient by the divisor.
7. How do you know that the quotient cannot be .25 or 25?
____
8. Look at this problem. .25|7.5
How do you know that 3.0 is wrong for the quotient?
How do you know that 300 is wrong for the quotient?
State which quotient is right for each of these:--
.021 or .21 or 2.1 or 21 or 210
______
9. 1.8|3.78
.021 or .21 or 21 or 210
______
10. 1.8|37.8
.03 or .3 or 3 or 30 or 300
______
11. 1.25|37.5
.03 or .3 or 3 or 30 or 300
______
12. 12.5|37.5
.05 or .5 or 5 or 50 or 500
______
13. 1.25|6.25
.05 or .5 or 5 or 50 or 500
______
14. 12.5|6.25
15. Is this rule true? If it is true, learn it.
#In a correct result, the number of decimal places in
the divisor and quotient together equals the number
of decimal places in the dividend.#
These and similar exercises excite the problem attitude in children _who
have a general interest in getting right answers_. Such a series
carefully arranged is a desirable introduction to a statement of the
rule for placing the decimal point in division with decimals. For it
attracts attention to the general principle (divisor x quotient should
equal dividend), which is more important than the rule for convenient
location of the decimal point, and it gives training in placing the
point by inspection of the divisor, quotient, and dividend, which
suffices for nineteen out of twenty cases that the pupil will ever
encounter outside of school. He is likely to remember this method by
inspection long after he has forgotten the fixed rule.
It is well for the pupil to be introduced to many arithmetical facts by
way of problems about their common uses. The clockface, the railroad
distance table in hundredths of a mile, the cyclometer and speedometer,
the recipe, and the like offer problems which enlist his interest and
energy and also connect the resulting arithmetical learning with the
activities where it is needed. There is no time cost, but a time-saving,
for the learning as a means to the solution of the problems is quicker
than the mere learning of the arithmetical facts by themselves alone. A
few samples of such procedure are shown below:--
GRADE 3
To be Done at Home
Look at a watch. Has it any hands besides the hour hand and the
minute hand? Find out all that you can about how a watch tells
seconds, how long a second is, and how many seconds make a minute.
GRADE 5
Measuring Rainfall
=Rainfall per Week=
(=cu. in. per sq. in. of area=)
June 1-7 1.056
8-14 1.103
15-21 1.040
22-28 .960
29-July 5 .915
July 6-12 .782
13-19 .790
20-26 .670
27-Aug. 2 .503
Aug. 3-9 .512
10-16 .240
17-23 .215
24-30 .811
1. In which weeks was the rainfall 1 or more?
2. Which week of August had the largest rainfall for that month?
3. Which was the driest week of the summer? (Driest means with
the least rainfall.)
4. Which week was the next to the driest?
5. In which weeks was the rainfall between .800 and 1.000?
6. Look down the table and estimate whether the average rainfall for
one week was about .5, or about .6, or about .7, or about .8, or
about .9.
Dairy Records
=Record of Star Elsie=
Pounds of Milk Butter-Fat per Pound of Milk
Jan. 1742 .0461
Feb. 1690 .0485
Mar. 1574 .0504
Apr. 1226 .0490
May 1202 .0466
June 1251 .0481
Read this record of the milk given by the cow Star Elsie. The first
column tells the number of pounds of milk given by Star Elsie each
month. The second column tells what fraction of a pound of butter-fat
each pound of milk contained.
1. Read the first line, saying, "In January this cow gave 1742 pounds
of milk. There were 461 ten thousandths of a pound of butter-fat
per pound of milk." Read the other lines in the same way.
2. How many pounds of butter-fat did the cow produce in Jan.?
3. In Feb.? 4. In Mar.? 5. In Apr.? 6. In May? 7. In June?
GRADE 5 OR LATER
Using Recipes to Make Larger or Smaller Quantities
I. State how much you would use of each material in the following
recipes: (_a_) To make double the quantity. (_b_) To make half the
quantity. (_c_) To make 1-1/2 times the quantity. You may use pencil
and paper when you cannot find the right amount mentally.
1. PEANUT PENUCHE
1 tablespoon butter
2 cups brown sugar
1/3 cup milk or cream
3/4 cup chopped peanuts
1/3 teaspoon salt
2. MOLASSES CANDY
1/2 cup butter
2 cups sugar
1 cup molasses
1-1/2 cups boiling water
3. RAISIN OPERA CARAMELS
2 cups light brown sugar
7/8 cup thin cream
1/2 cup raisins
4. WALNUT MOLASSES SQUARES
2 tablespoons butter
1 cup molasses
1/3 cup sugar
1/2 cup walnut meats
5. RECEPTION ROLLS
1 cup scalded milk
1-1/2 tablespoons sugar
1 teaspoon salt
1/4 cup lard
1 yeast cake
1/4 cup lukewarm water
White of 1 egg
3-1/2 cups flour
6. GRAHAM RAISED LOAF
2 cups milk
6 tablespoons molasses
1-1/2 teaspoons salt
1/3 yeast cake
1/4 cup lukewarm water
2 cups sifted Graham flour
1/2 cup Graham bran
3/4 cup flour (to knead)
II. How much would you use of each material in the following recipes:
(_a_) To make 2/3 as large a quantity? (_b_) To make 1-1/3 times as
much? (_c_) To make 2-1/2 times as much?
1. ENGLISH DUMPLINGS
1/2 pound beef suet
1-1/4 cups flour
3 teaspoons baking powder
1 teaspoon salt
1/2 teaspoon pepper
1 teaspoon minced parsley
1/2 cup cold water
2. WHITE MOUNTAIN ANGEL CAKE
1-1/2 cups egg whites
1-1/2 cups sugar
1 teaspoon cream of tartar
1 cup bread flour
1/4 teaspoon salt
1 teaspoon vanilla
In many cases arithmetical facts and principles can be well taught in
connection with some problem or project which is not arithmetical, but
which has special potency to arouse an intellectual activity in the
pupil which by some ingenuity can be directed to arithmetical learning.
Playing store is the most fundamental case. Planning for a party, seeing
who wins a game of bean bag, understanding the calendar for a month,
selecting Christmas presents, planning a picnic, arranging a garden, the
clock, the watch with second hand, and drawing very simple maps are
situations suggesting problems which may bring a living purpose into
arithmetical learning in grade 2. These are all available under ordinary
conditions of class instruction. A sample of such problems for a higher
grade (6) is shown below.
Estimating Areas
The children in the geography class had a contest in estimating
the areas of different surfaces. Each child wrote his estimates
for each of these maps, A, B, C, D, and E. (Only C and D are
shown here.) In the arithmetic class they learned how to find
the exact areas. Then they compared their estimates with the
exact areas to find who came nearest.
[Illustration]
Write your estimates for A, B, C, D, and E. Then study the
next 6 pages and learn how to find the exact areas.
(The next 6 pages comprise training in the mensuration of
parallelograms and triangles.)
In some cases the affairs of individual pupils include problems which
may be used to guide the individual in question to a zealous study of
arithmetic as a means of achieving his purpose--of making a canoe,
surveying an island, keeping the accounts of a Girls' Canning Club, or
the like. It requires much time and very great skill to direct the work
of thirty or more pupils each busy with a special type of his own, so as
to make the work instructive for each, but in some cases the expense of
time and skill is justified.
GENERAL PRINCIPLES
In general what should be meant when one says that it is desirable to
have pupils in the problem-attitude when they are studying arithmetic is
substantially as follows:--
_First._--Information that comes as an answer to questions is better
attended to, understood, and remembered than information that just
comes.
_Second._--Similarly, movements that come as a step toward achieving an
end that the pupil has in view are better connected with their
appropriate situations, and such connections are longer retained, than
is the case with movements that just happen.
_Third._--The more the pupil is set toward getting the question answered
or getting the end achieved, the greater is the satisfyingness attached
to the bonds of knowledge or skill which mean progress thereto.
_Fourth._--It is bad policy to rely exclusively on the purely
intellectualistic problems of "How can I do this?" "How can I get the
right answer?" "What is the reason for this?" "Is there a better way to
do that?" and the like. It is bad policy to supplement these
intellectualistic problems by only the remote problems of "How can I be
fitted to earn a higher wage?" "How can I make sure of graduating?" "How
can I please my parents?" and the like. The purely intellectualistic
problems have too weak an appeal for many pupils; the remote problems
are weak so long as they are remote and, what is worse, may be deprived
of the strength that they would have in due time if we attempt to use
them too soon. It is the extreme of bad policy to neglect those personal
and practical problems furnished by life outside the class in arithmetic
the solution of which can really be furthered by arithmetic then and
there. It is good policy to spend time in establishing certain mental
sets--stimulating, or even creating, certain needs--setting up problems
themselves--when the time so spent brings a sufficient improvement in
the quality and quantity of the pupils' interest in arithmetical
learning.
_Fifth._--It would be still worse policy to rely exclusively on
problems arising outside arithmetic. To learn arithmetic is itself a
series of problems of intrinsic interest and worth to healthy-minded
children. The need for ability to multiply with United States money or
to add fractions or to compute percents may be as truly vital and
engaging as the need for skill to make a party dress or for money to buy
it or for time to play baseball. The intellectualistic needs and
problems should be considered along with all others, and given whatever
weight their educational value deserves.
DIFFICULTY AND SUCCESS AS STIMULI
There are certain misconceptions of the doctrine of the
problem-attitude. The most noteworthy is that difficulty--temporary
failure--an inadequacy of already existing bonds--is the essential and
necessary stimulus to thinking and learning. Dewey himself does not, as
I understand him, mean this, but he has been interpreted as meaning it
by some of his followers.[22]
[22] In his _How We Think_.
Difficulty--temporary failure, inadequacy of existing bonds--on the
contrary does nothing whatsoever in and of itself; and what is done by
the annoying lack of success which sometimes accompanies difficulty
sometimes hinders thinking and learning.
Mere difficulty, mere failure, mere inadequacy of existing bonds, does
nothing. It is hard for me to add three eight-place numbers at a glance;
I have failed to find as effective illustrations for pages 276 to 277 as
I wished; my existing sensori-motor connections are inadequate to
playing a golf course in 65. But these events and conditions have done
nothing to stimulate me in respect to the behavior in question. In the
first of the three there is no annoying lack and no dynamic influence at
all; in the second there was to some degree an annoying lack--a slight
irritation at not getting just what I wanted,--and this might have
impelled me to further thinking (though it did not, and getting one
tiptop illustration would as a rule stimulate me to hunt for others more
than failing to get such). In the third case the lack of the 65 does not
annoy me or have any noteworthy dynamic effect. The lack of 90 instead
of 95-100 is annoying and is at times a stimulus to further learning,
though not nearly so strong a stimulus as the attainment of the 90 would
be! At other times this annoying lack is distinctly inhibitory--a
stimulus to ceasing to learn. In the intellectual life the inhibitory
effect seems far the commoner of the two. Not getting answers seems as a
rule to make us stop trying to get them. The annoying lack of success
with a theoretical problem most often makes us desert it for problems to
whose solution the existing bonds promise to be more adequate.
The real issue in all this concerns the relative strength, in the
pupil's intellectual life, of the "negative reaction" of behavior in
general. An animal whose life processes are interfered with so that an
annoying state of affairs is set up, changes his behavior, making one
after another responses as his instincts and learned tendencies
prescribe, until the annoying state of affairs is terminated, or the
animal dies, or suffers the annoyance as less than the alternatives
which his responses have produced. When the annoying state of affairs is
characterized by the failure of things as they are to minister to a
craving--as in cases of hunger, loneliness, sex-pursuit, and the
like,--we have stimulus to action by an annoying lack or need, with
relief from action by the satisfaction of the need.
Such is in some measure true of man's intellectual life. In recalling a
forgotten name, in solving certain puzzles, or in simplifying an
algebraic complex, there is an annoying lack of the name, solution, or
factor, a trial of one after another response, until the annoyance is
relieved by success or made less potent by fatigue or distraction. Even
here the _difficulty_ does not do anything--but only the annoying
interference with our intellectual peace by the problem. Further,
although for the particular problem, the annoying lack stimulates, and
the successful attainment stops thinking, the later and more important
general effect on thinking is the reverse. Successful attainment stops
our thinking _on that problem_ but makes us more predisposed later to
thinking _in general_.
Overt negative reaction, however, plays a relatively small part in man's
intellectual life. Filling intellectual voids or relieving intellectual
strains in this way is much less frequent than being stimulated
positively by things seen, words read, and past connections acting under
modified circumstances. The notion of thinking as coming to a lack,
filling it, meeting an obstacle, dodging it, being held up by a
difficulty and overcoming it, is so one-sided as to verge on phantasy.
The overt lacks, strains, and difficulties come perhaps once in five
hours of smooth straightforward use and adaptation of existing
connections, and they might as truly be called hindrances to
thought--barriers which past successes help the thinker to surmount.
Problems themselves come more often as cherished issues which new facts
reveal, and whose contemplation the thinker enjoys, than as strains or
lacks or 'problems which I need to solve.' It is just as true that the
thinker gets many of his problems as results from, or bonuses along
with, his information, as that he gets much of his information as
results of his efforts to solve problems.
As between difficulty and success, success is in the long run more
productive of thinking. Necessity is not the mother of invention.
Knowledge of previous inventions is the mother; original ability is the
father. The solutions of previous problems are more potent in producing
both new problems and their solutions than is the mere awareness of
problems and desire to have them solved.
In the case of arithmetic, learning to cancel instead of getting the
product of the dividends and the product of the divisors and dividing
the former by the latter, is a clear case of very valuable learning,
with ease emphasized rather than difficulty, with the adequacy of
existing bonds (when slightly redirected) as the prime feature of the
process rather than their inadequacy, and with no sense of failure or
lack or conflict. It would be absurd to spend time in arousing in the
pupil, before beginning cancellation, a sense of a difficulty--viz.,
that the full multiplying and dividing takes longer than one would like.
A pupil in grade 4 or 5 might well contemplate that difficulty for years
to no advantage. He should at once begin to cancel and prove by checking
that errorless cancellation always gives the right answer. To emphasize
before teaching cancellation the inadequacy of the old full multiplying
and dividing would, moreover, not only be uneconomical as a means to
teaching cancellation; it would amount to casting needless slurs on
valuable past acquisitions, and it would, scientifically, be false.
For, until a pupil has learned to cancel, the old full multiplying is
not inadequate; it is admirable in every respect. The issue of its
inadequacy does not truly appear until the new method is found. It is
the best way until the better way is mastered.
In the same way it is unwise to spend time in making pupils aware of the
annoying lacks to be supplied by the multiplication tables, the division
tables, the casting out of nines, or the use of the product of the
length and breadth of a rectangle as its area, the unit being changed to
the square erected on the linear unit as base. The annoying lack will
be unproductive, while the learning takes place readily as a
modification of existing habits, and is sufficiently appreciated as soon
as it does take place. The multiplication tables come when instead of
merely counting by 7s from 0 up saying "7, 14, 21," etc., the pupil
counts by 7s from 0 up saying "Two sevens make 14, three sevens make 21,
four sevens make 28," etc. The division tables come as easy selections
from the known multiplications; the casting out of nines comes as an
easy device. The computation of the area of a rectangle is best
facilitated, not by awareness of the lack of a process for doing it, but
by awareness of the success of the process as verified objectively.
In all these cases, too, the pupil would be misled if we aroused first a
sense of the inadequacy of counting, adding, and objective division, an
awareness of the difficulties which the multiplication and division
tables and nines device and area theorem relieve. The displaced
processes are admirable and no unnecessary fault should be found with
them, and they are _not_ inadequate until the shorter ways have been
learned.
FALSE INFERENCES
One false inference about the problem-attitude is that the pupil should
always understand the aim or issue before beginning to form the bonds
which give the method or process that provides the solution. On the
contrary, he will often get the process more easily and value it more
highly if he is taught what it is _for_ gradually while he is learning
it. The system of decimal notation, for example, may better be taken
first as a mere fact, just as we teach a child to talk without trying
first to have him understand the value of verbal intercourse, or to keep
clean without trying first to have him understand the bacteriological
consequences of filth.
A second inference--that the pupil should always be taught to care about
an issue and crave a process for managing it before beginning to learn
the process--is equally false. On the contrary, the best way to become
interested in certain issues and the ways of handling them is to learn
the process--even to learn it by sheer habituation--and then note what
it does for us. Such is the case with ".1666-2/3 x = divide by 6,"
".333-1/3 x = divide by 3," "multiply by .875 = divide the number by 8
and subtract the quotient from the number."
A third unwise tendency is to degrade the mere giving of information--to
belittle the value of facts acquired in any other way than in the course
of deliberate effort by the pupil to relieve a problem or conflict or
difficulty. As a protest against merely verbal knowledge, and merely
memoriter knowledge, and neglect of the active, questioning search for
knowledge, this tendency to belittle mere facts has been healthy, but as
a general doctrine it is itself equally one-sided. Mere facts not got by
the pupil's thinking are often of enormous value. They may stimulate to
active thinking just as truly as that may stimulate to the reception of
facts. In arithmetic, for example, the names of the numbers, the use of
the fractional form to signify that the upper number is divided by the
lower number, the early use of the decimal point in U. S. money to
distinguish dollars from cents, and the meanings of "each," "whole,"
"part," "together," "in all," "sum," "difference," "product,"
"quotient," and the like are self-justifying facts.
A fourth false inference is that whatever teaching makes the pupil face
a question and think out its answer is thereby justified. This is not
necessarily so unless the question is a worthy one and the answer that
is thought out an intrinsically valuable one and the process of thinking
used one that is appropriate for that pupil for that question. Merely
to think may be of little value. To rely much on formal discipline is
just as pernicious here as elsewhere. The tendency to emphasize the
methods of learning arithmetic at the expense of what is learned is
likely to lead to abuses different in nature but as bad in effect as
that to which the emphasis on disciplinary rather than content value has
led in the study of languages and grammar, or in the old puzzle problems
of arithmetic.
The last false inference that I shall discuss here is the inference that
most of the problems by which arithmetical learning is stimulated had
better be external to arithmetic itself--problems about Noah's Ark or
Easter Flowers or the Merry Go Round or A Trip down the Rhine.
Outside interests should be kept in mind, as has been abundantly
illustrated in this volume, but it is folly to neglect the power, even
for very young or for very stupid children, of the problem "How can I
get the right answer?" Children do have intellectual interests. They do
like dominoes, checkers, anagrams, and riddles as truly as playing tag,
picking flowers, and baking cake. With carefully graded work that is
within their powers they like to learn to add, subtract, multiply, and
divide with integers, fractions, and decimals, and to work out
quantitative relations.
In some measure, learning arithmetic is like learning to typewrite. The
learner of the latter has little desire to present attractive-looking
excuses for being late, or to save expense for paper. He has no desire
to hoard copies of such and such literary gems. He may gain zeal from
the fact that a school party is to be given and invitations are to be
sent out, but the problem "To typewrite better" is after all his main
problem. Learning arithmetic is in some measure a game whose moves are
motivated by the general set of the mind toward victory--winning right
answers. As a ball-player learns to throw the ball accurately to
first-base, not primarily because of any particular problem concerning
getting rid of the ball, or having the man at first-base possess it, or
putting out an opponent against whom he has a grudge, but because that
skill is required by the game as a whole, so the pupil, in some measure,
learns the technique of arithmetic, not because of particular concrete
problems whose solutions it furnishes, but because that technique is
required by the game of arithmetic--a game that has intrinsic worth and
many general recommendations.
CHAPTER XV
INDIVIDUAL DIFFERENCES
The general facts concerning individual variations in abilities--that
the variations are large, that they are continuous, and that for
children of the same age they usually cluster around one typical or
modal ability, becoming less and less frequent as we pass to very high
or very low degrees of the ability--are all well illustrated by
arithmetical abilities.
NATURE AND AMOUNT
The surfaces of frequency shown in Figs. 61, 62, and 63 are samples. In
these diagrams each space along the baseline represents a certain score
or degree of ability, and the height of the surface above it represents
the number of individuals obtaining that score. Thus in Fig. 61, 63 out
of 1000 soldiers had no correct answer, 36 out of 1000 had one correct
answer, 49 had two, 55 had three, 67 had four, and so on, in a test with
problems (stated in words).
Figure 61 shows that these adults varied from no problems solved
correctly to eighteen, around eight as a central tendency. Figure 62
shows that children of the same year-age (they were also from the same
neighborhood and in the same school) varied from under 40 to over 200
figures correct. Figure 63 shows that even among children who have all
reached the same school grade and so had rather similar educational
opportunities in arithmetic, the variation is still very great. It
requires a range from 15 to over 30 examples right to include even nine
tenths of them.
[Illustration: FIG. 61.--The scores of 1000 soldiers in the
National Army born in English-speaking countries, in Test 2 of the
Army Alpha. The score is the number of correct answers obtained in
five minutes. Probably 10 to 15 percent of these men were unable to
read or able to read only very easy sentences at a very slow rate.
Data furnished by the Division of Psychology in the office of the
Surgeon General.]
It should, however, be noted that if each individual had been scored by
the average of his work on eight or ten different days instead of by his
work in just one test, the variability would have been somewhat less
than appears in Figs. 61, 62, and 63.
[Illustration: FIG. 62.--The scores of 100 11-year-old pupils
in a test of computation. Estimated from the data given by Burt
['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the
number of correct figures.]
It is also the case that if each individual had been scored, not in
problem-solving alone or division alone, but in an elaborate examination
on the whole field of arithmetic, the variability would have been
somewhat less than appears in Figs. 61, 62, and 63. On the other hand,
if the officers and the soldiers rejected for feeblemindedness had been
included in Fig. 61, if the 11-year-olds in special classes for the very
dull had been included in Fig. 62, and if all children who had been to
school six years had been included in Fig. 63, no matter what grade they
had reached, the effect would have been to _increase_ the variability.
[Illustration: FIG. 63.--The scores of pupils in grade 6 in city
schools in the Woody Division Test A. The score is the number of
correct answers obtained in 20 minutes. From Woody ['16, p. 61].]
In spite of the effort by school officers to collect in any one school
grade those somewhat equal in ability or in achievement or in a mixture
of the two, the population of the same grades in the same school system
shows a very wide range in any arithmetical ability. This is partly
because promotion is on a more general basis than arithmetical ability
so that some very able arithmeticians are deliberately held back on
account of other deficiencies, and some very incompetent arithmeticians
are advanced on account of other excellencies. It is partly because of
general inaccuracy in classifying and promoting pupils.
In a composite score made up of the sum of the scores in Woody
tests,--Add. A, Subt. A, Mult. A, and Div. A, and two tests in
problem-solving (ten and six graded problems, with maximum attainable
credits of 30 and 18), Kruse ['18] found facts from which I compute
those of Table 13, and Figs. 64 to 66, for pupils all having the
training of the same city system, one which sought to grade its pupils
very carefully.
[Illustration: FIGS. 64, 65, and 66.--The scores of pupils in
grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a
composite of tests in computation and problem-solving. The time
was about 120 minutes. The maximum score attainable was 196.]
The overlapping of grade upon grade should be noted. Of the pupils in
grade 6 about 18 percent do better than the average pupil in grade 7,
and about 7 percent do better than the average pupil in grade 8. Of the
pupils in grade 8 about 33 percent do worse than the average pupil in
grade 7 and about 12 percent do worse than the average pupil in grade 6.
TABLE 13
RELATIVE FREQUENCIES OF SCORES IN AN EXTENSIVE TEAM OF ARITHMETICAL
TESTS.[23] IN PERCENTS
==============================================
SCORE | GRADE 6 | GRADE 7 | GRADE 8
------------+-----------+-----------+---------
70 to 79 | 1.3 | .9 | .4
80 " 89 | 5.5 | 2.3 | .4
90 " 99 | 10.6 | 4.3 | 2.9
100 " 109 | 19.4 | 5.2 | 4.4
110 " 119 | 19.8 | 18.5 | 5.8
120 " 129 | 23.5 | 16.2 | 16.8
130 " 139 | 12.6 | 17.5 | 16.8
140 " 149 | 4.6 | 13.9 | 22.9
150 " 159 | 1.7 | 13.6 | 17.1
160 " 169 | 1.2 | 4.8 | 9.4
170 " 179 | | 2.5 | 3.3
==============================================
[23] Compiled from data on p. 89 of Kruse ['18].
DIFFERENCES WITHIN ONE CLASS
The variation within a single class for which a single teacher has to
provide is great. Even when teaching is departmental and promotion is by
subjects, and when also the school is a large one and classification
within a grade is by ability--there may be a wide range for any given
special component ability. Under ordinary circumstances the range is so
great as to be one of the chief limiting conditions for the teaching of
arithmetic. Many methods appropriate to the top quarter of the class
will be almost useless for the bottom quarter, and _vice versa_.
[Illustration: FIGS. 67 and 68.--The scores of ten 6 B classes in
a 12-minute test in computation with integers (the Courtis Test 7).
The score is the number of units done. Certain long tasks are
counted as two units.]
Figures 67 and 68 show the scores of ten classes taken at random from
ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of
computation done in 12 minutes. Observe the very wide variation present
in the case of every class. The variation within a class would be
somewhat reduced if each pupil were measured by his average in eight or
ten such tests given on different days. If a rather generous allowance
is made for this we still have a variation in speed as great as that
shown in Fig. 69, as the fact to be expected for a class of thirty-two 6
B pupils.
[Illustration: FIG. 69.--A conservative estimate of the amount of
variation to be expected within a single class of 32 pupils in
grade 6, in the number of units done in Courtis Test 7 when all
chance variations are eliminated.]
The variations within a class in respect to what processes are
understood so as to be done with only occasional errors may be
illustrated further as follows:--A teacher in grade 4 at or near the
middle of the year in a city doing the customary work in arithmetic will
probably find some pupil in her class who cannot do column addition even
without carrying, or the easiest written subtraction
(8 9 78)
(5 3 or 37)
(- - --),
who does not know his multiplication tables or how to derive them, or
understand the meanings of + - x and /, or have any useful ideas
whatever about division.
There will probably be some child in the class who can do such work as
that shown below, and with very few errors.
Add 3/8 + 5/8 + 7/8 + 1/8 2-1/2 1/6 + 3/8
6-3/8
3-3/4
-----
Subtract 10.00 4 yd. 1 ft. 6 in.
3.49 2 yd. 2 ft. 3 in.
----- ----------------------
Multiply 1-1/4 x 8 16 145
2-5/8 206
------ ---
_______ _____
Divide 2)13.50 25)9750
The invention of means of teaching thirty so different children at once
with the maximum help and minimum hindrance from their different
capacities and acquisitions is one of the great opportunities for
applied science.
Courtis, emphasizing the social demand for a certain moderate
arithmetical attainment in the case of nearly all elementary school
children of, say, grade 6, has urged that definite special means be
taken to bring the deficient children up to certain standards, without
causing undesirable 'overlearning' by the more gifted children. Certain
experimental work to this end has been carried out by him and others,
but probably much more must be done before an authoritative program for
securing certain minimum standards for all or nearly all pupils can be
arranged.
THE CAUSES OF INDIVIDUAL DIFFERENCES
The differences found among children of the same grade in the same city
are due in large measure to inborn differences in their original
natures. If, by a miracle, the children studied by Courtis, or by Woody,
or by Kruse had all received exactly the same nurture from birth to
date, they would still have varied greatly in arithmetical ability,
perhaps almost as much as they now do vary.
The evidence for this is the general evidence that variation in original
nature is responsible for much of the eventual variation found in
intellectual and moral traits, plus certain special evidence in the case
of arithmetical abilities themselves.
Thorndike found ['05] that in tests with addition and multiplication
twins were very much more alike than siblings[24] two or three years
apart in age, though the resemblance in home and school training in
arithmetic should be nearly as great for the latter as for the former.
Also the young twins (9-11) showed as close a resemblance in addition
and multiplication as the older twins (12-15), although the similarities
of training in arithmetic have had twice as long to operate in the
latter case.
[24] Siblings is used for children of the same parents.
If the differences found, say among children in grade 6 in addition,
were due to differences in the quantity and quality of training in
addition which they have had, then by giving each of them 200 minutes of
additional identical training the differences should be reduced. For the
200 minutes of identical training is a step toward equalizing training.
It has been found in many investigations of the matter that when we make
training in arithmetic more nearly equal for any group the variation
within the group is not reduced.
On the contrary, equalizing training seems rather to increase
differences. The superior individual seems to have attained his
superiority by his own superiority of nature rather than by superior
past training, for, during a period of equal training for all, he
increases his lead. For example, compare the gains of different
individuals due to about 300 minutes of practice in mental
multiplication of a three-place number by a three-place number shown
in Table 14 below, from data obtained by the author ['08].[25]
[25] Similar results have been obtained in the case of arithmetical
and other abilities by Thorndike ['08, '10, '15, '16], Whitley
['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and
Thorndike ['13], Hahn and Thorndike ['14], and on a very
large scale by Race in a study as yet unpublished.
TABLE 14
THE EFFECT OF EQUAL AMOUNTS OF PRACTICE UPON INDIVIDUAL DIFFERENCE IN
THE MULTIPLICATION OF THREE-PLACE NUMBERS
====================================================================
| AMOUNT | PERCENTAGE OF
| |CORRECT FIGURES
|----------------+---------------
| Initial | | Initial |
| Score | Gain | Score | Gain
-----------------------------------+---------+------+---------+-----
Initially highest five individuals | 85 | 61 | 70 | 18
next five " | 56 | 51 | 68 | 10
next six " | 46 | 22 | 74 | 8
next six " | 38 | 8 | 58 | 12
next six " | 29 | 24 | 56 | 14
====================================================================
THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES
Achievement in arithmetic depends upon a number of different abilities.
For example, accuracy in copying numbers depends upon eyesight, ability
to perceive visual details, and short-term memory for these. Long
column addition depends chiefly upon great strength of the addition
combinations especially in higher decades, 'carrying,' and keeping one's
place in the column. The solution of problems framed in words requires
understanding of language, the analysis of the situation described into
its elements, the selection of the right elements for use at each step
and their use in the right relations.
Since the abilities which together constitute arithmetical ability are
thus specialized, the individual who is the best of a thousand of his
age or grade in respect to, say, adding integers, may occupy different
stations, perhaps from 1st to 600th, in multiplying with integers,
placing the decimal point in division with decimals, solving novel
problems, copying figures, etc., etc. Such specialization is in part due
to his having had, relatively to the others in the thousand, more or
better training in certain of these abilities than in others, and to
various circumstances of life which have caused him to have, relatively
to the others in the thousand, greater interest in certain of these
achievements than in others. The specialization is not wholly due
thereto, however. Certain inborn characteristics of an individual
predispose him to different degrees of superiority or inferiority to
other men in different features of arithmetic.
We measure the extent to which ability of one sort goes with or fails to
go with ability of some other sort by the coefficient of correlation
between the two. If every individual keeps the same rank in the second
ability--if the individual who is the best of the thousand in one is the
best of the group in the other, and so on down the list--the correlation
is 1.00. In proportion as the ranks of individuals vary in the two
abilities the coefficient drops from 1.00, a coefficient of 0 meaning
that the best individual in ability A is no more likely to be in first
place in ability B than to be in any other rank.
The meanings of coefficients of correlation of .90, .70, .50, and 0 are
shown by Tables 15, 16, 17 and 18.[26]
[26] Unless he has a thorough understanding of the underlying
theory, the student should be very cautious in making
inferences from coefficients of correlation.
TABLE 15
DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .90
======================================================================
|10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST
----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
1st tenth | | | | | .1 | .4 | 1.8 | 6.6 |22.4 |68.7
2d tenth | | | .1 | .4 | 1.4 | 4.7 |11.5 |23.5 |36.0 |22.4
3d tenth | | .1 | .5 | 2.1 | 5.8 |12.8 |21.1 |27.4 |23.5 | 6.6
4th tenth | | .4 | 2.1 | 6.4 |12.8 |20.1 |23.8 |21.2 |11.5 | 1.8
5th tenth | .1 | 1.4 | 5.8 |12.8 |19.3 |22.6 |20.1 |12.8 | 4.7 | .4
6th tenth | .4 | 4.7 |12.8 |20.1 |22.6 |19.3 |12.8 | 5.8 | 1.4 | .1
7th tenth | 1.8 |11.5 |21.2 |23.8 |20.1 |12.8 | 6.4 | 2.1 | .4 |
8th tenth | 6.6 |23.5 |27.4 |21.1 |12.8 | 5.8 | 2.1 | .5 | .1 |
9th tenth |22.4 |36.0 |23.5 |11.5 | 4.7 | 1.4 | .4 | .1 | |
10th tenth|68.7 |22.4 | 6.6 | 1.8 | .4 | .1 | | | |
======================================================================
TABLE 16
DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .70
======================================================================
|10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST
----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
1st tenth | | .2 | .7 | 1.5 | 2.8 | 4.8 | 8.0 |13.0 |22.3 |46.7
2d tenth | .2 | 1.2 | 2.6 | 4.5 | 7.0 | 9.8 |13.4 |17.3 |21.7 |22.3
3d tenth | .7 | 2.6 | 5.0 | 7.3 |10.0 |12.5 |14.9 |16.7 |17.3 |13.0
4th tenth | 1.5 | 4.5 | 7.3 | 9.8 |12.0 |13.7 |14.8 |14.9 |13.4 | 8.0
5th tenth | 2.8 | 7.0 |10.0 |12.0 |13.4 |14.0 |13.7 |12.5 | 9.8 | 4.8
6th tenth | 4.8 | 9.8 |12.5 |13.7 |14.0 |13.4 |12.0 |10.0 | 7.0 | 2.8
7th tenth | 8.0 |13.4 |14.9 |14.8 |13.7 |12.0 | 9.8 | 7.3 | 4.5 | 1.5
8th tenth |13.0 |17.3 |16.7 |14.9 |12.5 |10.0 | 7.3 | 5.0 | 2.6 | .7
9th tenth |22.3 |21.7 |17.3 |13.4 | 9.8 | 7.0 | 4.5 | 2.6 | 1.2 | .2
10th tenth|46.7 |22.3 |13.0 | 8.0 | 4.8 | 2.8 | 1.5 | .7 | .2 |
======================================================================
TABLE 17
DISTRIBUTION OF ARRAYS OF SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .50
======================================================================
|10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST
----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
1st tenth | .8 | 2.0 | 3.2 | 4.6 | 6.2 | 8.1 |10.5 |13.9 |18.0 |31.8
2d tenth | 2.0 | 4.1 | 5.7 | 7.3 | 8.8 |10.5 |12.2 |14.1 |16.4 |18.9
3d tenth | 3.2 | 5.7 | 7.4 | 8.9 |10.0 |11.2 |12.3 |13.3 |14.1 |13.9
4th tenth | 4.6 | 7.3 | 8.8 | 9.9 |10.8 |11.6 |12.0 |12.3 |12.2 |10.5
5th tenth | 6.2 | 8.8 |10.0 |10.8 |11.3 |11.5 |11.6 |11.2 |10.5 | 8.1
6th tenth | 8.1 |10.5 |11.2 |11.6 |11.5 |11.3 |10.8 |10.0 | 8.8 | 6.2
7th tenth |10.5 |12.2 |12.3 |12.0 |11.6 |10.8 | 9.9 | 8.8 | 7.5 | 4.6
8th tenth |13.9 |14.1 |13.3 |12.3 |11.2 |10.0 | 8.8 | 7.4 | 5.7 | 3.2
9th tenth |18.9 |16.4 |14.1 |12.2 |10.5 | 8.8 | 7.3 | 5.7 | 4.1 | 2.0
10th tenth|31.8 |18.9 |13.9 |10.5 | 8.1 | 6.2 | 4.6 | 3.2 | 2.0 | .8
======================================================================
TABLE 18
DISTRIBUTION OF ARRAYS, IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .0
======================================================================
|10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST
----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
1st tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
2d tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
3d tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
4th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
5th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
6th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
7th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
8th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
9th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10
10th tenth|10 |10 |10 |10 |10 |10 |10 |10 |10 |10
======================================================================
The significance of any coefficient of correlation depends upon the
group of individuals for which it is determined. A correlation of .40
between computation and problem-solving in eighth-grade pupils of 14
years would mean a much closer real relation than a correlation of .40
in all 14-year-olds, and a very, very much closer relation than a
correlation of .40 for all children 8 to 15.
Unless the individuals concerned are very elaborately tested on several
days, the correlations obtained are "attenuated" toward 0 by the
"accidental" errors in the original measurements. This effect was not
known until 1904; consequently the correlations in the earlier studies
of arithmetic are all too low.
In general, the correlation between ability in any one important feature
of computation and ability in any other important feature of computation
is high. If we make enough tests to measure each individual exactly
in:--
(_A_) Subtraction with integers and decimals,
(_B_) Multiplication with integers and decimals,
(_C_) Division with integers and decimals,
(_D_) Multiplication and division with common fractions, and
(_E_) Computing with percents,
we shall probably find the intercorrelations for a thousand 14-year-olds
to be near .90. Addition of integers (_F_) will, however, correlate less
closely with any of the above, being apparently dependent on simpler and
more isolated abilities.
The correlation between problem-solving (_G_) and computation will be
very much less, probably not over .60.
It should be noted that even when the correlation is as high as .90,
there will be some individuals very high in one ability and very low in
the other. Such disparities are to some extent, as Courtis ['13, pp.
67-75] and Cobb ['17] have argued, due to inborn characteristics of the
individual in question which predispose him to very special sorts of
strength and weakness. They are often due, however, to defects in his
learning whereby he has acquired more ability than he needs in one line
of work or has failed to acquire some needed ability which was well
within his capacity.
In general, all correlations between an individual's divergence from the
common type or average of his age for one arithmetical function, and his
divergences from the average for any other arithmetical function, are
positive. The correlation due to original capacity more than
counterbalances the effects that robbing Peter to pay Paul may have.
Speed and accuracy are thus positively correlated. The individuals who
do the most work in ten minutes will be above the average in a test of
accuracy. The common notion that speed is opposed to accuracy is correct
when it means that the same person will tend to make more errors if he
works at too rapid a rate; but it is entirely wrong when it means that
the kind of person who works more rapidly than the average person is
likely to be less accurate than the average person.
Interest in arithmetic and ability at arithmetic are probably correlated
positively in the sense that the pupil who has more interest than other
pupils of his age tends in the long run to have more ability than they.
They are certainly correlated in the sense that the pupil who 'likes'
arithmetic better than geography or history tends to have relatively
more ability in arithmetic, or, in other words, that the pupil who is
more gifted at arithmetic than at drawing or English tends also to like
it better than he likes these. These correlations are high.
It is correct then to think of mathematical ability as, in a sense, a
unitary ability of which any one individual may have much or little,
most individuals possessing a moderate amount of it. This is
consistent, however, with the occasional appearance of individuals
possessed of very great talents for this or that particular feature of
mathematical ability and equally notable deficiencies in other features.
Finally it may be noted that ability in arithmetic, though occasionally
found in men otherwise very stupid, is usually associated with superior
intelligence in dealing with ideas and symbols of all sorts, and is one
of the best early indications thereof.
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INDEX
Abilities, arithmetical, nature of, 1 ff.;
measurement of, 27 ff.;
constitution of, 51 ff.;
organization of, 137 ff.
Abstract numbers, 85 ff.
Abstraction, 169 ff.
Accuracy, in relation to speed, 31;
in fundamental operations, 102 ff.
Addition, measurement of, 27 ff., 34;
constitution of, 52 f.;
habit in relation to, 71 f.;
in the higher decades, 75 f.;
accuracy in, 108 f.;
amount of practice in, 122 ff.;
interest in 196 f.
Aims of the teaching of arithmetic, 23 f.
AMES, A. F., 89
Analysis, learning by, 169 ff.;
systematic and opportunistic stimuli to, 178 f.;
gradual progress in, 180 ff.
Area, 257 f., 275
Arithmetic, sociology of, 24 ff.
Arithmetical abilities. _See_ Abilities.
Arithmetical language, 8 f., 19, 89 ff., 94 ff.
Arithmetical learning, before school, 199 ff.;
conditions of, 227 ff.;
in relation to time of day, 227 ff.;
in relation to time devoted to arithmetic, 228 ff.
Arithmetical reasoning. _See_ Reasoning.
Arithmetical terms, 8, 19
Averages, 40 f.; 135 f.
BALLOU, F. W., 34, 38
Banking, 256 f.
BINET, A., 201
Bonds, selection of, 70 ff.;
strength of, 102 ff.;
for temporary service, 111 ff.;
order of formation of, 141 ff.
_See also_ Habits.
BRANDELL, G., 211
BRANDFORD, B., 198 f.
BROWN, J. C., xvi, 103
BURGERSTEIN, L., 103
BURNETT, C. J., 202
BURT, C., 286
Cardinal and ordinal numbers confused, 206
Catch problems, 21 ff.
CHAPMAN, J. C., 49
Class, size of, in relation to arithmetical learning, 228;
variation within a, 289 ff.
COBB, M. V., 299
COFFMAN, L. D., xvi
Collection meaning of numbers, 3 ff.
Computation, measurements of, 33 ff.;
explanations of the processes in, 60 ff.;
accuracy in, 102 ff.
_See also_ Addition, Subtraction, Multiplication, Division,
Fractions, Decimal numbers, Percents.
Concomitants, law of varying, 172 ff.;
law of contrasting, 173 ff.
Concrete numbers, 85 ff.
Concrete objects, use of, 253 ff.
Conditions of arithmetical learning, 227 ff.
Constitution of arithmetical abilities, 51 ff.
Copying of numbers, eyestrain due to, 212 f.
Correlations of arithmetical abilities, 295 ff.
Courses of study, 232 f.
COURTIS, S. A., 28 ff., 43 ff., 49, 103, 291, 293, 299
Crutches, 112 f.
Culture-epoch theory, 198 f.
Dairy records, 273
Decimal numbers, uses of, 24 f.;
measurement of ability with, 36 ff.;
learning, 181 ff.;
division by, 270 f.
DE CROLY, M., 205
Deductive reasoning, 60 ff., 185 ff.
DEGAND, J., 205
Denominate numbers, 141 f., 147 f.
Described problems, 10 ff.
Development of knowledge of number, 205 ff.
DE VOSS, J. C., 49
DEWEY, J., 3, 83, 150, 205, 207, 208, 219, 266, 277
Differences in arithmetical ability, 285 ff.;
within a class, 289 ff.
Difficulty as a stimulus, 277 ff.
Drill, 102 ff.
Discipline, mental, 20
Distribution of practice, 156 ff.
Division, measurement of, 35 f., 37;
constitution of, 57 ff.;
deductive explanations of, 63, 64 f.;
inductive explanations of, 63 f., 65 f.;
habit in relation to, 72;
with remainders, 76;
with fractions, 78 ff.;
amount of practice in, 122 ff.;
distribution of practice in, 167;
use of the problem attitude in teaching, 270 f.
DONOVAN, M. E., 295
Elements, responses to, 169 ff.
Eleven, multiples of, 85
ELLIOTT, C. H., 228
Equation form, importance of, 77 f.
Explanations of the processes of computation, 60 ff.;
memory of, 115 f.;
time for giving, 154 ff.
Eyestrain in arithmetical work, 212 ff.
Facilitation, 143 ff.
Figures, printing of, 235 ff.;
writing of, 214 f., 241
FLYNN, F. J., 196
Fractions, uses of, 24 f.;
measurement of ability with, 36 ff.;
knowledge of the meaning of, 54 ff.
FREEMAN, F. N., 259, 261
FRIEDRICH, J., 103
Generalization, 169 ff.
GILBERT, J. A., 203
Graded tests, 28 ff., 36 ff.
Greatest common divisor, 88 f.
Habits, importance of, in arithmetical learning, 70 ff.;
now neglected, 75 ff.;
harmful or wasteful, 83 ff.; 91 ff.;
propaedeutic, 117 ff.;
organization of, 137 ff.;
arrangement of, 141 ff.
HAHN, H. H., 295
HALL, G. S., 200 f.
HARTMANN, B., 200 f.
HECK, W. H., 227
Heredity in arithmetical abilities, 293 ff.
Highest common factor, 88 f.
HOKE, K. J., 49
HOLMES, M. E., 103
HOWELL, H. B., 259
HUNT, C. W., 196
Hygiene of arithmetic, 212 ff., 234 ff.
Individual differences, 285 ff.
Inductive reasoning, 60 ff., 169 ff.
Insurance, 256
Interest as a principle determining the order of topics, 150 ff.
Interests, instinctive 195 ff.;
censuses of, 209 ff.;
neglect of childish, 226 ff.;
in self-management, 223 f.;
intrinsic, 224 ff.
Interference, 143 ff.
Inventories of arithmetical knowledge and skill, 199 ff.
JESSUP, W. A., xvi
KELLY, F. J., 49
KING, A. C., 103, 227
KIRBY, T. J., 76 f., 104, 295
KLAPPER, P., xvi
KRUSE, P. J., 289, 293
Ladder tests, 28 ff., 36 ff.
Language in arithmetic, 8 f., 19, 89 ff., 94 ff.
LASER, H., 103
LAY, W. A., 259, 261
Learning, nature of arithmetical, 1 ff.
Least common multiple, 88 f.
LEWIS, E. O., 210 f.
LOBSIEN, M., 209 f.
MCCALL, W. A., 49
MCDOUGLE, E. C., 85 ff.
MCKNIGHT, J. A., 210
MCLELLAN, J. A., 3, 83, 89, 205, 207
Manipulation of numbers, 60 ff.
Meaning, of numbers, 2 ff., 171;
of a fraction, 54 ff.;
of decimals, 181 f.
Measurement of arithmetical abilities, 27 ff.
Mental arithmetic, 262 ff.
MESSENGER, J. F., 202
Metric system, 147
MEUMANN, E., 261
MITCHELL, H. E., 24
MONROE, W. S., 49
Multiplication, measurement of, 35, 36;
constitution of, 51;
deductive explanations of, 61;
inductive explanations of, 61 f.;
with fractions, 78 ff.;
by eleven, 85;
amount of practice in, 122 ff.;
order of learning the elementary facts of, 144 f.;
distribution of practice in, 158 ff.;
use of the problem attitude in teaching, 267 ff.
NANU, H. A., 202
National Intelligence Tests, 49 f.
Negative reaction in intellectual life, 278 f.
Number pictures, 259 ff.
Numbers, meaning of, 2;
as measures of continuous quantities, 75;
abstract and concrete, 85 ff.;
denominate, 141 f., 147 f.;
use of large, 145 f.;
perception of, 205 ff.;
early awareness of, 205 ff.;
confusion of cardinal and ordinal, 206.
_See also_ Decimal numbers _and_ Fractions.
Objective aids, used for verification, 154;
in general, 243 ff.
Oral arithmetic, 262 ff.
Order of topics, 141 ff.
Ordinal numbers, confused with cardinal, 206
Original tendencies and arithmetic, 195 ff.
Overlearning, 134 ff.
Percents, 80 f.
Perception of number, 202 ff.
PHILLIPS, D. E., 3, 4, 205, 207
Pictures, hygiene of, 246 ff.;
number, 259 ff.
POMMER, O., 212
Practice, amount of, 122 ff.;
distribution of, 156 ff.
Precision in fundamental operations, 102 ff.
Problem attitude, 266 ff.
Problems, 9 ff.;
"catch," 21 ff.;
measurement of ability with, 42 ff.;
whose answer must be known in order to frame them, 93 f.;
verbal form of, 111 f.;
interest in, 220 ff.;
as introductions to arithmetical learning, 266 ff.
Propaedeutic bonds, 117 ff.
Purposive thinking, 193 ff.
Quantity, number and, 85 ff.;
perception of, 202 ff.
RACE, H., 295
Rainfall, 272
Ratio, 225 f.;
meaning of numbers, 3 ff.
Reaction, negative, 278 f.
Reality, in problems, 9 ff.
Reasoning, arithmetical, nature of, 19 ff.;
measurement of ability in, 42 ff.;
derivation of tables by, 58 f.;
about the rationale of computations, 60 ff.;
habit in relation to, 73 f., 190 ff.;
problems which provoke false, 100 f.;
the essentials of arithmetical, 185 ff.;
selection in, 187 ff.;
as the cooeperation of organized habits, 190 ff.
Recapitulation theory, 198 f.
Recipes, 273 f.
Rectangle, area of, 257 f.
RICE, J. M., 228 ff.
RUSH, G. P., 49
SEEKEL, E., 212
SELKIN, F. B., 196 f.
Sequence of topics, 141 ff.
Series meaning of numbers, 2 ff.
Size of class in relation to arithmetical learning, 228
SMITH, D. E., xvi, 224
Social instincts, use of, 195 f.
Sociology of arithmetic, 24 ff.
Speed in relation to accuracy, 31, 108
SPEER, W. W., 3, 5, 83
Spiral order, 141, 145
STARCH, D., 49, 295
STERN, W., 210, 212
STONE, C. W., 27 ff., 42 ff., 228 ff.
Subtraction, measurement of, 34 f.;
constitution of, 57 f.;
amount of practice in, 122 ff.
Supervision, 233 f.
SUZZALLO, H., xvi
Temporary bonds, 111 ff.
Terms, 113 f.
Tests of arithmetical abilities, 27 ff.
THORNDIKE, E. L., 34, 38 ff., 227, 294
Time, devoted to arithmetic, 228 ff.;
of day, in relation to arithmetical learning, 227 f.
Type, hygiene of, 235 ff.
Underlearning, 134 ff.
United States money, 148 ff.
Units of measure, arbitrary, 5, 83 f.
Variation, among individuals, 285 ff.
Variety, in teaching, 153
Verification, 81 f.;
aided by greater strength of the fundamental bonds, 107 ff.
WALSH, J. H., 11
WELLS, F. L., 295
WHITE, E. E., 5
WHITLEY, M. T., 295
WIEDERKEHR, G., 212
WILSON, G. M., 24, 49
WOODY, C., 29 ff., 52, 287, 293
Words. _See_ Language _and_ Terms.
Written arithmetic, 262 ff.
Zero in multiplication, 179 f.
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