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SUPPLEMENTARY EDUCATIONAL MONOGRAPHS 

Published in conjunction with 
THE SCHOOL REVIEW and THE ELEMENTARY SCHOOL JOURNAL 

No. 18 June 1922 



HOW NUMERALS ARE READ 

AN EXPERIMENTAL STUDY OF THE READING OF 

ISOLATED NUMERALS AND NUMERALS 

IN ARITHMETIC PROBLEMS 



HOW NUMERALS ARE READ 

AN EXPERIMENTAL STUDY OF THE READING OF 

ISOLATED NUMERALS AND NUMERALS 

IN ARITHMETIC PROBLEMS 



By 
PAUL WASHINGTON TERRY 




THE UNIVERSITY OF CHICAGO 

CHICAGO, ILLINOIS 



Copyright 1922 By 
The University of Chicago 



All Rights Reserved 



Published June 1922 



TABLE OF CONTENTS 



List of Plates ix 

List of Tables xi 

List of Selections xiii 

Chapter I. Introduction i 

PART I. PRELIMINARY STUDIES OF THE READING OF 
NUMERALS— BY INTROSPECTIVE METHODS 
Chapter II. Numer.a.ls in Arithmetical Problems — First Pre- 
liminary Stltdy 2 

1. Description of the Study 2 

2. First Reading and Re-reading Distinguished . . 3 

3. Partial First Reading and WTiole First Reading of 
Numerals 3 

4. Individual Subjects Classified as Partial First Readers 

and as WTiole First Readers 6 

5. Re-reading of the Several Numerals .... 7 

6. Re-reading by Individual Subjects .... 9 

7. Summary 10 

Chapter III. R.ange of Correct Recall of Numer.\ls after the 

First Re.ading — Secont) Preliminary Study . . 12 

Description of the Study 12 

Range of Recall of the Several Numerals . . . 14 

Range of Recall — by the Several Subjects ... 16 

Further Evidence as to the Purpose of First Reading . 17 

Items of Recall Not Included in the Classifications . 17 

Summar>^ of Conclusions 17 

Chapter IV. Analysis of the Re-reading of Numerals in Arith- 
metical Problems — Third Preliminary Stutjy . . 19 

1. Description of the Study 19 

2. Objects and Nature of the Re-readings . . . 22 

3. Duration of Re-readings 22 

4. Summary' 23 

Chapter V. Reading Numer.als in Columns — Fourth Pre- 
liminary Stltdy 24 

1. Description of the Study 24 

2. General Description of the Three Main Groups Used 25 

3. Main-Group Patterns for Numerals of Like Length . 27 



I. 
2. 
3- 
4- 
5- 
6. 



VI 



TABLE OF CONTENTS 



PAGE 

4. Variations in Numerical Language .... 29 

5. Influence of Punctuation on the Grouping of^Digits 

of Longer Numerals 31 

6. Persistence of Patterns from the First Reading 
through a Second Reading 33 

7. Summary of Conclusions ^3 

PART II. STUDIES OF THE READING OF NUMERALS— 
BY USE OF PHOTOGRAPHIC APPARATUS 

Chapter \T. Description of the Eye-Movement Studies . . 35 

1. Apparatus Described 35 

2. Three Types of Reading-Materials Used . . . 35 

3. Instructions to Subjects and Description of Subjects 38 

4. Procedure on the Part of the Observer. ... 39 

5. Guide for Reading the Plates 39 

6. Plates I-XV 41-54 

Chapter \TI. First Reading of Numerals in Problems ... 56 

1. Introduction 56 

2. Comparison between the Reading of Numerals and 
Words in Problems 56 

3. Partial and Whole First Reading of Numerals . . 59 

4. The Several Subjects as Partial and Whole First 
Readers 64 

5. Relative Value of Partial and of Whole First Reading 64 

6. Development of the Method of Partial First Reading 66 

7. Summary of Conclusions 67 

Chapter VIII. Re-reading and Computation 69 

1. Two Types of Re-reading of Numerals. ... 69 

2. Methods and Procedures Used in the Process of 
Computation 72 

3. Summary 76 

Chapter IX. The Reading of Isolated Numerals in Lines . . 78 

1. Introduction 78 

Previous Investigations by Gray and Dearborn . 78 

Descriptions of Plates 79 

Plates XVI-XXV 80-84 

2. Two Types of Pauses 85 

3. Differences in the Readings of Numerals of Different 
Lengths 86 

4. The Special Numerals 90 

5. Two Methods of Attack in Reading Numerals . . 91 

6. Summary 92 



TABLE OF CONTEXTS Vll 

PAGE 

Chapter X. Comparisons of Rates of Readixg .... 94 

1. Comparison of the Subjects of the Present Investiga- 
tion with the Subjects of Schmidt's Investigation in 
Respect to Rates of Reading 94 

2. Comparisons of Rates of Reading the Three Types of 
Reading-Materials 96 

3. Summary 97 

Chapter XI, Practical Application to Classroom Teaching . . 98 

1. The Question of Reading in Arithmetic ... 98 

2. Preliminar>' Analytical Reading of Problems . . 99 

3. Application of Partial Reading 100 

4. AppHcation of Re-reading 102 

5. Miscellaneous Applications 104 

Index 107 



LIST OF PLATES 

PLATE AGE 

I. First Reading of Problem 4 by Subject B and His Pro- 
cedure in Solving the Problem 41 

II. First Reading of Problem 4 by Subject G and Re-reading 

the Numeral 1000 42 

III. First Reading of Problem i by Subject W and Multiplica- 

tion Direct from the Problem Card with One Numeral 
Used as the "Base of Operations" 43 

IV. First Reading of Problem 3 by Subject Hb and Re- 

reading the Numerals for Copying 44 

V. First Reading of Problem 2 by Subject Hb, Re-reading 
the First Numeral, and Subsequent Re-reading of 

Both Numerals for Copying 45 

\I. First Reading of Problem 5 by Subject ]\I and Re-reading 

and Copying the Two Numerals 46 

\H. First Reading of Problem i by Subject Hb and Re- 
reading the First Numeral for Copying .... 47 
VIII. First Reading of Problem i by Subject B and the Process 
of Computation with the First Numeral as the "Base 

of Operations" 48 

IX. First Reading of Problem 2 by Subject G and the Pro- 
cess of Adding the Two Numerals 49 

X. First Reading of Problem i by Subject G and the Process 

of Computation 50 

XL First Reading of Problem i by Subject M, Re-reading 

Words, and the Process of Computation ... 50 
XII. First Reading of Problem 3 by Subject G and the Process 

of Computation 51 

XIII. First Reading of Problem 5 by Subject H and the Process 

of Computation 52 

XIV. First Reading of Problem 3 by Subject W with Partial 

Reading of Numerals 53 

XV. First Reading of Problem 5 by Subject G and the Process 

of Computation 54 

XVI-XVII. Reading of Isolated Numerals by Subject G . . .80 

XVIII-XIX. Reading of Isolated Numerals by Subject H . . . 81 

XX-XXI. Reading of Isolated Numerals by Subject ]\I . . .82 

XXII-XXIII. Reading of Isolated Numerals by Subject B . . .83 

XXIV-XXV. Reading of Isolated Numerals by Subject W . . .84 

ix 



LIST OF TABLES 

TABLE PAGE 

I. Partial First Readings and Whole First Readings by Ten 

Subjects in Seven Problems 4 

II. Ranks of Numerals according to Percentages of Partial First 

Readings 5 

III. Subjects Arranged according to Number of Partial and Whole 

First Readings 7 

IV. Number of Re-readings 7 

V. Ranks of Numerals according to Percentages of Re-readings . 8 

VI. Number of Re-readings by V^arious Subjects .... 9 

VII. Range of Correct Recall of Numerals from First Reading of 

Problems 14 

VIII. Varying Ranges of Correct Recall of Three- to Seven-Digit 

Numerals by the Several Subjects 16 

IX. Numerals and Words Read at Each Re-reading Together with 

Number of Seconds Required . .' 21 

X. Number of Seconds Required for First Reading and for 

Re-reading of Problems 23 

XL Main Group Patterns Used in Reading One- to Seven-Digit 

Numerals in Columns 26 

XII. Number of One-. Two-, and Three-Digit Groups Used in 

Reading Numerals of the Several Digit-Lengths in Columns 28 

XIII. Number of Simple (3) and of Complex (1-2) Three-Digit 

Groups Used in Reading Five-, Six-, and Seven-Digit 
Numerals in Columns 28 

XIV. Numerical Language Patterns Used by Subject G in Reading 

Numerals in Columns 30 

XV. Effect of Punctuation on the Number of Two- and Three- 
Digit Groups Used in Reading Five-, Six-, and Seven-Digit 
Numerals in Columns 32 

XVI. Description of the Five Problems Read before the Photo- 
graphic Apparatus 36 

XVII. Average Number of Digits Included in a Pause on Numerals 
Contrasted with Average Number of Letters Included in 
a Pause on Words during First Reading 57 

xi 



xii LIST OF TABLES 

TABLE PAGE 

XVIII. Average Duration of Pauses in Fiftieths of a Second on 
Numerals Contrasted with Average Duration of Pauses on 
Words during First Reading 58 

XIX. Percentage of Regressive Pauses on Numerals Contrasted 
with Percentage of Regressive Pauses on Words during 
First Reading 58 

XX. Duration in Fiftieths of a Second, and Serial Order of the 
Several Pauses Used in Whole and Partial First Readings of 
Numerals 60 

XXI. First Reading of Numerals in Problems Contrasted with the 

Reading of Isolated Numerals of Corresponding Lengths . 62 

XXII. Reading of Numerals in Problems by Partial First Readers 
Contrasted with Reading of Numerals in Problems by 
Whole First Readers 64 

XXIII. Reading of Words in Problems by Partial First Readers 

Contrasted with Reading of Words in Problems by Whole 
First Readers 65 

XXIV. Type of Re-reading Given to Numerals, or to Numerals and 

Words, before Beginning of Computation .... 70 

XX\'. Two Methods of Proceeding with Numerals for Purposes of 

Computation after the First Reading 72 

XXVI. Analysis of the Process of Computation in Which One Numeral 

Is Used as the "Base of Operations" 75 

XXVII. Average Number of Pauses and Average Reading-Time per 

Numeral for Isolated Numerals 87 

XXVIII. Average Pause-Duration of Isolated Numerals of the Several 

Digit-Lengths 88 

XXIX. Number of Isolated Numerals of the Several Digit-Lengths 
Which Were Read with Various Numbers of Pauses per 
Numeral 89 

XXX. Reading of Special Numerals Compared with Reading of 

Other Isolated Numerals of Corresponding Digit-Lengths . 90 

XXXI. Speed and the Two Methods of Attack Used in Reading 

Isolated Numerals 91 

XXXII. Subjects of the Present Investigation Compared with Those 

of Schmidt's Investigation in Respect to Speed of Reading . 95 

XXXIII. Comparative Data from Readings of Five Problems, Ordinary 

Prose, and Isolated Numerals 96 



LIST OF SELECTIONS 

SELECTIO;^ PAGE 

1. Five Problems Read before Photographic Apparatus . . . . 36 

2. Isolated Numerals Read before Photographic Apparatus . . . 37 

3. Ordinary Prose Read before Photographic Apparatus . . . . 38 



CHAPTER I 



INTRODUCTION 



Numerous investigations in the psychology of reading and in the 
measurement of reading abiUty have developed a valuable body of 
scientific information about the methods of reading words and sentences 
of the ordinary kind, but the reading of numerals has had only 
occasional attention in these studies and that merely in an incidental 
way. Such work on reading as has been done in the field of arithmetic 
has concerned itself with isolated numerals rather than with numerals 
set in sentences or problems. 

The present investigation is concerned \sdth the reading of numerals 
both in separate lines and in the context of arithmetical problems. The 
first part of this report describes a series of studies, based on introspective 
observations, of some of the relatively definite and highly developed 
habits of graduate students in reading numerals both isolated and in 
problems. The second part of the report deals with this same class of 
readers and with the same kinds of reading materials, but employs 
objective methods. 

For the first part of the report the data were obtained by recording 
introspective observations which were made by the subjects after they 
had read a set of arithmetical problems in which numerals occurred. 
The introspections were supplemented by directly observing and report- 
ing the results of the reading of isolated numerals. The information 
secured in this preliminary work serves as a basis for the interpretation 
of the data obtained in the second part of the investigation in which 
photographic records of eye-movements were secured. The whole 
investigation is only an introduction to the study of the methods 
employed by children in their gradual acquisition of the power of reading 
nurnerals. This large genetic study was the original aim of the present 
investigation. The intricacies of the problem turned what was originally 
thought of as an introductory investigation into an elaborate detailed 
study. Yet educational implications are present even in this preHminary 
work. Through the study of adults, a body of facts has been discovered 
which throws light on methods of reading problems in arithmetic to 
which children must ultimately attain, whatever be the initial habits 
through which they pass in the course of their development. 



PART I. PRELIMINARY STUDIES OF THE READING OF 
NUMERALS— BY INTROSPECTIVE METHODS 

CHAPTER II 

NUMERALS IN ARITHMETICAL PROBLEMS— FIRST 
PRELIMINARY STUDY 

I. DESCRIPTION OF THE STUDY 

For the first preliminary study seven simple arithmetical problems 
were used. These were so formulated that each included a set of from 
one to four numerals. The problems were so made up that while the 
numerals in each one were similar, those which were used in the different 
problems exhibited variations in length. The numerals in problems 
I, 3, 5, and 6 are two in number in each case, but vary in digit-length 
from one to seven digits. Problem 2 includes a set of four numerals, 
each numeral being made up of from one to two digits; Problem 4 has 
four numerals made up of from three to four digits; and Problem 7 
uses a familiar date and two numerals of exceptional character, namely 
100 and 1000. 

The problems which were used in the first preliminary study are as 
follows: 

1. At 65 cents a dozen, what will 8 dozen eggs cost ? 

2. A man buys 5 tons of coal at 9 dollars a ton, and 3 cords of wood at 
12 dollars a cord. What is the total cost of both of them? 

3. A farmer owns one farm of 286 acres, and another of 1754 acres. How 
many acres does he own all together ? 

4. A wholesale grocer has 4375 cases of canned corn. To three customers 
he shipped 286, 2567, and 615 cases respectively. How many did he have 
left? 

5. If electricity travels on a wire at the rate of 288,106 miles per second, 
how long will it take to travel 144,053 miles? 

6. If one railroad uses 2,191,504 cross ties during the year, and another 
railroad 617,450 in the same period of time, how many more tics does the one 
use than the other ? 

7. During 1918 a citizen bought four $100 Liberty bonds, and two $1000 
bonds. What is the total of the sum he invested in these bonds ? 

Ten graduate students of the School of Education of the University 
of Chicago were asked to solve all of the problems. They were instructed 



NUMERALS IN ARITHMETICAL PROBLEMS 3 

to work the problems rapidly and accurately, and with pencil and paper 
or without, as they preferred. They were urged to observe faithfully 
the arithmetic problem-solving attitude from the beginning of a problem 
to its solution. After the answer of each problem was recorded the 
subjects were asked to describe in detail their experiences while reading 
the problem with special reference to the numerals. After the lirst 
problem had been solved and the reading of its numerals described, 
the subjects began to note the kinds of experiences they were asked to 
observe. As a result they were able to give the desired description 
more promptly and easily with each successive problem. These were 
recorded and condensed into tables I-VI. 

2. FIRST READING AND RE-READING DISTINGUISHED 

The most obvious and general fact noted in the records of the several 
subjects was their clear and unmistakable differentiation of the reading 
of a problem into two definite and distinct phases differing in time and 
in purpose, namely, the first reading and the re-reading. Subsequent 
sections of the investigation emphasize the importance of this observation 
concerning the distinction between two phases of the reading of a problem. 
The general procedure of each subject was, first, to read the problem 
through "to get the sense" or "to see what was to be done with the 
numbers," and secondly, to re-read one or all of the numerals, and 
sometimes also a few of the accompanying words. These re-readings of 
the numerals were for such purposes as "verification" of their first 
reading, or the "cultivation of assurance" before copying the figures on 
paper for computation. The subject with one or two exceptions was 
not aware that he habitually followed such a procedure until he began 
to make introspective observations of his habits. 

3. PARTIAL FIRST RESIDING AND WHOLE FIRST READING OF NUMERALS 

The knowledge gained during the first reading was found to be very 
different in different cases. Subjects sometimes perceived numerals as 
merely numerals; sometimes they noted only the first digit or the first 
two digits. At times they noted the number of digits but did not attend 
to any one in particular. Again they reported a numeral as large or as 
small, or as larger or smaller than some other numeral. Sometimes 
they noted its location in the typewritten line. Frequently two or 
more of these items were included in the general perception. In all 
such cases as have been described, the perception lacks detail and preci- 
sion. It is evidently a kind of cursory preliminary recognition of the 



4 HOW NUMERALS ARE READ 

general character and setting of the numeral. Its value consists in the 
fact that it permits the subject to think about the problem without 
entering at first into the minute details of solution. 

There were cases, however, even in the first readings, in which the 
subjects gave attention to the identity and place of every component 
digit. In addition these careful readers noted also the character of the 
numeral, observing whether it was a whole number or a decimal. They 
also gained a notion of the magnitude of the numeral as determined by 
the number of digits. With each of the subjects, cases were found in 
which the recognition was full and detailed. Such cases were recorded 
as "whole first readings." Any reading which fell short of complete 
detail was recorded as a "partial first reading." 

The results of the introspections are given in full in Table I. Begin- 
ning at the top of the left-hand column, this table should be read verti- 
cally as follows: Problem i contains the two numerals 65 and 8, and these 

TABLE I 

Partial First Readings and Whole First Readings by Ten Subjects in 
Seven Problems 





Problems 




I 


2 


3 


4 


S 


6 


7 


1 


65 
8 

20 
3 
16 


s 

9 
3 
12 
40 
12 
28 


386 
I7S4 

20 
10 
10 


4375 

286 

2567 

61S 

40 

31 

9 


288,106 
144,053 


2,191,504 
617,450 


1918 


100 


Numerals given in problems < 




1000 


[ 










Total number of readings 


20 
13 

7 


20 
II 
9 


10 

10 


20 




I 




19 























were read altogether a total of twenty times by the ten subjects. Of the 
twenty readings, three were partial first readings, sixteen were whole 
first readings, and one could not be classified. 

Examination of Table I shows that the numerals in problems 4, 5, 
and 6, which were all longer ones with three to seven digits, were read 
partially more than half of the times and accordingly have percentages 
of partial first readings of 50 or more. The numerals in problems 1,2, 
and 7, on the other hand, were read wholly more than half of the times 
and have percentages of partial first readings of only 30 or less. The 
longer numerals are seen to have been read partially more frequently, 
while the shorter numerals are read in detail more frequently. Attention 
should be called at this point, however, to the fact that the whole first 
reading of a numeral does not necessarily mean that the numeral will 



NUMER.\LS IN ARITHMETICAL PROBLEMS 5 

not be re-read. On the contrar}^ later discussions in this report will 
show that almost all numerals were re-read after the first reading, 
including even those which were read in detail during the first reading. 

In order to bring out the relation between partial and whole readings 
and the character of the numerals, Table II was compiled. This table 
shows the ranks of the various numerals with reference to the frequency 
of partial readings. Percentages were calculated by dividing the number 
of times a numeral was partially read by the total number of readings 
of that numeral. 

Among the longer numerals a greater digit-length appears to cause 
a large percentage of partial readings. Such a comparison between 
numerals of different lengths is significant when the same number of 
numerals is used in the various problems compared. Problems 3, 5, 

TABLE II 
Ramks of Numerals according to Percentages of Partial First Re.adings 



Ranks 


Percentage 
of Partial 

First 
Readings 


Description of 
Numerals 


Numerals Read 


I 

2 

2 


77 
65 
55 
50 
30 
15 
5 



Four three- to four-digit 
Two six-digit 
Two six- to seven-digit 
Two three- to four-digit 
Four one- to two-digit 
Two one- to two-digit 
Two familiar 
Date 


4375; 286; 2567; 615 
288,106; 144,053 
2,191004; 617,450 
386; 1754 
5; 9; 3; 12 
65; 8 


4 


c 


6 


7 


8 


1918 





and 6 each have two numerals. The sLx digit numerals of Problem 5, 
and the six- and seven-digit numerals of Problem 6 were given respectively 
65 per cent and 55 per cent of partial first readings. The three- and four- 
digit numerals of Problem 3, on the other hand, were given only 50 per 
cent of partial first readings. 

Of the numerals which were usually given a whole first reading, the 
date 1918 stands out as different in character from other numerals of 
like length. It is the only one which was never partially read. The 
very familiar numerals 100 and 1000 with the dollar sign attached were 
like the date for the most part, in that they were read partially only 
once. In this case, the partial reading was revealed by a mistake made 
by the reader — the numeral 100 was read as i.oo instead of as 100. 

The inclusion of several numerals in the same problem appears to 
induce a greater proportion of partial first readings. In Problem 4, 



6 HOW NUMERALS ARE READ 

where there are four numerals, the percentage of partial readings is 77, 
whereas in Problem 3, where only two numerals appear, the percentage 
of partial readings is only 50, although the numerals in both problems 
are of the same lengths. The explanation of this fact seems to be that 
the subject loses interest in the numerals when many of them appear 
together. Consequently he does not make the radical adjustments in 
rate of reading which would be necessary for the careful reading of a 
series of several mmierals. 

The validity of this explanation is supported by the results of another 
comparison of a similar type which can be made from the tables. The 
numerals in Problems i and 2 are all one or two digits in length. There 
are two numerals in Problem i, and four in Problem 2. The percentage 
of partial readings in Problem i is 15, whereas in Problem 2 the percentage 
of partial readings is 30, or twice as great as that in Problem i. In this 
comparison, as in that above, where four numerals of a certain digit- 
length appear in a problem, they were more frequently read partially 
than when only two such numerals appear. 

The first numeral in a problem tends to be given a more careful and 
thorough reading than any of the other numerals in the same problem. 
The basis for this statement is found in the fact that in three of the four 
problems which employ the longer numerals, the first numeral receives 
a greater number of whole first readings than any of the numerals that 
follow. According to the original tabulations, the first numeral in 
Problem 5, 288,106, was given five whole first readings whereas the 
second numeral, 144,053, was given only two whole first readings. 
A similar preponderance of whole readings appears in favor of the first 
numeral in both problems 3 and 4. A comparison of the foregoing kind 
cannot be drawn, however, between shorter numerals, since they were 
almost invariably given whole first readings regardless of their position 
within the problem. 

4. INDIVIDUAL SUBJECTS CLASSIFIED AS PARTIAL FIRST READERS AND 
AS WHOLE FIRST READERS 

In Table III the ten subjects are arranged in order from left to right 
according to the number of their partial readings. They range from 14 
partial first readings by G to no partial first readings by Subject H. 
The total in each case is 19 readings. G, Bl, and S show a significant 
preponderance of partial readings. H, T, D, and K, on the other hand, 
exhibit a preponderance of whole first readings, each of the latter showing 
12 or more such readings out of a possible 19. These seven subjects 



NUMERALS IN ARITHMETICAL PROBLEMS 



can accordingly be classified into two groups as partial first readers and 
whole first readers. The partial readers read partially not only the 
three- to seven-digit numerals usually thus read, but also several of the 
other numerals which are usually read in detail. Similarly the whole 

TABLE III 

Subjects Arranged according to Number of Partial and 

Whole First Re.adings 













Subjects 












G 


Bl 


S 


P 


De 


K 


Ko 


D 


T 


H 


Partial first readings 


14 
5 


13 
6 


12 

7 


9 

10 


8 
II 


7 
12 


7 
II 

I 


6 
13 


3 
16 



19 


Doubtful 























readers read in detail not only the nine one- and two-digit numerals and 
the familiar numerals which are usually so read but also several of the 
other numerals which are usually read only partially. 

Subjects Ko, De, and P are not so distinctly marked off as the seven 
discussed above. However, since they show a preponderant number of 
whole first readings, they may be classified as whole first readers. When 
they are so classified there are seven whole first readers and only three 
partial first readers. There were, therefore, more than twice as many 
whole first readers as partial first readers among the subjects of this 
study. 

5. RE-READING OF THE SEVERAL NUMERALS 

After the first reading of a problem it was left entirely to the choice 
of the subject whether he should or should not re-read the numerals in the 
problem. In all but a few cases, which are classified as " Doubtful, " the 
reports of every subject show when he re-read any individual numeral. 
Table IV gives, for each set of munerals, the number of times they were 

TABLE IV 

Number of Re-readings 





Problems 




I 


2 


3 


4 


5 


6 


7 


Numerals given in problems 


65 
8 

II 
9 


5 

9 

.3 

12 

38 


386 
1754 

20 


4375 

286 

2567 

61S 

39 

I 


288,106 
144,053 


2,191,504 
617,450 


1918 


100 
1000 














17 

I 
2 


20 



10 


13 
7 


Numerals not re-read 


Doubtful 


2 














1 





8 



HOW NUMERALS ARE READ 



re-read, the number of times they were not re-read, and the number of 
doubtful cases. Table V gives the ranks of the several sets of numerals 
according to the percentages of re-readings. The percentage of re- 
readings for any set of numerals was found by dividing the total num- 
ber of re-readings which the numerals of the set received, by the total 
number of re-readings which it was possible for them to have received. 
Examination of Table IV reveals the fact that the numerals of every 
set but one were very generally re-read. The longer numerals were 
re-read almost without exception. From Table V it is seen that the 
four sets of numerals from three to seven digits in length which are found 
in problems 3, 4, 5, and 6, received 85 per cent, 97.5 per cent, and 100 
per cent, respectively, of the numbers of possible re-readings. In two 
cases only were numerals of this length reported as not re-read, and only 

TABLE V 
Ranks of Numerals according to Percentages of Re-readings 



Ranks 


Percentage 

of 
Re-readings 


Description 
of Numerals 


Numerals Read • 


I c 


100 
100' 

97-5 
95 
85 
65 
55 



Two six- to seven-digit 
Two three- to four-digit 
Four three- to four-digit 
Four one- to two-digit 
Two six-digit 
Two familiar 
Two one- to two-digit 
Date 


2,191,504; 617,450 


I c 


386; 1754 


Z 


4375; 286; 2567; 615 


4. 


5; 9; 3', 12 


e 


288,106; 144,053 


6 


100; 1000 


7 


65; 8 


8 


1918 







two cases were reported as doubtful. Since there are ten ordinary 
numerals of three- to seven-digit lengths, it was possible for these 
numerals to have been re-read a total of one hundred times by the ten 
subjects. Of this number of possible re-readings the three- to seven-digit 
numerals actually were re-read ninety-six tunes. 

Even the short one- and two-digit numerals and the familiar numeri- 
cals were very generally re-read, although usually they had been given 
whole first readings. Each of these sets of numerals, with the excep- 
tion of the familiar date numeral, was re-read 50 per cent or more 
of the possible times. The one- and two-digit numerals, in problems 
which contain only two numerals, were re-read 55 per cent of the 
possible times, but the numerals of this same length, in problems 
which contain four numerals, were re-read 95 per cent of the possible 
times. The familiar numerals with the dollar sign attached were re-read 
65 per cent of the possible times. The only numeral never re-read is the 



NUMER.\LS IX ARITHMETICAL PROBLEMS 



familiar date 19 iS, which was not necessary in any way to the solution 
of the problem in which it appears. 

6. RE-READING BY INDIVIDUAL SUBJECTS 

The ver}' large percentages of re-readings of the several sets of 
numerals imply that most of the individual subjects are persistent 
re-readers. Examination of Table VI, in which the facts for each indi- 
vidual subject are displayed, proves that such is the case. Subjects 
G, Bl, and S who had read most of the numerals only partially during 
the first reading, proceeded to re-read the numerals before they began to 
solve the problems. They re-read not only the numerals at which they 
had merely glanced the first time, but also those which they had read in 
detail at first reading. None of these three subjects showed a number 
of numerals not re-read equal to the number of numerals which he had 
read in detail at first reading. 

TABLE VI 

Number of Re-re.\dings by Various Subjects 





Subjects 




G 


Bl 


S 


P 


De 


K 


Ko 


D 


T 


H 


Number of re-readings 

Numerals not re-read 

Doubtful 


16 
3 


18 

I 


16 

3 


13 
6 


17 

I 


17 

2 


18 
I 


16 
3 


12 

S 

2 


18 
I 























The seven subjects who were classified as whole first readers re-read 
practically as many of the numerals as the partial first readers. Subject 
H, who gave all of the numerals a whole first reading, re-read as many 
numerals as any partial first reader. The smallest numbers of numerals 
re-read are found in the records of the whole first readers, T and P. 
These same subjects have the largest numbers of numerals not re-read. 
Besides the familiar numerals and the one- and two-digit numerals, T 
did not re-read the numeral 4375 and P did not re-read 288,106. 

At this time attention should be called to the fact, which will be 
elaborated in chapter viii, that the re-reading of numerals appears to 
be very closely connected with copying them on paper for computation. 
With rare exceptions the subjects, when solving the problems of this 
study, followed the procedure of copying the numerals and computing 
with pencil. In the chapter referred to above, however, it was found 
that several subjects, including G and H of the present study, solved the 
problems which were read in the second part of the investigation "men- 



lO HOW NUMERALS ARE READ 

tally" and directly from the text. Such subjects when solving problems 
"mentally" did not re-read the numerals. 

After each subject had solved all of the problems, he was asked to 
describe any individual attitudes or previous experiences which he 
believed had affected in an important way his methods of reading 
numerals in arithmetical problems. Three of the subjects gave descrip- 
tions which threw light upon their procedures as reported above. 
Subject T stated that he has had from his early school days unusual ability 
in retaining by visual memory both long and short numerals which he had 
read. This ability had recently undergone intensive training in the 
form of much reading and copying of army serial numbers, which his 
work as company clerk in the army required. He believes his habit is 
to read numerals in detail wherever he sees them, and that he could have 
solved perhaps all of the seven problems immediately after the first 
reading without looking at the numerals again. He goes on to say, 
however, that notwithstanding his careful first reading of the numerals, 
he is in the habit of re-reading them before beginning computation 
"in order to be sure." 

Subject H, who gave every numeral a whole first reading, states 
that early difficulties with arithmetical problems caused him to develop 
habits of great caution in reading and solving them. He describes his 
procedure as beginning with a careful, complete reading of every numeral 
in a problem when he first comes to it. He then returns to re-read 
every numeral and sometimes some of the words. 

Subject G, whose record presents the largest number of partial first 
readings of the numerals, reports that he intends to obtain "only a 
general idea" of the numerals from the first reading, especially if they 
are long ones. He explains that he chose this attitude toward the 
numerals after he had learned through experience that he was unable 
to recall them for computation and "had to go back for them anyhow." 

7. SUMMARY 

A summary of the results of the first preliminary study on how 
numerals in arithmetical problems are read includes the following 
points: 

I. The subjects distinguished two phases in the reading of problems, 
namely, a first reading and a re-reading. The purpose of the first 
reading is to discover the conditions of the problem, while that of the 
re-reading is to perceive the numerals accurately for use in computation. 



NUMER.\LS IN ARITHMETICAL PROBLEMS II 

2. Two ranges of perception of numerals during the first reading 
are distinguished, namely, whole first reading and partial first reading. 

3. Shorter numerals and very familiar numerals more frequently 
receive whole first readings, whereas longer numerals more frequently 
receive partial first readings. 

4. The first munerals in problems which have numerals of three to 
seven digits in length, are commonly given whole first readings. 

5. When as many as four numerals appear in a problem they receive 
a greater proportion of partial first readings than in those cases in which 
only two numerals of the same digit-length appear. 

6. Subjects differ widely in their habits. Some are predominantly 
whole first readers, others are partial first readers. 

7. Numerals of all lengths and types, when they are used in computa- 
tion, are very generally re-read for computation. 

8. All subjects persistently re-read numerals when they begin to use 
them in computation. 



CHAPTER III 

RANGE OF CORRECT RECALL OF NUMERALS AFTER THE FIRST 
READING— SECOND PRELIMINARY STUDY 

I. DESCRIPTION OF THE STUDY 

The purpose of the second preliminary study was to obtain further 
information as to the nature of the readings of the numerals during the 
first reading of a problem. The general plan followed for the accomplish- 
ment of this purpose was to have the subjects report all of the details 
of the numerals, which they were able to recall immediately after the 
first reading. 

The subjects were seven graduate students in the School of Education 
of the University of Chicago. Of the seven subjects, only one. Subject 
G, had served in the first preliminary study. The materials which they 
read were twelve simple arithmetical problems similar to those used in 
the first preliminar\' study. SLx of the problems were work problems 
in the sense that each of them was worked until an answer was found. 
No questions, however, were asked concerning the numerals of these 
problems. Their function was to help the subjects maintain throughout 
the study the problem-solving attitude by actually solving problems. 
The readings of the other six problems were the bases of the data which 
were used in the study. In the context of these problems a total of 
sixteen numerals varying in length from one to seven digits was included. 
Two or three numerals were placed in each of five of the problems. 
In Problem D, however, four numerals appeared tpgether. Numerals 
of similar length were placed in the same problem, as was done in the 
problems of the first preliminary study. All of the problems were 
presented in typewriting. 

The problems which were read for data are as follows: 

Problem A 

At 43 cents a dozen, what will 2 dozen eggs cost ? 

Problem B 

If a man buys 5 tons of coal for 45 dollars, what will he have to pay for 
8 tons ? 

Problem C 

A farmer owns one farm of 246 acres, and another of 1754 acres. How 
many acres does he owti altogether ? 



CORRECT RECALL OF NUMERALS AFTER FIRST READING 13 

Problem D 

A wholesale merchant had on hand 1000 cases of canned corn. From 
three factories he bought 1276, 91 and 718 cases respectively. How many did 
he then have ? 

Problem E 

If one railroad uses 2,981.534 cross ties during the year, and another rail- 
road 617,453 in the same period of time, how many more ties does the one use 
than the other ? 

Problem F 

If one man can do a piece of work in 10 days, and another can do the same 
work in 9 days, what should be the wages of the second, if the wages of the 
first are 3 dollars ? 

The subjects were informed that they would be expected to solve 
most of the problems, but that they would be interrupted in their proce- 
dure in some of them by the placing of a cardboard screen over the text 
which was being read. They were told that they would not be informed 
beforehand as to which problems were to be solved and which were to 
be covered with the screen. Accordingly they were urged to maintain 
their normal problem-solving attitude toward every problem that was 
presented. 

The time chosen for the placing of the screen, when it was to be 
used, was the moment when the subject completed the first reading of 
the problem. The subjects were therefore instructed to signal by a 
slight movement of the left hand the moment at which they were 
completing the first reading. This they were able to do without embar- 
rassment after a brief training period with practice problems. The 
screen was used only with those six problems that carried in their 
context the numerals which were to be studied. 

Immediately after the screen was placed over one of these problems 
the subject was asked to give in detail all of the items of the numerals 
which he could recall. After having proceeded in this manner with the 
first problem which was to be studied in this way, the subject understood 
what kinds of details concerning the numerals were desired, and with 
subsequent problems easily and promptly gave such of the details as he 
could recall. 

The results are reported in tables VII and VIII under five classifica- 
tions. The classification, complete, signifies that every digit of the 
numeral was recalled in its true identity and place. The second classifica- 
tion includes correct recall of the digit-length of the numeral and the 
identity and place of at least its first two digits. The third classification 



14 



HOW NUMER.\LS ARE READ 



includes correct recall of the digit-length and the identity of the first 
digit; and the fourth classification includes correct recall of the digit- 
length of the numeral. The fifth classification includes those cases in 
which the subjects were able to recall nothing more of a numeral than its 
presence in the problem. 

2. RANGE OF RECALL OF THE SEVERAL NUMERALS 

The range of correct recall from the first readings of each individual 
numeral is displayed in Table VII. The point which stands out most 
strikingly after a survey of the table is that almost invariably some item 
from the first reading of ever}' numeral is correctly recalled. A glance 

TABLE VII 
Range of Correct Recall of Numerals from First Re.\ding of Problems 





One- and Two-Digit 

NuifERALS 


Three- to Seven-Digit Numerals 


2 

< 

M 




S 


Prob- 
lem 
B 


Prob- 
lem 
F 


1=^ 


Prob- 
lem 
C 


Problem 
D 


Problem 
E 




a 
s 

< 


Numerals read in 






































problems 43 

Total number of 




S 


45 


S 




9 


3 






1754 








7T« 




































readings given 






































each numeral by 






































all subjects 7 


7 


7 


7 


7 


6 


6 


6 


S^ 


6 


b 


b 


b 


6 


b 


7 


7 


50 


103 


Range of correct 






































recall of numer-, 






































als: Complete . . s 


6 


6 


6 


7 


6 


6 


6 


48 


b 


I 


5 


I 





I 








14 


62 


First two digits: 






































and digit-length 5 






6 




6 






17 


6 


4 


.S 


?. 





I 


I 


I 


21 


38 


First digit and 






































digit-length.... 6 


6 


6 


7 


7 


6 


6 


6 


no 


6 


5 


5 


4 


2 


I 


6 


3 


32 


82 


Digit-length .... 6 


6 


6 


7 


7 


6 


6 


6 


.SO 


6 


b 


5 


6 


4 


S 


7 


7 


44 


04 


Merely noticed i 























I 








I 








2 








3 


4 



at the "totals" column at the extreme right shows that some item of the 
numerals in 94 of the 103 total number of readings was recalled. In 
five cases only, the numeral was not even "merely noticed." Such 
frequency of recall implies that in the minds of subjects confronted with 
arithmetical problems to be solved the numerals hold a place of unique 
significance among the other elements of the problems, and in con- 
sequence are noticed almost invariably. 

The range of recall which most frequently follows this habitual 
notice of the numerals also stands out clearly from this table. This 
item is the correct digit-length of the numerals. It was recalled from a 
very great majority of the readings, namely, from 94 of the total 103 
cases. All of the exceptions occur in Problem 4 where four numerals of 



CORRECT REC.\LL OF NUMERALS AFTER FIRST READING 15 

three different digit-lengths occur. In such a situation it was an easy- 
matter for the length of one numeral to be confused with that of another. 
Evidently the number of digits in a numeral is a point of exceptional 
interest to the readers. This is not difficult to understand in view of the 
fact that the number of digits in a numeral is certainly one of the most 
highly significant indications of its value. 

With the shorter numerals, complete recall was achieved in almost 
every instance. In the left-hand section of Table VII it is shown that 
such is the case in 48 instances of a total of 53. Apparently no greater 
demands were made upon the attention of the subjects by the reading 
of one- and two-digit numerals than practically all of them were both 
able and willing to meet. It is also apparent that the effort which was 
habitually used in reading the short numerals so firmly fixed them in 
memory that they were able to be recalled completely. 

The longer numerals on the contrary do not exhibit such large 
preponderances in the higher ranges of recall as are exhibited by the 
shorter numerals. In comparatively few instances the longer numerals 
were completely recalled. The explanation is believed to be due to two 
facts: first, that longer numerals according to common experience are 
more difficult to recall than shorter numerals; and second, that in most 
cases the longer numerals were only partially read during the first 
reading of the problems. 

When reading the longer numerals the subjects evidently gave more 
emphatic attention to the first one or two digits than to any other 
digits. In more than half of the cases the first digit was recalled, and 
in slightly less than half of the cases the first two digits were recalled. 
In only three instances were three or more digits retained, when recalls 
of the familiar numeral 1000 and of the numeral 246, which is found in 
the peculiar place of "first" numeral in Problem C, are excepted. The 
chief explanation of this greater emphasis on the first digits probably 
lies in the general habit of attacking printed matter from the left. It is 
also possible that the adult subjects of this study had learned empirically 
the greater value of the first digit of a numeral and its greater significance 
to the solving of the problem, and therefore had formed the habit of 
paying special attention to it when reading problems. 

The first numeral of a problem seems to receive more careful attention 
during the first reading than any of the other numerals in the problem. 
More details of the first numeral are correctly recalled than of other 
numerals. Such a comparison can be made between the longer numerals 
only, because, as has been pointed out in previous paragraphs of this 



i6 



HOW NUMERALS ARE READ 



Study, the shorter numerals almost invariably — and therefore quite 
without regard to position in the problem — are read closely enough for 
complete recall. The longer numerals appear in problems C, D, and E. 
In each of these problems more details of the first numeral are recalled. 
Problem C will serve as an illustration. Here the first numeral, 246, is 
completely recalled in all cases, while the second numeral is so recalled 
only once. In Problem D the numeral 1276 is considered the first 
numeral rather than 1000 because of the peculiar character of the latter 
numeral. The higher ranges of recall of first numerals are probably 
due in part to the advantages of "initial" position. By virtue of this 
position they would tend to be more vividly impressed upon the memories 
of the readers. In additon to this advantage it appears probable that 
the adult subjects of this study have learned empirically to pay greater 
attention to the first numeral. By so doing they would be able to make 
special use of it, not only in determining the conditions of the problem, 
but also as a base in reaching decisions concerning the relations of the 
numerals to each other. 

The peculiar quality of the familiar numeral 1000 again distinguishes 
it from other numerals of the same length. In every case but one it was 
recalled completely. This single exception was due to an unusual case 
of confusion on the part of the subject, which caused him to forget even 
the sense of the problem. 

3. RANGE OF RECALL— BY THE SEVERAL SUBJECTS 

Certain subjects recall much more of the numerals than others. 
The significant differences between them occur in the higher ranges of 
recall and with the longer numerals. These differences are displayed in 
detail in Table VIII. The several individuals divide themselves into 
two general groups according as they reported relatively many or 

TABLE VIII 

Varying Ranges of Correct Recall of Three- to Seven-Digit Numerals by 
THE Several Subjects 



Subjects 



Hb 



Bak 



Th 



Number of numerals read by subjects 
Range of correct recall of numerals: 

Complete 

First two digits and digit-length . . . 

First digit and digit-length 

Digit-length 

Merely noticed 



CORRECT RECALL OF NUMERALS AFTER FIRST READING 17 

relatively few of the longer numerals in the higher ranges of recall. 
Subjects Hb, R, Bak, and Th are included in the first group, and L, G, 
and C constitute the second group. The contrast between Hb of the 
first group and G of the second is striking. The former completely 
recalls half of the longer numerals and the first two digits of seven of the 
eight longer numerals read, while the latter recalls completely only one 
longer numeral and the first two digits of only one. 

Differences between the two individuals in first-reading attitudes 
seem to account for the large differences exhibited by them in the 
ranges of recall. Subject Hb is a pronounced whole first reader. He 
intends to "grasp" all of a numeral when he first reads it. Subject G, 
on the other hand, is a striking example of the type of partial first 
readers. His purpose during the first reading, in so far as the numerals 
are concerned, is to obtain only a "general idea." 

4. FURTHER EVIDENCE AS TO THE PURPOSE OF FIRST READING 

Further evidence is found in this study in support of the conclusion 
presented in the first preliminary study that the main purpose of the 
first reading is to find the conditions of the problem in order to know 
how to proceed with solving. In nearly all instances the subjects in 
this study were able to indicate a correct procedure for the solution of 
any problem after the first reading. They were able to do this even in 
the many instances where they could recall nothing more of the numerals 
than their digit-lengths. 

5. ITEMS OF RECALL NOT INCLUDED IN THE CLASSIFICATIONS 

The classification scheme used in this study does not include every 
item concerning the numerals which was reported. In many cases 
subjects reported correctly the line of the problem in which a numeral 
appeared. In several cases they recalled its approximate location 
within the hne. No items incorrectly recalled are included in the 
classifications. Several such items were reported. In general they 
followed the t3rpes of errors which would be found in any study of errors 
in the reading of numerals in arithmetical problems. 

6. SUMMARY OF CONCLUSIONS 

The results of the second preliminary study on the range of correct 
recall of numerals after first reading may be summarized as follows: 
(i) Some item of almost every numeral is recalled. (2) The digit length 
of numerals is recalled almost invariably. (3) The shorter numerals 
and the familiar numeral 1000 are completely recalled almost invariably. 



l8 HOW NUMERALS ARE READ 

(4) The first one or two digits of longer numerals are recalled in a majority 
of cases. (5) The first numeral, in problems which include numerals of 
the greater lengths, is more frequently recalled than any other numeral 
in the problem. (6) The subjects divided themselves into two groups 
according as they recalled in the higher ranges large or small proportions 
of the longer numerals. (7) Further evidence appears in support of the 
previous conclusion that the main purpose of the first reading of a 
problem is to learn its conditions. 



CHAPTER IV 

ANALYSIS OF THE RE-READING OF NUMERALS IN ARITHMETICAL 
PROBLEMS— THIRD PRELIMINARY STUDY 

I. DESCRIPTION OF THE STUDY 

The distinction was drawn between the first-reading and the re- 
reading phases of the reading of arithmetical problems in the first 
preliminary study. The general purpose of re-reading as stated was 
"to perceive the numerals accurately for computation." The present 
study was designed to give further description of the purposes of the 
subjects and of their activities with the numerals during the re-readings. 
The general method which was used to obtain the data was that of 
introspective observation on the part of adult readers. 

The readers were four graduate students in the School of Education 
of the University of Chicago. One of them, Subject S, had read the 
problems of the first preliminary study. None of the others served as 
subjects in any other study of the investigation. They were asked to 
solve the five simple arithmetical problems which were later used as 
reading materials in the eye-movement studies and which are described 
in detail in chapter vi. Each subject was given pencil and paper and 
told that he might use them in solving the problems or not use them, as 
he chose. Before the beginning of the experiment the subjects were 
informed concerning the first-reading and re-reading phases of the 
reading of problems. At the conclusion of the experiment each of the 
subjects was of the opinion that his reading of arithmetical problems 
habitually followed these phases, and that the information given con- 
cerning them had not caused him to vary from his normal procedure. 

The subjects were instructed to attack each problem immediately 
when it was presented and to proceed with it in accordance with their 
normal problem-solving attitude. They were to press a conveniently 
placed telegraph key at the instant of beginning to read and continue 
the pressure throughout the first reading. Immediately at the con- 
clusion of the first reading the key was released. Thereafter whenever 
the attention of the subject was directed to the re-reading of any item 
from the text of the problem, the key was pressed and held, until atten- 
tion was directed away from the text whereupon the key was immediately 
released. The effect of this practice was to secure a separate record for 
each of the one or more acts of re-reading from the text of a problem. 

19 



20 HOW NUMERALS ARE READ 

Every pressure and release of the key was recorded on a smoked- 
paper record sheet which was moving on two kymograph drums. The 
duration of each pressure on the key was measured in seconds by the use 
of a chronometer which was so placed that its marker recorded the time 
intervals on the record sheet side by side with the records from the key. 
A brief period of training with practice problems in this procedure was 
necessary in order to enable the subjects to follow the procedure correctly 
and easily. Immediately after the solving of a problem, and with its 
text before them for reference, the subjects were asked to report the 
words or numerals in the text of the problem, upon which their attention 
was directed at each separate re-reading. This they were able to do with 
promptness and certainty. The reports of the subjects and the time 
records from the kymograph are presented in tables IX and X. 

The reading of Problem 2 by Subject Ba will serve as an illustration 
of the experimental procedure. Ba began to read the problem imme- 
diately when it was placed before him and at the same moment he pressed 
the key. The instant he finished the first reading of the problem, 
which required a time interval of 7.6 seconds, he released the key. 
Without delay he turned his attention to the numeral 357 in the text of 
the problem and immediately pressed the key. During an interval of 
1.4 seconds he re-read this numeral. He then directed his attention to 
the sheet of paper on which he intended to copy the numeral and at the 
same time released the key. When 357 was copied he turned his atten- 
tion to the numeral 1643, pressing the key at the same instant. During 
an interval of 2.4 seconds he re-read this numeral. When the re-reading 
of 1643 was completed he looked to the copy sheet to copy the numeral 
and at that moment released the key. This done, once more he glanced 
at the problem, simultaneously pressing the key, and fixed his attention 
upon the last sentence for .2 of a second. At the conclusion of this 
interval he released the key and was ready to proceed with solving the 
problem. 

The numerals or words read at each re-reading from the problems 
are given for every subject in Table IX. The time in seconds required 
for re-reading the numerals or words is given under the numerals or words 
in every case. Beginning at the top of the table the first left-hand 
column reads that two re-readings were given to items from Problem i 
which contains the numerals 47 and 2. The item from the text of the 
problem which was read by the first re-reading was the numeral 47, and 
the duration of this re-reading was 2.4 seconds. At the second re-reading 
the numeral 2 was read and the time required for this second re-reading 



RE-READING OF NUMERM.S IN ARITHMETICS PROBLEMS 21 



Pi 





H 




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3 











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2; 






12; 


i 



2 2 HOW NUMER.\LS ARE READ 

was again 2.4 seconds. The sum of the durations of the two re-readings 
from the problem, i.e., the total re-reading time for the problem was, 
therefore, 2.4 seconds+2.4 seconds or 4.8 seconds. This last detail of 
information appears in Table X, in the first left-hand column, and in 
the upper row. 

2. OBJECTS AND NATURE OF THE RE-READINGS 

The point which stands out most clearly in Table IX is that numerals 
were the objects of the re-readings almost invariably. Only three 
instances appear in the entire number of re-readings in which the objects 
of the re-readings were words. It is also clearly apparent that the 
numerals of the problems were almost invariably re-read. In five 
instances only, numerals were not re-read. 

These facts serve as additional evidence in support of the conclusion, 
which was drawn in a previous section, that the numerals are of a nature 
which clearly distinguishes them from the other contextual elements of 
arithmetical problems and which causes them to make unusual demands 
upon the attention of readers. 

Further light is thrown upon the procedures of subjects with the 
numerals by a review of the original reports. These reports show that 
every numeral which was re-read was copied on the computation paper 
inunediately after it was re-read. In all of these cases paper and pencil 
were used in solving the problems. In such cases copying the numerals 
is one of the earlier moves in the total process of solving the problem. 
Re-reading the numerals appears in most cases to have been a necessary 
step preliminary to copying them. It is, therefore, proper to speak of 
such re-reading as re-reading for copying. 

The number of readings which was required for the re-reading of 
a numeral for copying was usually one. Numerals varying in length 
from one to seven digits were thus re-read at one reading. To none of 
the numerals of one- to four-digit lengths was more than one re-reading 
given. To several of the six- and seven-digit numerals, on the other 
hand, two re-readings each were given. In these cases the first three or 
four digits of the numeral were re-read and copied as a group, after which 
the remaining three digits were re-read and copied as a second group. 
In two instances, the two numerals of Problem 2 were copied from one 
re-reading. 

3. DURATION OF RE-READINGS 

The duration of the first reading and the sum of the durations of 
the re-readings are given for the reading of each of the several problems 



RE-READING OF NUMERALS IN ARITHMETICAL PROBLEMS 23 



by each of the individual subjects in Table X. Examination of the 
data shows that numerals of greater lengths required greater total 
re-reading times than numerals of lesser lengths. The total re-reading 
time in the data for each individual subject increases gradually, in most 
cases, from the relatively small total time required to re-read the short 
numerals of Problem i to the relatively greater time required to re-read 
the long numerals of problems 3 and 5. The numerals of Problem 4, 
which offers four nimierals for re-reading, received in most cases a greater 
total re-reading time than the numerals of any other of the five problems, 
none of which offers more than two numerals for re-reading. 

TABLE X 

Number of Seconds Reqihred for First Reading and for 
Re-reading of Problems 



Problems . 



Numerals read . 



357 
1643 



243,987 
21,765 



1000; 1276 
gi; 817 



,918,564 
617,453 



Subject Ba — 

First reading time . . . 

Total re-reading time . 
Subject Gl — 

First reading time . . . 

Total re-reading time . 
Subject S — 

First reading time . . . 

Total re-reading time . 
Subject Wm — 

First reading time ... 

Total re-reading time . 



2.4 
4.8 



2.0 
0.8 



1.8 



3.8 



7.6 
4.0 



10.4 
18.6 



5.» 
6.2 



12.8 
3-0 



20.0 
10. o 



The total re-reading time of a problem is in most cases shorter ihan 
the first reading time of the problem. The obvious explanation lies in 
the comparatively small amount of work to be done during the re-reading. 
At this time as a rule the numerals only are included in the reading, 
whereas during the first reading all of the contextual elements of the 
problem, both words and numerals, are included. 



4. SUMMARY 

The following conclusions may be drawn from this study: (i) The 
objects of the re-readings from the problems were almost invariably 
numerals. (2) The numerals were re-read for copying on the computation 
sheets. (3) One re-reading for copying was sufficient for most of the 
numerals. Some of the longer numerals, however, were re-read in two 
parts. (4) The numerals of greater length required longer times for re- 
reading, on the part of a majority of the subjects. (5) The total time 
required for re-reading the numerals was less than the time required 
for the first reading, with three of the four subjects. 



CHAPTER V 

READING NUMER-\LS IX COLUMNS— FOURTH PRELIMINARY STUDY 

I. DESCRIPTION OF THE STUDY 

The purpose of this study was to give some description of the reading 
of numerals, when nimierals only appeared as the material to be read and 
when each individual numeral was placed in a separate line. The 
general plan followed in procuring the data was to have adult subjects 
copy numerals from the pages on which they appeared onto other sheets, 
and at the same time articulate the numerals in an easy natural way. 
This articulation was recorded by the author of this report. By means 
of a system of notes which will be described later, it was possible to get 
a fairly full account of what was said and, as experience in making the 
records accumulated, it was possible to distinguish clearly the various 
types of reading. 

The subjects were four graduate students in the School of Education 
of the University of Chicago. Subjects R, L, and G had each served in 
the second preUminary study. Subjects G and H had read the problems 
of the first preliminar}^ study and photographic records of the eye- 
movements of both H and G appear in the second part of this report. 
The materials which they read were forty-eight ordinary whole numerals 
varying in length from one to seven digits, and including seven numerals 
for each different digit-length except that there were only six numerals 
of seven-digit length. Punctuation in the form of commas was used in 
the customary way with some of the five-, six-, and seven-digit numerals; 
and with some of these numerals it was not used. Four numerals of 
both the five- and six-digit lengths, and three numerals of the seven-digit 
length were punctuated, while three numerals of each of the five-, six-, 
and seven-digit lengths were not punctuated. The numerals were type- 
written on separate hnes and so arranged that the tens, hundreds, etc., 
places of the numeral above were not exactly above the same places of 
the numeral in the line below. Subjects R and G each read the whole 
set of numerals twice, while the other two subjects read each set only 
once. 

The subjects copied the mmierals from the text sheet on to the copy 
sheet at normal speed. At the same time they articulated the numerals 
in an easy low voice which the observer was able to hear at a distance of 

24 



READING NUMERALS IN COLUMNS 25 

approximately two feet. That part of the instructions which called for 
copying the numerals was inserted in order to provide a genuine working 
purpose for reading them. At the same time this purpose required an 
exact reading of every numeral. The kind of reading done, therefore, 
in compliance with these instructions was of a relatively clearly defined 
functional type, quite similar in obvious ways to the re-reading of 
numerals for copying, which was described in the preceding study. The 
provision for articulation enabled the observer to report the readings in 
so far as the grouping of the digits of the numerals and the numerical 
language used in reading them were concerned. At the same time the 
articulation did not seem to interfere with the reading. 

Numerals are said to be read by digit groups when certain successive 
digits are so closely associated with each other in the reading as to form 
units of reading, which units are at the same time clearly distinguished 
from other similar units. The digits which constitute a group are bound 
together by being pronounced in quick succession as one series. The 
pronunciations of the several digit groups are separated from each other 
by time intervals distinctly longer than the time intervals which separate 
the pronunciations o^ the individual digits. 

For reporting the digit groups and the numerical language used in 
reading the numerals a simple code system was devised which was 
based on the symbols i, 2, and 3 signifying respectively the grouping 
of the digits of a numeral in groups of one, two, and three digits. A few 
other signs were necessarily added to indicate various modifications of 
these groups. A brief period of training with sets of practice numerals 
was undergone by the observer and by each of the subjects. After this 
training, they were able to proceed with the experiment in full conformity 
with the instructions. The data which were obtained during the course 
of this stiidy are arranged in tables XI-XV. 

2. GENERAL DESCRIPTION OF THE THREE MAIN GROUPS USED 

The most striking feature of the reading of numerals, which was 
discovered in this study, was the fact that the subjects habitually 
divided the numerals into digit groups. Three different sizes of groups 
were clearly distinguished in the readings, namely, those that were made 
up of one, two, and three digits respectively. The one-digit groups 
appeared more frequently in the one-, three-, and seven-digit numerals 
as is shown in Table XII. The two- and three-digit groups appeared 
in the readings of numerals of all the greater digit-lengths. Relatively 
large numbers of three-digit groups appear in readings of the five-, six-, 



26 



HOW NUMERALS ARE READ 



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Hfa 






(N M M M tH M 



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ro ro fO CN CO M M 



CO CS M (N M (N 


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M W CO 04 M Ht 


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M <N M O) 



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.12 o 



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READING NUMERALS IN COLUMNS 27 

and seven-digit numerals. In the cases of the six- and seven-digit nu- 
merals the explanation of this condition lies in the fact that numerals of 
these lengths present just twice as many opportunities for the employ- 
ment of three-digit groups as numerals of four- or five-digit lengths. In 
the case of the five-digit numerals, on the other hand, the large number 
of three-digit groups is attributable to the remarkable uniformity with 
which the habit of reading five-digit numerals in two groups of two and 
three digits respectively was followed by all of the subjects. 

The three-digit groups exhibited two distinct types which are 
referred to as the simple and complex types. The three digits of the 
simple type of three-digit groups are pronounced individually and with 
equal time intervals between them. The three digits of the complex 
type on the other hand are pronounced in two distinct subgroups, the 
first of which includes one digit and the second, two digits. Both types 
appear in the readings of the five-, six-, and seven-digit numerals, as 
is seen in the readings of Subject G, which are reported in detail in 
Table XIV. 

Habit on the part of the individual subject appears as the most 
conspicuous factor in determining which of the two types of three-digit 
group was chosen, when a three-digit group was used. The last right- 
hand column of Table XIII discloses the fact that Subject H used the 
simple type of three-digit group only, while Subject G used it in a 
preponderant number of cases. On the other hand Subject R used the 
complex type lalmost invariably. 

3. MAIN-GROUP PATTERNS FOR NUMERALS OF LIKE LENGTH 

It became apparent early in the course of this study that the digits 
of numerals of any one length were being grouped in much the same way 
by all of the subjects. This observation is strikingly confirmed by 
the data which are presented in Table XI. The fact that appears most 
strikingly after an examination of this table is that the digits of numerals 
of any particular length are divided into a certain number of groups, 
which groups are made up of certain numbers of digits and stand in a cer- 
tain order of succession. The case of the seven-digit punctuated numerals 
will serve for illustration. The digits of these numerals are seen to have 
been divided into three groups by all of the subjects. In the first group, 
one digit is found almost invariably, in the second group three digits, 
and in the third group three digits. To this succession of digit groups 
of certain sizes only on€ exception occurs. Such an arrangement of the 
digits of a numeral is designated as a main-group pattern. 



28 



HOW NUMERALS ARE READ 



The one- and two-digit numerals were each read as single groups of 
one and two digits respectively. The first variation from one main- 
group pattern as representative of the reading of numerals of the same 
length occurs in the three-digit numerals, which exhibit two patterns. 



TABLE XII 

Number of One-, Two-, and Three-Digit Groups Used in Reading Numerals 
OF THE Several Digit-Lengths in Columns 



Digit-Length of Numerals 



7t 



Number of readings given numerals. 
Number of: 

One-digit groups 

Two-digit groups 

Three-digit groups 



36* 



3 

45 
3Q 



36 

35 
30 
SI 



* There are 36 readings of four-digit numerals when the 6 readings of the numeral 1000, which 
was always read as "one thousand," are omitted. 

t One four-digit group was used bj' Subject L in reading one of the seven-digit numerals. 



TABLE XIII 

Number of Simple (3) and of Complex (1-2) Three-Digit Groups Used in 
Reading Five-, Srx-, and Seven-Digit Numerals in Columns 





Digit-Length of NtrnERALS 




5 


6 


7 


S-7 


S-7 




Punc- 
tuated 


Non- 
Punc- 
tuated 


Punc- 
tuated 


Non- 
Punc- 
tuated 


Punc- 
tuated 


Non- 
Punc- 
tuated 


Punc- 
tuated 


Non- 
Punc- 
tuated 


Punc- 
tuated 
and 
Non- 
Punc- 
tuated 


Subject R— 






5 
II 

13 

3 

8 


12 
2 


7 
5 

6 
5 

6 


10 
3 

2 


12 

24 

25 

10 
18 


"28" 

7 
3 

4 


12 


Complex three-digit groups 

Subject G— 


8 

6 
2 

4 


6 

2 

3 

2 


52 

32 


Complex three-digit groups 

Subject H— 


13 

22 






Subject L — 


I 
3 


I 


5 


3 


2 
3 


2 


8 
9 


6 

I 


14 


Complex three-digit groups 


I 


3 


10 



The four-digit numerals appear almost invariably in a pattern of two 
groups of two digits each. A conspicuous exception to this regular main- 
group pattern of the four-digit numerals is found in the familiar numeral 
1000 which was regularly read as "one thousand." This exception is 



READING NUMERALS IN COLUMNS 29 

further evidence of the fact, to which attention has been called in other 
sections of this report, that this numeral is different in quality from other 
numerals of the same length. 

The five-digit numerals show in a preponderant number of cases a 
pattern of two groups of two and three digits respectively. The domi- 
nant pattern for the six-digit numerals is that of two groups of three 
digits each. The first main-group patterns made up of three groups to 
appear among punctuated numerals are found in the seven-digit numer- 
als. When numerals of any of the greater digit-lengths are written 
without punctuation, other patterns than the main-group pattern for 
that digit-length make their appearance. In the case of non-punctuated, 
seven-digit numerals as many as ten different main-group patterns 
were found. With all of the other digit-lengths, however, conformity to 
main-group pattern is clearly the rule with all subjects. Evidently such 
arrangements of the digits of numerals, when they are being read for copy- 
ing, are procedures which have been very thoroughly conventionalized 
by long practice, or else such procedures rest closely upon certain funda- 
mental laws of mental action. 

4. VARIATIONS IN NUMERICAL L.ANTGUAGE 

The fact that all numerals of a certain digit-length were usually 
read in the same main-group pattern does not mean that the language 
used in reading them was the same. Variations in language were found 
in the readings of both two- and three-digit groups. By the use of 
symbols to represent each of these variations, the detailed numerical 
language which was used by one subject, namely by Subject G, is given 
in Table XIV for the readings of all the numerals. The patterns which 
appear in this table may, therefore, be designated as numerical-language 
patterns. 

Four different numerical-language patterns are found in the readings 
by Subject G of the punctuated numerals of five-digit lengths. Any 
five-digit numeral may be pronounced according to each of these 
patterns. The numeral 76,184, which was one of the numerals read by 
the subjects, may be taken as an example of the five-digit numerals. 
When this numeral is pronounced successively according to each of the 
four numerical-language patterns, as they are represented from top to 
bottom in the column of punctuated five-digit numerals, the following 
four different pronunciations result: " seven ty-sLx — one eight four"; 
''seven six— one, eight four"; "seven six — one eight four"; ''seventy- 
six — one, eighty-four." 



3° 



HOW NUMERALS ARE READ 



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1-1 C/2 











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ml 






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, _-, 




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00 " r V 


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•J- S^ 












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C 








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READING NUMERALS IN COLUMNS 31 

A second examination of the four numerical-language patterns, the 
pronunciations of which are presented immediately above, reveals the 
fact that all four of the patterns are modifications of one fundamental 
main-group pattern. This fundamental pattern contains two groups, 
the first of which is a two-digit group, while the second is a three-digit 
group. It is the variations that appear in the pronunciations of both 
of these groups that distinguish the four different numerical-language 
patterns. In the two-digit groups the differences in language are 
merely those between the words, "seven six" and '* seventy-six," or 
again between "eight four" and "eighty-four." In the three-digit 
groups the words used to pronounce the simple type differ from those 
used to pronounce the complex type, as "one eight four" differs from 
"one, eighty-four." 

The habits of individual subjects were discovered in a previous 
paragraph to be the most conspicuous factors in determining which of 
the two types of three-digit groups was used. The original records show 
that such individual habits similarly were the chief factors in determining 
what language was used in pronouncing the two-digit groups. On the 
other hand, the fundamental main-group pattern for any length of numeral 
appeared consistently in the readings of all subjects. It is, therefore, 
apparent that the selection of the digit-length of groups and the selection 
of the order of their appearance are more fundamental phases of the 
reading of the numerals than the selection of the particular tj-pe of 
three-digit group and the choice of the particular words by which the 
two- and three-digit groups are to be pronounced. 

5. INFLUENCE OF PUNCTUATION ON THE GROUPING OF DIGITS OF 
LONGER NUMERALS 

Great differences appear between the readings of punctuated and 
non-punctuated numerals in the number of both two- and three-digit 
groups which are employed. Punctuation apparently has the effect of 
increasing the number of three-digit groups used, and conversely of 
decreasing the number of two-digit groups. In the extreme right-hand 
column of Table XV it is seen that each of the last three subjects used 
a much greater proportion of three-digit groups for the punctuated 
numerals, and on the other hand a much greater proportion of two-digit 
groups among the non-punctuated numerals. 

The preponderance of three-digit groups in the punctuated numerals 
and conversely the preponderance of two-digit groups in the non- 
punctuated numerals are each relatively much greater for the six- and 



32 



HOW NUMERALS ARE READ 



seven-digit numerals than for five-digit numerals. Such a situation may 
be partly explained by the fact to which attention was called above, 
that it is possible to use just twice as many three-digit groups in reading 
a six- or seven-digit numeral as in reading a five-digit numeral. Fewer 
main-group patterns appear in the columns for punctuated numerals in 
Table XI than in the columns for non-punctuated numerals. Examina- 
tion of the patterns in both columns shows that there are greater 
numbers of groups in the non-punctuated patterns, and this is due 
mainly to the more frequent use of the smaller group of two digits. 

The readings of one subject, R, exhibited practically no differences 
in selection of two- and three-digit groups, which may be attributed to 

TABLE XV 

Effect of Punctuation on the Number of Two- and Three-Digit Groups Used 
IN Re.ading Five-, Six-, and Seven-Digit Numerals in Columns 





Digit-Length of Numerals 




5 


6 


7 


5-7 




Punc- 
tuated 


Non- 
Punc- 
tuated 


Punc- 
tuated 


Non- 
Punc- 
tuated 


Punc- 
tuated 


Non- 
Punc- 
tuated 


Punc- 
tuated 


Non- 
Punc- 
tuated 


Subject R— 


8 
8 

8 
8 

4 
4 

4 
4 


6 
6 

7 
5 

4 

2 

4 
2 








3 

10 

12 

4 

8 

2 

4 
2 


8 
36 

10 

35 

4 
18 

5 
17 


9 

28 

34 
II 

21 

4 

12 
7 


Three-digit groups 

Subject G— 

Two-digit groups 

Three-digit groups 

Subject H— 

Two-digit groups 

Three-digit groups 

Subject L — 

Two-digit groups 

Three-digit groups 


i6 

■■'la'" 
"■"s"' 


12 

IS 

2 

9 

4 
3 


12 

2 
II 

""e""' 

I 

5 



punctuation. With the larger numerals he used three-digit groups 
consistently, wherever it was possible to use them, in both punctuated 
and non-punctuated numerals. The exception in this case is probably 
attributable to his having attained to a relatively high stage of proficiency 
in the mechanical processes of reading numerals by means of a large 
amount of special practice in a kind of reading of numerals which is very 
similar to that used in this study. This practice he had gained while 
earning his living in the capacity of railroad rate clerk. A large part of 
his work was to read numerals and call them off to a colleague, who 
copied them on other paper. 

The easy use of larger digit groups seems to give greater facility and 
greater speed to the reading of numerals. The value of punctuation 



READING NUMERALS IN COLUMNS ^3 

to the subjects in large part lies in the fact that its employment encour- 
aged the use of the larger group of three digits. The subject confronted 
with the necessity of reading a large but unknown number of unspaced 
digits is in a difficult situation. Such situations are not frequently encoun- 
tered in the experiences of the ordinary reader. In consequence he pro- 
ceeds with caution and with the smaller groups of one and two digits. 
The great number of small groups and the large number of group patterns 
which were employed in the readings of non-punctuated numerals of 
seven-digits length, as shown in Table XI, are evidently results of 
procedure under difficulty and with uncertainty. The same situation 
produced the great number and variety of numerical-language patterns 
in the readings by Subject G of the same numerals. In situations such 
as these, employment of the symbols of punctuation appears to afford 
great and immediate relief. 

6. PERSISTENCE OF PATTERNS FROM THE FIRST READING THROUGH 
A SECOND READING 

Opportunity to study the persistence of the main-group and numerical- 
language patterns, which were found in the first reading of the numerals, 
through the second reading of the same numerals was given in the cases 
of two subjects. Subjects R and G each read all of the numerals at two 
separate readings. The interval of time between the two readings was 
approximately 30 minutes with each subject. It was found that both 
R and G read the same numerals in the same patterns at both readings 
with very few exceptions. Subject R, who was more highly trained in 
the reading of numerals than any other subject, made fewer changes 
than G. More changes were made in numerical-language patterns than 
in main-group patterns. Most of the changes, which were made, were 
found in the non-punctuated numerals of seven-digits length. 

The fact that the same main-group patterns so consistently 
reappeared at the different readings of these subjects gives further 
evidence in support of the conclusion, which was advanced in a paragraph 
above, that the arrangement of digits in main-group patterns has been 
very thoroughly conventionalized, or else that such procedure rests 
closely upon certain fundamental laws of mental action. 

7. SUMMARY OF CONCLUSIOJNS 

The following conclusions are drawn from the data presented in 
this study concerning the articulated reading of numerals for copying. 
(i) The digits of numerals are grouped in the process of reading. The 



34 HOW NUMERALS ARE READ 

groups of digits are of three sizes, namely, of one, two, and three digits 
respectively. (2) The numerals of each of several digit-lengths are read 
almost invariably in a main-group pattern which is peculiar to that 
digit-length. (3) Various numerical-language patterns are used in 
pronouncing numerals of the same length. (4) The employment of 
punctuation with the longer numerals encourages the use of three-digit 
groups and, conversely, discourages the use of two-digit groups in the 
reading. A larger group unit is thus secured. (5) The main-group and 
numerical-language patterns which are used in the first reading of 
numerals persist for the most part through a second reading of the 
same numerals. 



PART II. STUDIES OF THE READING OF NUMERALS— 
BY USE OF PHOTOGRAPHIC APPARATUS 

CHAPTER VI 

DESCRIPTION OF THE EYE-MOVEMEXT STUDIES 
I. APPARATUS DESCRIBED 

The data which are presented in the remaming sections of this 
report were obtained through the use of an apparatus designed to record 
the movements of the eyes in reading by means of photography. The 
apparatus is described and its use explained in a monograph by Dr. C. T. 
Gray/ and excellent photographs and diagrams of the same are found 
in a magazine article by Gilliland.^ A few slight adaptations of the 
apparatus and procedure, which are described in these references, were 
necessary in view of the materials and purposes of the present investiga- 
tion. The materials which were read by the subjects of this investigation 
were printed on separate cards eight and one-half by four inches in size. 
These cards were placed on the stand immediately before the lenses of 
the camera and directly before the eyes of the readers. A flood of light 
reflected from the overhead mirror gave bright illumination to any 
materials which were placed upon the stand. A convenient elbow-rest 
was provided for the right arm in such a manner that computation with 
a pencil could be undertaken easily and comfortably, and directly upon 
the problem card, whenever the subject chose to do so. With this 
arrangement it was possible for the pencil of light which is reflected 
from the eye, to register continuously upon the film during periods of 
computation, as well as during periods of reading from the problem. 

2. THREE TYPES OF READING-MATERIALS USED 

The reading-materials which were selected for this part of the 
investigation were of three different types, namely, simple arithmetical 
problems, numerals isolated in lines, and a paragraph of ordinary 
expository prose. 

' C. T. Gray, "Tj'pes of Reading Abilitj- as Exhibited Through Tests and Labora- 
toty Experiments," Supplementary Educational Monographs, Vol. I, Xo. 5 (1917), 
pp. 83-91. 

^ A. R. Gilliland, "Photographic Methods for Studying Reading," Visual Educa- 
tion, Vol. II, Xo. 2 (February, 192 1), pp. 21-26. 

35 



36 



HOW NUMERALS ARE READ 



The arithmetical problems were so designed as to provide a simple 
and genuine problem-setting for the numerals which had been selected 
for further study. The numerals thus selected included representatives 
from each of the several lengths of from one to seven digits, the familiar 
numeral looo, and a group of three numerals placed closely together in 
one problem. Further details concerning the problems are given in 
Table XVI and the problems exactly as they were read by the subjects 
appear as Selection i. 

TABLE XVI 

Description of the Five Problems Re.\d in the Photogr.\phig Apparatus 





Ordinal 
Number 
OF Each 
Line in 
Problem 


Length 

OF 

Line in 
Milli- 
meters 


The Number in Each Line of 


Number of 
Problem 


Words 


Letters 


Numerals 


Digits 


Words 

and 

Numerals 


Letters 

and 

Digits 




ist 

fist 
\2d 

ist 

2d 

bd 

[ist 

Ud 

l3d 
fist 

^2d 

Ud 


76 

9S 
88 

95 
102 
86 

95 
102 
90 

95 
102 
94 


9 

II 

8 

II 

7 
9 

10 
8 
8 

8 
10 
12 


34 

40 
37 

44 
34 
37 

42 
56 
30 

40 
43 

45 


2 

I 
I 


2 


I 

3 

I 
I 



3 

3 

4 


II 


4 

9 

7 
6 



II 

12 

II 
9 
9 

n 
8 
II 

9 
11 
12 


37 




43 




41 

44 




37 

46 
S6 
39 

47 






45 


Total for all 
problems. . 


12 




III 


482 


12 


47 


123 


529 






Average num- 
ber per line 




93-33 





























Note. — The number of spaces between words is not counted. 

SELECTION 1 
Five Problems Read before Photographic Apparatus 



1. At 47 cents a dozen what will 2 dozen eggs cost ? 

2. A timber man owns one plot of 357 acres, and another of 
1643 acres. How much ground does he own altogether ? 

3. A wholesale grain firm had at the beginning of the day 
243,987 bushels of wheat. During the day's trading 21,765 
bushels were sold. How much did they then have ? 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 37 

4. A commission house liad on hand 1000 cases of canned corn. 
From three different canning factories they bought respectively 
1276, 91, and 817 cases. How man}- did they then have? 

5. If one telephone company uses 1,918,564 cross bars during 
the j'ear, and another company in the same period uses 617,453 
cross bars, how many more does the one use than the other ? 

The numerals isolated in lines included a list of thirty-four numerals. 
Twenty-eight of the list of thirty-four consisted of ordinary numerals 
which were selected by taking four numerals from each of the seven- 
numeral lengths of one to seven digits. In addition, the list included the 
six special form numerals, namely, 1000, 7,7,7^, 25,000, o, 99, and 637,637. 
They were presented to the readers on two different cards. The line 
space between any two numerals was 16 mm., and the lines were placed 
two spaces apart. These two details of arrangement were followed in 
order that the reading of any numeral might be entirely separate from 
the reading of any other. The numerals are reproduced as Selection 2 
and in the same form in which they were presented to the readers. 

SELECTION 2 

Isolated Numerals Read before Photographic Apparatus 



(Card One) 



836 


3 


5489 756,352 


46 


4,325,986 


85,974 


239 


1 16,789 


1024 


354,908 


12 


2,374,957 


1000 333 
(Card Two) 


25,000 




5,184 


9317 


17 ' 2 


5,236,795 


256 


13,819 


1928 


365 8 


93,548 


3,984,673 


)7,308 


52 


99 


637,637 





38 HOW NUMERALS ARE READ 

The selection of ordinary expository prose was taken from Judd's 
Psychology of High-School Subjects. The subjects read directly from the 
book. Data were tabulated from the reading of ten lines by each subject- 
Because of defects in the records of subjects W and G only five and seven 
lines respectively were tabulated from their readings. The regularity 
in number and duration of pauses found in the data for the few lines, 
which were tabulated for these subjects, however, give evidence that the 
data for these few Unes represent the normal reading of these subjects 
in these materials. The record which represented the reading of Hb 
was totally unsatisfactory for use. The ten lines of the text were each 
93 mm. in length and included loi words and 452 letters. They are 
reproduced as Selection 3. 

SELECTION 3 
Ordix.\ry Prose Re.\d before Photographic .\pparatus 
"Anyone who has struggled with the German language has an 
appreciation of the satisfaction which the novice feels in watching 
the way an expert in this language manages a separable verb. The 
moment the verb is used in a sentence, there arises a feeling of craving 
for the remainder of the verb. The skillful German places between the 
verb and the prefix a long series of phrases and words, but ultimately 
arrives with perfect precision at the end of the sentence, and gives the 
satisfaction which comes from a proper closing of the feeling which was 
started when the. . . ."' 

3. INSTRUCTIONS TO SUBJECTS AND DESCRIPTION OF SUBJECTS 

The instructions given the readers were varied for each of the three 
kinds of materials which were read. When the problems were being 
read the subjects were asked to attack them with the normal problem- 
solving attitude. Each individual was provided with a pencil which he 
was told he might, or might not, use in computation, as he chose. The 
instructions which the readers received for the isolated numerals were 
to read the numerals successively in the lines, to read all of them accu- 
rately, and to proceed at the normal rate of speed. Each numeral was 
to be articulated in an easy, natural manner and in a voice which was 
barely audible to the observer, who stood at a distance of approximately 
three feet. Provision for this slight articulation was included as a means 
of encouraging the complete reading of all numerals. For the expository 
prose selection the instructions were to read silently for a clear under- 

' C. H. Judd, Psychology of High-School Subjects. Boston: Ginn & Co., 1915. 
Pp. 190. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 39 

Standing of the paragraph and at normal speed. The volume from which 
the selection was taken was famihar in a general way to all of the readers. 
They were given its title in advance and the subject-matter of the passage 
to be read was described as relating to the psychology of language. 

The six subjects, records of whose readings appear in this part of the 
investigation, were all male graduate students in the School of Education 
of the University of Chicago. Three of them had served in various 
preUminary studies. Subject G had read the problems of the first and 
second preliminary studies and was classified as a pronounced partial 
first reader of numerals in problems. Subjects H and Hb had served 
in the first and second studies respectively and were both found to be 
pronounced whole first readers. The original plan of the investigation 
specified that the subjects whose types of reading had been studied in 
the preliminary sections should act as subjects for the eye-movement 
experiments. Of the photographic records which were made of the 
subjects of pre\dous studies, however, only those of G, H, and Hb were 
entirely satisfactory. None of the subjects reported past experiences 
which seemed likely to have had important influence upon his reading 
of the materials of this study. 

4. PROCEDURE ON THE PART OF THE OBSERVER 

The instructions were given to the subjects before they took their 
seats at the camera, and samples of each of the three kinds of materials, 
which were to be read, were examined by them. When the readers 
were properly seated before the camera they were given a brief training 
with practice problems and with practice sets of isolated numerals until 
procedure according to the instructions was mastered. After the 
solving of each problem, several of the subjects were asked to make 
brief introspective observations concerning whole and partial first reading 
of numerals, the re-reading of numerals and the steps used in computa- 
tion. Their reports were recorded and later served as a basis for inter- 
pretation of the corresponding eye-movement records. 

5. GUIDE FOR READING THE PLATES 

The photographic films, upon which the hnes of dots representing 
pauses of the eyes were recorded, were used as slides in a projection 
lantern. The records of the photographs were in this manner projected 
upon a screen, which, at the same time, held the texts of the various 
reading materials. The photographic picture of the subject's reading 
was thus superimposed upon the exact text of the materials which he had 
read. It was, therefore, possible to locate directly upon the lines of the 



40 HOW NUMERALS ARE READ 

text itself the letter or digit about which the attention of the subject 
was centered at any pause of the eye. By counting the number of dots 
in the lines of dots, which represented the pauses, the exact durations 
of the pauses were ascertained. 

The plates, which describe readings of the problems, are numbered 
I-XV as presented in this chapter. The remaining plates, which 
describe readings of the isolated numerals, are numbered XVI-XXV and 
are found in chapter ix. The lines of reading materials which are found 
in the plates are reproductions of the lines which were read by the 
subjects. The short straight vertical lines which cross the lines of print 
represent pauses of the eye. The particular letter, digit or space which 
is crossed by a vertical Hne represents the approximate center of the field 
of perception which was included in that pause. The arable numbers, 
I, 2, 3, etc., above each of the vertical lines indicate the serial order of 
each pause among the pauses which were used in reading the problem. 
When the serial number of a pause moves to the left of the serial number 
of the previous pause a backward or regressive movement of the eye is 
indicated. The number at the lower end of a pause line gives the dura- 
tion of the pause in units of i '50 of a second. 

The vertical lines which mark the pauses used in the first reading of 
the problem are located on the lines of the reproduced text. All pauses 
used in re-reading or in copying numerals or in computation are recorded 
below the last line of the problem. For convenience in interpretation 
the numerals which were read during such pauses are typewritten below 
the last line of the problem and directly below the several digit spaces 
occupied by these numerals in the lines below. A straight horizontal 
hne below the vertical lines that indicate the computation pauses, 
describes the location of numerals which have been copied, or of answers 
which have been recorded. 

The reading of the plates may be illustrated by the reading of 
Plate XII, which is as follows: Pause i, which falls in the first line of 
the problem, begins the first reading of the problem. It is located on the 
letter "a" of the word "wholesale" and the duration of the pause is 
10/50 of a second. Pause 2, which is a backward or regressive movement 
from Pause i, is located on the letter "o" of "wholesale" and the 
duration of this pause is also 10/50 of a second. There are seven pauses 
in line i. Pause 9, which is in line 2, falls on digit "9" of the numeral 
"243,987" and its duration is 20/50 of a second. It is a regressive 
movement from Pause 8. Pause 22 completes the first reading of the 
problem. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 



41 



The pauses which follow represent the process of computation and 
are recorded below the text of the problem. Pause 23, which is the 
first pause used in the process of computation, is apparently a locating 
pause. It is followed by Pause 24, by which digit ''7'' of the numeral 

PLATE I 

5 4 5 «. r 

A conimission house hati on har|d IC 00 capes of cannep corn 



Fiom tl: 



ee different 



f 3 



13 



14- 



anning "actories they )ought resjectively 



I? rfc 17 13 15 20 21 



1273, 



)1, 



2 3 25 /4 I J 9 



22 



nd (^7 cases. How man ^' did they thei ha^ 



X3 24 



3S 36 70 28 3j; 27 4° 4t 3 9 



:.; 7 >,9 a]Ld 817 



n t3 H 30 7 14 

43 ZS lb 19 7 9 



IDOO 



13 4 X 



(«. 



First reading of Problem 4 by Subject B and his procedure in solving the problem. 
X indicates that it was impossible to determine with precision the duration of the 
pause. 

" 243,987 " is read in 41/50 of a second. Attention then passes immedi- 
ately with Pause 25 to digit ''5" of "21,765," which digit is to be 
subtracted from the afore-mentioned digit "7.'' Pauses 26, 27, etc., 
continue the process of computation, which ends with Pause 32. The 
subject then shut his eyes and the record was finished. 

Plate I records the reading of Problem 4 by Subject B. In the first 
reading of the problem, which is represented by the pauses in the lines 



42 



HOW NUMER.\LS ARE READ 



of the text itself, an initial regression is noted in line i. Pause i evidently 
was not located closely enough to the left end of the line for a satisfactory 
beginning. Similar initial regressive movements appear in Unes 2 and 
3. During pauses 16 to 19, inclusive, the two numerals 1276 and 91 were 
given whole first readings. With Pause 24 the first reading of the 
problem was finished. 

PLATE II 



A comr lission house lad on ha 



la 



id 10 )0 cases of ca: med cc 



IS 



rn. 



From tl 



ree 



iifferen 



12 



■anning fac 



ories they 



15 



la 



14 



bought 'espee ively 



13 



1276, 



}<o 



14 



IT 



)1, and 517 ca ;es. ] low m my di^ \ they th ;n have ? 



%i 



1300 



First reading of Problem 4 by Subject G and re-reading the numeral 1000 



The remaining pauses, which represent subsequent procedure with 
the problem, are placed below the lines of the text. The numeral 1000 
was re-read with Pause 25. The records do not give sufficiently detailed 
information for the identification of the purposes of the individual pauses 
subsequent to Pause 25. The computation was, however, conducted 
directly from the problem card and apparently the subject added the 
numerals 1276 and 91 first, and then added 817 to that result. The 
answer was recorded during, or immediately after. Pause 41. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 



43 



Plate II contains the record of Subject G for Problem 4. He 
proceeded rapidly with the first line, but read the second line with a 
number of pauses which is relatively large as compared with the number 
of pauses on his other lines. The numeral 1000 was read in detail ■v\dth 
Pause 4 in a time interval of 11/50 of a second. With pauses 15 and 
16 he gave partial first readings to the three numerals, 1276, 91, and 817. 
Such partial first readings of these numerals were in this instance suf- 
ficient preparation for computation with them. The first reading of 
the problem ^vas completed with Pause 21. 



4 2. 



PLATE III 
S t b 7 » 



At 4 r cents i . do 



l(> l(> 35 



eii what 



will 2 



10 13 



dozen 3ggs cost ? 



24 







94* 

*The answer, 94, was recorded during Pause 16 at the point indicated. 

First reading of Problem i by Subject W and multiplication direct from the 
problem card with one numeral used as the "base of operations." 



Immediately after the first reading, the subject quickly re-read looo. 
After Pause 22, the record was unsatisfactory. 

Plate III shows the first reading of Problem i , the re-reading of both 
of the numerals, and the process which was followed in solving the 
problem. The first pause fell much too far to the right of the beginning 
of the line and two regressive movements were necessary as shown by 
pauses 2 and 3. The numeral 47 was evidently carefully read in the two 
pauses of 16/50 and 35/50 of a second which it received. The first 
reading of the problem was completed with Pause 9. 

The re-reading began immediately with Pause 10 on the numeral 
47. It was a long pause of greater than average duration notwith- 
standing the fact that 47 had been carefully read during the first reading. 



44 



HOW NUMERALS ARE READ 



Pause 1 1 probably served as a guiding pause in the long move from 47 
to 2. This last numeral was re-read during Pause 12. After this the 
subject returned to the numeral 47 and made it the base of the operation 

PLATE IV 

I Z 3 4 5 6 T 8 

A wiole^ale grain irn. hac at the beginning o" the day 

tZ II 9 15 'I 7 6 7 



243, 



n 



19 14 



r 

22 17 



\(b 17 



bushels 



41 Co 



of whea 



during 



s 15 




22 23 £4 25 24. 27 

bushdls were sold How mdch did they thin ha^ ^e ? 



II 



8 



15 



19 



3T 



38 31 1 

2|p:5. 

10 IS 



n 



35 2J 34 



(2, 



33 
33 30 






4^, +5 



2> 



42. 



43 



47 ,48 




T 'la 10 10 



The numerals were copied at the place indicated by the horizontal line. 

First reading of Problem 3 by Subject Hb and re-reading the numerals for copying. 
X indicates that it was impossible to determine with precision the duration of the 
pauses. 

of multipUcation, which took place during pauses 13 and 14. When the 
computation was completed, the eye returned to the vicinity of the 
numeral 2, where the answer was recorded during Pause 16. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 



45 



Plate IV illustrates the first reading of Problem 3 and the re-reading 
of both of its numerals for copying by Subject Hb. The numeral 
243,987 was given a detailed whole first reading, while the numeral 
21,765 in the same line was passed by with a rapid partial first reading. 
Of the large number of pauses used in reading Hne 2, six were placed on 
the digits of one numeral. The first reading was concluded with a 
relatively rapid reading of the last Hne. 

Immediately after the first reading, pauses 28 and 29 were used 
apparently in locating the first numeral and Pause 30 in locating the 

PLATE V 
3 f 5 4* T 8 9 10 



A tinber mai own? 



ir 



on ; plj)t bf $57 afcres, ajid ajiother of 

w 7 19 12 17 M, 14. JO 



l£ \3 11 



1342 



ZS 9 



acres 



IS* 



How 



much groi nd doea he owij altogether ? 

8 8 I'o }z 



23 21 24. 20 Zt. 25 26 i9 

1543 I sjJ? 

First reading of Problem 2 by Subject Hb,- re-reading the first numeral and 
subsequent re-reading both numerals for copying. 

place where it was to be copied. During pauses 31 and 32, the numeral 
was re-read. During pauses 2,2> and 34 reference was again made to the 
place of copying, while during pauses 35 and 36 the numeral was located 
again. With pauses 37 and 38 apparently the first group of digits was 
re-read, and the second group of digits was re-read with Pause 40. The 
second numeral appears to have been located with pauses 42 and 43, and 
it was then re-read. During Pause 45 or Pause 49 (or during both 
pauses) 21,765 was copied. After Pause 49, the record could not be 
followed accurately. 

Plate V shows that Subject Hb gave a very detailed and cautious 
first reading to Problem 2. Ten pauses, none of which represented a 



46 



HOW NUMER.\LS ARE READ 



regressive movement, were required to read the first line. In the second 

line, the numeral 1643 was given a whole first reading with pauses 11, 

12, and 13. 

PLATE VI 



If one 



telepho 16 



company uses 1,918,364 cross 



bars luring 



!2. 13. 14 15 '6 '?■ I* 

thfe yeaj", and anothdr compahy in the same period uses ( 17,433 



i& 



16 4 



cross bars, h|)w many mor^ d(()es the one u se than the ot ler ? 

3 9 



3fc 



^^ 29 34- 



19 12. Zl 



24. 26 



27 2.5 ^8 53 30 3< 3 5 



I.' <, 3 «J 40 W J7 '3 8 '« 



The numerals were copied at the place indicated by the horizontal line. 

First reading of Problem 5 by Subject M and re-reading and copying the two 
numerals, x indicates that it was impossible to determine with precision the duration 
of the pause. 

When the first reading was finished Hb proceeded immediately to 
re-read 357, apparently moved by some such purpose as the re-location 
of the numeral or the verification of one of its digits. He then re-read 
and copied 1643 with pauses 20 to 24, inclusive, whereupon he passed to 
357, which he re-read with Pause 26. After this pause, the subject's 
attention was directed to the margin at the left of the text of the problem. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 



47 



' During the first reading of the first two lines of Problem 5, as shown 
in Plate VI, Subject ]M used a large number of pauses of relatively short 
durations. The last line of the problem was read with much greater 
rapidity. Each of the long numerals was given a partial first reading. 
M approached the second numeral with a short pause and left it with 
another short pause. This method of reading long numerals was 
followed by him in the case of isolated numerals in several instances to 
which attention is called in the comment on plates XX-XXI. 

Immediately at the end of the first reading he re-read the numeral 
1,918,564 with pauses 24 to 27, inclusive, and copied it at the same time 
at the point indicated without moving his eyes from the numeral. The 
second numeral was re-located with Pause 28, and the first numeral 
which had now been copied was located with Pause 29 in order to deter- 
mine where to copy the second numeral. This numeral was then re-read 
and copied at the place indicated in the plate. 

Plate VU shows Subject Hb reading the first problem very cautiously. 
With only two exceptions every word and numeral in the problem was 



PLATE \TI 

4 5. 6 7 

At 4i7 cfent^ a doztn whkt vJjiW I do 



T 13 13 



en eg 5s co st ? 



19 12 5 2 3 10 21 



} 



4T 

First reading of Problem i by Subject Hb and re-reading the first numeral for 
copying. 



read individually. Such a large number of pauses is in sharp contrast 
with the relatively small number of pauses which were used by B and G 
in reading the same problem as shown in plates VIII and X. 

After the first reading, which was finished with Pause 10, the numeral 
2 was not re-read. The first numeral, however, was re-read with Pause 
13 and immediately copied on the problem card in the margin to the 
left of the text. 

In Plate VIII a rapid first reading of the text of the problem by 
Subject B is observed. Immediately at the conclusion of the first reading, 



48 



HOW NUMERALS ARE READ 



the subject proceeded to the first numeral which was used as the "base 
of operations" for the multipHcation process in pauses 7 and 8. The 
numeral 2 meanwhile was retained in memory. The answer was recorded 
during Pause 9 immediately below the word "what" in the text of the 
problem. 

PLATE VIII 



At 4 7 cen ;s i dozen what will ' dozei 



eggf cost? 



s T 

47 



^k 



7 II 80 

* The answer, q4, was recorded at the point indicated during Pause g. 

First reading of Problem i by Subject B and the process of computation with 
the first numeral as the "base of operations." 



In Plate IX Subject G is shown reading Problem 2 with a relatively 
small number of pauses. Only four pauses were used in the last line. 
The first numeral was read partially while the second numeral, 1643, was 
read in detail. G is the only subject who, when the isolated numerals 
were being read, was able to read four digits in detail with one pause. 
The instances in which he did this are described in the comment concern- 
ing plates XVI and XVII. 

When the first reading was completed, the subject turned his atten- 
tion to the numeral 357 and used it as the "base of operations" during 
the process of addition. This operation was carried on "mentally" 
and directly from the problem card. One or two digits were taken at a 
time, the computation starting from the right with Pause 12. The 
answer was recorded during Pause 13 or immediately thereafter. At 
the end of Pause 13 the subject closed his eyes. 

The plate itself supplies ample internal evidence of the fact that 
the nmneral 1643 was wholly read with the first pause which was 18/50 
of a second in duration. Such is undoubtedly the case since the subject 
was able to produce the correct answer without ever looking at the 
numeral again. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 



49 



The reading and solving of Problem i by Subject G is described in 
Plate X. The conditions of the problem and the identity of the numerals 
were evidently grasped during the first five or six pauses. The numerals 
were not re-read and the answer was recorded during either Pause 8 or 
Pause 9, or during both. 

During pauses 6 and 8 the subject may have been occupied with 
' ' mental ' ' computation. This suggestion is offered as a possible explana- 
tion of the fact that Pause 6 was not located on any reading material, 
but was nevertheless the longest of all pauses used in connection with the 

PLATE IX 

I 2. 3 4 5 6 



A tiriber man 



^1 



)wns on 



; plot of c 57 acres, and a lothei of 



5 « T 

16^ 3 aces. How mu^h ground does h 
a 10 



i own altogether ? 



]{ 



First reading of Problem 
numerals. 



i2 

I' 

l5- 25 

by Subject G and the process of adding the two 



problem. It does not seem probable that Pause 7 was needed by this 
subject as a re-reading pause in this problem. If computation was 
proceeding during pauses 6 and 7, evidently the eyes were roving around 
without direction. Such undirected roving occurred very rarely, if at 
all. In most cases, the eyes of the subjects were fixed on the numerals 
which were involved in the computation. 

The first reading, the re-reading, and the solution of Problem i by 
Subject M are shown in Plate XL Apparently the subject became 
confused on the first few words of the line as is indicated by the backward 
and forward movements of the pauses. Such confusion in the reading 
of problems was found in but very few instances. 



so 



HOW NUMERALS ARE READ 



After the first reading, some of the last words of the problem and 
the numeral 2 were re-read. Such re-reading of words in a problem 
was a very rare occurrence on the part of the subjects of this investigation. 
The answer was recorded immediately after Pause 18 in the margin of 
the card to the left of the text of the problem. 

PLATE X 

4 s 



At 4t cen is a dozen what 



% 10 



mWi 



dozen eggs cost? 



9 8 



12 



y 



>4' 

6' 8 

*The answer, 94, was recorded during Pause 8 or 9. 

First reading of Problem i by Subject G and the process of computation 



PLATE XI 

32+65IT « 9 

At -^ 7 ( e] its I c Dzei i what will fe dozeh eggs cosfl ? 



15 9 5 10 T T 9 



<4 



12. 



47 



1) 



II. IS 



13 14 



IZ t£> II 13 fO 10 

First reading of Problem i by Subject M, re-reading words and the process of 
computation. 

During the first reading of Problem 3, as shown in Plate XII, 
Subject G gave both of the numerals partial first readings. Immediately 
at the end of the first reading, which was finished with Pause 22, he 
began the process of subtracting the second numeral from the first. 
With Pause 23 he located the first numeral and with Pause 24 perceived 
its first right-hand digit. He then quickly glanced at the first right-hand 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 



51 



digit of the second numeral with Pause 25. The movements back and 
forth between the two numerals continue steadily, one-digit place being 
computed at each movement, until the answer was recorded immediately 
after Pause 32. 

PLATE XII 

g I 3 4 5 6 T 

A wh)lesile graii firm had at the beginning of the d&f 



243,38 

20 15 



bushels of wieat. Daring the 



day's tra- 



img 21,765 

2&> 



If 16 1% 19 80 

bushels vere soli. How much did 



they thei have? 



22 



32 



28 £<i M 



24:: .98 



2T Zt 5^ 41 3 



3» 21. ?T 2 5- 




j« z$ 30 ^^ 



First reading of Problem 3 by Subject G and the process of computation 

Attention should be called to the fact that since all of the digits of 
the answer were the digit 2, it was easier for the subject to hold the 
answer in memory as long as he did before recording. The larger numeral 
is seen to have given one more pause, not counting Pause 23, and the 
average duration of its pauses was greater than that of the smaller 
numeral. The computation began and ended with the digits of the 
longer numeral. 



52 



HOW NUMERALS ARE READ 



In Plate XIII are found illustrations of pronounced whole first 
reading of numerals by Subject H. Even the text of the problem seems 
to have been read and re-read with very short spans of attention and 
with meticulous care. 

PLATE XIII 



If one 



2 3 

telephone dompan; 



24 



"/ uses f, 

9 



4 8 



( 



)18, 

9 14. 



10 4> le. II 



•(- crojs bMi; during 



14 10 lit 



the yeix, and aiothei 



compar y 



19 24 

in the samj peifod uses 

13 13 




?8 



12 



e« 



cross b irs, 1 low r lany moi e doe 



the 



.31 35. 

one u 

II 8 



se 



^4 3fc 

1 han the othdr ' 



37 



<2 I E 




i*' so 47 44 




13 IJ 
H 13 '3' 10 i/2/,;j,2 



1301111 * 

*The answer, 1301111, was recorded at the point indicated. 

First reading of Problem 5 by Subject H and the process of computation 

After the first reading, which is concluded with Pause 37, the compu- 
tation began immediately and proceeded in a manner similar to that 
described in the comment concerning Plate XII. The numeral 1,918,564 
was used as the "base of operations"; the computation both began 
and ended with its digits. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 



53 



The figures of the answer, 1,301,111, were recorded one digit at a 
time, as they were produced by the computation, and immediately below 
the words, "use than the other," in the text of the problem. Several of 
the pauses were used in directing the hand as it recorded the digits of 
the answer. This was true of pauses on each of the two numerals. An 
effort is made to give the numbers of such pauses in Table XX\T. 

In Plate XIV two excellent cases of pronounced partial first readings 
are found. Although the numerals are five and six digits in length, 
respectively, nevertheless, each one was read ^dth a single pause. In 



A whol 



PLATE XIV 

? 3 4 5 



;sale gran fir 



11 h id a ; the begir 



ning of t 



le day 



243,987 bu; 



23 



hels of whe 

/4 



t. Durint the day 

»3 



s tradiig 21, 



14 



65 



ti 



busiels were sold. How riuch did they then hive? 



First reading of Problem 3 by Subject W with partial reading of numerals 



this plate a clear illustration of rapid reading of the last Une of a problem 
is also found. Only five pauses were required for reading line 3 and 
their durations were less than this subject's average pause-duration on 
words as given in Table XVIII. 

Plate XV exhibits the process of solving Problem 5 as it was carried 
on by Subject G. His procedure was similar to that of Subject H which 
is described in the comment accompanying Plate XIII. 

The use to which Subject G put each individual pause in the compu- 
tation is described in Table XXVI. An important difference should be 
noted between the procedures of subjects G and H in solving Problem 5. 



54 



HOW NUMERALS ARE READ 



Subject G, as is shown by the location of pauses 23 and 24 of Plate XV, 
began the computation by taking the first right-hand digit of 617,453 
and proceeded to relate it to the corresponding digit of 1,918,564. 
He continued the process by moving from right to left. Subject H, on 



PLATE XV 



If one t^Iejhone compajiy usps I,pi8,5p4 cros| bars ^uring 
r2 10 



the year, 



14 



md another c|)mpany tn the same per 



14- IS 

ibd uses 6117,453 
3 



13 IT Id ts 

cross bars , ho' v man y more 

4 IS 10 M 



£0 2/ 

ioes the one use 1 

9 <8 



22 

han th^ ; other ? 

3 



44 
4? 



3T 



•f« 33 



S4- 



Z4 11' 



'7| 

24 3 



43 



5 3£ S9 25 XS 45 



9LiJ,5Sl 



4< 3« 33 23 30 2t 



et> 19 (0 29 9 >4 



3<. 



23 24 2 «8 9 



1301111 

*The answer, 1301111, was recorded at the point indicated. 

First reading of Problem 5 by Subject G and the process of computation 

the other hand, began the computation by taking the first right-hand 
digit of 1,918,564, and proceeded to find the corresponding digit of 
617,453. He continued the process, as did Subject G, by moving from 
right to left. Both subjects, however, appear to have emphasized the 
larger numeral as the "base of operations." The details are given in 
Table XXVI. 



DESCRIPTION OF THE EYE-MOVEMENT STUDIES 55 

This concludes the general description of the photographic records. 
In the following divisions of the report various phases of the reading of 
numerals will be discussed in greater detail. In the next chapter a 
description is given of the first reading of the problems material. Chapter 
viii provides a discussion of the re-reading of problems and the processes 
of computation. The reading of isolated numerals is described in chapter 
ix. In the last chapter the performance of the subjects of this investiga- 
tion is compared with that of the subjects of an important investigation 
by another author, and finally the report concludes with a discussion of 
the differences in the demands which are made upon the attention of 
readers by the three different t3rpes of reading-materials. 



CHAPTER VII 

FIRST READING OF NUMERALS IN PROBLEMS 
I. INTRODUCTION 

When examining plates I-XV, which reproduce the lines of the prob- 
lems as they were read and which locate within the lines the pauses 
as they occurred in the readings, the unusually large number of pauses 
per Une stands out very conspicuously. The average number of pauses 
per Hne for all subjects is 8.08; and there are individual Hnes in which 
as many as 10, 11, 12, 13, and even 14 pauses are found. The large 
number of pauses appears all the more remarkable when it is remembered 
that all of the readers were advanced graduate students, who are entirely 
famiUar with simple arithmetical problems, and who would be expected 
to quahfy as better than average readers. 

Attention should be called at this point to the fact, which is given 
more detailed treatment in a later section, that the subjects of this study 
were not slow readers. It appears that there is good ground for assuming 
that the reading of arithmetical problems is more difficult than the 
reading of ordinary prose. The question suggests itself, therefore : Did 
the two elements of which the problems are composed, namely, the 
numerals and the accompanying words, make equal demands upon the 
attention of the individuals who read them in this study? The data, 
by means of which comparisons may be drawn between the numerals 
and the words, with respect to average duration of pauses, average 
number of letters or digits included per pause, and the percentage 
which the regressions are of the total number of pauses, are presented 
in tables XVII-XIX. 

2. COMPARISON BETWEEN THE READING OF NUMERALS AND 
WORDS IN PROBLEMS 

It is evident from a glance at Table XVII that there is a very great 
difference in the average ranges of acts of perception according as digits 
in numerals or letters in words are read. In the readings of all of the 
subjects the average number of digits included by a pause on numerals 
was less than the average number of letters included by a pause on words. 
The disparity between these averages is slightly greater in the cases of 
the three whole first readers B, H, and Hb, all of whom show shorter 

S6 



FIRST READING OF NUMERALS IN PROBLEMS 



57 



ranges of perception of digits than the three other subjects, who are par- 
tial first readers. Even the partial first readers, however, in every case, 
perceived on the average less than half as many digits as letters per 
pause. 

The explanation of this shorter range of perception for numerals 
than for words, when both occur in the same arithmetical problem is 
probably the same as that given by Dearborn in accounting for the short 
"number span of attention," which he had noted.' The digits in 
numerals do not appear constantly in the same combinations as do the 
letters in words. In consequence, the numerals in their continually new 
combinations of digits make larger demands upon the attention of 
readers. Every individual digit is significant in itself and must be noted ; 
and all of the digits must be viewed in combination before the numeral 

TABLE XVII 

Average Number of Digits Included in a Pause on Nuiierals Contrasted with 

Average Number of Letters Included in a Pause on Words during 

First Reading 





Subjects 


Average 

FOR 




G 


M 


w 


B 


H 


Hb 


All 

Subjects 


Average number of digits included in a pause on 
numerals 


3 40 
7.30 


2.3S 
S-QS 


2.93 
6.79 


1. 81 
7.90 


1.88 
S.18 


1.88 
5-74 


2.38 
6.47 


Average number of letters included in a pause on 





Note. — Each subject read the five problems which included 12 numerals totaUng 47 digits, and 
in words totaling 482 letters. 

is completely read. Words, however, as several investigations of the 
span of perception have shown, are perceived as wholes. The letters 
appear and reappear in the same regular combinations, which become 
familiar in the earlier years of schooling. Readers have become accus- 
tomed to them as words and are able to proceed easily with whole words 
as units of perception. 

As noted in the foregoing paragraph, the pauses on numerals are 
more concerned with analysis and combination of the component digits 
than the pauses on words are with similar processes with the letters. 
Such a difference would be expected to make itself evident in a greater 
average duration for the pauses on numerals than for the pauses on 
words. The data which are displayed in Table XVIII justify such an 

^ W. F. Dearborn, "The Psychology of Reading, An Experimental Study of 
the Reading Pauses and Movements of the Eye," Columbia University Contributions 
to Philosophy and Psychology, Vol. XIV, No. i (1906), pp. 70-71. New York: The 
Science Press. 



58 



HOW NUMERALS ARE READ 



expectation. With each of the several subjects, it is seen that the average 
duration of the pauses on numerals was decidedly greater than the 
average duration of the pauses on words. The average for all subjects 
of the average pause-durations, when numerals were read, is approxi- 
mately 40 per cent greater than the same average duration when words 
were being read. 

TABLE XVIII 

Average Duration of Pauses ix Fiftieths of a Second on Numerals Contrasted 
WITH Average Duration of Pauses on Words during First Reading 



Subjects 



VV 



Hb 



Average 

FOR 

All 
Subjects 



Total number of pauses INumerals. . 
used by subject in reading/ Words .... 
I. Average duration of pauses on nu- 
merals 

Average variation 



13-92 
3-92 



2. Average duration of pauses on words 10.72 
Average variation 1 2.27 



15-2° 

4-54 

9.87 
1.41 



15-31 
5-35 



13-02 
3-38 



18.46 
5-92 



9.18 
1. 16 



13 30 
3-97 



11.74 
4.13 



13.48 
S-o 



10.99 
2.94 



10.92 
2-55 



A comparison between the numerals and the words in respect to 
the percentage which the number of regressive pauses is of the total 
number of pauses for a subject, yields further evidence of the greater 
reading-demands made by nimierals. In the cases of subjects M, B, H, 
and Hb, as found in Table XIX, decidedly larger percentages of regres- 

TABLE XIX 

Percentage of Regressive P.auses on Numerals Contrasted with Percentage 
OF Regressive Pauses on Words during First Reading 





Subjects 




G 


M 


W 


B 


H 


Hb 


Total number of regressive pauses/Numerals 

located by subject on \ Words 

Percentage which the number of regressive pauses on: 

1. Numerals is of the total number of pauses on 
numerals 

2. Words is of the total number of pauses on words 


2 
II 

14.28 
16.6 


6 
10 

30-0 
12.3 


I 

4 

6.25 
S.63 


7 
8 

26.9 
13 -I 


10 
22 

40.0 
23.6s 


5 

I 

20.0 
1. 19 



sive pauses appear in the case of the numerals than upon the accompany- 
ing words. The explanation of such differences probably Ues in the 
difficulty of reading in the same hnes, materials which call for such differ- 
ent ranges of attention and durations of pauses as did the numerals 
and the words in these problems. When proceeding at the rate of 
reading and with the range of perception which is adapted to words, 



FIRST READING OF NUMER.\LS IN PROBLEMS 59 

the subject apparently passes over some of the numerals with a reading 
which does not satisfy him, and he immediately returns to read or to 
re-read all or a part of the numeral. 

3. PARTIAL AND \\-HOLE FIRST READING OF NUMERALS 

Whole first reading was defined in the first preliminary study of 
the investigation to include such readings of numerals during the first 
reading of a problem as noted the character of the numeral and the 
identity and place in the numeral of each individual digit. Any reading 
of a numeral which did not include these items was called a partial first 
reading. In Table XX the kind of reading given each of the twelve 
numerals in the problems by each of the several subjects is described 
in detail. The data which are included in this table are based upon 
introspective observations concerning their readings by several of the 
subjects, and upon inferences which were drawn directly from the 
plates. With the longer numerals a very small number of pauses of 
short duration in some instances gave indisputable evidence of partial 
reading. In several of the records answers to problems which included 
shorter numerals were computed and recorded when the numerals had 
been read only on the first reading; obviously such readings were whole 
first readings. A few readings could not be placed with certainty in 
either category and are, therefore, marked D, which means doubtful. 
Plates II, IV, VI, XII, and XIV present instances in which numerals 
received partial first readings, and plates I, IV, IX, and XIII show 
other instances in which numerals received whole readings. 

There are marked differences between partial and whole first readings 
of the longer numerals in respect to the number of pauses per numeral 
and the total time required for reading the numeral. These differences 
are quickly apparent when Table XX and the plates which bear illustra- 
tions of the two methods are studied in detail. Illustrations of both 
methods of reading are found in Plate IV which represents the reading of 
Problem 3 by Subject Hb. In this problem one of the numerals, 243,987, 
was given a whole first reading which included six pauses and measured 
a total reading-time of 101/50 seconds, while the other numeral, 21,765, 
was given a partial reading which included only one pause with the 
duration of 14/50 of a second. 

Further emphasis is given to the difference between partial and whole 
reading of numerals by a comparison of the first readings of the numerals 
in the problems with the readings of the numerals isolated in lines. As 
prescribed by the conditions, the readings of the isolated numerals 



6o 



HOW NUMERALS ARE READ 



^ 



a 

7. 


^ 


o 




o 


Ph 


H 




tn 


o 




Jg 


< 


<; 


^ 




o 




tn 




U 




H 




M 





a 

Id 

n 

Hi 


3 
CO 

"o 

E 

3 


5_ ifi 
3 3 S 




00 M 00 












^ Lo -^. -yj f- r- 




t^ 0* M Tf W 


TT 


00 


^£ - 5*2 £"2 § s £ Q-i^ 


^ HI M 


c. 




! 







t « 




§ 


::§ :?| :?§-2 1 ;:'!:: 1 









^ 


00 


T? 

00 . 0^ ^^^"^ i^ ^-^ 

-a- -a* :?&, 0^ „-^ 1Q 

M 0> 0* 

OO 




to « M 






o._ t_ o,~ i- ^- S"^ 

M <0 


•* « 




t^ 


00 


1- 


,^ 00" '^ ^^ 

".fl^ ^S^ ^E' 5;^ « a^ j;^-' 

00" r^ 


•* w 





r* 


00 

"SS °Q «£ I^B "-£ M§! 


1- 


- 


" 






«H ^ 




* 


t^ 




««« 


H 




°°i"i|i"i-i"a 




10 M 


« 


^i-i^i-i-i""! 









s 

3 

is 
i 
2 


Si : 

it 

e1 
3 >- 
Is 


Durations of pauses 

in serial order 

Durations of pauses 

in serial order 

Durations of pauses 

in serial order 

Durations of i)auses 

in serial order 

Durations of pauses 

in serial order 

Durations of pauses 

in serial order 


h 
C 

.. r. 
3ftH 


II 

5 ° 






sioaTaas 


S ^ 


« S ^ 


SXO 




an 
V 


s 



^!i 



S2 ^ 

> o 
9 » 









FIRST READING OF NUMERALS IN PROBLEMS 6 1 

were of the quality of whole readings. The data are arranged in Table 
XXI for such a comparison between the two sets of numerals in each 
of the several digit-lengths with respect to the average number of pauses 
per numeral, the average pause-duration, and the average time required 
for reading individual numerals. In respect to each of the three points 
named for comparison the isolated numerals are found to have larger 
averages than the problem numerals in each of the several digit-lengths. 
If only those problem numerals which were given partial first readings 
were included in the comparison, the differences between the two sets 
of numerals would be appreciably greater than they are. 

On the other hand a certain degree of qualification is attached 
necessarily to the significance of the large differences found because of 
the effect probably produced on the reading of the isolated numerals 
by one of the conditions under which they were read. The slight 
articulation used in reading the isolated numerals presumably acted 
to diminish the speed with which they were read. It should be noted 
also that all of the subjects gave e\'idence of greater interest in reading 
and solving the problems than in reading the numerals isolated in lines. 
The probable effect of such great interest in the problems was to stimulate 
the readers to a more rapid rate of reading the numerals in problems 
than the numerals of corresponding digit-length which were isolated 
in lines. 

Proper allowances should be made for the differences which were 
produced in the readings of the two sets of numerals by such variations 
in conditions as are named above. When this allowance is made, it is 
found that the whole first readings which were given the longer numerals 
in the problems by the whole first readers include numbers of pauses and 
total reading-times per numeral which are similar to those found in the 
readings of the isolated numerals. The whole first reading of nimierals 
in problems is evidently similar in kind to the reading of numerals 
isolated in fines. 

As was found in the first prefiminary study, marked differences 
appear between the shorter and longer numerals in respect to the number 
of times in which they were partially and wholly read during first read- 
ings. The one- and two-digit numerals were read in detail in most 
instances in the present study as they were in pre\dous studies. Only 
one pause, for the most part, was required for the reading of the shorter 
numerals and the durations of such pauses ran as low as 5/50 and 8/50 
of a second. The longer numerals on the other hand received a sHghtly 
greater number of partial readings than whole readings. The partial 



62 



HOW NUMERALS ARE READ 



< Q S 

G 5 w 

g d, tn 

CJ « " 

S o o 



I ^ 

Ph CD 





- 


-a 
1 


t 


^ 


00 


o 




E 
2 


" 






>o 




•* 


OOO >^ 
1^00 O 




E 
o 


. 


CO « r^ 
O ^ " 




« 


•a 


■* 


rn 'n 




E 
o 


" 


vO o o 

O ■* so 




■<* 


•a 


t 


OOO "1 




E 


w 


00 -t « 
« O O 




t^ 


1 


^ 


^ >« 

a o o 

M r-^ 00 




E 

►5:3 

o 


" 


M so ^ 




- 




t 


OsO 

«oo r~ 

M C) 00 




E 
o 


" 


w >o 00 




M 




■^ 


MO M 




i 


M 


SO SO 
OsO SO 




1 
I 

c 

c 
'5 


) 
) 


•fl 

3 C. 

If 

2; 


>> • 

y 

:: 0) 


*-• 

a 

3 

a 
"o 

E 

3 

2 - 
< 


c 

i 

c 

> 

< 


Oj2 

1e 

.i: 3 
3 c 

2''3 

u'> 

E^ 

. M 

< 



FIRST READING OF NUMERALS IN PROBLEMS 63 

readings for the five-digit numerals included one pause for the most 
part ; and in the six- and seven-digit numerals two pauses were included 
in most instances. In respect to proportion of partial readings received, 
the compact group of three numerals in Problem 4, which are placed 
one immediately after the other, may be classified with the longer 
numerals. 

The familiar numeral 1000 has received a kind of reading which 
distinguishes it from other numerals of the same length, wherever it 
has appeared in the preliminary studies. In the present study 1000 
was given whole first readings without exception. In no instance was 
more than one pause required for the reading, and the lengths of the 
pauses closely approximate average pause-durations. For whole first 
readings of other numerals of four-digit length, which are found in the 
problems, two or three pauses were required. Unlike the other four- 
digit numerals, 1000 is evidently read as a whole, as words are read. 

It is probable that the form of 1000, which is that of the digit " i," 
followed by the very obvious group of three "o" digits is easier of 
perception than any ordinary group of four digits. It is probably 
easier of perception than any numeral which is made up of a group of 
four digits all of which are the same digit. The frequency of appearance 
of 1000 in the experience of the average reader is probably greater than 
that of any other single combination of four digits. By virtue, therefore, 
of the obviousness of its structure and by virtue of the frequency of its 
use the numeral 1000 has become familiar to the average reader in the 
same sense that words are familiar, and in consequence is read as words 
are read. 

The number of digits which it is possible to read partially in one 
pause of partial reading is relatively large. The five digits of the five- 
digit numerals were read with one pause in three of the four instances 
when they were partially read. One of the sLx-digit numerals was read 
one time with one pause, as is shown in Table XX. The six and seven 
digits of the six- and seven-digit numerals were, however, read with two 
pauses in most instances when they were partially read. The usual 
number of digits read at one pause when such longer numerals were 
read partially is, therefore, either three or four. 

In the preliminary study which was concerned with the range of 
recall from the first reading of numerals in problems it was shown that 
in a great preponderance of instances the subjects were able to recall 
at least the number of digits in a numeral. In view of this fact it seems 
reasonable to assume that one or more of the subjects who read the 



64 



HOW NUMERALS ARE READ 



five-digit numerals partially and with a single pause, were able to read 
all five of the five digits at one pause, at least to the extent of noting 
that a numeral was there and that its length was five digits. Apparently, 
therefore, it is possible for subjects to read to this extent as many as 
five digits at one act of perception. 

4, THE SEVERAL SUBJECTS AS PARTIAL AND ^\^^OLE FIRST READERS 

The largest factor in determining whether the longer numerals shall 
be given partial or whole first readings is found in the attitude of indi- 
vidual subjects toward these nimierals. The same numerals were given 
partial readings by some of the subjects and whole readings by others. 
An examination of the data in Table XX makes it evident that some of 
the subjects gave partial readings with noticeable consistency to the 
longer numerals, while other subjects with approximately equal con- 
sistency gave whole readings to these numerals. Subjects G, M, and 
W clearly exhibit the former tendency and may, therefore, be classified 
as partial first readers, while B, H, and Hb illustrate the latter tendency 
and may be called whole first readers. 

5. RELATIVE VALUE OF PARTIAL AND OF \VHOLE FIRST READING 

Any determination of the relative value of whole and partial reading 
as methods of reading numerals during the first reading of a problem 
will be concerned to a large extent with the question as to which of the 
methods is more economical of the reader's time. This question resolves 
itself in large part into a comparison between the partial and whole first 
readers in respect to the total time required to read all of the numerals 
of the problems. The three partial readers G, M, and W, as is readily 
seen by inspection of Table XXII, used decidedly shorter total times 

TABLE XXII 

Reading of Numeraxs in Problems by Partial First Readers Contrasted with 

Reading of Numerals in Problems by Whole First Re.aders 

(Time unit = 1/50 of a second) 





Subjects 




Partial First Readers 


Whole First Readers 




G 


M 


W 


B 


H 


Hb 




195 

14 

13.92 


304 

20 

15-20 


245 

16 

15-31 


480 

26 

18.46 


359 

27 

13-30 


337 


Total number of pauses required to read all numerals 


25 
13.48 







Note. — Each subject read all five problems which included twelve numerals. 



FIRST READING OF NUMERALS IN PROBLEMS 



65 



for the first reading of all of the numerals than the whole readers B, H, 
and Hb. Partial first reading, therefore, in so far as time required for 
reading the numerals is concerned, was undoubtedly the more economical 
of the two methods. 

The words of the problems were also read more rapidly by the 
partial first readers, although the case is not as clear for the words as 
it is for the numerals. The details are given in Table XXIII. The 
fastest reader of the words is the whole first reader, Subject B, who is 
also the slowest reader of the numerals. His case appears to be very 
exceptional and no adequate explanation is found in the records, nor 
was the subject himself able to account for his relatively high speed 
with words. After B, the two partial readers, G and M, have the 
fastest records. The slowest record was made by the whole first reader 
H. By virtue of relatively higher speeds with both numerals and words 
the partial first readers completed the first reading of the problems in 
shorter total reading-times than the whole readers. 

TABLE XXIII 

Reading of Words in Problems by Partial First Readers Contrasted with 

Re.^ding of Words in Problems by Whole First Readers 

(Time unit = 1/50 of a second) 





Subjects 




Partial First Readers Whole First Readers 




G 


M 


W B 


H 


Hb 


Total time required to read all words of problems. . . 
Total number of pauses required to read all words of 


708 

66 
10.72 


Soo 

81 
9.87 


925 
71 


560 

61 
9.18 


1092 

93 
11.74 


923 

84 
10.99 











Note. — Each subject read all five problems which included iii words. 



Another basis may be found upon which significant inferences can 
be drawn as to the relative value of the two methods of first reading of 
numerals. It is possible to draw a comparison between the several 
individuals, who use the one or the other of the two methods, in respect 
to their rates of reading with reading materials other than arithmetical 
problems. Stated in other terms the point in question is: WTiich of 
the two methods was used by the readers who exhibit higher rates of 
reading in other materials. The data concerning the rates of speed 
with which the subjects read the ordinary prose selection may be used in 
this comparison. With these materials the average reading-time per 



66 HOW NUMERALS ARE READ 

line for subjects M and G were 44.88/50 and 52.52/50 seconds, respec- 
tively, and for subjects W, H, and B they were 75.21/50, 75/50, and 
80.78/50 of a second, respectively. It is clear, therefore, that the faster 
readers of the ordinary prose selection used the partial method of reading 
numerals on the first reading, and two of the three slower readers of the 
ordinary prose used the whole method of reading numerals. 

The more rapid reading of the partial first readers is due to the fact 
that for the most part they used fewer pauses in reading the same 
materials than the whole readers. Such is the explanation of the greater 
speeds whether the materials which were used were the numerals or 
words of a problem, or the lines of the ordinary prose selection. A 
smaller number of pauses also explains the exceptionally short total 
reading-time of Subject B on the words of the problems. 

6. DEVELOPMENT OF THE METHOD OF PARTIAL FIRST READING 

The large differences between the methods of partial and whole 
first reading in the several important respects to which attention has 
been directed in this immediate study and in previous studies of the 
investigation can be satisfactorily explained only in the light of large 
differences in attitude on the part of the individuals who use respectively 
the one or the other of the two methods. The evidence is strong that 
partial and whole first readers entertain very different attitudes toward 
the numerals when the problem is being read for the first time. The 
fact that such differences between the two groups do exist does not 
necessarily imply that the members of either group are conscious, either 
of their own peculiar attitude, or of the existence of a different attitude. 
Most of the subjects of the investigation were not conscious of their 
attitudes. The impHcation is, rather, that through long experience with 
problems these subjects have developed in an empirical way two widely 
differing sets of habits of reading numerals during the first reading of a 
problem. 

When first learning to read numerals in problems, these subjects, 
as most individuals probably do, proceeded to read the numerals very 
slowly, as they came to them, and one digit at a time. The whole 
reading attitude toward numerals would, therefore, be the attitude 
more natural for beginners in reading arithmetical problems. In the 
course of extensive experience, apparently, some of the individuals were 
more impressed than others, with the differences between words and 
numerals in respect to the rates of speed which were found practicable 
in reading them. These individuals were thus stimulated to learn a new 



FIRST READINGS OF NUxMERALS IN PROBLEMS 67 

method of procedure with the numerals, which would make for a quicker 
disposition of them in reading problems. Such a new procedure could 
not consist simply of a radical increase of the span of perception to include 
larger units of recognition, as had been done with word materials when 
whole words and phrases came to be taken as units of perception. 
Perception of the larger numerals by numeral wholes appears to be 
quite impracticable because of the nature of numerals as continually 
varying combinations of digits. 

The plan that was learned consists of a rapid passage over the 
numeral during which time details are skipped and only the most 
outstanding facts concerning it are gathered. Such is the procedure 
which is designated as partial first reading and in preliminary sections 
of the investigation the identity of the first digit of a numeral and 
recognition of the number of its digits were found to be facts of the 
outstanding nature referred to. Various ranges of partial first reading 
came to find empirical acceptance in the course of the development of 
the habit of partial reading by the natural trial-and-error method of 
learning. Such changes in procedure were probably able to be 
accomplished with little or no embarrassment to individuals in the 
practical solving of problems because of the almost invariable habit of 
re-reading the numerals after the first reading of a problem. 

7. SUMMARY OF CONCLUSIONS 

The chapter may be summarized as follows: 

1. The numerals of problems make greater demands upon the 
attention of readers than do the accompanying words, as is shown by 
the following facts : (a) The average number of digits included by a pause 
on numerals is decidedly smaller than the average number of letters 
included by a pause on words, (b) The average duration of pauses on 
numerals is greater than the average duration of pauses on words in the 
cases of all subjects, (c) The percentage of regressive pauses on numerals 
is greater than the percentage of such pauses on words. The explanation 
of the greater demands of the numerals probably lies in the fact that the 
combinations of digits in numerals are continually different, whereas 
combinations of letters in words remain stable. 

2. More pauses per numeral and greater total reading-times per 
numeral are required for whole first reading of numerals than for partial 
first reading. 

3. Shorter numerals are given whole first readings almost invariably. 
Longer numerals, on the other hand, are given whole or partial readings 
according as the subjects who read them are whole or partial readers. 



68 HOW NUMERALS ARE READ 

4. The numeral 1000 is regularly read as if it were a word rather than 
a numeral. 

5. The subjects divide themselves into groups of partial first readers 
and of whole first readers according as they read the longer numerals 
by the partial or whole method. 

6. The partial method is more economical than the whole method 
in point of total time required to read the numerals during the first 
reading of a problem. 

7. The subjects who read the ordinary prose selection more rapidly 
use the partial method of reading the longer numerals for the most part. 

8. The more rapid rates of reading in both the words and numerals 
of problems and in the ordinary prose selection are exhibited by subjects 
who use the smallest number of pauses in such readings. 

9. Partial first reading of numerals is probably learned empirically 
as a method of more rapidly disposing of the numerals in reading. The 
essential characteristics of the method are skipping the details of a 
numeral and recognizing only the most outstanding facts concerning it. 



CHAPTER VIII 

RE-READING AND COMPUTATION 

I. TWO TYPES OF RE-READING OF NUMERALS 

Re-reading from a problem takes place immediately upon completion 
of the first reading. Two distinct types of re-reading of numerals appear. 
The two types are distinguished by differences in function. In some 
cases such differences are disclosed directly by reports which were made 
by the subjects themselves upon the basis of introspective observations 
as to what numerals were re-read and why they were so re-read. In 
other cases definite evidence as to which tx-pe of re-reading was used is 
found by careful examination of the readings as they are described in 
the plates. 

The first type, which may be called simple re-reading, has as its 
function, apparently, the gathering of further information concerning 
the numerals before a decision has been reached as to what plan will be 
followed in solving the problem. Only one of the numerals of a problem 
is re-read after this fashion. A single case of exception was found in 
the reading of Problem 5 by Subject Hb when two numerals were 
re-read in this way. The implication is that such re-reading is under- 
taken only when the subject is definitely interested in some specific 
detail of a certain numeral. According to the reports of subjects these 
specific details include various items of verification of numerals such as 
the identity of certain digits, the number of digits in the numeral, and 
the location of the numeral within the line. 

The type of re-reading which was given a numeral is indicated in 
Table XXIV in all cases but two. The two exceptions are with the 
readings of problems 2 and 3 by Subject M. In these instances several 
of the words of the problems were re-read along with the numerals. 
Because of this fact interpretation of the records was complicated to 
such an extent as to make it impossible to distinguish with certainty 
which type of re-reading was given the numerals. 

Instances of simple re-reading are described in detail in plates I, 
II, III, V, and XL The numeral 1000 was given such a re-reading by 
four of the subjects; illustrations are found in plates I and II. Subject 
Hb gave a re-reading of the simple type to numerals in each of the 
last four problems, and an illustration of his procedure in the case of 

69 



70 



HOW NUMERALS ARE READ 





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RE-READING AND COMPUTATION 71 

Problem 2 appears in Plate V. In Plate XI is found an instance of simple 
re-reading of the numeral 2 along with certain words of the problem by- 
Subject M. 

The second type of re-reading of numerals is that for the purpose 
of copying the numerals on to the problem card. In the case of each 
re-reading so classified the records show that the subject after he had 
re-read the numeral, proceeded immediately to copy it on the problem 
card. Such re-reading was the normal procedure for subjects Hb and 
M as is shown in Table XXIV. More detailed descriptions of 
re-readings of this ty^e are found in plates V, VI, and VII. The record 
of Subject M, which is presented in Plate VI, shows a distinct variation 
from the procedure of re-reading as described above. Subject M 
re-read the numeral 1,918,564 by the four pauses, 24 to 27, inclusive, 
and at the same time copied the numeral in the vicinity of pauses 32 to 34, 
without moving his eyes from it during the process of re-reading and 
copying. 

Whether a numeral was or was not re-read depended upon the habits 
of individual subjects rather than upon either the length of the numeral 
or upon the quality of first reading which it had received. Numerals 
of all digit-lengths, including both those which had received whole first 
readings and those which had received partial first readings, were re-read. 
Re-reading was practiced systematically both by subjects Hb and M, 
the former of whom was classified as a whole first reader and the latter 
as a partial first reader. Of the four subjects who almost invariably 
proceeded without re-reading, two were classified as whole first readers 
and two as partial first readers. 

Attention should be called at this point to the marked differences 
between the small proportion of re-readers of numerals, as reported 
immediately above, and the fact that all subjects re-read most of the 
numerals of the problems which were used in the first prehminary study. 
Subjects G and H who with only one exception did not re-read the 
nimierals in the present study did, however, persistently re-read the 
numerals of the problems of the first prehminary study. In subsequent 
paragraphs, re-reading is found to be closely connected with the pro- 
cedure of copying the numerals on paper for computation with pencil. 
With only occasional exceptions this procedure was followed regularly 
by all of the subjects in the first prehminary study, including subjects 
G and H. It is probable, therefore, that their persistent re-reading in 
the first study was done for the most part in order to insure accuracy 
in copying the numerals on the computation paper. 



72 



HOW NUMERALS ARE READ 



2. METHODS AND PROCEDURES USED IN THE 
PROCESS OF COMPUTATION 

Two distinct methods of computation were exhibited by the subjects 
in respect to their procedure with the numerals immediately after the 
first reading of a problem. In one case two of the subjects re-read and 
copied the numerals on the problem card, and wrote out on the card 
with pencil the figures used in the process of computation. In the other 
case, four of the subjects computed "mentally" and directly from the 
numerals as they were printed on the problem card. The former may 
be called computation from copied figures and the latter, computation 
direct from the problem card. Table XXV shows which procedure was 
followed with each problem by each of the several subjects. 

TABLE XXV 

Two Methods of Proceeding with Numerals for the Purposes of Computation 
AFTER the First Reading 





Problem 


Subject 














I 


2 


3 


4 


5 


Hb 


X 


X 


X 


X 


X 


M 


Re-read 2 


Re-read 

numerals 






X 





X 





X 





X 


w 





H 





B 











Re-read 1000 






G 











Re-read 1000 







Explanation of symbols — 

"X" =re-read and copied numerals, and computed from copied numerals. 
"0"=computed immediately and directly from the problem card without re-readi; 



It is important at the beginning of the discussion of the two methods 
of procedure that the most essential difference between them be set 
forth clearly. The difference lies primarily in the number of mental 
steps involved in the two methods. At least two additional mental 
steps are required by the method of computation from copied figures, 
namely, re-reading of numerals and copying them on the computation 
card. The method of computation direct from the problem card avoids 



RE-READING AND COMPUTATION 73 

these two steps. A large pedagogical significance attaches to the elimi- 
nation of two such mental steps. By their ehmination it would appear 
that many opportunities for error are avoided and valuable economies 
of time and of mental and physical effort may be effected. 

The description of the procedures which were used in computation 
that follows, is concerned only with such procedures as were exhibited 
by the subjects who computed directly from the problem cards. It was 
found impossible to interpret precisely the records of eye-movements 
over copied figures in respect to the location of the pauses on the figures. 
The processes of computation could be followed to the solution of the 
problem in only a Umited number of the records even of those subjects 
who used the method of direct computation. 

The procedure of direct computation from the problem card appears 
to have been followed without regard to the quahty of first reading which 
the numerals had received. Cases of direct coinputation are found as 
sequels both to partial first readings and to whole first readings of the 
numerals. In Plate XI direct computation is observed to have followed 
upon whole first readings of both numerals of the problem. Similar 
illustrations are found in plates VIII and X. 

In Plate XII, however, the two numerals had been only partially 
read during the first reading. In this instance the numerals never were 
read completely at any reading. Similar cases were found with other 
subjects. Evidently for some subjects only a partial reading of the 
numerals of a problem is necessary for the successful use of such numerals 
in computation. 

It is important to notice in this connection that the same subjects 
do not always use the same procedure in computing with different sets 
of problems under different conditions. Illustrations are found in the 
cases of subjects G and H. These two subjects used the method of 
direct computation with the "five problems,'' as reported immediately 
above. With the "seven problems" of the first preliminary study, 
however, they followed the procedure of computation from copied 
figures. 

An explanation of these facts should begin with the very probable 
premise that these two subjects, who were adults and advanced graduate 
students, were able to follow successfully either procedure in sohdng 
such simple arithmetical problems as were used in this investigation. 
The method of direct computation was followed with greater faciUty 
with the "five problems" than with the problems of the first prehminary 
study because of the greater ob\dousness of the answers to the "five 



74 HOW NUMERALS ARE READ 

problems." The numerals of most of these problems had been so 
selected as to make the answers even numbers, or numbers with every 
digit the same, in order to minimize the labor of computation. It is 
probable also that the subjects worked at somewhat higher tensions 
when seated before the camera for solving the "five problems" than 
when seated at an ordinary desk for solving the "seven problems." 
Such higher tensions might well have influenced the workers to select 
the quicker direct method rather than the slower copying method of 
solving. 

During the process of computation, the numerals, which are in the 
context of the problem and with which the computation is concerned, 
do not receive the same quahty of attention from the subjects. This 
fact is most clearly in evidence when two numerals appear in the context 
of a problem. The records show that more pauses and pauses of greater 
average duration were located on the digits of one of the numerals than 
on the digits of the other numeral. In the cases of shorter numerals, 
pauses were located upon only one of the numerals, while the other 
numeral was retained in memory. By such unequal distribution of 
attention one of the numerals was, in effect, made the "base of opera- 
tions" during the computation. Three examples of this procedure were 
found in the solving of the problems in which the shorter numerals 
appeared. The records appear in detail in plates III, VIII, and IX. 
In Plate IX, which will serve as an illustration, the numeral 357 was used 
as the "base of operations, " while the numeral 1643 was held in memory. 

An example of the same procedure with the longer numerals of 
Problem 3 is found in Plate XII. In this instance the numeral 243,987 
was used as the "base of operations." Five pauses with an average 
pause-duration of 38/50 of a second were located on this numeral during 
the process of computation. On the other hand, on the numeral 21,765 
only four pauses were located during the computation, and their average 
duration was only 28.75/50 of a second. 

Two further examples of the procedure are found in the solution of 
Problem 5 by subjects H and G in plates XIII and XV. Interpretation 
of plates XIII and XV is compHcated to a large extent by the presence 
in the records of a number of pauses which were not used strictly in the 
processes of computation. Such additional pauses were used apparently 
in locating the digits next in order for computation, or else they were 
used in directing the hand when it was engaged in recording figures of 
the answer. Such pauses, therefore, may be referred to as locating- 
and as recording-pauses respectively. 



RE-READING AND COMPUTATION 



75 





s 


HH 


t3 


> 

X! 


^ 


X 


z 


w 


O 


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a 


pq 


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O O PS tAr-- 



76 HOW NUMERALS ARE READ 

It was found impossible to distinguish the locating- and recording- 
pauses in plates XIII and XV from the computation pauses with abso- 
lute certainty. An effort was made, however, after a detailed study of 
the original reports of subjects H and G on their solutions of Problem 5, 
to separate the locating- and recording-pauses from the computation 
pauses. The result appears in Table XXVI. An inspection of the table 
shows that both subjects H and G used the numeral 1,918,564 as the 
"base of operations" during computation. 

The longer numeral in five of the six cases, which are described above, 
was taken as the "base of operation." Such selection of the longer 
nimieral is probably in keeping with the practice common in the solving 
of arithmetical problems which, when two numerals are arranged for 
computation, places the greater numeral first in order and relates the 
second numeral to the greater. The larger number of pauses upon the 
longer numeral, which at the same time is the "base of operations," 
is due to the fact that computation both begins and ends with the longer 
numeral, and to the further fact that in the longer numeral an additional 
digit appears for reading. 

The large difference between the average duration of the pauses on 
the numeral which was used as the "base of operations" and the average 
duration of the pauses on the other numeral is significant of a difference 
in function between the two sets of pauses. The pauses on both numerals 
necessarily must use such time as is sufficient for recognition on the part 
of the reader of the digits with which the pauses are severally concerned. 
In addition to the work of recognitions, however, some of the pauses 
on the "base of operations" numeral evidently perform service in the 
more strictly arithmetical processes. For such service a greater pause- 
duration would undoubtedly be necessary. 

3, SUMMARY 

1. Two types of re-reading of numerals are distinguishable. Simple 
re-reading is concerned with verification of details of the numerals. 
Re-reading for copying is concerned with reading the numerals for 
copying on the computation card. 

2. Two of the subjects normally re-read all of the numerals. The 
four other subjects normally do not re-read the numerals. Whether 
the numerals are or are not re-read depends upon the habits of individual 
subjects. 

3. Two methods of proceeding with the numerals after the first 
reading are distinguishable. In the one, computation begins immedi- 



RE-READING AND COMPUTATION 77 

ately and is carried on directly from the context of the problem. In the 
other case the numerals are re-read and copied, and the computation 
proceeds from the copied figures. By the former method, two mental 
steps are saved. 

4. The method of immediate computation direct from the context of 
the problem is used without regard to whether the numerals have 
received a partial or a whole first reading. 

5. During computation one numeral is taken as the "base of opera- 
tions." A large number of pauses and pauses of greater average duration 
are located on the digits of this numeral than on the digits of the other 
numeral. The significance of the greater duration of such pauses 
probably lies in the additional work of the more strictly arithmetical 
processes which seems to have been done during these pauses. 



CHAPTER IX 

THE READING OF ISOLATED NUMERALS IN LINES 

I. INTRODUCTION 

As was Stated in the introductory paragraphs of this report such 
attention as has been given to the reading of numerals in previous 
experiments in the psychology of reading has been incidental to other 
purposes. The numerals which were read in previous experiments 
were in each case isolated numerals in lines. Gray," when investigating 
the perception span of good and poor readers, had a number of individuals 
read short Unes of unspaced digits and groups of the same digit as well 
as selections of words with meaning. A summarizing paragraph at the 
end of his discussion contains the conclusion that differences between 
the span of attention of the good and poor reader disappear in a very 
large measure when digits or groups of the same digit are read. 

Dearborn,^ while interested chiefly in the span of attention and in 
the question as to whether perception proceeds by number wholes or by 
individual digits, had several subjects read lines of digits which were 
printed consecutively without spacing, and lines of numerals varying 
in length from two to six digits. In the records which were obtained 
from these readings he observed that the time required for reading the 
same number of unspaced digits in a line was greater when the subjects 
grouped the digits by fours than when they grouped the digits by threes. 
He noticed also that the time required for reading numerals increased 
with increases in the digit-lengths of the numerals. Attention was 
called, at the same time, to the larger number of "shifts" or "breaks" 
in the fixations on numerals than in the fixations on words, and the 
opinion was expressed that a single digit was probably sometimes the 
unit of perception in the two-digit numerals. 

In a preliminary study of the present investigation, data were 
presented concerning the reading of numerals arranged in columns. 

I C. T. Gray, "Types of Reading Ability as Exhibited through Tests and 
Laboratory Experiments," Supplementary Educational Monographs, Vol. I, No. 5 
(1917), p. 146. 

^W. F. Dearborn, "Psychology of Reading: An Experimental Study of the 
Reading Pauses and Movements of the Eye," Columbia University Contributions to 
Philosophy atid Psychology, Vol. XIV, No. i. New York: The Science Press, 1906. 
See chapter x, "The Number Span of Attention," pp. 67-73. 

78 



THE READING OF ISOLATED NUMERALS IN LINES 79 

In that study special attention was called to the fact that the digits 
of the numerals were read in groups. The purpose of the present study 
is to examine in detail the readings of a representative number of numerals 
of each of the several digit-lengths of from one to seven digits. Varia- 
tions in the readings of the numerals of the several lengths are reported 
and the reading habits, which were exhibited by individual readers, 
are described. The records of the movements of the eyes of the subjects 
while engaged in reading numerals are given in plates XVI-XXV, and 
the data from the records are condensed and 'arranged in tables XXVII- 
XXXI. 

In the plates the variations in length of the vertical lines above and 
below the lines of printed numerals were provided merely for convenience 
in drawing in the numbers of the various pauses. 

When the initial pause of a line did not fall on one of the digits of 
the first numeral in the line, such a pause was not included in the tables. 
When an initial pause fell on the first numeral of a line and was followed 
by a regressive movement, such a pause was not counted in the tables. 
It is obvious that counting pauses of the latter sort would have given 
in each case an additional pause to the first numeral in the line merely 
because of its position as first numeral in the line. 

In plates XVI and XVII the reading of isolated numerals by Subject 
G is represented. At the beginning of each line of numerals an initial 
regressive movement was found necessary in the effort to locate the first 
digits of the first numeral. Subject G reads with relatively few pauses, 
but with pauses of relatively long duration. The pauses vary widely 
in duration; the range of variation extends from 4/50 to 90/50 seconds 
with the average duration at 33.88/50 of a second. Single pauses, when 
they are located on the longer numerals, tend to perceive two or three 
digits rather than one or two. In three instances the subject 
accomplished the remarkable feat of reading four digits during a single 
pause. The three instances are found in Plate XVII; two are in the 
first line with the numerals 9317 and 5,236,795; and the third is in the 
second line with the numeral 1928. 

In plates XVIII and XIX appear the readings of isolated numerals 
by Subject H. This subject read with a relatively large number of 
pauses, but with pauses of relatively short duration. The pauses 
varied in duration from 4/50 to 56/50 seconds. The average dura- 
tion, which was 19.32/50 of a second, was shorter than that of any other 
subject. Many short guiding-pauses appear. Single pauses, even 
when they are located on the longer numerals, tend to include only 



8o 



HOW NUMERALS ARE READ 



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2,37^,9 



30 29 16 



5 4> 



10( 







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13 5 

24 '3 



r4. w 
Reading of isolated numerals by Subject G 



17 



PLATE XVII 

■I ) 



2 5M0 



7 9 

5, J36,7' 15 



65 



42. 



2)6 

31 



2 5 

74;;,8[9 

3£ 33 9 



4- 

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3'/ 28 23 



2 3 

10r,3)8 

2& 33 M 



4 

52 

32 



6 

9^ 
39 



6J7,( 



37 



21 4 34 



Reading of isolated numerals by Subject G 



THE READING OF ISOLATED NUMERALS IN LINES 



8c 6 

5^ 



4 5 6 

54811 



(o 25 21 



PLATE XVIII 

cr 8 
7 ip 9 IE 



7se 



I? Z\ 13 'I 
■33 13 



35:5 



46 



«4 I5 l4 IT it 



4,125 



983 



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8 ),9: 4 
45 sz 



3 
2; 19 



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5 £R ^5 5 



3 

l}2tt 

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54,908 

4/ KO 3 5 



I 

12 

£4- 



75,1; 4 

^2 33 4, 



2 

2J374,9 



)i 



40 43 43 



r (. 
100} 

25 9 



ft 

313 

48 



9 10 



25,000 



»5 



Reading of isolated numerals by Subject H 



fl3 



«„4 

7 
3 



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22 



PLATE XrX 

(0 li 

I 

•4 



IP 13 14- |5 

5,2:16,735 
29 £0 3fc I* 



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2)e 



74 J, 



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20 



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54^ 



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iT BO 21 

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2 3 14 

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7 4. 
515 

8 24 



9 

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S5 



6 



57J 
3a 



(2 /3 

6J7 



R4 ^"^ 



14 



Reading of isolated numerals by Subject H 



82 



HOW NUMERALS ARE READ 



PLATE XX 

\Z i3 9 



3 -a. » 4- 5 



use 



■5 5 IS a 12 9 



6 T 8 

at 13 (> 



M 



lO 



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12 



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3 21 



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2 39 

•31 7 



40 



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13 



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J,: 174,; 5: 

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ICOO 

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45 



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2r ar 



Reading of isolated numerals by Subject M 



4 5 

{31! 

29 14 



PLATE XXI 

C 7 8 






T 7 



10 1 1 



2y^,:9 



12. 



IS lb 

2 5G 

3& 4 



■^ 5 



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10 i9 g 
9 >+ 



112 1 



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15 



9 10 
3£ <> 



12 



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7 



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19 n 



li \7 18 



3,934,5:: 



4' 23 



245 



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(31 



2 £ 10 13 23 3 (/ 



3 10 

9) 
10 J 



ir 



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Reading of isolated numerals by Subject M 



THE READING OF ISOLATED NUMER.\LS IN LINES 



83 



3 ^ 

8:56 
30 5 



3i. (o 



5 (o 



5t8) 



PLATE XXII 

S 9 7 »o 



75 ), 55 2 



31 24 'S 41 



^6 



(3 l£ 14 i5 



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20 15 fc3 -37 



Z 3 



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55 4<i 8 7 II 



P39 

43 



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11)789 
39 44 



ti 10 t2 
28 \t 21 



>3 14 15 



3 54, 



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12 



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10)0 

30 



51 



30 3 48 25 £2. 

Reading of isolated numerals by Subject B 



76 



18 t 



II 

J5,0(jl0 

r 40 



+ I 7 
3U 



20 34 fc 10 41 II T 



PLATE XXIII 

8 3 

\ \ 

3T £"3 



iO II l£ 

),2 56,7)5 

21 30 51 



9c 



6 
53 



743,8 



7 I? 



U25 

«l 31 



S 3 .10 

365 

32 (5 "'2 



21 



12 14 15 13 



<I3 



54^ 



2b (i Zb Z4 



16 ir (8 
5,9n4,(.73 

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137 

43 2'"* 



^ 5 



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6 

52 



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8 

S9 



26 14 34 

Reading of isolated numerals by Subject B 



9 II .10 

637,63- 
33 30 II 



84 



HOW NUMERALS ARE READ 



3 < Z 



I 5 



T 4T 



PLATE XXIV 



)48|l 

28 /o 7 



7^6,3)2 

4% 35 



|3 

46 
22 



K (5 i(« iT 

4,32 3,956 

»0 ZJ Zt> 39 



2 3 4 

S5,37^ 



5 <i> T 

250 



3 II lit lO 

16,7:^1 



"9 72 14 10 23 3t « 15 K 40 £T T 



/3 (4 

1(124 

4£ II 



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3)4, 



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2 / 

12 

25 8 



2 3 

73,H4 

38 34 8 



3 4 5 (i 

2,374,)o'' 
;9 3i, ;9 'I 



7 

loop 

iC "52, 



3 10 

;J3:} 



9 43 

Reading of isolated numerals by Subject W 



12. 

2|5,qoo 

24 4"3 



4 5 

1317 

22 33 



PLATE XXV 




17 

38 



31 



9 II |0 

).^36,79;i 

19 40 21 3T 



12 13 

25 3 

12 21 



2 3 4 1 

713,8.9 

45 T5 27 7 



5 (o 

U25 



Jt 



(.5 



i6^ 

10 34 



2i 



10 11 

95,5 :8 

30 22 



12 



I3..I5 (4 ( 



1% 24 5 



67B 

12 



2 3 4-1 



]07 



,30^ 






P" ' 



45 24 29 " 25 



2« 



« 9 

13 14 



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(2 13 



6 3: 



23 20 31 



14 



T4 12. 



Reading of isolated numerals by Subject W 



THE READING OF ISOLATED NUMERALS IN LINES 85 

one or two digits at a pause. The readings, which were given the 
numerals 756,352 in Plate XVIII and 93,548 in Plate XIX, are illustra- 
tions of very detailed reading. The readings given the last three 
numerals on Plate XVIII, the special numerals 1000, t,;^;^, and 25,000 
show fewer pauses than H gave to other numerals of like lengths. He 
evidently read 1000 as a whole. A relatively large number of regressive 
pauses is found on these plates. 

The readings of Subject M are described in plates XX and XXI. 
His methods of reading the numerals were similar to those of Subject H, 
to which attention was called in connection with plates XVIII and XIX. 
The total reading-time for all the numerals was less for Subject M than 
for any other subject, despite the relatively large number of pauses 
which he used. Several instances appear in these plates of short initial 
and final pauses on the same longer numerals. The readings of the 
numerals 4,325,986 and 16,789 in Plate XX and of 5,236,795 and 743,819 
in Plate XXI illustrate such use of the guiding-pauses. The special 
numerals were read in the same manner as other numerals of like lengths. 
Plates XXII and XXIII record the reading of isolated numerals by 
Subject B. It appears that this subject's readings are irregular in 
respect to number of pauses on numerals of the greater lengths. Some 
of the longer numerals were read with few pauses. The numerals 
5,236,795 and 3,984,673 in Plate XXIII are illustrations of this type, 
while the numerals 9317 and 743,819 on the same plate were read with a 
comparatively large number of pauses. The numeral 1000 was evidently 
read as a whole. Initial regressions appear consistently in each line. 

The readings of isolated numerals by Subject W are presented in 
plates XXIV and XXV. A persistent use of two pauses appears in 
the readings of numerals of from three- to five-digit lengths. Two 
instances are seen in the numerals 5489 and 16,789 in Plate XXIV when 
even the re-readings of the numerals were done with pairs of pauses. 
Initial regressions occur consistently. 

2. TWO T\TES OF PAUSES 

When a detailed examination is made of the pauses with which the 
numerals were read, it appears that they represent two distinct types. 

The two types are distinguished by differences in function. Pauses 
of the first type, which may be called strictly reading-pauses, were 
probably used in recognizing the identity of the digits of the numerals 
and the relations between the digits. Such pauses are invariably located 
on the numerals. Their durations are approximately equal to, or greater 



86 HOW NUMERALS ARE READ 

than, the average duration of the pauses of the subject whose records 
are under consideration. A preponderant number of the pauses of any 
subject are of this first type. 

Pauses of the second type, which may be called guiding-pauses, 
were probably used in locating the first digits or the last digits of the 
numerals. They are found on the initial or final digits of numerals, 
and more frequently on numerals of greater digit-lengths. Some of 
these pauses appear on the Lines between the numerals. The first pause 
in any Une was ahnost invariably of this type and of very brief duration 
when compared with other pauses. Subjects H and M used larger 
numbers of pauses of this type than any of the other subjects. 

3. DIFFERENCES IN THE READINGS OF NUMERALS OF 
DIFFERENT LENGTHS 

Upon inspection of the last row of Table XXVII it is found that the 
average total reading-time per numeral increases steadily from an 
average of 21.45/50 of a second for the one-digit numerals to an average 
of 104.54/50 of a second for the seven-digit numerals. The same 
continual increase is found almost without exception in the rows of the 
several subjects. Likewise the average number of pauses per numeral, 
when the records of all subjects are averaged, increases steadily from 
the average of 1.15 pauses on one-digit numerals to the average of 4.15 
pauses on seven-digit numerals. It is clear therefore that the total 
reading-times and the number of pauses which were required to read 
a numeral, depended upon its digit-length. 

The average duration of the pauses, on the other hand, does not 
depend upon the length of the several numerals in the same consistent 
fashion as does the average number of pauses per numeral. The details 
may be found in Table XXVIII, where it is seen that both in the rows 
for individual subjects as weU as in the row which presents averages for 
all of the subjects, the average pause-duration not only fails to increase 
steadily with increasing lengths of the numerals but in several cases 
actually decreases. 

With three of the subjects, M, H, and G, on the other hand, the 
average duration of the pauses increases steadily from that of the 
numerals of one digit to that of the numerals of three digits. Two of 
these three subjects, M and G, however, as may be observed in Table 
XXIX, read the one, two, and three digits of the one-, two-, and three- 
digit numerals, respectively, for the most part with a single pause. 
In the case of Subject B, also, a steady increase in pause-duration is 
found when the one-, two-, and three-digit numerals were read at single 



THE READING OF ISOLATED NUMERALS IN LINES 



87 





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HOW NUMERALS ARE READ 






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THE READING OF ISOLATED NUMERALS IN LINES 



89 





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3 3 3 3 3 
CA) C/2 CO CAl C/2 



go 



HOW NUMERALS ARE READ 



pauses. It is evident, therefore, that with certain subjects and to this 
hmited extent, the pause-duration does increase with increases in the 
number of digits read. 

Numerals of the same length exhibited a notable consistency in the 
number of pauses with which they were read by the several subjects. 
As may be observed in Table XXIX two different numbers of pauses 
include the number of pauses used in reading a preponderant number of 
the numerals of each length. The one- and two-digit numerals were 
read for the most part by single pauses. One and two pauses were 
used with the three-digit numerals and two and three pauses with the 
four- and five-digit numerals in most cases. The longer numerals 
exhibit more variety in the number of pauses with which they were 
read. With these numerals, the habits of individuals seem larger 
deciding factors than the tendency to a conventional number of pauses. 

4. THE SPECIAL NUMERALS 

The six special numerals, o, 99, 333, 1000, 25,000, and 637,637, were 
included among the other numerals, in order that data might be obtained 
concerning such variations in the readings of numerals, as might be due 
to famiharity and to regularity in form on the part of the numerals 
read. The data for the two special numerals 1000 and 25,000 show 
consistent and significant variations from the data which represent the 
readings of ordinary numerals of like lengths. As may be observed in 
Table XXX the two numerals 1000 and 25,000 exhibit markedly shorter 
average reading-time per numeral, average number of pauses per numeral, 
and average pause-durations, than the ordinary numerals of the same 
lengths with which they are compared. 

TABLE XXX 

Reading of Speciax Numeraxs Compared with Reading of Other Isolated 

Numerals of Corresponding Digit-Lengths 

(Time unit = 1/50 of a second) 



Special numerals. 



637,637 



Corresponding digit-lengths of other isolated numer- 
als with which the special numerals are compared 



Average reading-time per numeral: 

Special numerals 

Other ntimerals 

Average number of pauses per niuneral: 

Special numerals 

Other numerals 

Average pause-duration: 

Special numerals 

Other numerals 



16.4 
21. 45 



16.4 
19-3 



30.8 
28.7 



20.86 



46.6 
48.05 

1 .40 
1.9 

33-29 
27.04 



33-6 
59.6s 



1.6 



26.28 



53-8 
77.75 

2.6 
2.9 

24.46 
28.43 



93 -o 
94.05 

4.50 
3.7 

25.82 
26.88 



Note. — Each subject read each special numeral once, and at the same reading with the other isolated 
numerals. 



THE READING OF ISOLATED NUMER.\LS IX LINES 



91 



Each of the two numerals represents the special quality of regularity 
of form and the quality of familiarity which were described in the 
discussion of the numeral 1000 in chapter vii. Regularity of form, in 
what may be called common fashion as distinguished from the special 
fashion of regularity which has been described for the numeral 1000, 
is a quahty of the other special numerals 99, 7,7,7,, and 637,637. Such 
regularity of form, however, did not seem to be significant enough to 
affect important variations in the readings of those numerals. This 
immediate fact and the facts which are noted above appear in 
re-enforcement of the conclusion reached regarding the numeral 1000 
in chapter vii. Evidently the quahties of special regularity of form and 
famiharity are influential factors in the reading of numerals, even when 
the numerals are isolated in lines. 

5. TWO METHODS OF ATTACK IN RE.\DING NUMERALS 

Two distinct methods of attack were used by different subjects in 
reading the numerals. By one method a relatively large number of 
pauses of relatively short average duration were used in the readings, 
and by the other relatively few pauses of relatively long average duration 
were used. The contrast between the two methods is sharply outlined 
by differences in the data for subjects H and G, who represent the first 
and second methods respectively, as such data are arranged in Table 
XXXI. Subject M also read by the first method and subjects W and 
B by the second method. In plates XVIII and XIX a relatively large 
number of guiding-pauses are found, while comparatively few appear 
in plates XVI and XVII. The shortest total reading-times for the whole 



TABLE XXXI 

Speed and the Two ISIethods of Attack Used in Reading Isolated Xumer-Als 
(Time unit = 1/50 of a second) 





Many Pauses of 


Fewe 
A\ 


r Pauses of Longer 




Duration 


•erage Duration 


Subjects 


M 


H 


G 


W B 










Total reading-time for aU nu- 
merals 

Total number of pauses used in 
reading all numerals 

Average pause-duration 


1549 

76 
20.38 


1642 

85 
19.32 


1728 

51 
33 38 


1797 

69 
26.04 


1937 

67 
28.92 



Note. — Each subject read four each of the one-, two-, three-, 
a total of twenty-eight numerals. 



and seven-digit numerals or 



92 HOW NUMERALS ARE READ 

set of isolated numerals are found in the records of subjects M and H 
who used the many-short-pauses method of attack in their readings. 

The number of digits which are read at a single pause depends upon 
the method of attack which the subject uses, and upon the number of 
digits which a numeral offers for reading. Subjects M and H, who use 
the method of many short pauses, tend to perceive one and two digits 
at a pause. Subjects G, B, and W, on the other hand, who use the 
method of few and long pauses, tend to perceive two and three digits 
at a pause. The one and two digits of the one- and two-digit numerals 
were almost invariably read by single pauses by all subjects. In all 
of the plates which describe the readings of M, G, W, and B instances 
are to be found among the readings of the longer numerals in which as 
many as three digits were read at a pause. 

In only three instances, however, when the special form numeral 
looo is not brought into consideration, as many as four digits are known 
to have been read at a single pause. All three of the instances are in 
the readings of Subject G and are reported on Plate XVII. Two 
instances occur on the four-digit numerals 9317 and 1928, and the other 
on the seven-digit numeral, 5,236,795. Evidently the reading of four 
digits at a single pause is very exceptional and it is significant that it 
occurs only in the records of the subject, who used the smallest number 
of pauses with the greatest average duration. 

6. SUMMARY 

1. Two t^pes of pauses are distinguished by differences in duration. 
The strictly reading-pauses are probably used to recognize the identity 
of the digits of the numerals, and the relations between them. The 
guiding-pauses are used probably in locating the initial and final digits 
of numerals. 

2. The average total reading-time per numeral and the average 
number of pauses per numeral increase gradually from the averages of 
the one-digit numerals to the averages of the seven-digit numerals. 

3. With three subjects the average duration of pauses increased 
with increases in the number of digits read by the pauses. 

4. Two different numbers of pauses include the number of pauses 
used in reading a preponderant number of the numerals of each length. 
The greatest consistency in this respect is found in the numerals of from 
one to five digits in length. 

5. The quality of familiarity, rather than the quality of regularity 
of form reduced the average number of pauses and the total reading- 



THE READING OF ISOLATED NUMERALS IN LINES 93 

times of the numerals 1000 and 25,000 below the averages of ordinary 
numerals of like lengths. 

6. Two distinct methods of attack are used in reading numerals. 
By the one method a relatively large number of pauses of relatively 
short average duration is used, and by the other method relatively 
few pauses of relatively long average duration are used. The subjects 
who employed the former method read the numerals in shorter total 
reading-time. 

7. The usual number of digits read per pause is one or two, or two 
or three according to the habits of the subject who is reading. One 
subject is able to read as many as four digits at single pauses. 



CHAPTER X 

COMPARISONS OF RATES OF READING 

I. COMPARISON OF THE SUBJECTS OF THE PRESENT INVESTIGATION 

WITH THE SUBJECTS OF SCHMIDT'S INVESTIGATION IN 

RESPECT TO RATES OF READING 

Attention has been called in previous sections of this report to the 
fact that larger numbers of pauses of relatively greater average duration 
were used in reading the arithmetical problems than are commonly- 
used in reading ordinary prose materials. It was decided, therefore, 
to test the reading-speeds of the several subjects of the present investiga- 
tion with a different t\'pe of reading-material. The data, which were 
thus obtained, could then be compared with other data which represented 
he reading-speeds of similar individuals with similar materials. 

The type of material, which was selected as most appropriate for 
this purpose, was that of ordinary expository prose. The text of the 
selection was taken from Judd's Psychology oj High-School Subjects, 
a volume which was famiUar in a general way to all of the readers. 
The data obtained with this material were to be compared with the 
results reported by Schmidt' in a previous investigation of the reading 
of "Ught passages from James's Psychology'^^ by 45 "adults, mostly 
graduate and undergraduate students."^ 

The conditions which governed the readings of the two selections 
were similar for the most part. In one respect, however, an impor- 
tant difference obtained. In Schmidt's investigation the subjects were 
instructed to "read rapidly for the thought."'' In the present investi- 
gation, out of deference to purposes which are stated in the latter part of 
this chapter, the subjects were instructed to read for a " clear understand- 
ing" and at "normal speed." Further details concerning this material 
and the conditions under which it was given appear in chapter vi. 

In order to facilitate the comparison certain rearrangements of the 
data as originally reported by Schmidt were made. The time unit 
used in his investigation was the sigma, or i/iooo of a second. Figures 

' W. A. Schmidt, "An Experimental Study in the Psychology of Reading," 
Supplementary Educational Monographs, Vol. I, No. 2 (191 7), p. 42. 

^ Ibid., pp. 32; 50. ^Ihid., p. 34. * Ibid., p. 28. 

94 



COMPARISONS OF RATES OF READING 



95 



which represented duration of time in this unit were converted into other 
figures representing the same durations in units of 1/50 of a second. 
By multiplying average pause-duration by average number of pauses 
per line figures were obtained for the average reading-time per line. 
In this way a single number is found to represent rates of reading. 
The data from the two investigations are arranged in Table XXXII. 

When the data in Table XXXII are compared it is found that the 
subjects of the present investigation read at conspicuously higher rates, 
as judged by average reading-time per line, than the subjects of Schmidt's 
investigation. The superiority in speed on the part of the former 
subjects is due in largest measure to the decidedly shorter average dura- 
tions of their pauses, although they also used fewer pauses per line, 

TABLE XXXII 

Subjects of the Present Investig.atiox Compared with Those of Schmidt's 

IxVESTIG.\TION IN ReSPECT TO SPEED OF READING 

(Time unit = 1/50 of a second) 





Subjects 




Average 
Number 
of Pauses 
per Line* 


Average 

Pause- 

Duratdon 


Average 
Deviation 


Average 
Reading- 
Time 
per Linet 


Of the present investigation (5 adults) 
Of Schmidt's investigation (45 adults) 


6.05 
6.50 


10.75 
i5-4lt 


2.87 
3-93 


65.04 
100.17 



* The line^lengtii of the materials read by Schmidt's subjects was go mm.; that of the materials 
read by the subjects of the present investigation was 93 mm. 

t Schmidt's results, which are reported by him in time units of sigma (i/iooo of a second), are here 
presented in time units of 1/50 of a second. 

+ The average reading-time per line is obtained by multipljing the average number of pauses per 
line by the average pause-duration. 



The advantage in favor of the subjects of the present investigation is 
emphasized by the fact that they were reading under instructions which 
called for only "normal speed," whereas the other subjects read under 
the instructions, "read rapidly for the thought." 

So large a number of adult cases was included in the investigation 
by Schmidt that the figures reported by him may be taken as representing 
rehable averages of the performances of such subjects. Upon the basis 
of comparison with the results thus reported it is concluded that the 
subjects of the present investigation may be classified as decidedly 
better than average adults in respect to speed of reading. The implica- 
tion of the foregoing paragraphs and of this conclusion is obvious. 
The larger number of pauses of relatively greater duration, which was 



96 



HOW NUMERALS ARE READ 



found in the readings of the problems, is due to the nature of the problems 
as a type of reading-material rather than to slow speeds on the part of 
the readers. 



2. 



COMPARISONS OF RATES OF READING THE THREE TYPES OF 
READING-MATERIALS 



Significant evidence of differences between types of reading-materials, 
and concerning the nature of the differences as well, may be secured by 
comparisons between the rates with which the subjects read the different 
types of materials. With such a purpose in view, comparisons were 
arranged between the data from the readings of the problems, of the 
isolated numerals, and of the ordinary expository prose selection. The 
conditions which governed the readings of all three types of materials 
were essentially similar in that they called for such kinds of reading as 
are most customary for the several types, and for "normal speed" on 
the part of the subjects. The data which were obtained are, therefore, 
representative of normal performances on the parts of the readers with 
these three types of materials. Table XXXIII was designed to facilitate 
the comparison. 

TABLE XXXIII 

Comparative Data from Readings of Five Problems, Ordinary Prose, and 

Isolated Numerals 
(Time unit= 1/50 of a second) 



Subjects 



M 



W 



Hb 



Average 

FOR 

All 
Subjects 



Average number of pauses per line on: 

The five problems (first reading) 

The ordinary prose selection 

Isolated numerals 

Average number of letters, or digits, per pause 



The five problems (first reading) . 
The ordinary prose selection .... 

Isolated numerals 

Average duration of pauses on: 
The five problems (first reading) . 
The ordinary prose selection .... 
Isolated numerals 



6.66 

4-71 



7-25 
7 -SO 



8.41 
5-20 



7-25 
6.20 



9.83 
6.66 



9.08 



8.08 
6.0s 



6.61 
9.61 

2.28 

11.28 
II. IS 
30. S7 



6.08 
6.03 
1.66 

11-95 
10.77 
28.62 



S-24 
8.69 



10.93 

8.63 

19.8s 



6.08 
7.38 
1 .62 

13-44 
12.13 
26.11 



4-48 
6-75 
I-31 

12. II 
11.25 
18. i8 



4-8s 



11.58 



S-56 
7.69 
1 .67 



10. 75 
23-80 



Note. — Satisfactory data on the reading of ordinary prose and isolated numerals were not obtained 
from Subject Hb. 

The average line-length of the five problems materials was 93.33 mm., of the ordinary prose selection 
93.0 mm. 



The conspicuous feature of Table XXXIII is the marked superiority 
in the speed with which the ordinary prose selection was read. The 
greatest differences are found between the isolated numerals and the 
prose selection. All of the subjects exhibit these differences, both in 



COMPARISONS OF RATES OF READING 97 

respect to average number of digits or letters read per pause and in 
respect to average duration of pauses as well. The differences indicating 
greater speed with the prose are in large proportion. Such differences 
in speed undoubtedly reflect the great and obvious differences between 
materials for prose and for isolated numerals in respect to mechanical 
form, context, and the attitudes which they inspire in the minds of 
readers. E\ddently isolated numerals are much more difficult as reading- 
material than ordinary expository prose. 

The differences in rates between the five problems and the ordinary 
prose materials are decidedly significant although not as great in propor- 
tion as those found in the comparison just drawn. Four of the five 
subjects read the ordinary prose at higher rates than they had used with 
the five problems. The higher speeds with the prose are due for the 
most part to fewer pauses per fine, although the durations of the pauses 
on the prose are somewhat shorter than those on the problems. The 
obvious interpretation of the differences is found in the greater difficulty 
of the problems as reading-materials. To a large extent, as has been 
shown in previous sections of this report, the greater difficulty is due 
to the exactions of the numerals in the problems. It is probable also 
that greater exertions were undergone by the subjects in their efforts 
to grasp accurately the terms of arithmetical problems than to learn 
the facts contained in a selection of ordinary, expository prose. 

3. SUMMARY 

The following conclusions may be drawn from the discussion of 
this chapter: (i) The subjects of the present investigation read at 
decidedly higher rates than the subjects of Schmidt's investigation. 
They are, therefore, classified as decidedly better than average adults in 
respect to speed of reading. (2) The ordinary', expository prose material 
was read at significantly higher rates than either the arithmetical 
problems or the isolated numerals materials. The conclusion was 
drawn, therefore, that arithmetical problems and isolated numerals are 
decidedly more difficult as types of reading-materials than ordinary 
expository prose. 



CHAPTER XI 

PRACTICAL APPLICATION TO CLASSROOM TEACHING 
I. THE QUESTION OF READING IN ARITHMETIC 

Arithmetic is commonly reputed to be one of the most difficult sub- 
jects in the curriculum of the elementary school. The meagerness of 
the results which are obtained at the cost of such large amounts of time 
as are devoted to the study of the subject is a matter of continuous 
complaint. The changes which have come about as a consequence of 
efforts to improve the situation have resulted for the most part in a new 
selection of subject-matter for textbooks and in rearrangements of the 
sequence of topics. The problem exercises have become more practical in 
character and there is evidence of a tendency to employ in problems only 
such classes and magnitudes of numerals as are used in everyday affairs. 

When the work which has been done in these directions is duly 
recognized, the significant fact remains that some of the most important 
divisions of the field have not yet been occupied. The number of 
scientific studies in the psychology of arithmetic is still surprisingly 
small. As compared with reading, arithmetic has been seriously neglected. 
Fortunately, however, much of the information which has been made 
available in reports on reading can be used in the study of arithmetic. 
A number of recent reports of experiments in the teaching of reading 
has served to emphasize the existence of numerous distinct types of 
reading materials each of which calls for the development of an appro- 
priate t>Tpe of reading abihty on the part of the student. Arithmetical 
problems and isolated numerals as well, when considered as reading 
situations, represent distinct types of reading materials. The special 
types of procedure which adult subjects used in reading such materials 
have been described in previous chapters of this report. 

Whatever the nature of the materials, however, reading is essentially 
the process of getting meaning from the printed page. When the subject- 
matter is difficult, patient and systematic search for the thought is 
necessary. Instead of taking pains in this fashion, children frequently 
resort to mechanical pronunciation of the words or mere scanning of the 
hues.' In such cases no adequate idea of the meaning is obtained, and 

' Estaline Wilson, "Improving the Ability to Read Arithmetic Problems," Ele- 
mentary School Journal, Vol. XXII, No. 5 (January, 1922), pp. 380-86. 



PRACTICAL APPLICATION TO CLASSROOM TEACHING 99 

if the materials read be arithmetical problems, a correct solution is 
impossible. It is scarcely an exaggeration to say that few children, 
including those with the experience of the upper grades, have developed 
a method of reading problems which could be described as a rapid and 
skilful attack on the reading situation. 

The extent to which the appearance of numerals along with words 
in the text of a problem is responsible for this condition is but slightly 
understood. Numerals as a distinct and significant object of study 
have received very little attention in the literature of arithmetic. Occa- 
sional reference is made to the abstract character of the numerical con- 
cept and the difficulty of teaching children the meaning of the symbols. 
Discussions of classroom experience also are reported occasionally. Of 
these the question: What magnitudes of numerals should be used when 
new problems are introduced ? is fairly illustrative. Numerals in the 
context of problems, but for exceptions of this character, have been 
almost completely neglected. 

2. PRELIMINARY ANALYTICAL READING OF PROBLEMS 

The presence of numerals among words, however, is only one of 
the features which distinguish arithmetical problems as a specialized 
t\^e of reading materials. The existence of other equally complicating 
features is implied when teachers express the opinion that it is more 
difficult to teach the interpretation of problems than the mechanical 
skills of computation. Despite general acceptance of this opinion, 
scientific studies in the psychology of arithmetic have been concerned 
almost entirely with the field of operations. As a consequence, exact 
knowledge of the nature of the processes of interpretation is lacking for 
the most part. • 

Efforts have been made, notwithstanding, upon the basis of common 
observation to describe the reading situation which a problem offers and 
the preliminary thinking about it which is necessary before the reader is 
prepared for the work of computation. The most essential character- 
istic of a problem is the fact that it presents a series of conditions which 
describe a certain state of affairs. Some of the conditions appear in 
precise quantity. The quantities stand in definite relationships with 
each other and are stated in abstract terms. 

Each of the elements of this complex situation must be compre- 
hended by the student during the preliminary reading. He must draw in 
his imagination an accurate picture of the situation, which is described, 
and take account of each of the facts of relation. Following this, comes 



lOO HOW NUMERALS ARE READ 

a canvass of the plans for solving which suggest themselves, and the 
passing of judgment on the appropriateness of each plan to the con- 
ditions of the problem. The reading processes which are carried on in 
this manner are analytical in character and call for a high degree of skill 
and patience, as well as for a certain amount of practical acquaintance 
with the facts described. It is doubtful if teachers generally have acquired 
anything like an adequate appreciation of the confusion which children 
feel when confronted with a situation to be studied in this fashion. 

With the intention of facilitating the formation of working habits 
which will bring relief to this confusion, authors of teaching manuals 
undertake to outline the steps which should be followed in reading and 
solving a problem. The first two or three steps usually provide for the 
preliminary analytical reading. The following directions, which appear 
in a recent manual as the first three of five steps, will serve as an illustra- 
tion: ''First, the pupil must read the problem and understand what it 
means. Second, he must state in his own words what the problem calls 

for Third, he must find the material that the problem gives to 

work with.'" 

The reader of this report will notice immediately that no suggestions 
are given concerning the proper treatment of numerals. As far as the 
first three steps are concerned it would seem that the directions were 
designed for use in connection with that special type of problem only 
in which numerals are not included. The author appears to assume 
that children are able to make as rapid and prompt a disposition of the 
numerals as the adult subjects of the present investigation. It is clear 
that a rapid and thoughtful reading of problems containing numerals is 
not practicable under the directions as given above, unless the reader is 
relieved of the details of the numerals and is free to devote his entire 
attention to the conditions of the problem. This desirable result, as 
has been shown in a previous chapter of this report, can be obtained by 
employing the partial method of reading numerals. 

3. APPLICATION OF PARTIAL READING 

The importance of conducting the difficult processes of analytical 
reading in the most direct and unhampered manner is great enough to 
warrant the inclusion with the directions quoted above of any additional 
directions that may be necessary. Undoubtedly supplementary direc- 
tions which prescribed partial reading of the numerals would greatly 
facilitate the preliminary reading of problems. Such a prescription 

' Kendall and Mirick, How to Teach the Fundamental Subjects, p. 169. Boston: 
Houghton Mifflin Co. 



PRACTICAI. APPLICATION TO CLASSROOM TEACHING loi 

would also serve to emphasize the significance of partial reading, 
and the extensive possibilities of improvement which lie in its use could 
begin to be realized. At this point, the reader should be reminded of 
the fact that the problems which were employed in the present investiga- 
tion were both brief and simple. Since the partial method facilitated 
the reading of such problems, its value for the larger and more involved 
problems which are found in the ordinary textbook would be corre- 
spondingly greater. And likewise, a relief measure which was desirable 
for adult graduate students should prove all the more helpful to children 
in the elementary school. 

An effort has been made, therefore, to prepare recommendations con- 
cerning the reading of problems which should be considered as additional 
to the directions that are quoted above. The recommendations are as 
follows: 

1. Pupils should be taught to distinguish between the first reading 
and the re-reading phases in their attack on problems. 

2. They should learn to consider numerals and the accompanying 
descriptive conditions as different elements of a problem and separable 
for reading purposes. 

3. During the first reading, they should devote their entire attention 
to the conditions of the problem. 

4. x\t the same time skill should be developed in partial reading of 
numerals. 

5. While this skill is being acquired, pupils should be apprised of the 
essential similarity between the conditions of the problem and such 
details of the numerals as are perceived by partial reading. 

Although the preceding recommendations were derived for the most 
part from the findings of this report, their validity does not rest on this 
basis alone. They derive additional support by comparison with the 
findings of other investigations in the field of reading. Gray, in sum- 
marizing the principles of method which were deduced from recent 
studies of reading, states that emphasis of the elements on which mean- 
ing depends, improves comprehension.^ A closely related principle is 
stated by Freeman: "Rate of reading is increased by attending to the 
meaning as distinguished from the mechanics."^ In the case of arith- 
metical problems, the elements referred to by Gray, obviously, are the 

' W. S. Gray, "Principles of Method in Teaching Reading, as Derived from 
Scientific Investigation," Eighteenth Yearbook of the National Society for the Study of 
Education, Part II, p. 42. 

^ F. N. Freeman, The Psychology of the Common Branches, p. 93. Houghton 
Mifflin Co. , ' ' 



I02 HOW NUMERALS ARE READ 

conditions of the problem and such details of the numerals as identity of 
the first digit and the number of digits. Attention to these items and 
to these alone is obtained by the use of partial reading as recommended 
above. The remaining details of the numerals are of the nature of 
"mechanics," as described by Freeman, and when the partial method of 
reading is used, the attention of the student is relieved of the "mechanics" 
and is free to search out the meaning. 

Further comparisons with the results of other investigations empha- 
size the fact that the use of partial reading as recommended above repre- 
sents a more progressive type of reading. Progress in reading, according 
to Freeman, consists in a decrease of the number of pauses of the eye 
and an increase in the scope of recognition at each pause. ^ Gray con- 
cludes that "regular rhythmical movements of the eyes are prerequisite 
to rapid silent reading."^ When partial reading is used, the numerals of 
a problem are not read in detail and it has been shown that fewer pauses 
are required to get the meaning from the printed line. As a conse- 
quence, the average amount of material perceived at a pause is increased. 
It is clear, therefore, that both of the conditions which Freeman describes 
as representative of progress in reading are encouraged by the method 
of partial reading. At the same time, by employment of this method, 
the eye is relieved of the most severe exactions of the numerals and does 
not suffer the delay in movement which, owing to the nature of numerals 
as reading materials, is unavoidable when the numerals are read in detail. 
By virtue of this relief, the eye is able to approximate more nearly 
the rhythm of movement which is customary for lines of words, and 
which is "prerequisite to rapid silent reading." 

4. APPLICATION OF RE-RE.\DING 

The recommendations which appear above are concerned exclusively 
with the first reading of the problem. When the first reading is com- 
pleted ordinarily the pupil is ready for the re-reading. It is possible, 
as was shown in chapter viii, to omit the work of re-reading by employing 
the method of mental computation while reading the problem as printed. 
The latter is a more economical procedure than the alternative method 
of computation from copied figures. So rapid a procedure as direct 
computation, however, is practicable only with easy problems and for 
pupils of unusual arithmetical ability. Probably the great majority of 
pupils solve the largest proportion of their problems with the aid of 
pencil and paper. 

'F. N. Freeman, op. cil., p. 83. ^ W. S. Gray, op. cil., p. 39. 



PRACTICAL APPLICATION TO CLASSROOM TEACHING 103 

For all such pupils the work of re-reading and copying the numerals 
is an essential step in the process of solving. Extensive use of a method, 
however, does not guarantee successful accomphshment. By experi- 
ence, and by careful investigation as well, teachers have learned that 
pupils do not copy numerals accurately. The situation, as it exists, is 
too serious to be neglected. Intelligent and vigorous efforts should be 
made to eliminate errors of this kind completely. Substantial improve- 
ment undoubtedly could be effected, by relieving the pupil's mind of every 
avoidable distraction during the copying. Setting apart a definite place 
for the step of copying, in the total procedure of solving, would consti- 
tute an important move in this direction. For this reason, the teacher's 
plan of instruction for arithmetical problems should include a special 
period of drill in re-reading and copying numerals. 

Wlien the plan of instruction has been amended, in accordance with 
this suggestion, it is important that the most efficient methods of reading 
the numerals be selected for use during the drill period. The methods 
of re-reading which were employed by the adult subjects of the present 
investigation are suggestive of the type of drill which should be given. 
For this reason the conclusions which were derived in chapter v were 
taken as the basis of recommendations concerning re-reading which are 
submitted as follows: (i) the numerals of any digit length should be 
read according to their dominant main-group pattern; and (2) the 
simplest possible numerical language should be used. 

The effect of these recommendations and the advantages that lie 
in their adoption may be illustrated with the numerals 56,283 and 497. 
When the numeral 56,283 is treated as directed, it is read as follows: 
five six, two eight three. In this way the main-group pattern for five- 
digit numerals, which calls for two groups of two and three digits respec- 
tively, is followed. No words are used except such as are required to 
name the digits in order of succession. In the case of the numeral 497 
the proper pronunciation is four, nine seven. In this instance the 
dominant main-group pattern for three-digit numerals is followed and 
no superfluous language is included. 

In the reading of both numerals, advantage is taken of the strong 
natural tendency of the mind to arrange the members of a series of 
stimuli in groups. With the verbal description as brief as possible, no 
unnecessary waste of energy is incurred in pronunciation. In addition 
to the value of economy, an important reduction is effected in oppor- 
tunity for errors by reading the numerals exactly as they are printed 
and precisely in the form in which they are to be copied. The feasi- 



I04 HOW NUMERALS ARE READ 

bility of the recommendations is beyond doubt since they are observed 
regularly in practice by various vocational groups, such as telephone 
operators and bookkeepers, who use numerals extensively in their 
daily work. 

Nor are the directions as given applicable only to oral reading. It 
is a matter of common knowledge among students of reading that a 
very close connection exists between the inner speech of silent reading 
and the behavior of oral reading. Numerous elements of procedure are 
common to both. So far as the foregoing recommendations are con- 
cerned, there appears no necessity for drawing a distinction between oral 
and silent reading. 

5. MISCELLANEOUS APPLICATIONS 

Although the findings of the present investigation are not concerned 
extensively with the processes of computation, important data were 
presented in chapter viii concerning "computation direct from the 
problem card." The records show that this method is a very direct 
road to the answer and its use undoubtedly reduces the number of oppor- 
tunities for error. The great rapidity with which the abler students 
can solve suitable problems in this manner would tend to increase the 
pupils' interest and concentration. For these reasons it is recommended 
that with abler students the process of direct computation be more 
extensively employed in rapid drill with problems. 

The question will be raised as to the grade at which the methods of 
partial reading and re-reading should be introduced. The findings of 
the present investigation do not bear directly on this question. Nor 
can other than provisional conclusions be drawn until the reading of 
problems by children in the various grades has been studied. Signifi- 
cant inferences, however, can be drawn in the light of certain principles 
of method in teaching reading which were derived from scientific study 
and which are now generally accepted. 

Partial reading and re-reading are methods of skilful and rapid 
silent reading. They represent a degree of achievement which is more 
advanced than mere ability to recognize the words. It is reasonably 
certain, therefore, that children are not prepared to read in this fashion 
before the teaching emphasis has been shifted from oral to silent reading, 
which, as is now generally known, should be done between the second 
and fourth grades. Beginning with the latter grade, progress in reading 
consists in large part in ability to master increasingly difficult materials. 
Arithmetical problems with several conditions and with longer numerals 



PRACTICAL APPLICATION TO CLASSROOM TEACHING 105 

constitute such materials and it is this t\^e of problem which is attacked 
to advantage by the use of partial reading. In the nature of the case, 
partial reading and re-reading are highly specialized types of procedure. 
It is during the fourth, fifth, and sixth grades that pupils should be 
trained to use different types of reading ability. In view of this con- 
sideration, and of such others as are named above, it appears that the 
fourth grade is the appropriate time for the introduction of the new 
methods. 

The discussion of applications ought not to be concluded without 
pointing out the fact that a body of experimental data in many instances 
is valuable for other purposes than those which were originally respon- 
sible for the investigation. Diagnosis of individual difficulties is an 
increasingly important feature of instruction in modern school systems. 
For such work, well-trained teachers and supervisors rely to as great 
an extent as is possible upon the materials which are available in scientific 
reports. No other body of material affords an equally detailed and 
illuminating description of the intellectual processes which are carried 
on in the ordinary work of the school. The description of first reading 
and re-reading and of certain computational processes, which is avail- 
able in the present report, will prove useful for diagnosis of difficulties 
with arithmetical problems. Significant but less detailed descriptions 
of the range of correct recall of numerals, the grouping of digits, numeri- 
cal-language patterns, effect of punctuation, and of such other items as 
a perusal of the table of contents will disclose, are also available for the 
same purpose. 



INDEX 



Articulation of numerals, 24, 38, 61 

"Base of operations": one numeral used 
as, 74-76 

Combinations of digits, 57, 67 
Complex three-digit groups, 27 
Computation from copied figures: de- 
scribed, 72; wide use of, 102 
Copying numerals, 103 

Dearborn, W. F., 57, 78 
Diagnosis of pupil difficulties, 105 
Digit groups: code used in reporting, 25; 
description of, 25; and habits of 
subjects, 27; influence of punctuation 
on, 31; of longer and shorter numer- 
als, 27-29; types of three-digit groups, 
27 
Direct computation: contrasted with 
computation from copied figures, 72; 
defined, 9; pedagogical significance of, 
73; recommendation concerning, 104; 
when practicable, 102; where found, 73 
Directions for reading problems: five 
recommendations, loi ; by Kendall 
and ^Slirick, 100 

Essential characteristics of a problem, 99 

Familiar numerals: explanation of, 63; 
main-group patterns of, 28; partial 
reading of. 5; and range of recall, 15; 
special treatment of, 85 

Familiarity of form in numerals, 90 

Final pauses on numerals, 85 

First digits of numerals: correctly re- 
called, 15, 67; pauses located on, 86 

First numeral in line, 79 

First numerals: and partial reading, 15; 
and range of recall, 15 

First reading: defined, 3; purpose of , 17; 
recommendations concerning, loi ; 
time required, 23 

Freeman, F. X., loi, 102 

Gilliland, A. R., 35 
Gray, C. T., 35, 78 



Graj', W. S., loi, 102 
Guiding pauses, 79, 86, 91 

Initial pause on a line of numerals, 79, 85 

Interest in problems, 61 

Introspective observ^ations, i, 39, 59, 69 

Instructions to subjects for: eye-move- 
ment studies, s^; first preliminary 
study, 3; fourth preliminarj- study, 24; 
second prelim inar}' study, 13; third 
preliminary study, 19 

Isolated numerals: average number of 
pauses on, 90; compared with problem 
numerals, 60; of eye-movement studies, 
37; of fourth preliminary' study, 24; 
nature of, as reading materials, 96; 
reading of the special, 90 

James, William, 94 
Judd, C. H., 38, 94 

Kendall and Mirick, 100 

Last digits of numerals, 86 

Locating pauses, 74-76 

Longer numerals: influence of punctua- 
tion on grouping of digits, 31, s^', 
number of pauses used in reading, 90; 
partial and whole readings of, 59, 61, 
62; partial reading of, 5, range of 
recall of, 15; re-reading for copj'ing, 
23; used as "base of operations," 76 

Main-group patterns: definition of, 27; 

recommendations concerning use of, 

103; of various sizes of numerals, 28 
]Many-short-pauses method of reading 

numerals, 91 
Methods of computation: two types of, 

72 
Methods of reading numerals, two, 91 

Numerals: articulation of, 24; classes 
studied, i; non-punctuated, 31; as 
peculiar reading materials, 22; previ- 
ous investigations of, i; shorter range 
of perception of, 56; the special, 90; 
total reading time of, 86; used in eye- 
movement studies, 86; used in first 



io8 



NOW NUMERALS ARE READ 



preliminary study, 2; used in fourth 
preliminary study, 24 
Numerical language: code used in re- 
porting, 25; and habits of subjects, 31; 
persistence of patterns, ss^ a-^d 
punctuation, 33; recommendations 
concerning use of, 103; variations in, 
29-31 

Ordinary prose selection, 38; rate of 
reading, 96 

Partial first readings, 59 
Partial readers: attitude of, 66; classi- 
fication of, 7, 64; prose reading rate 
of, 65; range of perception, 56; and 
range of recall, 17; reading of words 
by, 65; re-reading of, 9, 71; use fewer 
pauses, 66 
Partial reading: of the date, 191 8, 5; 
description of, 3, 67; development of, 
66; and direct computation, 73; and 
the essential elements of a problem, 
loi; of familiar numerals, 5; of first 
numeral, 6; frequency of, 4; of 
individual subjects, 6; introduce at 
fourth grade, 104; of larger and shorter 
numerals, 5; number of digits included 
in a pause, 63; and progress in reading, 
102; of several numerals in one 
problem, 5; value of, 4, 64, 100 
Pauses on numerals: average duration of, 
57, 86, 90; duration of, on isolated 
numerals, 79; final pauses, 85; greater 
duration of "base of operations" 
pauses, 76; guiding pauses, 79, 86, 91; 
initial pause on a line of numerals, 79, 
85; locating pauses, 74-76; pairs of, 
85; range of perception of, 57; recogni- 
tion pauses, 76; recording pauses, 
74-76; regressive pauses, 58, 79; two 
types of, 85 
Photographic apparatus: description of, 

35; films, 39; use of, i 
Practice numerals, 25 
Practice problems, 13, 20, 39 
Problem-solving attitude, 13, 38 
Problems as reading materials, 95-97, 99 
Problems used in : eye-movement studies, 
36, 73; first preliminary study, 2; 
second preliminar}' study, 12; third 
preliminary study, 19, 36 
Problems without numerals, 100 
Progress in reading by partial reading. 



Punctuation: and digit groups, 31; and 
main-group patterns, 32; and numer- 
ical language, S3', value of, 2,2 

Range of correct recall: classifications of, 
13; complete, 15; extent of, 14; of 
first numerals, 15; most frequent, 14; 
of shorter numerals, 15; according to 
subjects, 16; unclassified items, 17 

Range of perception, 56; and develop- 
ment of partial reading, 66, 67; of 
digits, 63, 79, 92; of Subject G, 92 

Rate of reading: different materials, 96; 
explanation of higher rates, 97; Free- 
man on, loi; prose by partial and 
whole readers, 65; of Schmidt's sub- 
jects, 95; of subjects of present 
investigation, 95 

Reading materials: different types of, 98; 

isolated numerals as, 96; ordinary 

prose as, 96; problems as, 95-97 
Recognition pauses, 76 
Recommendations concerning : reading 

of problems, loi; re-reading and 

copying numerals, 103 
Recording pauses, 74-76 
Regressive pauses: in detailed reading, 

85; ex-planation of, 58; initial, 79, 85; 

on words and numerals, 58 
Regularity of form in numerals, 90 
Re-reading: dependent on habits of 

subjects, 71; on length of numerals, 8; 

distinguished from first reading, 3; 

following whole reading, 5, 9; and 

mental computation, 9, 10; numerals, 

objects of, 22; percentages of, 8; 

purpose of, 3, 71; two types of, 69 
Re-reading for copying: description of, 

22, 71; introduce at fourth grade, 104; 

recommendations concerning, 103; 

several numerals in one problem, 23; 

shorter and longer numerals, 23 
Re-reading time, 23 
Rythmical movements of the eye and 

partial reading, 102 

Schmidt, W. A., 94, 95 

Several numerals in one problem: 

partial reading of, 5, 6, 63; re-reading 

of, 23 
Shorter numerals: partial and whole 

reading of, 61; partial reading of, 5; 

range of recall of, 15; re-reading for 

copying, 23 



INDEX 



log 



Simple re-reading, 69-71 

Simple three-digit groups, 27 

Special numerals, 90 

Subjects who ser\^ed in: eye-movement 
studies, 39; first preliminary study, 2; 
fourth preliminary study, 24; second 
preliminary study, 12; third pre- 
liminary study, 19 

Total reading time of numerals, 86; 
shortest, 91 

Visual memory, 10 



Whole first readings, 59; of isolated 
numerals, 61 

Whole readers: attitude of, 66; classi- 
fication of, 7, 64; prose reading, rate 
of, 65; range of perception, 56; range 
of recall, 17; reading of words by, 65; 
re-reading of, 9, 71 

Whole reading: description of, 4; and 
direct computation, 73; explanation 
of, 10; relative value of, 64; used by 
beginners, 66 

Wilson, Estaline, 98 

Words: range of perception of, 56; 
reading of, by partial readers, 65 



UNIVERSITY OF BRITISH COLUMBIA LIBRARY 

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Terry, P.W. 

How numerals are read. 







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