BF UC-NRLF of Cbicago HOW NUMERALS ARE READ AN EXPERIMENTAL STUDY OF THE READING OF ISOLATED NUMERALS AND NUMERALS IN ARITHMETIC PROBLEMS A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF ARTS AND LITERATURE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF EDUCATION BY PAUL WASHINGTON TERRY Private Edition, Distributed By THE UNIVERSITY OF CHICAGO LIBRARIES CHICAGO, ILLINOIS Reprinted from SUPPLEMENTARY EDUCATIONAL MONOGRAPHS, No. 18, June, 1922 EDUCATION DEP Ube ZHniversfts of Gbtca'aicL v -X :*': !/ HOW NUMERALS ARE READ AN EXPERIMENTAL STUDY OF THE READING OF ISOLATED NUMERALS AND NUMERALS IN ARITHMETIC PROBLEMS A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF ARTS AND LITERATURE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF EDUCATION BY PAUL WASHINGTON TERRY Private Edition, Distributed By THE UNIVERSITY OF CHICAGO LIBRARIES CHICAGO, ILLINOIS Reprinted from SUPPLEMENTARY EDUCATIONAL MONOGRAPHS, No. 18, June, 1922 COPYRIGHT 1922 BY THE UNIVERSITY OF CHICAGO All Rights Reserved Published June 1922 EDUCATION DEPfT TABLE OF CONTENTS PAGE 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. NUMERALS IN ARITHMETICAL PROBLEMS FIRST PRE- LIMINARY STUDY 2 1. Description of the Study. ... . . . 2 2. First Reading and Re-reading Distinguished . . 3 3. Partial First Reading and Whole First Reading of Numerals . . . . .3 4. Individual Subjects Classified as Partial First Readers and as Whole First Readers 6 5. Re-reading of the Several Numerals . . . . 7 6. Re-reading by Individual Subjects . . . . 9 7. Summary 10 CHAPTER III. RANGE OF CORRECT RECALL OF NUMERALS AFTER THE FIRST READING SECOND PRELIMINARY STUDY . . 12 1. Description of the Study 12 2. Range of Recall of the Several Numerals . . . 14 3. Range of Recall by the Several Subjects 16 4. Further Evidence as to the Purpose of First Reading . 17 5. Items of Recall Not Included in the Classifications . 17 6. Summary of Conclusions 17 CHAPTER IV. ANALYSIS OF THE RE-READING OF NUMERALS IN ARITH- METICAL PROBLEMS THIRD PRELIMINARY STUDY . . 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 NUMERALS IN COLUMNS FOURTH PRE- LIMINARY STUDY 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 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 33 PART II. STUDIES OF THE READING OF NUMERALS BY USE OF PHOTOGRAPHIC APPARATUS CHAPTER VI. 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 VII. 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 CONTENTS vil PAGE CHAPTER X. COMPARISONS or RATES OF READING 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. Preliminary Analytical Reading of Problems . . 99 3. Application of Partial Reading 100 4. Application 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 VI. First Reading of Problem 5 by Subject M and Re-reading and Copying the Two Numerals . . . . .46 VII. 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 XI. 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 M . . .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 Various 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 XI. 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 XXV. 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 SELECTION 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 xiii CHAPTER I INTRODUCTION Numerous investigations in the psychology of reading and in the measurement of reading ability 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 with 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 numerals. 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 preliminary 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 / 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 ties 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 first 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 ah 1 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 READING 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 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 5 6 7 Numerals given in problems \ 6 I 5 9 386 I7S4 4375 286 288,106 144,053 2,191,504 617,450 1918 100 IOOO [ 12 615 Total number of readings 20 4O 2O 40 20 20 IO 20 Partial first readings 3 12 IO 3 1 13 ii o I Whole first readings T6 28 IO 9 7 9 IO 19 Doubtful I 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 NUMERALS IN ARITHMETICAL PROBLEMS not be re-read. On the contrary 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 RANKS OF NUMERALS ACCORDING TO PERCENTAGES OF PARTIAL FIRST READINGS Ranks Percentage of Partial First Readings Description of Numerals Numerals Read I 77 Four three- to four-digit 437C,; 286: 2<67; 6l< 2 6c Two six-digit 288,106; I44,CXa 2 ec Two six- to seven-digit 2,101X04; 617,4^0 4" CO Two three- to four-digit 386; 1754 c 3O Four one- to two-digit <; o; 31 12 6 I? Two one- to two-digit 65; 8 7 c Two familiar 100; 1000 8 o Date 1918 and 6 each have two numerals. The six 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 numerals. 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 or PARTIAL AND WHOLE FIRST READINGS SUBJ ECTS G Bl S P De K Ko D T H Partial first readings Whole first readings 14 5 I3 6 12 7 9 10 8 ii 7 12 7 6 3 16 o Doubtful I 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 OP 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 numerals, the number of times they were TABLE IV NUMBER OF RE-READINGS Problems i 2 3 4 5 6 Numerals given in problems 6 I 5 9 3 386 1754 4375 286 2567 288,106 144,053 2,191,504 617,450 1918 IOO IOOO 12 615 Number of re-readings Numerals not re-read ii g 38 20 39 i 17 i 20 IO 13 1 Doubtful 2 2 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 r IOO Two six- to seven-digit 2.IQI ^04.' 6l7 4.^0 I t IOO Two three- to four-digit 386' 17^4 07 Hb R Bak Th L G C Number of numerals read by subjects Range of correct recall of numerals: Complete 8 4 8 2 8 2 4 2 8 2 6 i 8 i First two digits and digit-length 7 4 4 2 2 i i First digit and digit-length 7 7 6 3 3 2 4 Digit-length 8 7 8 4 5 5 7 Merely noticed o i o I o i 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 line. No items incorrectly recalled are included in the classifications. Several such items were reported. In general they followed the types 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. 1 8 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 NUMERALS IN ARITHMETICAL PROBLEMS 21 - to vO I II I CO t CO . ' 00 vO * t^ O vO g V) 1 BO CO to vO vo 1 cfi Q M H N n M H H rt ij S to 00 vo* CO i-T - \r> in vo ^ *-t 5- . ^ t M Tt ^M O>H M CO 3 s S M ^ , ^ 5^ ^ ^ M _^ M M , QJ * * " PH CO ^ 8 o' ' ' : ' : ' - v-^i . . . . <; H r>. to H <; M vO ^ vJ H ' ; . ; t- to t^ to 3 rS H " H ^ M ^ ^J 4 tJ ; d II I I Q M * ? -d M M < a 1 Numerals read in problems I Ordinal number of re-readings I 13 '"o ' TJ ' *a O w -4) -4) F Ot3'Ot5i3i3 T3 !3 ' S'| S'g 8 '& N C 6 N C OT C | | || || | | COT COT COT C ' t^. i 3 o cr * -52 ill *i*lij| "c .t! > rt ^ O rt ^O 1 "i II xJ"iJ 8 '" a v g_ B ^"T^ ~ ^ ' Z M M 4-J < ^ & 8-8 1 1 O c "" "?>^ ' a> U >2 3 "* \Q | 4J ta ^ IB , e 1 I C/3 ^O ^^ "^ a JJ i i ^ 11 "III \o 5i "H & ^ & 1 fe^ "S^l 1 i ; ^ fc S ^-, i* 1 o h " ii"S 42 ^ V JE c-g V^ *P M CS be "^ ^* i 1 &-2 "1* .0 s " a c r 'c M - P* 5 x-. ^x w "" * g o Q 11 00 1 W J M *J^i o. -a" B D C s i i f " 3 > FH C4 M M B k^ pq V 1 rt42 w ^ & ! 1 w g * M C^ -^ s. S ITr d - ^ "" 1 |--- ^ | H M :1 i 1 || w o ^j- o* c<4D T3 o S "g en t/3 w 1 | f*3 M M 8 *j M n-rt-Q H W M 11 "2 1 1 Si i 5 5 *j ^ *J 0) tB PH j- S *& 'aHp W t ^ ^ O t^H*^ ^ ^ Jc < b o H 2" H lH I& g s H < fcc*^ 8 tc i3 .'" *~. ' +J.& 43 uJjf.g .j 'S)" rt 3" 3 "c NUMERICA] C li i III S i Hi rt Jj *^ .2 S * rt c c^ In O< : E .y &j^ " J2 , rt "^ _, t/ i N ' w 5 60 ^ s S '"S'o 1 1 ^| lull's ^ - &~ ^^^"g w *0 "rt _fi _C ^ -G *^ -C 1 1 B O S G ^ ^ 4 1 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 type 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 OP 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 3 2 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 READING 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 Two-digit groups Three-digit groups Subject G Two-digit groups Three-digit groups Subject H Two-digit groups Three-digit groups Subject L Two-digit groups 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 ,1 34 II 21 4 12 7 16 '"io"' ........ ...... 12 15 2 9 4 3 12 2 II 6 I 5 Three-digit groups 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 33 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 CONCLUSIONS 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-MOVEMENT STUDIES I. APPARATUS DESCRIBED The data which are presented in the remaining 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, 1 and excellent photographs and diagrams of the same are found in a magazine article by Gilliland. 2 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. 1 C. T. Gray, "Types of Reading Ability as Exhibited Through Tests and Labora- tory Experiments," Supplementary Educational Monographs, Vol. I, No. 5 (1917), pp. 83-91. 2 A. R. Gilliland, "Photographic Methods for Studying Reading," Visual Educa- tion, Vol. II, No. 2 (February, 1921), 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 1000, 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 READ IN THE PHOTOGRAPHIC APPARATUS NUMBER OF PROBLEM ORDINAL NUMBER OF EACH LINE IN PROBLEM LENGTH OF LINE IN MILLI- METERS THE NUMBER IN EACH LINE OF Words Letters Numerals Digits Words and Numerals Letters and Digits i ISt fist \2d fist 2d U fist 2d l3d fist J 2 d Ud 76 y 95 IO2 86 95 102 90 95 102 94 9 ii 8 ii 7 9 10 8 8 8 10 12 34 4 37 44 34 37 42 56 30 40 43 45 2 i i 2 I o 3 i i o 3 3 4 ii o 4 9 7 6 ii 12 9 II 9 9 II 8 ii 9 II 12 37 43 4i 44 45 37 46 56 39 47 49 45 2 3 4 5 Total for all problems. . Average num- ber per line 12 III 482 12 47 123 529 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 had on hand 1000 cases of canned corn. From three different canning factories they bought respectively 1276, 91, and 817 cases. How many did they then have? 5. If one telephone company uses 1,918,564 cross bars during the year, 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, 333, 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 25,000 (Card Two) 76,184 9317 17 2 5,236,795 256 743,819 1928 365 8 93,548 3,984,673 107,308 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 lines 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 101 words and 452 letters. They are reproduced as Selection 3. SELECTION 3 ORDINARY PROSE READ BEFORE PHOTOGRAPHIC APPARATUS "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 " x 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- 1 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 familiar 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 preliminary 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 previous 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 lines 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 line represents the approximate center of the field of perception which was included in that pause. The arabic 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 1/50 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 line 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 readmg of the problem. DESCRIPTION OF THE EYE-MOVEMENT STUDIES 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 z )ir|missi A commission ho ise hati on hand 1 of canned corn, i ning factories they Bought respectively /* IS ffe IT IS 15 20 21 22 23 24 276, l, nd $17 dases. How many did they thei ha\e? 25 ipoo 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 NUMERALS 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 lines 2 and 3. During pauses 1 6 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 irhissi< [ I A comrhission house had on hahd 1 18 12 10 II 7 .> II IZ 13 14. From trjree tiifTerent banning factories they tespectr bought respectively 9 r 12 9 ro 13 s. JIow mp,ny did they thfn have? 10 9 iboo 9 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 with 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 was completed with Pause 21. PLATE III IT J 11 I J 1 At 4F cents d doted whafl will 2 dozen eggs cost ? It I*. 35 7 10 19 i 24 f 5l l 4j3ZZ '34 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 1000. 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 A wholesale graii is o dng of at the beginning of the day 767 l.4 (5 ijb 17 bushels of whea 9 * '5 ,17 I? I . During the da ZO 's trafling 2J,765 r 14 38.31 \(o 1 1 Y If T'Tf 9 r T p| ia to '10 1015 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 multiplication, 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 line 2, six were placed on the digits of one numeral. The first reading was concluded with a relatively rapid reading of the last line. 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 5 < A timber owns on pllt bf p7 apres, ajid another of 19 '2 IT 16 >+ JO How much groi ad 18 he owij altogether ? as. ia 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 33 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 HOW NUMERALS 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 n, 12, and 13. PLATE VI 1 1 f J If ond telephone company uses G.5 < 7 3 II. 10 thfe yea j ,, ,3 , and another company in the 1 "I 'J VI sahie period uses 617,4p3 10 10 J 15 4 20. 19 2 4 , ht cross bars w many mor does the one use than the other? f 4 33 37 3 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 VII shows Subject Hb reading the first problem very cautiously. With only two exceptions every word and numeral in the problem was a dozfcn what 12. 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 multiplication 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 /I dozen what will 2 dozeil eggi cost ? is 7 H 8< *The answer, 94, was recorded at the point indicated during Pause 9. 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 numeral 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. f ? 4 timber man owns onfe plot of 35 1 fci 3 '2, 13 A tiipber man f>wns on^ plot of 2(57 acres, and ahothej of II 7 1643 acl-es. 1643 acres. How mufh ground does hf own altogether? a uf First reading of Problem 2 by Subject G and the process of adding the two numerals. 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 XI. 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. 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 1 *l '[ "J At 47 cents a dozen what kvill 2 dozen eggi cost ? 1C.' 20 J 3 te4* 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 lHVlTl At 47 dents a dozen what will JJ dozefi eggs cosl|? 15 9 S 7 T 9 14 12. 47 14 II 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 A whi (0 [ 1 I 1 )lesble graih firm haffl 10 lj at the beginning of ' 1 1 tihe dakr Vl ' "L 1 " 1 1 243,987 bushels of wheat. Dbring the day's trading 2U765 zo 15 ii io t >a '<> zd 1 'I "I "I busUels were solti. Ho\s| much did hell 13* they then have? 14' Z5 jo 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. 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 (telephone dompani uses 6j '5. T '*. 19 24 20 the yeJr, and ajiotheij comparjy in the Jam ; peri ; period uses pl7 20 13 *Tt " If 1 one ufee flhan the oth^r ? 7 .I J Z9 50,47 43,, 40 u'es Z. 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 XXVI. 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 with a single pause. In PLATE XIV 3 \ 1 1 .'I. 1 A wholesale grajn firpi hp,d ap the beginning of tne day 10 IS \ ] 11 1 alesale grain firm had at IT 14 I* 3 9 1 '] "I "I '] "I '1 243,987 bushels of whedt. DurinJ; the dayfs trading 21J65 Z3 32 '4 's' iJ 14 I* '1 "1 1 1 "1 bushels were sold. How much did they then rmve? 13 /o' ii ' n'l 3 First reading of Problem 3 by Subject W with partial reading of numerals this plate a clear illustration of rapid reading of the last line 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 *Ll 1 1 \ \ 1 \ 'J If one telephone company uses 1,D18,564 cros| bars jhmngj 12 W 9 J 3 J ' J T' ,. "1 I "I the year, and another company tn the same peribd uses 6 J 13' ?' J 1 1 5^7,453 iJ 3 cross 1 1 '1 1 T 1 T rsL how mankr more poes the bne use flhan thfe other ? V \i to* M' s a 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 types 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 line stands out very conspicuously. The average number of pauses per line for all subjects is 8.08; and there are individual lines in which as many as 10, n, 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 familiar with simple arithmetical problems, and who would be expected to qualify 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 56 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. 1 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 NUMERALS CONTRASTED WITH AVERAGE NUMBER OF LETTERS INCLUDED IN A PAUSE ON WORDS DURING FIRST READING SUBJ ECTS AVERAGE FOR G M w B H Hb ALL SUBJECTS Average number of digits included in a pause on numerals i 81 i 88 i 88 2 *8 Average number of letters included in a pause on words 7 30 e xg 6 4.7 NOTE. Each subject read the five problems which included 12 numerals totaling 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. 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 IN FIFTIETHS OF A SECOND ON NUMERALS CONTRASTED WITH AVERAGE DURATION OF PAUSES ON WORDS DURING FIRST READING SUBJECTS AVERAGE FOR ALL SUBJECTS G M W B H Hb Total number of pauses ^Numerals. . used by subject in reading/ Words .... i. Average duration of pauses on nu- merals 14 66 13-92 3-92 10.72 2.27 20 81 15.20 4-54 9-87 1.41 16 7i 15-31 5-35 13.02 3.38 26 61 18.46 5-92 9.18 1.16 27 93 13.30 3-97 11.74 4-13 11 13.48 5-0 10.99 2-94 14.98 4-77 10.92 2-55 Average variation 2. Average duration of pauses on words Average variation 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 numerals. 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 PAUSES ON NUMERALS CONTRASTED WITH PERCENTAGE OF REGRESSIVE PAUSES ON WORDS DURING FIRST READING 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: i. Numerals is of the total number of pauses on 2 II 14.28 6 10 30.0 I 4 6.25 7 8 26.9 10 22 4O.O 5 i 20. o 2. Words is of the total number of pauses on words 16.6 12.3 5.63 13-1 23.65 1.19 SUBJECTS sive pauses appear in the case of the numerals than upon the accompany- ing words. The explanation of such differences probably lies in the difficulty of reading in the same lines, 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 NUMERALS 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 WHOLE 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 60 HOW NUMERALS ARE READ II i | I N MM CO M CO 2 a c I 3 ^ ~-i M PH s Si-- 1 4 ^ ** S c .' 3 P 00 -'8&sSea5E .MM i V c. ^^ ^^"i'^^^i H ^^ . M*"' "^ rO^^ 00 ^^ rJ-N i 1 M i Tj- M 5- " oo" f^ 11 a o 10 00 53 -3 *^ ^Pn Q 2^ "-Q -^ M"^ . >"o O M ss M 11 - -i^^-i-i-i ':-> ': f! Problem Number. . . Numerals which were read in problems . . :i : i : ea : : : C. I- , u Q.IH O.hl O. I- O. fc, O O O O O O c"rt c"c3 c"rt fi"c3 c 2 c .2 o'C o'C o'C o'C o'C o u Q Q p Q Q Q Number of: Partial readings . . . Whole readings. . . . Doubtful cases HOTE. The capital let tful cases which the rec sioafsng K g sioafans ny 1 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 evidence 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 numerals in problems is evidently similar in kind to the reading of numerals isolated in lines. As was found in the first preliminary 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 previous 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 slightly greater number of partial readings than whole readings. The partial 62 HOW NUMERALS ARE READ READING OF H B 52 g o % ^ w 1 o ^ < Q 2; O S 1 & a 5 C/3 S rt w 2 PQ fH 10 M PQ || ^4 S H Pn ^ 'S g ^ NUMERA 1 I Q 1 fe O O co I i O 1 Tfr ** 2 -o GJ H 00 >O 1-1 compu 0, Z "cd ^ S3. o i i M Ov J M 3 "o^ G CO 1 Isolated - Ov 5 tC 1 1 to J vOO 1 - " H M X VO "c ^" 10 "t ^ u C/3 1 ^t _r ^* 3 3 i .1 M " 2 ^ H N "3 j 1 * d; o" c ^ m " tS M o - * >0 O -C M ^^^ oc" N ^ t ^ c* 3 1 I I! vO to IO H i-T o" g-pauses, t 1 P ^ 5 t "I ". vO 12 _G $ H "3 VO T? o" i w O ^ JO 1 II M i H it *3 ^ ^ 3 /- _^ 00 vO ^ ^ a "in ^ r - rt 1 E | _O v| | *? ^ " TD 1 3 1 "* M" to oo" "* M ^ cS M" tO H putation ] S in 1 i i= 1 : 3 : 3 : : : : 3 g 3 Pi .2 ^ : 8 : i ' d : : O * f3 J3 ' U3 y^ S 1 a |f 1 1 B 1 'S o 8 U |l ? 1 1 o "g .S-G rt g | -g ft 22 B ll -^l 8 -"s ;-s|| B hj * tH 2 U-^^W^_)4-< 5 PH < "S Icn^ - ^2^JS 5^ ^ ^'-g I. IS ^ fi "^ C c/3 C O**t3 ^^j ^ i-i H ^ ^ rt S G ^fl > j-2 S'S | C rt 4-j a _rt rt 111 l|l|l|||| PH !z; 6 6 P^ WH< 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 numeral 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, 1 when investigating the perception span of good and poor readers, had a number of individuals read short lines 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, 2 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. 1 C. T. Gray, "Types of Reading Ability as Exhibited through Tests and Laboratory Experiments," Supplementary Educational Monographs, Vol. I, No. 5 (1917), p. 146. 2 W. F. Dearborn, "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. 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 836 fcs 7 1 )i 4, ,5 9 28" "29 PLATE XVI v 756, *,9F4 4,2 21 k 24 33 31 90 33 15 VI 354,908 42 17 J2 4441 Tl II 74 30 29 16 8 14. II 24 I: 9. ,10 3 T f 1 1 fl 1 ftt H I 5 T r -ii 7 10 3f 23 13 4 15 24 24 If. /A .1? .19 & 40 } }ft . 1 T Reading of isolated numerals by Subject M PLATE XXI },i|4 Jsir ip I sbbjAJ el 4 29 4 'l9 T T I ' 4 " 14 f 2< I 15 to 3 a 3 ! ' M u 1 'e 14 Reading of isolated numerals by Subject M THE READING OF ISOLATED NUMERALS IN LINES 83 1 T 50 5 PLATE XXII 3 T .10 TT 32 31 TT 31 24 '* 4' 36, 20 15 t>3 17 "llf 8,7 55 44 8 r " 33 44 21 14 IS 9? 13 27 I 'Ultf I 9 J T rf 3 ! 4 '? 5 ! T &T ? 309 4 JiW L 51 7 40 Reading of isolated numerals by Subject B PLATE XXIII * 1 ' HT 1 I 7618tt pplH J7 B 2o 34 (o 10 41 it T 3T fc* 21 30 51 53 51 3 5 I 12 14 J5 .13 |6 IT f 39 35 43 f41 26 14 34 33 3o -Jl Reading of isolated numerals by Subject B 8 4 HOW NUMERALS ARE READ PLATE XXIV 2 I 5 1 3 ,4 15 (* IT IS 5>5 23 T 47 W 73 756,35! 4* 35 :,B2B,9*6 JO 21 -2i 39 >H I '3 32 4 to 23 3k 5, H, 'V ' 3 , ,'4 IS , 6 '.T 15 48 ;H if 40 2T T 42. 11 354, ) 5? 3^ 34- 1 H" 12, 25 S 19 3k /9 'I t& ^2 9 43 24 Reading of isolated numerals by Subject W PLATE XXV 38 34 frn 14 33 38 31 19 40 21 37 ** 45 T52T? H 10 .11 12. 13.15 .14 16 30 IT r i ,8467 it af 'i* 2 e, ^ t, 45 25' '3 a 13 14 23 20 31 T4 ^ 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, 333, 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 TYPES 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 line was almost 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 well 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 w " W flJ ^ QJ s I < b A m ^ . O K 10 O M 00 10 10 Oi O O 10 O 00 a o> ', o . !> 10 * 10 10 O O 2 sfi '??' $ T, i ^ yi .P 2.8 , 1 1 a| g.| 4-i O 4J "S V3 "o '* "o '+3 *0 T3 o n i.s 1.1 1! il ::: ^ g^ S-S 6-3 S-H g- Si 3 1 88 HOW NUMERALS ARE READ O 10 O Ov I 01 M ^ M ^ O OO ro t^ to 00 ^ f 1O a <+) Q t*\ O CO I s * O* CO w vd CO N rC. o 00 Ol O O 00 Ov N W ^3. i3 V3 M M M M CO H >0 ro TJ- Oi 00 IN O a M o vd 4 o q B hi s 3 >0 ON 00 O M 3 J5 t^ oo" o M M "? H C/2 1 l CO CO * t- 8 W | O> M rT !>. M M Tl- M CS W x-s s Q ^0 00 00 M 1 in & M M M Q in < ^ 10 vo 2 .0 2 t>. g p4 j^ CO 00 vo O O p a g Oi 00 t^ M oo' I ^ > < t tt * V 1 * "^ Oi i o ^ CO \O O CO s M g h -S t^ \O >0 ro M : B b CO CO * 10 c^ .5 s SB O M M O O 10 a 1 M 10 IO 10 Tf rf Oi * M O 00 a 10 O H 'NJ I O M M CO 10 S i G w O < 1 PL, a' : 1 G 1 c 1 c' ' 1 G' ' 2 . o : | .2 : | .2 : | . s : | .3:3 g .2 3 ir each of < ill III 111 111 111 2 1 in in in in in 1 1 "1