done. Improvement shows itself either by a decrease in time for doing
Plata Til. Laamijag oxurvea of 4th grada ohlldran In BatlpUoatlon.
fha laft hand cnrrm ahowa Vbm noiobor of problamB aolvad In two mlii>
«t«a on IB different ftaya. The right band onrre showe the aToraga
tliM reiialred to do a sln^la problem on tha 16 different days. Tha
toznar raoorda progTaeB In anount dona, tha latter In tlaa aoniiuiual*
Il8 INTRODUCTORY PSYCHOLOGY FOR TUACllKRS
the saine Usk (as in the mirror-drawing experiment) or by an increase
in what is accomplished in the same work-period (as in the arithmetic
tests). Now either of these curves can be transniute.i so as to appear
m the other form. Take, for example, the curve of learning of the
4th Grade children in multiplication (shown in the left hand curve of
Plate VII). Here we see that the children performed 11 problems
correctly on the first occasion, 15.5 problems on the second, etc. They
accomplished that much in 60 seconds. At that rate it required 5.5
seconds to do one problem on the first occasion (i. e., 6cK-i 1=5.5) >
3.9 seconds to do one problem on the second occasion (i. e.,
60-^-15.5=3.9) ; etc. When these quotients are plotted for the
trials we obtain the right hand curve in Plate VII. The two curves,
then, both record the same facts, altho one goes up and the other
comes down. With a little practice in thinking in terms of curves this
seeming paradox will no longer bother one.
EXPLANATION OF INDIVIDUAL DIFFERENCES IN TERMS OF "HEREDITy"
In the case of the mirror-drawing experiment, or the simple arith-
metical work, the situation is the same for all the individuals. All the
individuals are confronted with the same apparatus or the same blank
of 80 problems. In one sense this is not strictly true, as we have al-
ready seen, since different individuals respond to different details in
the entire situation. But these differences are not due to actual physi-
cal differences in the situation, but rather to differences in the indi-
viduals themselves. We may then properly speak of the situations con-
fronting the individuals as being exactly the same in all ten cases. It
then remains to explain the differences we find among the ten indi-
viduals in terms of "original nature" or "training."
The Effect of Previous Training. We have learned that all indi-
viduals show greater improvement at the commencement of practice
than at the end. This being the case the learning curves of those who
have had no previous practice zvill rise more rapidly and slozv up more
gradually than in the case of those who have had previous practice'
This fact may be illustrated in Plate VIII by saying that the person
who has had no previous practice (training) would have the learning
curve marked B. The person with previous training might have in-
stead a curve similar to A. The former's curve would show very
marked gains at the start and would show a large improvement alto-
gether. The latter's curve would not show such a marked gain at the
start and would not show such a large total improvement. We may
think of A's curve as not being complete — that the first 15 trials are
not shown here (having been performed before) and that what is
rq>resented is trials 16 to 41. This is on the assumption that A and B
are exactly identical in every respect. This is further shown in the
two curves by representing B's progress in trials 16 to 26 as exactly
Plat* VIII. Showing laaLznln^ cmma of two Indl-riait&ls
who ar* Idantloal In all r«sp«ota save In the asonnt
of trainlae In tha azltfaDatloal ooahmatlons.
equal to A's progress in trials i to 11. And if the curves were •on-
tinued, B's progress in trials 26 to 41 would be identical to A's records
in trials 11 to 26. Previous training, then affects an individnaYs learn-
ing curve by raising its starting-point and by eliminating to some ex-
tent at least the ordinary big rise at the start.
It was stated above that B would show apparently greater improve-
ment than A. The word "apparently" should be emphasized. Plate
VIII is so drawn as to indicate that altho B's curve shows a g^reater
gain than A's curve when measured in terms of improvement in prob-
lems performed correctly (i. e., 5 problems to 33.0 problems as against
I20 INTRODUCTORY PSYCHOLOGY FOR TEACHERS
29.2 problems to 35.9 problems) yet in terms of number of trials B
has not gained over A. He started out 15 trials behind and remained
15 trials behind to the end. If B's curve were extended for 15 trials
more it would then reach the point reached by A at his 41st trial — the
end of his practice period. It is an extremely difficult matter to meas-
ure relative improvement in terms of time or amount of work done,
because as one approaches his limit each unit of effort will produce a
smaller and smaller gain in time saved or work accomplished.'
The Effect of Differences in Hereditary Endozvment. How do differ-
ences in sheer hereditary endowment affect learnins^ curves? Plate IX
illustrates this point. The individual with the best endowment will
show the greatest improvement, the person with the least endowment
will show the least improvement. Curves B, C, and D represent the
learning curves of three persons ; curve B being the curve of the best
endowed, curve C being of a poorer endowed person, and curve D be-
ing of the poorest endowed person of the three. The better the original
tiature of the individual the greater will be the improvement resulting
from practice. These three individuals with equal training and varying
degrees of hereditary endowment would not even do equally well, of
course, on the first trial, because the better endowed person would do
better than the others right from the start.
One warning should be given here. The degree of efficiency of the
original nature of the individual must be considered as it applies to
the particular task being tested. For example, a great musician (hav-
ing superior original nature in musical lines) may not necessarily have
superior endowment in mirror-drawing. The musician's curve in
mirror-drawing will show great improvement or not ; depending not
upon endowment in general, but upon the endowment which he has
that pertains to mirror-drawing.
The Effect of Differences in Training and Heredity Combined. Now
let us consider, third, some combinations of these two factors. We
may have four individuals, (i) A having good heredity and previous
training, (2) B having good heredity but no previous training, (3)
E having poor heredity and previous training, and (4) D having poor
heredity and no previous training. (Poor heredity is to be under-
stood as endowment having to do with the trait under discussion ;
training to be considered in terms of so many units of time devoted to
learning specific material.) Then their learning curves would take
more or less the forms illustrated in Plate X. A and E can be thought
of as having had 15 units of instruction, and B and D as having had
( I ) This point is discussed further in the writer's monograph, Effects of Hookworm
Disease on the Mental and Physical Development of Children. International Heialth
Commission, 1916, pp. 22-39.
Plat* II. Showing loaming curves of thrss Indirliuals
viUi Aiffer«nt Itatttdltary eodomnonta.
none. As B is superior to D by hereditary endowment he will do better
than the latter at the start and will rapidly leave him behind. (See
Plate IX, where this point is alone considered.) The more training
they receive the more different will they become as far as this trait is
considered, because of the difference in their ability. In the same way
A and E, who have had some previous training become more and more
unlike as they continue their training. These curves illustrate, then, the
principle that continued training makes individuals of different heredi-
tary endowment more and more unlike. We shall return to this point
a little later.
The curves of A and B are symmetrical. A's curve actually being
the same as B's from the latter's i6th trial on to what would be his 41st
trial. The curves of E and D are also symmetrical in the same way.
Because of their previous training A and E will maintain their supe-
122 IXTRODLCTOUY PSYCHOLOGY FOR TlCACIIERS
riority over B and D, respectively. This superiority seemingly grows
smaller and smaller with practice. It actually does if measured in
terms of problems performed, but it does not if measured in
terms of effort, for A always remains ahead of B to the extent of what
15 units of time will produce, and likewise E remains ahead of D to that
The difference between the good heredity of A and B and the
poor heredity of E and D is meant to be a considerable difference. Yet
it is not exaggerated at all in comparison with the differences found in
most any class room. The differences between the average of the
4th Grade and the group of retarded children is about equal to that
shown here between A and E. In Plate XI are shown the curves of a
child from the 4th Grade and another from the retarded group. The
former is not the brightest in that grade (actually rated nth in a class
of 28) and the latter is not the dullest among these unfortunate chil-
dren. The retarded child's record was, o problems, 0,0.0, 1,0, i, 2, 2, 2,
and after 170 minutes drill, 5, 5, and 4. Here measles intervened to
spoil our record. In fairness to the records it should be stated that
undoubtedly the 4th Grade child practiced on these combinations out-
side of school. But the dull child had also this opportunity. The
curves do represent consequently the learning that followed equal
stimulations in the school. One child could respond in an adequate
manner and did so and the other child could not and so did not. Some
children can learn mathematics so that they eventually master calculus
and its applications to engineering, while others never get beyond the
fundamentals. Some children master the principles of art and de-
sign and become skilled in dressmaking, millinery, architecture, paint-
ing, etc., while others are oblivious to the most atrocious combinations
of color or form in their clothes, their home surroundings, etc. The
gifted child learns rapidly and improves tremendously, the child who is
lacking learns slowly and learns very little.
INDIVIDUAL DIFFERENCES IN SOLVING SIMPLE .\RITHMETICAL
Let US now more or less review what has been discussed in this les-
son but consider the matter in terms of the data studied in Lesson 23.
These data are plotted in Plate XII. The curves do not
bring out the points so clearly as do the theoretically constructed
curves of Plates VIII, IX, and X. Nevertheless they bear witness to all
of those points.
I. The greater the amount of practice the higher the curves start.
This point needs no further discussion.
t, •' K
3 »< a *»
0.0 c -H a-H
^ ♦> e »
C B C p.
a ■• u
» ♦» t*
iH rH T< 3 C
•H « c
+^ "H « -• l<
s) >< to 0. tiO-lJ
CL( j3 ^ ja w
' •W -tH -vH
\ ^ pa M *> oj
124 INTKODL'CTOKV PSYCHOLOGY FOR TEACHERS
2. The greater the aviount of practice the less rapid the gain. This
point is true but it does not always appear, due to the presence of con-
flicting factors. Altho none of these groups had had any previous
training with the particular tests under discussion, yet we naturally
would expect the adults to have had more practice and so to show
less improvement than the 4th Grade children. The real cause, how-
ever, as to why the curves do not clearly illustrate the point made at
the commencement of this paragraph is due to the differences in the
groups in terms of heredity. Not only are the adults superior to the
4th Grade children because they have a mature development of their
hereditary nature, but also without question a class of college men and
women are superior to a class of 4th Grade children. That is, the 4th
Grade class will not average as high an endowment when they be-
come adults as do the college students. This class of 43 college students
is probably composed of the brightest students from 43 4th Grade
classes. The great differences in heredity cover up then the effect of
much practice versus little practice.
3. The greater the hereditary endowment the greater the improve-
ment from training. This point is clear from the curves and from
what has just been stated.
4. The greater the training the more a group of individuals be-
come unlike. At the commencement of the training recorded here the
three groups could perform as follows :
College students solve 59 plroblems per minute.
4th Grade Children solve 19 problems per minute.
Defective Children solve 4 problems per minute.
A. D. 21. 1
and at the end of ten practice periods they performed as follows : —
College students solve 76 problems per minute.
4th Grade children solve 30 problems per minute.
Defective Children solve 7 problems per minute.
A. D. 25.6
As the A. D. has increased we know the groups are less alike than
before. This fact is shown also in this way.
College students are superior to 4th Grade Children at start by 40
College students are superior to 4th Grade Children at end by 46
t/umber at Tf»bUmS
Plat* HI. Showing learning oorres
in solving eimple arithmotloaJ. eom-
blnatlona from adults, Curve A.
(B-Test) and Ourro B iBX-Teet) : 4th
grade ohildren. Curve C (B-T«st)
and Curve C (BI-Teat): and from da-
faotlvs ohildren. Curves E and F (B
Teat, — Curve ? prior to and Curve B
after 170 minutes of speoial drill
en addition ooobinations.)
College students are superior to Defective Children at start by 55
College students are superior to Defective Children at end by 69
4th Grade Children are superior to Defective Children at start by 15
4th Grade Children are superior to Defective Children at end by 2^
126 IXTUODUCTORV PSYCHOLOGY FOR TKACIIERS
This fourth fact, that training causes a group to "fly apart," to be-
come niore and more unlike, due to the inherent differences in the
hereditary equipment of the members of the group, affects our school
work most profoundly. It makes clear that no grade can be taught as
a class without some members very shortly doing such good work as
to tempt the authorities to promote them into the next grade and some
other children doing such poor work as to lead to their being put back
into the grade below or to force the teacher to give them individual
instruction. No mechanical administrative scheme for holding a class
together will ever work satisfactorily because the members of that
class cannot advance at the same rate. The solution to this difficulty
has not been evolved, but if it ever is, in the writer's opinion, it will in-
clude a yery flexible scheme of promotion by subject-matter, cfiuulcil
with extensive provision for individual coaching of children that are
markedly behind and markedly ahead of their class. This point will
be taken up again later. But right now it should be realized that the
main point of the whole problem is that children cannot progress in
their learning at the same rate: — that some go last, some go slow.
and some advance at average speed.
LESSON 25. THE GENERAL LAW AS TO HOW INDIVIDUALS
We know that people are different almost before we realize that
there are people. We distinguish between tall people and short peo-
ple, fat people and thin people, clever people and silly people, and
most of us would agree fairly well in our classifications. But how do
we draw these distinctions? Do we have hard and fast lines, enclosed
between which one class is set off from another? Should we say that
all men between o inches and 62 inches in height, for instance, are
short, and those between 62 inches and 84 inches are tall ? Or that any
one less than 125 lbs. is thin and anyone more than 125 lbs. is fat?
.\nd eren if we decide to be so definite in these cases, (tho certainly our
standard is artificial) where shall we draw the line in the case of men-
tal attainments? Are we all talented or stupid for example? Or are
most of us merely average people without special qualifying adjectives.
and the rest of us simply either better or worse than the average?
That is, instead of having separate little groups of idiots, normal folks,
and geniuses, the members of each class keeping carefully to them-
selves, do we perhaps have but one class of individuals, all typified by
the average, yet all varying from the average in greater or less degree ?
We are about to perform an experiment in throwing dice. This is
as purely a chance performance as we can get. Let us see if the
LESSON 25 127
Number of Throws
2 15 II
21 25 I 6 4 2 10 7 3 20 9 23
46 8 10 12 14 16 18
Total Amount of Throws.
Plate XIII. Illustrating by means of a "surface of distribution"
twenty-five throws of three dice.
throws are distinctly different or whether they follow one general law.
For example, can we divide the throws into two groups — high and
low, or must we think in terms of one group with variations from its
average? In any case the results may apply to our biological problem
as given above.
Problem. In throning dice are the totals distinctly different or do
they approach a general type?
Apparatus. Coordinate paper ; 3 dice.
Procedure. Part i. Lay off on your coordinate paper a base line,
and number the squares from o to 20, as is done in Plate XIII. Lay off
a vertical axis and number the squares from o to 35. Now commence
and throw your three dice. Count up the total of the three dice and
record that total on your coordinate paper in its proper place. (The
writer threw first a 4, 3, and i, making a total of 8. A little square
was then drawn as indicated by the i in Plate XIII. An 11 was
thrown next and it is indicated by the 2 in the Plate. A 14 was thrown
third, etc. Twenty-five throws are indicated in this Plate, the twenty-
fifth throw being a 7. Plate XIII shows then that the writer threw
one 6 two 12s
one 7 one 13
three 8s two 14s
three 9s one 15
six los one 16, and
three lis one 17.
Thus 25 throws are distributed or indicated in the plate.
Record in this way 100 throws. Show your completed diagram to
the instructor before proceeding further.
128 IXTRODUCroUV I'SYCriOLOGY FOR TEACH KRS
Such a diagram is called a surface of distribution as it shows just
how all the throws were distributed among the possible totals.
Part 2. Now determine how many different totals can be obtained
by throwing three dice. (In Plate XIII are indicated 12 different
totals, i. e., from a total of 6 to a total of 17, inclusive.) Present your
answer to your instructor before proceeding further.
Part 3. Now figure out (a) all the possible dilTcrcnt combinations*
it is possible to obtain by throwing three dice.
(This assignment is independent of Part i and can be worked out
without any reference to it). The writer threw first a 4, 3, and i ; next
time he threw a 3, 5, and i ; the third he threw a 6, 5, and 3. Here are
three different combinations. The question is, how many different
combinations are there? (Consider in this connection that a throw of
4, 3, and 2 is different from a 2, 4, and 3, and both of these arc dif-
ferent from a 3, 2, and 4.)
Also figure out (b) how many of each total you will obtain when
every possible combination is considered. (For example, throws of 2,
4, and 6 ; 5, 5, and 2 ; 5, 6, and i ; are three different combinations, but
they all give the same total, i. e., 12.) In Plate XIII are indicated 12 dif-
ferent totals, i. e., from a total of 6 to a total of 17, inclusive. On the
preceding page are listed how many of each of these totals the writer
obtained in his 25 throws.
Part 4. Suppose instead of getting the 100 throws you did get, you
had thrown the dice as many times as there are different combinations
and in throwing the dice that number of times had got each and all
of these different combinations. Plot a surface of distribution to illus-
trate just this.
Part 5. What relation do you think there exists between the sur-
face of distribution you actually obtained by throwing the dice 100
times and the surface of distribution obtained in the preceding para-
What relation do you think there exists between the findings in this
experiment of throwing dice and the general problem of how individuals
differ? Can throws be divided into two or more groups; can in-
Hand in your report at the next class-hour.
•Mathematica)ly speaking what is wanted here is permutations, not combinations.
That is, in forming combinations we are only concerned with the number of things
each selection contains, whereas in forming permutations, we have also to consider
the order of the things which make up each arrangement; for instance, if from six
nuTnbers, 1, 2, 3, 4, 5, 6, we make a selection of three, such as 123, this single
combination admits of being arr&nged in the following ways: — 123. 132, 213, 231.
312, and 321. and so gives rise to six different permutations.
LESSON 26.— GENERAL LAW AS TO HOW INDIVIDUALS
THE NORMAL SURFACE OF DISTRIBUTION.
If one should take three dice and throw them 216 times, each time
counting up the total score and plotting this score, one might obtain a
surface of distribution somewhat like the three surfaces shown in
Plate XIV. The first and third were actually so obtained, the middle
one is the perfect surface which chance theoretically should give.
One may figure out this theoretically perfect surface in this way.
Count up all the throws that are possible and record how many times
each total appears. You may have
and I and i,
total of 3
When you have so obtained all the 216 totals you will find that you have
When these data are plotted you have the ideal surface of distribu-
tion in Plate XIV. All this means that when you throw three dice you
are just as likely to get any one combination as any other. But you
are more likely to get a total of 10 or 11 than 3 or 18. You can ex-
press this likelihood by the expression 27 to i. for there are 27 combi-
nations that will give a total of 10 or 11, whereas there is only one
combination that will give 3 or 18. Our normal curve of distribution
represents then that surface most likely to be obtained by 216 throws.
Actually we seldom get exactly that ideal surface, but we do get sur-
faces that approximate it in general appearance.
Discuss, Lesson 25
Exper., Lesson 27
Lesson 2 7
Lesson 26 i
INTRODUCTORY PSYCHOLOGY FOR TEACH tRS
., _ Three aurfaoea of dlatrlbutlon obtained from thro\7lng three dice 816
uase. Tho flrat and third aurfaoea were obtained from 216 actual throwa. The
aeoond la baao4 on what theorotlo&lly ahould be obtained from that number of
One may think of this matter of throwing three dice as being con-
ditioned on three independent factors, each one of which may vary in-
dependently in six different ways. When the three independent factors
with their six possible variations are considered as a whole, we realize
that there are 216 independent combinations possible. But the 216
independent combinations do not give 216 different final scores. They