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,XACT MEASUREMENTS

IN

EDUCATION

JAMES LEROY STOCKTON, A. M. (Columbia)

SUPERINTENDENT ELEMENTARY DEPARTMENT

N<AMAL SCHOOL, WINONA, MINN.

CHICAGO NEW YORK

ROW, PETERSON & COMPANY

EXACT MEASUREMENTS

IN

EDUCATION

JAMES LEROY STOCKTON, A. M. (Columbia)

SUPERINTENDENT ELEMENTARY DEPARTMENT

NORMAL SCHOOL, WINONA, MINN.

CHICAGO NEW YORK

ROW, PETERSON & COMPANY

LB\\3\

o Q

COPYRIGHT, 1915

BY

JAMES LEROY STOCKTON

EXACT MEASUREMENTS

IN

EDUCATION

THESES

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

331230

4 EX.ACT MEASUREMENTS

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-

tention.]

EXACT MEASUREMENTS IN EDUCATION

I

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.

6 EXACT MEASUREMENTS

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

IN EUUCATIDN 7

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.

8 EXACT MEASUREMENTS

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,

IN EDUCATION 9

" 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-

10 EXACT MEASUREMENTS

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

IX EDUCATION H

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-

12 EXACT MEASUREMENTS

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/'

IN EDUCATION 13

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-

14 EXACT MEASUREMENTS

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-

IN EDUCATION 15

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

16 EXACT MEASUREMENTS

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,

IN EDUCATION 17

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

disregarded.

18 EXACT MEASUREMENTS

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-

sidered.

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

IN EDUCATION 19

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

20 EXACT MEASUREMENTS

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

IN EDUCATION 21

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

22 EXACT MEASUREMENTS

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 -

IN EDUCATION 23

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

LB

U3I

IO

o

CD

,XACT MEASUREMENTS

IN

EDUCATION

JAMES LEROY STOCKTON, A. M. (Columbia)

SUPERINTENDENT ELEMENTARY DEPARTMENT

N<AMAL SCHOOL, WINONA, MINN.

CHICAGO NEW YORK

ROW, PETERSON & COMPANY

EXACT MEASUREMENTS

IN

EDUCATION

JAMES LEROY STOCKTON, A. M. (Columbia)

SUPERINTENDENT ELEMENTARY DEPARTMENT

NORMAL SCHOOL, WINONA, MINN.

CHICAGO NEW YORK

ROW, PETERSON & COMPANY

LB\\3\

o Q

COPYRIGHT, 1915

BY

JAMES LEROY STOCKTON

EXACT MEASUREMENTS

IN

EDUCATION

THESES

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

331230

4 EX.ACT MEASUREMENTS

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-

tention.]

EXACT MEASUREMENTS IN EDUCATION

I

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.

6 EXACT MEASUREMENTS

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

IN EUUCATIDN 7

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.

8 EXACT MEASUREMENTS

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,

IN EDUCATION 9

" 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-

10 EXACT MEASUREMENTS

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

IX EDUCATION H

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-

12 EXACT MEASUREMENTS

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/'

IN EDUCATION 13

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-

14 EXACT MEASUREMENTS

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-

IN EDUCATION 15

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

16 EXACT MEASUREMENTS

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,

IN EDUCATION 17

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

disregarded.

18 EXACT MEASUREMENTS

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-

sidered.

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

IN EDUCATION 19

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

20 EXACT MEASUREMENTS

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

IN EDUCATION 21

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

22 EXACT MEASUREMENTS

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 -

IN EDUCATION 23

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