James Leroy Stockton.

Exact measurements in education online

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I. Measurement in Education should have for
its goal the computation of work and rate-of-work
(power), in the sense in which these terms are
used in Mechanics.

II. Scales of force, space, and time, exist, or
can be made, for school subjects; and the stand-
ard units of these scales of force, and space, and
time, should be combined into standard units of
work and rate-of-work (power), such units
directly corresponding to the foot-pound and the
horse-power. (In this paper units are worked
out for penmanship, and illustrated by experi-
mental work involving certain applications of the
Thorndike Scale.)

III. Many units in many school subjects should



be supplemented by a single unit, making possible
the computation of mental work and rate-of-
mental-work (mental power) in all school sub-
jects. The force involved in this computation is
intelligence; the space is measured in elements of
expression. (As there is no adequate scale of
intelligence uncombined with any mechanical fac-
tor, a theory of the necessary scale is ventured.)

IV. In any case, to consider either force, space,
or time, alone, or to combine them in an arbitrary
manner, gives unreliable results. [This is shown,
for computations in school subjects, by the pen-
manship illustration. For computations of men-
tal work, and mental power, experience with the
Binet-Simon tests is cited in proof of the con-



Most persons do not any longer question the
possibility of measurement in Education, because
it has become apparent that measurements always
have been made, and are continuing to be made.
When it is said that a piece of work is good, bad,
or indifferent, a measuring scale of at least three
steps is evidently being used. If papers are
marked A, B, C, D, E, according to the judgment
of the examiner, a scale of five steps is being used.
This is clearly evident; measurement is a fact
in all departments of Education whenever the
value of the product is expressed.

There are, however, many conscientious think-
ers who still question the degree of exactness to
which the measurement should be carried. The
common rough measurements which are con-
stantly used do not seem so objectionable as the
more exact scientific measurements which are
being proposed. It is feared that too much exact-
ness will make Education formal or mechanical.


If this fear were justifiable it would furnish a
very strong foundation for a stand against meas-
urement, for modern Education cannot defend
formalism. Fortunately, however, the difficulty
can be met with the following statements :

(1) Education, in so far as it can be measured,
is a product,

(2) Mechanical methods of measuring a prod-
uct do not require mechanical methods of pro-
ducing that product. Handwriting might be
measured by the most mechanical means one could
imagine, and yet have been produced by the freest,
most spontaneous method that exists. The worst
that can be said is that mechanical measurement
may, in the careless and unthoughtful, tend to
produce mechanical methods of production; but
pre-supposing reasonable thoughtfulness in its
use, nothing promises more for Education than
does exact scientific measurement.

In this work progress has been made through
the establishment of relatively exact scales in
certain school subjects ; but the progress has been
slow, as it always is in a new field. Confusion,
also, is beginning to result, because the plunge
into this undiscovered country has naturally been


made with no very definite route marked out in
advance, and with no very adequate conception
of the extent of the territory to be explored.
There is not much evidence that it is realized that
the making of scales may be merely a scouting
on the frontier merely the beginnings of roads
whose end lies in a more remote country. If
this should prove to be true much wandering will
be prevented if a return is made to the starting
point, and an attempt made, in the light of all
past experience, to map the whole route from
the beginning to the end. Then if the map shows
districts to be traversed in which as yet no road
exists, the problem will at least be clear when
these sections are reached.

It is the purpose of this paper to suggest that
an unexplored district does exist in the field of
measurement in Education, and that the making
of scales takes the investigator only part way on
the road to the final goal. An attempt will be
made to show that even with the scales now avail-
able, or with other similar ones which may be
made, still another step must be taken or Educa-
tion remains in the same condition as was the
science of Mechanics before the time of Watt.


Before Watt the scales of feet, pounds, and min-
utes were in use, but there was no attempt to use
them in a computation of work and rate-of-work
by means of the composite units called the foot-
pound and the horse-power. The formulation of
these units opened a new realm in Mechanics.
From now on this discussion will deal with the
hypothesis that there is such a new realm in
measurement in Education, and that all of our
efforts in this field, including the making of scales,
will gain in definiteness and worth through being
directed toward this final goal the computation
of work and rate-of-work; work being used in its
technical meaning for the science of Mechanics.

Any hypothesis, in order to justify itself, must
show wherein it meets conditions unmet before;
it gains its adherents through its ability to clear
up existing confusions, and to present worthy
results. Therefore the problem squarely in view
is (1) to show that there is confusion, (2) to show
that this hypothesis clears up at least some of it,
and (3) to show that the results from the applica-
tion of the hypothesis are reasonable and valuable.

There are at least three points where confusion
exists. The first is clearly stated by Whipple,


" Manual of Mental and Physical Tests, " as fol-
lows. " The question arises: shall efficiency be
measured in terms of quality, excellence, delicacy,
or accuracy of work, or shall it be measured in
terms of quantity, rate, or speed of work? For
this question no general answer can be given."
Certain expedients are then suggested, but no
final and exact program is outlined. An attempt
will be made to show that the hypothesis of work
clears up the problem of the true relation between
quantitative and qualitative scales, which is the
real problem propounded in the foregoing quota-
tion. Another source of confusion, distinct, but
indirectly included by Whipple in the lines just
quoted, lies in the treatment of the time element
involved in testing. This, when considered at all,
is ordinarily carried as a separate index; but in
many cases there is a tendency to neglect it
entirely, often with grave results, as happens
when two schools are compared in handwriting,
without any consideration of the time involved
in the production of the specimens. The need
for a separate index vanishes under the hypothe-
sis of work, and time receives its legitimate and
necessary emphasis. The third source of confu-


sion is in the conception of efficiency itself. This
conception is vague and indefinite. Various defi-
nitions are contending for recognition. All school
measurement is supposed to be directed toward
the determination of relative efficiency, and yet
there is disagreement as to what constitutes true
efficiency. There can be no such disagreement
under the hypothesis of work.

These claims for the hypothesis must now be
more closely examined and tested. This task will
be furthered by an analysis of mechanical work
and rate-of-work. As already indicated, before
the time of Watt the scales of feet, of pounds, and
of minutes, were in use. It was therefore possible
to know that a force of 5047.00 pounds was at
work where it was found necessary to exert
another force of 5047.00 pounds against it as
in lifting against the force of gravity. It was
also easily seen that another valuable formulation
could be made if distance were included. To say
that one machine lifted a weight of 5047.00 pounds,
and another a weight of 5556.00 pounds, led
naturally to the idea that the second machine was
the stronger; but as soon as the distance was
taken into consideration a doubt was raised. If


the first machine raised 5047.00 pounds four feet,
and the second machine raised its 5556.00 pounds
four feet or more the doubt as to the greater

strength of the second madhine did not exist.

But if the first machine raised 5047.00 pounds
four feet and the second machine raised 5556.00
pounds three feet, indefiniteness as to strength
was apparent. It was possible to carry the two
indexes in each case (5047.00 pounds lifted four
feet, and 5556.00 pounds lifted three feet) and to
get certain rather valuable results. One could
say that he preferred the smaller amount lifted
the greater distance, or the larger amount lifted
the smaller distance; but the computation of work
from these data made a single index possible, put
definiteness into exact comparison of the two, and
so opened the new realm as previously mentioned.
Quoting from a modern text in physics:
" "When a body acted upon by a force moves in
the direction in which the force is acting, work is
said to be done. * * * The amount of work
done is measured by the product of the force by
the distance which the body moves along tlie line
of the action of the force. Thus when a two
pound weight is raised three feet, it moves a dis-


tance of three feet against a force of two pounds
and therefore six foot-pounds of work is done
against the force of attraction of the earth."*

Work, therefore, in Mechanics means force
acting through space, and is computed by the
formula W = F X S. Where work is to be con-
sidered, force alone means nothing and space
alone means nothing; but force acting through
space means ivork, and a certain unit of force
(the pound) acting through a certain unit of
space (the foot) means a certain unit of work
(the foot-pound). This unit of work may be
briefly expressed as unit force acting through
unit space. By means of this unit the two ma-
chines above referred to may be definitely com-
pared as to the work they do. One machine did
work equal to 5047.00X4.00, or 20188.00 foot-
pounds. The other did work equal to 5556.00 X
3.00, or 16668.00 foot-pounds. The relative work-
ing ability of the two machines is definitely ex-
pressed by the ratio of 20188.00 to 16668.00.

But there is still another element to be con-
sidered; viz., that of time. The amount of work

*Kimball " College Physics/'


is the same whether 5047.00 pounds be lifted 4.00
feet in one minute or in one hour or in one year;
but it is often important to know for various rea-
sons, at what rate this work can be delivered.
Hence another unit (a certain amount of work
delivered in a certain time) becomes necessary.
If a definite amount of work in a definite time is
taken, it is not important just what the amount
or the time may be, except for considerations of
convenience. But if there is no unit agreed upon,
two indexes must be carried as before, and com-
parisons are again cumbersome. 20188.00 foot-
pounds in five seconds, must perhaps be compared
with 16668.00 foot-pounds in 5y 2 seconds. In
order to do this it must all be put upon the basis
of amount delivered in one second by dividing
the number of foot-pounds of work by the time.
20188.00 foot-pounds divided by 5.00 = 4037.60
foot-pounds per second; 16668.00 foot-pounds

divided by 5.50 = 3030.54 foot-pounds per second.

These can now be compared with each other.

But it is still better to have a standard unit of
accomplishment per second and compare all other
accomplishments with the unit. Watt selected as
the unit of rate-of-work the number of foot-


pounds per second accomplished by the average
horse (550.00 foot-pounds per second). He could
have used any other number, but this number
proved convenient. Using it as a unit, it is seen
that the machine which did 4037.00 foot-pounds
per second was a 7.34 horse-power machine. The
machine which did 3030.55 foot-pounds per second
was a 5.51 horse-power machine. These two
results admit of immediate and perfect compari-
son, and the formulation of this method of com-
puting rate-of-work (or power, as the physicist
calls it) opened to Mechanics the second part of
the new realm, as the computation of work itself
opened the first part of that realm.

In attempting to appropriate for Education
this new field of work and rate-of-work (power)
it is necessary to formulate units of work and
rate-of-work (power) based upon either an anal-
ogy to, or an identity with, force acting through
space in time. Examination of the situation
seems to show a real identity. That which is
measured in Education is always some kind of
expression through movement occurring in space,
which movement is controlled (changed) either in
direction or magnitude by some agent. The dif-


ferences which we measure in handwriting are
differences in direction and magnitude of motion,
registered on paper in the form of letters. Even
thought itself becomes manifest and can be meas-
ured only in terms of expression, which expres-
sion is in movement, resolved in the last analysis
into changes in direction or magnitude. Now the
only name the world has ever had for that which
changes the motion of a body, either in direction
or amount, is force. There seems to be no reason
for calling the agent behind expression by any
other name than force. It meets the definition of
force, and is measured as all force must be ; i. e.
in terms of its products. There is therefore an
identity between one element in units of work
and rate-of-work (power) in Mechanics, and the
same element in Education. (This affirmation of
identity is meant to carry only so far as the
assertion that the agent behind achievement in
Education is a force. This force may differ from
other forces, just as electrical force probably
differs from gravitational force etc.)

But all of the movements which are initiated
and controlled by the force, take place in space
and time. That is, the force acts through the


space in the production of the given movement in
the given time. In handwriting when a word is
written, the force (or control) acts through the
space roughly measured by the linear arrange-
ment of letters, this measurement being exactly
parallel to the rough measurement of space by
paces or other such linear units, used before the
more accurate foot and inch where selected as
units. The addition of the time element here as
elsewhere, provides for the computation of rate-
of-work, or power. This relation between force,
space, and time is not an arbitrary but a natural
and necessary relation. Physics demonstrated
and adopted it; physics did not create it. The
relation between the factors is a universal rela-
tion which is found wherever the three factors
are involved.

Hence it seems inevitable to apply this prin-
ciple in Education in a manner similar to its use
in Mechanics.*

*Reference is made earlier in this paper (page . . ) to
certain attempts (see Whipple, Manual of Mental and
Physical Tests) to correlate these factors. Reference
should also be made to Brown's excellent article on
Beading in the Elementary School Teacher for June,


An attempt will now be made fully to illustrate
and to apply the idea in the field of handwriting,
since it is there that the most suitable scales nec-
essary to the formation of the units are found.
In handwriting there is motion under varying
degrees of control. This control which alters the
direction and magnitude of motion is a force.
But the force here presents a complication of two
factors; viz., conscious direction, which may be
called intelligent force, or intelligence ; and habit,
which is mechanical. It follows that the motion,
then, is a resultant of the action of more than
one force; but this does not alter anything in
relation to the computations. A resultant of two
or more forces is dealt with under the same laws
as are simple forces. The one thing which must
be remembered in this connection is that because
the force, intelligence, is combined with a mechan-
ical factor, the work computed cannot be called
purely mental work but mere penmanship work.

1914, and to others. In all cases, however, which have
come under the observation of the writer of this article,
arbitrary relations have been established among the fac-
tors, and the necessary and permanent relation has been


In the second part of this paper the discussion
of the computation of purely mental work, where
the force involved is intelligence alone, is con-

Now in order to make the formulation of units
possible, there must be a scale of the force and a
scale of the space. Then the standard unit of the
scale of force can be combined with the standard
unit of the scale of space into the standard unit of
penmanship work ; and the standard unit of pen-
manship work, complicated with the standard unit
of a scale of time, can be the standard unit of rate-
of -penmanship work, or penmanship power. But
can the force involved in penmanship work be
measured? Not directly, any more than the force
of gravity can be measured directly. But the
force of gravity is measured by its effects (ten-
sion of a spring), and the force involved in pen-
manship work can be measured by one of its
effects; viz., the amount of quality exhibited by
the handwriting produced. This amount of
quality is, roughly at least, measured by the
Thorndike handwriting scale, and the idea of
such a scale is apparently sound and capable of
refinement. Of this more will be said later. In


the meantime this scale will be used as a means
of continuing the illustration; and it should con-
tinue to be used for purposes of school measure-
ment until a better one takes its place, or until
it is further made more nearly perfect.

Let unit force (or control) be that control
which produces penmanship which exhibits the
amount of quality designated as No. 1 of the
Thorndike scale^ Let unit space be the space
measured by one letter. Then if a person writes
60.00 letters equal to No. 12.00 quality Thorndike
scale, the work involved is force X space or 60.00
X 12.00 or 720.00 units of work. These units
correspond to foot-pounds and should be desig-
nated by some name of similar significance.

It is necessary at this point to guard against
the idea that the plan as outlined above
identifies force with quality of handwriting, and
space with quantity of handwriting. The quality
of the writing is not the force, but it is the
measure of the force; the number of letters is
not the space, but it is the measure of the space.

Since quantity and quality are here mentioned,
it seems best to discuss them further in order to
show that the plan does give the combination of


quantitative and qualitative scales which solves
the vexed question (as claimed earlier in the
paper). When it is said that a person does 60.00
letters of No. 12.00 quality in a minute, and work
is computed by finding the product of 60.00 and
12.00 according to the formula W = F X S,
viewed superficially it seems as if force were
identified with quality and space with quantity,
and that the two (quantity and quality) were
merely multiplied together as a solution of the
quantity-quality difficulty. But force is not iden-
tified with quality nor space with quantity; and
when 60.00 is multiplied by 12.00 force is not
being multiplied by space (as the formula F X S
would seem to imply) but a measure of force is
multiplied by a measure of space, as previously
indicated. Neither when 60.00 is multiplied by
12.00 is quality multiplied by quantity; but a
quantity of quality, used as a measure of force,
is multiplied by another quantity of quality, used
as a measure of space. The Thorndike scale is
a quantity-quality scale. No. 1 handwriting as
measured by the scale exhibits a certain amount
(quantity) of handwriting quality; No. 12.00
handwriting, following the assumption of the


author of the scale, exhibits an amount of hand-
writing quality 12.00 times as great as that
exhibited by No. 1 handwriting. That is to say
that what we designate as No. 12.00 quality is not
quality alone, but quantity of quality. It is the
same with space. The unit of space in writing
is the letter. This is rough, as has been admit-
ted, but letters arranged in linear fashion meas-
ure the space much as it might be measured by
more or less irregular paces. 60.00 paces means
60.00 movements of pace quality. Spaces and
paces have many qualities all of which are not
held in common, but one quality is common
to both; viz., extension. Hence the extension
involved in paces is often used to measure the
extension of space. In like manner it is pro-
posed to use the quality of extension involved in
letters as a measure of the extension of space.
One letter, therefore, is equal to a unitary
amount (quantity) of the space quality known
as extension. Therefore the multiplication of
60.00 by 12.00 in the problem above cited, and in
all similar problems, while it seems to be a mul-
tiplication of quantity by quality, and actually
settles our confusion as to the relation of these


scales, is really a multiplication of a quantity of
quality by another quantity of quality, or in other
words a multiplication of quantity by quantity.
A summary of points thus far made follows :
Exact measurement in Education is desirable
and much has been done; but there is a realm
into which it has not been extended; this is the
realm of work. Computation of work requires
the consideration of force acting through space.
There must be a quantitative scale of some meas-
ure of the force, made in definite standard units
which can be counted, and the steps of the scale
must bear a definite and known relation to one
another. There must also be a definite scale of
the space, meeting the same conditions as does
the scale for the measurement of the force. Then
the standard units of these scales must be com-
bined into a composite unit of work, comparable
to the foot-pound. So far it has been shown how
the conditions can be met for handwriting: the
Thorndike scale is used as the measure of the
force, No. 1 handwriting being the unit; letters
are used to measure the space, one letter being
the unit. Combining these standard units into
a composite unit of work gives One Letter -


No. 1.00 T scale as the result; the 60.00 letters
No. 12.00 T scale equal 720.00 units of work
(using the formula W = F X S).

Now it becomes necessary to compute rate-of-
work, and a unit must be found. When Watt
wished to compute rate-of-work (power) he had
to settle upon a representative number of foot-
pounds per unit of time as a unit. So for hand-
writing there must be selected a certain number
of letters No. 1.00 T scale per unit of time. Any
number would do, provided that it was definite
and agreed upon, and used by every one. But
for comparative purposes (in order that the unit
may stand as a sort of goal of achievement) it
is desirable that the number be put at some point
near, probably slightly above, the average combi-
nation of speed and control possible for the
average seventh and eighth grade public school
pupil. However, since all seventh and eighth
grade public school pupils write above No. 1.00
T scale handwriting, it is most feasible to get
the average of both speed and control for such

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Online LibraryJames Leroy StocktonExact measurements in education → online text (page 1 of 3)